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--- abstract: 'We describe the accelerated propagation wave arising from a non-local reaction-diffusion equation. This equation originates from an ecological problem, where accelerated biological invasions have been documented. The analysis is based on the comparison of this model with a related local equation, and on the analysis of the dynamics of the solutions of this second model thanks to probabilistic methods.' author: - 'N. Berestycki, C. Mouhot, G. Raoul' title: 'Existence of self-accelerating fronts for a non-local reaction-diffusion equations' --- Introduction and results {#sec:introduction} ======================== Biological invasions happen when a species recently introduced in a location succeeds to establish and to spread in this new environment. These introduction are usually either a consequence of human transportation systems [@Carlton], or a consequence of the climate change [@Kovats]. Biological invasions are occuring at an unprecedented rate [@Hulme], and have an important impact on e.g. biodiversity [@Sakai] and human well-being [@Pejchar; @Juliano]. Predicting the dynamics of those invasion is an issue, that requires (among other approaches) the development of new mathematical methods and results [@Clark; @Kolar]. In this study, we are interested in a particular phenomenon that may happen during biological invasions [@Thomas2; @Mack; @Edmonds]: the dispersion of the individuals increases during the invasion. As a result, the speed of the invasive front increases, and often keeps accelerating as long as the invasion progresses [@Mack]. The best documented case is a biological invasion of Cane Toads in Australia [@Phillips; @Urban]. The amphibians have been introduced in Australia in 1935 as a (failed) attempt to control beetles populations in cane plantations. Since then, cane toads have been invading large coastal areas, at an accelerated speed: the invasion started with a speed of 10 kilometres a year, and continuously accelerated to the impressive speed of 55 kilometres a years today [@Urban]. The mechanism for this acceleration is documented [@Lindstrom]: the individuals close to the invasion fronts have an anomalously high dispersion rate, and drive the invasion front. A model introduced in 1937 by Fisher in [@Fisher] (and simultaneously in [@KPP]) has proven very useful to describe biological invasions [@Shigesada; @Hastings]. This model describes the dynamics of the density of a population. In a homogeneous environment, a population which is initially present on a limited set only will propagate at an asymptotically constant speed [@Bramson], with a certain profile, called travelling wave [@KPP]. The study of travelling waves and related propagation phenomena has prompted a large mathematical literature, we refer to [@Xin] for a review on this active field of research. Recently, more surprising dynamics have been uncovered: in [@Roques], it has been shown that a slowly decaying initial condition may lead to accelerating invasion fronts. Similar dynamics can be observed for compactly supported initial populations if the diffusion operator of the Fisher-KPP equation is replaced by a nonlocal dispersal operator with fat tails [@Kot; @Garnier], or by a fractional diffusion operator [@Coulon]. Finally, in [@Bouin], it was proven that a similar dynamics can be observed when the diffusion operator is replaced by a kinetic operator modelling a run and tumble dynamics. The phenomena that we want to describe here is different from the ones described above: in our case, the acceleration dynamics is due to the continual selection of individual with enhanced dispersion abilities. To model such phenomena, involving both a spatial dynamics of the population and evolutionary phenomena (see [@Hairston; @Lambrinos]), the population should be structured by a phenotypic trait as well as a spatial variable. Starting from an Individual Based model of such a population, a large population limit can be performed [@Fournier] to obtain a non-local parabolic equation. Related models have been studied in e.g. [@Prevost; @Alfaro]. The case where the phenotypic trait structuring the population is the dispersion rate of the population has been introduced in [@Benichou]. The model {#subsec:model} --------- We will consider a population described by its density $v=v(t,x,\theta)$, where $t\geq 0$ is the time variable, $x\in\mathbb R$ a spatial location, and $\theta\in (1,\infty)$ a phenotypic trait. The dynamics of the population is given by the following model: Model (NLoc): $$\begin{aligned} \left\{ \begin{array}{l} {\displaystyle}\partial_t v = \frac \theta 2 \Delta_x v + \frac 1 2\Delta_\theta v + v \left( 1- \langle v \rangle \right) \\[3mm] {\displaystyle}v = v(t,x,\theta), \ t \ge 0, \ x \in {{\mathbb R}}, \ \theta \ge 1, \\[3mm] {\displaystyle}\langle v \rangle(t,x,\theta) := \int_{\max(\theta- A,1)} ^{\theta+A} v(t,x,\omega) {{\, \mathrm d}}\omega , \\[4mm] {\displaystyle}v(0,x,\theta) = v_0(x,\theta) \ge 0,\\[3mm] {\displaystyle}\partial_\theta v(t,x,1)=0,\ t \ge 0, \ x \in {{\mathbb R}}. \end{array} \right.\end{aligned}$$ In this model, we assume that individuals diffuse through space at a rate given by the phenotypic trait $\theta$. This phenotypic trait $\theta\in[1,\infty)$ is itself submitted to mutations, which appears in the model as a diffusion term in the variable $\theta$, at a rate constant rate $1$ independent from $x$ and $\theta$. We assume that the growth rate of the population in the absence of intra-specific competition is $1$, and is in particular independent of the spatial location $x$ and phenotypic trait $\theta$. We assume that the individuals are in competition with the individuals present in the same location, provided their phenotypic traits are not different, which is quantified by $A>0$. Note that (NLoc) would correspond to the model introduced in [@Benichou] if $A=\infty$; we will however always consider here that $A>0$ is finite. We also assume that the individuals reproduce asexualy: during sexual reproductions, recombinations of the DNA strains happen, which leads to very different mathematical models [@MR]. From a modelling point of view, assuming that the phenotypic trait $\theta$ can take arbitrarily large values may appear surprising. It seems however to be a reasonable assumption in this context: an artificial selection experiment [@Weber] has shown that it is possible to increase the dispersion rate of flies a hundred folds in just a hundred generations, with little impact on the reproduction rate of the individuals. The field data obtained in [@Urban] suggest that the set of possible phenotypic traits does not have a limiting effect on the evolution of dispersal in cane toad populations. The data collected in [@Lindstrom] provides some indications on how rapid evolution of the dispersion rate is possible: tracking data of the cane toads show that the animals alternate resting phases and ballistic motion, and the individuals at the front of the invasion simply have longer ballistic phases, and a higher directional persistence. These simple modifications of individual motion has limited energetic cost, while greatly increasing individuals dispersion rate. The main results {#sec:main-results} ---------------- We make the following assumptions on the initial condition: 1. (Compact support in $\theta$.) We have that $u_0(x, \theta) = 0$ unless $\theta_{\min} \le \theta \le \theta_{\max}$ for some $\theta_{\max} >\theta_{\min} \ge 1$; 2. (Thin tail) We have, for some $C,c>0$ $u_0(x, \theta) \le C \exp( - cx )$ uniformly over $x$ and $\theta$, and $\inf_{\mathbb R_-\times [\theta_{\min}', \theta_{\max}']}u_0>0$, for some $\theta_{\max}' >\theta_{\min}' \ge 1$; 3. (Regularity.) We assume that $\left((x,\theta)\mapsto \bar v_0(x,\theta=v_0(x,|\theta|+1)\right)\in C^{3}(\mathbb R^2)$, that is $$\sum_{0\leq k+l\leq 3}\|\partial_x^k\partial_\theta^l \bar v_0\|_{L^\infty(\mathbb R^2)}<\infty.$$ We can now state the main result of this study, which describes the acceleration of the invasion front: \[T:toads-nonlocal\] Let $v_0\in C^{2+\delta}(\mathbb R\times [1,\infty))$ with compact support in $\theta$, thin tail in $x$ and regular, as described in Subsection \[sec:main-results\]. Let $v(t,x, \theta)$ denote the corresponding solution of (NLoc). For $x \in {{\mathbb R}}$, let $S(t,x) = \sup_{\theta} v(t,x, \theta)$ and let $$\gamma_0 = \frac{2}{3}2^{1/4}.$$ We have for all $\gamma > \gamma_0$, $$\label{toads:UB-nonlocal} \lim_{t\to\infty} \sup_{x > \gamma t^{3/2}} S(t, x) \to 0$$ while for $\gamma < \gamma_0$ $$\label{toads:LB-nonlocal} \liminf_{t\to\infty} \inf_{x < \gamma t^{3/2} } S(t,x) >0.$$ In other words the population spreads in space as $\gamma_0 t^{3/2}$. An example of an initial condition which satisfies the assumptions (1) and (2) is given by $u_0(x, \theta) = H(x) {\mathbf{1}_{\{\theta \in (1,2)\}}}$, where $H$ is the Heavyside function. This is a good example to keep in mind for this result, and as a matter of fact much of the proof relies on the analysis of this example, for a modified model (where the non local competition is replaced by a local term, see (Loc)). Note however that the thin tail condition we make here is much weaker than the condition that is usually made for propagation front problems: for the Fisher-KPP equation $\partial_t n(t,x)-\frac 12\Delta_x n(t,x)=n(t,x)(1-n(t,x))$, the solution propagates at speed $\sqrt 2$ provided the tail of initial condition satisfies $n(0,x)\leq C e^{\sqrt 2 x}$ for some $C>0$. If the tail of the initial condition decreases slower than that, the solution can propagate much faster than that, and indeed, for any $c\geq \sqrt 2$ there exist a travelling wave propagating at speed $c$ (see e.g. [@Xin]). Note that if tail of the initial condition of (NLoc) decreases polynomially only, we expect that the population could propagate faster than what we describe here, just as it happens for the Fisher-KPP equation [@Roques]. \[Rmk:generalised-fronts\] This description of the propagation of the population is close to the notions of *spreading speed* (see e.g. [@Aronson-Weinberger; @Hamel]), and generalized travelling wave (see [@Berestycki-Hamel]). Indeed, the solution $u=u(t,x):\mathbb R_+\times\mathbb R\to\mathbb R_+$ of the Fisher-KPP equation is said to spreading at speed $2$, in the sense that if $u(0,\cdot)\neq 0$, $u(0,\cdot)\geq 0$ is compactly supported, then for any $c_-<2<c_+$, $$\lim_{t\to\infty} u(t,c_-t)=1,\quad \lim_{t\to\infty} u(t,c_+t)=0.$$ The description of the solution’s dynamics in Theorem \[T:toads-nonlocal\] can thus be seen as an extension of this spreading speed. Note that the estimates and provide a precise description on the acceleration of the invasion front: it does provide the exponent $t^{\frac 23}$ of the acceleration (see e.g. [@Bouin] for a result of this type), but it is indeed much more precise: We provide the exact multiplicative constant $\gamma_0$ in front of the leading term $\gamma_0 t^{\frac 32}$. Before describing the key ideas of the proof, let us discuss a natural generalizations of Theorem \[T:toads-nonlocal\], where the dispersion rate of the population is not given by the phenotypic trait $\theta$, but by $\theta^\alpha$, for some $\alpha>0$. The equation on $v$ would then become: $$\label{model-alpha} \partial_t v=\frac{\theta^\alpha}2\Delta_x v+\frac 12\Delta_\theta v+v(1-\langle v\rangle).$$ We believe the framework of our analysis could be used to show that $$\lim_{t\to\infty}\sup_{x\geq t^{\kappa}}\int v(t,x,\theta)\,d\theta =0,\quad \liminf_{t\to\infty}\sup_{x\geq t^{\kappa'}}\int v(t,x,\theta)\,d\theta >0,$$ for any $\kappa'<\frac{2+\alpha}2<\kappa$, this analysis is however beyond the scope of this study. Note that the case of (NLoc), leads to an acceletarion $x\sim t^{\frac{2+\alpha}2}=t^{\frac 32}$, that is close to the observation from [@Urban] on the invasion of Cane toads in Australia, which justifies our particular focus on this case. Another possible generalization of this model is to consider a competition term that is non local in both trait and phenotype. If we assume that the spacial non-locality of this competition is related to individual dispersal, a natural model to consider is $$\partial_t w = \frac{\theta}2 \Delta_x w +\frac 12 \Delta_\theta w + w \left( 1- \langle w \rangle \right),$$ where $\langle w \rangle(t,x,\theta) := \frac{1}{\sqrt \theta} \int_{x-\alpha \sqrt \theta}^{x+\alpha \sqrt \theta}\int_{\min(\theta- A,1)} ^{\theta+A} w(t,y,\omega) {{\, \mathrm d}}\omega {{\, \mathrm d}}y$. The dynamics of this other model can indeed be described with an approach similar to the one presented here: simple a priori estimates show that $\langle w \rangle $ is uniformly bounded. A De Giorgi-Moser iteration scheme (see [@Moser]) can then be used to show that $w$ is indeed uniformly bounded. One can then compare the dynamics of this model with the local model (Loc), as done in this study (see Section \[sec:comparison-models\]). Finally, let us mention that in (NLoc), it would be natural to consider the case where $A=\infty$. This is actually the model that was introduced in [@Benichou]. In Figure \[fig:4\], numerical simulations show that the description of the dynamics of (NLoc) seem to apply to the case where $A=\infty$ also. Proving this result would however require additional estimates. Discussion on the dynamics of the solutions {#subsec:discussion} ------------------------------------------- In Theorem \[T:toads-nonlocal\], we show that the position of the invasion front is well approximated by $$\label{eq:dyn-x} x(t)= \frac {2^{5/4}}3 t^{3/2}.$$ Indeed, the proof also provides some information on the phenotypic trait present at this front: at $x(t) = \gamma t^{3/2}$, $v(t,x(t), \theta(t)) >C>0$ for $$\label{eq:dyn-theta} \theta(t) = \frac {\sqrt{2}} 2 t.$$ Moreover it would be relatively easy to show that near the edge of the invasion front, that is in $x\sim \gamma_0 t^{3/2}=({2^{5/4}}/3) t^{3/2}$, *all* particles have a mobility of approximately $\theta (t)= (\sqrt{2}/2)t$. If we considered a linearisation of (NLoc), and a situation where $v$ is independent of $x$, then the solution would propagate towards large $\theta>1$ at speed $\sqrt 2$. It is worth noticing that the mobility $\theta$ found at the edge of the propagation front increases at only half this speed, $\theta = (\sqrt{2} / 2)t$. This dynamics is then indeed the effect of a combination of evolutionary and spatial dynamics. To validate the quantitative approximations and , we performed some numerical simulations of (NLoc), (Loc) and (NLoc) with $A=\infty$. The simulations are based on a finite difference scheme, with some additional Neuman boundary conditions at the edge of the $x$ interval we consider. The numerical results are in good agreement with the theoretical results and in each of the three cases: (NLoc) (Figure \[fig:1\] which corresponds to Theorem \[T:toads-nonlocal\]), (Loc) (Figure \[fig:3\] which corresponds to Theorem \[T:toads\]), and (NLoc) with $A=\infty$ (Figure \[fig:4\] for which we do not have theoretical results). For (NLoc), which is the main focus of this study, we provide in Figure \[fig:2\] a more precise comparison of the numerical position and phenotype at the front with the theoretical approximations and . The approximations developed in this study seem to provide a good description of the dynamics of solutions. ![Numerical simulation of (NLoc). The first graph represents $v(53,\cdot,\cdot)$. In the second graph, the red curve represents the theoretical position of the front, $x(t):t\mapsto (2^{5/4}/3)t^{3/2}$ (see ), while the blue curve represent the result of the numerical simulation. Similarly, in the last graph, the red curve represents the theoretical phenotypic trait present at the front, $\theta(t):t\mapsto (\sqrt 2/2)t$ (see ), while the blue curve represent the result of the numerical simulation.[]{data-label="fig:1"}](xthetaK.png "fig:"){width="70mm"}![Numerical simulation of (NLoc). The first graph represents $v(53,\cdot,\cdot)$. In the second graph, the red curve represents the theoretical position of the front, $x(t):t\mapsto (2^{5/4}/3)t^{3/2}$ (see ), while the blue curve represent the result of the numerical simulation. Similarly, in the last graph, the red curve represents the theoretical phenotypic trait present at the front, $\theta(t):t\mapsto (\sqrt 2/2)t$ (see ), while the blue curve represent the result of the numerical simulation.[]{data-label="fig:1"}](xK.png "fig:"){width="70mm"}![Numerical simulation of (NLoc). The first graph represents $v(53,\cdot,\cdot)$. In the second graph, the red curve represents the theoretical position of the front, $x(t):t\mapsto (2^{5/4}/3)t^{3/2}$ (see ), while the blue curve represent the result of the numerical simulation. Similarly, in the last graph, the red curve represents the theoretical phenotypic trait present at the front, $\theta(t):t\mapsto (\sqrt 2/2)t$ (see ), while the blue curve represent the result of the numerical simulation.[]{data-label="fig:1"}](thetaK.png "fig:"){width="70mm"} ![Numerical simulation of (NLoc). The first graph represents the quotient of the theoretical position $x(t):t\mapsto (2^{5/4}/3)t^{3/2}$ of the front (see ) divided by the corresponding quantity obtained by simulation. Similarly, the second graph represents the quotient of the theoretical phenotypic trait $\theta(t):t\mapsto (\sqrt 2/2)t$ present at the front (see ) divided by the corresponding quantity obtained by simulation.[]{data-label="fig:2"}](xxK.png "fig:"){width="70mm"}![Numerical simulation of (NLoc). The first graph represents the quotient of the theoretical position $x(t):t\mapsto (2^{5/4}/3)t^{3/2}$ of the front (see ) divided by the corresponding quantity obtained by simulation. Similarly, the second graph represents the quotient of the theoretical phenotypic trait $\theta(t):t\mapsto (\sqrt 2/2)t$ present at the front (see ) divided by the corresponding quantity obtained by simulation.[]{data-label="fig:2"}](thetathetaK.png "fig:"){width="70mm"} ![Numerical simulation of (Loc). The first graph represents $v(53,\cdot,\cdot)$. In the second graph, the red curve represents the theoretical position of the front, $x(t):t\mapsto (2^{5/4}/3)t^{3/2}$ (see ), while the blue curve represent the result of the numerical simulation. Similarly, in the last graph, the red curve represents the theoretical phenotypic trait present at the front, $\theta(t):t\mapsto (\sqrt 2/2)t$ (see ), while the blue curve represent the result of the numerical simulation.[]{data-label="fig:3"}](xthetan.png "fig:"){width="70mm"}![Numerical simulation of (Loc). The first graph represents $v(53,\cdot,\cdot)$. In the second graph, the red curve represents the theoretical position of the front, $x(t):t\mapsto (2^{5/4}/3)t^{3/2}$ (see ), while the blue curve represent the result of the numerical simulation. Similarly, in the last graph, the red curve represents the theoretical phenotypic trait present at the front, $\theta(t):t\mapsto (\sqrt 2/2)t$ (see ), while the blue curve represent the result of the numerical simulation.[]{data-label="fig:3"}](xn.png "fig:"){width="70mm"}![Numerical simulation of (Loc). The first graph represents $v(53,\cdot,\cdot)$. In the second graph, the red curve represents the theoretical position of the front, $x(t):t\mapsto (2^{5/4}/3)t^{3/2}$ (see ), while the blue curve represent the result of the numerical simulation. Similarly, in the last graph, the red curve represents the theoretical phenotypic trait present at the front, $\theta(t):t\mapsto (\sqrt 2/2)t$ (see ), while the blue curve represent the result of the numerical simulation.[]{data-label="fig:3"}](thetan.png "fig:"){width="70mm"} ![Numerical simulation of (NLoc) with $A=\infty$. The first graph represents $v(53,\cdot,\cdot)$. In the second graph, the red curve represents the theoretical position of the front, $x(t):t\mapsto (2^{5/4}/3)t^{3/2}$ (see ), while the blue curve represent the result of the numerical simulation. Similarly, in the last graph, the red curve represents the theoretical phenotypic trait present at the front, $\theta(t):t\mapsto (\sqrt 2/2)t$ (see ), while the blue curve represent the result of the numerical simulation.[]{data-label="fig:4"}](xthetaI.png "fig:"){width="70mm"}![Numerical simulation of (NLoc) with $A=\infty$. The first graph represents $v(53,\cdot,\cdot)$. In the second graph, the red curve represents the theoretical position of the front, $x(t):t\mapsto (2^{5/4}/3)t^{3/2}$ (see ), while the blue curve represent the result of the numerical simulation. Similarly, in the last graph, the red curve represents the theoretical phenotypic trait present at the front, $\theta(t):t\mapsto (\sqrt 2/2)t$ (see ), while the blue curve represent the result of the numerical simulation.[]{data-label="fig:4"}](xI.png "fig:"){width="70mm"}![Numerical simulation of (NLoc) with $A=\infty$. The first graph represents $v(53,\cdot,\cdot)$. In the second graph, the red curve represents the theoretical position of the front, $x(t):t\mapsto (2^{5/4}/3)t^{3/2}$ (see ), while the blue curve represent the result of the numerical simulation. Similarly, in the last graph, the red curve represents the theoretical phenotypic trait present at the front, $\theta(t):t\mapsto (\sqrt 2/2)t$ (see ), while the blue curve represent the result of the numerical simulation.[]{data-label="fig:4"}](thetaI.png "fig:"){width="70mm"} Key ideas of the proofs {#sec:key-ideas-proofs} ----------------------- The first difficulty in the analysis of the model (NLoc) is to derive a uniform $L^\infty$ bound on the solution. This difficulty already appeared in [@Bouin], where an $L^\infty$ bound was constructed for travelling waves of (NLoc) (that is steady-states of (NLoc), with an additional drift term), provided the set of phenotypic traits is bounded: $\theta\in[1,\bar \theta]$. A bound of this type was also derived for a parabolic model in [@Turanova], still for a bounded set of phenotypic traits. In Section \[sec:comp-(II)\], we will prove that an $L^\infty$ bound on the solution $v$ of (Nloc) can be established, even when the set of phenotypes is unbounded, that is $\theta\in(1,\infty)$. The proof will be based on a generalization, and a simplification of the argument of [@Turanova]. Equipped with this uniform in time estimate, we will be able to show, in Section \[sec:comparison-models\] that the dynamics of solutions of (NLoc) is similar to the dynamics of the following parabolic model, where the non-local competition kernel $-\langle v\rangle$ is replaced by a local competition $-u$: Model (Loc): $$\begin{aligned} \left\{ \begin{array}{l} {\displaystyle}\partial_t u = \frac\theta 2 \Delta_x u + \frac 12\Delta_\theta u + u \left( 1- u \right) \\[3mm] {\displaystyle}u = u(t,x,\theta), \ t \ge 0, \ x \in {{\mathbb R}}, \ \theta \ge 1, \\[3mm] {\displaystyle}u(0,x,\theta) = u_0(x,\theta) \ge 0,\\[3mm] {\displaystyle}\partial_\theta u(t,x,1)=0,\ t \ge 0, \ x \in {{\mathbb R}}. \end{array} \right.\end{aligned}$$ To description of the dynamics of the solutions of this parabolic equation, we will use a probabilist representation of those solutions through branching brownian motions. This idea was introduced by McKean in [@mckean], who showed that the solution of the Fisher-KPP equation, that is $$\partial_t u-\frac 12\Delta_x u=u(1-u),$$ with the initial condition $u(0,x):=1_{x\leq 0}$ is given by $$u(t,x)=1-\E^x\left[\prod_{i\in I(t)}1_{X_i\leq 0}\right],$$ where $I(t)$ is the number of vertex at time $t$ in a branching Brownian tree, stated at time $0$ at location $x\in\mathbb R$, and $X_i(t)$ is the location of the vertex $i$ at time $t$. Notice that this stochastic representation of the solutions of the Fisher-KPP equation is very different from the Individual-Based Model underlying this equation. We will show in Section \[sec:preliminaries-proba\] that this representation of McKean can be extended to the solutions of (Loc), using a branching Brownian tree in $2D$. This Branching Brownian Motion is not standard: to represent the fact that the dispersion in $x$ is given by $\theta$ in (NLoc), the dispersion of the particles of the BBM in the two directions will be coupled. The result of this analysis for (Loc) can be summed up in Theorem \[T:toads\], where we will use a slightly modified assumption on the initial condition: 1. (Thin tail) We have, for some $C,c>0$ $u_0(x, \theta) \le C \exp( - cx )$ uniformly over $x$ and $\theta$, and $u_0(x,\theta)\to 1$ as $x \to - \infty$, uniformly over $\theta \in [\theta_{\min}, \theta_{\max}]$. \[T:toads\] Let $u_0\in L^\infty(\mathbb R\times\mathbb R_+)$ with compact support in $\theta$ as described in Subsection \[sec:main-results\], and a thin tail (see (2’) above). Let $u(t,x, \theta)$ denote the corresponding solution of (Loc) with either Neuman or Dirichlet boundary condition on $\{\theta=0\}$. For $x \in {{\mathbb R}}$, let $S(t,x) = \sup_{\theta} u(t,x, \theta)$ and let $$\gamma_0 = \frac{2}{3}(2)^{1/4}.$$ We have for all $\gamma > \gamma_0$, $$\label{toads:UB} \sup_{x > \gamma t^{3/2}} S(t, x) \to 0$$ while for $\gamma < \gamma_0$ $$\label{toads:LB} \inf_{x < \gamma t^{3/2} } S(t,x) \to 1$$ as $t \to \infty$. Proving this result will be the core of our study. We will first provide in Section \[sec:upper-bound\] an upper bound on the propagation of solutions of (Loc). We will then describe some *optimal* trajectories of the BBM in Section \[sec:trajectories\], which will in then allow us to derive a precise lower bound on the propagation of solutions in Section \[sec:lower-bound\]. In Section \[sec:conclusion-proof\], we combine those estimates to conclude the proof of Theorem \[T:toads\]. Probabilistic preliminaries {#sec:preliminaries-proba} =========================== We will assume without loss of generality that $\theta_{\min} = 0, \theta_{\max} = 1$. McKean representation --------------------- To start the proof we will use a McKean representation for the equation (Loc). To this end we recall the general idea of this representation. Let $L$ be the generator of some continuous Markov process $X$ taking values in ${{\mathbb R}}^d$. (Thus if $L = (1/2) \Delta$, $X$ is nothing but ordinary Brownian motion). Let $f_0$ be an initial measurable data with $0 \le f_0 \le 1$. Let $(X_t^i, i \in I_t, t \ge 0)$ be a system of branching diffusions based on $L$: that is, each particle branches at rate 1, and move according to the diffusion specified by $L$. All motions and branching events are independent of one another, and note that no particle ever dies. In these notations, $I_t$ is the set of indices of particles alive at time $t$ (note that $I_t$ is thus never empty). We label the positions of the particles at some time $t\ge 0$ by $(X_t^i)_{i \in I_t}$. Let $\P^x$ denote the law of this system when there is initially one particle at $x \in {{\mathbb R}}^d$. \[McKean representation [@mckean]\] \[P:mckean0\] Let $$u(t,x) = 1 - \E^{x} \left[ \prod_{i \in I(t)} (1 - f_0(X_t^i ) )\right]$$ solves the problem: $$\begin{cases} \frac{\partial u}{\partial t} (t,x) &= Lu + u(1-u)\\ u(0,x) & = f_0(x) \end{cases}$$ In fact, McKean’s result is stated for Brownian motion but it is straightforward to extend the result to a general diffusion. Applying the above result to our setting, we are led to the following representation. Introduce a branching Brownian motion in ${{\mathbb R}}^2$, where a general element for the first and second coordinates respecitvely will be labelled $x$ and $\theta$. We denote by $I_t$ the set of indices of particles alive at time $t$, and let $N_t = |I_t|$. We label the positions of the particles by $(W_t^i, \theta_t^i)_{i \in I_t}$. In the case of *Dirichlet* boundary conditions, we further kill the particle if it ever touches zero: that is, we consider $\tilde I(t) = \{ i \in I(t) : \inf_{s\le t} \theta^i_{s,t} >0 \}$. For a fixed $t\ge 0$ and $i \in I_t$, let $(W^i_{s,t}, \theta_{s,t}^i )$ denote the position of the ancestor of the particle $i \in I_t$ at time $s \le t$. We use this to build a new process $(X_t^i, i \in \tilde I_t)_{t\ge 0}$ as follows: we set $$\label{X} X_t^i = \int_0^t \sqrt{ \theta_{s,t}^i } dW_{s,t}^i$$ where the integral above is Itô’s stochastic integral with respect to the Brownian motion $(W^i_{s,t}, 0 \le s \le t)$. \[P:mckean\] Let $u(t,x, \theta)$ be as in Theorem \[T:toads\]. We have $$u(t,x, \theta) = 1- \E^{(x, \theta)} \left[ \prod_{i \in \tilde I_t} (1 - u_0 (X_t^i, \theta_t^i )) \right].\label{mckeanIC}$$ In particular, if $u_0(x, \theta) = H(x) {\mathbf{1}_{\{\theta \in (0,1)\}}}$, where $H$ is the Heavyside function, then $$u(t,x, \theta) = \P^{(0, \theta)} (\exists {i \in I_t}: X_t^i > x \text{ and } \theta_t^i <1 )$$ The only thing which needs to be noted that if $ \theta, W$ are two independent Brownian motions, and if $X_t = \int_0^{t\wedge \tau} \sqrt{\theta_s}dW_s$, then $\{(X_t, \theta_t), t \ge 0\}$, where $\tau$ is the first hitting of zero by $\theta$, forms a Markov process with generator $$L u (x, \theta) = \frac{|\theta|}{2} \frac{\partial^2 u}{\partial x^2} + \frac12 \frac{\partial^2 u}{\partial \theta^2}$$ and Dirichlet boundary conditions. Note that by convention, when $\tilde I_t$ is empty, the product $\prod_{i\in \tilde I_t}$ is set equal to 1. In the case of Neumann boundary conditions, the McKean representation is similar, except that we set $X_t^i = \int_0^t \sqrt{ |\theta_{s,t}^i | } dW_{s,t}^i$ for all $i \in I_t$ and the product is over all $i \in I_t$ in the formula . Many-to-one lemma ----------------- We will use repeatedly the so-called many-to-one lemma, which is a trivial but useful way of relating expected sum of functions of particle trajectories in a branching Brownian motion to the expected value of the same function applied to a single Brownian trajectory. \[L:many\_to\_one\] Let $T$ be a random stopping time of the filtration $ {\mathcal F}_t = \sigma(\bar W_i(s), \theta_i(s), i \in I(s), s\leq t)$, and assume that $T$ is almost surely finite. For any bounded measurable functional $g$ on the path space $C([0, \infty)^2)$, $$\E \left[ \sum_{i \in I_T} g( (W_{s,T}^i, \theta_{s,T}^i)_{s \leq T}) \right] = \E [e^T g((W_s, \theta_s))_{s \leq T})],$$ where $(W_s, \theta_s)_{s \geq 0}$ is a standard planar Brownian motion. Proof of upper bound for the local equation {#sec:upper-bound} =========================================== From now on, and until almost the very end of the proof, we will assume that $n_0(x, \theta) = H(x) {\mathbf{1}_{\{\theta \in (0,1)\}}}$. With the use of Proposition \[P:mckean\] the upper bound from Theorem \[T:toads\] is easy to prove. We fix $\theta$ some arbitrary initial value. We fix $\gamma > \gamma_0$ and call $Z_t = |\mathcal{Z}_t|$, where $$\mathcal{Z}_t = \{ i \in \tilde I_t: X_t^i \ge \gamma t^{3/2} \text{ and } \theta_t^i <1\}.$$ From Proposition \[P:mckean\] we get that $$u(t,x_t(1+\eps), \theta) = \P^{(0, \theta)} ( \mathcal{Z}_t \neq \emptyset ) \le \E^{(0, \theta)} (Z_t).$$ Hence it suffices to show that this expectation tends to 0 as $t \to \infty$. We will principally focus on the case of Dirichlet boundary conditions for readability, and make brief comments along the way on how to adapt the arguments to the case of Neumann boundary conditions. The idea will be to consider the particles such that $\int_0^t \theta_s ds$ has a fixed order of magnitude, namely $a t^2$ (and satisfy $|\theta_t| <1$). More precisely we introduce, for a fixed $h>0$, the set of particles $\tilde I_t(a)$ such that $$\tilde I_t(a) = \left\{ i \in \tilde I_t: a t^2 \le \int_0^t \theta_{s,t}^i ds \le (a+h) t^2 ; \theta_t^i < 1\right\}$$ In the case of Neumann boundary conditions, it is instead the set $ I_t(a) = \{ i \in I_t: a t^2 \le \int_0^t | \theta_{s,t}^i |ds \le (a+h) t^2 ; |\theta_t^i| <1 \}$ which is of interest. We first have the following lemma: \[L:velocity\_theta\] Suppose $a>0$. We have $$\label{tildeIa} \E^{\theta_0} [|\tilde I_t(a) | ] \le \frac{c}{t^{1/2}} e^{t(1- 3a^2/2)}$$ for some constant $c$, uniformly over $\theta_0>0$. In the case of Neumann boundary conditions, the same estimate holds with $\tilde I_t(a)$ replaced by $I_t(a)$. We start by $\tilde I_t(a)$. By the many to one lemma, we immediately get $$\begin{aligned} \E^{\theta_0}(| \tilde I_t(a)| ) & = e^t \P^{\theta_0}\left( a t^2 \le \int_0^t \theta_s ds \le (a+h) t^2 ; \inf_{s\le t} \theta_s > 0 ; \theta_t<1 \right)\\ &\le e^t \P^{\theta_0}\left( a t^2 \le \int_0^t \theta_s ds \le (a+h) t^2 ; \theta_t<1 \right)\end{aligned}$$ We will see that $$\P^{\theta_0}\left( \int_0^t \theta_s ds \ge at^2 ; \theta_t <1 \right)\sim \frac{c}{t^{1/2}} e^{t(1- 3a^2/2)}$$ and will then follow. Note that, by time-reversibility, under $\P^{\theta_0}$, $(\tilde \theta_s: = \theta_t - \theta_{t - s}, 0 \le s \le t)$ is a Brownian motion started from 0 and run for time $t$. Furthermore, if $\theta$ did not hit zero, and $\theta_t \le 1$, and $ \int_0^t \theta_s ds \ge at^2 $, then we certainly have that $ at^2 - t \le \int_0^t \tilde \theta_s ds$. Therefore, $$\P^{\theta_0}\left( a t^2 \le \int_0^t \theta_s ds ; \theta_t<1 \right) \le \P^0 \left( at^2 - t \le \int_0^t \theta_s ds \right).$$ Observe that under $\P^0$, $(\theta_s,s\le t)$ is a centred Gaussian process with covariance $\E(\theta_s \theta_t ) = s\wedge t$. We immediately get that $ \int_0^t \theta_s ds$ is a centred Gaussian random variable with variance $\sigma^2$ which can be explicitly computed: $$\label{variance_int} \sigma^2 = \int_0^t \int_0^t (s\wedge u ) ds du = \frac{t^3}{3}.$$ Consequently, using the easily established fact about standard normal random variables that as $x \to \infty$, $$\label{tail} \P(X \ge x) \sim x^{-1} \frac{e^{-x^2/2}}{\sqrt{2\pi}}$$ we obtain $$\begin{aligned} \P^0\left( \int_0^t \theta_s ds \ge at^2 - t \right) &= \P( X \ge (at^2 - t)/\sigma) \\ & \sim \frac{c}{t^{1/2}} \exp \left( - \frac{a^2 t^4}{2 t^3/3} \right) \\ & = \frac{c}{t^{1/2}} \exp \left( - \frac{3 a^2 t}{2} \right)\end{aligned}$$ as desired. This proves . For the case of Neumann boundary conditions, the proof is essentially similar, except that in the reversibility argument, we observe that if $\int_0^t | \theta_s | ds \ge at^2$ we also have $\int_0^t |\tilde \theta_s | ds \ge at^2 - t$ if we also know that $|\theta_t | \le 1$. The rest of the proof is similar, with the estimate for the tail $\int_0^1 | \theta_s| ds$ coming from a theorem of Tolmatz [@tolmatz] (see also, for a related question, [@svante]). From now on we focus on the case of Dirichlet boundary conditions, but the case of Neumann boundary conditions can be treated in exactly the same way thanks to Lemma \[L:velocity\_theta\]. Let $a_1 = \sqrt{2/3}$ and fix $a >a_1$. Let $B_1$ be the event that there is an $i\le N_t$ such that $$\label{apriori} \int_0^t \theta_{s,t}^i ds \ge at^2 \text{ and } |\theta_t^i| <1$$ Then $\sup_{\theta_0}\P^{(0, \theta_0) }(B_1) \to 0$. Apply the many-to-one Lemma and Lemma \[L:velocity\_theta\] to find that the expected number of particles satisfying tends to 0, and then use Markov’s inequality. Consider the filtration ${\mathcal G}_t = \sigma( \theta^i_{s,t}, i \in I_t, s\le t)$. When we condition on ${\mathcal G}_s$, $X_t^i$ is a Gaussian random variable with variance $\int_0^t \theta_{s,t}^i ds$. Fix $a = 2a_1$. We get, applying the many-to-one Lemma again, on the event $B_1^c$, $$\begin{aligned} \E^{(0, \theta_0)}(Z_t | {\mathcal G}_t) & \le \sum_{i \in I_t} \frac{\sqrt{\int_0^t \theta^i_{s,t} ds } }{x} \exp \left( - \frac{x^2}{2 \int_0^t \theta_{s,t}^i ds } \right) \\ & \le \sum_{n=0}^{2a_1/h} |I_t( n h) | \frac{\sqrt{nh t^2}}{ct^{3/2}} \exp \left( - \frac{(\gamma t^{3/2})^2}{2 nh t^2 } \right) \\ & = \sum_{n=0}^{2a_1/h} |I_t( n h) | t^{-1/2}\sqrt{nh} \exp \left( - \frac{\gamma^2}{2nh} t \right) .\end{aligned}$$ Taking expectations, we get $$\begin{aligned} \E^{(0, \theta_0)}(Z_t) & \le \sum_{n=0}^{2a_1/h} t^{-1/2}\sqrt{2a_1} t^{-1/2} \exp( t(1- 3(nh)^2/2)) \exp \left( - \frac{\gamma^2}{2nh} t \right) \nonumber \\ & \le \frac{(2a_1)^{3/2} }{ht} \exp ( t \max_{a\in[0,2a_1]} \left\{1- \frac{3a^2}{2} - \frac{\gamma^2}{2a} \right\} ) \label{ubExp}\end{aligned}$$ We claim that for $\gamma> \gamma_0$, this maximum, call it $M(\gamma)$, is negative. Indeed, let $$\phi(a) = 1- \frac{3a^2}{2} - \frac{\gamma^2}{2a}.$$ At a maximum point of $\phi$ we also have $\phi'(a) = 0$ or $-3a + \gamma^2/ (2a^2) = 0$ or $$\gamma^2 = 6a^3,$$ in which the maximum is given by $$M(\gamma) = 1- \frac{3a^2}{2} - 3 a^2 = 1- \frac{9a^2}{2}.$$ But since $\gamma>\gamma_0 = (2/3) 2^{1/4}$ we get that $a > a_0 = \sqrt{2}/3$. It follows that $M(\gamma) < M(\gamma_0) = 0$. We deduce that $$\P^{(0, \theta_0)}(Z_t \ge 1) \le \E^{(0, \theta_0)}(Z_t \mathbf{1}_{B_1^c}) + \P^{(0, \theta_0)}(B_1) \to 0, \label{conclUB}$$ uniformly over $\theta_0$, as desired. Identifying the relevant stochastic trajectories {#sec:trajectories} ================================================ In this short section, we discuss the intuitive idea on which the rest of the proof is based. For a lower bound, the idea is to apply a second moment argument to say that there are indeed particles at distance $x_t = \gamma t^{3/2} $ with high probability for $\gamma < \gamma_0$. In order to do so, we need to identify the relevant trajectories which make this event possible. It can be guessed that a particle satisfying $\theta_0 = 0$ and $\int_0^t \theta_s ds \ge at^2$ fluctuates around a deterministic function $\bar f$ on the interval $[0,t]$ given by $$\label{GoodFunction} \bar f(s) = 3a\left(s - \frac{s^2}{2t}\right) ; \ \ s\in[0,t].$$ Note that $\bar f$ is not linear. This is somewhat surprising as optimal Brownian paths which get to a large distance are roughly linear; as can be seen by trying to minimise the Dirichlet energy of functions going from 0 to a far away point $x$ in a given amount of time $t$. (Indeed, optimal paths in the usual KPP equation are roughly linear). To identify the optimal trajectories here, recall that the ’probability’ of a given path $f$ is roughly $\exp( - (1/2) \int_0^t \bar f'(s)^2 ds )$. (This can be made rigorous for instance using the theory of large deviations, see Schilder’s theorem.). Hence the function $\bar f$ is obtained as the minimiser of the Dirichlet energy subject to a constraint: $$\bar f \equiv \arg\min \left\{ \frac12 \int_0^t \bar f'(s)^2ds: \text{ subject to } \int_0^t \bar f(s)ds = at^2\right\}.$$ By a standard calculus of variations argument, (i.e., $\bar + \eps \phi$ has a bigger Dirichlet norm than $\bar f$ for any function $\phi$ with $\int \phi = 0$) we deduce that $\int_0^t \bar f '' \phi = 0$ for any such function after integration by parts. We deduce $\bar f '' $ is constant. So $\bar f$ is a parabola, of the form $$\bar f(s) = bs + cs^2$$ (there is no constant term as $f(0) = 0$. We claim that an optimal trajectory must further satisfy $\bar f'(t) = 0$. This can be justified on a heuristic level but can also be taken as an ansatz otherwise; the fact that the upper bound and lower bound match up then justify it a posteriori. At any rate, this leads to the equation $2ct + b = 0$. Finally, plugging $\int_0^t \bar f(s) ds = at^2$ gives the value of the coefficients: $b = 3a$, $c = -3a/(2t)$. We then find $$\label{cost} \int_0^t \bar f'(s)^2 ds = 3a^2 t$$ consistent with the above lemma. The proof of the lower bound relies crucially on the identification of the function $\bar f$ above. Indeed our strategy will be to show that there are particles at the desired distance $x = \gamma t^{3/2}$ by also requiring that $\theta_s$ stays close to the time-reversal of $\bar f$. In particular, this will explain the requirement on $\theta_0 = \bar f (t) = 3at/2 = (\sqrt{2} /2) t$. (Note that this is only half of the maximal value of $\theta^i_t$ among all particles $i \in I_t$, since the particles $\theta^i$ are performing standard BBM). The particle will then reach its final position, $x = \gamma t^{3/2}$ by moving linearly *on the correct timescale*, that is the timescale which turns $X$ into a Brownian motion. Consequently, the position at time $s$ of a particle ending up at $\gamma t^{3/2}$ will be approximately given by $$X_s = W\left( \int_0^s \theta_u du\right) \approx \mu \int_0^s \theta_u du ,$$ where $\mu = \gamma t^{3/2} / \left( \int_0^t \theta_u du \right) = \gamma t^{3/2} / (at^2) = \gamma t^{-1/2} /a$. Therefore we find, $$X_s \approx \mu \int_0^s \bar f(t-u) du$$ and making the relevant calculation, $$\label{OptTraj} X_s \approx \frac{3 \gamma}{2} \left( s \sqrt{t} - \frac{s^3}{3 t^{3/2}}\right).$$ It would be interesting to know whether the above guesses can give rise to a simplified and purely analytic proof of the main theorem of this paper, in the spirit of the recent PDE proof of Bramson’s logarithmic delay in the KPP equation by Hamel, Nolen, Roquejoffre and Ryzhik [@HNRR]. Open questions -------------- From a probabilistic perspective, this raises several questions of interest. Do particles near the maximum have a trivial correlation structure, as in the usual branching Brownian motion? (In the terminology of spin glasses, this would correspond to a 1-step replica symmetry breaking for the associated Gibbs measure). Secondly, can the shape of the front be described in more detail? We believe that the effective size of the front (say, the spacing between the first and third quartiles of $S(t,x)$, or some other function of $u$) does not stay of order 1 as in the case of the usual branching Brownian motion. Instead we believe that the front spreads over time with a spacing of size roughly $\sqrt{t}$. Supporting evidence for this comes from the proof of the lower bound, where a fluctuation of size $\sqrt{t}$ is inherent in the identification of relevant trajectories. Scaling by $\sqrt{t}$, does one obtain a limiting shape for the front? Proof of lower bound for local equation {#sec:lower-bound} ======================================= For this we rely on our understanding of the optimal trajectories in the previous section. Fix $\gamma<\gamma_0$ and set $a$ such that $6a^3 = \gamma^2$, so $a<a_0$ and $\phi(a) = M(\gamma) >0$. Set $$\bar f(s) = 3a\left(s - \frac{s^2}{2t}\right),$$ and set $f$ to be a $C^2$ function which is always greater than $1/2$ and which coincides with $\bar f (t-s)$ on $[0, t-1]$. Thus for $s \le t-1$, $$\label{f} f(s) = \bar f(t-s) = \frac{3a}{2} \left(t - \frac{s^2}t\right).$$ Now set $$\begin{aligned} M_s &= ( t-s+1/4)^{3/4}, \ \ 0 \le s\le t \\ m & = 10 t^{1/2} \\ \tau &= at^2 \\ \mu& = \frac{\gamma t^{3/2}}{\tau}. \end{aligned}$$ Recall that by choice, $\int_0^t \bar f(s)ds = at^2 = \tau$. Let $\Omega = C([0,\infty), {{\mathbb R}})$ be the set of continuous trajectories, equipped with the Borel $\sigma$-field induced by the topology of uniform convergence on compacts. For $\theta \in \Omega$ set $$J_s = \int_0^s \theta_s ds; \ \ J = J_t$$ Good events ----------- Our goal will be to estimate $u(t,x, \theta)$ for values of $x$ close to $x= \gamma t^{3/2}$ where $\gamma$ is as above of $\theta$ satisfies: $| \theta - 3at/2| \le 1$. Hence we will assume that our initial condition is such that there is initially one particle at $x_0, \theta_0$ with $$\label{IC} |x_0| \le 1; \ \ \left|\theta_0 - \frac{3a}{2} t \right| \le 1.$$ We introduce the event $$\label{Ai} {\mathcal A}= \left\{ \theta \in \Omega: \left| \theta_s - f(s) \right| \le M_s \text{ for all } s \le t \right\} \cap \{ \tau - t \le J \le \tau \}.$$ In other words the event ${\mathcal A}$ is the event that $\theta $ remains not too far away (at most $M_s$) from the function $ f$, throughout the interval $[0,t]$, and has a total integral which is smaller than that of $f $ but by no more than $O(t)$. Recall that the function $ f$ is, apart from an unimportant additive constant $(1/2)$, the optimal trajectory identified in the previous section, which will guarantee that $\int_0^t \theta_s ds \approx at^2 = \tau$ and satisfies $\theta_t \le 1$. The error bound $M_s$ is decreasing from approximately $O(t^{3/4})$ initially to $M_t < 1/2$ at the end. In particular, note that if ${\mathcal A}$ holds, then $0\le \theta_t \le 1$ and moreover $\theta$ never hits zero on $[0,t]$, and so there is no difference between $\int_0^t \theta_s ds$ and $\int_0^t |\theta_s|ds$. (In particular, the following proof works for both Dirichlet or Neumann boundary conditions.) We introduce a second event ${\mathcal B}$ which deals with the $W$ coordinate, and which is defined as follows. First observe that given Define the event ${\mathcal B}$ by $$\label{Bi} {\mathcal B}: = \left\{ W \in \Omega: \sup_{u \le \tau} W_s - s \mu \le m , \sup_{u\in [\tau - t, \tau] }\left| W_{u} - \gamma t^{3/2} \right|\le m \right\}$$ for some small enough $\delta$. In other words, the event ${\mathcal B}$ is that the Brownian path progresses linearly towards its target position $\gamma t^{3/2}$ up to time $\tau$, is always below the corresponding straight line (shifted by about $m = 10\sqrt{t}$), and lies within $m = 10 \sqrt{t}$ of that target throughout the interval $[\tau - t, \tau]$. Now return to the branching Brownian motion $(\theta^i_t, W^i_t)$ from the previous section. For $i \in \tilde I_t$, set $$J^i_t = \int_0^t \theta^i_{s,t}ds,$$ and set $K^i(\cdot)$ to be the cad lag inverse of $J^i_\cdot$. We will sometimes write $J^i$ for $J^i_t$ in order to lighten to the notations. Set $\tilde W^i(s) = X^i(K^i(s))$, and note that by Dubins–Schwarz theorem, for a fixed $i \in \tilde I_t$, $\tilde W^i$ is just a Brownian motion over $[0, J^i]$. For $i \in \tilde I_t$, let ${\mathcal A}_i = \{ \theta_{\cdot, t}^i \in {\mathcal A}\}$ and ${\mathcal B}_i = \{ W^i \in {\mathcal B}\}$. We shall consider the good event ${\mathcal G}_i = {\mathcal A}_i \cap {\mathcal B}_i$ and set $$Z= \sum_{i \in I(t) } 1_{ {\mathcal G}_i }$$ the number of particles which satisfy this good event. Note that if with high probability there is a particle satisfying ${\mathcal G}_i$, (i.e. if $Z >0$) then at time $t$ the position of this particle is $ \gamma t^{3/2} (1+ o(1))$. Indeed, the position $X_t^i$ of particle $i$ at time $t$ is by definition $\tilde W_{J^i}^i $. On ${\mathcal A}_i$, $|J^i - \tau | \le t$ so if ${\mathcal B}_i$ also holds, $|\tilde W^i_{J^i} - \gamma t^{3/2} | \le m = 10 t^{1/2}$. Thus if $Z>0$ there is a particle such that ${\mathcal A}_i \cap {\mathcal B}_i$ holds and hence the maximal particle is greater than $ct^{3/2} - m$. We conclude using the McKean representation. Hence it suffices to prove that $Z>0$ with probability tending to 1 as $t \to \infty$. To this end we use the Payley–Zigmund inequality: $$\label{PZ} \P( Z>0) \ge \frac{ \E( Z)^2}{ \E(Z^2)}$$ We will thus compute the first and second moment of $Z$, and show that $\E(Z^2) \le C \E(Z)^2$. This will show that $\P(Z>0) \ge p$ for some uniform $p>0$. A simple argument will then show that in fact we can bound this probability from below by something arbitrarily close to 1. First moment of $Z$ ------------------- We will first establish the following lower bound on the first moment of $Z$. \[P:EZ\] $$\label{EZ} \E(Z) \ge \frac{C}{t^{5/2}} \exp\left( t \left( 1 - \frac{3a^2}{2} - \frac{\gamma^2 }{2a} \right) \right) .$$ Note that, up to a polynomial factor, this matches the upper bound in . This polynomial term will be tracked carefully over the course of the proof, to match the upper bound on the second moment later on, so that the bound in is bounded away from zero. By the many-to-one Lemma, $\E(Z) = e^t \P({\mathcal A}\cap {\mathcal B})$. Observe that ${\mathcal A}$ and ${\mathcal B}$ are independent, so we can estimate $\P({\mathcal A})$ and $\P({\mathcal B})$ separately. Let ${{\mathbb Q}}$ denote the law of Brownian motion (started from $\theta_0 = f(t) = 3at/2$ with drift $-\bar f(s)$, that is, the distribution on $C[0,t]$ of the process $(\theta_s - \bar f(s), s \le t )$ under $\P^{\theta_0}$. Then by Girsanov’s theorem, $$\begin{aligned} \P( {\mathcal A}) & = {{\mathbb Q}}\left( |\theta_s| \le M_s \text{ for all } s\le t ; - t \le J \le 0 \right) \\ & = \E^0 \left ( 1_{\{ | \theta_\cdot |\le M_\cdot , -t \le J \le 0 \} } \exp ( - \int_0^t \bar f'(s) d\theta_s - \frac12 \int_0^t \bar f'(s)^2 ds ) \right).\end{aligned}$$ Now, by stochastic integration by parts (or Itô’s formula), since $ \bar f$ is differentiable and is thus a finite variation process, we deduce that it has zero cross-variation with the Brownian motion $\theta$, and hence $$\begin{aligned} \int_0^t \bar f'(s) d\theta_s &= [ \bar f'(t) \theta_t - \bar f'(0) \theta_0] - \int_0^t \theta_s f''(s) ds \\ & = \frac{3a}t \int_0^t \theta_s ds + O(1)\end{aligned}$$ since $\bar f$ satisfies $\bar f'(0) = 0$, $\theta_t = O(1)$ on ${\mathcal A}$ and $\bar f'(t) = O(1)$, and $f''(s) = (-3a/t)$ for all $s\le t$. Therefore, using definition of the event ${\mathcal A}$, and the relation , $$\begin{aligned} \P({\mathcal A}) &= \exp \left( - \frac12 \int_0^t f'(s)^2ds \right) \E^0 \left ( 1_{\{ | \theta_\cdot |\le M_\cdot , -t \le J \le 0 \} } \exp \left( - \frac{3a}t \int_0^t \theta_s ds\right) \right) \nonumber \\ & \ge \exp \left( - \frac{3a^2}{2}t \right) \P^0 \left( | \theta_\cdot |\le M_\cdot , -t \le J \le 0 \right) \exp( - 3a)\label{PA1} \end{aligned}$$ Now, we will need the following Lemma: \[L:theta\_int\_dev\] $$\P^0 \left( | \theta_\cdot |\le M_\cdot , -t \le J \le 0 \right) \ge \frac{C}{t} $$ Let $F_{a,b}$ be the event that $|\theta_s| \le M_s$ for all $a\le s\le b$. We also introduce the stronger event $F_{a,b}^+$ that $|\theta_s| \le M_s/2$ for all $a\le s\le b$ We will denote $\bar \theta_s = \theta_{t-s}$ and also introduce $\bar F_{a,b} = \{|\bar \theta_u |\le \bar M_u \text{ for all } u \in [a,b]\}$ where $\bar M_u = M_{t-u}$. Fix $s_0 = \lambda (\log t)^2$ where $\lambda$ is a big constant to be chosen later. We will work conditionally given $\bar \Theta_{s_0} = \sigma( \bar \theta_s, s\le s_0)$ and assume that $F^+_{t-s_0, t}$ holds. Note that, conditionally on $\bar \Theta_{s_0}$, the process $(\theta_s, 0 \le s\le t - s_0)$ is a Brownian bridge of duration $t-s_0$ from 0 to $z = \bar \theta_{s_0}$, a distribution which we will denote by ${\mathbb{P}^{0 \stackrel{t-s_0}{\rightarrow} z} }$. Its time-reversal is hence a Brownian bridge of duration $t-s_0$, from $z$ to 0. Observe that a Brownian bridge from $0$ to $z$, of duration $t-s_0$ , when restricted to the interval $[0,3t/4]$, is absolutely continuous with respect to a Brownian motion started from 0, and furthermore the density is uniformly bounded by a constant $C$ (see e.g. (6.28) in [@KS]), since $z = o(\sqrt{t})$ on $F_{t-s_0, t}^+$. Therefore, for $s\le 3t/4$, $$\begin{aligned} {\mathbb{P}^{0 \stackrel{t}{\rightarrow} z} } ( F_{0, 3t/4}) & \le C \P^0(F_{0,3t/4}) \le \P( \sup_{s \le 3t/4} \theta_s \ge c t^{3/4}) \le 2 e^{- c t^{1/2}} \label{hitM0}\end{aligned}$$ by the reflection principle and . Denote $\bar E_s = \bar F^c_{s,2s}$. Note that once again the law of ${\mathbb{P}^{z \stackrel{t-s_0}{\rightarrow} 0} }$, restricted to $[0, t/2]$, is absolutely continuous to a Brownian motion started from $z$ and the density is uniformly bounded. Hence for $s\le t/4$, by scaling, since $|z| \le (1/2) \bar M_{s_0}$ on $F_{t-s_0, t}^+$, $$\begin{aligned} {\mathbb{P}^{z \stackrel{t-s_0}{\rightarrow} 0} }(\bar \theta_u = \bar M_u \text{ for some } u \in [s, 2s]) &\le C \P^z ( \bar \theta_u = \bar M_u \text{ for some } u \in [s, 2s]) \nonumber \\ &\le C \P^0 ( \sup_{u \le 2} \theta_u \ge C' s^{1/4} ) \nonumber \\ & \le C e^{- C' s^{1/2}} \label{hitM}\end{aligned}$$ by the reflection principle and again. By combining with we deduce that $$\label{hitM2} \P^0( F_{0, t - s_0} ^c | \bar \Theta_{s_0}) \le t^{-2}$$ for $\lambda$ sufficiently large, on $F_{t-s_0, t}^+$. Now let $J_1 = \int_0^{s_0} \bar \theta_u du$ and let $J_2 = J - J_1 = \int_{s_0}^t \bar \theta_u du$. Note that $$\begin{aligned} \P( -t \le J \le 0 ; F_{s_0,t} | \bar \Theta_{s_0}) & = \P(-t - J_1 \le J_2 \le -J_1 | \bar \Theta_{s_0}) - \P( F^c_{s_0,t} | \Theta_{s_0}) .\end{aligned}$$ Note that, given $\bar \Theta_{s_0}$, $J_2$ is the integral of a certain Gaussian process, namely a Brownian bridge of duration $t- s_0$ from $\bar \theta_{s_0}$ to 0. This has mean $m_2 = z ( t - s_0)/2$ and a computation similar to shows that it has a variance of order at least $C t^3$. Hence $$\P(-t - J_1 \le J_2 \le -J_1 | \bar \Theta_{s_0}) = \P\left( \mathcal{N}(0,1) \in \left[ \frac{-m_2 - t - J_1}{Ct^{3/2}}, \frac{ - m_2 - J_1}{Ct^{3/2}} \right] \right).$$ On the event we are considering, $0 \le m_2 \le t (\log t)^{3/4}$ and $0\le J_1\le s_0^{7/4} \le (\log t)^2$, so the interval in the right hand side has a size at least $O(1/\sqrt{t})$ and is located within $[-1,1]$ where the density of $\mathcal{N}(0,1)$ is bounded away from zero. Hence, $$\P(-t - J_1 \le J_2 \le -J_1 | \bar \Theta_{s_0}) \ge \frac{C}{\sqrt{t}}.$$ Consequently, taking expectations so as to remove the conditional expectation given $\Theta_{s_0}$, we deduce that $$\begin{aligned} \P( - t \le J \le 0; F_{0,t}) & \ge \E(\mathbf{1}_{F_{t- s_0,t}^+} \P( -t \le J \le 0 ; F_{0,t - s_0} | \bar\Theta_{s_0}) )\nonumber \\ &\ge \E(\mathbf{1}_{F_{t- s_0,t}^+} [\P( -t \le J \le 0 | \bar\Theta_{s_0}) - \P ( F_{0,t - s_0} | \bar \Theta_{s_0}) ] ) \nonumber \\ &\ge \P( F_{t - s_0, t}^+ ) C ( t^{-1/2} - t^{-2} ) \label{hitM3} $$ Note also that $\P(F_{t,t}^+) \ge C t^{-1/2}$ and, by following the argument in , $\P(F_{t-s_0, t}^+ | F_{t,t}^+) \ge p$ for some $p>0$. It follows that $\P( F_{t - s_0, t}^+ ) \ge C t^{-1/2}$. Plugging this into we get $$\P( - t \le J \le 0; F_{0,t}) \ge Ct^{-1}$$ as desired. Altogether, combining Lemma \[L:theta\_int\_dev\] and , we get $$\label{PA} \P({\mathcal A}) \ge \frac{C}{t} \exp \left( - \frac{3a^2}{2}t \right).$$ We now turn to $\P( {\mathcal B})$, which is somewhat easier. Recall the value of the drift $$\mu = \frac{ \gamma t^{3/2} }{\tau} = O( t^{-1/2}),$$ so that $\mu \tau = \gamma t^{3/2}$ (i.e., $\mu$ is the slope of the line involved in the definition of ${\mathcal B}$). From Girsanov’s theorem, $$\begin{aligned} \P( {\mathcal B}) & = \P\left( \sup_{u \le \tau} (W_u - u \mu) \le m ; \sup_{u \in [\tau - t, \tau] } | W_{u} - \gamma t^{3/2} | \le m\right) \\ & = \E^0 \left( 1_{\{ \sup_{u \le \tau} (W_u - u \mu) \le m ; \sup_{u \in [\tau - t, \tau] } | W_{u} - \gamma t^{3/2} | \le m \} } \exp ( - \mu W_{\tau} - \frac12\mu^2 \tau ) \right) \\ & \ge \exp ( - \mu m - \frac12 \mu^2 \tau ) \P^0 \left( \sup_{u \le \tau} W_u \le m ; \sup_{u \in [\tau - t, \tau] } | W_{u} | \le m\right) \\ & \ge O(1) \exp ( - \frac{c^2 t }{2a} ) \P^0 \left( \sup_{u \le \tau} W_u \le m ; \sup_{u \in [\tau - t, \tau] } | W_{u} | \le m \right).\end{aligned}$$ $$\P^0 \left( \sup_{u \le \tau} W_u \le m ; \sup_{u \in [\tau - t, \tau] } | W_{u} | \le m \right) \ge c m^3 \tau^{-3/2}$$ We can use the reflection principle to compute this. Indeed, we know that for $0\le a \le b$, letting $S_T = \sup_{s\le T} W_s $, $$\P_0(S_T \ge b, W_T \le a) = \P( W_T \ge 2b - a) =\int_{2b -a}^\infty \frac1{\sqrt{2\pi T}} e^{ - x^2 / (2T)} dx$$ and hence the joint density of $(S_T, W_T)$ at the point $0 \le a\le b$ is, after differentiating twice the above expression, is $$\label{reflection} \frac{2(2b- a)}{2\sqrt{2\pi} T^{3/2}} e^{- (2b- a)^2/(2T)}$$ If $T \ge C m^2$, by integrating, we find $$\begin{aligned} \P( \sup_{u \le T} W_u \le m ; W_{T} \in [- 3m/4, - m/4] ) & \ge \int_{-3m/4}^{-m/4}\int_0^m \frac{C(2b-a)}{T^{3/2}} e^{ - (2b- a)^2/(2T)} db da\\ & \ge \frac{C}{T^{3/2}} \int_{-3m/4}^{- m/4} \int_0^m (2b-a ) db da \ge \frac{C m^3}{T^{3/2}}. \end{aligned}$$ We apply this result with $T = \tau - t$ and then apply the Markov property at time $T$, to find that $$\begin{aligned} & \P^0 \left( \sup_{u \le \tau} W_u \le m ; \sup_{u \in [\tau - t, \tau] } | W_{u} | \le m \right) \\ &\ge \P( \sup_{u \le T} W_u \le m ; W_{T} \in [- 3m/4, - m/4] ) \P^0 (\sup_{u \in [0,t]} W_u \le m /4)\ge \frac{C m^3}{T^{3/2}} p\end{aligned}$$ for some $p>0$, by scaling. The result follows. Putting things together, we find $$\label{PB} \P({\mathcal B}) \ge O(1) t^{-3/2} \exp ( - \frac{\gamma^2 t }{2a} ) ,$$ Now, Proposition \[P:EZ\] follows by combining and . Second moment of $Z$ -------------------- In this section we prove the following upper bound on the second moment of $Z$. \[P:Z2\] $$\E(Z^2) \le \frac{C}{t^5} \exp( 2t ( 1 - \frac{3a^2}{2} - \frac{\gamma^2 }{2a} ) ) .$$ To compute the second moment we use a modified version of the many-to-one Lemma, which can be called the many-to-two lemma. We begin with a useful definition for what follows. Let $B_1$ be a real Brownian motion and $T\ge 0$ a possibly random time. We say that $B_2$ branches from $B_1$ at time $T$ if there exists another Brownian motion $W$, independent of $B_1$ and $T$, such that $$B_2(u) = \begin{cases} B_1(u) & \text{ for } u \le T;\\ B_1(u) + W(u-T) & \text{ for } u \ge T. \end{cases}$$ This definition is in fact symmetric: if $B_2$ branches from $B_1$ at time $T$, then $B_1$ branches from $B_2$ at time $T$. We will sometime simply say that $B_1$ and $B_2$ branch from each other at time $T$. \[L:many\_to\_two\] Let $F$ be a measurable functional on $C[0, t]$, and set $$Z = \sum_{i \in I_t} F(X_{s,t}^i, 0 \le s \le t).$$ Let $T$ be an exponential random variable with parameter $2$, and let $B_1, B_2$ be Brownian motions branching from each other at time $T$. Then we have $$\E(Z^2) = e^{2t} \E [ e^{T\wedge t} F(B_1(s), 0 \le s \le t ) F(B_2(s), 0 \le s \le t)].$$ See [@manytofew] for a slightly more general result. We will use it as follows. For a fixed $0< s < t$, let $(B_1, B_2)$ denote Brownian motions branching from each other at time $T=s$. (Technically we should indicate the dependence on $s$ within the notation, but we will avoid doing this in order to ease readability). We then have, if $F(X) = {\mathbf{1}_{\{ X\in {\mathcal A}\}}}$ for some Borel set ${\mathcal A}$ on $C[0,t]$ $$\label{many2} \E(Z^2) = \E(Z) + 2 \int_0^t e^{2(t-s)} e^s \P(B_1\in {\mathcal A}, B_2 \in {\mathcal A}) ds.$$ This can be interpreted as follows: write $Z^2 = \sum_{i,j} {\mathbf{1}_{\{X_i, X_j \in {\mathcal A}\}}}$. If we decompose on the time at which the particles $i$ and $j$ have their most recent common ancestor, say $s$, then there are $e^s$ expected potential ancestors at generation $s$, and each produces $e^{t-s}$ expected descendants at generation $t$. Hence the number of pair of descendants descending from a given individual at generation $s$ is $e^{2(t-s)}$ if we order them, and each pair is counted twice (hence an additional factor 2). Hence in view of , our task will be to show that if ${\mathcal A}$ and ${\mathcal B}$ are the events defined in and , and ${\mathcal G}= {\mathcal A}\cap {\mathcal B}$, then $$\label{goal} \int_0^t e^{2(t-s)} e^s \P(B_1\in {\mathcal G}, B_2 \in {\mathcal G}) ds \le \frac{C}{t^6} \exp( 2t ( 1 - \frac{3a^2}{2} - \frac{\gamma^2 }{2a} ) ) .$$ This will imply Proposition \[P:Z2\] as it is clear that $\E(Z) \to \infty$ and thus $\E(Z)\le \E(Z)^2$ for $t$ large enough. We thus turn to the Let $0 \le s \le t$. We begin by spelling out the event $\{B_1, B_2 \in {\mathcal G}\}$ more explicitly. Take two independent real Brownian motions $\theta$ and $W$. We can then introduce a proces $(\theta'_u, 0\le u \le t)$ which branches from $\theta$ at time $s$, and a process $W'$ which is a Brownian motion branching from $W$ at time $J_s$, where $$J_s = \int_0^s \theta_u du = \int_0^s \theta'_u du.$$ Thus $W'_u = W_u$ for $u \le J_s$ and $W'_u = W_{J_s} + \tilde W_{u - J_s}$ for $u \ge J_s$, where $\tilde W$ is independent from $(\theta, W)$. **Event ${\mathcal B}$**. The event $\{B_1, B_2 \in {\mathcal G}\} $ can be reformulated as $\{ \theta, \theta' \in {\mathcal A}\} \cap \{ W, W' \in {\mathcal B}\}$. We will first condition on the entire processes $\theta, \theta'$ and compute the conditional probability that $(W, W') \in {\mathcal B}$. Recall our notation $J_s = \int_0^s \theta_u du$. For this computation, as before, the key result we use is Girsanov’s theorem. The joint law of $(W_u - u \mu, W'_u - u \mu, u \le \tau)$ has density $$\begin{aligned} Z_B &= \exp\left( \left\{- \mu W(J_s) - \frac12 \mu^2 J_s \right\} - \{\mu(W(\tau) - W(J_s)) + \frac12 \mu^2 (\tau - J_s)\} \right.\\ & \ \ \ \ \ \ \ \ \ \left. -\left\{ \mu(W'(\tau) - W'(J_s)) + \frac12 \mu^2 (\tau - J_s)\right\} \right). \label{Girsanov2}\end{aligned}$$ We have grouped the terms in brackets so that they are independent of one another (conditionally on $(\theta, \theta')$), but in fact we will reorder them in a convenient way later on. Consequently, $$\begin{aligned} \P( W, W' \in {\mathcal B}| ( \theta, \theta') ) & = \E\left( \left. Z_B {\mathbf{1}_{\{ \sup_{u \le \tau} (W_u, W'_u ) \le m ; \sup_{u \in [\tau - t, \tau] } | W_{u}, W'_u | \le m\}}} \right| (\theta, \theta') \right) \\ & \le \E\left( \left. Z_B {\mathbf{1}_{\{ \sup_{u \le J_s}W_{u} \le m\}}} {\mathbf{1}_{\{\sup _{u \in [J_s, \tau]}(W_u, W'_u ) \le m\}}} {\mathbf{1}_{\{W_\tau, W'_\tau \ge - m \}}} \right| ( \theta, \theta') \right)\\ &\le \exp\left( - \frac12 \mu^2 \tau - \frac12 \mu^2 (\tau - J_s) \right) \times\\ & \times \E \left( e^{ \mu W(J_s) } {\mathbf{1}_{\{W(J_s) \le m\}}} e^{ - \mu W(\tau) - \mu W'(\tau) } {\mathbf{1}_{\{\sup _{u \in [J_s, \tau]}(W_u, W'_u ) \le m\}}} {\mathbf{1}_{\{W_\tau, W'_\tau \ge - m \}}} \right) \end{aligned}$$ Now, we need the following lemma: \[L:3/2\] For all $x, y >0$, $$\P( \inf_{u\le 2T} W_u \ge 0 ; W_{2T} \in [ 0, y]| W_0 = x) \le \frac{Cx y^2 }{T^{3/2}}.$$ where $C$ is independent of $x$, $y$. The cases where $x\le \sqrt{T}$ and $x \ge \sqrt{T}$ can be treated similarly. For simplicity we focus on the case $x \le \sqrt{T}$ which is slightly more interesting. We first note that $\P(\inf_{u\le T} W_u \ge 0 | W_0 = x) \le c x/\sqrt{T}$ by well known (and very simple) estimates. By the Markov property of Brownian motion at time $T$, conditionally given $\inf_{u\le T} W_u \ge 0 $ and $W_{T}=z$, the probability that $\inf_{u\in [T, 2T]} W_u \ge 0$ and $W_{2T} \in [0, y]$ can be expressed as $$\begin{aligned} \label{32} \P_x( \inf_{u\in [0, 2T]} W_u \ge 0 ; W_{2T} \in [0,y] | W_{T} = z; \inf_{u \le T} W_u \ge 0 ) & = \P_0( \sup_{u \le T} W_u \le z; W_{T} \in [z-y, z] )\end{aligned}$$ The latter can be computed using the reflection principle. Indeed, we know that for $0\le a \le b$, letting $S_T = \sup_{s\le T} W_s $, $$\P_0(S_T \ge b, W_T \le a) = \P( W_T \ge 2b - a) =\int_{2b -a}^\infty \frac1{\sqrt{2\pi T}} e^{ - x^2 / (2T)} dx$$ and hence the joint density of $(S_T, W_T)$ at the point $0 \le a\le b$ is, after differentiating twice the above expression, is $$\frac{2(2b- a)}{2\sqrt{2\pi} T^{3/2}} e^{- (2b- a)^2/(2T)}$$ Integrating over $0\le b \le z$ and $0 \le z-y \le a\le b$ gives us, after applying Fubini’s theorem, making a change of variable, $$\begin{aligned} \P_0( \sup_{u \le T} W_u \le z; W_{T} \in [z-y, z] ) & = \int_{z-y}^z {da} \int_{a}^z \frac{(2b -a)}{ \sqrt{2\pi } T^{3/2} } e^{- (2b- a)^2/(2T)}db \\ & \le \frac{C}{T^{3/2}} \int_{z-y}^z {da} \int_a^z (2b-a) db\\ & = \frac{C}{T^{3/2}} \int_{z-y}^z z(z-a) {da} = \frac{C}{T^{3/2}} zy^2.\end{aligned}$$ Taking expectations in this yields $$\begin{aligned} \P_x( \inf_{u\in [0, 2T]} W_u \ge 0 ; W_{2T} \in [0,y] ) &\le \frac{Cxy^2 }{T^2} \E_x ( W_T | \inf_{u \le T} W_u \ge 0 ) \\ & \le \frac{Cxy^2}{T^{3/2}}. \end{aligned}$$ Indeed, the position at time $T$ of $W_T$, given $\inf_{u \le T} W_u \ge 0$, is dominated stochastically from above by a three-dimensional Bessel process started from $x$ at time $T$, whose expectation is $\E_x(R_T) \le c_3 \sqrt{T}$ by scaling, for some constant universal constant $c_3$, since we assumed $x \le \sqrt{T}$. The case $x\ge \sqrt{T}$ is similar (but easier). We condition on $W(J_s)$ and apply Lemma \[L:3/2\]. Since $W(\tau) \ge - m$ and $\mu m \le O(1)$, we thus obtain $$\begin{aligned} & \E \left(e^{ - \mu W(\tau) - \mu W'(\tau) } {\mathbf{1}_{\{\sup _{u \in [J_s, \tau]}(W_u, W'_u ) \le m\}}} {\mathbf{1}_{\{W_\tau, W'_\tau \ge - m \}}} | W(J_s) \right) \\ & \le O(1) \P \left( {\sup _{u \in [J_s, \tau]}(W_u, W'_u ) \le 1} ; {W_\tau, W'_\tau \ge - m } | W(J_s) \right) \\ & \le O(1) \frac{(m+|W(J_s)|)^2 m^4}{(\tau - J_s)^3} \le O(1) \frac{(m^2+W(J_s)^2) m^4}{(\tau - J_s)^3}.\end{aligned}$$ Taking the expectation again, we find $$\begin{aligned} \P( W, W' \in {\mathcal B}| ( \theta, \theta') ) & \le O(1) \exp\left( - \frac12 \mu^2 \tau - \frac12 \mu^2 (\tau - J_s) \right)\frac{m^4}{(\tau - J_s)^3}\times \\ & \ \ \ \times \E( e^{\mu W(J_s)} (m^2 + W(J_s)^2) {\mathbf{1}_{\{W(J_s) \le m\}}} | ( \theta, \theta') )\nonumber \\ & \le O(1) \exp\left( - \frac12 \mu^2 \tau - \frac12 \mu^2 (\tau - J_s) \right)\frac{m^4}{(\tau - J_s)^3} (m^2 + \E( W(J_s)^2 | ( \theta, \theta')) )\nonumber \\ & = O(1) \exp\left( - \mu^2 \tau + \frac12 \mu^2 J_s \right)\frac{m^4}{(\tau - J_s)^3} (m^2 + J_s) = : B. \label{Bdom}\end{aligned}$$ **Event ${\mathcal A}$**. We now turn to the event $\theta, \theta' \in {\mathcal A}$. As in , we have that the joint law of $(\theta_u - f(u), \theta'_u - f(u), 0 \le u \le t) $ has a density (with respect to a pair of driftless Brownian motions branching from each other at $s$) which is $$\begin{aligned} Z_A & = \exp \left( - f'(s) \theta_s -\frac{3a}{t} \int_0^s \theta_u du - \frac12 \int_0^t f'(u)^2 du + \right. \nonumber \\ & \ \ \ \ \ \ \ \ \ \ \left. + \{ f'(s) \theta_s - \frac{3a}{t} \int_s^t \theta_u du - \frac12 \int_s^t f'(u)^2 du \} + \right.\nonumber \\ & \ \ \ \ \ \ \ \ \ \ \left. + \{ f'(s) \theta'_s - \frac{3a}{t} \int_s^t \theta'_u du - \frac12 \int_s^t f'(u)^2 du \} \right) \nonumber \\ & \le \exp \left( - \frac12 \int_0^t f'(u)^2 du - \int_s^t f'(u)^2 du + \theta_s f'(s) + \frac{3a}{t } \int_0^s \theta_u du\right) . \label{Girsanov3}\end{aligned}$$ where the inequality holds on the event $\theta, \theta' \in {\mathcal A}$. Furthermore, $f'(s) = -3as/t $, $|\theta_s |\le M_s \le t^{3/4}$, hence $\int_0^s \theta_u du \le C s t^{3/4}$, so that $$Z_A\le z^+_A : = \exp \left( - \int_0^t f'(u)^2 du + \frac12 \int_0^s f'(u)^2 du + \frac{Cs}{t^{1/4} }\right)$$ Note that $z^+_A$ is nonrandom. Also, on the event $\theta, \theta' \in {\mathcal A}$, and while $s\le t /2$, the random variable $B$ on the right hand side of is bounded above by $$\begin{aligned} B& \le O(1) \frac{m^4}{t^6} ( t + st) \exp\left( - \mu^2 \tau + \frac12 \mu^2 J_s \right)\\ & \le O(1) \frac1{t^3} ( 1+ s) \exp\left( - \mu^2 \tau + \frac12 \mu^2 J_s \right).\end{aligned}$$ Therefore, we may rewrite, letting $j(s) = \int_0^s f(u) du$, $$\begin{aligned} \P ( \theta, \theta' \in {\mathcal A}; W, W' \in {\mathcal B}) & \le \E ( B {\mathbf{1}_{\{\theta, \theta' \in {\mathcal A}\}}} ) \\ & \le O(1) \frac{s+1}{t^3} \exp\left( - \mu^2 \tau \right) \E\left( e^{\frac12 \mu^2 J_s} {\mathbf{1}_{\{\theta, \theta' \in {\mathcal A}\}}} \right) \\ & \le O(1) \frac{s+1}{t^3} \exp\left( - \mu^2 \tau + \frac12 \mu^2 j(s) \right) \E \left( z^+_A {\mathbf{1}_{\{ | \theta_\cdot|, |\theta'_\cdot| \le M_u; -t \le J, J' \le 0 \}}} e^{\frac12 \mu^2 J_s}\right) \\ & \le O(1) \frac{s+1}{t^3} \exp\left( - \mu^2 \tau + \frac12 \mu^2 j(s) \right) z^+_A \exp\left(\frac12 \mu^2 \frac{3as}{t^{1/4}} \right)\times\\ & \ \ \ \ \ \ \ \ \ \times \P( - t \le J, J' \le 0 ; | \theta_\cdot|, |\theta'_\cdot| \le M_\cdot)\end{aligned}$$ where we have used that (as above) $J_s \le \frac{3as}{t^{1/4}}$ on $|\theta_\cdot|, |\theta'_\cdot| \le M_\cdot$. Since $\mu^2 =O(1/t)$, we deduce $$\P ( \theta, \theta' \in {\mathcal A}; W, W' \in {\mathcal B}) \le O(1) \frac{s+1}{t^3} \exp\left( - \mu^2 \tau + \frac12 \mu^2 j(s) \right) z^+_A \P( - t \le J, J' \le 0 ; | \theta_t|, |\theta'_t| \le 1 ).$$ We will need the following lemma: \[L:bridge\] For $x,j,a\in {{\mathbb R}}$, $$\P_x( j - a\le J \le j ; |\theta_t| \le 1 ) \le \frac{C a }{(t +1 )^{2}}$$ for some constant $C$ independent of $x$, $j$, $a$ and $t$. Observe first that the position $\theta_t$ is normally distributed with mean $x$ and variance $t$, so its density is uniformly bounded by $Ct^{-1/2}$ and $\P_x( |\theta_t| \le 1) \le C t^{-1/2}$. When we condition on the value of $\theta_t = z$, $J$ is a normal random variable, as the integral of a certain Gaussian process (namely, a Brownian bridge of duration $t$ from $x$ to $z$) with a certain mean and a variance at most $t^3/3$. Consequently the probability density function of $J$ is uniformy bounded by $C /(t+1)^{3/2}$. Integrating over the interval $[j-a, j]$ gives $${\mathbb{P}^{x \stackrel{t}{\rightarrow} z} } ( J \in [j-a, j]) \le \frac{C a }{(t +1 )^{3/2}},$$ uniformly in $z$. Hence $$\P_x( j - a\le J \le j ; |\theta_t| \le 1 )\le \frac{C a }{(t +1 )^{3/2}} \P_x ( |\theta_t| \le 1) \le \frac{Ca}{t^2}.$$ Since this probability is also trivially bounded by 1, we get the desired upper bound. From Lemma \[L:bridge\] it follows, after conditioning by $(\theta_u, u \le s)$, that $$\P( - t \le J, J' \le 0 ; |\theta_t|, |\theta'_t| \le 1) \le \frac{Ct^2 }{(t+1-s)^4}.$$ Consequently, $$\begin{aligned} \P ( \theta, \theta' \in {\mathcal A}; W, W' \in {\mathcal B}) & \le O(1) \frac{s+1}{t^3} \exp\left( - \mu^2 \tau + \frac12 \mu^2 j(s) \right) z^+_A \frac{t^2}{(t+1-s)^4}\\ & \le \frac{O(1)(s+1) }{t (t+1 - s)^4} \exp\left( - \mu^2 \tau + \frac12 \mu^2 j(s) \right) z^+_A .\end{aligned}$$ We return to the integral that we wish to estimate in , which is $$\int_0^{t} e^{2t} e^{-s} \P( B, B' \in {\mathcal G}) ds.$$ On the interval $[0, t/2]$ we may write that $t+1 -s \ge t/2$ so we deduce, after writing out the various terms, $$\begin{aligned} e^{2t} \int_0^{t/2} e^{-s} \P( B, B' \in {\mathcal G}) ds& \le e^{2t - \mu^2 \tau - \int_0^{t} f'(u)^2 du} \frac{O(1)}{t^5} \int_0^{t/2} e^{-s} (s+1) e^{\frac12 \mu^2 j(s) + \frac12 \int_0^s f'(u)^2du + Cs/t^{1/4}}ds \label{corbound}\end{aligned}$$ Over the interval $[t/2, t]$, we use the crude bound $t+1-s \le 1$ and we get $$\begin{aligned} e^{2t} \int_{t/2}^{t} e^{-s} \P( B, B' \in {\mathcal G}) ds& \le e^{2t - \mu^2 \tau - \int_0^{t} f'(u)^2 du} \frac{O(1)}{t} \int_{t/2}^t e^{-s} (s+1) e^{\frac12 \mu^2 j(s) + \frac12 \int_0^s f'(u)^2du + Cs/t^{1/4}}ds \label{corbound2}\end{aligned}$$ Note that the exponential prefactor in front of the two integrals above is precisely $e^{2t (1 - 3a^2 - c^2/2a)}/t^5$, as desired in . Hence it suffices to show that the total integral above (from $0$ to $t$) is bounded. We compute the exponential term inside the integrand. Recall that for $s\le t-1$, $f(s) = (3a/2)( t - s^2/t)$, that $\mu = c t^{-1/2}/a$. Thus, $$\begin{aligned} j(s) = \int_0^s f(u)du & = \int_0^s \frac{3a}{2} (t - \frac{u^2}{t}) du = \frac{3a}{2}(ts - \frac{s^3}{3t})\end{aligned}$$ so that $$\label{j} \frac12 \mu^2 j(s) = \frac{3\gamma^2}{4a}( s - \frac{s^3}{3t^2}).$$ Also, $$\begin{aligned} \label{f'} \int_0^s f'(u)^2 du & = \frac{9a^2}{t^2} \int_0^s u^2 du = \frac{3a^2}{t^2}s^3.\end{aligned}$$ Hence returning to the exponential term in the integral in the right hand side of and , and making a change of variable $s = xt, x\in (0,1)$, $$\exp\left\{ -s + \frac12 \mu^2 j(s) + \frac12 \int_0^s f'(u)^2du + Cst^{-1/4} \right\} = \exp\left\{ t \psi (x) +C t^{3/4} x \right\}$$ where $$\psi(x) = x ( -1 + \frac{3\gamma^2}{4a}) + x^3 ( \frac{3a^2}{2} - \frac{\gamma^2}{4a}).$$ Using the relation $\gamma^2 = 6 a^3 $ we see that all the cubic terms cancel, and we are left with $$\begin{aligned} \psi(x) & = - x(1 - \frac{9a^2}{2} )\end{aligned}$$ and note that since $a< \sqrt{2}/3$, $1- 9a^2/2 >0$. Thus returning to the integral in , this becomes: $$\int_0^{t/2} (s+1) \exp\left( - s( 1- \frac{9a^2}{2}) + {C s}{t^{-1/4} } \right) ds .$$ It is easy to see that this is less than $C$ for some universal constant $C>0$. We deduce that $$\E(Z^2) \le \frac{C}{t^5} \exp( 2t ( 1 - \frac{3a^2}{2} - \frac{\gamma^2 }{2a} ) )$$ which concludes the proof of Proposition \[P:Z2\]. End of proof of Theorem \[T:toads\] {#sec:conclusion-proof} =================================== It is now not hard to finish the proof of Theorem \[T:toads\]. First suppose that $u_0(x, \theta) = H(x) {\mathbf{1}_{\{\theta\in (0,1)\}}}$. In this case we have already obtained the upper bound $\sup_{\theta_0 >0} \sup_{x> \gamma t^{3/2}} u(t,x, \theta_0) \to 0$ in for $\gamma > \gamma_0$. For the lower bound, combining Propositions \[P:EZ\] and \[P:Z2\] and the Payley–Zygmund inequality we see that for $\gamma <\gamma_0$, $$\label{liminf} \liminf_{t \to \infty} \P^{x, \theta}( \max X_i(t) > \gamma t^{3/2}) \ge C$$ for some $C>0$ and for $ \theta_0 \in (\theta_-, \theta_+) $ and $x_0 \in (-1,1)$ where $ \theta_\pm = \frac{3a }{2}t \pm 1,$ as per . Supposer that initially there is one particle at $x_0 = 0$ and $\theta_0 = 3a t /2$. Fix a large constant $T>0$ and condition on the population at time $T$ (given ${\mathcal F}_T)$. Let $I$ be the set of particles $i \in \tilde I(T)$ such that $\theta^i_T \in [\theta_-, \theta_+]$ and $X^i_T \in (-1,1)$. For each particle $i \in I$, let $x_i = X^i_T$ and $\theta_i = \theta^i_T$. Then $$\begin{aligned} \P^{0,\theta_0} \left( \max_{i\in \tilde I_t} X_t^i \le c t^{3/2} | {\mathcal F}_T\right) \le \prod_{i \in I} \P^{x_i,\theta_i}\left( \max_{i \in \tilde I_{T-t}} X_t^i \le \gamma (t - T)^{3/2} - x_i \right) \end{aligned}$$ so $$\liminf_{t \to \infty} \P^{0,\theta_0}\left( \max_{i\in \tilde I_t} X_t^i \le c t^{3/2} | {\mathcal F}_T\right) \le (1- C)^{| I | }$$ Also, the random variable on the left hand side (before taking the liminf) is bounded by one, so by the dominated convergence theorem, $$\liminf_{t\to \infty} \P^{0,\theta_0}\left( \max_{i\in \tilde I_t} X_t^i \le \gamma t^{3/2} \right) \le \E( (1-C)^{|I|} )$$ uniformly in $T$. Since it is clear that $|I | \to \infty$ in probability as $T \to \infty$, we deduce that $$\label{LB_heavyside} \liminf_{t\to \infty} \P^{0,\theta_0}\left( \max_{i\in \tilde I_t } X_t^i \le \gamma t^{3/2} \right) = 0,$$ which completes the proof of Theorem \[T:toads\] in the case of the Heavyside initial condition. We now turn to the general case of initial conditions subject to our assumptions, and use the general case of the McKean representation (Proposition \[P:mckean\]) together with the result obtained above in the particular case of Heavyside data. We first consider the upper bound on the speed. Fix $\gamma > \gamma_0$ and let $\gamma_0 < \gamma' < \gamma$ and let $\delta = \gamma' - \gamma_0$. Note that for $x \ge \gamma t^{3/2}$, $$\prod_{i\in \tilde I_t} ( 1- u_0(X_t^i)) \ge {\mathbf{1}_{\{\text{all particles are greater than $\delta t^{3/2} $ } ; |\tilde I_t| \le e^{2t}\}}} \times (1- C\exp(- c \delta t^{3/2} ))^{e^{2t}}$$ thanks to our assumption on the behaviour at $+ \infty$ of $u_0$. Therefore, $$u(t,x,\theta) \le 1- (1+ o(1)) \P^{x, \theta}(\text{all particles are greater than $\delta t^{3/2} $ } ; |\tilde I_t| \le e^{2t})$$ The probability of the event above tends to one by the above observations and Markov’s inequality as $\E(|\tilde I_t| ) \le \E ( | I_t| ) = e^t$. Lower bound. Note that for all $\delta>0$, we can find $A>0$ chosen sufficiently large so that $f(x) \ge 1- \delta$ if $x \le - A$. Therefore, $$\prod_{i \in \tilde I_t} ( 1- f(X_t^i)) \le \delta + {\mathbf{1}_{\{X_t^i \ge - A \text{ for all } i \in \tilde I_t\}}}$$ and it follows that $ u(t,x, \theta) \ge 1- \delta - \P_{(x, \theta)}( X_t^i \ge - A \text{ for all } i \in \tilde I_t). $ Fix $x \le x_0 = \gamma t^{3/2}$ with $\gamma < \gamma_0$, and $\theta = 3at /2$. Since $$\P^{(0, \theta)} ( \exists i: X_t^i \ge x +A ) \ge \P^{(0, \theta)} ( \exists i: X_t^i \ge x_0 +A ) \to 1$$ as $t \to \infty$ (uniformly in $x \le x_0 = \gamma t^{3/2}$) by using with $x = \gamma't^{3/2}$ where $\gamma < \gamma'< \gamma_0$, we deduce $$\liminf_{t\to \infty} \inf_{x\le \gamma t^{3/2} }u(t,x, \theta) \ge 1 - \delta$$ for $x = \gamma t^{3/2}$, $\theta = 3at/2$. Thus $\inf_{x \le \gamma t^{3/2}} S(t,x) \to 1$ as $t \to \infty$, as desired. Global-in-time estimate {#sec:comp-(II)} ======================= \[theo:unif-pointwise-II\] Let us consider a solution $v$ to (NLoc) with initial data $v_0$ satisfying the compact support in $\theta$ assumption (1) and the regularity assumption (3), described in Subsection \[sec:main-results\]. Then the unique global non-negative $L^\infty$ solution satisfies the global pointwise bound $$\begin{aligned} \label{eq:3} {\forall \,}t \ge 0, \ x \in {{\mathbb R}}, \ \theta \ge 1, \quad 0 \le w(t,x,\theta) \le M \end{aligned}$$ for an explicit constant $M$ independent of the solution. \[lem:sursol\] Let $v=v(t,x,\theta)$ the solution of (NLoc) with an initial condition satisfying the compact support in $\theta$ and regularity assumptions of Subsection \[sec:main-results\]. Then, for any $\tau>0$, there exist $C>0$ such that for $k,\,l\in\mathbb N$, $k+l\leq 3$, $$\forall (t,x,\theta)\in [0,\tau]\times \mathbb R\times[1,\infty),\quad |\partial_x^k\partial_\theta^l v(t,x,\theta)|\leq Ce^{\frac{\theta^2}{2\tau+1}},$$ and $$\forall (t,x,\theta)\in [0,\tau]\times \mathbb R\times[1,\infty),\quad |\partial_t v(t,x,\theta)|+|\partial_x\partial_t v(t,x,\theta)|+|\partial_\theta\partial_t v(t,x,\theta)|\leq Ce^{-\frac{\theta^2}{2\tau+1}}.$$ In order to avoid dealing with boundaries in $\theta$, we shift $[1,+\infty)$ to $[0,+\infty)$ and then do mirror symmetry, i.e. we define $\bar v(t,x,\theta) := v(t,x,|\theta|+1)$, which solves $$\begin{aligned} \label{model-2-sym} \left\{ \begin{array}{l} {\displaystyle}\partial_t \bar v = \frac{|\theta| +1}2\Delta_x \bar v + \frac 12\Delta_\theta \bar v + \bar v \left( 1- \langle \bar v \rangle \right) \\[3mm] {\displaystyle}\bar v = \bar v(t,x,\theta), \ t \ge 0, \ x \in {{\mathbb R}}, \ \theta \in {{\mathbb R}}, \quad \bar v(t,x,-\theta)=\bar v(t,x,\theta) \\[3mm] {\displaystyle}\langle \bar v \rangle(t,x,\theta) := \int_{\min(\theta- A,0)} ^{\theta+A} \bar v(t,x,\omega) {{\, \mathrm d}}\omega {{\, \mathrm d}}y, \ \theta \ge 0,\\[4mm] {\displaystyle}\bar v(0,x,\theta) = \bar v_0(x,\theta) \ge 0. \end{array} \right.\end{aligned}$$ Let $\varepsilon>0$, and $\varphi$ a regular approximation of $\theta\mapsto\frac{|\theta|+1}2$. More precisely, we assume that $\varphi(\theta)=\varphi(-\theta)$ for $\theta\in\mathbb R$, $\varphi(\theta)=\frac{|\theta|+1}2$ for $|\theta|\geq \varepsilon>0$, and that for some $C>0$, $$\|\varphi'\|_{L^\infty}\leq C,\quad \|\varphi''\|_{L^\infty}\leq \frac C{\varepsilon},\quad\|\varphi'''\|_{L^\infty}\leq \frac C{\varepsilon^2}.$$ Such functions can be constructed for any $\varepsilon>0$, for instance through the rescaling of such a function for $\varepsilon=1$. Let $w$ a solution of $$\label{eq:homogene0} \partial_t w-\varphi(\theta)\Delta_x w-\frac 12\Delta_\theta w=w\left(1-\langle w\rangle\right),$$ with a regular initial condition $w_0=w_0(x,\theta)$ (in the sense that $\|w_0\|_{C^3(\mathbb R^2)}<\infty$) with a compact support in $\theta$ (that is $w_0(x,\theta)=0$ if $|\theta|>\bar\theta$, for some $\bar\theta>0$). Then, the comparison principle shows that $w(t,x,\theta)\leq \|w_0\|_{L^\infty}e^t$ for $t\geq 0$. More precisely, we notice that $$\label{eq:homogene0bis} \partial_t w-\varphi(\theta)\Delta_x w-\frac 12\Delta_\theta w-w\leq 0,$$ while $\bar w_1(t,x,\theta):= Ce^{t-\frac{\theta^2}{2t+1}}$ is a solution of . If $C>0$ is large enough, $w(0,\cdot,\cdot)\leq \bar w_1(0,\cdot,\cdot)$, and the comparison principle then implies that $0\leq w\leq \bar w$. We notice that for $k\in\{1,2,3\}$, $\partial_x^k w$ satisfies , the same argument then shows that for some $C>0$, $|\partial_x^k w|\leq Ce^{t-\frac{\theta^2}{2t+1}}$. We notice next that $\partial_\theta w$ satisfies: $$\begin{aligned} \partial_t \left(\partial_\theta w\right)-\varphi(\theta)\Delta_x \left(\partial_\theta w\right)-\frac 12\Delta_\theta \left(\partial_\theta w\right)-\partial_\theta w&=&\varphi'(\theta)\Delta_x w+\mathcal O(1)\|w\|_{L^\infty}w\nonumber\\ &\leq& C\left(\|\varphi'\|_{L^\infty}+1\right)e^{t-\frac{\theta^2}{2t+1}}.\label{eq:homogene1}\end{aligned}$$ Let $\bar w_1(t,x,\theta):=Ce^{\lambda_1 t-\frac{\theta^2}{2t+1}}$, which satisfies $$\partial_t \bar w_1-\varphi(\theta)\Delta_x \bar w_1-\frac 12\Delta_\theta \bar w_1-\bar w_1\geq C(\lambda_1-1) e^{t-\frac{\theta^2}{2t+1}} ,$$ so that $\bar w_1$ is a super-solution of for $t\in[0,\tau]$, as soon as $\lambda_1>0$ is chosen large enough. Just as above, for $k\in\{1,2\}$, it is possible to repeat this method to show that for some $C>0$ and $\lambda_1>0$, $|\partial_x^k\partial_\theta w|\leq Ce^{\lambda_1 t-\frac{\theta^2}{2t+1}}$. We now turn to $\partial_\theta^2 w$, which satisfies: $$\begin{aligned} &\partial_t \left(\partial_\theta^2 w\right)-\varphi(\theta)\Delta_x \left(\partial_\theta^2 w\right)-\frac 12\Delta_\theta \left(\partial_\theta^2 w\right)\nonumber\\ &\quad=\varphi''(\theta)\Delta_x w+2\varphi'(\theta)\Delta_x \partial_\theta w+\mathcal O(1)\left(\|\partial_\theta w\|_{L^\infty}w+\|w\|_{L^\infty}\partial_\theta w\right)\nonumber\\ &\quad \leq C\left({\mathbf{1}_{\{\theta\in[-\varepsilon,\varepsilon]\}}}\frac 1\varepsilon+1\right)e^{\lambda_1 t-\frac{\theta^2}{2t+1}}.\label{eq:homogene2}\end{aligned}$$ Let $\bar w_2(t,x,\theta):=Ce^{\lambda_2 t-\frac{\psi_2(\theta)}{2t+1}}$, where $\psi_2(\theta)=\frac 1\varepsilon \theta^2+1-\varepsilon$ on $[-\varepsilon,\varepsilon]$, and $\psi_2(\theta)=\left(|\theta|+1-\varepsilon\right)^2$ for $|\theta|\geq\varepsilon$. Then, $\psi_2\in C^1(\mathbb R)$ (in particular, $\|\psi'\|_{L^\infty}<2$), and $\psi_2''(\theta)=2 \left(\frac 1\varepsilon{\mathbf{1}_{\{\theta\in[-\varepsilon,\varepsilon]\}}}+{\mathbf{1}_{\{\theta\in[-\varepsilon,\varepsilon]^c\}}}\right)$. Then, for $t\in[0,\tau]$, $$\begin{aligned} \partial_t \bar w_2-\varphi(\theta)\Delta_x \bar w_2-\frac 12\Delta_\theta \bar w_2&=&\left(\lambda_2 +\frac{2\psi_2(\theta)}{(2t+1)^2}+\frac{\psi_2''(\theta)}{4t+2}-\frac{(\psi_2'(\theta))^2}{2(2t+1)^2}\right)\bar w_2\\ &=&\left(\lambda_2+\frac{2\psi_2(\theta)}{(2t+1)^2}+\frac{\frac 1\varepsilon{\mathbf{1}_{\{\theta\in[-\varepsilon,\varepsilon]\}}}+{\mathbf{1}_{\{\theta\in[-\varepsilon,\varepsilon]^c\}}}}{2t+1}-\frac{(\psi_2'(\theta))^2}{2(2t+1)^2}\right)\bar w_2\\ &\geq&\left((\lambda_2-2)+\frac {1}\varepsilon {\mathbf{1}_{\{\theta\in[-\varepsilon,\varepsilon]\}}}\right)Ce^{\lambda_2 t-\frac{\psi_2(\theta)}{2t+1}}.\end{aligned}$$ Since $\psi_2(\theta)\leq \theta^2+1$, if we chose $C,\,\lambda_2>0$ large enough, then $\bar w_2$ is a super solution of for $t\in[0,\tau]$, and then $|\partial_\theta^2 w|\leq \bar w_2\leq Ce^{\lambda_2 t-\frac{\theta^2}{2t+1}}$ for some $C>0$ (for $t\in[0,\tau]$). Here also, a similar estimate applies to $\partial_x\partial_\theta^2 w$, to show that $|\partial_x\partial_\theta^2 w|\leq Ce^{\lambda_2 t-\frac{\theta^2}{2t+1}}$. Note that this estimate is sufficient to prove the well posedness of the problem for $t\in[0,\tau]$, and thus in particular the symmetry of $w$ in $\theta$: $w(t,x,\theta)=w(t,x,-\theta)$ for $(t,x,\theta)\in[0,\tau]\times\mathbb R^2$, which implies in particular that $\partial_\theta^3 w(t,x,0)=0$ for $(t,x)\in\mathbb R_+\times\mathbb R$. This property will be important to estimate $\partial_\theta^3 w$. We have $$\begin{aligned} &\partial_t \left(\partial_\theta^3 w\right)-\varphi(\theta)\Delta_x \left(\partial_\theta^3 w\right)-\frac 12\Delta_\theta \left(\partial_\theta^3 w\right)\nonumber\\ &\quad=\varphi'''(\theta)\Delta_x w+3\varphi''(\theta)\Delta_x \partial_\theta w+3\varphi'(\theta)\Delta_x \partial_\theta^2 w\nonumber\\ &\qquad+\mathcal O(1)\left(\|w\|_{L^\infty}+\|\partial_\theta w\|_{L^\infty}+\|\partial_\theta^2 w\|_{L^\infty}\right)\left(w+\partial_\theta w+\partial_\theta^2 w\right)\nonumber\\ &\quad\leq C {\mathbf{1}_{\{\theta\in[-\varepsilon,\varepsilon]\}}}\left(|\varphi'''(\theta)|+|\varphi''(\theta)|\right)+C\|\varphi'\|_{L^\infty}e^{\lambda_1^2 t-\frac{\theta^2}{2t+1}}\nonumber\\ &\quad\leq C\left(\frac 1{\varepsilon^2}{\mathbf{1}_{\{\theta\in[-\varepsilon,\varepsilon]\}}}+1\right)e^{\lambda_1^2 t-\frac{\theta^2}{2t+1}}. \label{eq:homogene3}\end{aligned}$$ Since $\partial_\theta^3 w(t,x,0)=0$ for $\theta=0$, $\partial_\theta^3 w(t,x,\theta)$ is a sub-solution of the problem on $\mathbb R_+\times\mathbb R\times [0,\infty)$ with a Dirichlet boundary condition in $\theta=0$. We will build a super-solution for this half-space problem: $$\bar w_2(t,x,\theta)=\min\left(C_1e^{\lambda_2 t-\frac{\theta^2}{2t+1}},C_2\left(1+\sqrt{\theta/\varepsilon}\right)\right)={\mathbf{1}_{\{\theta\in [\bar\theta(t),\infty)\}}}C_1e^{\lambda_2 t-\frac{\theta^2}{2t+1}}+{\mathbf{1}_{\{\theta\in (0,\bar\theta(t))\}}}C_2\sqrt{\theta/\varepsilon},$$ for $t\in[0,\tau]$. Moreover, if $C_1$ is chosen sufficiently larger than $C_2$, then $\varepsilon<\theta(t)<C\varepsilon$, for some constant $C>0$. Since a minimum of two super-solutions is a super-solution we simply need to check that $\bar w_{2,1}(t,x,\theta)=C_1e^{\lambda_2 t}$ is a super-solution of on $\mathbb R_+\times\mathbb R\times[\varepsilon,\infty)$, and $\bar w_{2,2}(t,x,\theta)=C_2\left(1+\sqrt{\theta/\varepsilon}\right)$ is a super-solution of on $\mathbb R_+\times\mathbb R\times(0,C\varepsilon)$. The argument for $\bar w_{2,1}$ is similar to earlier cases, while $\bar w_{2,2}$ satisfies $$\partial_t \bar w_{2,2}-\varphi(\theta)\Delta_x \bar w_{2,2}-\frac 12\Delta_\theta \bar w_{2,2}-\bar w_{2,2}=\frac{C_2}{4\sqrt\varepsilon}\theta^{-3/2}-C_2\left(1+\sqrt{\theta/\varepsilon}\right)\geq \frac {C_2}{8\varepsilon^2},$$ provided $\varepsilon>0$ is small enough. $\bar w_{2,2}$ is thus indeed a super-solution of on $[0,\tau]\times\mathbb R\times(0,C\varepsilon)$, provided $C_2$ is chosen large enough. The comparison principle then shows that $|\partial_\theta^3 w(t,x,\theta)|\leq \bar w_2(t,x,|\theta|)\leq Ce^{\lambda_2 t-\frac{\theta^2}{2t+1}}$. We notice next that $\partial_t w(0,\cdot,\cdot)=\varphi(\theta)\Delta_x w(0,\cdot,\cdot)-\frac 12\Delta_\theta w(0,\cdot,\cdot)-w(0,\cdot,\cdot)$ is in $C^1(\mathbb R)$ with a compact support in $\theta$, and $\partial_t w$ is a solution of an equation similar to . The argument above (see and ) can then be reproduced to show that $|\partial_t w(t,x,\theta)|+|\partial_x\partial_t w(t,x,\theta)|+|\partial_\theta\partial_t w(t,x,\theta)|\leq Ce^{\lambda_t t-\frac{\theta^2}{2t+1}}$. We have proven that there exists $C>0$ independent of $\varepsilon>0$ (small enough) such that for $k,\,l\in\mathbb N$, $k+l\leq 3$, $$\forall (t,x,\theta)\in [0,\tau]\times \mathbb R^2,\quad |\partial_x^k\partial_\theta^l w(t,x,\theta)|\leq Ce^{-\frac{\theta^2}{2\tau+1}},$$ and $$|\partial_t w(t,x,\theta)|+|\partial_x\partial_t w(t,x,\theta)|+|\partial_\theta\partial_t w(t,x,\theta)|\leq Ce^{-\frac{\theta^2}{2\tau+1}}.$$ We can thus pass to the limit $\varepsilon\to 0$ in these estimates to obtain similar estimates on solutions of , which conclude the proof. We will now prove Theorem \[theo:unif-pointwise-II\]. Note that this proof draws a lot of inspiration from [@Turanova]. *Step 1. Definitions and rescaling.* We define the cylinder around a point $\bar z := (\bar t, \bar x, \bar \theta)$: $$\label{def:Qr} Q_{\bar z,R}:=(\bar t-R^2,\bar t)\times B((\bar x,\bar\theta),R).$$ We might omit the base point and/or the size $R$ when obvious from the context. We define for any cylinder $Q=Q_{\bar z,R}$ the norms $$[u]_{\delta/2,\delta, Q}=\sup_{(t,x)\neq (s,y)\in Q}\frac{|u(t,x)-u(s,y)|}{\left(|x-y|+|t-s|^{1/2}\right)^\delta},$$ $$|u|_{\delta/2,\delta, Q}=\|u\|_{L^\infty(Q)}+[u]_{\delta/2,\delta, Q},$$ $$[u]_{1+\delta/2,2+\delta, Q}=[\partial_t u]_{\delta/2,\delta, Q}+\sum_{i,j=1}^2[\partial_{x_ix_j}^2 u]_{\delta/2,\delta, Q},$$ $$\begin{aligned} |u|_{1+\delta/2,2+\delta, Q}&=&\|u\|_{L^\infty(Q)}+\sum_{i=1}^2\|\partial_{x_i} u\|_{L^\infty(Q)}+\|\partial_{t} u\|_{L^\infty(Q)}\\ &&+\sum_{i,j=1}^2\|\partial_{x_ix_j}^2 u\|_{L^\infty(Q)}+[u]_{1+\delta/2,2+\delta, Q}.\end{aligned}$$ We recall the definition of $\bar v$, and introduce an additional notation: for a given base point $\bar z$, we rescale the problem, $\tilde v = (t,x,\theta) = \bar v\left(t,\sqrt{\bar \theta+1} x, \theta\right)$, to get $$\begin{aligned} \label{model-2-sym-resca} \left\{ \begin{array}{l} {\displaystyle}\partial_t \tilde v = \frac{|\theta| +1}{2\left(|\bar \theta|+1\right)} \Delta_x \tilde v + \Delta_\theta \tilde v + \tilde v \left( 1- \langle \tilde v \rangle \right) \\[3mm] {\displaystyle}\tilde v = \tilde v(t,x,\theta), \ t \ge 0, \ x \in {{\mathbb R}}, \ \theta \in {{\mathbb R}}, \quad \tilde v(t,x,-\theta)=\tilde v(t,x,\theta) \\[3mm] {\displaystyle}\langle \tilde v \rangle(t,x,\theta) := \int_{\min(\theta- A,0)} ^{\theta+A} \tilde v(t,x,\omega) {{\, \mathrm d}}\omega {{\, \mathrm d}}y, \ \theta \ge 0,\\[4mm] {\displaystyle}\tilde v(0,x,\theta) = \tilde v_0(x,\theta) \ge 0. \end{array} \right.\end{aligned}$$ *Step 2. Relating $|u|_{1+\delta/2,2+\delta, Q}$ to $\|\cdot\|_{L^\infty}$.* We use (in the particular framework needed here) the following two results from [@Krylov]: \[lem:interior-est\] Let $\delta\in (0,1)$, $(\bar t,\bar x,\bar\theta)\in{{\mathbb R}}_+\times{{\mathbb R}}^2$. If $S \in C^{\delta/2}_t C^{\delta}_{x,\theta}(Q_2)$ (i.e. $|S|_{\delta/2,\delta,Q_2}<\infty$) and $V(t,x,\theta)$ is a solution of $$\partial_t V-\frac{|\theta|+1}{2\left(|\bar \theta|+1\right)}\Delta_x V-\Delta_\theta V=S,\textrm{ on }Q_2,$$ then, there exist a universal constant $C>0$ such that $$|V|_{1+\delta/2,2+\delta,Q_1}\leq C\left(|S|_{\delta/2,\delta,Q_2}+\|V\|_{L^\infty(Q_2)}\right).$$ \[lem:interpolation\] Let $(\bar t,\bar x,\bar \theta)\in{{\mathbb R}}_+\times{{\mathbb R}}^2$ and $\delta>0$. There exists a constant $N>0$ such that for any $\varepsilon>0$, and any $V\in C^{1+\delta/2}_t C^{2+\delta}_{x,\theta}(Q_3)$, $$[V]_{\delta/2,\delta, Q_2}\leq \varepsilon [V]_{1+\delta/2,2+\delta,Q_3}+N\varepsilon^{-\delta/2}\|V\|_{L^\infty(Q_3)}.$$ We recall the definition  of $Q_{(\bar t,\bar x,\bar\theta),R}$, and $\tilde v$ as in . Thanks to Lemma \[lem:sursol\], if $\bar t\in[1,(3+A)^2+2]$, then $$\begin{aligned} |\tilde v|_{1+\delta/2,2+\delta, Q_{(\bar t,\bar x,\bar \theta),1}}&\leq& C\left(|\bar \theta|^{2+\delta}+1\right)\bigg(\sum_{k+l\leq 3}\|\partial_x^k\partial_\theta^l v(t,x,\theta)\|_{L^\infty([0,\bar t]\times\mathbb R\times [-\bar\theta-A,\bar \theta+A])}\nonumber\\ &&+\|\partial_t v(t,x,\theta)\|_{L^\infty([0,\bar t]\times\mathbb R\times [-\bar\theta-A,\bar \theta+A])}+\|\partial_x\partial_t v(t,x,\theta)\|_{L^\infty([0,\bar t]\times\mathbb R\times [-\bar\theta-A,\bar \theta+A])}\nonumber\\ &&+\|\partial_\theta\partial_t v(t,x,\theta)\|_{L^\infty([0,\bar t]\times\mathbb R\times [-\bar\theta-A,\bar \theta+A])}\bigg)\leq C_0,\label{est-cond-ini}\end{aligned}$$ where the constant $C_0>0$ is independent of $(\bar t,\bar x,\bar \theta)\in [1,(3+A)^2+2)\times {{\mathbb R}}^2$. Let $T>(3+A)^2+1$, and $M$ such that $$\label{assM} M>2C_0,$$ where $C$ is here the constant appearing in . We assume also that $\|v\|_{L^\infty([0,T]\times\mathbb R\times\mathbb R_+)}\leq M$. For $(\bar t,\bar x,\bar \theta)\in [(3+A)^2,T)\times {{\mathbb R}}^2$, we can apply Lemma \[lem:interior-est\] to obtain$$|\tilde v|_{1+\delta/2,2+\delta,Q_{1}}\leq C\left(|\left(1-\langle \tilde v\rangle\right)\tilde v|_{\delta/2,\delta,Q_{2}}+\|\tilde v\|_{L^\infty(Q_{2})}\right).$$ We estimate further $$\begin{aligned} &|\left(1-\langle \tilde v\rangle\right)\tilde v|_{\delta/2,\delta,Q_{2}\cap ({{\mathbb R}}_+\times{{\mathbb R}}^2)}\\ &\quad=[\left(1-\langle \tilde v\rangle\right)\tilde v]_{\delta/2,\delta,Q_{2}\cap( {{\mathbb R}}_+\times{{\mathbb R}}^2)}+\|\left(1-\langle \tilde v\rangle\right)\tilde v\|_{L^\infty(Q_{2}\cap ({{\mathbb R}}_+\times{{\mathbb R}}^2))}\\ &\quad\leq\|1-\langle \tilde v\rangle\|_{L^\infty(Q_{2}\cap ({{\mathbb R}}_+\times{{\mathbb R}}^2))}[\tilde v]_{\delta/2,\delta,Q_{2}\cap {{\mathbb R}}_+\times{{\mathbb R}}^2}\\ &\quad+[1-\langle \tilde v\rangle]_{\delta/2,\delta,Q_{2}\cap ({{\mathbb R}}_+\times{{\mathbb R}}^2)}\| \tilde v\|_{L^\infty(Q_{2}\cap ({{\mathbb R}}_+\times{{\mathbb R}}^2))}\\ &\qquad+\|1-\langle \tilde v\rangle\|_{L^\infty(Q_{2}\cap ({{\mathbb R}}_+\times{{\mathbb R}}^2))}\|\tilde v\|_{L^\infty(Q_{2}\cap ({{\mathbb R}}_+\times{{\mathbb R}}^2))}\\ &\quad\leq C\left(M[\tilde v]_{\delta/2,\delta,Q_{2+A}\cap ({{\mathbb R}}_+\times{{\mathbb R}}^2)}+M^2\right),\end{aligned}$$ and then, $$|\tilde v|_{1+\delta/2,2+\delta,Q_{1}\cap ({{\mathbb R}}_+\times{{\mathbb R}}^2)}\leq C M\left([\tilde v]_{\delta/2,\delta,Q_{2+A}\cap ({{\mathbb R}}_+\times{{\mathbb R}}^2)}+M\right).$$ Using Lemma \[lem:interpolation\], we get $$|\tilde v|_{\delta/2,\delta,Q_{2+A}}\leq CM\left(\varepsilon [\tilde v]_{1+\delta/2,2+\delta,Q_{3+A}}+\left(1+\varepsilon^{-\delta/2}\right)M \right).$$ We select $\varepsilon \sim \alpha M^{-1}$, to get $$\label{eq-step2} |\tilde v|_{1+\delta/2,2+\delta,Q_{1}}\leq \alpha|\tilde v|_{1+\delta/2,2+\delta,Q_{3+A}} + C M^{2+\delta/2},$$ for some new $C>0$. Let now $\tilde v,\,\tilde v_*$ as in , for $(\bar t,\bar x,\bar \theta)$ and $(\bar t_*,\bar x_*,\bar \theta_*)$ respectively, where $|\bar\theta-\bar \theta_*| \leq 3+A$. Let also $r>0$. Then, $$\tilde v(t,x,\theta)=\bar v\left(t,\sqrt{|\bar\theta|+1}x,\theta\right)=\tilde v_*\left(t,\sqrt{\frac{|\bar\theta|+1}{|\bar \theta_*|+1}}x,\theta\right).$$ Let $\phi_{(\bar t_*,\bar x_*,\bar \theta_*)}:(t,x,\theta)\mapsto \left(t,\sqrt{\frac{|\bar\theta|+1}{|\bar \theta_*|+1}}x,\theta\right)$. Since we assumed that $|\bar\theta-\bar \theta_*| \leq 3+A$, we have $\sqrt{\frac{|\bar\theta|+1}{|\bar \theta_*|+1}}\leq 3+A$, which implies $$|\tilde v|_{1+\delta/2,2+\delta,Q_{(\bar t_*,\bar x_*,\bar \theta_*),r}}\leq C|\tilde v_*|_{1+\delta/2,2+\delta,\phi_{(\bar t_*,\bar x_*,\bar \theta_*)}\left(Q_{(\bar t_*,\bar x_*,\bar \theta_*),r}\right)}.$$ Moreover, $\phi_{(\bar t_*,\bar x_*,\bar \theta_*)}\left(Q_{(\bar t_*,\bar x_*,\bar \theta_*),r}\right)\subset Q_{\phi_{(\bar t_*,\bar x_*,\bar \theta_*)}(\bar t_*,\bar x_*,\bar \theta_*),(3+A)r}$, and then, $$|\tilde v|_{1+\delta/2,2+\delta,Q_{(\bar t_*,\bar x_*,\bar \theta_*),r}} \leq C|\tilde v_*|_{1+\delta/2,2+\delta,Q_{\phi_{(\bar t_*,\bar x_*,\bar \theta_*)}(\bar t_*,\bar x_*,\bar \theta_*),(3+A)r}}.$$ We notice now that there exists $\left((\bar t_i,\bar x_i,\bar \theta_i)\right)_{i=1,\dots,N}$, (where $N$ is a function of $A$ only, and $\bar t_i\in (1,\bar t]$) such that $Q_{(\bar t,\bar x,\bar \theta),3+A}\subset \cup_{i=1}^N Q_{(\bar t_i,\bar x_i,\bar \theta_i),1/(3+A)}$. Thus, $$\begin{aligned} |\tilde v|_{1+\delta/2,2+\delta,Q_{3+A}}&\leq& C\sum_{i=1}^N |\tilde v|_{1+\delta/2,2+\delta,Q_{(\bar t_i,\bar x_i,\bar \theta_i),1/(3+A)}}\nonumber\\ &\leq&C\sum_{i=1}^N |\tilde v_i|_{1+\delta/2,2+\delta,Q_{\phi_{(\bar t_i,\bar x_i,\bar \theta_i)}(\bar t_i,\bar x_i,\bar \theta_i),1}},\label{eq-step3}\end{aligned}$$ where $\tilde v_i$ is the equivalent of $\tilde v$, with $(\bar t_i,\bar x_i,\bar \theta_i)$ instead of $(\bar t,\bar x,\bar \theta)$. We define now, for some $T>0$, $$\|v\|_{1+\delta/2,2+\delta,T}:=\max_{(\bar t,\bar x,\bar \theta)\in [1,\bar t]\times \mathbb R^2} |\tilde v|_{1+\delta/2,2+\delta,Q_{(\bar t,\bar x,\bar \theta),1}}.$$ If $\bar t\in [(3+A)^2+1,T]$, we can apply and to show that$$\begin{aligned} |\tilde v|_{1+\delta/2,2+\delta,Q_{1}}&\leq& \alpha |\tilde v|_{1+\delta/2,2+\delta,Q_{3+A}} + C M^{2+\delta/2}\\ &\leq& \alpha C\sum_{i=1}^N |\tilde v_i|_{1+\delta/2,2+\delta,Q_{\phi_{(\bar t_i,\bar x_i,\bar \theta_i)}(\bar t_i,\bar x_i,\bar \theta_i),1}} + C M^{2+\delta/2}\\ &\leq& \alpha CN \|v\|_{1+\delta/2,2+\delta,T}+ C M^{2+\delta/2}.\end{aligned}$$ This estimate holds for any $(\bar t, \bar x,\bar \theta)\in[ (3+A)^2+1,T]\times\mathbb R^2$, and thanks to the assumption on $M$ and , it also holds for $(\bar t, \bar x,\bar \theta)\in[0, (3+A)^2+1]\times\mathbb R^2$. Thus, $$\|v\|_{1+\delta/2,2+\delta,T}\leq \alpha CN\|v\|_{1+\delta/2,2+\delta,T}+ C M^{2+\delta/2},$$ and we chose $\alpha:= \frac 1{2CN}$ to obtain that $$\|v\|_{1+\delta/2,2+\delta,T}\leq C M^{2+\delta/2}.$$ *Step 3. Maximum principle.* Thanks to and , we know that $\|v\|_{L^\infty([0,T]\times\mathbb R\times\mathbb R_+)}\leq M$. Our goal is to show that indeed, $\|v\|_{L^\infty([0,T]\times\mathbb R\times\mathbb R_+)}< M$. Assume that there exists $(\bar t,\bar x,\bar \theta)\in (0,\infty)\times\mathbb R\times [1,\infty)$ such that $v$ reaches the value $M$. Then $v(\bar t,\bar x,\bar \theta+1)=M$, while $$\forall (t,x,\theta)\in [0,\bar t]\times \mathbb R\times [1,\infty),\quad v(t,x,\theta)\leq M.$$ If we define as before $\bar v(t,x,\theta):=v(t,x,|\theta|+1)$ (see ), then $$\partial_t \bar v(\bar t,\bar x,\bar \theta)\geq 0,\; \Delta_x \bar v(\bar t,\bar x,\bar \theta)\leq 0,\;\Delta_\theta \bar v(\bar t,\bar x,\bar \theta)\leq 0,$$ which, combined to , implies $$0\leq v(\bar t,\bar x,\bar \theta)\left(1-\langle v\rangle(\bar t,\bar x,\bar \theta)\right),$$ and since $v(\bar t,\bar x,\bar \theta)>0$, $$\label{eq:ineq1} 0\leq 1-\langle v\rangle(\bar t,\bar x,\bar \theta).$$ For any $a\in(0,A)$ $$\begin{aligned} \langle \bar v\rangle (\bar t,\bar x,\bar \theta)&\geq&\frac 12\int_{\bar\theta-a}^{\bar\theta+a}v(t,x,\theta')\,d\theta'\nonumber\\ &\geq& a v(\bar t,\bar x,\bar \theta)-a^2\|\partial_\theta v(\bar t,\bar x,\cdot)\|_{L^\infty([\bar\theta-a,\bar\theta+a])}.\label{eq:ineq2}\end{aligned}$$ We can now use the Gagliardo-Nirenberg interpolation inequality and the previous step with $T:=\bar t$, to estimate the last term of : $$\begin{aligned} \|\partial_\theta \bar v(\bar t,\bar x,\cdot)\|_{L^\infty([\bar\theta-a,\bar\theta+a])}&\leq& \|\partial_\theta \bar v(\bar t,\bar x,\cdot)\|_{L^\infty([\bar\theta-a,\bar\theta+a])}\\ &\leq&C \|\bar v\|_{L^\infty([\bar\theta-a,\bar\theta+a])}^{\frac 12}\|\Delta_{\theta}\bar v\|_{L^\infty([\bar\theta-a,\bar\theta+a])}^{\frac 12}+C\|\bar v\|_{L^\infty([\bar\theta-a,\bar\theta+a])}\nonumber\\ &\leq&CM^{\frac 12}\left(CM^{2+\delta/2}\right)^{\frac 12}+CM=CM^{\frac {3+\delta/2}2},\end{aligned}$$ where $C>0$ is a universal constant. Thanks to , and the last estimate, we get $$0\leq 1-\left(aM-Ca^2M^{\frac {3+\delta/2}2}\right).$$ If we select $a:=2M^{-1}$, we get $$0\leq 1-\left(2-\frac {4C}{M^{\frac {1-\delta /2}2}}\right),$$ in which we chose e.g. $\delta=1/2$. We then obtain a contradiction as soon as $M>(4 C)^4$. This contradiction implies that $\|v\|_{L^\infty([0,T]\times\mathbb R\times\mathbb R_+)}< M$. Let now a constant $M$ satisfying . For $T\in((3+A)^2+1,(3+A)^2+2)$, we have $\|v\|_{L^\infty([0,T]\times\mathbb R\times\mathbb R_+)}< M$. We can then define the largest $T>(3+A)^2+1$ such that $\|v\|_{L^\infty([0,T]\times\mathbb R\times\mathbb R_+)}\leq M$. If $T\neq\infty$, the argument above leads to a contradiction, which proved the Theorem. Comparison of the models {#sec:comparison-models} ======================== We show two comparison principles between the models (Loc) and (NLoc). First, we construct a solution of (Loc) that will provide a lower bound for solutions of (NLoc): \[prop:nonlocal\] Let $v_0\in C^{2+\delta}(\mathbb R\times [1,\infty))$ with compact support in $\theta$, thin tail in $x$ and regular, as described in Subsection \[sec:main-results\]. Let $v(t,x,\theta)$ the corresponding solution of (NLoc). For any $\eta>0$ small, there exists $\varepsilon>0$ such that For any $\eta>0$, there exists $\varepsilon>0$ such that $$\forall (t,x,\theta)\in\mathbb R_+\times \mathbb R\times [1,\infty),\quad \varepsilon u\left((1-\eta)t,\sqrt{1-\eta}\,x,\sqrt{1-\eta}(\theta-1)+1\right)\leq v(t,x,\theta),$$ where $u$ is the solution of (Loc) with initial condition $$\label{condini-comp-inf} u_0(t,x,\theta)={\mathbf{1}_{\{\mathbb R_-\times (1,2)\}}}.$$ Let $\eta>0$. We recall the definition of $\bar v$. Thanks to Theorem \[theo:unif-pointwise-II\], $\bar v$ satisfies $$\partial_t \bar v-\frac{1+|\theta|}2\Delta_x \bar v-\frac 12\Delta_\theta \bar v\geq (1-2AM)\bar v,$$ and since $\bar v(0,x,\theta)>C>0$ for $(x,\theta)\in\mathbb R_-\times (\theta_{\min},\theta_{\max})$, there exists $C_0>0$ such that $\bar v(1,x,\theta)>C_0$ for $(x,\theta)\in\mathbb R_-\times [-1,1]$. Then, $$\label{ineq-ini} C_0 \bar u_0(0,x,\theta)<v_0(1,x,\theta),\textrm{ for }(x,\theta)\in\mathbb R_-\times [-1,1].$$ For any $(x,\bar\theta)\in [1,\infty)\times\mathbb R$, $\bar v(t,x,\theta)=\tilde v\left(t,\sqrt{\bar \theta+1},\theta\right)$ is solution of , which is a parabolic equation, and the coefficients of this equation are bounded for $(t,x,\theta)\in \mathbb R\times \mathbb R\times [\bar \theta-2A,\bar \theta +2A]$, with a bound on those coefficients that is uniform in $(\bar x,\bar \theta)$. Thanks to this property and the $L^\infty$ bound $\|\bar v\|_{L^\infty}\leq M<\infty$ provided by Theorem \[theo:unif-pointwise-II\], we can apply the Harnack-type inequality Theorem 2.6 from [@Alfaro] with $\delta:=\frac\eta 2$. There exists then $C_H>0$ such that for any $t\geq 1$, $$\langle \tilde v\rangle (t,\bar x,\bar \theta)\leq C_H\tilde v(t,\bar x,\theta)+\frac\eta 2,$$ and then $\langle \bar v\rangle (t,\bar x,\bar \theta)\leq C_H\bar v(t,\bar x,\bar \theta)+\frac\eta 2$, where the constant $C_H$ is independent of $(\bar x,\bar\theta)$. $\bar v$ then satisfies $$\label{ineq-Harnack} \partial_t\bar v -\frac{|\theta|+1} 2\Delta_x\bar v-\frac 12\Delta_\theta\bar v\geq \left(1-\frac \eta 2-2AC_H\bar v\right)\bar v.$$ Let now $\bar u(t,x,\theta):= u(t,x,|\theta|+1)$, and notice that $\hat u(t,x,v):=\varepsilon u\left((1-\eta)t,\sqrt{1-\eta}\,x,\sqrt{1-\eta}\,(\theta-1)+1\right)$ satisfies $$\partial_t\hat u -\frac{|\theta|+1} 2\Delta_x\hat u-\frac 12\Delta_\theta\hat u= (1-\eta)\left(1-\frac 1\varepsilon\hat u\right)\hat u.$$ If we chose $\varepsilon=\min(4AC_H,C_0)$, then $\hat u$ is a sub-solution of , which, combined to and the comparison principle, implies that for $(t,x,\theta)\in\mathbb R_+\times \mathbb R\times [1,\infty)$, $$\varepsilon u\left((1-\eta)t,\sqrt{1-\eta}\,x,\sqrt{1-\eta}\,(\theta-1)+1\right)\leq v(t,x,\theta).$$ The second step is to construct a solution of (Loc) that will provide an upper bound for solutions of (NLoc): \[prop:nonlocal2\] Let $v$ a solution of (NLoc) such that its initial condition $v_0$ satisfies (1) and (2) of Subsection \[sec:main-results\]. For any $\eta>0$ small, there exists $\varepsilon>0$ such that $$\forall (t,x,\theta)\in\mathbb R_+\times \mathbb R\times [1,\infty),\quad v(t,x,\theta)\leq \frac 1\varepsilon u\left((1+\eta)t,\sqrt{1+\eta}\,x,\sqrt{1+\eta}\,\theta\right),$$ where $u$ the solution of (Loc) with initial condition $$\label{cond_ini_u} u_0(x,\theta)=\left\{\begin{array}{l}\varepsilon v_0\left(\frac x{\sqrt{1+\eta}},\frac{\theta-1}{\sqrt{1+\eta} }+1\right),\textrm{ for }(x,\theta)\in \left(\mathbb R_-\times (1,2)\right)^c\\ 1,\textrm{ for }(x,\theta)\in \mathbb R_-\times (1,2). \end{array}\right.$$ We notice that $v$ satisfies $$\partial_t v-\frac \theta 2\Delta_x v-\frac 12 \Delta_\theta v=v(1-\langle v\rangle )\leq v< v\left(1+\eta-\frac\eta {2M}v\right).$$ If $u$ is a solution of (Loc), then $\hat u=\frac{4M}\eta u\left((1+\eta)t,\sqrt{1+\eta}x,\sqrt{1+\eta}(\theta-1)+1\right)$ satisfies $$\label{eqsup} \partial_t \hat u-\frac \theta 2\Delta_x \hat u-\frac 12 \Delta_\theta \hat u=\hat u\left(1+\eta-\frac\eta {2M}\hat u\right).$$ Moreover, if we assume that the initial condition of $u$ is given by with $\varepsilon=\frac \eta{2M}$, then $v_0\leq \hat u(0,\cdot,\cdot)$ (notice that $\|v_0\|_{L^\infty}\leq M<\frac{4M}\eta$), and the comparison principle applied to the (local) parabolic equation implies that $v(t,x,\theta)\leq \hat u(t,x,\theta)$ for $(t,x,\theta)\in\mathbb R_+\times\mathbb R\times[1,\infty)$, which proves the result. Let us first consider the upper bound on the propagation of $v$. Thanks to Proposition \[prop:nonlocal2\], for any $\eta>0$ there exists $\varepsilon>0$ such that $$v\left(\frac t{1+\eta},\frac x{\sqrt{1+\eta}},\frac\theta{\sqrt{1+\eta}}\right)\leq \frac 1\varepsilon u(t,x,\theta),$$ where $u$ is the solution of (Loc) with initial condition . Since the initial condition satisfies the conditions (1) and (2’) (see Subsection \[sec:key-ideas-proofs\]), Theorem \[T:toads\] applies, and for any $\tilde \gamma>\gamma_0$, there exists $\tilde \varepsilon>0$ such that $$\lim_{t\to\infty} \sup_{\theta\geq 1}v\left(\frac t{1+\eta},\frac {\tilde\gamma}{\sqrt{1+\eta}}t^{\frac 32},\theta\right)=0,$$ that is $\lim_{t\to\infty} \sup_{\theta\geq 1}v\left(t, \tilde \gamma(1+\eta)t^{\frac 32},\theta\right)=0$, which proves , provided we chose $\eta>0$ small enough, and $\tilde \gamma>\gamma_0$ small enough. Proving the lower bound is very similar: Thanks to Proposition \[prop:nonlocal\], for any $\eta>0$ there exists $\varepsilon>0$ such that $$\varepsilon u(t,x,\theta)\leq v\left(\frac t{1-\eta},\frac x{\sqrt{1-\eta}},\frac \theta{\sqrt{1+\eta}}\right),$$ where $u$ is the solution of (Loc) with initial condition .Since the initial condition satisfies the conditions (1) and (2’), Theorem \[T:toads\] applies, and for any $\tilde \gamma<\gamma_0$, there exists $\tilde \varepsilon>0$ such that for any $t\geq 1$, $$\tilde\varepsilon \leq \sup_{\theta\geq 1}v\left(\frac t{1-\eta},\frac {\tilde \gamma}{\sqrt{1-\eta}}t^{\frac 32},\theta\right),$$ that is, for any $t\geq 2$, $$\tilde\varepsilon\leq \sup_{\theta\geq 1}v\left(t, \tilde \gamma(1-\eta)t^{\frac 32},\theta\right),$$ which proves , provided we chose $\eta>0$ small enough, and $\tilde\gamma<\gamma_0$ large enough. Acknowledgements {#acknowledgements .unnumbered} ================ The first author acknowledges the financial support of EPSRC grants EP/L018896/1 and EP/I03372X/1. The second author’s work is supported by the ERC starting grant MATKIT. The third author was partially supported by the ANR grant MODEVOL, ANR-13-JS01-0009, and by the CNRS/Royal Society exchange project CODYN. [100]{} M. Alfaro, J. Coville, G. Raoul, Travelling waves in a nonlocal equation as a model for a population structured by a space variable and a phenotypic trait, Comm. Partial Differential Equations, 38(12), 2126–2154 (2013). D. G. Aronson, Non-negative solutions of linear parabolic equations. *Ann. Scuola Norm. Sup. Pisa* 22(3), 607–694 (1968). D. G. Aronson, H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math. 30, 33–76 (1978). M. Q. Benedict, Q. Mark, R. S. Levine, W. A. Hawley, L. P. Lounibos, Spread of the tiger: Global risk of invasion by the mosquito Aedes albopictus, Vector Borne Zoonotic Dis. 7(1), 76–85 (2007). O. Bénichou, V. Calvez, N. Meunier, R. Voituriez, Front acceleration by dynamic selection in Fisher population waves, Phys. Rev. E 86(4), 041908 (2012). H. Berestycki, F. Hamel, Generalized travelling waves for reaction-diffusion equations, Perspectives in Nonlinear Partial Differential Equations, in honor of H. Brezis, H. Berestycki et al. (eds), Contemp. Mathematics 446 (2007), Amer. Math. Soc., Providence, RI. H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: travelling waves and steady states, Nonlinearity 22(12), 2813–2844 (2009). F. Blanquart, The demography of a metapopulation in an environment changing in time and space, Theor. Pop. Biol. 94, 1–9 (2014). E. Bouin, V. Calvez, G. Nadin, Propagation in a kinetic reaction-transport equation: travelling waves and accelerating fronts, accepted in Arch. Ration. Mech. Anal. E. Bouin, V. Calvez, N. Meunier, S. Mirrahimi, B. Perthame, G. Raoul, and R. Voituriez, Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration, C. R. Math. Acad. Sci. Paris 350, 761–766 (2012). E. Bouin, V. Calvez, Travelling waves for the cane toads equation with bounded traits, Nonlinearity 27, 2233–2253 (2014). E. Bouin, S. Mirrahimi, A Hamilton-Jacobi limit for a model of population stuctured by space and trait, accepted in Comm. Math. Sci. M.D. Bramson, Maximal displacement of branching Brownian motion, Comm. Pure Appl. Math 31, 531-581 (1978). R. S. Cantrell, C. Cosner, Y. Lou, Evolutionary stability of ideal free dispersal strategies in patchy environments, J. Math. Biol. (2011). J. T. Carlton, J. B. Geller, Ecological Roulette: The Global Transport of Nonindigenous Marine Organisms, Science 261(5117), 78–82 (1993). J. S. Clark, S. R. Carpenter, M. Barber, S. Collins, A. Dobson, J. A. Foley, D. M. Lodge, M. Pascual, R. Pielke, W. Pizer, C. Pringle, W. V. Reid, K. A. Rose, O. Sala, W. H. Schlesinger, D. H. Wall, D. Wear, Ecological forecasts: An emerging imperative, Science 293, 657–660 (2001). A. C. Coulon, J. M. Roquejoffre, Transition between linear and exponential propagation in Fisher-KPP type reaction-diffusion equations, Comm. Partial Differential Equations 37, 2029–2049 (2012). J. Dockery, V. Hutson, K. Mischaikow, M. Pernarowski, The evolution of slow dispersal rates: a reaction diffusion model, J. Math. Biol. 37 (1) 61–83 (1998). C. A. Edmonds, A. S. Lillie, L. L. Cavalli-Sforza, Mutations arising in the wave front of an expanding population, Proc. Natl. Acad. Sci. 101(4), 975–979 (2004). L. C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. R. A. Fisher, [*The wave of advance of advantageous genes*]{}, Ann. of Eugenics [**7**]{} (1937), 355–369. N. Fournier, S. Méléard, A microscopic probabilistic description of a locally regulated population and macroscopic approximations, Ann. Appl. Probab. 14(4),1880–1919 (2004). J. Garnier, Accelerating solutions in integro-differential equations, SIAM J. Math. Anal. 43, 1955–1974 (2011). D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag: Berlin, 1998. N. G. Hairston, S. P. Ellner, M. A. Geber, T. Yoshida, J. A. Fox, Rapid evolution and the convergence of ecological and evolutionary time, Ecol. Lett. 8(10), 1114–1127 (2005). F. Hamel, G. Nadin, Spreading properties and complex dynamics for monostable reaction-diffusion equations, Comm. Part. Diff. Equations 37, 511–537 (2012). F. Hamel, J. Nolen, J.-M. Roquejoffre, L. Ryzhik, A short proof of the logarithmic Bramson correction in Fisher-KPP equations, Networks and Heterogeneous Media, vol. 8 no. 1 (2013), pp. 275-289. S. Harris and M. Roberts. The many-to-few lemma and multiple spines. arXiv:1106.4761. A. Hastings, Can Spatial Variation Alone Lead to Selection for Dispersal? Theor. Popul. Biol. 24, 244–251 (1983). A. Hasting, K. Cuddington, K.F. Davies, C.J. Dugaw, S. Elmendorf, A. Freestone, S. Harrison, M. Holland, J. Lambrinos, U. Malvadkar, et al, The spatial spread of invasions: new developments in theory and evidence, Ecol. Lett. 8(1), 91–101 (2005). C. L. Hughes, C. Dytham, J. K. Hill, Modelling and analysing evolution of dispersal in populations at expanding range boundaries, Ecol. Entomol. 32, 437–445 (2007). P. E. Hulme et al., eds (2009) Delivering Alien Invasive Species Inventories for Europe (DAISIE) Handbook of Alien Species in Europe, Springer. S. Janson and G. Louchard. Tail estimates for the Brownian excursion area and other Brownian areas. *Electr. J. Probab.*, Vol. 12 (2007), Paper no. 58, pp. 1600–1632. S. A. Juliano, L. P. Lounibos, Ecology of invasive mosquitoes: effects on resident species and on human health, Ecol. Lett. 8(5) 558–574 (2005). Karatzas, I. and Shreve, S. *Brownian motion and stochastic calculus.* Springer, second edition. S. Klopfstein, M. Currat, L. Excoffier, The Fate of Mutations Surfing on the Wave of a Range Expansion, Mol. Biol. Evol. 23(3), 482–490 (2006). C. S. Kolar, D. M. Lodge, Progress in invasion biology: predicting invaders, Trends Ecol. Evol. 16(4), 199–204 (2001). A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, *Étude de l’équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique*, Bulletin Université d’État à Moscow (Bjul. Moskowskogo Gos. Univ), Série Internationale, (1937), Section A, 1–26. M. Kot, M. Lewis, P. Van den Driessche, Dispersal data and the spread of invading organisms, Ecology 77, 2027–2042 (1996). R. S. Kovats, D. H. Campbell-Lendrum, A. J. McMichael, A. Woodward, J. St H. Cox, Early effects of climate change: do they include changes in vector-borne disease? Philos. Trans. R. Soc. Lond. B Biol. Sci. 356(1411), 1057–1068 (2001). N. V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces. Graduate Studies in Mathematics, 12. American Mathematical Society, Providence, RI, 1996. J. G. Lambrinos, How Interactions between Ecology and Evolution Influencence Contemporary Invasion Dynamics, Ecology 85(8), 2061–2070 (2004). T. Lindström, G. P. Brown, S. A. Sissonb, B. L. Phillips, R. Shine, Rapid shifts in dispersal behavior on an expanding range edge, Proc. Natl. Acad. Sci. 110(33), 13452–13456 (2013). E. Lombaert, A. Estoup, B. Facon, B. Joubard, B. J.C. Gregoire, A. Jannin, A. Blin, T. Guillemaud, Rapid increase in dispersal during range expansion in the invasive ladybird Harmonia axyridis, J. Evol. Biol. 27(3) 508–517 (2014). L. P. Lounibos, Invasions by insect vectors of human disease, Annu. Rev. Entomol., 47:233–66 (2002). R. N. Mack, D. Simberloff, W. M. Lonsdale, H. Evans, M. Clout, F. A. Bazzaz, Biotic invasions: Causes, epidemiology, global consequences, and control, Ecol. Appl. 3(10) 689–710 (2000). F. Massol, A. Duputier, P. David, P. Jarne, Asymetric patch size distribution leads to disruptive selection on dispersal, Ecology 65(2), 490–500 (2010). H. P. McKean (1975). Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov. *Comm. Pure Appl. Math.*, XXVIII, 323–331. S. Mirrahimi, G. Raoul, Population structured by a space variable and a phenotypical trait, Theor. Pop. Biol. 84, 87–103 (2013), A. Monty, G. Mahy, Evolution of dispersal traits along an invasion route in the wind-dispersed Senecio inaequidens (Asteraceae), Oikos 119, 1563–1570 (2010). V. C. Moran, H.G. Zimmermann, Biological control of jointed cactus, Opuntia aurantiaca (Cactaceae), in South Africa, Agric. Ecosyst. Environ. 37, 5–27 (1991). J. Moser, A harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17(1), 101–134 (1964). L. Pejchar, H. A. Mooney, Invasive species, ecosystem services and human well-being, Trends Ecol. Evol. 24(9), 497–504 (2009). B. L. Phillips, G. P. Brown, J. Webb, and R. Shine, Invasion and the evolution of speed in toads, Nature 439, 803 (2006). D. Pimentel, L. Lach, R. Zuniga, D. Morrison, Environmental and economic costs of nonindigenous species in the United States. BioScience 50 53–65 (2000). B. V. Purse, P. S. Mellor, D. J. Rogers, A. R. Samuel, P. P. C. Mertens, M. Baylis, Climate change and the recent emergence of bluetongue in Europe, Nature Rev. Microbiol. 3, 171–181 (2005). C. Prevost, Applications of partial differential equations and their numerical simulations of population dynamics, PhD Thesis, University of Orleans (2004). L. Roques, F. Hamel Fast propagation for KPP equations with slowly decaying initial conditions, J. Differ. Equations, 249, 1726–1745 (2010). O. Ronce, How Does It Feel to Be Like a Rolling Stone? Ten Questions About Dispersal Evolution A. K. Sakai, F. W. Allendorf, J. S. Holt, D. M. Lodge, J. Molofsky, K. A. With, S.Baughman, R. J. Cabin, J. E. Cohen, N. C. Ellstrand, D. E. McCauley, P. O’Neil, I. M. Parker, J. N. Thompson, S. G. Weller, The population biology of invasive species, Annu. Rev. Ecol. Syst. 32, 305–332 (2001). S. J. Schreiber, The Evolution of Patch Selection in Stochastic Environments, Am. Nat. 180(1), 17–34 (2012). N. Shigesada, K. Kawasaki, Biological invasions: theory and practice, Oxford university press 1997. D. Simberloff J. L. Martin, P. Genovesi, V. Maris, D. A. Wardle, J. Aronson, F. Courchamp, B. Galil, E. Garcıa-Berthou, M. Pascal, P. Pysek, R. Sousa, E. Tabacchi, M. Vila, Impacts of biological invasions: what’s what and the way forward, Trends Ecol. Evol 28(1), 58–66 (2013). C. D. Thomas, Dispersal and extinction in fragmented landscapes, Proc. Biol. Sci. 267(1439), 139–145 (2000). C.D. Thomas, E. J. Bodsworth, R. J. Wilson, A. D. Simmons, Z. G. Davies, M. Musche, L. Conradt, Ecological and evolutionary processes at expanding range margins, Nature 411, 577-581 (2001). L. Tolmatz, The saddle point method for the integral of the absolute value of the Brownian motion. *Discrete random walks* (Paris, 2003), Discrete Math. Theor. Comput. Sci. Proc. AC, Nancy, 2003, pp. 309–324. MR2042397. O. Turanova, On a model of a population with variable motility, submitted (http://math.uchicago.edu/ turanova/motility.pdf) M. C. Urban, B. L. Phillips, D. K. Skelly, R. Shine, A toad more traveled: the heterogeneous invasion dynamics of cane toads in Australia, Am. Nat. 171(3), 134-148 (2008). P. M. Vitousek, C. M. D’Antonio, L. L. Loope, M. Rejmanek, R. Westbrooks, Introduced species: A significant component of human-caused global change, New Zeal. J. Ecol. 21(1), 1–16 (1997) K.E. Weber, Large Genetic Change at Small Fatness Cost in Large Populations of Drosophila Zumgmter Selected for Wind Tunnel Flight:Rethinking Fitness Surfaces, Genetics 144, 205–213 (1996). K. D. Whitney, C. A. Gabler, Rapid evolution in introduced species, ‘invasive traits’ and recipient communities: challenges for predicting invasive potential, Divers. Distrib. 14(4), 569–580 (2008). J. Xin, *Front propagation in heterogeneous media*, SIAM Rev. [**42**]{}, (2000), no. 2, 161–230.

--- abstract: 'We present new observations from Z-Spec, a broadband 185 - 305 GHz spectrometer, of sub-millimeter bright lensed sources recently detected by the Herschel Astrophysical Terahertz Large Area Survey (H-ATLAS). Four out of five sources observed were detected in CO, and their redshifts measured using a new redshift finding algorithm that uses combinations of the signal-to-noise of all the lines falling in the Z-Spec bandpass to determine redshifts with high confidence, even in cases where the signal-to-noise in individual lines is low. Lower limits for the dust masses ($\sim$a few 10$^{8}$ M$_{\odot}$) and spatial extents ($\sim$1 kpc equivalent radius) are derived from the continuum spectral energy distributions, corresponding to dust temperatures between 54 and 69 K. The dust and gas properties, as determined by the CO line luminosities, are characteristic of dusty starburst galaxies, with star formation rates of 10$^{2-3}$ M$_{\odot}$ yr$^{-1}$. In the LTE approximation, we derive relatively low CO excitation temperatures ($\lesssim 100$ K) and optical depths ($\tau\lesssim1$). Using a maximum likelihood technique, we perform a non-LTE excitation analysis of the detected CO lines in each galaxy to further constrain the bulk molecular gas properties. We find that the mid-$J$ CO lines measured by Z-Spec localize the best solutions to either a high-temperature / low-density region, or a low-temperature / high-density region near the LTE solution, with the optical depth varying accordingly.' author: - | R. E. Lupu,$^{1\ast}$ K. S. Scott,$^{1}$ J. E. Aguirre,$^{1}$ I. Aretxaga,$^{2}$ R. Auld,$^{3}$ E. Barton,$^{4}$ A. Beelen,$^{5}$ F. Bertoldi,$^{6}$ J. J. Bock,$^{7,8}$ D. Bonfield,$^{9}$ C. M. Bradford,$^{7,8}$ S. Buttiglione,$^{10}$ A. Cava,$^{11,12}$ D. L. Clements,$^{13}$ J. Cooke,$^{4,8}$ A. Cooray,$^{4}$ H. Dannerbauer,$^{14}$ A. Dariush,$^{3}$ G. De Zotti,$^{10,15}$ L. Dunne,$^{16}$ S. Dye,$^{3}$ S. Eales,$^{3}$ D. Frayer,$^{17}$ J. Fritz,$^{18}$ J. Glenn,$^{19}$ D. H. Hughes,$^{2}$ E. Ibar,$^{20}$ R. J. Ivison,$^{20,21}$ M. J. Jarvis,$^{9}$ J. Kamenetzky,$^{19}$ S. Kim,$^{4}$ G. Lagache,$^{22,23}$ L. Leeuw,$^{24,25}$ S. Maddox,$^{16}$ P. R. Maloney,$^{19}$ H. Matsuhara,$^{26}$ E. J. Murphy,$^{27}$ B. J. Naylor,$^{7}$ M. Negrello,$^{28}$ H. Nguyen,$^{7}$ A. Omont,$^{29}$ E. Pascale,$^{3}$ M. Pohlen,$^{3}$ E. Rigby,$^{16}$ G. Rodighiero,$^{30}$ S. Serjeant,$^{28}$ D. Smith,$^{16}$ P. Temi,$^{31}$ M. Thompson,$^{9}$ I. Valtchanov,$^{32}$ A. Verma,$^{33}$ J. D. Vieira,$^{8}$ J. Zmuidzinas$^{7,8}$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ title: 'MEASUREMENTS OF CO REDSHIFTS WITH Z-SPEC FOR LENSED SUBMILLIMETER GALAXIES DISCOVERED IN THE H-ATLAS SURVEY' --- INTRODUCTION ============ Galaxies detected by their thermal dust emission at submillimeter (submm) and millimeter (mm) wavelengths ($\lambda \approx 250-2000\,\mu$m) comprise an important population of massive systems in the early Universe that are thought to be undergoing an early phase of intense star formation in their evolution [@blain02]. Dust grains within star-forming regions in these galaxies are heated by incident optical and ultraviolet (UV) radiation from young stars and thermally re-radiate this energy at far-infrared (far-IR) to mm wavelengths, with the peak of dust emission occurring at $\sim60-200\,\mu$m in the rest-frame [@soifer91]. It is estimated that about half of all star-formation in the Universe is heavily obscured by dust and therefore difficult to identify in even the deepest surveys at optical/ultraviolet wavelengths [@Puget1996]. Observations at submm/mm wavelengths sample the Rayleigh-Jeans tail of the thermal dust spectrum, which rises steeply with frequency $\sim\nu^{3.5}$ [@dunne00]. For observations at $\lambda > 500\,\mu$m, the climb up this steep spectrum with increasing redshift roughly cancels the effect of cosmological dimming with increasing distance [e.g., @blain02], making galaxies with a fixed luminosity have roughly the same observed flux density at submm/mm wavelengths for redshifts between $1 < z < 10$. This allows a distance-independent study of dust-obscured star-formation and galaxy evolution spanning the epoch of peak star formation activity in the Universe [$z\sim2-3$, e.g., @chapman05]. Although first predicted by @Low68, the population of high-redshift and heavily dust-obscured galaxies (submillimeter galaxies, SMGs) was first revealed a decade ago [@smail97], and several wide-area surveys at 850$\mu$m – 1.2mm have been carried out since then [e.g., @weiss09b; @austermann10; @Coppin2006; @Bertoldi2007; @scott08], mapping a total of $\sim4$deg$^2$ of sky. More recently, much larger area surveys have been undertaken with the South Pole Telescope [SPT, @vieira09] at $\lambda=1.4-2$mm, the Balloon-borne Large Aperture Submillimeter Telescope [BLAST, @pascale08] at $\lambda=250-500\,\mu$m, and the [*Herschel Space Observatory*]{} [@Pilbratt10] at $\lambda=55-670\,\mu$m. Mapping a total area of $\sim200$deg$^2$ to date [@pascale08; @vieira09; @Eales10], these surveys have uncovered a population of rare, and unusually bright, distant galaxies. Their inferred IR luminosities and high redshifts are consistent with a significant fraction of these extremely bright submm/mm galaxies being gravitationally lensed [@negrello07], but proof requires extensive multi-wavelength follow-up campaigns. Their observed flux densities can be magnified by factors $> 10$ due to lensing by intervening foreground galaxies or clusters, as observed in similarly bright systems [e.g., @Swinbank2010; @Solomon2005]. By targeting lensed objects, we can study the typical properties of the star forming galaxies in the early Universe that would otherwise be inaccessible in a blank survey due to sensitivity limitations and source confusion. The ongoing Herschel-Astrophysical Terahertz Large Area Survey [H-ATLAS, @Eales10] in the Science Demonstration Phase (SDP) has already covered 14.4 deg$^{2}$ out of the $\sim$550 deg$^{2}$ planned, resulting in $\sim$6600 sources [@Clements10; @Rigby10] with fluxes measured at 250, 350, and 500 $\mu$m using the Spectral and Photometric Imaging Receiver [SPIRE, @Griffin10; @Pascale10], and fluxes at 100 and 160 $\mu$m obtained with the Photodetector Array Camera and Spectrometer [PACS, @Poglitsch10; @Ibar2010]. Given the large areal coverage, H-ATLAS can detect the brightest (i.e. rarest) distant submm galaxies and is the first example where the efficient selection of lensed galaxies at submm wavelengths has been demonstrated [@Negrello10]. To understand the nature of these galaxies, in particular whether they represent a previously undiscovered population of intrinsically bright sources [e.g., @devriendt10] or are relatively normal starburst galaxies lensed by foreground structures [e.g., @negrello07], requires both complementary data at other wavelengths and measurements of their redshifts. However, measuring spectroscopic redshifts for these sources is challenging: their positional accuracy from submm/mm imaging is often poor due to diffraction limitations at these long wavelengths, and they tend to be highly extincted by dust, making spectroscopic measurements from optical ground-based telescopes difficult [e.g., @chapman05]. Photometric redshifts obtained using submm bands are very useful for estimating the high redshift nature of such sources, but suffer from errors due to the degeneracy between the dust temperature and the redshift, which limit their precision to $\Delta z \approx 0.3$ [@Aretxaga07; @Hughes02]. When the photometric redshift estimates involve SED template fitting, errors can also arise from our limited knowledge of the intrinsic SMG SED from FIR to radio, and its evolution with redshift. Combined with multi-wavelength data, training sets of spectroscopic redshifts may prove useful for reducing these errors for application to the large photometric datasets from ongoing and future surveys. SMGs contain large reservoirs of molecular gas [$10^{10-11}$M$_{\odot}$, @tacconi08], whose cooling is dominated by the rotational lines of CO, almost equally spaced by $\sim$115 GHz in the rest frame. Thus, the CO line detections at wavelengths beween 1 cm and 1 mm (30-300 GHz) offer the most direct measurement of their redshifts. However, with the exception of only three other CO redshifts measured to date [@daddi09a; @weiss09a; @Swinbank2010], CO detections have largely been limited to SMGs whose redshifts were already known from optical spectroscopy [e.g., @Frayer98], as a consequence of the small instantaneous bandwidth of typical mm-wavelength receivers. Z-Spec overcomes this limitation due to its large bandwidth, covering the entire 1-1.5 mm atmospheric window, allowing simultaneous observations of multiple CO lines for galaxies at redshifts $z > 0.5$. Although the potential of using the CO ladder for redshift determination is well known [e.g., @silk1997], due to sensitivity limitations of current instruments, only large area submm surveys can provide a significant number of sources bright enough for such measurements. These spectra can be used not only for an efficient redshift determination, but also to constrain the physical properties of the gas and dust (e.g., mass, density, temperature) in these galaxies [e.g., @bradford09], by measuring the CO line strengths and the continuum slope. The analysis of the CO properties requires measurements of multiple CO lines, often involving the use of multiple instruments. To date, several spectral line energy distributions (SLEDs) for the CO molecule have been constructed for small mixed samples of galaxies and quasars [@Papadopoulos2010b; @Wang2010; @Bayet2009], or individual objects. Relatively well sampled CO SLEDs have been constructed from the ground for bright quasars [@Weiss2007; @bradford09; @Danielson2010], while complete CO SLEDs have been measured by the $Herschel$ $Space$ $Observatory$ in low redshift galaxies [@Panuzzo2010; @vanderWerf2010]. Most SMGs have been observed in only one or two CO lines [see e.g., @Harris10; @Ivison2010; @Aravena2010; @tacconi08; @greve05; @Solomon2005], and their physical properties remain largely unknown. This situation has improved in recent years, with observations of multiple CO lines in individual SMGs [@Ao2008; @Carilli2010; @Lestrade2010; @Riechers2010; @Danielson2010]. The best sampled CO SLEDs show that multiple CO components are required to explain the full line luminosity distribution, where most of the mid-$J$ CO emission can generally be fit by a warm component, with kinetic temperatures of 40-60 K and gas volume densities of 10$^{3}$-10$^{4}$ cm$^{-3}$. However, solutions with kinetic temperatures of a few$\times$100 K and lower densities are also allowed by the data [@Ao2008; @Weiss2007; @Bayet2009], and this region of the parameter space has been insufficiently explored. Z-Spec has an important advantage over other instruments, since it covers a large portion of the CO SLED in a single observation (depending on the redshift) and with a common calibration for the entire bandpass. This paper describes observations of five H-ATLAS sources undertaken with Z-Spec. Based on the CO emission detected by Z-Spec, we successfully determined the redshifts of four out of five lensed galaxies. The Z-Spec observations are described in Section \[sobs\], followed by the description of the algorithm for redshift determination in Section \[sredshift\]. We use the measured redshift to constrain the spectral energy distribution (SED) of these galaxies, estimating the dust temperature and emissivity index, as well as the total infrared luminosity. We perform an analysis of the partial CO SLEDs, constructed from the lines observed by Z-Spec, to constrain the physical conditions of the molecular gas. The analysis of the galaxy SEDs and CO emission lines is presented in Section \[sgalaxy\], and a summary of our results can be found in Section \[sconcl\]. Throughout the paper we assume a standard $\Lambda$CDM cosmology, with H$_{0}$=71 km s$^{-1}$ Mpc$^{-1}$, $\Omega_{\mathrm{M}}$=0.27, $\Omega_{\Lambda}$=0.73 [@Spergel07]. OBSERVATIONS AND DATA REDUCTION\[sobs\] ======================================= We selected five high-z candidates among submm-bright galaxies with F(500$\mu$m)$>$100 mJy (Table \[tab:obs\]) from the H-ATLAS survey for follow-up observations with Z-Spec on the 10-m Caltech Submillimeter Observatory (CSO). Z-Spec is a single spatial pixel grating spectrometer with 160 silicon-nitride micro-mesh bolometer detectors (i.e. channels) operating from $190-308$GHz [@naylor03; @earle06; @bradford09]. The frequency response of the Z-Spec channels is approximately gaussian, with a variable FWHM from 720 to 1290 km s$^{-1}$ over the bandpass, that is roughly equal to the channel separation [@earle06]. This follow-up was undertaken to confirm that these sources are lensed by lower redshift galaxy clusters or individual galaxies by measuring their redshifts directly at submm wavelengths. The redshifts of the foreground objects have been measured in the optical, and found to be much lower than the redshifts of the submm galaxies, consistent with the lensing scenario [@Negrello10]. For convenience, throughout the paper we identify our targets by their names used in the SDP H-ATLAS catalogue (SDP.9, SDP.11, SDP.17, SDP.81, and SDP.130). We carried out the Z-Spec observations of H-ATLAS sources at the CSO from 2010 March 07 - May 14 under generally good to excellent observing conditions, accumulating from 6.8 to 22.5 hour integrations on each target. The zenith opacity at 225GHz (monitored by the CSO tau meter) was $\tau_{225\mathrm{GHz}} = 0.06$ on average, and $\tau_{225\mathrm{GHz}} \le 0.07$ for 75% of the observations. A summary of the observations, including the total integration time on each source is given in Table \[tab:obs\]. The Z-Spec data were taken using the standard “chop-and-nod” mode in order to estimate and subtract the atmospheric signal from the raw data. The secondary mirror was chopped on- and off-source at a rate of 1.6Hz with a chop throw of 90arc-sec while stepping through a 4-part nod cycle which position switches the primary mirror, integrating for 20sec at each nod position. The chopping removes atmosphere fluctuations and the nodding removes instrumental offsets due to imperfect match between the two chopped positions. We checked the pointing every $2-4$ hrs by observing quasars and other bright targets located close in elevation to the H-ATLAS targets, making small (typically $<$ 10arc-sec) adjustments to the telescope pointing model in real-time. We analyze the data using customized software in the same manner as described in @bradford09. For each channel, the nods are calibrated and averaged together, weighting by the inverse variance of the detector noise. Absolute calibration is determined by observations of Mars once per night, which we use to build a model of the flux conversion factor (from instrument Volts to Jy) as a function of each detector’s mean operating (“DC”) voltage [@bradford09]. Since the DC voltage depends on the combination of the bath temperature and the total optical loading on the detectors, we use these curves to determine appropriate calibration factors to apply to each nod individually. Based on the root mean square (rms) deviations of the Mars measurements from the best-fit curves, the channel calibration uncertainties are $3-8$%, excluding the lowest frequencies for which a clean subtraction of the atmosphere is hindered by the pressure-broadened 183GHz atmospheric water line. These uncertainties are propagated through the data reduction. The median rms uncertainties on the final co-added spectra for the H-ATLAS galaxies are listed in Table \[tab:obs\]. These errors do not include the $\sim5$% uncertainty on the brightness temperature of Mars [@wright07]. The calibrated Z-Spec spectra of the five ATLAS galaxies are shown in Figures \[spectra\] and \[spectra17\]. The redshifts of these sources are determined using a custom algorithm, tailored specifically for multiple lines observed simultaneously in the same bandpass. This algorithm is presented in the next section. REDSHIFT DETERMINATION\[sredshift\] =================================== Algorithm Description --------------------- The redshift determination relies on the simultaneous detection of multiple CO and atomic lines. Using the multiple CO lines falling in the Z-Spec bandpass, we developed a redshift-finding algorithm that is capable of handling cases where the signal-to-noise in individual lines is low. The number of CO lines redshifted in the Z-Spec bandpass grows from 2 at $z=0.51$ (CO(3-2) and CO(4-3)) to 4 or more at $z>2$ (starting at CO(5-4) through CO(8-7)). The measurement of redshifts greater than $\sim3$ requires the presence of high-excitation, warm CO gas, where transitions between higher rotational levels become important. As the width of the Z-Spec channels varies from 720 to 1290 km s$^{-1}$ over the bandpass, larger than most observed line widths, most of the signal from one line will be concentrated in a single channel. The redshift finding algorithm uses two test statistics, $E_{1}$ and $E_{2}$ (Eqs. \[e1eq\] and \[e2eq\]), constructed from combinations of the detection significance in those channels in which a reference line would be observed by Z-Spec at a given redshift $z$. The reference line list contains all CO rotational lines up to the CO(17-16) transition, the \[\] 492.16 GHz line, the \[\] 1458.8 GHz line, and the \[\] 1900.569 GHz line. For redshifts $z>1$ the \[\] 809.342 GHz and CO(7-6) 806.651 GHz lines can fall in the same Z-Spec channel, and are therefore degenerate for the purpose of this procedure. The values of the test statistics are related to the probability that the lines from the reference list, redshifted by a factor of $(1+z)$, are present in the spectrum. The null hypothesis is that there is no signal detected in the spectrum for any value of the redshift, and the distribution of the test statistics for the null hypothesis can be obtained using blank sky spectra, flat continuum sources, or noise simulations. Both statistics are maximized when the redshifted frequencies of the reference lines match the frequencies where the largest signal is present in the spectra, after continuum subtraction. The continuum subtraction uses a fourth degree polynomial to better account for local deviations from a power-law. The use of two statistics instead of one helps reduce the number of false redshifts that can be due to random noise fluctuation in the spectra. We search a redshift range between 0.6 and 5.6 in steps of 0.001. Let $n(z)$ be the number of reference lines that would fall in the Z-Spec bandpass at redshift $z$. The algorithm loops through all the $z$ values, redshifting all the lines in the line list, and finding the set of $n(z)$ Z-Spec channels corresponding to the lines in the bandpass at that redshift. The two test statistics, $E_{1}$ and $E_{2}$, are evaluated for each redshift using the signal and noise in the set of $n(z)$ channels determined in the previous step. The first test statistic, $E_{1}(z)$, is defined as the ratio of the signal to the noise summed only over the Z-Spec channels that correspond to a line in our list when redshifted to redshift $z$, $$\label{e1eq} E_{1}(z)=\frac{\sum_{i} S_{i}}{\sqrt{\sum_{i} \sigma_{i}^{2}}},$$ where the sum is taken from 1 to $n(z)$, and $S_{i}$ and $\sigma_{i}$ are the continuum subtracted signal and noise, respectively, for the channel corresponding to line $i$. The second test statistic, $E_{2}(z)$, is defined as $$\label{e2eq} E_{2}(z)=\mathrm{median}\{ f_{ij} | f_{ij}=0.5(S_{i}/\sigma_{i}+S_{j}/\sigma_{j}), 1\leq i,j\leq n(z), i < j\},$$ where the set contains all possible pairs of lines in the Z-Spec bandpass at the corresponding redshift. The $E_{2}$ statistic can be easily understood as a rejection criterion. Setting $E_{2}=\epsilon$ as a lower limit is equivalent to discarding all redshifts for which the median value of the average S/N in any pair of channels is less than $\epsilon$. Note that $E_{1}$ would be a reasonable statistic for single line detections, but $E_{2}$ relies on multiple lines being present in the Z-Spec bandpass at a given redshift. For redshifts $<0.6$, $E_{1}$ is still a useful statistic, though the presence of only one line does not make for an unambiguous redshift determination. No a priori knowledge of the relative line strengths is assumed, and therefore the algorithm gives equal weight to all the lines in the search list. As such, the significance of the determined redshift is dependent on which lines are detected in the spectrum relative to the lines that were expected based on the reference list. This is the case for SDP.17b (see Section \[iz\]), where the redshift significance increases greatly if we include the water line on our line list. However, such an extension of the reference line list is not always justified, since the significance for the redshift of another galaxy where the water line is not detected would be unnecessarily diminished. Care must also be taken in using the current line list at higher redshifts, where the high $J$ CO lines are likely to have much lower significance relative to the \[\] and \[\] lines. For example, the strength of the CO lines drops with increasing $J$ for starburst galaxies [e.g., @Danielson2010], but remains relatively constant for AGN-dominated galaxies and quasars [e.g., @bradford09; @vanderWerf2010]. Current knowledge of line strengths and the nature of the sources at high redshift does not warrant including this information in the redshift search. Testing the Method: Simulations on Blank Sky -------------------------------------------- The distributions of $E_{1}$ and $E_{2}$, as well as the effect of noise on the positions of their maxima, have been confirmed by simulating multiple realizations of [*blank sky*]{} spectra. The simulations take an input spectrum and generate new realizations by choosing random values of the signal in each channel, normally distributed around the value of the input signal in that channel, with a standard deviation equal to the measurement error in the same channel. The error array associated with the new realization is scaled accordingly, such that the signal-to-noise in each channel is equal to that of the input spectrum. As the input spectrum we used a 6.5 hour integration on blank sky, recorded on May 11 and 12, 2010. Other featureless spectra can also be used for this test. For each realization, we calculate the values of both test statistics and determine the redshifts corresponding to their maxima. The distribution of these redshifts is shown in Figure \[simskyzearch\], derived from a simulation containing 1000 realizations of the sky spectrum. Ideally, this distribution should be uniform, but the input sky spectrum is not perfectly smooth, possibly due to small instrumental artefacts, which are reflected by the probability variations. The noise cross-correlation between channels are present in the initial sky spectrum, but not in the simulated spectra. We find that although the $values$ of $E_{1}$ and $E_{2}$ for each redshift are mostly correlated, the [*locations of their maxima*]{} are not (Pearson correlation coefficient 0.15). Figure \[simskyzearch\] conveys this fact by comparing the maximum joint probability that $E_{1}$ and $E_{2}$ select the same redshift, relative to the maximum probability of a redshift found by $E_{1}$ independently ($\sim$3% vs. $\sim$17%). This shows that the combination of two test statistics is more robust against random fluctuations, considerably reducing the noise floor across the redshift range. A realization of the two test statistics using a blank sky spectrum is shown in Figure \[skyzearch\]. In this case, the maxima of the two statistics occur for different redshifts, and no value of $E_{2}$ is above the threshold of 2. For noise-dominated spectra, such as blank sky, $E_{1}(z)$ is well fit by a normal distribution with $\sigma_{E1}$=0.97, and $E_{2}(z)$ by a t distribution with 10 degrees of freedom (Figure \[estdist\]). For our redshift determination, we first ask that the maximum value of $E_{1}$ have a significance $\geq3\sigma$. Equivalently, the null is rejected at 99.73% confidence level. However, as noise fluctuations can still lead to random, albeit significant, peaks in these statistics even for blank sky spectra, the redshift confirmation also requires that the maxima of the two test statistics occur at the same redshift. To summarize, a redshift $z_{0}$ is accepted when the following conditions hold: $$\label{acceq} \begin{aligned} E_{1}(z_{0})\geq 3\sigma_{E1},\\ E_{1}(z_{0})=max(E_{1}),\\ E_{2}(z_{0})=max(E_{2}). \end{aligned}$$ For a line pair, the $E_{1}$ 3$\sigma$ threshold corresponds to an average S/N per channel of 2.12. Z-Spec reaches a sensitivity of about 0.5 Jy s$^{1/2}$ per channel for an atmospheric optical depth $\tau_{225}$=0.068 [@inami08]. Combining the 3$\sigma$ threshold criterion with the measured sensitivity of Z-Spec, we estimate that a redshift can be determined in less than 1.5 hours of integration time if the line flux densities per channel are on the order of 15 mJy, but can require more than 12.5 hours if the flux density is less than 5 mJy. For our galaxy sample, the mean integrated CO line flux (Table \[tab:lco\]) is $\sim$18 Jy km s$^{-1}$, while the average width of the channels is 950 km s$^{-1}$. However, the flux density per channel could be only $\sim$10 mJy if the line flux happens to be split between two adjacent channels. In this case, the typical integration time for obtaining a redshift with Z-Spec would be at least 3.5 hours. Comments on Individual Redshifts \[iz\] --------------------------------------- The results of applying this algorithm to our galaxy sample are shown in Figure \[A981zearch\]. The redshift value and its uncertainty (Table \[tab:cont\]) are determined from the position and width of the peak of the $E_{1}$ test statistic, and the significance of the redshift peak is tested against the null hypothesis (Figure \[e1hist\]). We fit a gaussian to the peak of the $E_{1}(z)$ statistic, and define the redshift error as the upper limit for the standard deviation of this gaussian. When lines are present in the spectrum, aside from the main peak due to the true redshift, secondary peaks will arise in the $E_{1}$ and $E_{2}$ distributions, corresponding to redshifts where some of the lines in the line list fall on the same channels as the observed lines. The secondary peaks are marked by blue asterisks for each source in Figure \[A981zearch\]. The real redshift will have higher significance than these secondary peaks, since the largest number of lines add their contribution to the total signal in this case. Excluding the true redshift and all the redshifts corresponding to secondary peaks from the test statistics, we are left with the null distribution, as shown in Figure \[e1hist\]. This figure also shows the significance of the redshift determination for each source, by marking the position of the peak value of the test statistic relative to the 3$\sigma$ threshold. [*SDP.81*]{} The redshift for SDP.81, z=3.037$\pm$0.01, obtained by this method at 3.8$\sigma$ significance on 19 March 2010, was confirmed (z=3.042$\pm$0.001) with follow-up observations with the IRAM Plateau de Bure Interferometer on 23 March 2010 [IRAM/PdBI, @Negrello10; @Neri10] and with an independent blind search on 25 March 2010 by the Zpectrometer instrument at the Green Bank Telescope [GBT/Zpectrometer, @Negrello10; @Frayer10], informed by a concurrent photometric redshift estimate (2.9$^{+0.2}_{-0.3}$). The significance of the test statistics for our redshift determinations shows that the integration time necessary to secure a redshift can be much shorter than the values listed in Table \[tab:obs\], and future submm instruments with better sensitivity will be able to obtain the redshifts of such galaxies even faster. [*SDP.17*]{} Given the size of the Z-Spec beam (FWHM$\approx$30) and the possible presence of lensing or other foreground structures in the same beam, the observed spectrum could be a combination of features from multiple objects. We choose this interpretation for the spectrum of SDP.17, best described by two components at different redshifts (both listed in Table \[tab:cont\]). The first redshift found by our algorithm is 2.308 (SDP.17b). After fitting the CO lines at this redshift and subtracting them from the spectrum, we perform a second redshift determination, identifying a second component with a redshift of 0.942 (SDP.17a). This combination explains all the features present in the spectrum (see Figure \[spectra17\]), and is consistent with the interpretation of the 299 GHz feature as the restframe 987 GHz water line at a redshift of 2.308. This water line has been seen to be very strong in other AGN and star-forming galaxies at low redshift, such as Mrk231 and Arp 220 [@Gonzales2010], and it has been tentatively detected in the Cloverleaf quasar at z=2.56 by @bradford09. The redshifts of SDP.9 and SDP.17b have recently been confirmed by follow-up observations of the CO(2-1) and (3-2) lines, respectively (L. Leeuw, private communication), with the Combined Array for Research in Millimeter-wave Astronomy (CARMA). The second redshift (SDP.17a) has a much lower significance, but it is in agreement with the photometric and spectroscopic optical redshifts [0.77$\pm$0.13 and 0.9435$\pm$0.0009, respectively @Negrello10]. Alternatively, the peak now identified with the CO(5-4) line at z=0.94 could be a wing of this water line, due to a $\sim$1600 km s$^{-1}$ outflow. The presence of multiple ULIRGs in a single line of sight is intriguing, and is an example of discoveries that can be made possible by Z-Spec’s broad bandwidth. [*SDP.130*]{} SDP.130 has a redshift of 2.6260$\pm$0.0003, measured by GBT/Zpectrometer [z=6.625$\pm$0.001, @Frayer10], and made more precise with PdBI/IRAM [@Negrello10; @Neri10]. So far, three CO lines have been measured in this galaxy at this redshift, namely the CO(1-0) line observed with the Zpectrometer, and the CO(3-2) and CO(5-4) lines observed with PdBI [@Negrello10], on a tuning that was successfully guided by the sub-mm photometric redshift of $z=2.6^{+0.4}_{-0.2}$ [@Negrello10]. However, we do not detect any of the higher $J$ transitions ($J_{u}>$6) that would fall in the Z-Spec bandpass at this redshift. This non-detection, which places upper limits on the integrated fluxes of the CO (6-5) through (9-8) lines at $<$ 12.5 Jy km s$^{-1}$, suggests a low ($<$ 50 K) gas temperature in the z=2.626 galaxy. We attempted to identify the line at 277 GHz, marked in red in Figure \[spectra\], with the CO (3-2) transition at z=0.25, but that would be inconsistent with the optical spectroscopic redshift of the lensing galaxy [0.220$\pm$0.002, @Negrello10] by more than 7000 km s$^{-1}$, as well as inconsistent with the other observed properties of the foreground galaxy. Similarly, identifying this feature with the 987 GHz water line at z=2.626 would require a velocity offset of $\sim$4200 km s$^{-1}$, and usually the presence of highly excited CO gas, which is not observed. This feature remains unidentified. [*SDP.9 and SDP.11*]{} The redshifts for these galaxies have been measured at 6.5$\sigma$ and 5.3$\sigma$ significance, respectively. The redshift of SDP.9 has been confirmed by CARMA observations, and more follow-up observations are currently planned for both SDP.9 and SDP.11. GAS AND DUST PROPERTIES\[sgalaxy\] ================================== A model including the lines and power-law continuum is fit to each spectrum in Figures \[spectra\] and \[spectra17\], allowing the line intensities, redshift, and continuum slope to vary. The best fit power-law index $\alpha$ for each galaxy is listed in Table \[tab:cont\]. The initial estimate for the redshift is provided by the algorithm described above, and the fit is constrained by the requirement that all the lines be at the same redshift. In cases where some of the lines are blended, we first fit only the unblended lines to obtain a more precise value for the redshift, and then we fit all the lines simultaneously, with the redshift kept fixed, to get the integrated line strengths, listed in Table \[tab:lco\]. Although the lines are not resolved, the signal from one line can be spread among adjacent channels due to the overlap of their frequency responses. We measure only the integrated line strengths, taking into account the frequency response of each Z-Spec channel, weighted according to the line width. On average, line widths below $\sim$1000 km s$^{-1}$ are not resolved by Z-Spec, and we choose a value of 300 km s$^{-1}$ in fitting the integrated line strengths. This value closely matches the width of the lines for SDP.81 and SDP.130 at PdBI [@Neri10], but is relatively low compared to the range found by interferometric measurements of other lensed high-redshift galaxies [@greve05; @Knudsen2009]. However, the determination of the integrated line fluxes is not sensitive to the choice of the line width up to values of the order of the channel width. The largest uncertainties in the integrated line strengths arise in the case of line blending, such as the CO(7-6) and \[\] $^{3}P_{2}\rightarrow^{3}$$P_{1}$ lines, or the overlapping lines at different redshifts in SDP.17 (blended lines are indicated in Table \[tab:lco\]). Continuum Spectral Energy Distributions\[sseds\] ------------------------------------------------ The continuum data for all 5 galaxies is shown in Figure \[seds\]. The measured continuum flux from Z-Spec is found to be in good agreement with the MAMBO 1.2 mm photometry [@Negrello10], except for SDP.9. Estimates of the total amount of dust and star formation rates in each galaxy can be obtained by fitting their far-infrared (far-IR) to submillimeter spectral energy distribution (SED). In this fit we include the Z-Spec data along with the Herschel-SPIRE and Herschel-PACS photometric points, as well as the Submillimeter Array (SMA) measurements at 880 $\mu$m for SDP.81 and SDP.130 [@Negrello10]. The far-IR rest frame SED can be described by a modified blackbody function, defined as $$\label{bbody} F_{\nu}=Q_{\nu}(\beta)B_{\nu}(T_{d})\Omega_{d}=(1-e^{-\tau{(\nu_{0})}(\nu/\nu_{0})^{\beta}})\frac{2h\nu^{3}}{c^{2}}\frac{1}{e^{h\nu/kT_{d}}-1}\Omega_{d}=\frac{L_{IR}}{4\pi d^{2}}\frac{Q_{\nu}(\beta)B_{\nu}(T_{d})}{\int Q_{\nu}(\beta)B_{\nu}(T_{d})d\nu},$$ where $B_{\nu}$($T_{d}$) is the Planck function, $\tau(\nu_{0})=1$ is the optical depth at $\nu_{0}$, $\Omega_{d} $ represents the observed solid angle of the dust emitting region, $d$ is the (known) distance to the source, and $h$ and $k$ denote the Planck and Boltzmann constant, respectively. The fit can be performed with three parameters: $T_{d}$, $\beta$, and a scale factor. The overall scale of the SED can be parametrized either in terms of the solid angle $\Omega_{d}$, or the total infrared luminosity ($L_{IR}$), defined as the integral of the SED from 8 to 1000 $\mu$m (rest frame). The $L_{IR}$ derived in this manner underestimates the true total infrared luminosity, due to the likely presence of warmer dust components that contribute at shorter wavelengths. Including $\nu_{0}$ as a fourth parameter in the fit leads to a value of 1251$\pm$130 GHz for SDP.9, but no strong constraints are found for the rest of the sample. The low value found for $\nu_{0}$, and the observed flattening of the peak of the SEDs in the far-IR suggest that the SEDs of the galaxies in our sample can be modeled either as combinations of multiple graybodies with different temperatures, or as a single graybody with a large optical depth at far-IR wavelengths. The simplest model that can reproduce the data for the entire sample has fixed $\beta=2$ [@Priddey2001] and $\nu_{0}$=1300 GHz, in agreement to the value found for SDP.9. The dust emissivity index $\beta=2$ is also consistent with the error bars of the Z-Spec spectra. The best-fit models are shown in Figure \[seds\], and the corresponding values for $T_{d}$ are listed in Table \[tab:cont\]. With dust temperatures between 54 and 69 K, the peak of the dust SED is found in a narrow range of wavelengths (73 to 92 $\mu$m) for all lensed galaxies in the sample. It is important to bear in mind that this fitting function for the SED is largely empirical, and the degree to which $T_{d}$ and $\beta$ represent physical quantities is complicated by the spatial averaging over the entire galaxy and the degeneracy between a distribution in dust temperature and a distribution of dust types (represented by $\beta$). The formal errors for the fitted parameters ($T_{d}$) should not be interpreted as errors on physical quantities, due to these caveats. Even though such SED fits could be obtained using just the photometric points, the addition of Z-Spec data not only strongly constrains the continuum slope, but also breaks the degeneracy between $T_{d}$ and redshift [@Blain:1999], by independently determining the latter. Since Z-Spec has determined the redshift, we are able to obtain $T_{d}$ from the continuum fit, which otherwise would constrain the quantity $T_{d}/(1+z)$ [e.g., @Amblard10]. This degeneracy can lead to significant variations in the derived $T_{d}$ if the redshift is not measured independently. The implications of the continuum slope measured by Z-Spec for the dust composition will be analyzed in a future work. Using Eq. \[bbody\] corrected for redshift and the derived $T_{d}$, we can estimate the observed size of the dust emitting region. This solid angle will be affected by the lensing magnification factor. If the dust optical depth at submm wavelengths is low, as is often the case, $\Omega_{d}$ will be correlated with $\tau$, and therefore with $\beta$ [@Hughes1993]. However, we break this degeneracy by fixing $\beta$. The resulting $\Omega_{d}$ ranges between 0.30 arcsec$^{2}$ for SDP.17b and 1.44 arcsec$^{2}$ for SDP.17a. These values may underestimate source sizes that are resolved in the SMA images with a resolution of $\sim$0.8 arcsec at 340 GHz. Using the magnification factors from Table \[tab:cont\], the intrinsic size of the dust emitting region will have an equivalent radius of 0.7 kpc for SDP.81 and 1.3 kpc for SDP.130. Note however, that $\Omega_{d}$ corresponds to the effective solid angle of the dust emitting region, such as the total area of small clumps spread over a larger region. In an image where these clumps are unresolved, the total observed solid angle can appear to be larger. Having estimated the source size, the total dust mass follows from the relationship $\tau(\nu)=\kappa(\nu)M_{d}/D_{A}^{2}\Omega_{d}$, with $\kappa(\nu)=0.4(\nu/250GHz)^{\beta}$ [e.g., @Weiss2007], where D$_{A}$ is the angular diameter distance. The dust mass can also be estimated in the optically thin limit ($1-e^{-\tau}\simeq\tau$) without the additional step of deriving $\Omega_{d}$, by substituting $\tau(\nu)$ directly in Equation \[bbody\]. This is a good approximation at 250 GHz (1.2 mm), in the middle of the Z-Spec bandpass. Calculated in the optically thin limit, the dust mass is a robust estimate of the lower limit for the total dust mass in the galaxy, $M_{d,lim}$. Using the optically thin approximation and the 250 GHz flux density measured by Z-Spec, we derive values for the magnified $M_{d,lim}$ of a few$\times$10$^{9}$ M$_{\odot}$, as listed in Table \[tab:cont\]. If the dust is optically thick, as suggested by $\nu_{0}$=1300 GHz, the calculated $M_{d,lim}$ will underestimate the true dust mass for our galaxy sample by at most 30%. The dust mass is also inversely correlated with the assumed temperature, and will be underestimated when using the dust temperature corresponding to the peak of the SED. This temperature is likely too large to represent the bulk of the dust. Assuming that the 250 GHz flux is partially due to a dust component with a temperature as low as 20 K, and taking into account the optical depth corrections, we estimate that the total dust mass could be larger than $M_{d,lim}$ by up to a factor of $\sim$4. To summarize, with good approximation, the true dust masses for these galaxies will be found in the interval $[1,4]\times M_{d,lim}$. The uncertainties in $M_{d,lim}$ are mostly due to uncertainties in the expression for $\kappa(\nu)$. Note that the quantity $M_{d}/\Omega_{d}$ is proportional to $\tau$ and independent of temperature; for a given $\tau$, a lower limit for $M_{d}$ implies a lower limit for $\Omega_{d}$, but this limit for $\Omega_{d}$ will decrease with increasing $\tau$. A more realistic approach is to fit the photometric and continuum points with a library of SEDs, taking into account the transmission curve of each instrument. We apply this method to our galaxy sample, using the SED libraries of @chary01 (CE01), and @dale02 (DH02). We find that the IR luminosities derived from the modified blackbody fitting are at most factor of $\sim$2 lower than those when we use the SED libraries, and within 20% from the $L_{IR}$ obtained assuming the models of @dacunha08, calibrated for ULIRGs, with $A_{V}>2$ [@Negrello10]. The variations between the values of $L_{IR}$ obtained by different methods reflect the systematic uncertainties in deriving this quantity. Similar underestimates have been found by others, and are due to the fact that the submm photometry does not measure the warm dust component of the SED, if one is present [@Swinbank2010; @Ivison10]. This is emphasized by the poor fit of the modified blackbody curve to the Herschel-PACS data points, and the significant flux at shorter wavelengths predicted by the CE01 and DH02 models (Figure \[seds\]). Any derivation of $L_{IR}$ is model dependent, with the largest differences arising from the presence of a warm dust component in the SED libraries. In Table \[tab:cont\] we list the $L_{IR}$ values derived from the SED template fitting method, as they represent a more accurate description of the total infrared energy output than the modified blackbody. On average, our $L_{IR}$ values are about 25% lower than those found by @Negrello10, but these differences are difficult to judge without data shortward of 100 $\mu$m. Note that these values are rather smaller than typical IR luminosities of classical SMGs and unbiased sources detected by Herschel surveys. Except for SDP.17, we attribute all the submm flux density to the high redshift galaxy. The foreground lenses for the other galaxies in our sample have optical properties consistent with being quiescent elliptical galaxies, and are therefore unlikely to have a significant submm emission. We have attempted a decomposition of the SDP.17 SED, using the two measured redshifts and a wavelength-independent scaling factor for each of the two components. The $\chi^{2}$ value for the SED template fits is minimized when the observed flux density is split in half between the two components. This factor has been taken into account in Figure \[seds\] and in deriving the $L_{IR}$ for SDP.17a and SDP.17b, as listed in Table \[tab:cont\]. However, the large dust mass inferred for SDP.17a could be an indication that the SED decomposition between SDP.17a and SDP.17b overestimates the contribution of SDP.17a. We estimate the star formation rates for our galaxy sample using the conversion factor $SFR $($M_{\odot}$ yr$^{-1}$)=1.5$\times$10$^{-10}$($L_{IR}$/$L_{\odot}$) [@Solomon1997]. Since the selected galaxies are lensed by foreground objects with magnification factors $\sim$10 [@Negrello10], the intrinsic IR and CO line luminosities will be $\sim$10 times lower than the direct conversion from the measured fluxes. SDP.81 and SDP.130 have magnification factors of 25 and 6, respectively, as derived from the best-fit lens model to the high-resolution sub-mm images available for these two objects [@Negrello10]. In Tables \[tab:cont\] and \[tab:sfr\], we left the quantities affected by gravitational lensing magnification unmodified, for reference, but the presence of this contribution is indicated by the letter $\mu$ in front. Based on model predictions [@negrello07], a typical amplification factor of 10 can be applied to these values. Once corrected for magnification, the infrared luminosities and corresponding SFRs are those typical of Ultra Luminous Infrared Galaxies (ULIRGs). CO Line Luminosities and Spectral Energy Distributions\[sco\] ------------------------------------------------------------- The measurements of CO lines reveal important information about the physical properties and excitation conditions of the molecular gas, as well as the total gas budget in these galaxies. These parameters can be used to investigate the link between star formation and gas properties. Higher gas temperatures and lower densities would suggest that the star formation has been quenched, while lower temperatures associated with higher densities would show that most of the gas in the galaxy is still collapsing and forming stars. This follows from the fact that the Jeans mass increases rapidly at higher temperatures and low densities. A useful quantity describing the CO lines is the velocity-integrated brightness temperature scaled by the area of the source, $L_{CO}^{'}$, in units of K km s$^{-1}$ pc$^{2}$. In what follows, the brightness temperatures are computed in the Rayleigh-Jeans limit. If the CO is thermalized and the lines are optically thick, $L_{CO}^{'}$ will be the same for all rotational transitions for which the Rayleigh-Jeans approximation holds. This quantity is traditionally derived from the CO(1-0) transition and related to the total molecular mass via the empirical relation $M_{gas}=\alpha L_{CO}^{'}$, where $\alpha$ is 4.6 M$_{\odot}$ (K km s$^{-1}$ pc$^{2}$)$^{-1}$ for the Galaxy [@Solomon1997], and 0.8 M$_{\odot}$ (K km s$^{-1}$ pc$^{2}$)$^{-1}$ for ULIRGs [@Solomon2005; @tacconi08]. Following @Solomon2005, we can use the latter value for $\alpha$ and the $L_{CO}^{'}$ for the lowest observed CO transition to determine the gas masses. This procedure assumes that all transitions from CO(1-0) up to the lowest observed are thermalized, which might not necessarily be the case. A recent comparison of the CO(3-2) and (1-0) lines [@Harris10] shows that the ratio of the brightness temperatures for these two lines averages to 0.6 rather than 1, due to the presence of multi-phase CO gas. Moreover, the mid-$J$ CO transitions do not account for the possible presence of a colder gas component, making the $M_{gas}$ derived in this manner a lower limit for the total gas mass in the galaxy. Assuming that the lines with $J_{u}>$3 are thermalized, corresponding to a warmer gas component, we apply this correction factor to the lowest CO transition measured, and obtain the gas masses listed in Table \[tab:sfr\]. These values result in an average molecular gas-to-dust ratio for the lensed galaxies of 34$\pm$10, independent of magnification. The mean does not include the foreground SDP.17a, and is in agreement with the values found for other SMG samples [@Kovacs2006; @Michalow2010; @Santini2010]. Using the SFRs derived from the IR luminosities, the gas reservoir probed by CO implies a gas depletion time \[$M_{gas}$/$L_{IR}$\] in these objects of $\sim10^{7}$ years, similar to other known SMGs [@Solomon2005; @greve05]. This can be interpreted as the starburst lifetime under the assumptions of constant SFR and no gas inflow. Note that this estimate of the gas depletion time is independent of lensing magnification. The star formation efficiency can be expressed directly in terms of $L_{IR}/L_{CO}^{'}$, without the need for a gas mass conversion factor. After accounting for the lensing magnification factor, $L_{IR}$ and $L_{CO}^{'}$ for our sample follow the same relationship as other SMGs and ULIRGs [@greve05; @Wang2010], within the scatter. In order to derive the physical characteristics of the gas in these galaxies, including the gas temperature, density, pressure, and CO column density, we need measurements of multiple CO transitions, sampling the rotational ladder as well as possible. The spectral line energy distribution (SLED) for the CO molecule has been constructed in a few cases for nearby and low redshift galaxies [e.g., @Panuzzo2010]. In Figure \[sleds\] we show the partial SLEDs for our galaxy sample, constructed from the lines detected in the Z-Spec bandpass. This plot favors a distribution with the brightest lines between CO(5-4) and (7-6), similar to the distribution observed for other SMGs and starburst galaxies [@Weiss07; @Danielson2010]. The shape of the line luminosity distribution does not reflect only the CO excitation temperature, but also the gas density, and the effects of the optical depth at the line frequency [@Goldsmith1999; @Papadopoulos2010b]. In the optically thin limit, the CO column density scales with the absolute value of the line intensity, assuming that the source size and the magnification factor are known. Under the assumption of local thermodynamic equilibrium (LTE), all CO transitions have the same excitation temperature, $T_{ex}$, also equal to the gas kinetic temperature $T_{kin}$, signifying that all rotational levels are populated according to the Maxwell-Boltzmann distribution at temperature $T_{ex}$. In Section \[slte\] we estimate these parameters by fitting the partial SLEDs, using the relationship between the integrated line brightness temperature, column density and excitation temperature, under LTE. Although this case is limiting due to the assumption of constant $T_{ex}$ for all levels, it is interesting to compare the predictions of this model to the more general non-LTE models, given its simple physical interpretation. In the non-LTE case, presented in Section \[snlte\], the models involve a larger number of parameters, and are less well constrained. We use RADEX [@vanderTak07] to compute the brightness temperatures of the CO lines and estimate the likelihood distribution over the parameter space. These distributions allow us to assess if the available data are able to distinguish between the LTE and non-LTE models. ### LTE models\[slte\] The integrated line flux $S_{\nu}\Delta$v (in Jy km s$^{-1}$) in the observer’s frame is related to the velocity-integrated Rayleigh-Jeans source brightness $W(J)$ by [e.g., @Solomon1997] $$\label{ttojy} W(J)=\frac{\lambda^{2}_{J,J-1,rest}(1+z)^{3}}{2k \Omega_{a}}S_{\nu}\Delta\mathrm{v}\frac{\Omega_{a}}{\Omega_{s}}$$ where $\Omega_{s}$ and $\Omega_{a}$ are the solid angles of the source and the antenna, respectively. $W(J)$ is in units of K km s$^{-1}$. The last fraction represents the inverse of the beam filling fraction. The contribution of the gravitational lensing magnification should cancel out in this expression, as it contributes to both $S_{\nu}$ and $\Omega_{s}$, but the true $\Omega_{s}$ is not known. In principle, the same approach taken for the continuum (Section \[sseds\]) could be used to determine the source size. However, such a fit requires a minimum of three parameters, and will not be well constrained by the number of CO lines in our SLEDs. In addition, the optical depth depends directly on the column density, and cannot be estimated independently, in the same way that the dust optical depth was determined by the continuum slope. We assume an intrinsic source size of $\sim$2 kpc, consistent with the angular diameter of 0.2 found for an SMG at a redshift of 2.3259 [@Swinbank2010; @Negrello10], and similar to the size of the dust emitting region found in Section \[sseds\]. The corresponding beam filling fractions are listed in Table \[tab:sfr\]. As this source solid angle now represents the intrinsic size, and not the magnified one, we must correct the observed flux densities by the lensing magnification factors. We use the values listed in Table \[tab:cont\], when available, and assume a value of 10 in all other cases. The case of SDP.17a is treated differently, as it is assumed to be a foreground galaxy, not affected by gravitational lensing. For the intrinsic size of SDP.17a, we use a value of 1.54 arcsec$^{2}$, which approximates the size of the optical image. @Negrello10 identify two galaxies in the i-band image of SDP.17 and fit both light distributions with the GALFIT software. As the presence of two galaxies could indicate a possible merger, we choose the source size of SDP.17a to be the sum of the areas of these two galaxies. The distribution of the velocity-integrated brightness temperatures for the CO lines can be constructed starting from the CO column density and gas temperature, under the assumption of LTE. Following @Goldsmith1999, the velocity-integrated Rayleigh-Jeans source brightness is given by $$\label{btemp} W(J)=N_{J}\frac{hc^{3}A_{J,J-1}}{8\pi k\nu^{2}}\frac{1-e^{-\tau_{J,J-1}}}{\tau_{J,J-1}},$$ where $\tau_{J,J-1}$ is the line center optical depth, and $A_{J,J-1}$ is the Einstein $A$ coefficient for the transition. In LTE, the column density of molecules in the upper level, $N_{J}$, is related to the total column density $N$, by $$\label{nupper} N_{J}=\frac{N}{Z}g_{J}e^{-E_{J}/kT_{ex}},$$ where $Z$ is the partition function, $E_{J}$ is the energy of level $J$, and $g_{J}=2J+1$ is the degeneracy of level $J$. The line center optical depth can be expressed as a function of column density, temperature, and line width $\Delta$v as $$\label{tauline} \tau_{J,J-1}=A_{J,J-1}\frac{c^{3}}{8\pi\nu^{3}\Delta\mathrm{v}}N_{J}(e^{h\nu/kT_{ex}}-1).$$ We fit Equation \[btemp\] to the measured $W(J)$ distribution, with the column density and gas temperature as free parameters, and $\Delta$v = 300 km s$^{-1}$. We find that the best fit models have relatively low optical depths ($\lesssim 1$) such that the choice of the line width has only a small effect on the fitted parameters. For the lensed galaxies, the measured CO SLEDs and the range of SLEDs allowed by the formal 1$\sigma$ interval for the gas temperature are are shown in Figure \[sleds\]. The SLEDs can be characterized by an overall scale and line ratios. The scale of the observed SLEDs is mainly a result of the CO column density and the beam filling fraction, while the line ratios depend on the CO temperature and gas (H$_{2}$) density. The parameters in each pair are therefore largely degenerate and anti-correlated. This degeneracy is characteristic to CO and other molecular SLEDs, regardless of galaxy type. The last correlation (between temperature and gas density) only exists until LTE is reached, and the temperature becomes fixed. By making assumptions on the beam filling fraction and gas density, we can place limits on the remaining parameters. The error bars on the column densities derived in this manner are correlated with the errors in the beam filling fraction, which are not known. Similarly, by making the assumption of LTE for all transitions up to CO (7-6), we are constraining the gas density to be greater than the critical density for this transition ($n$\[H$_{2}$\] $\gtrsim$ 3$\times$10$^{5}$ cm$^{-3}$). At densities $n$\[H$_{2}$\] $\gtrsim10^{6}$ cm$^{-3}$, considerably larger than the average value observed in Galactic molecular clouds, all observed lines should be in LTE. Values of the gas density more typical for Galactic molecular clouds (10$^{3}$-10$^{4}$ cm$^{-3}$) correlate with higher gas temperatures, of a few hundred degrees, in order to reproduce the observed line ratios. The best-fit LTE CO column densities are $\sim$few$\times$10$^{18}$ cm$^{-2}$, and the gas temperature ranges between 41 and 115 K, as listed in Table \[tab:sfr\]. Taking into account the assumed source size, we estimate total CO masses of a few$\times$10$^{6}$ M$_{\odot}$, only $\sim$10$^{-4}$ of the total gas mass. Together with the large pressure associated with this model ($\sim$10$8$ K cm$^{-3}$), this suggests that the LTE scenario can only describe a small portion of the gas. Other regions of the parameter space are associated with non-LTE gas excitation, explored with the RADEX modeling in the next section. ### Non-LTE radiative transfer models of CO line excitation\[snlte\] In general, the rotational levels of the CO molecule might not be populated according to a single temperature, and the gas is not necessarily in equilibrium with the radiation field. By dropping the LTE assumption, we allow the excitation temperature to be a function of transition, being determined by the level populations for each line, while the kinetic temperature will be the global quantity describing the thermal energy of the gas. The level populations are found by solving the detailed balance equations including both radiative and collisional rates, and the output intensities are calculated by solving the radiative transfer equations. Usually, these equations are strongly coupled, involving large spatial and frequency grids, and further complicated by the number of molecules and transitions involved. Simplifying assumptions are usually made to reduce the computing time, depending on the problem at hand. We use RADEX to estimate the range of physical parameters consistent with the measured line strengths when dropping the LTE assumption. RADEX is a one dimensional, non-LTE radiative transfer code, that solves for the level populations iteratively, employing the escape probability approximation for the radiative transfer [@vanderTak07]. The medium is assumed homogeneous and isothermal, and the number, type, and abundance of the participating molecules is selectable by the user. The input parameters are the kinetic temperature, $T_{kin}$, the number density of molecular hydrogen, $n$\[H$_{2}$\], as the collisional partner, and the column densities per unit line width of the participating molecules, only CO in our case. The background radiation field is the CMB, redshifted according to the redshift of each galaxy. The output contains the predicted line excitation temperatures, optical depths, and line intensities. The output line fluxes are scaled by an additional factor $\phi$, that represents fractional corrections to the size of the emitting region and to the gravitational lensing magnification factor. It would correspond to the area filling fraction of the emitting region, if the size and lensing magnification factor of the source were known precisely. A value $\phi>1$ would suggest that the assumed source size was underestimated. We compare the measured flux densities with the line intensities output by RADEX using the values for source sizes, line widths, and lensing magnification factors assumed in Section \[slte\] for the LTE model. For the case $\phi=1$ and $n$\[H$_{2}$\]$\gg n_{crit}$ for all transitions, RADEX should recover the LTE SLED as determined from $T_{ex}$ and N\[CO\] in Section \[slte\]. This set of parameters represents the model m01, listed in Table \[tab:radex\] for each galaxy. The high $n$\[H$_{2}$\] chosen for this model (10$^{6}$ cm$^{-3}$) insures that all measured lines will be in LTE. The SLED computed by RADEX for m01 is shown by the blue line in Figure \[sleds\], showing a good agreement with the LTE method described in the previous section. We run RADEX for a range of input models, parametrized by $T_{kin}$, $N$\[CO\], $n$\[$H_{2}$\] and $\phi$, and compute the likelihood density function for all models following the method described in @Ward03. Weak priors are set to rule out unphysical solutions, keeping the total molecular mass smaller than the dynamical mass, and the length of the CO column smaller than the physical size of the galaxy [@Ward03; @Panuzzo2010]. The dynamical mass cut-off is estimated choosing the line width of 300 km s$^{-1}$. We also impose a limit for the kinetic temperature at 3000 K, where collisional dissociation of CO starts to dominate, weakly dependent on the gas density. We map the surface of the likelihood distribution and determine the location of its maximum by running a Markov chain Monte Carlo (MCMC) algorithm, described in detail in @Scott10. The 2D marginal probability contours obtained from the MCMC algorithm are shown in Figures \[tnmcmc\] and \[nnmcmc\], with the position of the 4D maximum likelihood indicated by the dotted line, and the parameters of the m01 model, equivalent to the LTE solution, shown by the dashed lines. Note that due to projection effects the parameters corresponding to the maximum of the likelihood in the 4D parameter space are not necessarily the same as the coordinates of the maximum for the 2D marginal probability distributions. The set of parameters that maximizes the 4D likelihood for each galaxy is listed in Table \[tab:radex\] as model m02, and the line luminosities predicted by this model are shown in blue in Figure \[sleds\]. The 68% credible regions are calculated as the smallest intervals containing 68% of the 1D marginal probability for each parameter, around the value corresponding to the 4D maximum likelihood. The 4D probability distributions are highly non-gaussian, and the maxima of the marginalized distributions are not a good representation of the true maximum of the 4D distribution. Due to the aforementioned degeneracies (see Section \[slte\]), the product between the kinetic temperature and gas density on one hand, and CO column density and $\phi$ on the other hand, are better constrained than individual parameters. These products lead to the values for gas pressure and total gas mass listed in Table \[tab:radex\]. Taking into account the lensing magnification factor, the inferred gas mass for non-LTE models is in good agreement with the mass derived from $L_{CO}^{'}$ (Table \[tab:sfr\]). The credible regions for $\phi$ suggest that the size of the emitting region could be larger than assumed for the LTE models, which would also correspond to a larger area characterized by lower gas density and pressure than the LTE case. The properties of these models can be compared by calculating the $total$ CO luminosity, summed over all transitions in the model, as a proxy for the indicators of the star formation rate and efficiency. The ratios $L_{IR}$/$L_{CO}^{'}$ for each model listed in Table \[tab:radex\] suggest that the star formation efficiency is higher for the LTE models. Using the correlation between total $L_{CO}$ and $L_{IR}$ derived for a mixed sample of nearby and high redshift galaxies by @Bayet2009, we find that the total $L_{CO}$ obtained by integrating the models will overpredict the measured $L_{IR}$ in both cases, but are consistent within the scatter. The deviation between the predicted and measured $L_{IR}$ is larger for model m02, slightly disfavoring this solution. While the LTE models tend to underpredict the total gas masses, the non-LTE models potentially underpredict the total infrared luminosity. The region of the parameter space that is most consistent with the observed line strengths is enclosed by the likelihood contours in Figures \[tnmcmc\] and \[nnmcmc\]. We see that the LTE solution implies large gas densities ($n$\[H$_{2}$\]), as expected, but a class of non-LTE solutions can be found in the region defined by the likelihood contours in parameter space. The likelihood space roughly splits into high density/low temperature, and low density/high temperature solutions. One possible additional complication to the interpretation can arise from high dust optical depths @Papadopoulos2010b, not included in our models, that can lead to the suppression of mid-$J$ CO lines and to an underestimate of the excitation temperature. However, the likelihood distribution is relatively shallow over the whole region, reflecting the insufficient amount of information in our data. Comparatively, other studies of high redshift SMGs find a warm CO component with $n$\[H$_{2}$\] around 10$^{4}$ cm$^{-3}$ and temperatures between $\sim$40 and 60 K [@Riechers2010; @Carilli2010; @Danielson2010], a region marginally allowed by our contours. The constraints on the parameter space for the non-LTE models are weak, as expected given the average sampling of the SLED, and cannot distinguish at this point between the LTE and non-LTE scenarios. To emphasize the insight gained by including additional lines in the fit, we add to the SLED of SDP.81 the CO(1-0) integrated flux from @Frayer10. A likelihood analysis for the new set of lines results in the best fit parameters listed in Table \[tab:radex\] as m03. The 2D marginalized likelihoods for this case are shown by the light grey contours in Figures \[tnmcmc\] and \[nnmcmc\]. The tightening of the likelihood contours is substantial with just one line added to the data, and the LTE region of the parameter space becomes less favored. However, the limitation of this model is that it assumes a single gas component, while most of the emission in the CO(1-0) line could be originating from cold molecular gas. The brightness temperatures predicted by the RADEX models m01 and m02 are shown in Figure \[tempsone\], to emphasize the large deviations beween the predictions of the two models, especially for lower $J$ transitions. The measured data points have been scaled by the lensing magnification factors listed in Table \[tab:cont\] when available, and by a factor of 10 in all other cases. This figure shows that the constraints on the model parameters can be tightened by measurements of lower $J$ transitions, especially the CO(1-0) line. Even if most of the CO(1-0) emission comes from a colder gas component, using this value as an upper limit will help rule out some regions of the parameter space, as in our example for SDP.81. Distinguishing between the different regions in parameter space will clarify the state of the ISM in these galaxies, and thus their star formation histories. Specifically, hot/low-density gas may signal the action of a feedback process quenching further star formation, by increasing the Jeans mass. This high temperature/low density solution has not been fully investigated, but recent studies show that other CO SLEDs can be consistent with it [@Panuzzo2010; @Weiss2007; @Ao2008; @Bayet2009]. The CO SLED in M82 is fit by a small CO component with a kinetic temperature of almost 600 K [@Panuzzo2010], while solutions with $T_{kin}$ of a few$\times$100 K are found by @Bayet2009, and can be allowed by the LVG models for IRAS F10212+4724 [@Ao2008] and APM 08279+5255 [@Weiss2007]. Such temperatures suggest energy input from outflows or AGN activity. The presence of an AGN component in SDP.17b is supported by the relatively flat SLED from CO(6-5) to CO(8-7), similar to the Cloverleaf quasar or Mrk231 [@bradford09; @vanderWerf2010], and the emission line of water, also observed in galaxies with an AGN component, such as Mrk231 [@Gonzales2010]. CONCLUSIONS\[sconcl\] ===================== Far-IR / submillimeter-wave surveys are revealing submillimeter-bright galaxies from the first half of the Universe by the tens of thousands, but their detailed study requires spectroscopic redshift measurements. We have studied a sample of the brightest sources and have demonstrated a new redshift-measurement technique with our broadband millimeter-wave grating spectrometer, Z-Spec. Z-Spec measures multiple rotational transitions of carbon monoxide, the dominant coolant of molecular gas in galaxies, and thus is not dependent on optical counterparts which are often absent or hard to identify, as is the case for these galaxies. We find redshifts ranging roughly between 1 and 3, in line with previous determinations [@chapman05], reaching back to an era when the Universe was 15% is present age. Their fluxes are proven to be amplified by gravitational lensing [@Negrello10], making them ideal targets for spectroscopic follow-ups. From the observed CO line luminosities and integrated $L_{IR}$, typical conversion factors reveal that these galaxies each house roughly 10$^{10}$ M$_{\odot}$ of molecular gas, and have SFRs between 10$^{2}$ and 10$^{3}$ M$_{\odot}$yr$^{-1}$, after correcting for lensing magnification. Regardless of the magnification details, we are clearly witnessing a rare episode of rapid star formation in these galaxies, since the timescale over which the observed luminosity can be generated by converting the inferred mass of gas into stars is only a few tens of millions of years (depending on the details of the star formation and the accretion of more gas), which is a small fraction of the Universe’s age even at this early epoch. We estimate that the dust masses in our sample of lensed galaxies are around a few$\times$10$^{8}$ M$_{\odot}$, and the wavelengths corresponding to the peaks of their dust SEDs fall within a narrow range, between 73 and 92 $\mu$m in the rest frame. For this initial set of lensed submm galaxies both the dust properties derived from the IR SED, and the physical conditions of the molecular gas probed by the CO lines, are broadly comparable to those in known SMGs [@greve05; @Solomon2005; @Casey09], with excitation temperatures in the 30-120 K range, and $L_{CO}^{'}$/$L_{IR}$ between 1 and 3$\times$10$^{-3}$ K km s$^{-1}$ pc$^{2}$/L$_{\odot}$, as measured from the mid-$J$ CO lines. The partial SLEDs for the CO molecule constructed from the lines observed by Z-Spec cannot distinguish between different models of CO excitation. The simplest assumption is that of local thermodynamic equilibrium (LTE), under which we can derive the gas column density and excitation temperature. We find that the relative line strengths can be reproduced by relatively low excitation temperatures ($<$ 100 K), and optical depths ($<$ 1). In the non-LTE case, other parts of the parameter space are allowed, including higher optical depths, while measurements of the lower rotational transitions are essential in confirming such models. By being able to characterize galaxies that can be inaccessible at other wavelengths, the combination of large-area submm surveys and spectroscopic follow-ups of the CO emission lines will lead to substantial progress in our understanding of high redshift galaxies and their evolution. These results suggest the possibility of a rapid growth in our understanding of SMGs, independent of their optical or radio counterparts, but enabled by strong gravitational lensing magnification. We are indebted to the staff of the Caltech Submillimeter Observatory for their unflagging support. This work was supported by NSF grant AST-0807990 to J. Aguirre and by the CSO NSF Cooperative Agreement AST-0838261. Support was provided to J. Kamenetzky by an NSF Graduate Research Fellowship. Z-spec was constructed under NASA SARA grants NAGS-11911 and NAGS-12788 and an NSF Career grant (AST-0239270) and a Research Corporation Award (RI0928) to J. Glenn, in collaboration with the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. We acknowledge Peter Ade and his group for their filters and Lionel Duband for the 3He / 4He refrigerator in Z-Spec, and are grateful for their help in the early integration of the instrument. R. L. wishes to thank Tom Loredo for useful discussions regarding the significance of the redshift determination. We appreciate the help of Robert Hanni and Jon Rodriguez with observing. [cccccc]{} H-ATLAS J090740.0-004200 &SDP.9 & Apr 27 - May 14 & $0.05-0.21$ & 10.6 & 4.0\ H-ATLAS J091043.1-000322 &SDP.11 & Apr 28 - May 4 & $0.05-0.18$ & 6.8 & 5.5\ H-ATLAS J090302.9-014128 &SDP.17 & Mar 28 - Apr 1 & $0.04-0.08$ & 18.2 & 2.9\ H-ATLAS J090311.6+003905 &SDP.81 & Mar 7 - Mar 12 & $0.02-0.05$ & 22.5 & 2.3\ H-ATLAS J091304.9-005344 &SDP.130 & Mar 21 - Mar 22 & $0.04-0.08$ & 8.6 & 4.4\ [cccccccccc]{} SDP.9 & ... & 1.577$\pm$0.008 & 1.1e-11 & 4.4$\pm$0.6 & 57$\pm$1 & 3.8$\pm$0.2 & 2.5& 0.65 & 6.6$\pm$0.9\ SDP.11 & ... & 1.786$\pm$0.005 & 3.1e-8 & 7.8$\pm$0.9 & 69$\pm$1 & 5.7$\pm$0.4 & 1.7 & 0.43 & 11.7$\pm$1.4\ SDP.17a & ... & 0.942$\pm$0.004 & 1.0e-4 & 0.4$\pm$0.1 & 27$\pm$1 & 2.9$\pm$0.1 & 4.9 & 1.44 & 0.6$\pm$0.2\ SDP.17b & ... & 2.308$\pm$0.011 & 1.9e-7 & 3.9$\pm$1.0 & 66$\pm$1 & 2.9$\pm$0.1 & 1.1 & 0.30 & 5.8$\pm$1.5\ SDP.81 & 18-31 & 3.037$\pm$0.010 & 6.7e-5 & 6.4$\pm$0.3 & 58$\pm$1 & 3.2$\pm$0.1 & 2.2 & 0.69 & 9.6$\pm$0.5\ SDP.130 & 5-7 & 2.626$\pm$0.0003 & N/A & 4.3$\pm$0.2 & 55$\pm$1 & 2.7$\pm$0.3 & 1.6 & 0.47 & 6.5$\pm$0.3\ [ccccccc]{} CO (4-3) & 461.041 & ... & ... & 15$\pm$9 &... & ...\ CO (5-4) & 576.268 & 25$\pm$5 & 23$\pm$8 & 29$\pm$9 &... & ...\ CO (6-5) & 691.473 & 33$\pm$7 & 29$\pm$10 & ... &17$\pm$5 & ...\ CO (7-6) & 806.652 & ... & 18$\pm$14 & ... &11$\pm$7 & 12$\pm$4\ CO (8-7) & 921.800 & ... & ... & ... &16$\pm$6 & 5$\pm$3\ CO (9-8) & 1036.91 & ... & ... & ... &... & 6$\pm$3\ CO (10-9) & 1151.99 & ... & ... & ... &... & $<$ 6.5\ \[\] $^{3}P_{1}\rightarrow^{3}$$P_{0}$ & 492.160 & $<$ 11 & ... & $<$ 6 &... & ...\ \[\] $^{3}P_{2}\rightarrow^{3}$$P_{1}$ & 809.342 & ... & 31$\pm$14 & ... &13$\pm$7 & $<$ 6.5\ H$_{2}$O $2_{0,2}-1_{1,1}$& 987.914 & ... & ... & ... &19$\pm$7 & ...\ [ccccccccc]{} SDP.9 & 13$\pm$3 & 1.7 & 2.6 & 19$\pm$4 & 76$\pm$24 & 0.3 & 0.251 & 1.4$\pm$0.3\ SDP.11 & 15$\pm$5 & 2.0 & 1.7 & 47$\pm$14 & 41$\pm$6 & 0.3 & 0.983 & 3.5$\pm$1\ SDP.17a & 4$\pm$3 & 0.5 & 8.3 & 2.3$\pm$0.7 & 114$\pm$99 & 1.3 & 0.015 & 0.17$\pm$0.05\ SDP.17b & 12$\pm$3 & 1.6 & 2.8 & 19$\pm$6 & 65$\pm$10 & 0.3 & 0.298 & 1.4$\pm$0.4\ SDP.81 & 10$\pm$3 & 1.3 & 1.4 & 11$\pm$3 & 53$\pm$5 & 0.6 & 0.149 & 0.8$\pm$0.2\ [cccccccc]{} SDP.9 &&&&&&&\ m01 & 76. & 18.27 & 6.20 & 1.00 & 8.1 & 8.8 & 55\ m02 & 2628. & 20.50 & 3.37 & 0.05 & 6.8 & 9.7 & 29\ 68% credible region & 149-2631 & 19.03-21.10 & 3.37-6.06 & 0.03-0.58 & 6.29-8.64 & 8.71-10.14 & 12-114\ SDP.11 &&&&&&&\ m01 & 41. & 18.67 & 6.20 & 1.00 & 7.8 & 9.2 & 78\ m02 & 21. & 19.74 & 6.50 & 1.66 & 7.8 & 10.5 & 67\ 68% credible region & 21-611 & 18.42-20.17 & 3.61-6.55 & 0.19-1.82 & 5.85-7.83 & 9.14-10.50 & 27-520\ SDP.17b &&&&&&&\ m01 & 63. & 18.32 & 6.20 & 1.00 & 8.0 & 8.8 & 55\ m02 & 1988. & 20.03 & 1.99 & 0.85 & 5.3 & 10.5 & 21\ 68% credible region & 136-1992 & 19.11-21.03 & 1.99-5.14 & 0.04-1.02 & 5.29-7.59 & 8.93-10.51 & 0-626\ SDP.81 &&&&&&&\ m01 & 53. & 18.04 & 6.20 & 1.00 & 7.9 & 8.6 & 71\ m02 & 213. & 20.33 & 2.63 & 0.38 & 5.0 & 10.4 & 25\ 68% credible region & 63-855 & 18.53-20.43 & 2.55-5.17 & 0.05-1.12 & 4.95-7.24 & 8.44-10.44 & 8-640\ m03 & 481. & 16.71 & 3.20 & 130. & 5.89 & 9.35 & 18\ 68% credible region & 368-2749 & 14.85-17.08 & 2.56-3.42 & 35-6600 & 5.80-6.04 & 9.22-9.43 & 12-55\ ![The Z-Spec spectra of four submillimeter bright H-ATLAS galaxies. The fit to the continuum and CO lines at the measured redshift is overplotted in red, and the positions of the strongest lines falling in the Z-Spec bandpass are indicated by the vertical blue lines. The line indicated in red in the spectrum of SDP.130 is unidentified. \[spectra\]](f1.eps) ![The Z-Spec spectrum of the H-ATLAS source SDP.17. The fit to the continuum and CO lines at $z=0.94$ is overplotted in red in the upper panel, and the rotational CO lines are indicated by the vertical blue lines. These lines have been subtracted from the spectrum shown in the lower panel. The red line in the lower panel shows the fit including the lines identified at $z=2.308$. \[spectra17\]](f2.eps) ![ Blank sky probability distributions for the positions of the $E_{1}$ and $E_{2}$ maxima as a function of redshift. The redshift bin size is 0.003, corresponding to the average width of the Z-Spec channels. These probabilities have been obtained by running the redshift finding algorithm on 1000 realizations of the blank sky spectrum. The highest probability corresponds to the redshift most likely to be determined by each statistic. The last panel shows the probability distribution when the same redshift is found jointly by $E_{1}$ and $E_{2}$. The maximum probability that a false positive is obtained simultaneously by the two test statistics is $\sim$3%. \[simskyzearch\]](f3.eps) ![Null test for the redshift finding algorithm, using a blank sky spectrum. Note that the values of the $E_{2}$ test statistic are systematically below 1.5, and the positions of the maxima of the two test statistics do not coincide. \[skyzearch\]](f4.eps) ![Distributions of the two test statistics derived from blank sky spectra. The histograms for $E_{1}$ and $E_{2}$ are shown with a dashed and continuous line, respectively. The fits with a normal distribution for $E_{1}$ and a t distribution for $E_{2}$ are overplotted as the smooth red curves. \[estdist\]](f5.eps) ![Results of running the redshift-finding algorithm for all the H-ATLAS sources in our sample. The $E_{2}$ test statistic has been offset vertically by 7 units, for clarity. The blue asterisks show the positions of the largest secondary peaks arising from coincidences with the lines from the actual redshift (see text). These peaks contain the same information as the main peak. In the SDP.17 panel, we note the extra peaks that do not match the secondary peaks corresponding to the first selected redshift. The SDP.17a panel shows the determination of the second redshift from the same spectrum, after subtracting the high-redshift component. No redshift is determined for SDP.130. \[A981zearch\]](f6.eps) ![$E_{1}$ distribution for each galaxy for which we determined the redshift. The original $E_{1}$ distribution is shown in black, the red distribution being obtained after subtracting the main peak of $E_{1}$ and all its secondary peaks. The normal distribution fit to the null $E_{1}$ distribution is shown in blue. The red dashed line shows the 3$\sigma$ position for the null gaussian, and the black dashed lines show the positions of the maximum of the test statistic for the measured redshifts, to emphasize the significance of our detections. \[e1hist\]](f7.eps) ![The best-fit SED models for the five H-ATLAS galaxies in our sample. The continuous line shows the modified blackbody spectrum with $\nu_{0}=1300$ GHz and $\beta=2.0$, while the dotted and dashed lines show the SEDs obtained from the SED libraries of CE01 and DH02, respectively. The total infrared luminosities are calculated as the average between the CE01 and DH02 SED template fits, to account for emission above the blackbody spectrum at higher frequencies. The parameters for the modified blackbody fits are also listed in Table \[tab:cont\].\[seds\]](f8.eps) ![Spectral line energy distributions, uncorrected for gravitational lensing magnification. The Z-Spec measurements are shown connected by the black histogram. The data point for the CO(1-0) line measured by @Frayer10 in SDP.81 falls at the bottom of the panel, and is better seen in Figure \[tempsone\]. The red lines show the SLEDs predicted by the best-fit LTE model (continuous), and the LTE models corresponding to the limits of the 1$\sigma$ standard confidence interval for $T_{ex}$ determined from the fit (dashed). The blue line corresponds to the SLED predicted by RADEX using the same temperature and column density as the LTE model, while the green line shows the SLED predicted by RADEX with the parameters given by the 4D maximum likelihood solution. The parameters for these two RADEX models, m01 and m02, respectively, are listed in Table \[tab:radex\]. \[sleds\]](f9.eps) ![Contour plots of the ($T_{kin}$, $n$\[H$_{2}$\]) 2D marginal likelihood distributions, generated by a MCMC sampling of the parameter space for RADEX models. The contours are in n$\sigma$-equivalent steps, enclosing 68.3%, 95.4%, 99.7%, and 99.99% of the probability, respectively. The dashed lines correspond to parameters that reproduce the LTE solution (model m01), and the dotted lines indicate the parameters corresponding to the RADEX 4D maximum likelihood solution (model m02). Note that the 2D marginal distributions will not necessarily have the same maximum as the 4D distribution. The kinetic temperature is limited to 3000 K, where collisional dissociation of CO becomes important. In the SDP.81 panel, the lighter contours show the probability levels for a model including the CO(1-0) from @Frayer10. The parameters for this model are listed as model m03 in Table \[tab:radex\]. \[tnmcmc\]](f10.eps) ![Same as Figure \[tnmcmc\] for the ($N[CO]$, $n$\[H$_{2}$\]) 2D marginal likelihood distributions. \[nnmcmc\]](f11.eps) ![$W(J)$ as a function of transition for four of the galaxies in our sample. For clarity, the values for the same transition in different galaxies have been slightly offset around the position of the upper $J$ level. The triangle point represents the intensity of the CO(1-0) line for SDP.81 measured by @Frayer10. The $W(J)$ distributions predicted by the RADEX models m01 and m02 are shown with a dashed and continuous line, respectively. These lines emphasize the constraints on the allowed parameter space that can be gained by having measurements of both higher and lower $J$ transitions. \[tempsone\]](f12.eps)

--- abstract: 'In this paper we introduce a framework for computing upper bounds yet accurate WCET for hardware platforms with caches and pipelines. The methodology we propose consists of 3 steps: 1) given a program to analyse, compute an equivalent (WCET-wise) abstract program; 2) build a timed game by composing this abstract program with a network of timed automata modeling the architecture; and 3) compute the WCET as the optimal time to reach a winning state in this game. We demonstrate the applicability of our framework on standard benchmarks for an ARM9 processor with instruction and data caches, and compute the WCET with UPPAAL-TiGA. We also show that this framework can easily be extended to take into account dynamic changes in the speed of the processor during program execution.' author: - 'Franck Cassez[^1]' bibliography: - 'wcet.bib' title: | Timed Games for Computing\ Worst-Case Execution-Times --- Introduction ============ Embedded real-time systems are composed of a set of tasks (software) that run on a given architecture (hardware). These systems are subject to strict timing constraints and these constraints must be enforced by a scheduler. Designing an effective scheduler is possible only if some bounds are known about the execution times of each task. For simple scheduling algorithms non preemptive, the knowledge of the *worst-case execution-time* (WCET) is sufficient to design a scheduler. For more complex scheduling algorithms with preemption or shared resources, the WCET for each task might not give rise to the WCET for the entire system though. This is why most critical embedded systems rely on a rather simple scheduling algorithm. Performance wise, determining tight bounds for WCET is crucial as using rough over-estimates might either result in a set of tasks being wrongly declared non schedulable or a lot of computation time might be wasted in idling cycles and loss of energy/power. #### **The WCET Problem.** The execution time, $\et(p,d,H)$, of a program $p$, with input data $d$ on the hardware $H$, is measured as the number of cycles of the fastest component of the hardware the processor. Data take their values in a finite domain $\calD$. The program is given in binary code or equivalently in the assembly language of the target processor[^2]. The *worst-case execution-time* of program $p$ on hardware $H$ is defined by: $$\wcet(p,H)=\sup_{d \in \calD} \et(p,d,H) \mathpunct.$$ The WCET problem asks the following: Given $p$ and $H$, compute $\wcet(p,H)$. In general, the WCET problem is undecidable because otherwise we could solve the halting problem[^3]. However, for programs that always terminate and have a bounded number of paths, it is obviously (theoretically) computable. Indeed the possible runs of the program can be represented by a finite tree. Notice that this does not mean that the problem is tractable though. If the input data are known or the program execution time is indepedent from the input data, the tree contains a single path and it is usually feasible to compute the WCET. Likewise, if we can determine some input data that produces the WCET (this might be as difficult as computing the WCET), we can compute the WCET on a single-path program. If is not often the case that the input data are known or that we can determine an input that produces the WCET. Rather the (values of the) input data are unknown, and the number of paths to be explored might be extremely large: for instance, for a Bubble Sort program with $100$ data to be sorted, the tree representing all the runs of the (assembly) program on all the possible input data has more than $2^{50}$ nodes. Although symbolic methods (using BDDs) can be applied to analyse some programs with a huge number of states, they will fail to compute the exact WCET on Bubble Sort by exploring all the possible paths. Another difficulty of the WCET problem stems from the more and more complex architectures embedded real-time systems are running on. They usually feature a multi-stage *pipeline* and a fast memory component like a *cache*, and they both influence in a complicated manner the WCET. It is then a challenging problem to determine a precise WCET even for relativey small programs running on complex architectures. #### **Methods and Tools for the WCET Problem.** The reader is referred to [@wcet-survey-2008] for an exhaustive presentation of the WCET computation techniques and tools. There are two main classes of methods for computing WCET. - Testing-based methods. These methods are based on experiments running the program on some data, using a simulator of the hardware or the real platform. The execution time of an experiment is measured and, on a large set of experiments, a maximal and minimal bound can be obtained. The maximal bound computed this way is *unsafe* as not all the possible paths have been explored. These methods might not be suitable for safety critical embedded systems but they are versatile and rather easy to implement. RapiTime [@rapitime] (based on pWCET [@pWCET]) and Mtime [@mtime] are measurement tools that implement this technique. - Verification-based methods. These methods often rely on the computation of an *abstract* graph, the control flow graph (CFG), and an abstract model of the hardware. Together with a static analysis tool they can be combined to compute WCET. The CFG should produce a super set of the set of all feasible paths. Thus the largest execution time on the abstract program is an upper bound of the WCET. Such methods produce *safe* WCET, but are difficult to implement. Moreover, the abstract program can be extremely large and beyond the scope of any analysis. In this case, a solution is to take an even more abstract program which results in drifting further away from the exact WCET. Although difficult to implement, there are quite a lot of tools implementing this scheme: Bound-T [@bound-T], OTAWA [@otawa], TuBound [@tubound], Chronos [@chronos], SWEET [@sweet-2003] and aiT [@aiT; @wcet-ai-aswsd-ferdinand-04] are static analysis-based tools for computing WCET. The verification-based tools mentioned above rely on the construction of a control flow graph, and the determination of loop bounds. This can be achieved using user annotations (in the source code) or sometimes infered automatically. The CFG is also annotated with some timing information about the cache misses/hits and pipeline stalls, and paths analysis is carried out on this model by Integer Linear Programming (ILP). The algorithms implemented in the tools use both the program and the hardware specification to compute the CFG fed to the ILP solver. The architecture of the tools themselves is thus monolithic: it is not easy to adapt an algorithm for a new processor. This is witnessed by *WCET’08 Challenge Report* [@wcet-chal-report-08] that highlights the difficulties encountered by the participants to adapt their tools for the new hardware in a reasonable amount of time. #### **WCET and Model-Checking.** Surprisingly enough, only a few tools use model-checking techniques to compute WCET. Considering that ($i$) modern architectures are composed of *concurrent* components (the stages of the pipeline, caches) and ($ii$) these components *synchronize* and synchronization depends on *timing constraints* (time to execute in one stage of the pipeline, time to fetch a data from the cache), formal models like *timed automata* [@AD94] and state-of-the-art *real-time model-checkers* like UPPAAL[@uppaal-sttt-97; @uppaal-40-qest-behrmann-06] appear well-suited to address the WCET problem. It has previously been claimed [@Wilhelm-04] that *model-checking* was not adequate to compute WCET, but this statement has since been revised. In [@wcet-cav-metzner-04], A. Metzner showed that model-checkers could well be used to compute safe WCET on the CFG for programs running on pipelined processors with an instruction cache. In [@hubert-wcet-09], B. Hubert and M. Schoeberl consider Java programs and compare ILP-based techniques with model-checking techniques using the model-checker UPPAAL. Model-checking techniques seem slower but easily amenable to changes (in the hardware model). The recommendation is to use ILP tools for large programs and model-checking tools for code fragments. More recently, the TASM toolset [@tasm-cav-07] (M. Ouimet & K. Lundqvist) has been used to compute WCET with UPPAAL: the TASM machine is a high level machine not featuring pipelining nor caches and computing the WCET amounts to finding the longest path (timewise) in a timed automaton that specifies a tasks. Another use of timed automata (TA) and the model-checker UPPAAL for computing WCET on pipelined processors with caches is reported in [@metamoc-2009]. The framework METAMOC described in [@metamoc-2009] (A. E. Dalsgard *et al.*) consists in: 1) computing a flow graph (FG) from a binary program, 2) composing this FG with a (network of timed automata) model of the processor and the caches. Computing the WCET is then reduced to a safety (or dually a reachability) property $\mathsf{AG} \; (\text{Time} \leq k)$ (reads “on all paths, the variable , global time, is less than $k$”) that can be checked with UPPAAL. The previous framework is extremelly elegant yet has some shortcomings. Out of the $15$ programs[^4] of the benchmarks only $7$ can be analysed with a concrete instruction and data cache (Table .6.1, page 84 in [@metamoc-2009]). It is also surprising that some single-path programs could not be analysed with concrete caches. The tool chain relies on a value analysis tool which fails on $3$ of the $15$ programs. It requires a specialised version of UPPAAL (not available) to avoid a binary search for computing the WCET. #### **Our Contribution.** In this paper we use *timed game automata* (TGA) and UPPAAL-TiGA [@tiga-cav-07] (UPPAAL for timed games) to compute WCET. We model the WCET problem as a two-player timed game. Intuitively Player 1 is the program, and Player 2 is in charge of deciding the outcome of the *comparison* instructions (`cmp, tst` which set the branching conditions) that depend on the input data. As the choice of the input data is not controllable by Player 1, we obtain a two-player game. The problem we solve on this game is an *optimal time reachability problem*: “What is the optimal time for Player 1 to reach the end of the program ?” What is similar to the previously mentioned approach [@metamoc-2009] (A. E. Dalsgard *et al.*) is the timed automata models for the caches[^5] and pipeline stages the model of the architecture, but we use a totally different model for the program. We propose a new and very compact encoding of the program and pipeline stages’ states which enables us to compute the WCET for $13$ out of the previous $15$ programs[^6] (see Table \[tab-results\], page ). Moreover, compared to METAMOC that uses a computer with 32GB RAM, we can compute the results on a laptop computer (2Ghz Dual Core, 2GB RAM) within a few seconds. Using timed games instead of timed automata is also a major difference: the on-the-fly algorithm [@cassez-concur-05] implemented in UPPAAL-TiGA is different from the one running in UPPAAL, and it can also compute the *optimal time* (in the presence of adversary) to reach a designated state. Thus we do not need to do a binary search or use a tailored version of UPPAAL to compute the results. We also show that taking into account processor speed variations is easy in our framework. This can be important as it is possible to adjust the speed of the processor depending on the program to be run. For some programs, the saved power can be upto 22% (see Table \[tab-results\]). The advantages of our approach are many-fold (METAMOC [@metamoc-2009] shares 1–3): 1. it is very easy to implement as it consists of two separate and independant phases: 1) computation of a model of the program to be analysed; this only requires a (formal) semantics of the assembly language of the target processor[^7]; 2) computation of the WCET with UPPAAL-TiGA and the models for the caches, pipelines which specify the timing features. A model of a cache (always miss or FIFO) can be substitued by changing the cache component only (no need to recompute the model obtained in phase 1). 2. the design of the models for pipeline stages and caches can be stressed by simulating some simple samples programs; this enables us to get more confidence in the model of the hardware as this is not hidden in the analysis algorithm; this is especially important for concurrent architectures like pipelined processors that can be hard to describe; 3. UPPAAL or UPPAAL-TiGA can be used to simulate the program on the architecture. It is thus a quick way of obtaining a simulator for a given hardware; 4. we do not require annotations. Instead, we run a simulation of the program with some given bounds on the number of branching or a maximal number of states. If too many branchings are encountered, the user is required to provide a constraint for the corresponding instruction in the program to remove some infeasible paths; 5. we solve an *optimal time reachability problem* on the program $p$ of the form: “what is the optimal time to enforce *termination* of program $p$ ?”. This at once 1) proves that $p$ terminates on every input data, and 2) computes the WCET. This could not be achieved in METAMOC [@metamoc-2009] as the UPPAAL model contains priorities and deadlock freedom cannot be checked on models with priorities: thus if the safety property $\mathsf{AG} \; (\text{Time} \leq k)$ is satisfied, it does not mean that no deadlocks occurred; the deadlocks could be due to a flaw in the design of the pipeline model but in any case, it does not give a safe bound for the WCET as deadlocks have not been excluded. 6. it is easy to add *power* related constraints in the model processor speed variations; 7. we also show that not every program instruction is worth simulating and some *abstraction* on the effect of some instructions can be safely done. For example, in the Fibonnacci program, the content of the variable with the result is irrelevant for the computation of the WCET. It does not influence any branching nodes. We show how to check that an abstract program is *equivalent* to a concrete one and examplify this on some of the benchmarks from . #### **Outline of the Paper.** In Section \[sec-concrete-abs\], we briefly introduce the ARM9 architecture and the assumptions we make on the assembly programs to be analysed. Section \[sec-games\] describes how to encode an assembly program with non-deterministic choices into a game. In Section \[sec-architecture\] we give the timed automata models of the architecture we use to compute the WCET. Section \[sec-tool-chain\] gives an overview of the tool chain we propose and the components (compiler) we have designed together with some comments on the case studies presented in Table \[tab-results\]. Concrete and Abstract Programs {#sec-concrete-abs} ============================== #### **Program, Registers, Memory.** A *program* $p$ is a list of instructions $p=i_1,i_2,\cdots,i_k$ and $i_1$ is the initial instruction. The control usually goes from intruction $i_k$ to $i_{k+1}$ except for branching intructions that give the next instruction $i_j$ to be performed. Each instruction performs some basic operations (arithmetic, logic, memory load or store, branching) and has a duration which gives the amount of time it takes in each stage of the pipeline of the processor[^8]. We assume the duration is independant from the content of the operands of the instructions[^9]. In the sequel we use the variable $\iota$ to denote an instruction of $p$. The hardware on which $p$ runs has a pool of *registers* (different from the main memory and the caches). We let $\calR=\{r_0,\cdots,r_k\}$ be the set of registers. For example on the ARM9 [@arm9-ref] processor there are $16$ registers. A designated register **pc** contains the program counter and points to the next instruction to be performed (register $15$ on the ARM9). We let $\calM=\{m_1,m_2,\cdots,m_n\}$ be the set of memory cells’ addresses used by the program (we assume the program can access $\calM$). The content of the memory cells and registers is in a finite domain $\calD$ ($32$ bit integers). #### **Semantics.** When program $p$ runs on input data $d$, it generates a computation that changes the values of the registers and memory cells. A state (of the computation of $p$) is given by a mapping $v: \calR \cup \calM \rightarrow \calD$ and we let $\calV$ be the set of states. Performing an instruction results in a state change, and is deterministic. Given an instruction $\iota$ ($\iota$ including the operands and can be thought of as the *code* of an assembly instruction), the semantics of $\iota$ is a mapping ${[\![\iota]\!]}: \calV \xrightarrow{} \calV$. As the program counter is in one of the registers, the semantics of a program $p$ is completely determined by the current state of the computation. From a state $v$, the next state (in the computation of $p$) is $v'$ and we denote this $v \xrightarrow{} v'$. $v'$ is given by ${[\![\iota]\!]}(v)$ where and $\iota=i_\mathbf{pc}$ (we use **pc** both for the register and the content of this register to avoid hefty notations). For branching instructions, the control is determined by the *status bits* and we assume there are also part of the **pc** register. We assume **pc** is incremented by $1$ after each instruction (except for branching instruction). In an actual computer, it is incremented by the *word size* but these details are irrelevant at this stage. #### **Side Effects of an Instruction.** Each instruction reads from and writes to some subset of registers. We let $\iread(\iota)$ (resp. $\iwrite(\iota)$) be the set of “read from” (resp. “written to”) registers for instruction $\iota$. Each instruction can also read or write to main memory cells. We let $\mread(\iota)$ (resp. $\mwrite(\iota)$) be the set of memory cells addresses read from (resp. written to) by instruction $\iota$. 00000000 <main>: 0: e3a00009 mov r0, #9 ; 0x9 4: eaffffff b 8 <binary_search> 00000008 <binary_search>: 8: e92d4030 stmdb sp!, {r4, r5, lr} c: e59f4040 ldr r4, [pc, #64] ; 10: e3a0e000 mov lr, #0 ; 0x0 14: e3a0c00e mov ip, #14 ; 0xe 18: e3e05000 mvn r5, #0 ; 0x0 1c: e08e300c add r3, lr, ip 20: e1a020c3 mov r2, r3, asr #1 24: e7943182 ldr r3, [r4, r2, lsl #3] 28: e0841182 add r1, r4, r2, lsl #3 2c: e1530000 cmp r3, r0 %\textbf{/ eq le /}% 30: 05915004 ldreq r5, [r1, #4] 34: 024ec001 subeq ip, lr, #1 ; 0x1 38: 0a000001 beq 44 <binary_search+0x3c> 3c: c242c001 subgt ip, r2, #1 ; 0x1 40: d282e001 addle lr, r2, #1 ; 0x1 44: e15e000c cmp lr, ip %\textbf{/ le /}% 48: c1a00005 movgt r0, r5 4c: dafffff2 ble 1c <binary_search+0x14> 50: e8bd8030 ldmia sp!, {r4, r5, pc} 54: 00000158 andeq r0, r0, r8, asr r1 An example of an assembly program is given in Listing \[lis-bs\]. This program performs a binary search on an array of 14 elements. Line 24 loads register with a value of the array at address $v(\mathsf{r4})+(v(\mathsf{r2})*8)$. As we do not know the values of the array, the value of is unknown after this instruction. contains the value we are looking for, and is also unknown. As a consequence, the comparison of line 2c is undetermined as the value of in unknown. The outcome of the comparison is used later in conditional instructions (`ldreq r5, [r1, #4]` and `subgt ip,r2,#1`) and branching instructions `beq 44`. Two *status bits* are needed to encode the result of the comparison at line 24: whether is “lower or equal” than and whether is “equal” to . This is indicated by the two predicates and between . The address of the memory cell referenced at line 24 is determined by the previous outcomes of the comparison instruction at line 2c. #### **Runs.** A *run* of program $p$ from state $v_0$ (initial value of the input data) is the (unique) sequence of instructions performed by $p$ from $v_0$: $$\rho(p,v_0)= \iota_1 \quad \cdots \quad \iota_k \quad \cdots \quad \iota_n$$ with $\iota_1=i_1$. The length of the run $\rho(p,v_0)$ is $|\rho(p,v_0)|=n$. We assume that every run terminates, and that moreover, given $p$, there exists a contant $K_p$ $\forall v \in \calV, |\rho(p,v_0)| \leq K_p$. Intuitively, this means that all loops are bounded, and it implies that there is no run which encounters twice the same state. The state after the subsequence $\iota_1 \ \cdots \ \iota_k$ is determined by the composition of the semantics function of each instruction. If $v_{j}$ is the state after instruction $\iota_j$ then $v_{j+1}={[\![\iota_{j+1}]\!]}(v_j)$, and $v_0$ is the initial state. #### **Execution Time of a Run.** If each instruction was performed one after the other, the execution-time of a run would be the sum of the execution times of each instruction. On pipelined architectures with caches, the execution-time solely depends on: 1. the subsequences of instructions: pipeline *stalls* can occur, for instance because one instruction (in the execute stage) reads a register written to by the instruction in the next stage (memory stage). 2. the time to read or write a memory cell: instructions that require memory transfers (load and store) might take different durations if a *cache* is used, depending on whether the memory cell is already in the cache of not. We let $H$ denote the architecture of the system. $H$ refers to the pipeline structure and timing specifications, the cache initial state, size, replacement policy and timing specifications, and the timing specifications of the main memory. The *execution-time* of a run $\rho$ is completely determined by: - the architecture $H$, - the duration of each instruction of $\rho$ in each stage of the pipeline, - the registers read from and written to, and memory cells read from or written to by each instruction of $\rho$. The *duration* of a run $\rho$ on architecture $H$ is denoted $\et_H(\rho)$. This function might be rather complex but is yet well-defined. To formalize the previous informal definition, assume the architecture $H$ is fixed. Let $\rho=\iota_1 \, \cdots \, \iota_n$ and $\rho'=\iota'_1 \, \cdots \, \iota'_n$ be two runs of program $p$. We say that $\rho$ and $\rho'$ are (time-wise) *$H$-equivalent* and write $\rho \approx_H \rho'$ if for each $1 \leq k \leq n$: - the duration of $\iota_k$ in each stage of the pipeline is the same as the duration of $\iota'_k$; - the registers used as operands and memory cells referenced are also the same: $\phi(\iota_k)=\phi(\iota'_k)$ for $\phi \in \{ \iread,\iwrite,\mread,\mwrite\}$. \[fact-equiv\] If $\rho \approx_H \rho'$ then $\et_H(\rho)=\et_H(\rho')$. The *worst-case execution-time* for program $p$ on architecture $H$ is given by: $$\wcet(p,H) = \max_{v_0 \in \calD} \et_H(\rho(p,v_0))\mathpunct.$$ #### **Timing Anomalies.** *Timing anomalies* [@wcet-survey-2008] can occur because of the complex architecture of the hardware $H$. The term refers to counter-intuitive observations in the sense that larger *local* execution-times may not result in larger *global* execution-times. *Pre-fetching* instructions can lead to such observations on some processors. This can also be observed on complex pipeline architectures (*out-of-order* execution of instructions). On architectures that do not exhibit *timing anomalies*, the function $\et_H$ is in some sense *monotonic*. For instance an achitecture $H_\mu$ with an “always miss” cache (or equivalently no cache) will produce a WCET which is always greater than on an architecture $H$ with a cache of size more than $1$. As we consider worst-case execution-time, a *random* replacement policy for a cache is equivalent to an “always miss” cache. Let $H_{r}$ denote a cache with random replacement policy, and $H$ a regular cache (LRU, FIFO, semi-random replacement policy). The following holds: $\wcet(p,H) \leq \wcet(p,H_\mu) = \wcet(p,H_r).$ This implies that an over-approximation of $\wcet(p,H)$ can always be obtained using an equivalent architecture $H'$ with an “always miss” cache. The same remark applies for the pipeline of architecture $H$. If $H'$ is the same as $H$ with larger durations for each instruction at each stage, then $\wcet(p,H)\!\!\leq \wcet(p,H')$. If a pipeline *stall* in $H$ implies a pipeline stall in $H'$ for every program and every input data, then $\wcet(p,H) \leq \wcet(p,H')$. Another interesting case is when a *branch* instruction is executed. If it is not a loop, the program fragment has a diamond shape: both branches join at some future point in the computation. If the local worst-case execution time is obtained by taking one side of the branch instruction, we can safely ignore the other side as it does not contribute (more) to the global worst-case execution-time. The framework of this paper does handle timing anomalies, but some abstractions defined below are not safe for architecture exhibiting timing anomalies. #### **Abstractions.** In this section we introduce some simple abstractions that can be made on a program $p$. The aim of this abstraction is to reduce the space needed to encode the state of the computation. We examplify the usefulness of these abstractions on some benchmarks programs from . int fib(int n) { int i,Fnew,Fold,temp,ans; Fnew=1;Fold = 0; for(i=2;i<=30 && i<=n; i++) { temp=Fnew; Fnew=Fnew + Fold; Fold=temp; } ans=Fnew; return ans; } 0: mov r2, #2 ; 0x2 4: cmp r2, r0 8: mov ip, r0 c: mov r0, #1 ; 0x1 10: mov r1, #0 ; 0x0 14: movgt pc, lr 18: add r2, r2, #1 ; 0x1 1c: mov r3, r0 20: cmp r2, #30 ; 0x1e 24: cmple r2, ip 28: add r0, r0, r1 2c: mov r1, r3 30: ble 18 <fib+0x18> 34: mov pc, lr Listing \[fib-C\] (Fig.\[fig-fibo\]) gives a C function computing the Fibonacci number $n$. Its assembly language version is given in listing \[fib-as\]. The control flow of the assembly version is controlled by lines 20, 24 and 30: register contains the loop variable $i$ and is incremented at each round. Lines c, 10, 1c, 28 and 2c are not contributing to the program control flow. If we are only interested in the *execution-time* of this program, their *effects* can be safely abstracted away. We can replace them by equivalent instructions that modify only the **pc** register, with the same read/written registers (and memory cells if it happens to be a load/store instruction). For instance, instruction at line c, can be replaced by an *abstract* instruction with: - ${[\![\textsf{mov$^a$}]\!]}(v)=v'$ with $v'(r)=v(r)$ for each register different from **pc** and $v'(\mathbf{pc})=v(\mathbf{pc})+1$; - the duration of in each stage of the pipeline is the same as ; - the registers read from/written to by at line c are the same as the ones read from/written to by instruction at line c. In the end, we can abstract away the values of registers , and and assume they are always $0$ as no abstract instruction will modify them. The WCET of the abstracted program will be exactly the same as the concrete one. The goal of this abstraction is to reduce the space needed to encode a state of the computation. Instead of encoding $7$ registers, only $4$ are relevant for the computation of the WCET. A valid abstract program must simulate the execution tree of the concrete program. To be equivalent WCET-wise to the concrete program, it should also preserve the addresses of the referenced memory cells to ensure that cache hits/misses are preserved. To formalize the previous notions, we first define *critical* instructions. A *critical* instruction is an instruction that: - either sets some status bits; it can be a comparison or test () or an arithmetic instruction with the “s” flag on the ARM9 (a subtraction ); - or an instruction that references a memory cell (load register with the content of memory cell $\mathsf{r2} + (\mathsf{r3} \times 4)$). Next we define *abstract* instructions. As examplified for the instruction at line c previously, given an instruction $\iota$, the *abstracted instruction* $\iota^a$ is defined by: - the semantics of $\iota^a$ is ${[\![\iota^a]\!]}(v)=v'$ with $v'(x)=v(x)$ for each register $x$ different from **pc** and each memory cell $x$ in $\calM$, and $v'(\mathbf{pc})=v(\mathbf{pc})+1$; - the duration of $\iota^a$ in each stage of the pipeline is the same as the duration of $\iota$; - the registers read from/written to by $\iota^a$ are the same as the ones read from/written to by instruction $\iota$: $\phi(\iota)=\phi(\iota^a)$ for $\phi \in \{\iread,\iwrite,\mread,$ $\mwrite\}$. Let $p^a=i_1^a \cdots i_n^a$ be the abstract program that corresponds to $p=i_1 \cdots i_n$. An *abstraction mapping* $\alpha$ is a mapping that associates with each (concrete) instruction $\iota$ of $p$, either $\iota$ (identity) or $\iota^a$ ($\alpha$ determines whether $\iota$ is abstracted or not). We write $\iota^\alpha$ for $\alpha(\iota)$. Let $\rho(p,v_0)=\iota_1\iota_2 \cdots \iota_k$ be a run of $p$ from $v_0$ and $\rho(p^\alpha,v_0)=\iota^\alpha_1\iota^\alpha_2 \cdots \iota^\alpha_k$ the corresponding $\alpha$-abstracted run. Let $I_c(p,v_0)\subseteq \{1,2,\cdots,k\}$ be the set of indices $j+1 \in I_c(p,v_0) \iff \iota_{j+1}$ is a critical instruction in $\rho(p,v_0)$. Let $v_{j}$ be the state after executing instruction $j$ in $\rho(p,v_0)$ and $v_{j}^\alpha$ be the state after executing abstract instruction $j$ in $\rho(p^\alpha,v_0)$. The following Lemma states that, if the values of the registers read from/written to by any critical instruction (in $\rho(p,v_0)$), are equal to the values of the same registers in the abstract execution, the execution time of the concrete and abstract run is the same. \[lem-1\] If $\forall j+1 \in I_c(\rho(p,v_0))$, $v_j(r)=v^\alpha_j(r)$ for each $r \in \iread(\iota_{j+1}) \cup \iwrite(\iota_{j+1})$ then $\et_H(\rho(p,v_0))=\et_H(\rho(p^\alpha,v_0))$. If the values of the operand registers of each critical instruction $\iota_j$ are the same in the concrete and abstract runs before performing $\iota_j$ and $\iota_j^\alpha$, then: 1. the status bits that are set by the critical instruction have the same values in the concrete and abstract state; 2. the addresses of the memory cells referenced by the instruction are the same in the concrete and abstract run. The concrete and abstract run are thus $H$-rquivalent, $\rho(p,v_0) \approx_H \rho(p^\alpha,v_0)$. By Fact \[fact-equiv\], it follows that $\et_H(\rho(p,v_0))=\et_H(\rho(p^\alpha,v_0))$. If Lemma \[lem-1\] holds for each run $\rho(p,v_0)$ with $v_0 \in \calD$, we say that $p$ and $p^\alpha$ are $H$-equivalent and write $p \approx_H p^\alpha$. In this case, by definition of the WCET, we have: \[lem-2\] If $p \approx_H p^\alpha$ then $\wcet(p,H) = \wcet(p^\alpha,H)$. #### **Context Independence.** As we cannot simulate $p$ for every input data, we assume that the initial values of these data can be arbitrarily chosen. To formalize this, we use an extended domain for the values of the registers and memory cells: $\calD \cup \{\unk\}$ where $\unk$ is a special *unknown* value. At the beginning of the computation, every register (except **pc**) and memory cell has its value set to $\unk$. The initial state is thus $v_0$ with $v_0(x)=\unk$ for $x \in (\calR \setminus \{\mathbf{pc}\}) \cup \calM$ and $v(\mathbf{pc})=\start$ where $\start$ is the address of the first instruction of program $p$. We assume that for each program $p$, the addresses of the memory cells referenced during the course of the execution of the program, only depend on the current state and are independent from the input data values. By this, we mean that the address referenced at each point in a run of a program is determined by some registers values that are known. These values may depend on the actual content of some memory cells because they influence the branching instructions, but once a branch is chosen, the addresses can be computed. An example is a binary search program: we have to determine wether a sorted array $v$ contains a value $s$. The search continues as long as $s$ has not been found. The semantics of each instruction (next state) is extended to the extended domain $\calD \cup \{\unk\}$ as follows: - for arithmetic and logical instructions, the value of the result of an instruction is $\unk$ if the value of one of the operands is $\unk$; - for instructions that set the status bits, there might be more than one next state; if one operand is $\bot$, the next states are given by all the possible values of the status bits; - for memory transfer instructions (load, store with addresses in $\calM$) the result in memory or register is always $\bot$. Nevertheless, for transfers involving the *stack* (a subset of the addresses in $\calM$), we keep track of the values pushed or popped. The stack is quite often used on call/return of a function, and abstracting the content of the stack would result in some infeasible paths, or even to references to forbidden memory cells. - for branching instructions, there is one next state determined by the value of the target (unconditional branching) or by the status bits (conditional branching). From the previous extended definitions, there might be more than one run from the initial extended state $v_0$. We denote $p_\unk$ the non-deterministic program that corresponds to $p$ on the extended domain. The semantics of $p_\unk$ is a *tree*, $\stree(p_\unk)$ where the branches correspond to the choices of the status bits when required. Note that this tree might be unbounded. An important property of this tree, is that if $\rho(p,v_0)$ is a run of $p$ on input data $v_0$, there is a path $\rho'$ in $\stree(p_\unk)$ that satisfies $\rho(p,v_0) \approx_H \rho'$. Moreover, as we assume that the number of steps when running $p$ is bounded by $K_p$, we can safely truncate the tree $\stree(p_\unk)$ and prune all nodes that are more than $K_p$ steps apart from the root. Let $\runs(p_\unk)$ denote the set of rooted paths in the tree $\stree(p_\unk)$. We assume $\stree(p_\bot)$ has depth at most $K_p$. Let $$\wcet(p_\bot,H)=\max_{\rho \in \runs(p_\unk)} \et_H(\rho)\mathpunct .$$ As every run of $p$ is simulated by a run $p_\unk$, we have: $$\wcet(p,H) \leq \wcet(p_\bot,H) \mathpunct .$$ Moreover, we can also define an abstract version, $p_\bot^\alpha$, of $p_\bot$, given an abstraction mapping $\alpha$. The definitions are extended to te extended domain. As before we have: \[lem-3\] If $p_\bot \approx_H p_\bot^\alpha$, then $\wcet(p_\bot,H) = \wcet(p_\bot^\alpha,H)$. Combining Lemma \[lem-2\] and Lemma \[lem-3\], we have: If $p_\bot \approx_H p_\bot^\alpha$, $\wcet(p,H) \leq \wcet(p_\bot^\alpha,H)$. #### **Checking that $p^\alpha \equiv_H p$.** Checking whether $p_\bot \approx_H p_\bot^\alpha$ can be done by building a *synchronized product* of $p_\bot$ and $p_\bot^\alpha$ and checking wether each state preceeding a critical instruction satisfies the condition of Lemma \[lem-1\]. This is implemented in our framework (see Fig. \[fig-tool-chain\]) by generating a C++ file that performs this check. Table \[tab-results\], column *Abs* gives the ration of abstracted instructions for some programs (when we have chosen to abstract away some instructions). For some programs (`matmult` and `jfdcint`) the number of abstracted instructions is rather high. This indicates that the control flow is quite simple and governed by a small number of instructions. Notice that this abstraction does not change the WCET of the program. assumptions: - bounded loops, - bounded while equivalence: - projection, critical instruction Two equiv programs have the same WCET (thm) regardless of timing anomalies!! How to obtain an abstract program example computation of exact WCET on abstract program (cache and pipeline) From Programs to Games {#sec-games} ====================== In this section we describe how to encode an assembly program into a game. The encoding can be applied to any assembly language but we give examples for the ARM9 processor. Given a program $p$, we define a two-player game to model the runs of $p_\bot$ defined in the previous section. Player 1 executes the instructions of $p_\bot$. The role of Player 2 is to set the values of the status bits when an instruction that modifies them is encountered and some operands have unknown values, the result is undetermined. The outcome is thus picked up non-deterministically. On the ARM9 processor, there are $4$ status bits. A simple encoding would be to have $4$ boolean variables to model the value of each bit. As we let Player 2 choose the outcome, this corresponds to choosing four values for Player 2: N (negative), Z (zero), V (overflow) and C (carry). This could create $2^4=16$ different next states and thus as many new potential branches in the game. Most of the time, it is not necessary to know the actual values of the $4$ status bits. For instance the result of a comparison instruction with, say $\mathsf{r1}$ unknown, could be used later on only to check wether $\mathsf{r0}=\mathsf{r1}$. In this case the value of the Z-status bit is required but the values of the other status bits are irrelevant. To reduce the number of branches (choices of Player 2) in the game, we determine, for each instruction $\iota$ that sets a status bit, the next instructions that depend on the result of $\iota$. This can be computed on the program $p$. For each instruction $\iota$ that sets a status bits, we let $\flag(\iota)$ be the set of predicates used after $\iota$. For instance in the example code of Listing \[fib-as\], Fig. \[fig-fibo\] page , the result of the instruction line 4 is used at line 14, and the only predicate needed is (whether $\mathsf{r2}>\mathsf{r0}$). In the worst case we still need $4$ variables to encode the outcome of an instruction $\iota$ that sets the status bits, but we reduce the choices of Player 2 to the predicates in $\flag(\iota)$. In the previous examples, instead of having $16$ branches, there will be only $2$. To model program $p_\bot$ in UPPAAL we need: - an array, `val`, of $16$ variables for the registers of the ARM9 processor; - $4$ boolean variables for the status bits (we use `cmple`, `cmplt`, `cmpls`, `cmpeq` instead of the actual status bits N, Z, V and C, but this is equivalent); - a *stack* of size $K$ (the size of which has been determined in a previous stage). Although the model-checker UPPAAL that we use is extremely efficient, we have to be careful when encoding $p_\bot$: some information can be encoded using variables, but they will be part of the *state* of the network of TA we build, and will be encoded in the BDD representation of each state. Some information are not dynamic but rather static (the *type* of an instruction $\iota$, or the registers read/written $\iread(\iota)$ and $\iwrite(\iota)$) and can be encoded using UPPAAL *functions*. This saves space as functions are not part of the encoding of a state. Given a program $p_\bot$, we define the functions: - $\setNZ : p \rightarrow {\mathbb B}$ which, given an instruction $\iota \in p$, returns $\true$ if $\iota$ sets some status bits (comparison instructions and instructions with the “s” flag like etc); - $\cmpU: p \times \calV_\bot \rightarrow {\mathbb B}$ which returns $\true$ if the result of the instruction $\iota$ in state $v$ is unknown. As a shorthand we write $\NDcmp(\iota,v) = \setNZ(\iota) \wedge \cmpU(\iota,v)$ and this indicates whether instruction $\iota$, when executed from state $v$, should be played by Player 2 (the status bits should be set but an operand is unknown). In addition to this, we define another function $\update: \calV_\bot \rightarrow \calV_\bot$ which updates the values of the registers and the status bits if required: this function encodes the semantics of each instruction on the extended domain. The result for the Fibonnaci program of Listing \[fib-as-complete\] page  are given in Listings \[fib-C-setNZ\] and \[fib-C-update\]. These listings call for some comments: - Listing \[fib-as-complete\] contains the assembly code generated by `objdump` after compiling the C program with `gcc`; the instructions that set status bits have been annotated (lien 4 `/ le /`) by the predicates that should be set by the instruction (`le` in this case for instructions at lines 4, 20 and 24). - Listing \[fib-C-setNZ\] contains the functions that determine whether the result of an instruction that sets the status bits is undetermined. `UNKNOWN` is a special value[^10]. For instance, if the value of is unknown when executing instruction (hexadecimal) $20$ (decimal $32$), `cmpU` returns $\true$ and `SetStatusB` as well. - Listing \[fib-C-update\] contains the updates of the registers in the extended domain. The updates of an instruction are performed only if it is not abstracted away (`is_abstracted` function, not given here, but we can assume for now it always returns $\false$.) The instruction (UPPAAL translation lines 13 to 20) sets the `cmple` variable according to the values of and . If at least one of the values of and is unknown, the value of `cmple` will be chosen right after the update step by Player 2, overriding the previous value. The instruction is *unconditional*, and it has to be scheduled for execution. This is carried out by function `SET(-,-,-)` which sets $3$ values (in the first stage of the pipeline, see section \[sec-architecture\]): the label of the instruction ($4$), the memory addresses referenced by the instruction ($-1$ indicates no memory addresses), and wether the instruction is scheduled or not ($1$ in this case). For conditional instructions, , (UPPAAL translation lines 24 to 37), if the function `gt()` returns $\true$, the instruction is not scheduled (`SET(20,-1,0)`). Function `gt()` returns the complement value of `cmple` that has been set by the comparison instruction (or Player 2 if some operands were unknown) before. The last parameter of `SET(-,-,-)` has no meaning for conditional branching instructions as they are always scheduled. We use it to indicate whether the condition evaluates to $\true$ or $\false$. An example is instruction `ble 18` (UPPAAL translation lines 76 to 83 in listing \[fib-C-update\]). If the condition (function `le()`) evaluates to $\true$ this parameter is $\true$ and $\false$ otherwise. This information is used to simulate pipeline *flushes* when a branch prediction is wrong. <!-- --> 00000000 <fib>: 0: e3a02002 mov r2, #2 ; 0x2 4: e1520000 cmp r2, r0 / le / 8: e1a0c000 mov ip, r0 c: e3a00001 mov r0, #1 ; 0x1 10: e3a01000 mov r1, #0 ; 0x0 14: c1a0f00e movgt pc, lr 18: e2822001 add r2, r2, #1 ; 0x1 1c: e1a03000 mov r3, r0 20: e352001e cmp r2, #30 ; 0x1e / le / 24: d152000c cmple r2, ip / le / 28: e0800001 add r0, r0, r1 2c: e1a01003 mov r1, r3 30: dafffff8 ble 18 <fib+0x18> 34: e1a0f00e mov pc, lr 00000038 <main>: 38: e1a0c00d mov ip, sp 3c: e92dd810 stmdb sp!, {r4, fp, ip, lr, pc} 40: e3a0401e mov r4, #30 ; 0x1e 44: e24cb004 sub fp, ip, #4 ; 0x4 48: e1a00004 mov r0, r4 4c: ebffffeb bl 0 <fib> 50: e1a00004 mov r0, r4 54: e91ba810 ldmdb fp, {r4, fp, sp, pc} /* function to determine whether status bits should ne set */ bool SetStatusB(int i) { // i is the PC of instruction; function that tells whether status bits should be set // comparisons for function fib if (i==4) { // setting status bits for instruction cmp at 4 [0x4] return true ; } if (i==32) { // setting status bits for instruction cmp at 32 [0x20] return true ; } if (i==36) { // setting status bits for instruction cmp at 36 [0x24] return true ; } // comparisons for function main return false ; } /* comparisons for instructions used in the program */ bool cmpU(int i) { /* comparisons for function fib starting 0 ending 52 */ if (i==4) return val[r2]==UNKNOWN||val[r0]==UNKNOWN; // [0x4] if (i==32) return val[r2]==UNKNOWN; // [0x20] if (i==36) return val[r2]==UNKNOWN||val[ip]==UNKNOWN; // [0x24] /* comparisons for function main starting 56 ending 84 */ return false; // none if not found } // end comp of instruction /* setcmp for instructions used in the program */ void setcmp(int i,bool n1,bool n2) { /* res_comp for function fib starting 0 ending 52 */ if (i==4) { // instruction cmp r2, r0 at 4 [0x4] cmple=n1; } if (i==32) { // instruction cmp r2, #30 at 32 [0x20] cmple=n1; } if (i==36) { // instruction cmple r2, ip at 36 [0x24] cmple=n1; } /* res_comp for function main starting 56 ending 84 */ } // end setcmp of instruction bool NDcmp(int i) { return SetStatusB(i) && cmpU(i) ; } /* setcmp for instructions used in the program */ void setcmp(int i,bool n1,bool n2) { /* setcmp for function fib starting 0 ending 52 */ if (i==4) { // instruction cmp r2, r0 at 4 [0x4] cmple=n1; } if (i==32) { // instruction cmp r2, #30 at 32 [0x20] cmple=n1; } if (i==36) { // instruction cmple r2, ip at 36 [0x24] cmple=n1; } /* res_comp for function main starting 56 ending 84 */ } // end setcmp of instruction void update() { // update function int nextpc,nextfp,tmp; /* updates for function fib starting 0 ending 52 */ if (val[pc]==0) { // Instruction mov r2, #2 at 0x0 nextpc=val[pc]+4; if (!is_abstracted(val[pc])) { // effect of instruction is null if abstracted val[r2]=(2); } SET(0,-1,1); // instruction scheduled is 0, no memory access and scheduled } // end mov at 0x0 if (val[pc]==4) { // Instruction cmp r2, r0 at 0x4 nextpc=val[pc]+4; if (!is_abstracted(val[pc])) { // effect of instruction is null if abstracted // Should set the Z and N and C bits if ((val[r2]-(val[r0]))<=0) cmple=1 ; else cmple=0; } SET(4,-1,1); // instruction scheduled is 4, no memory access and scheduled } // end cmp at 0x4 ... if (val[pc]==20) { // Instruction movgt pc, lr at 0x14 nextpc=val[pc]+4; if (gt()) { if (!is_abstracted(val[pc])) { // effect of instruction is null if abstracted if (val[lr]==UNKNOWN) { val[pc]=UNKNOWN; } else { nextpc=(val[lr]); } } SET(20,-1,1); // instruction scheduled is 20, no memory access and scheduled } else SET(20,-1,0) ; // instruction not scheduled, no mem access } // end movgt at 0x14 if (val[pc]==24) { // Instruction add r2, r2, #1 at 0x18 nextpc=val[pc]+4; if (!is_abstracted(val[pc])) { // effect of instruction is null if abstracted if (val[r2]==UNKNOWN) { val[r2]=UNKNOWN; } else { val[r2]=(val[r2]+1); } } SET(24,-1,1); // instruction scheduled is 24, no memory access and scheduled } // end add at 0x18 ... if (val[pc]==32) { // Instruction cmp r2, #30 at 0x20 nextpc=val[pc]+4; if (!is_abstracted(val[pc])) { // effect of instruction is null if abstracted // Should set the Z and N and C bits if ((val[r2]-(30))<=0) cmple=1 ; else cmple=0; } SET(32,-1,1); // instruction scheduled is 32, no memory access and scheduled } // end cmp at 0x20 if (val[pc]==36) { // Instruction cmple r2, ip at 0x24 nextpc=val[pc]+4; if (le()) { if (!is_abstracted(val[pc])) { // effect of instruction is null if abstracted // Should set the Z and N and C bits if ((val[r2]-(val[ip]))<=0) cmple=1 ; else cmple=0; } SET(36,-1,1); // instruction scheduled is 36, no memory access and scheduled } else SET(36,-1,0) ; // instruction not scheduled, no mem access } // end cmple at 0x24 ... if (val[pc]==48 && (!le())) { // Instruction ble 18, at 0x30 nextpc=val[pc]+4; SET(48,-1,0) ; // instruction scheduled, no mem access, no branching } // end ble at 0x30 [cond false] if (val[pc]==48 && le()) { // Instruction ble 18, at 0x30 nextpc=24; // to 0x18 SET(48,-1,1) ; // instruction scheduled, no mem access, branching } // end ble at 0x30 [cond true] if (val[pc]==52) { // Instruction mov pc, lr at 0x34 nextpc=val[pc]+4; if (!is_abstracted(val[pc])) { // effect of instruction is null if abstracted if (val[lr]==UNKNOWN) { val[pc]=UNKNOWN; } else { nextpc=(val[lr]); } } SET(52,-1,1); // instruction scheduled is 52, no memory access and scheduled } // end mov at 0x34 /* end of updates for function fib */ /* updates for function main starting 56 ending 84 */ if (val[pc]==56) { // Instruction mov ip, sp at 0x38 nextpc=val[pc]+4; if (!is_abstracted(val[pc])) { // effect of instruction is null if abstracted if (val[sp]==UNKNOWN) { val[ip]=UNKNOWN; } else { val[ip]=(val[sp]); } } SET(56,-1,1); // instruction scheduled is 56, no memory access and scheduled } // end mov at 0x38 if (val[pc]==60) { // Instruction stmdb sp!,{r4,fp,ip,lr,pc,} at 0x3c nextpc=val[pc]+4; // push should first decrease val[pc] and then store in stack(val[pc]) push(val[pc]); push(val[lr]); push(val[ip]); push(val[fp]); push(val[r4]); SET(60,-1,1); // instruction scheduled is 60, no memory access } // end stmdb at 0x3c ... if (val[pc]==76) { // Instruction bl 0, (unconditional) at 0x4c nextpc=0; // to 0x0 val[lr]=80; SET(76,-1,1) ; // instruction scheduled, no mem access, branching } // end bl at 0x4c if (val[pc]==80) { // Instruction mov r0, r4 at 0x50 nextpc=val[pc]+4; if (!is_abstracted(val[pc])) { // effect of instruction is null if abstracted if (val[r4]==UNKNOWN) { val[r0]=UNKNOWN; } else { val[r0]=(val[r4]); } } SET(80,-1,1); // instruction scheduled is 80, no memory access and scheduled } // end mov at 0x50 if (val[pc]==84) { // Instruction ldmdb fp,{r4,fp,sp,pc,} at 0x54 nextpc=val[pc]+4; nextpc=stack(val[fp]-4); val[sp]=stack(val[fp]-8); nextfp=stack(val[fp]-12); val[r4]=stack(val[fp]-16); val[fp]=nextfp; SET(84,-1,1); // instruction scheduled is 84, no memory access } // end ldmdb at 0x54 /* end of updates for function main */ val[pc]=nextpc; } // end update The generic automaton to simulate a program $p_\unk$ is given in Fig. \[fig-prog-auto\]. We assume that the main function of the program $p_\unk$ is called by another program and a particular value `INIT_LR` gives the return point. The automaton *Prog* performs some initialization (`init_val()`) and then computes the next state until the end of the program is reached: this is when the value of the **pc** register is equal to the return point `INIT_LR` (guard `val[pc]=INIT_LR`). To simulate each instruction, the automaton *Prog* performs the following steps: 1. feed the current instruction $\iota$ to the first stage of the pipeline when it is empty (to do so it has to synchronize with the first stage of the pipeline, on the `fetch!` channel) and compute the next state (`update()` function). This also sets the next value of register **pc**. The result of `update()` is that the number of the current instruction is stored into the variable `pPC[FETCH_STAGE]` where `FETCH_STAGE` is the number of the first stage of the pipeline ($0$); 2. if the instruction $\iota$ in `pPC[FETCH_STAGE]` is an undetermined comparison (`NDcmp(pPC[FETCH_STAGE])` evaluates to $\true$), the upper dashed transition is taken: Player 2 chooses two values $n$ and $z$ and the predicates that must be set (`cmple`, `cmplt`, etc) are set by `setcmp` (Listing \[fib-C-setNZ\]). If $\iota$ does not set any flag or the outcome is determined by the current state (the operands are all known), the middle transtion is taken (Player 2 does not have to play). ![Generic Automaton *Prog* to Simulate a Program[]{data-label="fig-prog-auto"}](prog-auto-reduced.pdf) Model of the Hardware {#sec-architecture} ===================== In this section we give a UPPAAL model for the architecture of the pipelined processor ARM9 and for the caches. Model of the Pipeline --------------------- Each stage of the pipeline contains an instruction (and some other information). The information for each stage of the pipeline are stored in arrays: `pPC[k]` gives the number of the instruction in stage $k$; `Todo[k]` is a boolean value and indicates whether the instruction `pPC[k]` is scheduled (some instructions are conditional and are skipped); `dataAdr[k]` contains the address[^11] of the memory cell referenced by instruction `pPC[k]` ($-1$ if none). There are $5$ stages in the pipeline of the ARM9: - stage 1: this is the *fetch* stage. It fetches the next instruction (pointed to by the **pc** register) from the cache (or main memory) and this instruction becomes the current instruction of stage 1; - stage 2: *decode* stage. Decodes the instruction in stage 2; - stage 3: *execute* stage. Carries out the computation (addition, comparisons, etc) of the instruction in stage 3; - stage 4: *memory* stage. Carries out the transfers (from registers to main memory or main memory to registers) of the instruction in stage 4; - stage 5: *writeback* stage. Writes the value of registers that are (“writeback”) operands of the instruction in stage 5. An instruction $\iota$ enters the pipeline at stage 1. It is transfered from stage $i$ to $i+1$ as soon as possible. When it exits stage 5, it is completed. The execution of a program is completed when its last instruction is completed. #### **Pipeline Stalls.** The goal of *pipelining* is to split the execution of an instruction into different simple steps. The idea being that each step can be carried out concurrently for different instructions: while stage 1 fetches the next instruction $\iota_k$, stage 2 decodes instruction $\iota_{k-1}$, etc. It may happen that the simple steps of some sequences of instructions cannot be carried out concurrently. A *pipeline stall* is a situation when one stage $i$ of the pipeline cannot perform its computation because it has to wait for another stage $j>i$ to complete its computation. An example is when the execution of an instruction at stage 3 (execute) has an operand which is set in stage 4 (memory). The sequence of instructions of lines 0 and 4 will result in a pipeline stall at stage 3 for instruction $4$: when instruction $4$ ($\mathsf{r2}:=\mathsf{r0}-\mathsf{r1}$) is ready to execute at stage 3, it has to wait for instruction $0$ to complete (at stage 4) because instruction $0$ loads the value of memory cell $\mathsf{r1}$ into . 0: ldr r1, [r0] 4: sub r2, r0, r1 8: ... c: ldm r13, {r1,r2,r3} 10: add r4, r3, #1 14: ... Thus instruction $4$ stalls for one cycle[^12] at stage $3$. The situation for instructions c and 10 can even result in more than one cycle delay. The isntruction (line c) is a *multiple load* instruction. It loads the registers , and with the contents of memory cells pointed to by . Stage 4 performs the loads, but only one per cycle. Thus instruction 10 stalls for 3 cycles at stage 3. A pipeline stall may occur depending on: ($i$) the type of the instruction at stage 3, and the type of the instruction at stages 4 and 5; ($ii$) the registers (and memory addresses) used by the instructions at the corresponding stages. #### **Branch Prediction.** When a conditional branch instruction enters the pipeline, the next instruction to flow in is determined by the truth value of the condition. This value might not yet be available when the branch instruction is in the first stage of the pipeline. If the condition is determined by the value of a variable which is not in the cache, it might take a few cycles before the result becomes available. In this case, we should *stall* until the outcome of the comparison is computed. This might however be inefficient. Some heuristics can be applied to guess the most plausible next instruction after a conditional branching. After the *prediction*, the chosen instruction flows in the pipeline. If the guess was right the result is shortest execution time for this part of the program. If the guess was wrong, the computations of the mistakenly taken branch have to be undone, and the pipeline flushed which results in a longer execution time. We do not discuss here the choice of a good heuristics, but there are a few options that gve good results on *average*. In our model we follow [@metamoc-2009] and model the heuristics for branch prediction by: in a conditional branch, a branch is never taken (other heuristics can be accommodated for in our model). #### **UPPAAL Pipeline Model.** The timed automata models we introduce are close to the ones proposed in [@metamoc-2009]. However there are some differences as we do not have the same model for the program. The timed automata for each stage (ARM9, 5 stages) are depicted on Fig. \[fig-fetch-stage\] and Fig. \[fig-other-stage\]. The stage modelled by each automaton can be infered by the synchronization channel from the initial state (`decode?`). ![Timed Automata Model of the ARM9 Pipeline[]{data-label="fig-fetch-stage"}](fetch-stage-paper-cropped.pdf) The first stage of the pipeline is of particular importance as it models the case of a wrong guess in an branch prediction. The automaton of Fig. \[fig-fetch-stage\] models the following behaviour: 1. the automaton accepts a `fetch?` synchronization when it is idle; 2. after accepting an instruction (`fetch?` synchronizes with `fetch!` in the automaton *Prog* of Fig. \[fig-prog-auto\]), it actually fetches the instruction from main memory via the *instruction cache* (`CacheReadStart[INSTR_CACHE]!`, where `INSTR_CACHE` is the ID or the instruction cache); 3. when the instruction has been read from the cache or main memory, there are two options: 1. the instruction $\iota$ to be processed is a *conditional branch* (condition `type_of(pPC[me])==G4c`) and the variable `Todo[me]` indicates whether the condition was evaluated to $\true$ or $\false$. In case it is a conditional branch and the condition was $\true$, we simulate two “instruction read from the cache” steps: indeed our branch prediction algorithm is “never branch” and thus if it happened that we had to branch, we should simulate a pipeline `flush`. As we do not execute the instructions in the pipeline (but rather when we feed the first stage of the pipeline), this can be modelled by reading the next two instructions (the “never branch” prediction) without executing them, and then resuming the simulation from the target address of the branch instruction. 2. the instruction to be processed is not a conditional branching or the condition was evaluated to $\false$; in this case the prediction was right and nothing has to be undone. After an instruction has been fetched in the `fetch` stage, it is fed to the next stage of the pipeline. This is modelled by the `decode!` synchronization and the `copy(me,me+1)` transition. `copy(me,me+1)` copies the information in `pPC[me]`, `Todo[me]` and `dataAdr[me]` to the next stage `me+1`. ![Timed Automata Model of the ARM9 Pipeline[]{data-label="fig-other-stage"}](decode-stage-cropped.pdf){width="0.544\linewidth"} ![Timed Automata Model of the ARM9 Pipeline[]{data-label="fig-other-stage"}](memory-stage-cropped.pdf) ![Timed Automata Model of the ARM9 Pipeline[]{data-label="fig-other-stage"}](writeback-stage-cropped.pdf){width="0.54\linewidth"} The memory stage automaton is a bit more involved than the others as it has to take into account different options: if the instruction is a memory transfer (`type_of(pPC[me-1])==G2LDR` or `type_of(pPC[me-1])==G2STR`) and is scheduled (`Todo[me-1]` is $\true$) a synchronization with the data cache is requested. The type of the instructions is given by a UPPAAL function `type_of`. The duration is also given by a function `dur()` (used in the execute stage). Model of the Caches ------------------- A *cache* is a fast memory device. It is characterized by its size $K$ (usually in Kbytes), the length of a cache *line* ($B$ in Bytes) and the number of cache lines $L = \frac{K}{B}$. The main memory $\calM$ of a computer is divided into blocks equal to the length of the cache line. We let $\calM=\{m_0,m_1,\cdots,m_n\}$. The *associativity* of a cache determines where a memory block can reside. - *fully associative*: a block can be in any line; - *direct mapped*: a block can be in one line; - *$j$-way*: a block can be in $j$ different lines; in this case the cache is partitionned into $\frac{L}{j}$ different sets. Fully and direct mapped are particular instances of $j$-way caches. The partition induced by the $j$-way cache is denoted $\calP=\{P_1,\cdots,P_{\frac{L}{j}}\}$. The set of lines a memory can reside in is given by a mapping $\kappa : \calM \rightarrow \calP$. The replacement policy determines which block to eject from memory when the cache is full. The most common policies are: - LRU: least recently used; - FIFO: first-in first-out; - alternate and mixed and even random are permitted but not easily predictable. ![Timed Automata Model for the Caches[]{data-label="fig-cache"}](cache-auto-cropped.pdf){width="1\linewidth"} ![Timed Automata Model for the Caches[]{data-label="fig-cache"}](main-mem-cropped.pdf) Handling *writing* requests is also a distinctive feature of a cache. - handling write *hits*: - **write trough**: write cache and memory - **write back**: write cache; need for a dirty bit whihc is taken care of when ejecting a line from the cache; - handling write : - **write allocate**: write memory and fetch into cache; - **write no allocate**: write memory (no fetch). In this paper we model a cache with FIFO replacement policy and assume write allocate on a write/miss. #### **UPPAAL Cache Model.** The automaton modeling the behaviour of the cache (together with the model iof the main memory automaton) is given in Fig. \[fig-cache\]. After performing some initializations (`initCache()`, setting the initial state of the cache), it accepts either write or read requests. Depending on the request, and wether a cache line is dirty or not, a number of memory transactions (`PMT`) are needed to fetch the content of memory cell `m`. Each such transaction is performed one after the other. When it is completed the transfer from the cache to the register of the processor takes place and require `CACHE_SPEED` time units. Tool Chain and Case Studies {#sec-tool-chain} =========================== We have applied the previous framework to a number of benchmarks from . #### **Tool Chain.** The tool chain to compute WCET is depicted on Fig. \[fig-tool-chain\]. The component we have developed are `ARM2UPP` and `PATCH_UPP`: - `ARM2UPP` takes as input a program in assembly (`file.arm`) that has been annotated with the comparisons operators for each instruction that sets a status bit. It generates four files: - `file.{xml,q}` that contain respectively the UPPAAL network automata (and functions like `update()` etc) modeling the execution of the program on the architecture of the ARM9 and the UPPAAL queries to compute/check the WCET; - `file-reach` is an executable obtained by compiling `file-reach.cpp`; this latter file is a C++ program that simulates the program in `file.arm`. `file-reach` always terminates. However, early termination can be forced by passing some parameters (maximal number of states, maximal number of split cases). In case the number of split cases is too large ($2^{50}$ for Bubble Sort), it is possible to add some information in the file `file-reach.cpp` like constraints on the outcome of an unknown comparison. This step may be iterated several times. When it is completed the file `file.info` contains some useful information (like maximal stack size, etc). - `file-equiv` is an executable obtained by compiling `file-equiv.cpp`; this program checks whether an abstraction mapping (which is given by a function) is valid or not (implements the algorithm of section \[sec-concrete-abs\].). - `PATCH_UPP` modifies some constants in `file.xml` to incorporate the information from `file.info` (like stack size) and can also include the function of abstracted instructions (if it has been declared valid). (c-file) \[input\] [file.c]{}; (elf-file) \[inter, right=of c-file,xshift=-.2cm\] [file.elf]{}; (c-file) edge\[\] node [gcc]{} (elf-file) ; (arm-file) \[inter, below=of elf-file\] [file.arm]{}; (elf-file) edge\[swap\] node [objdump]{} (arm-file) ; (comp) \[mycomp,below=of arm-file\] [ARM2UPP]{}; (arm-file) edge\[swap\] (comp) ; (xml) \[output,above right=of comp,xshift=2.2cm\] ---------- file.xml file.q ---------- ; (reach) \[output,below right=of comp\] [file-reach]{}; (equiv) \[output,below=of comp,yshift=-1cm\] [file-equiv]{}; (comp) edge\[\] (xml); (comp) edge (reach); (comp) edge (equiv); (abstract) \[input,left=of equiv,xshift=-.5cm\] -------------- Abstracted Instructions -------------- ; (res-equiv) \[right=of equiv,color=red!70\] [YES/NO]{}; (abstract) edge (equiv); (equiv) edge (res-equiv); (reach-out) \[inter,right=of reach,xshift=1cm\] ------------------- stack size \# of split cases Unknown address ------------------- ; (ok) \[output,right=of reach-out,xshift=1cm\] [file.info]{}; (reach) edge (reach-out) (reach-out) edge node [ok]{} (ok); ($ (reach-out.north) $) – ($ (reach-out.north) + (0,2mm) $) node\[xshift=-1.5cm,yshift=2mm\] [not ok$^*$]{} -| ($ (reach.north) $) ; (patch) \[mycomp,above=of ok,yshift=.8cm\] [PATCH\_UPP]{}; (xml) edge (patch) ; (ok) edge (patch); (file-patch) \[output,above=of patch,yshift=-.5cm\] ---------------- file-patch.xml file-patch.q ---------------- ; (patch) edge (file-patch); ($ (abstract.south) $) |- ($ (abstract.south) - (0,3mm) $) -| ($ (patch.east) + (.4cm,0mm) $) – ($ (patch.east) $) ; #### **UPPAAL-TiGA Queries.** In order to compute the WCET of a program, we can check wether the program always terminates within $k$ time units. This can be computed using a binary search with UPPAAL. The drawback of this check is that some deadlock may occur in the system, yielding a biased value of the WCET. An alternative way of computing the WCET is check a *control* property: “Can Player 1 enforce termination of the program and if yes, what is the best duration he can guarantee?” This optimal time reachability control objective can be checked in one query (see [@cassez-concur-05]) with UPPAAL-TiGA, provided we know an upper of the WCET. This can be roughly over-estimated on the program (we have not implemented this part yet). Optimal reachability of a location `l` is then specified by the control objective: control(#n,0) : A [ true U l ] if `#n` is a rough upper bound of the WCET[^13]. Program termination in the UPPAAL model happens when the location `DONE` is reached in the `writeBackStage` automaton (last stage of the pipeline). Thus the control property we check is: control(#n,0) : A [ true U WriteBackStage.DONE ] #### **case Studies & Results** We have applied the framexork described in Fig. \[fig-tool-chain\] to a number of benchmark programs from . We could not analyse the full set of programs because of the current limitations of our tools: - floating point operations are not supported yet; - a few operators (`ror`) of the ARM9 assembly language are not supported yet. There are not many published results about the actual WCET of the benchmarks (or when there are, the hardware parameters, cache speed, etc are not given). To evaluate the relevance of our method, we compare our results to the ones obtained with the METAMOC method [@metamoc-2009]. There are $15$ programs that can be analysed by METAMOC using a concrete instruction cache and an “always miss” data cache. Only $7$ of the $15$ programs can be analysed with both a concrete instruction and data cache. Using our encoding and tool chain, we could analyse $13$ out of these $15$ programs (two of them contains unsupported operations) with concrete caches. Moroever, the time/space needed to compute the results is very small compared to the resources used in METAMOC (32GB RAM computer). Table \[tab-results\] give the values of WCET for each program, and the time for UPPAAL-TiGA to compute the result. The time needed to compute the intermediary files is negligible. The timing specification of the caches are: `CACHE_SPEED=1` (processor) cycle is the same as the processor speed, and a memory transaction takes $10$ processor cycles. The UPPAAL files are available from <http://www.irccyn.fr/franck/wcet>. #### **Energy/Power Consumption Optimization.** The last column of Table \[tab-results\] gives the percentage of time the processor can run at a slower clock rate ($1/4$th of its fastest speed) without any impact on the WCET: this is due to the initial transient phase of the execution of a program where instructions are loaded into the cache. For some small programs the result is impressive (22% for `janne-complex`). To do this we just add a automaton to the network that switches the rate from $4$ to $1$ after a certain amount of time. Another interesting and easy computation that can be done, is to fix the time the processor runs at a slower rate (in the initial phase) and compute the optimal time to reach the end the program (which is the WCET) under this constraint. [||l||c|c|c|c|c|c||]{}   **Program**   & --------------- loc$^\dagger$ --------------- : Results (C programs compiled with `gcc -O2`)[]{data-label="tab-results"} &   -------------- $N^\ddagger$ -------------- : Results (C programs compiled with `gcc -O2`)[]{data-label="tab-results"}   &   ------------- UPPAAL-TiGA time/space ------------- : Results (C programs compiled with `gcc -O2`)[]{data-label="tab-results"}   &  WCET  &    Abs$^\star$    &      ------- Low Power ------- : Results (C programs compiled with `gcc -O2`)[]{data-label="tab-results"}     \ \  fac & 26 & 0 & 0.35s/6.91MB & 1883 & 4/34 & 26/1.3%\  fib & 74 & 0 & 0.25s/5.68MB & 571 & 4/22 & 26/4.5%\  janne-complex$^\ast$  & 65 & 0 & 0.54s/7.76MB & 792 & 0/23 &**176/22%**\  matmult$^\ast$ & 162 & 0 & 119.2s/936.75MB & 614827 & **31/107** & 800/0.001%\  jfdcint & 374 & 0 & 7.13s/55.99MB & 49017 & **394/454** & 108/0.22%\  expint(50,1) & 81 & 0 & 6.08s/59.16MB & 65042 & 0/124 & 70/1.7%\  expint(50,21) & 81 & 0 & 3.65s/43.21MB & 41015 & 0/124 & 71/1.7%\  fdct & 238 & 0 & 2.83s/26.79MB & 26099 & 0/286 & 90/0.3%\  edn$^\ast$ & 284 & 0 & 22.28s/230.98MB & 62968 & 0/460 & 26/0.04%\  recursion$^\ast$ & 41 & 0 & 2.68s/28.82MB & 10335 & 0/38 & 32/0.3%\ \  bs & 174 & 5 & 0.52s/6.52MB & 366 & 0/22 & **30/8.2%**\  cnt$^\ast$ & 115 & 100 & 100.25s/377.02MB & 6483 & 0/82 & 40/0.06%\  insertsort$^\ast$ & 91 & 675 & 9.36s/81.27MB & 27061 & 0/53 & 400/1.4%\  ns$^\ast$ & 497 & 625 & 12.38s/110.92MB & 43239 & 0/41 & 32/0.0007%\ ------------------------------------------------------------------------ \ $^\dagger$lines of code in the C source file $^\ddagger N=$ Max number of Player 2 moves along a path\ $^\star$Abstracted Instr./Instr. $^\ast$Program selected for the WCET Challenge 2006 [@bench-malar] Conclusion ========== In this paper we have presented a framework based on timed games and the model checker UPPAAL-TiGA to compute WCET for programs running on architectures featuring pipelines and caches. The results we have obtained support the claim that model checking is adequate for computing WCET. Moreover UPPAAL-TiGA could be tuned to handle WCET computation more efficiently: *priorities* between processes can reduce unnecessary interleavings and there are not yet implemented in UPPAAL-TiGA (though they are in UPPAAL); a lot of time is spent *checking* whether a new state has already been encoutered: this will never be the case in the programs we check (otherwise they would be an infinite loop). Disabling this check would also reduce the time to compute the results. Of course, a program like Bubble Sort remains beyond the scope of analysis within our framework. Nevertheless, what we advocate is the combination of different techniques to solve the WCET problem: *abstract interpretation* (AI) combined with *Interger Linear Programming* (ILP) have given very good results [@wcet-ai-aswsd-ferdinand-04] but this method is yet to prove that: (1) it can be *easily* adapted to different processors and (2) it can take into account *power* related features (like change of speed of the processor). Our ongoing work focuses on two aspects: 1. extend the set of instructions supported by our compiler and provide models for other architectures (like ARM11); 2. add a *pre-processing* step to prune the execution tree of the program. The goal of this step is to reduce the number of paths of the program still preserving the paths giving the WCET. This step can be carried out using ILP techniques, or *counter-example guided abstraction refinement* (CEGAR) methods [@clarke-cegar-acm-03]. #### **Acknowledgements.** \ The author would like to thank Bernard Blackham (NICTA, Sydney) and Gernot Heiser (NICTA, Sydney) for their helpful comments and support. [^1]: Author supported by a Marie Curie International Outgoing Fellowship within the 7th European Community Framework Programme. [^2]: When we refer to the “source” code, we assume the program $p$ was generated by a compiler, and refer to the high-level program (in C) that was compiled into $p$. [^3]: Note this is true even for input data ranging over a finite domain, and can be proved using König’s Lemma. [^4]: The benchmarks contain 35 programs. In [@metamoc-2009], only 14 programs can be analysed with a concrete instruction cache and 7 with a concrete instruction and data cache. [^5]: Note that a similar model is reportedly due to A. P. Ravn in [@hubert-wcet-09]. [^6]: Say why $2$ fails ... [^7]: In contrast, the verification-based tools would need a description of the hardware to compute the CFG. [^8]: A particular case is a processor with one stage. [^9]: This is not always the case as for instance the duration of the instruction `mull` (multiplication on long integers) on the AMRM9 depends on how large one of the operand is. However, we can always take the longest duration to obtain a safe upper bound of the WCET. [^10]: We use an integer that is never used as an actual value in the content of any register. [^11]: For multiple loads and stores, this should be a range of addresses; this information is used only for determining whether a stall should occur in the pipeline. For multiple loads and stores, we force a stall in a pipeline until the end of the multiple loads/stores instruction. This is a safe encoding as the ARM9 does not exhibit timing anomalies. [^12]: We assume that the content of memory cell was in the cache and it takes one cycle to be fetched. [^13]: If `#n` is not large enough, UPPAAL-TiGA result will be “not controllable”.

--- abstract: 'We present ALMA observations of two moderate luminosity quasars at redshift 6. These quasars from the Canada-France High-z Quasar Survey (CFHQS) have black hole masses of $\sim 10^{8} M_\odot$. Both quasars are detected in the [\[C[ii]{}\]]{} line and dust continuum. Combining these data with our previous study of two similar CFHQS quasars we investigate the population properties. We show that $z>6$ quasars have a significantly lower far-infrared luminosity than bolometric-luminosity-matched samples at lower redshift, inferring a lower star formation rate, possibly correlated with the lower black hole masses at $z=6$. The ratios of [\[C[ii]{}\]]{} to far-infrared luminosities in the CFHQS quasars are comparable with those of starbursts of similar star formation rate in the local universe. We determine values of velocity dispersion and dynamical mass for the quasar host galaxies based on the [\[C[ii]{}\]]{} data. We find that there is no significant offset from the relations defined by nearby galaxies with similar black hole masses. There is however a marked increase in the scatter at $z=6$, beyond the large observational uncertainties.' author: - 'Chris J. Willott' - Jacqueline Bergeron and Alain Omont bibliography: - 'willott.bib' title: 'Star formation rate and dynamical mass of $10^{8}$ solar mass black hole host galaxies at redshift 6' --- Introduction ============ Improved astronomical observational facilities have enabled the discovery and study of many galaxies at an early phase of the Universe’s history. It is now possible to witness the majority of the stellar and black hole mass growth over cosmic time and identify how physical conditions at early times differ from now. One of the major relations to be determined as a function of time is the tight correlation between black hole mass and galaxy properties observed for nearby galaxies (see @Kormendy:2013 2013 for a review). Observations of this relation at high-redshift are critical to understanding the cause because most of the growth occurred at early times. Attempts to measure black hole and galaxy masses at high-redshift face a number of problems. Black hole mass measurements cannot be made directly by resolved kinematics of gas or stars within the black hole’s sphere of influence, nor by reverberation mapping. Instead black hole masses, $M_{\rm BH}$, of quasars can be measured at any redshift using the single-epoch virial mass estimator that involves measuring a low-ionization broad emission line, such as [Mg[ii]{}]{} or [H$\beta$]{}, and calibrating the location of the emitting gas with low-$z$ reverberation-mapped quasars [@Wandel:1999a]. For AGN with obscured broad lines $M_{\rm BH}$ can only be estimated from the luminosity making an assumption about the accretion rate relative to the Eddington limit. Measuring galaxy properties, such as luminosity or velocity dispersion, $\sigma$, of distant quasars is hampered by surface brightness dimming, the bright glare of the quasar and AGN (active galactic nuclei) emission line-contamination of spectral features. Up to $z\approx 1$ there has been considerable success in measuring AGN host galaxy luminosities, morphologies and in some cases velocity dispersions [@Cisternas:2011; @Park:2014]. At higher redshifts ($1<z<4$) the galaxy light is more difficult to separate from the quasar, which, combined with greater mass-to-light corrections, lead to larger uncertainties [@Merloni:2010; @Targett:2012]. The results of these studies are mixed with some evidence in favour of higher $M_{\rm BH}$ at a given galaxy mass. At yet higher redshifts it has proved impossible to measure the galaxy light of quasars [@Mechtley:2012] before launch of the  and instead the main method of determining galaxy mass is kinematics of cool gas in star-forming regions [@Carilli:2013]. Facilities such as the IRAM Plateau de Bure Interferometer, the Jansky Very Large Array and the Atacama Large Millimeter Array (ALMA) have sufficient sensitivity and resolution to resolve the gas in distant quasar hosts and provide dynamical masses [@Walter:2004; @Walter:2009; @Wang:2010; @Wang:2013]. In particular, ALMA has the sensitivity to probe $z=6$ quasar hosts with star formation rates, SFR, in the tens of solar masses per year, rather than only in the extreme starbursts previously observable [@Willott:2013]. The studies above focussed on $z\approx 6$ Sloan Digital Sky Survey (SDSS) and UKIRT Infrared Deep Sky Survey (UKIDSS) quasars with high UV and far-IR luminosities and found that their black holes are on average 10 times greater than the corresponding $\sigma$ for local galaxies, roughly consistent with a continuation of the evolution seen in lower redshift studies. Although observationally there appears to be an increase in $M_{\rm BH}$ with redshift at a given galaxy mass or $\sigma$, it has long been understood that there are selection biases that affect how closely the observations trace the underlying distribution. In particular, the steepness of the galaxy and dark matter mass functions combined with large scatter in their correlations with black hole mass mean that a high black-hole-mass-selected sample of quasars will have a systematic offset in $\sigma$ towards lower values. This effect, first identified by @Willott:2005b and @Fine:2006 was studied in detail in @Lauer:2007 and numerous studies thereafter. The magnitude of the effect depends upon the scatter in the correlation, which has not been conclusively measured at high-redshift, but appears to increase with redshift [@Schulze:2014]. @Willott:2005b and @Lauer:2007 showed that the bias is particularly strong for $M_{\rm BH}>10^9 M_\odot$ quasars such as those in the SDSS at $z \approx 6$ and therefore that the factor of 10 increase in $M_{\rm BH}$ at a given $\sigma$ first seen in the quasar SDSSJ1148+5251 [@Walter:2004] could be accounted for by the bias (see also @Schulze:2014 2014). In comparison, there would be little bias for a sample of high-$z$ quasars with black hole masses of $M_{\rm BH}\sim10^8 M_\odot$ [@Lauer:2007]. An alternative to measuring the evolution of the assembled galaxy and black hole masses is to determine the rate at which mass growth is occurring. For quasars the bolometric luminosity is a measure of the black hole mass growth rate. For galaxies, the star formation rate is proportional to the stellar mass growth. The star formation rate can be determined by the rest-frame far-infrared dust continuum luminosity. Additionally, the interstellar [\[C[ii]{}\]]{} far-infrared emission line is well-correlated with star-formation [@De-Looze:2014; @Sargsyan:2014] so can also be used as a star formation proxy. In @Willott:2013 (2013, hereafter Wi13) we presented Cycle 0 ALMA observations in the [\[C[ii]{}\]]{} line and 1.2mm continuum for two $z=6.4$ quasars from the Canada-France-High-z Quasar Survey (CFHQS, @Willott:2010a 2010b). These quasars have $M_{\rm BH}\sim10^8 M_\odot$, a factor of 10–30 lower than most SDSS quasars known at these redshifts. One quasar was detected in line and continuum and the other remained undetected in these sensitive observations placing an upper limit on its star formation rate of SFR$<40\,M_\odot\,{\rm yr}^{-1}$. In this paper we present ALMA observations of two further CFHQS quasars with similar redshift and black hole mass with the aim of providing a sample large enough to address the issue of how host galaxy properties such as SFR, $\sigma$ and dynamical mass depend upon black hole accretion rate and mass at a time just 1 billion years after the Big Bang. In particular, these quasars are not subject to the bias in the $M_{\rm BH} - \sigma$ relation discussed previously because of their moderate black hole masses. Cosmological parameters of $H_0=67.8~ {\rm km~s^{-1}~Mpc^{-1}}$, $\Omega_{\mathrm M}=0.307$ and $\Omega_\Lambda=0.693$ [@Planck-Collaboration:2014] are assumed throughout. Observations ============ CFHQSJ005502+014618 (hereafter J0055+0146) and CFHQSJ222901+145709 (hereafter J2229+1457) were observed with ALMA on the 28, 29 and 30 November 2013 for Cycle 1 project 2012.1.00676.S. Between 22 and 26 12m diameter antennae were used. The typical long baselines were $\sim 400$m providing similar spatial resolution to our Cycle 0 observations. Observations of the science targets were interleaved with nearby phase calibrators, J0108+0135 and J2232+1143. The amplitude calibrator was Neptune and the bandpass calibrators J2258-2758 and J2148+0657. Total on-source integration times were 4610s for J0055+0146 and 5490s for J2229+1457. The band 6 (1.3mm) receivers were set to cover the frequency range of the redshifted [\[C[ii]{}\]]{} transition ($\nu_{\rm rest}$=1900.5369 GHz) and sample the dust continuum. There are four $\approx 2$GHz basebands, two pairs of adjacent bands with a $11$GHz gap in between. The channel width is 15.625MHz (17kms$^{-1}$). The data were initially processed by North American ALMA Regional Center staff with the [CASA]{} software package. On inspection of these data it became clear that the [\[C[ii]{}\]]{} line of J0055+0146 was located right at the edge of the baseband, 1000kms$^{-1}$ from the targeted frequency defined by the broad, low-ionization [Mg[ii]{}]{} emission line ($z_{\rm MgII} = 5.983$; @Willott:2010 2010a). The [Mg[ii]{}]{} line redshift is usually close to the systemic redshift as measured by narrow optical lines with a dispersion of 270kms$^{-1}$ [@Richards:2002]. A large offset for this quasar was not particularly surprising for two reasons: firstly the signal-to-noise (SNR) of the [Mg[ii]{}]{} detection is not very high and the line appears double-peaked due to noise and/or associated absorption; secondly the [Ly$\alpha$]{} redshift ($z_{\rm Ly \alpha} = 6.02$) is offset from [Mg[ii]{}]{} by 1600kms$^{-1}$ (in the same direction as the [\[C[ii]{}\]]{} offset) and this would make the size of the [Ly$\alpha$]{} ionized near-zone negative, which is not physically sensible for a quasar with such a high ionizing flux and has not been observed in a sample of 27 $z\approx 6$ quasars [@Carilli:2010]. Due to this redshift uncertainty the receiver basebands were set up so that the adjacent band covered the [Ly$\alpha$]{} redshift with zero gap between the two bands. The default ALMA Regional Center reduction excluded 11 channels at each end of the 128 channel band. However, only the first 4 channels need to be excluded, so we re-reduced the data with [CASA]{} to include more spectral channels at the baseband edges. We checked that the noise does not increase in these extra channels, except for the very first and last channels to contain data so we excluded those. In summary our reduced product contains 118 of the original 128 channels per baseband, compared with 106 channels in the default reduction. ![ALMA spectrum of J0055+0146 covering two adjacent basebands. The gap between the bands with no data is shaded in gray. The higher frequency band is centred on the redshift determined from the [Mg[ii]{}]{} emission line whereas the lower frequency band covers the [Ly$\alpha$]{} redshift. The [\[C[ii]{}\]]{} line is found at the edge of the higher frequency band. The blue curve is a Gaussian plus continuum fit as described in the text. The red circle marks the continuum level independently measured in the three line-free basebands. The upper axis is the velocity offset from the best-fit [\[C[ii]{}\]]{} Gaussian peak.[]{data-label="fig:linespecj0055"}](fig1.pdf) Results ======= Figure \[fig:linespecj0055\] shows the reduced spectrum of J0055+0146 from the two adjacent basebands. The final gap between the bands is only $\approx$ 150 kms$^{-1}$ and crucially the peak of the [\[C[ii]{}\]]{} line is contained within the higher frequency band. The lower frequency band contains only a small amount of the line flux but provides an important constraint on the wings and hence the peak and width for a symmetric line. A single Gaussian plus flat continuum model was fit to the available data using a Markov-Chain Monte Carlo (MCMC). This process shows a good fit for a single Gaussian with FWHM=$359 \pm 27$ kms$^{-1}$. The formal uncertainty in the FWHM is very small considering that there is some missing data. This is because a symmetric line model is used and with the peak and wings covered by data there is little margin for deviation in the missing channels. We add in quadrature an extra 10% uncertainty in both the line flux and FWHM due to the missing channels. ![ALMA spectrum of J2229+1457 covering the single baseband containing the [\[C[ii]{}\]]{} line. The blue curve is a Gaussian plus continuum fit and the red circle the independent continuum level. The upper axis is the velocity offset from the best-fit [\[C[ii]{}\]]{} Gaussian peak.[]{data-label="fig:linespecj2229"}](fig2.pdf) The spectrum of J2229+1457 is plotted in Figure \[fig:linespecj2229\]. For this quasar the [\[C[ii]{}\]]{} line is centred in the band with no significant offset from the [Mg[ii]{}]{} redshift. The line is consistent with a single Gaussian with a best fit FWHM=$351 \pm 39$ kms$^{-1}$, similar to the value for J0055+0146. The continuum level of the fit is again consistent with the independent continuum level determined from the three line-free basebands. Measurements from the spectra are given in Table \[tab:data\]. ![The color scale shows the integrated [\[C[ii]{}\]]{} line maps for the two quasars. White contours of 1.2mm continuum emission from the three line-free basebands are over-plotted at levels of 2,4,6$\sigma$beam$^{-1}$. The quasar optical positions are shown with a black plus symbol. The positional offsets between the optical and millimeter are most likely due to astrometric mismatch, rather than a physical offset. J0055+0146 is well-detected in both continuum and line emission. J2229+1457 has only a $2\sigma$ continuum detection that is spatially co-incident with the line emission. The restoring beam is shown in yellow in the lower-left corner.[]{data-label="fig:contlinemaps"}](fig3.pdf) Figure \[fig:contlinemaps\] shows maps of the 1.2mm continuum (white contours) plus the [\[C[ii]{}\]]{} line (color scale) for the quasars. For both quasars there is no significant offset between any of the continuum or line centroids or the optical quasar position. The $< 1{^{\prime\prime}}$ mm-optical offset is within the relative uncertainty of the optical astrometry. The more accurate [\[C[ii]{}\]]{} positions for the two quasars are 00:55:02.92 +01.46.17.80 and 22:29:01.66 +14.57.08.30. For J2229+1457 there is only a marginal $2\sigma$ detection of the continuum located coincident with the peak of the line emission. As seen in Figure \[fig:contlinemaps\] there are several other continuum peaks of this magnitude or greater in the vicinity, so it is not considered a secure detection, however the measured flux and uncertainty are included in Table \[tab:data\]. The quasar 1.2mm continuum flux-densities were converted to far-infrared luminosity, $L_{\rm FIR}$, assuming a typical SED for high-redshift star-forming galaxies. As in we adopt a greybody spectrum with dust temperature, $T_{\rm d} =47$K and emissivity index, $\beta=1.6$. To convert from far-IR luminosity to star formation rate we use the relation SFR $(M_\odot\,{\rm yr}^{-1})=1.5\times10^{-10}L_{\rm FIR}\, (L_\odot)$ appropriate for a Chabrier IMF [@Carilli:2013]. We note that this assumes that all the dust contributing to the 1.2mm continuum is heated by hot stars and not by the quasar. An alternative estimate of the star formation rate comes from the [\[C[ii]{}\]]{} luminosity. We adopt the relation in @Sargsyan:2014 of SFR $(M_\odot\,{\rm yr}^{-1}) = 1.0\times10^{-7}L_{\rm [CII]} \, (L_\odot)$. For the remainder of this paper, uncertainties on $L_{\rm FIR}$ (and inferred SFR) only include the flux measurement uncertainties, not that of the dust temperature and luminosity to SFR conversion. [lll]{} & CFHQSJ0055+0146 & CFHQSJ2229+1457\ $z_{\rm MgII}\,^{\rm a}$ & $5.983 \pm 0.004 $ & $6.152 \pm 0.003 $\ $z_{\rm [CII]}$ & $6.0060 \pm 0.0008$ & $6.1517 \pm 0.0005$\ FWHM$_{\rm [CII]}$ & $359 \pm 45$ kms$^{-1} $& $351 \pm 39$ kms$^{-1} $\ $I _{\rm [CII]} ~($Jykms$^{-1}) $ & $0.839 \pm 0.132$ & $0.582 \pm 0.075$\ $L_{\rm [CII]} ~(L_\odot)$ & $(8.27 \pm 1.30) \times 10^8$ & $(5.96 \pm 0.77) \times 10^8$\ $f_{\rm 1.2mm}\ (\mu$Jy) & $211 \pm 34$ & $54 \pm 29$\ $L_{\rm FIR} ~ (L_\odot)$ & $(4.85 \pm 0.78) \times 10^{11}$ & $(1.24 \pm 0.67) \times 10^{11} $\ SFR$_{\rm [CII]}\,(M_\odot\,{\rm yr}^{-1})$ & $83 \pm 13$ & $60 \pm 8$\ SFR$_{\rm FIR}\,(M_\odot\,{\rm yr}^{-1})$ & $73 \pm 12$ & $19 \pm 10$\ $L_{\rm [CII]} / L_{\rm FIR}$ & $(1.70 \pm 0.38) \times 10^{-3}$ & $(4.80 \pm 2.67) \times 10^{-3}$\ [Notes.]{}—\ $^{\rm a}$ Derived from [Mg[ii]{}]{} $\lambda2799$ observations [@Willott:2010].\ Uncertainties in $L_{\rm FIR}$, SFR$_{\rm [CII]}$ and SFR$_{\rm FIR}$ only include measurement uncertainties, not the uncertainties in extrapolating from a monochromatic to integrated luminosity or that of the luminosity-SFR calibrations. The synthesized beam sizes are $0\farcs63$ by $0\farcs45$ for J0055+0146 and $0\farcs76$ by $0\farcs64$ for J2229+1457. The better resolution for J0055+0146 is mostly due to higher elevation of observation. We used the [CASA IMFIT]{} task to fit 2D gaussian models to these maps. For J0055 both the continuum and line are resolved with deconvolved source sizes of $0\farcs51 \pm 0\farcs13$ by $0\farcs35 \pm 0\farcs26$ at position angle 87 degrees and $0\farcs50 \pm 0\farcs14$ by $0\farcs18 \pm 0\farcs27$ at position angle 62 degrees, respectively. At the distance to this quasar the spatial extent of $0.5{^{\prime\prime}}$ is equal to a linear size of 2.9kpc. We note that the missing data in the red wing of the [\[C[ii]{}\]]{} line may cause a bias in the size and inclination if the emission comes from a rotating disk, but there is no evidence for this based on the similarity of the line and continuum sizes. For J2229+1457 the continuum is too poorly detected to attempt a size measurement and the line emission is only marginally more extended than the beam size. In several other $z\approx 6$ quasars velocity gradients across the sources are observed (; @Wang:2013 2013). Velocity gradients are not seen for either of these two quasars, although for J0055+0146 the missing data for 150kms$^{-1}$ of the red wing hampers our ability to detect such a gradient. Discussion ========== Evolution of far-IR luminosity ------------------------------ In we reported on the low far-IR luminosities of the two previously observed CFHQS quasars and implications for the relatively low SFR of these quasar host galaxies relative to the black hole accretion rate. We now revisit this issue with the sample of four $z\approx 6$ CFHQS quasars with ALMA observations. We note that this sample includes four of the six CFHQS quasars with measured black hole masses within the absolute magnitude range $-25.5<M_{1450}<-24$ at a declination low enough for ALMA observation. The two unobserved quasars have $7\times10^8<M_{\rm BH}<10^9 M_\odot$, and were not observed due to limited time available and the desire to study the lowest mass black holes from CFHQS. Therefore there is a slight bias to low black hole mass in this sample compared to pure UV-luminosity-selection. Two of the four quasars are well detected in the continuum with fluxes of $211 \pm 34$ (J0055+0146) and $120 \pm 35$ (J0210-0456) $\mu$Jy. J2229 has a marginal $2\sigma$ detection of $54 \pm 29\,\mu$Jy (Figure \[fig:contlinemaps\] and Table \[tab:data\]) and J2329-0301 is undetected with a $1\sigma$ rms of 30$\mu$Jy. We combine the four values of far-infrared luminosity derived from these measurements assuming that J2329-0301 has a flux equal to its $2\sigma$ upper limit of $60\,\mu$Jy. The mean and standard deviation of the sample is $L_{\rm FIR} = (2.6 \pm 1.4) \times 10^{11}\,L_\odot$. We note this is much lower than the values of $10^{12} - 10^{13} \,L_\odot$ typically discussed for $z\approx 6$ quasars due to two factors, firstly that the CFHQS sample here have lower AGN luminosity than most known $z\approx 6$ quasars and a correlation between AGN and far-IR luminosities is present [@Wang:2011; @Omont:2013], but also that our small sample is selected on quasar rest-frame UV luminosity and black hole mass, whereas previous studies have focussed on quasars with pre-ALMA millimeter continuum detections. The implication is that these quasars have very high black hole accretion rates as inferred from the AGN bolometric luminosity, $L_{\rm Bol}$, yet relatively low SFR. Such a scenario is consistent with the well-known evolutionary model whereby the optical quasar phase comes after the main star forming phase [@Khandai:2012; @Lapi:2014], possibly due to quasar feedback inhibiting gas cooling and star formation. The measured ratio of $L_{\rm FIR} / L_{\rm Bol}=0.035$ for the four CFHQS quasars is only found in the optical quasar phase of co-evolution at a time $\sim 1$Gyr after the onset of activity for the $z=2$ model of @Lapi:2014. Given that this is the age of the universe at $z=6$ the evolution must occur more rapidly at higher redshift. However, the effect of AGN variability may also be important leading to a selection effect whereby AGN luminosity-selected objects are observed to have lower ratios of $L_{\rm FIR} / L_{\rm Bol}$ than the time-averaged values [@Hickox:2014; @Veale:2014]. We have previously shown that the two $z=6.4$ CFHQS quasars observed with ALMA in Cycle 0 have $L_{\rm FIR}$ lower than quasars of similar AGN luminosity at lower redshift. On the other hand, at fixed AGN luminosity $L_{\rm FIR}$ is observed to rise from $z=0$ to $z=3$ [@Serjeant:2010; @Bonfield:2011; @Rosario:2012; @Rosario:2013]. We next analyze the evolution of $L_{\rm FIR}$ using our expanded ALMA sample at $z=6$ and comparable low-redshift data. At all redshifts we determine the mean $L_{\rm FIR}$ for optically-selected quasars and X-ray AGN in a narrow range of $L_{\rm Bol}$ corresponding to the mean $L_{\rm Bol}$ of the four CFHQS $z \approx 6$ quasars in this paper ($L_{\rm Bol} \sim 7 \times 10^{12}\,L_\odot$). ![Stacked mean far-infrared luminosity for samples of quasars at different redshifts. Details of the samples are described in the text, but all samples are selected to include roughly the same range in bolometric luminosity centred on $L_{\rm Bol} \sim 7 \times 10^{12}\,L_\odot$, the mean $L_{\rm Bol}$ of the four $z\approx 6$ CFHQS quasars plotted with the blue square. There is a clear rise in $L_{\rm FIR}$ up to a peak at $2<z<3$ followed by a decline to $z=6$. The magenta curve shows the mean $L_{\rm FIR}$ due to star formation for the model of @Veale:2014 including scaling by a factor of 2 to account for stellar mass loss.[]{data-label="fig:lfirevol"}](fig4.pdf) At $z<3$ we use three datasets based on imaging of AGN. @Serjeant:2010 stacked SPIRE data of optically-selected quasars and quoted their results as rest-frame 100$\mu$m luminosity. We adopt $L_{\rm FIR}= 1.43\,\nu L_{\nu} (100\,\mu{\rm m})$ [@Chary:2001] to convert to far-infrared luminosity. The absolute magnitude bin $-26<I_{\rm AB}<-24$ corresponds well to $L_{\rm Bol} \sim 7 \times 10^{12}\,L_\odot$ and we have trimmed the size of the highest redshift bin from $2<z<4$ to $2<z<2.7$ because inspection of the luminosity-redshift plane figure in @Serjeant:2010 shows all but one of the 52 quasars in this bin are at $z<2.7$. @Rosario:2013 analyzed the PACS data for optically-selected quasars in COSMOS. Due to the shorter wavelength of PACS than SPIRE they presented results in rest-frame 60$\mu$m luminosity. We adopt $L_{\rm FIR}= 1.5\,\nu L_{\nu} (60\,\mu{\rm m})$ [@Chary:2001] to convert to far-infrared luminosity. From this study we use only the highest redshift, highest luminosity bin as this compares well with $L_{\rm Bol} \sim 7 \times 10^{12}\,L_\odot$. @Rosario:2012 determined the mean infrared luminosity with PACS for X-ray-selected AGN from the COSMOS survey. We note that the X-ray selected AGN sample contains a mixture of broad-line, narrow-line and lineless AGN and these may have different evolutionary properties, but @Rosario:2013 showed that the mean $L_{\rm FIR}$ of quasars and X-ray-selected are similar at a given AGN luminosity and redshift. We use the @Rosario:2012 data from the AGN luminosity bin $8 \times 10^{11}< L_{\rm Bol}< 2 \times 10^{13}\,L_\odot$. Whilst most sources in this bin have $L_{\rm Bol}< 3 \times 10^{12}\,L_\odot$, we consider the results appropriate to compare to the $z\approx 6$ quasars as the correlation between $L_{\rm FIR}$ and $L_{\rm Bol}$ is very shallow at this luminosity in @Rosario:2012. Figure \[fig:lfirevol\] plots data from these three low-redshift studies with the CFHQS ALMA bin at $6<z<6.5$. The three low-redshift studies show a rise in $L_{\rm FIR}$ of a factor of 4 from $z=0.3$ to $z=2.4$. This rise is attributed to the general increase in massive galaxy specific star formation rate over this redshift range [@Hickox:2014]. The $6<z<6.5$ bin has a large dispersion due to the range in 1.2mm continuum flux measured for the 4 quasars. The mean $L_{\rm FIR}$ at $z \approx 6$ is a factor of about 2 lower than the $z=0.3$ bin and 6 lower than the $z=2.4$ peak at the so-called [*quasar epoch*]{}. There is clear evidence here for a turnaround that mimics the evolution of the quasar luminosity function [@McGreer:2013] and star formation rate density [@Bouwens:2014], albeit with a much less steep high-redshift decline due to the fact we are measuring star formation in special locations within the universe where dark matter halos must have collapsed much earlier than typical in order to build up the observed black hole masses of $\sim 10^8\,M_\odot$. What is the physical reason for this turnaround at $z>3$? In the evolutionary picture where the optical quasar phase follows the starburst phase one would expect the star formation and black hole accretion to be more tightly coupled at high-redshift where there is barely enough time for star formation to have decreased substantially. A clue may come from one of the few differences between quasars at these two epochs. @Willott:2010 showed that the Eddington ratios of matched quasar luminosity samples at $z=2$ and $z=6$ are significantly different with the $z=6$ quasars having a factor of 3$\times$ higher Eddington ratios and therefore 3$\times$ lower black hole masses than at $z=2$. Such a difference exists between the typical black hole mass of our CFHQS ALMA sample and that of the highest luminosity bin of @Rosario:2013. This Eddington ratio evolution is observed in other studies [@De-Rosa:2011; @Trakhtenbrot:2011; @Shen:2012] and predicted by many theoretical works due to the increase in gas supply to black holes at high-redshift [@Sijacki:2014]. In Figure \[fig:lfirevol\] we also plot a theoretical curve of mean $L_{\rm FIR}$ versus redshift for a simulated sample of rest-frame UV-selected quasars in the same $L_{\rm Bol}$ range as the observed quasar samples for the model of @Veale:2014. This model assumes an evolving linear relationship between star formation and black hole growth. The variant of the model plotted here is the “accretion” model where the quasar luminosity is proportional to the black hole growth rate and the Eddington ratio distribution is a truncated power-law with slope $\beta=0.6$ (dashed curve in Figure 8 of @Veale:2014 2014). The model is constrained by the observed evolving quasar luminosity function and the local ratio of black hole to galaxy mass. We have scaled this model with a factor of $2\times$ increase in $L_{\rm FIR}$ to account for stellar mass loss. As seen in Figure \[fig:lfirevol\] this curve increases from low redshift to the peak quasar epoch at $2<z<3$ by about the same factor as the data, although the total normalization of the curve is lower by a factor of 3 to 4. @Veale:2014 discuss some of the reasons why the normalization may be lower than the observations. The decrease in $L_{\rm FIR}$ with increasing cosmic time from $z=2$ to $z=0$ for fixed luminosity quasars is due to the fact that such quasars are rarer at lower redshift and on the steep end of the luminosity function where scatter is more important. This behaviour also follows from the general decrease in specific star formation rate with cosmic time. The high-redshift behaviour of a decline from $z=3$ to $z=6$ matches our observations, so it is instructive to understand why this occurs in the model. It is due to the assumed $(1+z)^2$ evolution of the ratio of accretion growth to stellar mass growth, but this assumed evolution is also degenerate with evolution in the Eddington ratio. As discussed previously there is observational evidence for positive evolution in the Eddington ratio from $z=2$ to $z=6$, meaning that the ratio of accretion growth to stellar mass growth may change more gradually than $(1+z)^2$ . A possible alternative explanation for the low $L_{\rm FIR}$ at $z=6$ is that at these early epochs insufficient dust has been generated so that star formation occurs more often within lower dust environments [@Ouchi:2013; @Tan:2013; @Fisher:2014; @Ota:2014]. In this case there could be a much smaller decline in the typical SFR of a luminous quasar hosting galaxy. However, two lines of evidence point towards this not being the main factor for our quasar sample. First, quasars at $z=6$ are known to have emission line ratios similar to lower redshift quasars inferring high metallicity at least close to the accreting black hole [@Freudling:2003]. Second, the [\[C[ii]{}\]]{}luminosities in three of the four quasars are high (see below), suggesting high carbon abundances throughout the host galaxies. The [\[C[ii]{}\]]{} – far-IR luminosity relation ------------------------------------------------ Three of the four CFHQS ALMA quasars are detected in both [\[C[ii]{}\]]{} line and 1.2mm continuum emission (two new detections in this paper and J0210$-$0456 in ). With the low $L_{\rm FIR}$ discussed in the previous section, these quasar host galaxies probe a new regime in $L_{\rm FIR}$ at high-redshift. In Figure \[fig:lciilfir\] we plot the ratio of [\[C[ii]{}\]]{} to far-IR luminosity as a function of $L_{\rm FIR}$. Also plotted are several samples from the literature which, due to ALMA at high-redshift and at low-redshift, are rapidly increasing in size and data quality. The low-redshift $z<0.4$ sample of galaxies is from @Gracia-Carpio:2011 (2011 and in prep.) and contains a mix of normal galaxies, starbursts and ultra-luminous infrared galaxies (ULIRGs), some of which contain AGN. The ULIRGs show a [*[\[C[ii]{}\]]{} deficit*]{} that has been widely discussed in the literature as due to possible factors including AGN contamination of $L_{\rm FIR}$ [@Sargsyan:2012], high gas fractions [@Gracia-Carpio:2011] or the dustiness, temperature and/or density of star forming regions [@Farrah:2013; @Magdis:2014]. Previous observations of high $L_{\rm FIR}$ $z>5$ SDSS quasars [@Maiolino:2005; @Wang:2013] showed a similar deficit. However many $0.5<z<5$ ULIRGs do not show this deficit and have $L_{\rm [CII]} / L_{\rm FIR}$ ratios comparable to low-redshift star-forming galaxies [@Stacey:2010]. This is visible in Figure \[fig:lciilfir\] for the $0.5<z<5$ compilation of @De-Looze:2014. ![The ratio of [\[C[ii]{}\]]{} to far-IR luminosity as a function of far-IR luminosity. High-redshift ($z>5$) sources, mostly quasar host galaxies, are identified with large symbols. Error bars are only plotted for the CFHQS ALMA sources to enhance the clarity of the figure. The solid and dotted blue lines show the best fit power-law and $1\sigma$ uncertainty for the $z>5$ sources. The $z>5$ relation is largely consistent with the distribution of data at lower redshift. []{data-label="fig:lciilfir"}](fig5.pdf) By adding three $z>6$ quasars with $10^{11}<L_{\rm FIR}<10^{12}\,{\rm L}_\odot$ to Figure \[fig:lciilfir\] we have greatly expanded the range of luminosities at the highest redshift. Large symbols on Figure \[fig:lciilfir\] identify $z>5$ sources. The three $z>5$ @De-Looze:2014 sources are HLSJ091828.6+514223 at $z=5.24$ [@Rawle:2014] , HFLS3 at $z=6.34$ [@Riechers:2013] and SDSSJ1148+5251 at $z=6.42$ [@Maiolino:2005]. We note that these sources were mostly selected for followup based on high $L_{\rm FIR} $. The quasar ULASJ1120+0641 at $z=7.1$ has a more moderate $L_{\rm FIR}$ and $L_{\rm [CII]} / L_{\rm FIR}$ ratio and lies in between the CFHQS and high $L_{\rm FIR}$ objects on the plot. Although we are wary of the selection effects in Figure \[fig:lciilfir\] and the as yet unknown cause for the change in $L_{\rm [CII]} / L_{\rm FIR}$ with $L_{\rm FIR}$ at high-redshift, the new data provide the opportunity to make the first measurement of the slope of this relation at $z>5$. We fit the 12 $z>5$ sources with a single power-law model of the dependence of $L_{\rm FIR}$ on $L_{\rm [CII]}$ incorporating the observational (but not systematic) uncertainties using a MCMC procedure. The best-fit relation is $$\log_{10} L_{\rm FIR}=0.59+1.27 \log_{10} L_{\rm [CII]}.$$ Therefore the $L_{\rm [CII]} / L_{\rm FIR}$ ratio line plotted in Figure \[fig:lciilfir\] has a logarithmic slope of $(1/1.27)-1=-0.21$. The MCMC $1\sigma$ uncertainties, based solely on the observational data, are plotted as dotted lines. These also favor a shallow negative slope, not nearly as steep as the slope that would be fit to the previous $z>0.5$ data that covers only a narrow range of $L_{\rm FIR}$. The $L_{\rm [CII]} / L_{\rm FIR}$ ratios of the CFHQS and ULAS $z>6$ quasars are not greatly different to those of similar $L_{\rm FIR}$ galaxies at low-redshift. @Wang:2013 note that the low $L_{\rm [CII]} / L_{\rm FIR}$ ratios for SDSS $z>5$ quasars may be at least in part due to AGN contamination of the far-IR emission. Future observations at higher spatial resolution will be critical to examine differences in the spatial distribution of the line and continuum emission (e.g. @Cicone:2014 2014). We expect to observe that the dust continuum is more compact than the [\[C[ii]{}\]]{} line in the high $L_{\rm FIR}$ $z>5$ quasars, due to either more centrally-concentrated starbursts with higher dust temperatures, like local ULIRGs, or AGN dust-heating. In contrast we expect the low $L_{\rm FIR}$ $z>5$ quasars have star formation spread more evenly throughout their host galaxies, with similar spatial distribution of line and continuum emission. The $z\approx 6$ $M_{\rm BH}-\sigma$ and $M_{\rm BH}-M_{\rm dyn}$ relationships ------------------------------------------------------------------------------- The combination of black hole mass estimates and [\[C[ii]{}\]]{} line host galaxy dynamics for these $z\approx 6$ quasars allows us to investigate the black hole - galaxy mass correlation at an early epoch in the universe. The evolution of this relationship is a critical constraint on the co-evolution (or not) of galaxies and their nuclear black holes. As discussed in the Introduction there are reasons to believe that past studies using only the most massive black holes from SDSS quasars (e.g. @Wang:2010 2010) were prone to a bias where one would expect the black holes to be relatively more massive than the galaxies [@Willott:2005b; @Lauer:2007], as observed. With new data on $M_{\rm BH} \sim 10^8\,M_\odot$ quasars we are able to test this hypothesis and determine any real offset from the local relationship. Additionally, most previous work in this area has used the molecular CO line to trace the gas dynamics. CO is usually more centrally concentrated than [\[C[ii]{}\]]{}, so [\[C[ii]{}\]]{} potentially probes a larger fraction of the total mass (although we note that @Wang:2013 2013 found similar dynamical masses using CO and [\[C[ii]{}\]]{} for their $z\approx 6$ quasars). In addition to the CFHQS and [@Wang:2013] quasars we add to our study two other $z>6$ quasars observed in the [\[C[ii]{}\]]{} line: SDSSJ1148+5251 at $z=6.42$ and the most distant known quasar, ULASJ1120+0641 at $z=7.08$. Both these quasars have [Mg[ii]{}]{}-derived black hole masses [@De-Rosa:2011; @De-Rosa:2014] with very low measurement uncertainties. The black hole masses for all three CFHQS quasars also come from [Mg[ii]{}]{} measurements [@Willott:2010]. Some of these spectra are of moderate SNR and have substantial measurement uncertainties on the black hole masses. To all the quasars with [Mg[ii]{}]{}-derived black hole masses we add a 0.3 dex uncertainty to the measurement uncertainties to account for the dispersion in the reverberation-mapped quasar calibration [@Shen:2008]. None of the [@Wang:2013] quasars have [Mg[ii]{}]{} measurements so black hole masses are estimated assuming that the quasars radiate at the Eddington limit, as observed for most $z\approx 6$ quasars [@Jiang:2007; @Kurk:2007; @Willott:2010; @De-Rosa:2011]. The dispersion in the lognormal Eddington ratio distribution at $z \approx 6$ is 0.3 dex [@Willott:2010]. We add 0.3 dex uncertainty from the observed dispersion in the Eddington ratio distribution in quadrature to the 0.3 dex due to the dispersion in the reverberation-mapped quasar calibration for a total uncertainty on the [@Wang:2013] quasar black hole masses of 0.45 dex. First we consider the $M_{\rm BH}-\sigma$ relationship. For nearby galaxies $\sigma$ is the velocity dispersion of the galaxy bulge. At high-redshift bulges are less common [@Cassata:2011] and we do not expect the $z\approx 6$ kinematics to match that of a pressure supported bulge. With the limited spatial resolution of current data we cannot be sure the [\[C[ii]{}\]]{} gas is distributed in a rotating disk, although there is evidence of this for some sources [@Wang:2013]. [@Ho:2007a] discusses the relationship and calibration of bulge velocity dispersion and disk circular velocity and concludes that although there is additional scatter one can relate molecular or atomic gas in a disk to stellar bulges. The major complication is the inclination of the disk. For a random sample of inclinations this can be modelled, however there is a possibility that quasars have disks oriented more often face-on, reducing the line-of-sight velocity dispersion [@Ho:2007]. [lccccl]{} Name & $z_{\rm [CII]}$ & $M_{\rm BH} ~(M_\odot)\,^{\rm a}$ & $\sigma ^{\rm b}$ & $M_{\rm dyn}~(M_\odot)$ & Refs.$^{\rm c}$\ CFHQSJ0055+0146 & $6.0060 \pm 0.0008$ & $(2.4^{+2.6}_{-1.4})\times 10^{8} $ & $207 \pm 45$ & $4.2\times 10^{10} $ & 1,2\ CFHQSJ0210$-$0456 & $6.4323 \pm 0.0005$ & $(0.8^{+1.0}_{-0.6})\times 10^{8} $ & $98 \pm 20$ & $1.3\times 10^{10} $ & 1,2,3\ CFHQSJ2229+1457 & $6.1517 \pm 0.0005$ & $(1.2^{+1.4}_{-0.8})\times 10^{8}$ & $241 \pm 51$ & $4.4\times 10^{10} $ & 1,2\ SDSSJ0129$-$0035 & $5.7787 \pm 0.0001$ & $(1.7^{+3.1}_{-1.1})\times 10^{8} $ & $112 \pm 21$ & $1.3\times 10^{10} $ & 4\ SDSSJ1044$-$0125 & $5.7847 \pm 0.0007$ & $(1.1^{+1.9}_{-0.7})\times 10^{10} $ & $291 \pm 76$ & — $^{\rm d}$ & 4\ SDSSJ1148+5251 & $6.4189 \pm 0.0006$ & $(4.9^{+4.9}_{-2.5})\times 10^{9} $ & $186 \pm 38$ & $1.8\times 10^{10} $ & 5,6,7\ SDSSJ2054$-$0005 & $6.0391 \pm 0.0001$ & $(0.9^{+1.6}_{-0.6})\times 10^{9} $ & $364 \pm 67$ & $7.2\times 10^{10} $ & 4\ SDSSJ2310+1855 & $6.0031 \pm 0.0002$ &$(2.8^{+5.1}_{-1.8})\times 10^{9} $ & $325 \pm 61$ & $9.6 \times 10^{10} $ & 4\ ULASJ1120+0641 & $7.0842 \pm 0.0004$ & $(2.4^{+2.4}_{-1.2})\times 10^{9} $ & $ 144 \pm 34$ & $2.4\times 10^{10} $ & 8,9\ ULASJ1319+0950 & $6.1330 \pm 0.0007$ &$(2.1^{+3.8}_{-1.4})\times 10^{9} $ & $381 \pm 91$ & $12.5\times 10^{10} $ & 4\ [Notes.]{}—\ $^{\rm a}$ Derived from [Mg[ii]{}]{} $\lambda2799$ observations if possible, else from Eddington luminosity assumption. Uncertainties include observational errors plus systematics based on calibrations.\ $^{\rm b}$ Derived from Gaussian FWHM fit to [\[C[ii]{}\]]{} spectrum using method of [@Ho:2007] including an inclination correction (see text for individual inclinations assumed). Uncertainties include observational errors plus systematics based on calibrations.\ $^{\rm c}$ References: (1) This paper, (2) @Willott:2010, (3) @Willott:2013, (4) @Wang:2013, (5) @Maiolino:2005, (6) @Walter:2009, (7) @De-Rosa:2011, (8) @Venemans:2012, (9) @De-Rosa:2014.\ $^{\rm d}$ This quasar does not have a dynamical mass calculation in @Wang:2013 due to the difference in the [\[C[ii]{}\]]{} and CO line profiles. We determine $\sigma$ using the method of [@Ho:2007], specifically setting the [\[C[ii]{}\]]{} line full-width at 20% equal to 1.5$\times$ the FWHM as expected for a Gaussian since most of the lines are approximately Gaussian. The [\[C[ii]{}\]]{} emitting gas is assumed to be in an inclined disk where the inclination angle, $i$, is determined by the ratio of minor ($a_{\rm min}$) and major ($a_{\rm maj}$) axes, $i=\cos^{-1}( a_{\rm min}/a_{\rm maj})$. The circular velocity is therefore $v_{\rm cir}=0.75 \,$FWHM$_{\rm [CII]} / \sin i$. For all of the quasars in this study we determine an inclination from the [\[C[ii]{}\]]{}data or assume an inclination if one or both the major and minor axes are unresolved. All the quasars in [@Wang:2013] were spatially resolved, although some had quite large uncertainties on $a_{\rm min}$ and $a_{\rm maj}$. We adopt the inclination angles from their paper. For the [\[C[ii]{}\]]{} emission of J0210$-$0456, $a_{\rm maj} =0\farcs52 \pm 0\farcs25$ (2.9kpc) with $i=64^{\circ}$ . For J0055+0146, $a_{\rm maj} =0\farcs50 \pm 0\farcs14$ (2.9kpc) with $i=69^{\circ}$ (Section 3, assuming no bias from missing red wing data). The [\[C[ii]{}\]]{} emission of J2229+1457 is only marginally spatially resolved ($0\farcs85$ versus beam size of $0\farcs76$) and we estimate an intrinsic FWHM of $\approx 0\farcs4$ (2.4kpc). An inclination of $i=55^{\circ}$ is assumed as this is the median inclination angle for the resolved sources in this paper and [@Wang:2013]. Neither SDSSJ1148+5251 nor ULASJ1120+0641 have published inclination angles, so we also assume $i=55^{\circ}$ for both of them. We adopt FWHM$_{\rm [CII]} =287 \pm 28$kms$^{-1}$ for SDSSJ1148+5251 [@Walter:2009] and FWHM$_{\rm [CII]} =235 \pm 35$kms$^{-1}$ for ULASJ1120+0641 [@Venemans:2012] . Values of black hole masses and $\sigma$ for this sample are provided in Table \[tab:masses\]. Figure \[fig:mbhsig\] shows the $M_{\rm BH} - \sigma$ relationship for the $z\approx 6$ quasar sample. Uncertainties on black hole masses include the scatter in the calibration as described previously. Uncertainties in $\sigma$ include FWHM measurement uncertainty plus a 10% uncertainty for the conversion from FWHM to $v_{\rm cir}$ and 15% for the conversion from $v_{\rm cir}$ to $\sigma$ as seen in the sample of [@Ho:2007a]. The black line is the local correlation of [@Kormendy:2013] with the gray band the $\pm 1\sigma$ scatter. The first thing to note is that the quasars are distributed around the local relationship rather than all being offset to low $\sigma$ as is commonly believed to be the case. As noted by [@Wang:2010], using the method of [@Ho:2007], rather than calculating $\sigma$ as FWHM$/2.35$, leads to much higher $\sigma$. Note that this is without adopting extreme face-on inclinations for most quasars. There are still several quasars, such as SDSSJ1148+5251 and ULASJ1120+0641, that have values of $\sigma$ considerably lower than the local relation. The main result of Figure \[fig:mbhsig\] is that whilst there is little mean shift between the $z=0$ and $z\approx 6$ data, there is a much larger scatter in the data at $z\approx 6$, well beyond the size of the error bars. This larger scatter at an early epoch is expected based on dynamical evolution, incoherence in AGN/starburst activity and the tightening of the relation over time from merging [@Peng:2007]. We note that our hypothesis that the bias described in the Introduction would lead to the lower $M_{\rm BH}$ quasars being located on the local relation with a lower scatter than the high $M_{\rm BH}$ quasars is not supported by these observations. The scatter in $\log_{10} \sigma$ at $M_{\rm BH}\approx 10^8\,M_\odot$ is about the same as that at $M_{\rm BH}> 10^9\,M_\odot$ We go one step further from $\sigma$ to determine dynamical masses using the deconvolved [\[C[ii]{}\]]{} sizes. For consistency, we follow the method of [@Wang:2013]. The dynamical mass within the disk radius is given by $M_{\rm dyn} \approx 1.16\times 10^5\, v_{\rm cir}^2 \,D \,M_\odot$ where $D$ is the disk diameter in kpc and calculated as $1.5 \times$ the deconvolved Gaussian spatial FWHM. The resulting dynamical masses are given in Table \[tab:masses\]. We note that there is considerable uncertainty on these values due to the unknown spatial and velocity structure of the gas, the marginal spatial resolution and limited sensitivity that means we may be missing more extended gas. Due to the these uncertainties we do not place formal error bars on the dynamical masses, following [@Wang:2013]. Higher resolution data in the future are required to confirm the derived masses. ![Black hole mass versus velocity dispersion calculated from the [\[C[ii]{}\]]{} line using the method of [@Ho:2007] for $z\approx 6$ quasars. Quasars from the CFHQS are shown as blue squares and the other symbols show quasars from the SDSS and ULAS surveys. The black line with gray shading is the local correlation $\pm 1\sigma$ scatter of black hole mass and bulge velocity dispersion [@Kormendy:2013]. The $z\approx 6$ quasars are distributed around the the local relationship, but with a much larger scatter and some quasars with significantly lower $\sigma$ for their $M_{\rm BH}$.[]{data-label="fig:mbhsig"}](fig6.pdf) ![Black hole mass versus host galaxy dynamical mass for $z\approx 6$ quasars. Symbols as for Figure \[fig:mbhsig\]. The black line with gray shading is the local correlation $\pm 1\sigma$ scatter from the work of [@Kormendy:2013] equating $M_{\rm dyn}$ to $M_{\rm bulge}$. The CFHQS quasars lie on the local relationship and do not show the large offset displayed by the most massive black holes. Uncertainties in $M_{\rm dyn}$ have not been calculated due to the reasons given in the text.[]{data-label="fig:mbhmdyn"}](fig7.pdf) SDSSJ1148+5251 has been extensively studied in [\[C[ii]{}\]]{} [@Maiolino:2005; @Walter:2009; @Maiolino:2012; @Cicone:2014]. The highest resolution observations by [@Walter:2009] revealed a very compact circumnuclear starburst with radius 0.75kpc and FWHM$_{\rm [CII]} =287 \pm 28$kms$^{-1}$. For an assumed inclination of $i=55^{\circ}$ this gives $M_{\rm dyn}=1.8 \times 10^{10}\,M_\odot$. For comparison [@Walter:2004] determined a dynamical mass from CO emission in this quasar of $5.5\times 10^{10}\,M_\odot$ within a larger radius of 2.5kpc, the larger radius being the main difference between the results. Recent observations have shown more complex [\[C[ii]{}\]]{} emission including evidence for gas extended over tens of kpc and at high velocities indicative of outflow [@Maiolino:2012; @Cicone:2014]. We adopt $M_{\rm dyn}=1.8 \times 10^{10}\,M_\odot$ for SDSSJ1148+5251 noting that the true value could be several times larger. The most distant known quasar, ULASJ1120+0641 at $z=7.08$, has been well detected in [\[C[ii]{}\]]{}, although not yet spatially resolved [@Venemans:2012]. Based on the published FWHM$_{\rm [CII]} =235 \pm 35$kms$^{-1}$ and assuming a spatial FWHM of 3kpc (similar to the other quasars resolved by ALMA) and $i=55^{\circ}$ we determine $M_{\rm dyn}=2.4 \times 10^{10}\,M_\odot$. In Figure \[fig:mbhmdyn\] we plot black hole mass versus galaxy dynamical mass for the most distant known quasars. The black line and gray shading represent the local correlation of $M_{\rm BH}$ with bulge mass $M_{\rm bulge}$ [@Kormendy:2013]. In the absence of gas accretion and mergers the present stellar bulge mass represents the sum of the gas and stellar mass at high-redshift, so it is a good comparison for the dynamical mass within the central few kpc. [@Kormendy:2013] note that their correlation (their equation 10) gives a black hole to bulge mass ratio of 0.5% at $M_{\rm bulge}=10^{11}\,M_\odot$ that is 2 to 4 times higher than previous estimates due to the omission of pseudobulges, galaxies with uncertain $M_{\rm BH}$ and ongoing mergers. The position of the high-$z$ data with respect to low redshift is fairly similar to Figure \[fig:mbhsig\], not surprising because $v_{\rm cir}$ derived from the [\[C[ii]{}\]]{} velocity FWHM is a major factor in both $\sigma$ and $M_{\rm dyn}$. The points are shifted somewhat further from the local bulge mass than for the local velocity dispersion. This shift is due to the smaller size of galaxies at high-redshift, as the size is the only term in the derivation of dynamical mass not in $\sigma$. We note the much greater dynamic range in black hole mass (2 dex) than in dynamical mass (1 dex) in our sample. This is likely due more to our selection over a wide range of quasar luminosity than to a non-linear relationship between these quantities at $z=6$. All three of the CFHQS quasars lie within the local $1\sigma$ scatter and the one $M_{\rm BH}\sim 10^8\,M_\odot$ quasar in [@Wang:2013] is only a factor of 4 greater than the local relationship. In contrast the $M_{\rm BH}> 10^9\,M_\odot$ quasars tend to show a larger scatter and larger offset above the local relationship as previously found [@Walter:2004; @Wang:2010; @Venemans:2012; @Wang:2013]. We caution that there are considerable uncertainties in some of these measurements as already discussed, but in dynamical mass the results look more like we would expect based on the quasar selection bias effect. Conclusions =========== During ALMA Early Science cycles 0 and 1 we have observed a complete sample of four $z>6$ moderate luminosity CFHQS quasars with black hole masses $\sim 10^8\,M_\odot$. Three of the four are detected in both far-IR continuum and the [\[C[ii]{}\]]{} emission line. The far-IR luminosity is found to be substantially lower than that of similar luminosity quasars at $1<z<3$. Assuming that far-IR luminosity traces star formation equally effectively at these redshifts this implies that at $z\approx 6$ quasars are growing their black holes more rapidly than their stellar mass compared to at the peak of the [*quasar epoch*]{} ($1<z<3$). The ratios of \[CII\] to far-IR luminosities for the CFHQS quasars lie in the range 0.001 to 0.01, similar to that of low-redshift galaxies at the same far-IR luminosity. This suggests a similar mode of star-formation spread throughout the host galaxy (rather than in dense circumnuclear starburst regions that have lower values for this ratio in local ULIRGs). Combining with previous $z>5.7$ quasar data at higher $L_{\rm FIR}$ we find that the far-IR luminosity dependence of the [\[C[ii]{}\]]{}/FIR ratio has a shallow negative slope, possibly due in part to an increase in $L_{\rm FIR}$ due to quasar-heated dust in some optically-luminous high-$z$ quasars. The three CFHQS quasars well-detected in the [\[C[ii]{}\]]{} emission line allow this atomic gas to be used as a tracer of the host galaxy dynamics. Combining with published data on higher black hole mass quasars we have investigated the $M_{\rm BH}-\sigma$ and $M_{\rm BH}-M_{\rm dyn}$ relations at $z\approx 6$. We show that the $z=6$ quasars display a $M_{\rm BH}-\sigma$ relation with similar slope and normalization to locally, but with much greater scatter. Similar results are obtained for the $M_{\rm BH}-M_{\rm dyn}$ relation with a somewhat higher normalization at $z=6$ and a higher scatter at high $M_{\rm BH}$. As discussed in Combining our results on the relatively low $L_{\rm FIR}$ for $ M_{\rm BH} \sim 10^8\,M_\odot$ $z\approx 6$ quasars with their location on the $M_{\rm BH}-\sigma$ relation leads to something of a paradox. The fact these quasars lie on the local $M_{\rm BH}-\sigma$ relation suggests that their host galaxies have undergone considerable evolution to acquire such a high dynamical mass. So why is it that this mass accumulation is not leading to a high star formation rate? As discussed in , simulations such as those of @Khandai:2012 and @Lapi:2014 predict that such low ratios of SFR to black hole accretion occur after episodes of strong feedback that inhibits star formation throughout quasar host galaxies. Another possibility mentioned in Section 4.1 is that $L_{\rm FIR}$ fails to trace star formation so effectively in these high-redshift galaxies, due to lower dust content (e.g. @Ouchi:2013 2013). Note that using $L_{\rm [CII]}$ as a star formation rate tracer instead of $L_{\rm FIR}$, would give higher SFR by a factor of three for one of the CFHQS quasars. Higher resolution follow-up [\[C[ii]{}\]]{} observations of these quasars are critical to measure more accurately the distribution and kinematics of the gas used as a dynamical tracer in order to reliably determine the location and scatter of the correlations between black holes and their host galaxies at high-redshift. Thanks to staff at the North America ALMA Regional Center for processing the ALMA data. Thanks to Melanie Veale for useful discussion and providing her models in electronic form. This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2012.1.00676.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. [*Facility:*]{} .

--- abstract: 'We give an explicit component Lagrangian construction of massive higher spin on-shell $N=1$ supermultiplets in four-dimensional Anti-de Sitter space $AdS_4$. We use a frame-like gauge invariant description of massive higher spin bosonic and fermionic fields. For the two types of the supermultiplets (with integer and half-integer superspins) each one containing two massive bosonic and two massive fermionic fields we derive the supertransformations leaving the sum of four their free Lagrangians invariant such that the algebra of these supertransformations is closed on-shell.' author: - | I.L. Buchbinder${}^{ab}$[^1], M.V. Khabarov${}^{cd}$[^2], T.V. Snegirev${}^{ae}$[^3], Yu.M. Zinoviev${}^{cd}$[^4]\ *[${}^a$Department of Theoretical Physics, Tomsk State Pedagogical University,]{}\ *[Tomsk, 634061, Russia]{}\ *[${}^b$National Research Tomsk State University, Tomsk 634050, Russia]{}\ *[${}^c$Institute for High Energy Physics of National Research Center “Kurchatov Institute”]{}\ *[Protvino, Moscow Region, 142281, Russia]{}\ *\ *[Dolgoprudny, Moscow Region, 141701, Russia]{}\ *[${}^e$National Research Tomsk Polytechnic University, Tomsk 634050, Russia]{}******** title: | Lagrangian formulation of the massive\ higher spin $N=1$ supermultiplets in $AdS_4$ space --- Introduction ============ The higher spin theory (see e.g. the reviews [@Vas04] [@Be04],[@BBS10]) has attracted significant interest for a long time and for many reasons. On the one hand, the theory of massless higher spin fields is a maximal extension of the Yang-Mills gauge theories and gravity including all spin fields. On the other hand, it is closely related to superstring theory which involves an infinite tower of higher spin massive fields. In principle, the higher spin field theory can provide the possibility to study some aspects of string theory in the framework of the field theory. It is also worth pointing out that the construction of Lagrangian formulations for the higher spin field models is extremely interesting itself since it allows to reveal the new unexpected properties to relativistic field theory in general. Beginning with work [@FV] it became clear that the nonlinear massless higher spin theory can only be realized in $AdS$ space with non-zero curvature. This raises the interest in studying the various aspects of field theory in $AdS$ space in the context of $AdS/CFT$-correspondence. Taking into account that the low-energy limit of superstring theory should lead to supersymmetric field theory we face the problem of constructing the supersymmetric massive higher spin models in the $AdS$ space. It is expected that the supersymmetry can be an essential ingredient of the consistent theory of all the fundamental interactions including quantum gravity. It is possible that such a theory should also involve the massless and/or massive higher spin fields. This paper is devoted to developing the $N=1$ supersymmetric Lagrangian formulation of free massive higher spin models in $AdS$ space in the framework of on-shell component formalism. In supersymmetric theories the massless or massive fields are combined into the corresponding supermultiplets. In the case of free field models containing the different spin fields, it is natural to expect that the Lagrangian should be the sum of the Lagrangians for each concrete spin field. To provide an explicit Lagrangian realization of the free supermultiplet one has to find supertransformations leaving the free Lagrangians invariant and show that the algebra of these supertransformations is closed at least on-shell. In the case of the $N=1$ supersymmetry the massless higher superspin-$s$ supermultiplets consist of the two massless fields with spins $(s,s+1/2)$. The task of constructing supertransformations for such supermultiplets in four dimensional flat space was completely solved in the metric-like formulation [@Cur79] and soon in the frame-like one [@Vas80]. In both cases the supertransformations have a simple enough structure and are determined uniquely by the invariance of the sum of the Lagrangians for two free massless fields with spins $s$ and $s+1/2$. Note that such a requirement allows to find only on-shell supersymmetry when supertransformations are closed on the equations of motion. In order to find off-shell supertransformations, it is necessary to introduce the corresponding auxiliary fields. A natural procedure to construct off-shell $N=1$ supersymmetric Lagrangian models is realized in terms of $N=1$ superspace and superfields (see e.g. [@BK98]), where all the auxiliary fields providing closure of the superalgebra are automatically obtained. In the framework of superfield formulation the $N=1$ supersymmetric massless higher spin models were constructed in the pioneer papers [@KSP93; @KS93]. Later, on the basis of these results, $N=1$ supersymmetric massless higher spin models were generalized for $AdS_4$ space [@KS94], [@GKS][^5]. In both cases the constructed superfield models, after eliminating the auxiliary fields, reduce to the sum of spin-$s$ and spin-$(s+1/2)$ (Fang)-Fronsdal Lagrangians [@Frons78; @FF78] in flat or $AdS$ spaces. The generalization for ${\cal N}=2$ massless higher spin supermultiplets was given in [@GKS], [@GKS96a]. There are much fewer results in the case of supersymmetric massive higher spin models even in the on-shell formalism, the reason being that when moving from the massless component formulation to the massive one very complicated higher derivative corrections must be introduced to the supertransformations. Moreover the higher the spin of the fields entering a supermultiplet the higher the number of derivatives one has to consider. The problem of the supersymmetric description of the massive higher spin supermultiplet was only explicitly resolved in 2007 for the case of $N=1$ on-shell 4D Poincare superalgebra [@Zin07a] (see also [@Zin02; @Zin07; @OT16]) using the gauge invariant formulation for the massive higher spin fields [@KZ97; @Zin01; @Met06][^6]. In such a formalism the description for the massive field is obtained in terms of the appropriately chosen set of the massless ones. It is assumed that the Lagrangian for massive higher spin supermultiplets is constructed as a sum of the corresponding Lagrangians for massless fields deformed by massive terms. However, it appeared [@Zin07a] that to realize such a program one has to use massless supermultiplets containing four fields $(k-1/2,k,k',k+1/2)$ as the building blocks, where two bosonic fields with equal spins have opposite parities, and this prevents us from separating them into the usual massless pairs. In [@Zin07a] it was shown that to obtain the massive deformation it is enough to add the non-derivative corrections to the supertransformations for the fermions only. Complicated higher derivative corrections to the supertransformations reappear when one tries to fix all local gauge symmetries, breaking the gauge invariance. Note however that in such construction the mass-like terms for the fermions in the Lagrangian take a complicated non-diagonal form making calculations rather cumbersome. Surprisingly however, in 4D the above results remain the main results in the massive supersymmetric higher spin theory until now[^7]. The aim of this paper is to extend and generalize the results of [@Zin07a] to the case of four dimensional $N=1$ $AdS_4$ superalgebra. We use the gauge invariant description of the massive higher spin bosonic and fermionic fields but in the frame-like version [@Zin08b; @PV10]. Recall that one of the attractive features of such a formalism is that it works nicely both in flat Minkowski space as well as in $(A)dS$ spaces. Our strategy differs from that of [@Zin07a]. For the Lagrangian we take just the sum of four free Lagrangians for the two massive bosonic and two massive fermionic fields entering the supermultiplet. Then for each pair of bosonic and fermionic fields (we call it superblock in what follows) we find the supertransformations leaving the sum of their two Lagrangians invariant. Next we combine all four possible superblocks and adjust their parameters so that the algebra of the supertransformations is closed on-shell. The paper is organized as follows. In section \[Section1\] we give all necessary descriptions of the frame-like formulation of massless bosonic and fermionic higher spin fields and alos we present the massless higher spin supermultiplets in $AdS_4$ in such a formalism. Massless models given in this section will serve as the building blocks for our construction of the massive higher spin models. In section \[Section2\] we give frame-like gauge invariant formulations for free massive arbitrary integer and half-inter spins. In section \[Section3\] we consider massive superblocks containing one massive bosonic and one massive fermionic field and find corresponding supertransformations. In section \[Section4\] we combine the constructed massive superblocks into one massive supermultiplet. [**Notations and conventions.**]{} In this work we use a technique of $p$-forms taking the values in the Grassmann algebra. The main geometrical objects are some $p$-forms $\Omega$ (p=0,1,2,3,4). They are defined as $$\Omega=dx^{\mu_1}\wedge...\wedge dx^{\mu_p}\Omega_{\mu_1...\mu_p},$$ where $\Omega_{\mu_1...\mu_p}$ is the antisymmetric tensor field. In particular, the partial derivative is defined as one-form $d=dx^\mu\partial_\mu$. In 4D it is convenient to use a frame-like multispinor formalism where all the Lorentz objects have local totally symmetric dotted and undotted spinorial indices (see a description of irreducible representations of 4D Lorentz group in terms of dotted and undotted spinors e.g. in [@BK98]). To simplify the expressions we will use the condensed notations for them such that e.g. $$\Omega^{\alpha(m)\dot\alpha(n)} = \Omega^{(\alpha_1\alpha_2 \dots \alpha_{m})(\dot\alpha_1\dot\alpha_2 \dots \dot\alpha_{n})}$$ We also always assume that spinor indices denoted by the same letters and placed on the same level are symmetrized, e.g. $$\Omega^{\alpha(m)\dot\alpha(n)} \zeta^{\dot\alpha} = \Omega^{\alpha(m)(\dot\alpha_1\dots \dot\alpha_{n}} \zeta^{\dot\alpha_{n+1})}$$ We work in the $AdS_4$ space which is described by pair 1-forms: background frame $e^{\alpha\dot\alpha}$ which enters explicitly in all constructions and background Lorentz connections $\omega^{\alpha\beta},\omega^{\dot\alpha\dot\beta}$ which are hidden in the 1-form covariant derivative $$D\Omega^{\alpha(m)\dot\alpha(n)}=d\Omega^{\alpha(m)\dot\alpha(n)} +\omega^\alpha{}_{\beta}\wedge\Omega^{\alpha(m-1)\beta\dot\alpha(n)} +\omega^{\dot\alpha}{}_{\dot\beta}\wedge\Omega^{\alpha(m)\beta\dot\alpha(n-1)\dot\beta}.$$ The covariant derivative satisfies the following normalization conditions: $$D\wedge D \Omega^{\alpha(m)\dot\alpha(n)} = - 2\lambda^2(E^\alpha{}_\beta\wedge\Omega^{\alpha(m-1)\beta\dot\alpha(n)}+E^{\dot\alpha}{}_{\dot\beta} \wedge\Omega^{\alpha(m)\dot\alpha(n-1)\dot\beta})$$ where $E^{\alpha\beta},E^{\dot\alpha\dot\beta}$ are basis elements of $2$-form spaces and defined below as the double product of $e^{\alpha\dot\alpha}$. Basis elements of $1,2,3,4$-form spaces are: $$e^{\alpha\dot\alpha},\quad E_2{}^{\alpha\beta},\quad E_2{}^{\dot\alpha\dot\beta},\quad E_3{}^{\alpha\dot\alpha},\quad E_4$$ They are defined as follows: $$\begin{aligned} e^{\alpha\dot\alpha}\wedge e^{\beta\dot\beta} &=& \varepsilon^{\alpha\beta} E^{\dot\alpha\dot\beta} + \varepsilon^{\dot\alpha\dot\beta }E^{\alpha\beta} \\ E^{\alpha\beta}\wedge e^{\gamma\dot\alpha} &=& \varepsilon^{\alpha\gamma} E^{\beta\dot\alpha}+ \varepsilon^{\beta\gamma} E^{\alpha\dot\alpha} \\ E^{\dot\alpha\dot\beta}\wedge e^{\alpha\dot\gamma} &=& -\varepsilon^{\dot\alpha\dot\gamma} E^{\alpha\dot\beta} -\varepsilon^{\dot\beta\dot\gamma} E^{\alpha\dot\alpha} \\ E^{\alpha\dot\alpha}\wedge e^{\beta\dot\beta} &=& \varepsilon^{\alpha\beta} \varepsilon^{\dot\alpha\dot\beta}E\end{aligned}$$ so the Hermitian conjugation laws look like $$(e^{\alpha\dot\alpha})^\dag=e^{\alpha\dot\alpha},\quad (E^{\alpha(2)})^\dag=E^{\dot\alpha(2)},\quad (E^{\alpha\dot\alpha})^\dag=-E^{\alpha\dot\alpha},\quad E^\dag=-E$$ We also write some useful relations for these basis elements $$e^{\alpha}{}_{\dot\beta}\wedge e^{\beta\dot\beta} = 2 E^{\alpha\beta},\qquad e_\beta{}^{\dot\alpha}\wedge e^{\beta\dot\beta} = 2 E^{\dot\alpha\dot\beta}$$ $$E^{\alpha}{}_\gamma \wedge e^{\gamma\dot\alpha} = 3 E^{\alpha\dot\alpha},\qquad E^{\dot\alpha}{}_{\dot\gamma}\wedge e^{\alpha\dot\gamma} = - 3E^{\alpha\dot\alpha}$$ $$E_\beta{}^{\dot\alpha}\wedge e^{\beta\dot\beta} = 2 \varepsilon^{\dot\alpha\dot\beta}E, \qquad E^{\alpha}{}_{\dot\beta} \wedge e^{\beta\dot\beta} = 2\varepsilon^{\alpha\beta} E$$ $$E^{\alpha\beta}\wedge E^{\dot\alpha\dot\beta} = 0,\qquad E^{\alpha}{}_\gamma \wedge E^{\beta\gamma} = 3 \varepsilon^{\alpha\beta} E$$ The spinor indices are raised and lowered with the help of the antisymmetric Lorentz invariant tensors $\epsilon^{\alpha\beta}, \epsilon^{\dot{\alpha}\dot{\beta}}$ and inverse $\epsilon_{\alpha\beta}, \epsilon_{\dot{\alpha}\dot{\beta}}$ respectively. All the products of $p$ - forms are understood in the sense of wedge-products. Henceforth the sign of wedge product $\wedge$ will be omitted. Massless higher spin models {#Section1} =========================== In this section we provide all necessary description of the massless bosonic and fermionic higher spin fields as well as massless higher spin supermultiplets in the frame-like multispinor formalism used in this work. In what follows they will serve as building blocks for our construction for massive supermultiplets. Integer spin $k$ ---------------- In the frame-like formulation a massless spin-$k$ field ($k\geq2$) is described by the physical one-form $f^{\alpha(k-1)\dot\alpha(k-1)}$ and the auxiliary one-forms $\Omega^{\alpha(k)\dot\alpha(k-2)},\Omega^{\alpha(k-2)\dot\alpha(k)}$, being the higher spin generalization of the frame and Lorentz connection in the frame-like formulation of gravity. Locally they are two-component multispinors symmetric on their dotted and undotted spinorial indices separately. These fields satisfy the following reality condition $$\label{RealCond1} (f^{\alpha(k-1)\dot\alpha(k-1)})^\dag = f^{\alpha(k-1)\dot\alpha(k-1)},\qquad (\Omega^{\alpha(k)\dot\alpha(k-2)})^\dag = \Omega^{\alpha(k-2)\dot\alpha(k)}$$ The free Lagrangian (a four-form in our formalism) for the massless bosonic field in the four-dimensional $AdS_4$ space looks like this: $$\begin{aligned} \label{MaslBosonLag} \frac{(-1)^k}{i} {\cal L}_k &=& k \Omega^{\alpha(k-1)\beta\dot\alpha(k-2)} E_\beta{}^\gamma \Omega_{\alpha(k-1)\gamma\dot\alpha(k-2)} - (k-2) \Omega^{\alpha(k)\dot\alpha(k-3)\dot\beta} E_{\dot\beta}{}^{\dot\gamma} \Omega_{\alpha(k)\dot\alpha(k-3)\dot\gamma} \nonumber \\ && + 2 \Omega^{\alpha(k-1)\beta\dot\alpha(k-2)} e_\beta{}^{\dot\beta} Df_{\alpha(k-1)\dot\alpha(k-2)\dot\beta} \nonumber \\ && + 2k\lambda^2 f^{\alpha(k-2)\beta\dot\alpha(k-1)} E_\beta{}^\gamma f_{\alpha(k-2)\gamma\dot\alpha(k-1)} + h.c.\end{aligned}$$ Here and henceforth $h.c.$ means hermitian conjugate terms defined by rules (\[RealCond1\]). This Lagrangian is invariant under the following gauge transformations $$\begin{aligned} \label{MaslGTB} \delta \Omega^{\alpha(k)\dot\alpha(k-2)} &=& D \eta^{\alpha(k),\dot\alpha(k-2)} + e_\beta{}^{\dot\alpha}\zeta^{\alpha(k)\beta\dot\alpha(k-3)} + \lambda e^\alpha{}_{\dot\beta} \xi^{\alpha(k-1)\dot\alpha(k-2)\dot\beta}\nonumber \\ \delta \Omega^{\alpha(k-2)\dot\alpha(k)} &=& D \eta^{\alpha(k-2),\dot\alpha(k)} + e^\alpha{}_{\dot\beta}\zeta^{\alpha(k-3)\dot\alpha(k)\dot\beta} + \lambda e_\beta{}^{\dot\alpha} \xi^{\alpha(k-2)\beta\dot\alpha(k-1)} \nonumber \\ \delta f^{\alpha(k-1)\dot\alpha(k-1)} &=& D\xi^{\alpha(k-1)\dot\alpha(k-1)} + e_\beta{}^{\dot\alpha}\eta^{\alpha(k-1)\beta\dot\alpha(k-2)} + e^\alpha{}_{\dot\beta}\eta^{\alpha(k-2)\dot\alpha(k-1)\dot\beta}\end{aligned}$$ where zero-forms $\xi^{\alpha(k-1)\dot\alpha(k-1)}$ and $\eta^{\alpha(k),\dot\alpha(k-2)}$ are the gauge parameters for the gauge fields $f^{\alpha(k-1)\dot\alpha(k-1)}$ and $\Omega^{\alpha(k),\dot\alpha(k-2)}$. The additional gauge parameter $\zeta^{\alpha(k+1)\dot\alpha(k-3)}$ leads to the introduction of the so-called extra field $\Upsilon^{\alpha(k+1)\dot\alpha(k-3)} + h.c.$ which in turn requires introduction of next extra symmetries and so on. The procedure stops at $$\begin{aligned} \label{ExtMaslGTB} \Upsilon^{\alpha(k+t)\dot\alpha(k-t-2)},\quad \Upsilon^{\alpha(k-t-2)\dot\alpha(k+t)},\qquad 1\leq t\leq k-2\end{aligned}$$ These extra gauge fields do not enter the free Lagrangian but play an important role in non-linear higher spin theory. One of the nice features of the frame-like formulation is that for all fields (physical, auxiliary and extra ones) one can construct a gauge invariant two-form (curvature) generalizing curvature and torsion for gravity. For the physical and auxiliary fields they have the form $$\begin{aligned} \label{MaslBosonCurv} {\cal R}^{\alpha(k)\dot\alpha(k-2)} &=& D \Omega^{\alpha(k),\dot\alpha(k-2)} + \lambda e^\alpha{}_{\dot\beta} f^{\alpha(k-1)\dot\alpha(k-2)\dot\beta} + e_\beta{}^{\dot\alpha} \Upsilon^{\alpha(k)\beta\dot\alpha(k-3)}\nonumber \\ {\cal R}^{\alpha(k-2)\dot\alpha(k)} &=& D \Omega^{\alpha(k-2),\dot\alpha(k)} + \lambda e_\beta{}^{\dot\alpha}f^{\alpha(k-2)\beta\dot\alpha(k-1)} + e^\alpha{}_{\dot\beta}\Upsilon^{\alpha(k-3)\dot\alpha(k)\dot\beta} \nonumber \\ {\cal T}^{\alpha(k-1)\dot\alpha(k-1)} &=& D f^{\alpha(k-1)\dot\alpha(k-1)} + e_\beta{}^{\dot\alpha}\Omega^{\alpha(k-1)\beta\dot\alpha(k-2)} + e^\alpha{}_{\dot\beta}\Omega^{\alpha(k-2)\dot\alpha(k-1)\dot\beta}\end{aligned}$$ In our construction for the massless and massive supermultiplets we use only the physical and auxiliary fields working in the so-called 1 and 1/2 order formalism which is very well known in supergravity. Namely, we do not consider any variations of the auxiliary fields but all calculations are done using the “zero torsion condition”: $$\begin{aligned} \label{MaslOn-Shell} {\cal T}^{\alpha(k-1)\dot\alpha(k-1)} \approx 0 \quad\Rightarrow\quad e_\beta{}^{\dot\alpha} {\cal R}^{\alpha(k-1)\beta\dot\alpha(k-2)} + e^\alpha{}_{\dot\beta}{\cal R}^{\alpha(k-2)\dot\alpha(k-1)\dot\beta}\approx 0\end{aligned}$$ At the same time the variation of the Lagrangian (\[MaslBosonLag\]) under the arbitrary variations of the physical fields can be written in the following simple form $$\begin{aligned} \label{MaslBosonVar} (-1)^k\delta{\cal L}_k&=&-i2{\cal R}^{\alpha(k-1)\beta\dot\alpha(k-2)}e_\beta{}^{\dot\beta} \delta f_{\alpha(k-1)\dot\alpha(k-2)\dot\beta}+h.c.\end{aligned}$$ One can see that in (\[MaslOn-Shell\]) and (\[MaslBosonVar\]), the curvatures ${\cal R}$ enter in such combinations that extra gauge field $\Upsilon$ is dropped out, therefore below they will be omitted. Half-integer spin $k+1/2$ ------------------------- In the frame-like formulation, the massless spin-$(k+1/2)$ field ($k\geq1$) is described by physical 1-forms $\Phi^{\alpha(k)\dot\alpha(k-1)},\Phi^{\alpha(k-1)\dot\alpha(k)}$. As in the bosonic case, these two-component multispinors are symmetric on their dotted and undotted spinorial indices separately and satisfy the reality condition $$(\Phi^{\alpha(k)\dot\alpha(k-1)})^\dag = \Phi^{\alpha(k-1)\dot\alpha(k)}$$ The free Lagrangian for such fields in $AdS_4$ space looks like this: $$\begin{aligned} \label{MaslFermLag} (-1)^k{\cal L}_{k+\frac12} &=& \Psi_{\alpha(k-1)\beta\dot\alpha(k-1)}e^\beta{}_{\dot\beta} D\Psi^{\alpha(k-1)\dot\alpha(k-1)\dot\beta}\nonumber \\ && + d_{k-1}[(k+1) \Psi_{\alpha(k-1)\beta\dot\alpha(k-1)} E^\beta{}_{\gamma} \Psi^{\alpha(k-1)\gamma\dot\alpha(k-1)} \nonumber \\ && - (k-1)\Psi_{\alpha(k)\dot\alpha(k-2)\dot\beta} E^{\dot\beta}{}_{\dot\gamma} \Psi^{\alpha(k)\dot\alpha(k-2)\dot\gamma}+ h.c.]\end{aligned}$$ and is invariant under the following gauge transformation $$\begin{aligned} \label{MaslFermGT} \delta \Psi^{\alpha(k)\dot\alpha(k-1)} &=& D \xi^{\alpha(k)\dot\alpha(k-1)} + e_\beta{}^{\dot\alpha} \eta^{\alpha(k)\beta\dot\alpha(k-2)} + 2d_{k-1} e^\alpha{}_{\dot\beta}\xi^{\alpha(k-1)\dot\alpha(k-1)\dot\beta} \nonumber \\ \delta \Psi^{\alpha(k-1)\dot\alpha(k)} &=& D \xi^{\alpha(k-1)\dot\alpha(k)} + e^\alpha{}_{\dot\beta}\eta^{\alpha(k-2)\dot\alpha(k)\dot\beta} + 2d_{k-1} e_\beta{}^{\dot\alpha} \xi^{\alpha(k-1)\beta\dot\alpha(k-1)}\end{aligned}$$ where $$d_{k-1} = \pm \frac{\lambda}{2}$$ Note that the transformations with the gauge parameters $\eta^{\alpha(k+1)\dot\alpha(k-2)},\eta^{\alpha(k-2)\dot\alpha(k+1)}$ lead to the introduction of extra fields that play the same role as in the bosonic case and do not enter the free Lagrangian. Up to these extra fields the gauge invariant curvatures for the physical fermionic fields have the following form $$\begin{aligned} \label{MaslFermCurv} {\cal F}^{\alpha(k)\dot\alpha(k-1)} &=& D \Psi^{\alpha(k)\dot\alpha(k-1)} + 2d_{k-1} e^\alpha{}_{\dot\beta}\Psi^{\alpha(k-1)\dot\alpha(k-1)\dot\beta} \nonumber \\ {\cal F}^{\alpha(k-1)\dot\alpha(k)} &=& D \Psi^{\alpha(k-1)\dot\alpha(k)} + 2d_{k-1} e_\beta{}^{\dot\alpha}\Psi^{\alpha(k-1)\beta\dot\alpha(k-1)}\end{aligned}$$ The variation of the free Lagrangian (\[MaslFermLag\]) under the arbitrary variations of the physical fields can be written as $$\begin{aligned} \label{MaslFermVar} (-1)^k\delta{\cal L}_{k+\frac12}&=&-{\cal F}_{\alpha(k-1)\beta\dot\alpha(k-1)}e^\beta{}_{\dot\beta} \delta\Psi^{\alpha(k-1)\dot\alpha(k-1)\dot\beta}+h.c.\end{aligned}$$ Here the curvature ${\cal F}$ also enters in such a combination with the frame $e$ that the extra fields drop out. In the following part of this section we combine massless higher spin fermionic and bosonic fields in the $N=1$ supermultiplet and construct an explicit form of the supertransformations leaving the sum of the free Lagrangians invariant. Supermultiplet $(k+1/2,k)$ -------------------------- This supermultiplet contains higher half-integer spin $k+1/2$ and integer spin $k$. They are described by $(\Phi^{\alpha(k)\dot\alpha(k-1)}, h.c.)$ and $(f^{\alpha(k-1)\dot\alpha(k-1)}, \Omega^{\alpha(k)\dot\alpha(k-2)},h.c.)$ respectively. We choose an ansatz for the supertransformations in the following form (as it was already mentioned, we consider supertransformations for the physical fields only): $$\begin{aligned} \label{MaslST1} \delta f^{\alpha(k-1)\dot\alpha(k-1)} &=& \alpha_{k-1}\Phi^{\alpha(k-1)\beta\dot\alpha(k-1)}\zeta_\beta - \bar\alpha_{k-1}\Phi^{\alpha(k-1)\dot\alpha(k-1)\dot\beta} \zeta_{\dot\beta} \nonumber \\ \delta \Phi^{\alpha(k)\dot\alpha(k-1)} &=& \beta_{k-1}\Omega^{\alpha(k)\dot\alpha(k-2)} \zeta^{\dot\alpha} + \gamma_{k-1} f^{\alpha(k-1)\dot\alpha(k-1)} \zeta^{\alpha} \nonumber \\ \delta \Phi^{\alpha(k-1)\dot\alpha(k)} &=& \bar\beta_{k-1}\Omega^{\alpha(k-2)\dot\alpha(k)}\zeta^{\alpha} + \bar\gamma_{k-1}f^{\alpha(k-1)\dot\alpha(k-1)} \zeta^{\dot\alpha}\end{aligned}$$ were we assume that the coefficients $\alpha_k,\beta_k,\gamma_k$ are complex. The parameters of the supertransformation $\zeta^\alpha, \zeta^{\dot\alpha}$ in $AdS_4$ satisfy the relations $$\label{ParamRealat} D \zeta^\alpha = - \lambda e^\alpha{}_{\dot\beta} \zeta^{\dot\beta}, \qquad D \zeta^{\dot\alpha} = - \lambda e_\beta{}^{\dot\alpha}\zeta^{\beta}$$ Using the expressions for Lagrangian variations (\[MaslBosonVar\]) and (\[MaslFermVar\]) as well as on-shell identity (\[MaslOn-Shell\]) the variation for the sum of the bosonic and fermionic Lagrangians can be written as follows: $$\begin{aligned} \label{MaslVar1} (-1)^k\delta({\cal L}_k+{\cal L}_{k+\frac12})&=&4i\alpha_{k-1}\Phi_{\alpha(k-2)\beta\gamma\dot\alpha(k-1)}e^\gamma{}_{\dot\gamma} {\cal R}^{\alpha(k-2)\dot\alpha(k-1)\dot\gamma}\zeta^{\beta}\nn\\ &&- (k-1)\bar\beta_{k-1}{\cal F}_{\alpha(k-1)\beta\dot\alpha(k-1)}e^\beta{}_{\dot\beta} \Omega^{\alpha(k-2)\dot\alpha(k-1)\dot\beta}\zeta^{\alpha}\nn \\ && -\bar\gamma_{k-1}({\cal F}_{\alpha(k-1)\beta\dot\alpha(k-1)}e^\beta{}_{\dot\beta} f^{\alpha(k-1)\dot\alpha(k-1)}\zeta^{\dot\beta}\nn\\ &&+ (k-1){\cal F}_{\alpha(k-1)\beta\dot\alpha(k-1)}e^\beta{}_{\dot\beta} f^{\alpha(k-1)\dot\alpha(k-2)\dot\beta}\zeta^{\dot\alpha})+h.c.\end{aligned}$$ Note that the invariance of the Lagrangian under the supertransformations can be achieved up to the total derivative only and this leads to a number of useful identities. For example, let us consider $$\begin{aligned} 0 &\approx& D[\Phi_{\alpha(k-1)\beta\dot\alpha(k-1)} e^\beta{}_{\dot\beta} \Omega^{\alpha(k-2)\dot\alpha(k-1)\dot\beta} \zeta^{\alpha}] \\ &=& D \Phi_{\alpha(k-1)\beta\dot\alpha(k-1)} e^\beta{}_{\dot\beta} \Omega^{\alpha(k-2)\dot\alpha(k-1)\dot\beta} \zeta^{\alpha} + \Phi_{\alpha(k-1)\beta\dot\alpha(k-1)} e^\beta{}_{\dot\beta} D \Omega^{\alpha(k-2)\dot\alpha(k-1)\dot\beta} \zeta^{\alpha} \\ && - \Phi_{\alpha(k-1)\beta\dot\alpha(k-1)} e^\beta{}_{\dot\beta} \Omega^{\alpha(k-2)\dot\alpha(k-1)\dot\beta} D \zeta^{\alpha}\end{aligned}$$ Using the explicit expressions for the bosonic (\[MaslBosonCurv\]) and fermionic (\[MaslFermCurv\]) curvatures as well as relation (\[ParamRealat\]) we obtain: $$\begin{aligned} 0 &=& {\cal F}_{\alpha(k-1)\beta\dot\alpha(k-1)} e^\beta{}_{\dot\beta} \Omega^{\alpha(k-2)\dot\alpha(k-1)\dot\beta} \zeta^{\alpha} + \Phi_{\alpha(k-2)\beta\gamma\dot\alpha(k-1)} e^\beta{}_{\dot\beta} {\cal R}^{\alpha(k-2)\dot\alpha(k-1)\dot\beta} \zeta^{\gamma} \\ && - 2{b_{k-1}} [(k+1)E^{\dot\beta}{}_{\dot\gamma} \Psi^{\alpha(k-2)\gamma\dot\alpha(k-1)\dot\gamma} \Omega_{\alpha(k-2)\dot\alpha(k-1)\dot\beta} \zeta_{\gamma} \\ && - (k-2) E_\beta{}^\alpha \Psi^{\alpha(k-3)\beta\gamma\dot\alpha(k)} \Omega_{\alpha(k-2)\dot\alpha(k)}\zeta_{\gamma} - E_\beta{}^\gamma \Psi^{\alpha(k-2)\beta\dot\alpha(k)} \Omega_{\alpha(k-2)\dot\alpha(k)} \zeta_{\gamma} ] \\ && - \lambda^2 [(k+1)E^\beta{}_\delta \Phi_{\alpha(k-2)\beta\gamma\dot\alpha(k-1)} f^{\alpha(k-2)\delta\dot\alpha(k-1)} - (k-1) E_{\dot\beta}{}^{\dot\alpha} \Phi_{\alpha(k-1)\gamma\dot\alpha(k-1)}f^{\alpha(k-1)\dot\alpha(k-2)\dot\beta} ]\zeta^{\gamma} \\ && + \lambda [E^{\alpha\beta} \Phi_{\alpha(k-1)\beta\dot\alpha(k-1)} \Omega^{\alpha(k-2)\dot\alpha(k-1)\dot\beta} \zeta_{\dot\beta}]\end{aligned}$$ Similarly, if one considers two relations: $$\begin{aligned} 0 &\approx& D[\Phi_{\alpha(k-1)\beta\dot\alpha(k-1)} e^\beta{}_{\dot\beta} f^{\alpha(k-1)\dot\alpha(k-1)} \zeta^{\dot\beta}] \\ 0 &\approx& D[\Phi_{\alpha(k-1)\beta\dot\alpha(k-2)\dot\gamma} e^\beta{}_{\dot\beta} f^{\alpha(k-1)\dot\alpha(k-2)\dot\beta}\zeta^{\dot\gamma}]\end{aligned}$$ then using the explicit expression for the fermionic curvature as well as zero torsion condition one obtains: $$\begin{aligned} 0 &=& {\cal F}_{\alpha(k-1)\beta\dot\alpha(k-1)} e^\beta{}_{\dot\beta} f^{\alpha(k-1)\dot\alpha(k-1)}\zeta^{\dot\beta} \\ && - 2{b_{k-1}} [(k+1)E^{\dot\beta}{}_{\dot\gamma} \Psi^{\alpha(k-1)\dot\alpha(k-1)\dot\gamma} f_{\alpha(k-1)\dot\alpha(k-1)} \zeta_{\dot\beta} \\ && - (k-1)E_\beta{}^\alpha \Psi^{\alpha(k-2)\beta\dot\alpha(k-1)\dot\beta} f_{\alpha(k-1)\dot\alpha(k-1)} \zeta_{\dot\beta} ] \\ && - [ (k-1)(- E_{\dot\beta}{}^{\dot\alpha} \Phi_{\alpha(k)\dot\alpha(k-1)} \Omega^{\alpha(k)\dot\alpha(k-2)} + E^\beta{}_\gamma \Phi_{\alpha(k-1)\beta\dot\alpha(k-2)\dot\beta} \Omega^{\alpha(k-1)\gamma\dot\alpha(k-2)}) \zeta^{\dot\beta} \\ && + (k-1)E^{\beta\alpha} \Phi_{\alpha(k-1)\beta\dot\alpha(k-1)} \Omega^{\alpha(k-2)\dot\alpha(k-1)\dot\gamma} \zeta_{\dot\gamma}] \\ && - 2\lambda E_\gamma{}^\beta \Phi_{\alpha(k-1)\beta\dot\alpha(k-1)} f^{\alpha(k-1)\dot\alpha(k-1)} \zeta^{\gamma}\end{aligned}$$ $$\begin{aligned} 0 &=& {\cal F}_{\alpha(k-1)\beta\dot\alpha(k-2)\dot\gamma} e^\beta{}_{\dot\beta} f^{\alpha(k-1)\dot\alpha(k-2)\dot\beta}\zeta^{\dot\gamma} \\ && - 2{b_{k-1}} [(k+1)E^{\dot\beta}{}_{\dot\delta} \Psi^{\alpha(k-1)\dot\alpha(k-2)\dot\gamma\dot\delta} f_{\alpha(k-1)\dot\alpha(k-2)\dot\beta} \zeta_{\dot\gamma} \\ && - (k-1)E_\beta{}^\alpha \Psi^{\alpha(k-2)\beta\dot\alpha(k-1)\dot\gamma} f_{\alpha(k-1)\dot\alpha(k-1)} \zeta_{\dot\gamma}] \\ && - [kE^\beta{}_\delta \Phi_{\alpha(k-1)\beta\dot\alpha(k-2)\dot\gamma} \Omega^{\alpha(k-1)\delta\dot\alpha(k-2)} -(k-2) E_{\dot\beta}{}^{\dot\alpha} \Phi_{\alpha(k)\dot\alpha(k-2)\dot\gamma} \Omega^{\alpha(k)\dot\alpha(k-3)\dot\beta} ]\zeta^{\dot\gamma} \\ && + \lambda[ E^{\dot\gamma}{}_{\dot\beta} \Phi_{\alpha(k-1)\gamma\dot\alpha(k-2)\dot\gamma} f^{\alpha(k-1)\dot\alpha(k-2)\dot\beta} \zeta^{\gamma} - E_\gamma{}^\beta \Phi_{\alpha(k-1)\beta\dot\alpha(k-1)} f^{\alpha(k-1)\dot\alpha(k-1)} \zeta^{\gamma}]\end{aligned}$$ Using these identities we obtain from the requirement $\delta({\cal L}_k+{\cal L}_{k+\frac12}) = 0$: $$\label{MaslSol1} \alpha_{k-1} = i\frac{(k-1)}{4} \bar\beta_{k-1}, \qquad \gamma_{k-1} = \lambda \bar\beta_{k-1}, \qquad 2d_{k-1}\bar\beta_{k-1} = \lambda \beta_{k-1}$$ The solution of the last relation depends on the sign of $d_{k-1}=\pm\frac{\lambda}{2}$. The parameter $\beta_{k}$ is real for the “+” sign and is imaginary for the “-”. These two solutions correspond to the parity-even and parity-odd bosonic fields entering the supermultiplet. This fact will be important for the construction of massive supermultiplets where two bosonic fields must have opposite parities. Supermultiplet $(k,k-1/2)$ -------------------------- This supermultiplet contains higher integer spin $k$ and half-integer spin $k-1/2$. They are described by $(f^{\alpha(k-1)\dot\alpha(k-1)},\Omega^{\alpha(k)\dot\alpha(k-2)},h.c.)$ and $(\Phi^{\alpha(k-1)\dot\alpha(k-2)}, h.c.)$ respectively. We choose an ansatz for the supertransformations in the following form $$\begin{aligned} \label{MaslST2} \delta f^{\alpha(k-1)\dot\alpha(k-1)} &=& \alpha'_{k-1}\Phi^{\alpha(k-1)\dot\alpha(k-2)} \zeta^{\dot\alpha} - \bar\alpha'_{k-1} \Phi^{\alpha(k-2)\dot\alpha(k-1)} \zeta^{\alpha} \nonumber \\ \delta \Psi^{\alpha(k-1)\dot\alpha(k-2)} &=& \beta'_{k-1}\Omega^{\alpha(k-1)\beta\dot\alpha(k-2)} \zeta_\beta + \gamma'_{k-1}f^{\alpha(k-1)\dot\alpha(k-2)\dot\beta} \zeta_{\dot\beta} \nonumber \\ \delta \Psi^{\alpha(k-2)\dot\alpha(k-1)} &=& \bar\beta'_{k-1}\Omega^{\alpha(k-2)\dot\alpha(k-1)\dot\beta} \zeta_{\dot\beta} + \bar\gamma'_{k-1} f^{\alpha(k-2)\beta\dot\alpha(k-1)} \zeta_\beta\end{aligned}$$ Here in the most general case, coefficients $\alpha'_k,\beta'_k,\gamma'_k$ are complex. Using the expressions for Lagrangian variations (\[MaslBosonVar\]) and (\[MaslFermVar\]) as well as on-shell relation (\[MaslOn-Shell\]) we get $$\begin{aligned} (-1)^k \delta ({\cal L}_k+{\cal L}_{k-\frac12}) &=& - 4i(k-1)\alpha'_{k-1} \Phi_{\alpha(k-2)\beta\dot\alpha(k-2)} e^\beta{}_{\dot\beta} {\cal R}^{\alpha(k-2)\dot\alpha(k-2)\dot\beta\dot\gamma} \zeta_{\dot\gamma} \\ && + \beta'^*_{k-1}{\cal F}_{\alpha(k-2)\gamma\dot\alpha(k-2)} e^\gamma{}_{\dot\gamma} \Omega^{\alpha(k-2)\dot\alpha(k-2)\dot\gamma\dot\beta} \zeta_{\dot\beta} \\ && + \gamma'^*_{k-1}{\cal F}_{\alpha(k-2)\gamma\dot\alpha(k-2)} e^\gamma{}_{\dot\gamma} f^{\alpha(k-2)\beta\dot\alpha(k-2)\dot\gamma} \zeta_\beta + h.c.\end{aligned}$$ As in previous case from two relations $$\begin{aligned} 0 &\approx& D [\Phi_{\alpha(k-2)\gamma\dot\alpha(k-2)} e^\gamma{}_{\dot\gamma} \Omega^{\alpha(k-2)\dot\alpha(k-2)\dot\gamma\dot\beta} \zeta_{\dot\beta}] \\ 0 &\approx& D [\Phi_{\alpha(k-2)\gamma\dot\alpha(k-2)} e^\gamma{}_{\dot\gamma} f^{\alpha(k-2)\beta\dot\alpha(k-2)\dot\gamma} \zeta_\beta]\end{aligned}$$ one can derive two identities: $$\begin{aligned} 0 &=& {\cal F}_{\alpha(k-2)\gamma\dot\alpha(k-2)} e^\gamma{}_{\dot\gamma} \Omega^{\alpha(k-2)\dot\alpha(k-2)\dot\gamma\dot\beta} \zeta_{\dot\beta} + \Phi_{\alpha(k-2)\gamma\dot\alpha(k-2)} e^\gamma{}_{\dot\gamma} {\cal R}^{\alpha(k-2)\dot\alpha(k-2)\dot\gamma\dot\beta} \zeta_{\dot\beta} \\ && + 2{b_{k-2}}[k E^{\dot\gamma}{}_{\dot\delta} \Psi^{\alpha(k-2)\dot\alpha(k-2)\dot\delta} \Omega_{\alpha(k-2)\dot\alpha(k-2)\dot\gamma\dot\beta} \zeta^{\dot\beta} \\ && - (k-2)E_\gamma{}^\beta \Psi^{\alpha(k-3)\gamma\dot\alpha(k-1)} \Omega_{\alpha(k-3)\beta\dot\alpha(k-1)\dot\beta} \zeta^{\dot\beta}] \\ && - \lambda^2 [- (k-2)E_{\dot\gamma}{}^{\dot\alpha} \Phi_{\alpha(k-1)\dot\alpha(k-2)} f^{\alpha(k-1)\dot\alpha(k-3)\dot\beta\dot\gamma} + (k+1) E^\gamma{}_\beta \Phi_{\alpha(k-2)\gamma\dot\alpha(k-2)} f^{\alpha(k-2)\beta\dot\alpha(k-2)\dot\beta}) \\ && - E_{\dot\gamma}{}^{\dot\beta} \Phi_{\alpha(k-1)\dot\alpha(k-2)} f^{\alpha(k-1)\dot\alpha(k-2)\dot\gamma}] \zeta_{\dot\beta} \\ && + \lambda [E_{\dot\beta\dot\gamma} \Phi_{\alpha(k-2)\gamma\dot\alpha(k-2)} \Omega^{\alpha(k-2)\dot\alpha(k-2)\dot\gamma\dot\beta} \zeta^\gamma]\end{aligned}$$ $$\begin{aligned} 0 &=& {\cal F}_{\alpha(k-2)\gamma\dot\alpha(k-2)} e^\gamma{}_{\dot\gamma} f^{\alpha(k-2)\beta\dot\alpha(k-2)\dot\gamma}\zeta_\beta \\ && + 2{b_{k-2}}[k E^{\dot\gamma}{}_{\dot\beta} \Psi^{\alpha(k-2)\dot\alpha(k-2)\dot\beta} f_{\alpha(k-2)\beta\dot\alpha(k-2)\dot\gamma} \zeta^\beta \\ && - (k-2)E_\gamma{}^\alpha \Psi^{\alpha(k-3)\gamma\dot\alpha(k-1)} f_{\alpha(k-2)\beta\dot\alpha(k-1)} \zeta^\beta] \\ && - [-(k-2) E_{\dot\gamma}{}^{\dot\alpha} \Phi_{\alpha(k-1)\dot\alpha(k-2)} \Omega^{\alpha(k-1)\beta\dot\alpha(k-3)\dot\gamma} \zeta_\beta \\ && + kE^\gamma{}_\delta \Phi_{\alpha(k-2)\gamma\dot\alpha(k-2)}\Omega^{\alpha(k-2)\beta\delta\dot\alpha(k-2)} \zeta_\beta + E_{\dot\gamma\dot\beta} \Phi_{\alpha(k-2)\gamma\dot\alpha(k-2)} \Omega^{\alpha(k-2)\dot\alpha(k-2)\dot\gamma\dot\beta} \zeta^\gamma] \\ && + \lambda[ E_{\dot\beta\dot\gamma} \Phi_{\alpha(k-1)\dot\alpha(k-2)} f^{\alpha(k-1)\dot\alpha(k-2)\dot\gamma} \zeta^{\dot\beta} + E_\beta{}^\gamma \Phi_{\alpha(k-2)\gamma\dot\alpha(k-2)} f^{\alpha(k-2)\beta\dot\alpha(k-2)\dot\gamma} \zeta_{\dot\gamma} ]\end{aligned}$$ Then the invariance of the Lagrangian under the supertransformations requires that $$\label{MaslSol2} \alpha'_{k-1} = \frac{i}{4(k-1)} \bar\beta'_{k-1}, \qquad \gamma'_{k-1} = \lambda \bar\beta'_{k-1}, \qquad 2d_{k-1} \bar\beta'_{k-1} = \lambda \beta'_{k-1}$$ As in the previous case, we see from last relation that parameter $\beta'_{k}$ can be real or imaginary. It depends on the sign of $d_{k-1}=\pm\frac{\lambda}{2}$ and is related to the parity of the fields entering the supermultiplet. Massive higher spin fields {#Section2} ========================== In this section we provide frame-like gauge invariant formulation for massive arbitrary integer and half-inter spins [@Zin08b] but with the multispinor formalism used for all local indices. Integer spin $s$ {#MasbBoson} ---------------- In the gauge invariant formalism a massive integer spin-$s$ field is described by a set of massless fields with spins $0\leq k\leq s$. Frame-like formulation of massless bosonic fields with spins $k\geq2$ were considered above, they are described by one-forms $(f^{\alpha(k-1)\dot\alpha(k-1)},\Omega^{\alpha(k)\dot\alpha(k-2)} + h.c.)$ while massless spin-$1$ is described by the physical one-form $A$ and auxiliary zero-forms $B^{\alpha(2)},B^{\dot\alpha(2)}$, and massless spin-$0$ is described by physical zero-form $\varphi$ and auxiliary zero-form $\pi^{\alpha\dot\alpha}$. The gauge invariant Lagrangian for the massive bosonic field has the form: $$\label{MasvBosonLag} {\cal L} = {\cal L}_{kin} + {\cal L}_{cross} + {\cal L}_{mass}$$ $$\begin{aligned} \frac{1}{i}{\cal L}_{kin} &=& \sum_{k=2}^{s}(-1)^k {\cal L}_k + 4 EB_{\alpha(2)} B^{\alpha(2)} +2 E_{\alpha(2)}B^{\alpha(2)} DA + h.c. \\ && - 6E \pi_{\alpha\dot\alpha} \pi^{\alpha\dot\alpha} -12 E_{\alpha\dot\alpha} \pi^{\alpha\dot\alpha} D\varphi \\ \frac{1}{i}{\cal L}_{cross} &=& \sum_{k=3}^{s}(-1)^{k+1}a_k[E_{\beta(2)} \Omega^{\alpha(k-2)\beta(2)\dot\alpha(k-2)} f_{\alpha(k-2)\dot\alpha(k-2)} \\ && + \frac{(k-2)}{k}E_{\beta(2)} f^{\alpha(k-3)\beta(2)\dot\alpha(k-1)} \Omega_{\alpha(k-3)\dot\alpha(k-1)} + h.c.] \\ && + a_0[\Omega^{\alpha(2)} E_{\alpha(2)} A - 2 B^{\beta\alpha} E_\beta{}^{\dot\beta} f_{\alpha\dot\beta} + h.c.] + \tilde{a}_0 E_{\alpha\dot\alpha} \pi^{\alpha\dot\alpha} A \\ \frac{1}{i}{\cal L}_{mass} &=& \sum_{k=2}^s(-1)^kb_k[ f^{\alpha(k-2)\beta\dot\alpha(k-1)} E_\beta{}^\gamma f_{\alpha(k-2)\gamma\dot\alpha(k-1)} + h.c.] \\ && + \frac{a_0\tilde{a}_0}{2} E_{\alpha\dot\alpha} f^{\alpha\dot\alpha}\varphi + 3a_0{}^2 E \varphi^2\end{aligned}$$ where $$\begin{aligned} b_k &=& \frac{2s(s+1)}{k(k-1)(k+1)}M^2, \qquad M{}^2 = m{}^2+s(s-1)\lambda^2 \nonumber \\ a_k{}^2 &=& \frac{4(s-k+1)(s+k)}{(k-1)(k-2)} [M^2-k(k-1)\lambda^2] \label{boson_date} \\ a_0{}^2 &=& 2(s-1)(s+2) [M^2-2\lambda^2], \qquad \tilde{a}_0{}^2 = 24s(s+1)M^2 \nonumber\end{aligned}$$ Here ${\cal L}_{kinetic}$ is just the sum of kinetic terms for all fields, that for $k \ge 2$ were defined in (\[MaslBosonLag\]), ${\cal L}_{mas}$ is the sum of the mass terms for them, while ${\cal L}_{cross}$ contains cross-terms gluing all these fields together. In what follows we assume that all parameters $a_k,a_0,\tilde{a}_0$ are positive. Explicit form of the coefficients (\[boson\_date\]) are determined by the invariance of the Lagrangian (\[MasvBosonLag\]) under the following gauge transformations $$\begin{aligned} \label{MasvBosonGT} \delta f^{\alpha(k-1)\dot\alpha(k-1)} &=& D \xi^{\alpha(k-1)\dot\alpha(k-1)} + e_\beta{}^{\dot\alpha}\eta^{\alpha(k-1)\beta\dot\alpha(k-2)} + e^\alpha{}_{\dot\beta}\eta^{\alpha(k-2)\dot\alpha(k-1)\dot\beta} \nonumber \\ && + \frac{(k-1)a_{k+1}}{2(k+1)} e_{\beta\dot\beta} \xi^{\alpha(k-1)\beta\dot\alpha(k-1)\dot\beta} + \frac{a_k}{2k(k-1)} e^{\alpha\dot\alpha} \xi^{\alpha(k-2)\dot\alpha(k-2)} \nonumber \\ \delta \Omega^{\alpha(k),\dot\alpha(k-2)} &=& D \eta^{\alpha(k),\dot\alpha(k-2)} + \frac{a_{k+1}}{2} e_{\beta\dot\beta} \eta^{\alpha(k)\beta\dot\alpha(k-2)\dot\beta} + \frac{a_k}{2k(k+1)} e^{\alpha\dot\alpha} \eta^{\alpha(k-1)\dot\alpha(k-3)} \nonumber \\ && + \frac{b_k}{2k} e^\alpha{}_{\dot\beta} \xi^{\alpha(k-1)\dot\alpha(k-2)\dot\beta} \\ \delta f^{\alpha\dot\alpha} &=& D\xi^{\alpha\dot\alpha} + e_\beta{}^{\dot\alpha} \eta^{\alpha\beta} + e^\alpha{}_{\dot\beta} \eta^{\dot\alpha\dot\beta} + \frac{a_{3}}{6} e_{\beta\dot\beta} \xi^{\alpha\beta\dot\alpha\dot\beta} - \frac{a_0}{4} e^{\alpha\dot\alpha}\xi^{} \nonumber \\ \delta B^{\alpha(2)} &=& \frac{a_0}{2} \eta^{\alpha(2)}, \qquad \delta A = D\xi - \frac{a_0}{2} e_{\alpha\dot\alpha} \xi^{\alpha\dot\alpha}\nonumber \\ \delta \pi^{\alpha\dot\alpha} &=& - \frac{a_0\tilde{a}_0}{24}\xi^{\alpha\dot\alpha},\qquad \delta\varphi = \frac{\tilde{a}_0}{12} \nonumber\end{aligned}$$ Compared to the massless case in the previous section, one can see that we still have all the gauge symmetries that our massless fields possessed modified so as to be consistent with the structure of the massive Lagrangian. Such gauge invariant formulation of the massive theory in $(A)dS_4$ space possesses some remarkable features. Firstly, we can consider a flat limit $\lambda\rightarrow0$ and immediately obtain the description of the massive fields in Minkowski space. Secondly, in anti-de Sitter space when $\lambda^2>0$ there is a correct massless limit $m\rightarrow0$ without the gap in the number of physical degrees of freedom. In such a limit our system decomposes into two systems describing the massless spin-$s$ and the massive spin-$(s-1)$ fields. Lastly, in de Sitter space when $\lambda^2<0$ one can consider the so-called partially massless limits $a_k\rightarrow0$. In such a limit, the system decomposes into the two subsystems describing the partially massless spin-$s$ field and the massive spin-$k$ field. As in the massless case, to construct a complete set of the gauge invariant objects one has to introduce a lot of extra fields which do not, however, enter the free Lagrangian. In the following, we restrict ourselves to the curvatures for the physical and auxiliary fields only. With the explicit expressions for the gauge transformations at our disposal (\[MasvBosonGT\]), it is rather straightforward to obtain (we omit all terms with the extra fields): $$\begin{aligned} \label{MasvBosonCurv} {\cal T}^{\alpha(k-1)\dot\alpha(k-1)} &=& D f^{\alpha(k-1)\dot\alpha(k-1)} + e_\beta{}^{\dot\alpha}\Omega^{\alpha(k-1)\beta\dot\alpha(k-2)} + e^\alpha{}_{\dot\beta}\Omega^{\alpha(k-2)\dot\alpha(k-1)\dot\beta} \nonumber \\ && + \frac{(k-1)a_{k+1}}{2(k+1)} e_{\beta\dot\beta} f^{\alpha(k-1)\beta\dot\alpha(k-1)\dot\beta} + \frac{a_k}{2k(k-1)} e^{\alpha\dot\alpha} f^{\alpha(k-2)\dot\alpha(k-2)} \nonumber \\ {\cal R}^{\alpha(k),\dot\alpha(k-2)} &=& D \Omega^{\alpha(k),\dot\alpha(k-2)} + \frac{a_{k+1}}{2} e_{\beta\dot\beta} \Omega^{\alpha(k)\beta\dot\alpha(k-2)\dot\beta} + \frac{a_k}{2k(k+1)} e^{\alpha\dot\alpha} \Omega^{\alpha(k-1)\dot\alpha(k-3)} \nonumber \\ && + \frac{b_k}{2k} e^\alpha{}_{\dot\beta} f^{\alpha(k-1)\dot\alpha(k-2)\dot\beta} \nonumber \\ {\cal R}^{\alpha(2)} &=& D \Omega^{\alpha(2)} + \frac{a_3}{2} e_{\beta\dot\beta} \Omega^{\alpha(2)\beta\dot\beta} + \frac{b_2}{4} e^\alpha{}_{\dot\beta} f^{\alpha\dot\beta} - \frac{a_0}{4} E^\alpha{}_\beta B^{\alpha\beta} + \frac{a_0\tilde{a}_0}{24}E^{\alpha(2)} \varphi \nonumber \\ {\cal T}^{\alpha\dot\alpha} &=& Df^{\alpha\dot\alpha} + e_\beta{}^{\dot\alpha} \Omega^{\alpha\beta} + e^\alpha{}_{\dot\beta} \Omega^{\dot\alpha\dot\beta} + \frac{a_{3}}{6} e_{\beta\dot\beta} f^{\alpha\beta\dot\alpha\dot\beta} - \frac{a_0}{4} e^{\alpha\dot\alpha} A \\ {\cal C}^{\alpha(2)} &=& DB^{\alpha(2)} - \frac{a_0}{2}\Omega^{\alpha(2)} - \frac{\tilde{a}_0}{24} e^\alpha{}_{\dot\beta}\pi^{\alpha\dot\beta} \nonumber \\ {\cal R} &=& DA + 2(E_{\alpha(2)} B^{\alpha(2)} + E_{\dot\alpha(2)}B^{\dot\alpha(2)}) - \frac{a_0}{2} e_{\alpha\dot\alpha} f^{\alpha\dot\alpha} \nonumber \\ {\cal C}^{\alpha\dot\alpha} &=& D\pi^{\alpha\dot\alpha} +\frac{a_0\tilde{a}_0}{24} f^{\alpha\dot\alpha} - \frac{\tilde{a}_0}{12} (e_\beta{}^{\dot\alpha} B^{\alpha\beta} + e^\alpha{}_{\dot\beta} B^{\dot\alpha\dot\beta}) + \frac{a_0{}^2}{8}e^{\alpha\dot\alpha} \varphi \nonumber \\ {\cal C} &=& D\varphi + e_{\alpha\dot\alpha} \pi^{\alpha\dot\alpha} -\frac{\tilde{a}_0}{12} A \nonumber\end{aligned}$$ In our construction of the massive supermultiplets we will consider supertransformations for the physical fields only. However, in all calculations we will heavily use the auxiliary fields equations (on-shell conditions) as well as corresponding algebraic identities: $$\begin{aligned} {\cal T}^{\alpha(k-1)\dot\alpha(k-1)} \approx 0 &\Rightarrow& e_\beta{}^{\dot\alpha} {\cal R}^{\alpha(k-1)\beta\dot\alpha(k-2)} + e^\alpha{}_{\dot\beta} {\cal R}^{\alpha(k-2)\dot\alpha(k-1)\dot\beta}\approx 0 \nonumber \\ {\cal R} \approx 0 &\Rightarrow& E_{\alpha(2)}{\cal C}^{\alpha(2)}+ E_{\dot\alpha(2)} {\cal C}^{\dot\alpha(2)} \approx 0 \label{MasvOn-Shell} \\ {\cal C} \approx 0 &\Rightarrow& e_{\alpha\dot\alpha} {\cal C}^{\alpha\dot\alpha} \approx 0 \quad\Rightarrow\quad E^{\alpha}{}_{\dot\gamma} {\cal C}^{\beta\dot\gamma} \approx \frac12 \varepsilon^{\alpha\beta} E_{\gamma}{}_{\dot\gamma} {\cal C}^{\gamma\dot\gamma} \nonumber\end{aligned}$$ The variation of the Lagrangian (\[MasvBosonLag\]) under the arbitrary variations for the physical fields takes the simple form $$\begin{aligned} \label{MasvBosonVar} \delta {\cal L} &=& - 2i\sum_{k=2}^s(-1)^k {\cal R}^{\alpha(k-1)\beta\dot\alpha(k-2)} e_\beta{}^{\dot\beta} \delta f_{\alpha(k-1)\dot\alpha(k-2)\dot\beta} \nonumber \\ && - 2i E_{\alpha(2)} {\cal C}^{\alpha(2)} \delta A + 12i E_{\alpha\dot\alpha} {\cal C}^{\alpha\dot\alpha} \delta \varphi + h.c.\end{aligned}$$ Let us stress once again that this expression is such that all extra fields drop out. Half-integer spin $s+1/2$ {#MasbFerm} ------------------------- In the gauge invariant formalism, the massive half-integer spin-$s+1/2$ field is described by the set of massless fields with spins $1/2\leq k+1/2\leq s+1/2$. Frame-like formulation for the massless fermionic fields with spins ($k\geq1$) were considered above, they are described by the one-forms $(\Phi^{\alpha(k)\dot\alpha(k-1)},h.c.)$, while massless spin-$1/2$ is described by a physical zero-form $(\phi^\alpha,h.c.)$. The Lagrangian for free massive field in $AdS_4$ have the form $$\begin{aligned} \label{MasvFermLag} {\cal L} &=& \sum_{k=1}^{s} (-1)^k \Phi_{\alpha(k-1)\beta\dot\alpha(k-1)} e^\beta{}_{\dot\beta} D\Phi^{\alpha(k-1)\dot\alpha(k-1)\dot\beta} - \phi_\alpha E^\alpha{}_{\dot\alpha} D\phi^{\dot\alpha} \nonumber \\ && + \sum_{k=2}^{s} (-1)^k c_k [E^{\beta(2)} \Phi_{\alpha(k-2)\beta(2)\dot\alpha(k-1)} \Phi^{\alpha(k-2)\dot\alpha(k-1)}] + c_0 \Phi_\alpha E^\alpha{}_{\dot\alpha} \phi^{\dot\alpha} + h.c. \nonumber \\ && + \sum_{k=1}^s (-1)^k d_{k}[(k+1) \Psi_{\alpha(k-1)\beta\dot\alpha(k-1)} E^\beta{}_{\gamma} \Psi^{\alpha(k-1)\gamma\dot\alpha(k-1)} \nonumber \\ && - (k-1)\Psi_{\alpha(k)\dot\alpha(k-2)\dot\beta} E^{\dot\beta}{}_{\dot\gamma} \Psi^{\alpha(k)\dot\alpha(k-2)\dot\gamma}] + 2d_1 E\phi_\alpha \phi^\alpha + h.c.\end{aligned}$$ where $$\begin{aligned} d_k &=& \pm \frac{(s+1)}{2k(k+1)}M_1, \qquad M_1{}^2 = m_1{}^2+s^2\lambda^2 \nonumber \\ c_k{}^2 &=& \frac{(s-k+1)(s+k+1)}{k^2}[M_1{}^2-k^2\lambda^2], \label{fermion_data} \\ c_0{}^2 &=& 2s(s+2)[M_1{}^2-\lambda^2] \nonumber\end{aligned}$$ In the following, we assume that the parameters $c_k,c_0,M_1$ are positive. Explicit form of the coefficients (\[fermion\_data\]) are determined by the invariance of the Lagrangian under the following gauge transformations $$\begin{aligned} \delta \Phi^{\alpha(k)\dot\alpha(k-1)} &=& D \xi^{\alpha(k)\dot\alpha(k-1)} + e_\beta{}^{\dot\alpha} \eta^{\alpha(k)\beta\dot\alpha(k-2)} + 2d_{k} e^\alpha{}_{\dot\beta} \xi^{\alpha(k-1)\dot\alpha(k-1)\dot\beta} \nonumber \\ && + c_{k+1} e_{\beta\dot\beta} \xi^{\alpha(k)\beta\dot\alpha(k-1)\dot\beta} + \frac{c_k}{(k-1)(k+1)}e^{\alpha\dot\alpha} \xi^{\alpha(k-1)\dot\alpha(k-2)} \\ \delta \phi^\alpha &=& c_0\xi^\alpha \nonumber\end{aligned}$$ The general structure of the Lagrangian (\[MasvFermLag\]) is the same as in the bosonic case. The first line is the sum of kinetic terms, the second line contains cross-terms and the last two lines are mass terms. In such a formulation we can take the correct massless limit $m_1\rightarrow0$ in $AdS$ ($\lambda^2>0$) and the correct partially massless limits $c_k\rightarrow0$ in $dS$ ($\lambda^2<0$). Taking a flat limit $\lambda\rightarrow0$ we obtain the description of the massive fermionic fields in Minkowski space. As in the bosonic case, we restrict ourselves with the gauge invariant curvatures for the physical fields only omitting all the extra fields: $$\begin{aligned} \label{MasvFermCurv} {\cal F}^{\alpha(k)\dot\alpha(k-1)} &=& D \Phi^{\alpha(k)\dot\alpha(k-1)} +2d_{k} e^\alpha{}_{\dot\beta}\Phi^{\alpha(k-1)\dot\alpha(k-1)\dot\beta} \nonumber \\ && + c_{k+1} e_{\beta\dot\beta} \Phi^{\alpha(k)\beta\dot\alpha(k-1)\dot\beta} + \frac{c_k}{(k-1)(k+1)}e^{\alpha\dot\alpha} \Phi^{\alpha(k-1)\dot\alpha(k-2)} \nonumber \\ {\cal F}^{\alpha} &=& D\Phi^{\alpha} + 2d_{1} e^\alpha{}_{\dot\beta}\Phi^{\dot\beta} + c_2 e_{\beta\dot\beta} \Phi^{\alpha\beta\dot\beta} - \frac{c_0}{3} E^\alpha{}_\beta \phi^\beta \\ {\cal C}^\alpha &=& D\phi^{\alpha} - c_0 \Phi^{\alpha} + 2d_1 e^{\alpha}{}_{\dot\beta} \phi^{\dot\beta} \nonumber\end{aligned}$$ The variation of the Lagrangian (\[MasvFermLag\]) under the arbitrary variations of the physical fields has the following form $$\label{MasvFermVar} \delta {\cal L} = - \sum_{k=1}^s(-1)^k {\cal F}_{\alpha(k-1)\beta\dot\alpha(k-1)} e^\beta{}_{\dot\beta} \delta \Psi^{\alpha(k-1)\dot\alpha(k-1)\dot\beta} - {\cal C}_\alpha E^\alpha{}_{\dot\alpha} \delta \phi^{\dot\alpha} + h.c.$$ Massive higher spin superblocks {#Section3} =============================== There are two types of massive $N=1$ supermultiplets, each one containing two massive bosonic fields (with opposite parities) and two massive fermionic ones: $$\xymatrix{ & \Phi_{s+\frac12} \ar@{-}[dr] & \\ f_s \ar@{-}[ur] & & f'_s \ar@{-}[dl] \\ & \Psi_{s-\frac12} \ar@{-}[ul] } \qquad \xymatrix{ & \Phi_{s-\frac12} \ar@{-}[dr] & \\ f_s \ar@{-}[ur] & & f'_{s-1} \ar@{-}[dl] \\ & \Psi_{s-\frac12} \ar@{-}[ul] }$$ To provide an explicit realization of such supermultiplets one has to find supertransformations connecting each bosonic field with each fermionic field so that: 1) the sum of the four free Lagrangians for these fields is invariant; 2) the algebra of the supertransformations is closed. In this work we use the following strategy. Firstly, for each pair of bosonic and fermionic fields (we call it superblock in what follows) we find the supertransformations leaving the sum of their two Lagrangians invariant. Then we combine all four fields together and adjust parameters of these superblocks so that the algebra of the supertransformations is closed. One can see from the diagrams above that there are only two non-trivial superblocks, namely $(s,s+1/2)$ and $(s-1/2,s)$. Such a strategy therefore greatly simplifies the whole construction. In the gauge invariant formalism the description of massive higher spin fields is constructed out of the appropriately chosen set of massless ones. It seems natural to expect that one can construct a description of massive higher spin supermultiplet out of an appropriately chosen set of massless ones. Indeed, if one decomposes all four massive fields into their massless components, the resulting spectrum of massless components does correspond to some set of massless supermultiplets. However, the explicit structure of the supertransformations (see below) shows that all massless components still remain connected with all their neighbours so that the whole system looks just like one big massless supermultiplet (similarly to what we obtained in the three dimensional case [@BSZ17]): $$\xymatrix{ f_{k-1} \ar@{-}[dr] & & f_k \ar@{-}[dr] & & f_{k+1} \\ \dots & (\Phi_{k-\frac12},\Psi_{k-\frac12}) \ar@{-}[ur] \ar@{-}[dr] & & (\Phi_{k+\frac12},\Psi_{k+\frac12}) \ar@{-}[ur] \ar@{-}[dr] & \dots \\ f'_{k-1} \ar@{-}[ur] & & f'_k \ar@{-}[ur] & & f'_{k+1} }$$ One can introduce new fermionic variables: $$\begin{aligned} \tilde{\Phi}_k &=& cos \Theta_k \Phi_k + \sin \Theta_k \Psi_k \\ \tilde{\Psi}_k &=& - sin \Theta_k \Phi_k + \cos \Theta_k \Psi_k\end{aligned}$$ and adjust mixing angles $\Theta_k$ so that the whole system decomposes into the sum of massless supermultiplets containing two bosonic and two fermionic fields: $$\xymatrix{ & & f_{k-1} \ar@{-}[dr] & & & & f_k \ar@{-}[dr] & & \\ \dots & \tilde{\Phi}_{k-\frac32} \ar@{-}[ur] \ar@{-}[dr] & & \tilde{\Psi}_{k-\frac12} & \oplus & \tilde{\Phi}_{k-\frac12} \ar@{-}[ur] \ar@{-}[dr] & & \tilde{\Psi}_{k+\frac12} & \dots \\ & & f'_{k-1} \ar@{-}[ur] & & & & f'_k \ar@{-}[ur] & & }$$ The separation of these supermultiplets into the usual pairs is impossible because the bosonic fields have opposite parities. However in this case the structure of cross and mass-like terms in the fermionic Lagrangian cease to be diagonal making the construction of massive supermultiplets more complicated. Note that it is this approach that was used in the previous works of one of the current authors [@Zin07a]. Supertransformations -------------------- We begin with a general discussion valid for the construction of both massive superblocks and consider the most general ansatz for the supertransformations. For the bosonic field variables we choose $$\begin{aligned} \label{MasvSTevBos} \delta f^{\alpha(k-1)\dot\alpha(k-1)} &=& \alpha_{k-1}\Phi^{\alpha(k-1)\beta\dot\alpha(k-1)} \zeta_\beta - \bar\alpha_{k-1}\Phi^{\alpha(k-1)\dot\alpha(k-1)\dot\beta} \zeta_{\dot\beta} \nonumber\\ && + \alpha'_{k-1} \Phi^{\alpha(k-1)\dot\alpha(k-2)} \zeta^{\dot\alpha} - \bar\alpha'_{k-1} \Phi^{\alpha(k-2)\dot\alpha(k-1)} \zeta^{\alpha} \nonumber \\ \delta A &=& \alpha_0 \Phi^\alpha\zeta_\alpha - \bar\alpha_0 \Phi^{\dot\alpha} \zeta_{\dot\alpha} + \alpha'_0 e_{\alpha\dot\alpha}\psi^\alpha\zeta^{\dot\alpha} - \bar\alpha'_0 e_{\alpha\dot\alpha}\psi^{\dot\alpha} \zeta^{\alpha} \\ \delta \varphi &=& \tilde\alpha_0 \phi^\alpha \zeta_{\alpha} - \bar{\tilde\alpha}_0 \phi^{\dot\alpha} \zeta_{\dot\alpha} \nonumber\end{aligned}$$ and for the fermionic ones $$\begin{aligned} \label{MasvSTevFerm} \delta \Phi^{\alpha(k)\dot\alpha(k-1)} &=& \beta_{k-1}\Omega^{\alpha(k)\dot\alpha(k-2)} \zeta^{\dot\alpha} + \gamma_{k-1}f^{\alpha(k-1)\dot\alpha(k-1)} \zeta^{\alpha} \nonumber \\ && + \beta'_{k-1} \Omega^{\alpha(k-1)\beta\dot\alpha(k-2)} \zeta_\beta + \gamma'_{k-1} f^{\alpha(k-1)\dot\alpha(k-2)\dot\beta}\zeta_{\dot\beta} \nonumber \\ \delta \Phi^\alpha &=& \beta_0 e_{\beta\dot\beta} B^{\alpha\beta}\zeta^{\dot\beta} + \beta'_{1} \Omega^{\alpha\beta} \zeta_\beta +\gamma_0 A \zeta^\alpha + \gamma'_{1}f^{\alpha\dot\beta} \zeta_{\dot\beta} + \hat\gamma_0 e^\alpha{}_{\dot\alpha} \varphi\zeta^{\dot\alpha} \\ \delta \phi^\alpha &=& \tilde\beta_0 \pi^{\alpha\dot\alpha} \zeta_{\dot\alpha} + \beta'_0 B^{\alpha\beta} \zeta_\beta + \tilde\gamma_0 \varphi \zeta^\alpha \nonumber\end{aligned}$$ where all coefficients are complex. One can see that the supertransformations for higher spin components are combinations of the massless supertransformations (\[MaslST1\]) and (\[MaslST2\]). $$\xymatrix{ f_{k-1} \ar@{-}[dr]^-{\alpha_{k-1}} & & f_k \ar@{-}[dr]^-{\alpha_k} & & f_{k+1} \\ \dots & \Phi_{k-\frac12} \ar@{-}[ur]^-{\alpha'_k} & & \Phi_{k+\frac12} \ar@{-}[ur]^{\alpha'_{k+1}} & \dots }$$ The ansatz for the supertransformations (\[MasvSTevBos\]), (\[MasvSTevFerm\]) has the same form for both massive superblocks $(s+1/2,s)$ and $(s,s-1/2)$, the only difference being in the boundary conditions. In the first case we have $$\label{InitCond1} \alpha_s = \beta_s = \gamma_s = 0, \qquad \alpha'_s = \beta'_s = \gamma'_s = 0$$ while in the second case $$\label{InitCond2} \alpha_{s-1} = \beta_{s-1} = \gamma_{s-1} = 0, \qquad \alpha'_s = \beta'_s = \gamma'_s = 0$$ The variation of the sum of the bosonic and fermionic Lagrangians (\[MasvBosonVar\]), (\[MasvFermVar\]) under the supertransformations (\[MasvSTevBos\]), (\[MasvSTevFerm\]) has the form $\delta{\cal L}+\delta{\cal L}'$, where $$\begin{aligned} \label{MasvVar} \delta {\cal L} &=& (-1)^k \sum_{k=2}^{s}[- (k-1)\bar\beta_{k-1} {\cal F}_{\alpha(k-1)\beta\dot\alpha(k-1)} e^\beta{}_{\dot\beta} \Omega^{\alpha(k-2)\dot\alpha(k-1)\dot\beta} \zeta^{\alpha} \nonumber \\ && + 4i\alpha_{k-1} \Phi_{\alpha(k-2)\beta\gamma\dot\alpha(k-1)} e^\gamma{}_{\dot\gamma} {\cal R}^{\alpha(k-2)\dot\alpha(k-1)\dot\gamma} \zeta^{\beta} \nonumber \\ && - \bar\gamma_{k-1} {\cal F}_{\alpha(k-1)\beta\dot\alpha(k-1)} e^\beta{}_{\dot\beta} (f^{\alpha(k-1)\dot\alpha(k-1)} \zeta^{\dot\beta} + (k-1)f^{\alpha(k-1)\dot\alpha(k-2)\dot\beta}\zeta^{\dot\alpha})] \nonumber \\ && - \bar\beta_0 E_{\dot\alpha\dot\beta} {\cal F}_{\alpha} B^{\dot\alpha\dot\beta} \zeta^{\alpha} + 4i\alpha_0 E_{\dot\beta(2)}\Phi_{\alpha} {\cal C}^{\dot\beta(2)} \zeta^{\alpha} \nonumber \\ && + \bar{\tilde\beta}_0 E^\alpha{}_{\dot\alpha} {\cal C}_\alpha \pi^{\beta\dot\alpha} \zeta_{\beta} + 12i\tilde\alpha_0 E_{\beta\dot\beta} \phi^\alpha {\cal C}^{\beta\dot\beta} \zeta_{\alpha} \nonumber \\ && + \bar\gamma_0 e^\alpha{}_{\dot\alpha} {\cal F}_{\alpha} A\zeta^{\dot\alpha} + \bar{\tilde\gamma}_0 E^\alpha{}_{\dot\alpha} {\cal C}_\alpha \varphi \zeta^{\dot\alpha} + 2\bar{\hat\gamma}_0 E^\alpha{}_\beta {\cal F}_{\alpha} \varphi \zeta^{\beta} + h.c. \nonumber \\ \delta {\cal L}' &=& \sum_{k=2}^{s} (-1)^k [\bar\beta'_{k-1} {\cal F}_{\alpha(k-2)\gamma\dot\alpha(k-2)} e^\gamma{}_{\dot\gamma} \Omega^{\alpha(k-2)\dot\alpha(k-2)\dot\gamma\dot\beta} \zeta_{\dot\beta} \nonumber \\ && - 4i(k-1)\alpha'_{k-1} \Phi_{\alpha(k-2)\beta\dot\alpha(k-2)} e^\beta{}_{\dot\beta} {\cal R}^{\alpha(k-2)\dot\alpha(k-2)\dot\beta\dot\gamma} \zeta_{\dot\gamma} \nonumber \\ && + \bar\gamma'_{k-1} {\cal F}_{\alpha(k-2)\gamma\dot\alpha(k-2)} e^\gamma{}_{\dot\gamma} f^{\alpha(k-2)\beta\dot\alpha(k-2)\dot\gamma}\zeta_\beta] \nonumber \\ && - \bar\beta'_0 {\cal C}_\alpha E^\alpha{}_{\dot\alpha} B^{\dot\alpha\dot\beta} \zeta_{\dot\beta} + 8i \alpha'_0 \psi_{\alpha}E^\alpha{}_{\dot\beta} {\cal C}^{\dot\beta(2)} \zeta_{\dot\beta} + h.c.\end{aligned}$$ Here we used the equations for the auxiliary bosonic fields (\[MasvOn-Shell\]). We now proceed as in the massless case, deriving the corresponding identities. Let us recall a general scheme. Lagrangian variation (\[MasvVar\]) has the structure $$\delta{\cal L} = ({\cal F}\Omega|_\beta \oplus \Phi {\cal R}|_\alpha \oplus {\cal F}f|_\gamma) \zeta$$ where $\Phi$, ${\cal F}$ are sets of all fields and curvatures for the fermion, $f$ is a set of physical fields for the boson and $\Omega$, ${\cal R}$ are sets of auxiliary fields and curvatures for the boson. Since a Lagrangian is defined up to a total derivative we have two type of identities $ D[\Phi\Omega\zeta]=0$, $D[\Phi f\zeta]=0$. They lead to $$\begin{aligned} \label{SchemIden1} {\cal F} \Omega \zeta &=& \Phi {\cal R} \zeta \oplus \Delta (\Phi\Omega,\Phi f) \zeta \\ \label{SchemIden2} {\cal F} f \zeta &=& \Phi {\cal T} \zeta \oplus \Delta (\Phi\Omega,\Phi f) \zeta\end{aligned}$$ Using the explicit form of identities (\[SchemIden1\]), (\[SchemIden2\]) (see Appendix A for details), we obtain expressions for the parameters $\alpha$ and $\gamma$ in terms of $\beta$: $$\begin{aligned} \alpha_{k-1} &=& i\frac{(k-1)}{4} \bar\beta_{k-1}, \qquad \alpha'_{k-1} = \frac{i}{4(k-1)} \bar\beta'_{k-1} \nonumber \\ \alpha_0 &=& -i\frac{\bar\beta_0}{4}, \qquad \tilde\alpha_0 = i\frac{\bar{\tilde\beta}_0}{24}, \qquad \alpha'_0 = - i\frac{\bar\beta'_0}{8} \label{SolutAlf}\end{aligned}$$ and $$\begin{aligned} \gamma_{k-1} &=& 2d_k \bar\beta_{k-1} ,\qquad \gamma'_{k-1} = 2d_{k-1} \bar\beta'_{k-1} \nonumber \\ \gamma_ 0 &=& - d_1 \bar\beta_0, \quad \tilde\gamma_0 = - 12\frac{c_0d_1}{\tilde{a}_0} \bar\beta_0, \quad \hat\gamma_0 = - \frac18 \tilde{a}_0 \beta_0 \label{SolutGam}\end{aligned}$$ and we also obtain recurrent equations on the parameters $\beta_k$ $$\begin{aligned} 2(k+1)\beta_{k-1} c_{k+1} &=& k\beta_{k} a_{k+1}, \qquad 2c_2 \beta_0 = a_0\beta_1, \qquad 2c_0\tilde\beta_0 = - {\tilde{a}_0} \beta_0 \label{RecurEq1} \\ \frac12 \beta'_{k-1} a_{k+1} &=& \beta'_kc_k, \qquad \frac12 a_0 \beta'_0 = c_0 \beta'_1, \qquad {\tilde{a}_0} \bar{\tilde\beta}_0 = - 24d_1 \beta'_0 \label{RecurEq2}\end{aligned}$$ as well as four independent equations which relate $\beta$ and $\beta'$ and the bosonic and fermionic mass parameters: $$\begin{aligned} 0 &=& \frac{\beta'_{k-1}c_k}{(k-1)} - \frac{\beta'_ka_{k+1}}{2(k+1)} + \lambda \beta_{k-1} - \gamma_{k-1} \label{Relation1} \\ 0 &=& (k-1)\beta_{k-1}c_{k} - \frac{(k-2)}{2} \beta_{k-2}a_{k} - \lambda \beta'_{k-1} + \gamma'_{k-1} \label{Relation2} \\ 0 &=& \frac{(k-1)}{2k} \bar\beta_{k-1}b_{k} - 2kd_{k}\gamma_{k-1} +\lambda \bar\gamma_{k-1} - \frac{\bar\gamma'_{k}a_{k+1}}{2k(k+1)}\label{Relation3} \\ 0 &=& \frac{\bar\beta'_{k-1}b_{k}}{2} - 2(k-1)d_{k-1}\gamma'_{k-1} -\lambda\bar\gamma'_{k-1} + \bar\gamma_{k-1}c_k \label{Relation4}\end{aligned}$$ The explicit solution of these equations depends on the concrete massive superblock. More spefically, it depends on the initial conditions (\[InitCond1\]), (\[InitCond2\]) and on the sign of $d_k$, i.e. on the sign before massive terms in Lagrangian for fermions. In the following we present exact solutions for two massive superblocks $(s+1/2,s)$ and $(s,s-1/2,)$. Superblock $(s+1/2,s)$ {#MasvSB1} ---------------------- Here we present our results for the massive superblock $(s+1/2,s)$. The massive boson spin-$s$ with the mass parameter $M$, as described in section \[MasbBoson\], we have $$\begin{aligned} b_k &=& \frac{2s(s+1)}{k(k-1)(k+1)}M^2, \qquad m{}^2 = M{}^2 - s(s-1)\lambda^2 \nonumber \\ a_k{}^2 &=& \frac{4(s-k+1)(s+k)}{(k-1)(k-2)} [M^2 - k(k-1)\lambda^2]\nonumber \\ a_0{}^2 &=& 2(s-1)(s+2)[M^2 - 2\lambda^2], \qquad \tilde{a}_0{}^2 = 24s(s+1)M^2\end{aligned}$$ The massive fermion spin-$(s+1/2)$ with mass parameter $M_1$ as described in section \[MasbFerm\], here $$\begin{aligned} d_k &=& \pm\frac{(s+1)}{2k(k+1)}M_1, \qquad m_1{}^2 = M_1{}^2-s^2\lambda^2 \nonumber \\ c_k{}^2 &=& \frac{(s-k+1)(s+k+1)}{k^2} [M_1{}^2 - k^2\lambda^2] \\ c_0{}^2 &=& 2s(s+2) [M_1{}^2 - \lambda^2]\end{aligned}$$ Supertransformations for massive superblock $(s+1/2,s)$ have the form (\[MasvSTevBos\]), (\[MasvSTevFerm\]) with the initial conditions (\[InitCond1\]). The parameters $\alpha_k$ and $\gamma_k$ are determined by (\[SolutAlf\]) and (\[SolutGam\]). From the equation (\[Relation3\]) one can obtain an important relation on the bosonic and fermionic mass parameters. Indeed, at $k=s$ we have $$(M^2-M_1{}^2) \bar\beta_{s-1} = \mp M_1\lambda \beta_{s-1}$$ where the sign corresponds to that of $d_k$. So we have four independent cases $$\begin{aligned} M^2 &=& M_1(M_1-\lambda), \qquad \bar\beta_{s-1} = \pm\beta_{s-1} \\ M^2 &=& M_1(M_1+\lambda), \qquad \bar\beta_{s-1} = \mp \beta_{s-1}\end{aligned}$$ The solution of other equations give, for $M^2=M_1(M_1-\lambda)$ $$\begin{aligned} \beta_{k-1} &=& \sqrt{\frac{(s+k+1)(M_1+k\lambda)}{(k-1)}}\beta, \quad \beta'_{k-1} = \sqrt{{(k-1)(s-k+1)(M_1-k\lambda)}}\beta \\ \beta_{0} &=& \sqrt{{2(s+2)(M_1+\lambda)}}\beta, \quad \beta'_{0} = 2\sqrt{{s(M_1-\lambda)}}\beta, \quad \tilde\beta_0 = - \sqrt{6(s+1)M_1}\beta\end{aligned}$$ and for $M^2=M_1(M_1+\lambda)$ $$\begin{aligned} \beta_{k-1}{} &=& \sqrt{\frac{(s+k+1)(M_1-k\lambda)}{(k-1)}}\beta,\quad \beta'_{k-1} = - \sqrt{{(k-1)(s-k+1)(M_1+k\lambda)}}\beta \\ \beta_{0} &=& \sqrt{{2(s+2)(M_1-\lambda)}}\beta, \quad \beta'_{0} = - 2\sqrt{{s(M_1+\lambda)}}\beta, \quad \tilde\beta_0 = - \sqrt{6(s+1)M_1}\beta\end{aligned}$$ Therefore, we see that in the two cases the parameters $\beta$ are real and in the other two they are imaginary. This means that one half of the solutions corresponds to the massive superblocks with the parity-even boson while another half corresponds to the massive superblocks with the parity-odd one. In order to present these four cases for the massive superblock $(s,s+1/2)$ in a more clear form let us introduce following notations. We denote integer spin $s$ with the mass parameter $M$ as $$\label{NotatBos} [s]_M^\pm$$ here $\pm$ corresponds to parity-even/parity-odd boson. We also denote half-integer spin $s+1/2$ with the mass parameter $M_1$ as $$\label{NotatFer} [s+\frac12]_{ M_1}^{\pm}$$ here $\pm$ corresponds to the sign of $d_k$. In these notations the four solutions for the massive superblock given above look as follows: $$1) \left( \begin{array}{c} {[s]}_{M}^+ \\ {[s+\frac12]}_{M_1}^+ \end{array} \right), \quad 2) \left( \begin{array}{c} {[s]}_{M}^- \\ {[s+\frac12]}_{M_1}^{-} \end{array} \right), \quad 3) \left( \begin{array}{c} {[s]}_{M'}^+ \\ {[s+\frac12]}_{M_1}^{-} \end{array} \right),\quad 4)\left( \begin{array}{c} {[s]}_{M'}^- \\ {[s+\frac12]}_{M_1}^{+} \end{array} \right)$$ where $$M^2 = M_1(M_1-\lambda), \qquad M'^2 = M_1(M_1+\lambda)$$ In the first and third cases we have $$\begin{aligned} \beta_{k-1} &=& \sqrt{\frac{(s+k+1)(M_1\pm k\lambda)}{(k-1)}}\rho,\quad \beta'_{k-1} = \pm\sqrt{{(k-1)(s-k+1)(M_1\mp k\lambda)}}\rho \\ \beta_{0} &=& \sqrt{{2(s+2)(M_1\pm\lambda)}}\rho, \quad \beta'_{0} = \pm2\sqrt{{s(M_1\mp\lambda)}}\rho, \quad \tilde\beta_0 = - \sqrt{6(s+1)M_1}\rho\end{aligned}$$ here the upper sign corresponds to 1) and the lower sign corresponds to 3). In the second and fourth cases we have $$\begin{aligned} \beta_{k-1} &=& i\sqrt{\frac{(s+k+1)(M_1\pm k\lambda)}{(k-1)}}\rho,\quad \beta'_{k-1} = \pm i\sqrt{{(k-1)(s-k+1)(M_1\mp k\lambda)}}\rho \\ \beta_{0} &=& i\sqrt{{2(s+2)(M_1\pm\lambda)}}\rho, \quad \beta'_{0} = \pm i2\sqrt{{s(M_1\mp\lambda)}}\rho, \quad \tilde\beta_0 = - i\sqrt{6(s+1)M_1}\rho\end{aligned}$$ here the upper sign corresponds to 2) and the lower sign corresponds to 4). Superblock $(s,s-1/2)$ {#MasvSB2} ---------------------- Here we collect our results for the massive superblock $(s,s-1/2)$. For the massive even or odd spin-$s$ boson with the mass parameter $M$ we use the same formulation as in the previous subsection, while for the massive spin-$(s-1/2)$ fermion with the mass parameter $M_2$ we use its description in section \[MasbFerm\] with the shift $s\rightarrow(s-1)$: $$\begin{aligned} d_k &=& \pm\frac{s}{2k(k+1)}M_2, \qquad m_2{}^2 = M_2{}^2-(s-1)^2\lambda^2 \\ c_k{}^2 &=& \frac{(s-k)(s+k)}{k^2}[M_2{}^2-k^2\lambda^2] \\ c_0{}^2 &=& 2(s-1)(s+1)[M_2{}^2-\lambda^2]\end{aligned}$$ Supertransformations for the massive superblock $(s-1/2,s)$ are the same as in the previous case (\[MasvSTevBos\]), (\[MasvSTevFerm\]) but with different initial conditions (\[InitCond2\]). The parameters $\alpha_k$ and $\gamma_k$ are still determined by (\[SolutAlf\]) and (\[SolutGam\]). From the equation (\[Relation2\]) one can relate bosonic and fermionic mass parameters, indeed at $k=s$ we have $$(M^2-M_2{}^2)\bar\beta'_{s-1} = \pm M_2\lambda\beta'_{s-1}$$ here the sign corresponds to that of $d_k$. So we again have four independent cases $$\begin{aligned} M^2 &=& M_2(M_2+\lambda), \qquad \bar\beta'_{s-1} = \pm\beta'_{s-1} \\ M^2 &=& M_2(M_2-\lambda), \qquad \bar\beta'_{s-1} = \mp\beta'_{s-1}\end{aligned}$$ The solution of other equations gives, for $M^2=M_2(M_2+\lambda)$ $$\begin{aligned} \beta_{k-1} &=& \sqrt{\frac{(s-k)(M_2-k\lambda)}{(k-1)}}\beta, \qquad \beta'_{k-1} = \sqrt{(k-1)(s+k)(M_2+k\lambda)}\beta \\ \beta_{0} &=& \sqrt{{2(s-1)(M_2-\lambda)}}\beta, \quad \beta'_{0} = 2\sqrt{{(s+1)(M_2+\lambda)}}\beta, \quad \tilde\beta_0 = - \sqrt{6sM_2}\beta\end{aligned}$$ and for $M^2=M_2(M_2-\lambda)$ $$\begin{aligned} \beta_{k-1} &=& \sqrt{\frac{(s-k)(M_2+k\lambda)}{(k-1)}}\beta, \quad \beta'_{k-1} = - \sqrt{(k-1)(s+k)(M_2-k\lambda)}\beta \\ \beta_{0} &=& \sqrt{{2(s-1)(M_2+\lambda)}}\beta, \quad \beta'_{0} = - 2\sqrt{{(s+1)(M_2-\lambda)}}\beta, \quad \tilde\beta_0 = - \sqrt{6sM_2}\beta\end{aligned}$$ Again we see that in two cases the parameters $\beta$ are real and in other two cases they are imaginary. They correspond to the massive superblocks with the parity-even and parity-odd bosons respectively. Using notations (\[NotatBos\]), (\[NotatFer\]) these four cases for the massive $(s,s-1/2)$ superblock can be presented as $$1) \left( \begin{array}{c} {[s]}_{M}^+ \\ {[s-\frac12]}_{M_2}^+ \end{array} \right), \quad 2) \left( \begin{array}{c} {[s]}_{M}^- \\ {[s-\frac12]}_{M_2}^{-} \end{array} \right), \quad 3) \left( \begin{array}{c} {[s]}_{M'}^+ \\ {[s-\frac12]}_{M_2}^{-} \end{array} \right), \quad 4)\left( \begin{array}{c} {[s]}_{M'}^- \\ {[s-\frac12]}_{M_2}^{+} \end{array} \right)$$ where $$M^2=M_2(M_2+\lambda),\qquad M'^2=M_2(M_2-\lambda)$$ For the first and third cases we have $$\begin{aligned} \beta_{k-1} &=& \sqrt{\frac{(s-k)(M_2\mp k\lambda)}{(k-1)}}\rho, \quad \beta'_{k-1} = \pm\sqrt{(k-1)(s+k)(M_2\pm k\lambda)}\rho \\ \beta_{0} &=& \sqrt{{2(s-1)(M_2\mp\lambda)}}\rho, \quad \beta'_{0} = \pm2\sqrt{{(s+1)(M_2\pm\lambda)}}\rho, \quad \tilde\beta_0 = - \sqrt{6sM_2}\rho\end{aligned}$$ here the upper sign corresponds to 1) and the lower sign corresponds to 3). In the second and fourth cases we have $$\begin{aligned} \beta_{k-1} &=& i\sqrt{\frac{(s-k)(M_2\mp k\lambda)}{(k-1)}}\rho, \quad \beta'_{k-1} = \pm i\sqrt{(k-1)(s+k)(M_2\pm k\lambda)}\rho \\ \beta_{0} &=& i\sqrt{{2(s-1)(M_2\mp\lambda)}}\rho, \quad \beta'_{0} = \pm i2\sqrt{{(s+1)(M_2\pm\lambda)}}\rho, \quad \tilde\beta_0 = - i\sqrt{6sM_2}\rho\end{aligned}$$ here the upper sign correspond to 2) and the lower sign correspond to 4). In all four cases $\rho$ is real. Massive higher spin supermultiplets {#Section4} =================================== In the previous section we constructed massive superblocks containing one massive fermion and one massive boson. For each individual superblock, we found supertransformations defined up to a one common parameter $\rho$. In this section we use these results to construct complete massive supermultiplets. For that we choose appropriate solutions for each superblock and adjust their parameters so that the algebra of these supertransformations is closed. In the next subsection, we consider general properties of such construction and then present our results for the case of integer and half-integer superspins. General construction -------------------- Any massive $N=1$ supermultiplet contains two massive fermions and two massive bosons. In the notations given in the previous section (\[NotatBos\]), (\[NotatFer\]) they have the following structure $$\xymatrix {& {[s+\frac12]_{M_1}^{+}} \ar@{-}[dl]_-{\rho_1} \ar@{-}[dr]^-{\rho_3} &\\ {[s]_{M}^+} \ar@{-}[dr]_-{\rho_2} & Y = s & {[{s}]_{M'}^-} \\ & {[s-\frac12]_{M_2}^-} \ar@{-}[ur]_-{\rho_4} & } \qquad \xymatrix {& {[s-\frac12]_{M_1}^+} \ar@{-}[dl]_-{\rho_1} \ar@{-}[dr]^-{\rho_3} &\\ {[s]_{M}^+} \ar@{-}[dr]_-{\rho_2} & Y = s-\frac12 & {[{s-1}]_{M'}^-}\\ & {[s-\frac12]_{M_2}^-} \ar@{-}[ur]_-{\rho_4} & }$$ As already mentioned, the two bosonic fields must have opposite parities and it appears that the two fermionic fields must have opposite signs of the mass terms. Let us introduce notations $(f_+,\Omega_+)$ for the parity-even boson and $(f_-,\Omega_-)$ for the parity-odd one. The fermions we denote as $\Phi_+,\Phi_-$ according to the sign of $d_k$. The ansatz for the supertransformations is a combination of four possible superblocks corresponding to the lines with the parameters $\rho_{1,2,3,4}$. For example, for the parity-even boson we take: $$\begin{aligned} \delta f_+^{\alpha(k-1)\dot\alpha(k-1)} &=& \alpha_{k-1}|_{\rho_1}\Phi_+^{\alpha(k-1)\beta\dot\alpha(k-1)} \zeta_\beta - \bar\alpha_{k-1}|_{\rho_1} \Phi_+^{\alpha(k-1)\dot\alpha(k-1)\dot\beta} \zeta_{\dot\beta} \\ && + \alpha'_{k-1}|_{\rho_1} \Phi_+^{\alpha(k-1)\dot\alpha(k-2)}\zeta^{\dot\alpha} - \bar\alpha'_{k-1}|_{\rho_1} \Phi_+^{\alpha(k-2)\dot\alpha(k-1)} \zeta^{\alpha} \\ && + \alpha_{k-1}|_{\rho_2} \Phi_-^{\alpha(k-1)\beta\dot\alpha(k-1)}\zeta_\beta - \bar\alpha_{k-1}|_{\rho_2} \Phi_-^{\alpha(k-1)\dot\alpha(k-1)\dot\beta} \zeta_{\dot\beta} \\ && + \alpha'_{k-1}|_{\rho_2} \Phi_-^{\alpha(k-1)\dot\alpha(k-2)}\zeta^{\dot\alpha} - \bar\alpha'_{k-1}|_{\rho_2} \Phi_-^{\alpha(k-2)\dot\alpha(k-1)} \zeta^{\alpha} \\ \delta \Phi_+^{\alpha(k)\dot\alpha(k-1)} &=& \beta_{k-1}|_{\rho_1} \Omega_+^{\alpha(k)\dot\alpha(k-2)} \zeta^{\dot\alpha} + \gamma_{k-1}|_{\rho_1} f_+^{\alpha(k-1)\dot\alpha(k-1)} \zeta^{\alpha} \\ && + \beta'_{k}|_{\rho_1} \Omega_+^{\alpha(k)\beta\dot\alpha(k-1)}\zeta_\beta + \gamma'_{k}|_{\rho_1} f_+^{\alpha(k)\dot\alpha(k-1)\dot\beta} \zeta_{\dot\beta} \\ \delta \Phi_-^{\alpha(k)\dot\alpha(k-1)} &=& \beta_{k-1}|_{\rho_2} \Omega_+^{\alpha(k)\dot\alpha(k-2)} \zeta^{\dot\alpha} + \gamma_{k-1}|_{\rho_2} f_+^{\alpha(k-1)\dot\alpha(k-1)} \zeta^{\alpha} \\ && + \beta'_{k}|_{\rho_2} \Omega_+^{\alpha(k)\beta\dot\alpha(k-1)} \zeta_\beta + \gamma'_{k}|_{\rho_2} f_+^{\alpha(k)\dot\alpha(k-1)\dot\beta} \zeta_{\dot\beta}\end{aligned}$$ (and similarly for the lower spin components), while the ansatz for the parity-odd one can be obtained by replacement $\rho_1 \to \rho_3$ and $\rho_2 \to \rho_4$. The commutator of the two supertransformations must produce a combination of translations and Lorentz transformations: $$\label{SA} \{Q_\alpha,Q_{\dot\alpha}\}\sim P_{\alpha\dot\alpha},\quad \{Q_\alpha,Q_{\alpha}\}\sim \lambda M_{\alpha\alpha},\quad \{Q_{\dot\alpha},Q_{\dot\alpha}\}\sim \lambda M_{\dot\alpha\dot\alpha}$$ The structure of the mass-shell condition (\[MasvOn-Shell\]) shows that, for example, the commutator on the bosonic field $f_+{}^{\alpha(k-1)\dot\alpha(k-1)}$ must contain fields $\Omega_+{}^{\alpha(k)\dot\alpha(k-2)}$, $\Omega_+{}^{\alpha(k-2)\dot\alpha(k)}$, $f_+^{\alpha(k)\dot\alpha(k)}$, $f_+{}^{\alpha(k-1)\dot\alpha(k-1)}$ and $f_+{}^{\alpha(k-2)\dot\alpha(k-2)}$ only. This gives a number of relations on the parameters: $$\alpha_{k-1}|_{\rho_1} \beta'_{k}|_{\rho_1} + \alpha_{k-1}|_{\rho_2} \beta'_{k}|_{\rho_2} = 0, \qquad \alpha'_{k-1}|_{\rho_1} \beta_{k-2}|_{\rho_1} + \alpha'_{k-1}|_{\rho_2} \beta_{k-2}|_{\rho_2} = 0$$ $$\alpha_{k-1}|_{\rho_1} \beta'_{k}|_{\rho_3} + \alpha_{k-1}|_{\rho_2} \beta'_{k}|_{\rho_4} = 0, \qquad \alpha'_{k-1}|_{\rho_1} \beta_{k-2}|_{\rho_3} + \alpha'_{k-1}|_{\rho_2} \beta_{k-2}|_{\rho_4} = 0$$ $$\alpha_{k-1}|_{\rho_1} \beta_{k-1}|_{\rho_3} + \alpha'_{k-1}|_{\rho_1} \beta'_{k-1}|_{\rho_3} + \alpha_{k-1}|_{\rho_2} \beta_{k-1}|_{\rho_4} + \alpha'_{k-1}|_{\rho_2} \beta'_{k-1}|_{\rho_4} = 0$$ $$\alpha_{k-1}|_{\rho_1} \gamma_{k-1}|_{\rho_3} - \bar\alpha'_{k-1}|_{\rho_1} \bar\gamma'_{k-1}|_{\rho_3} + \alpha_{k-1}|_{\rho_2} \gamma_{k-1}|_{\rho_4} - \bar\alpha'_{k-1}|_{\rho_2} \bar\gamma'_{k-1}|_{\rho_4} = 0$$ $$\alpha_{k-1}|_{\rho_1} \gamma'_{k}|_{\rho_3} - \bar\alpha_{k-1}|_{\rho_1} \bar\gamma'_{k}|_{\rho_3} + \alpha_{k-1}|_{\rho_2} \gamma'_{k}|_{\rho_4} - \bar\alpha_{k-1}|_{\rho_2} \bar\gamma'_{k}|_{\rho_4} = 0$$ $$\alpha'_{k-1}|_{\rho_1} \gamma_{k-2}|_{\rho_3} - \bar\alpha'_{k-1}|_{\rho_1} \bar\gamma_{k-2}|_{\rho_3} + \alpha'_{k-1}|_{\rho_2} \gamma_{k-2}|_{\rho_4} - \bar\alpha'_{k-1}|_{\rho_2} \bar\gamma_{k-2}|_{\rho_4} = 0$$ If these relations are fulfilled the resulting expression for the commutator has the form: $$\begin{aligned} \ [\delta_1, \delta_2 ] f_+^{\alpha(k-1)\dot\alpha(k-1)} &=& (\alpha_{k-1}|_{\rho_1} \beta_{k-1}|_{\rho_1} + \alpha'_{k-1}|_{\rho_1} \beta'_{k-1}|_{\rho_1} + \alpha_{k-1}|_{\rho_2} \beta_{k-1}|_{\rho_2} + \alpha'_{k-1}|_{\rho_2} \beta'_{k-1}|_{\rho_2}) \\ && \cdot [\Omega_+^{\alpha(k-1)\gamma\dot\alpha(k-2)} (\zeta_1^{\dot\alpha} \zeta_{2\gamma} - \zeta_2^{\dot\alpha}\zeta_{1\gamma}) + \Omega_+^{\alpha(k-2)\dot\alpha(k-1)\dot\gamma} (\zeta_1^{\alpha} \zeta_{2\dot\gamma} - \zeta_2^{\alpha} \zeta_{1\dot\gamma})] \\ && + (\alpha_{k-1}|_{\rho_1} \gamma'_k|_{\rho_1} + \alpha_{k-1}|_{\rho_2} \gamma'_k|_{\rho_2}) f_+^{\alpha(k-1)\gamma\dot\alpha(k-1)\dot\beta} (\zeta_{1\dot\beta} \zeta_{2\gamma} - \zeta_{2\dot\beta}\zeta_{1\gamma}) \\ && + (\alpha'_{k-1}|_{\rho_1} \gamma_{k-2}|_{\rho_1} + \alpha'_{k-1}|_{\rho_2} \gamma_{k-2}|_{\rho_2}) f_+^{\alpha(k-2)\dot\alpha(k-2)} (\zeta_1^{\alpha} \zeta_2^{\dot\alpha} - \zeta_2^{\alpha} \zeta_1^{\dot\alpha}) \\ && + (\alpha_{k-1}|_{\rho_1} \gamma_{k-1}|_{\rho_1} + \alpha'_{k-1}|_{\rho_1} \gamma'_{k-1}|_{\rho_1} + \alpha_{k-1}|_{\rho_2} \gamma_{k-1}|_{\rho_2} + \alpha'_{k-1}|_{\rho_2} \gamma'_{k-1}|_{\rho_2}) \\ && \cdot [ f_+^{\alpha(k-2)\gamma\dot\alpha(k-1)} (\zeta_1^{\alpha}\zeta_{2\gamma} - \zeta_2^{\alpha} \zeta_{1\gamma}) + f_+^{\alpha(k-1)\dot\alpha(k-2)\dot\gamma} (\zeta_1^{\dot\alpha}\zeta_{2\dot\gamma} - \zeta_2^{\dot\alpha} \zeta_{1\dot\gamma})]\end{aligned}$$ Let us stress that the coefficients in this expression must be $k$-independent. This gives additional restrictions on the parameters and also serves as a quite non-trivial test for our calculations. Thus to construct massive supermultiplets we start with the four suitable massive superblocks with four free parameters $\rho_{1,2,3,4}$. Then we require that the commutator of the two supertransformations on the bosonic fields be closed. In the next two subsections we apply this scheme to the massive supermultiplets with half-integer $Y = s-1/2$ and integer $Y=s$ superspins. Half-integer superspin $S-1/2$ ------------------------------ The massive superspin-(s-1/2) supermultiplet contains $$\xymatrix {& {[s-\frac12]_{M_1}^{+}} \ar@{-}[dl]_-{\rho_1} \ar@{-}[dr]^-{\rho_3} &\\ {[s]_{M}^+} \ar@{-}[dr]_-{\rho_2} & S-1/2 & {[{s-1}]_{M'}^-}\\ & {[s-\frac12]_{M_2}^{-}} \ar@{-}[ur]_-{\rho_4} & }$$ Firstly, we note that four superblocks give the following relations on the mass parameters: $$\begin{aligned} & M^2=M_1(M_1+\lambda) && {M'}^2=M_1(M_1+\lambda) \\ & M^2=M_2(M_2-\lambda) && {M'}^2=M_2(M_2-\lambda)\end{aligned}$$ Their solution is $$M^2={M'}^2=M_1(M_1+\lambda),\qquad M_2=M_1+\lambda$$ All the conditions for the closure of the superalgebra are fulfilled provided: $$\rho_1{}^2 = \rho_2{}^2 = \rho_3{}^2 = \rho_4{}^2, \qquad \rho_1\rho_3 = \rho_2\rho_4 \label{ConHI1}$$ If the relations (\[ConHI1\]) are satisfied then the commutators of the supertransformations on parity-even spin-$s$ $f_+$ and parity-odd spin-$(s-1)$ $f_-$ fields have the same form: $$\begin{aligned} \frac{1}{i\rho_0{}^2}{[\delta_1,\delta_2]} f_\pm^{\alpha(k-1)\dot\alpha(k-1)} &=& \Omega_\pm^{\alpha(k-1)\gamma\dot\alpha(k-2)} (\zeta_1^{\dot\alpha} \zeta_{2\gamma} - \zeta_2^{\dot\alpha} \zeta_{1\gamma}) + \Omega_\pm^{\alpha(k-2)\dot\alpha(k-1)\dot\gamma} (\zeta_1^{\alpha}\zeta_{2\dot\gamma} - \zeta_2^{\alpha} \zeta_{1\dot\gamma}) \\ && + \frac{(k-1)a_{k+1}}{2(k+1)} f_\pm^{\alpha(k-1)\gamma\dot\alpha(k-1)\dot\beta} (\zeta_{1\dot\beta}\zeta_{2\gamma} - \zeta_{2\dot\beta} \zeta_{1\gamma}) \\ && + \frac{a_{k}}{2k(k-1)} f_\pm^{\alpha(k-2)\dot\alpha(k-2)} (\zeta_1^{\alpha} \zeta_2^{\dot\alpha} - \zeta_2^{\alpha} \zeta_1^{\dot\alpha}) \\ && +\lambda[ f_\pm^{\alpha(k-2)\gamma\dot\alpha(k-1)}(\zeta_1^{\alpha} \zeta_{2\gamma} - \zeta_2^{\alpha} \zeta_{1\gamma}) + f_\pm^{\alpha(k-1)\dot\alpha(k-2)\dot\gamma} (\zeta_1^{\dot\alpha}\zeta_{2\dot\gamma} - \zeta_2^{\dot\alpha} \zeta_{1\dot\gamma})]\end{aligned}$$ $$\begin{aligned} \frac{1}{i\rho_0{}^2}{[\delta_1,\delta_2]} f_\pm^{\alpha\dot\alpha} &=& \Omega_\pm^{\alpha\gamma} (\zeta_1^{\dot\alpha} \zeta_{2\gamma} -\zeta_2^{\dot\alpha} \zeta_{1\gamma}) + \Omega_\pm^{\dot\alpha\dot\gamma} (\zeta_1^{\alpha} \zeta_{2\dot\gamma} - \zeta_2^{\alpha} \zeta_{1\dot\gamma}) \\ && + \frac{a_3}{6} f_\pm^{\alpha\gamma\dot\alpha\dot\beta} (\zeta_{1\dot\beta} \zeta_{2\gamma} - \zeta_{2\dot\beta}\zeta_{1\gamma}) - \frac{a_0}{4} A_\pm (\zeta_1^\alpha \zeta_2^{\dot\alpha} - \zeta_2^\alpha \zeta_1^{\dot\alpha}) \\ && + \lambda[ f_\pm^{\gamma\dot\alpha} (\zeta_1^{\alpha}\zeta_{2\gamma} - \zeta_2^{\alpha} \zeta_{1\gamma}) + f_\pm^{\alpha\dot\gamma} (\zeta_1^{\dot\alpha} \zeta_{2\dot\gamma} - \zeta_2^{\dot\alpha} \zeta_{1\dot\gamma})] \\ \frac{1}{i\rho_0{}^2}{[\delta_1,\delta_2]} A_\pm &=& - 2[e_{\beta\dot\beta} B_\pm^{\alpha\beta} (\zeta_1^{\dot\beta}\zeta_{2\alpha} - \zeta_2^{\dot\beta} \zeta_{1\alpha}) + e_{\beta\dot\beta} B_\pm^{\dot\alpha\dot\beta} (\zeta_1^{\beta}\zeta_{2\dot\alpha} - \zeta_2^{\beta} \zeta_{1\dot\alpha})] \\ && - \frac{a_0}{2} f_\pm^{\alpha\dot\beta} (\zeta_{1\dot\beta}\zeta_{2\alpha} - \zeta_{2\dot\beta} \zeta_{1\alpha}) \\ \frac{1}{i\rho_0{}^2}{[\delta_1,\delta_2]} \varphi_\pm &=& \pi_\pm^{\alpha\dot\alpha} (\zeta_{1\dot\alpha} \zeta_{2\alpha} - \zeta_{2\dot\alpha} \zeta_{1\alpha})\end{aligned}$$ where $a_k$ is determined by (\[boson\_date\]) for spin $s$ and spin $(s-1)$ respectively and $$\rho_0{}^2=\frac{s(2M_1+\lambda)}{2}\rho_1{}^2$$ Integer superspin $S$ --------------------- The massive superspin-s supermultiplet contains $$\xymatrix {& {[s+\frac12]_{M_1}^{+}} \ar@{-}[dl]_-{\rho_1} \ar@{-}[dr]^-{\rho_3} &\\ {[s]_{M}^{\uparrow}} \ar@{-}[dr]_-{\rho_2} & S & {[{s}]_{M'}^{\downarrow}}\\ & {[s-\frac12]_{M_2}^{-}} \ar@{-}[ur]_-{\rho_4} & }$$ First of all, we note that the four superblocks give the following relations on the mass parameters $$\begin{aligned} & M^2=M_1(M_1-\lambda) && {M'}^2=M_1(M_1+\lambda) \\ & M^2=M_2(M_2-\lambda) && {M'}^2=M_2(M_2+\lambda)\end{aligned}$$ Their solution is $$M^2 = M_1(M_1-\lambda), \qquad {M'}^2 = M_1(M_1+\lambda), \qquad M_2 = M_1$$ The requirement that the superalgebra be closed again leads to $$\rho_1{}^2 = \rho_2{}^2 = \rho_3{}^2 = \rho_4{}^2, \qquad \rho_1\rho_3 = \rho_2\rho_4\label{ConI1}$$ If the relations (\[ConI1\]) are satisfied then commutators of supertransformations on parity-even and parity-odd bosonic spin-$s$ fields have the same form: $$\begin{aligned} \frac{1}{i\rho_0{}^2}{[\delta_1,\delta_2]} f_\pm^{\alpha(k-1)\dot\alpha(k-1)} &=& \Omega_\pm^{\alpha(k-1)\gamma\dot\alpha(k-2)} (\zeta_1^{\dot\alpha}\zeta_{2\gamma} - \zeta_2^{\dot\alpha} \zeta_{1\gamma}) + \Omega_\pm^{\alpha(k-2)\dot\alpha(k-1)\dot\gamma} (\zeta_1^{\alpha}\zeta_{2\dot\gamma} - \zeta_2^{\alpha} \zeta_{1\dot\gamma}) \\ && + \frac{(k-1)a_{k+1}}{2(k+1)} f_\pm^{\alpha(k-1)\gamma\dot\alpha(k-1)\dot\beta} (\zeta_{1\dot\beta}\zeta_{2\gamma} - \zeta_{2\dot\beta} \zeta_{1\gamma}) \\ && + \frac{a_{k}}{2k(k-1)} f_\pm^{\alpha(k-2)\dot\alpha(k-2)}(\zeta_1^{\alpha} \zeta_2^{\dot\alpha} - \zeta_2^{\alpha} \zeta_1^{\dot\alpha}) \\ && +\lambda[ f_\pm^{\alpha(k-2)\gamma\dot\alpha(k-1)}(\zeta_1^{\alpha} \zeta_{2\gamma} - \zeta_2^{\alpha} \zeta_{1\gamma}) + f_\pm^{\alpha(k-1)\dot\alpha(k-2)\dot\gamma} (\zeta_1^{\dot\alpha}\zeta_{2\dot\gamma} - \zeta_2^{\dot\alpha} \zeta_{1\dot\gamma})] \\ \frac{1}{i\rho_0{}^2}{[\delta_1,\delta_2]} f_\pm^{\alpha\dot\alpha} &=& \Omega_\pm^{\alpha\gamma} (\zeta_1^{\dot\alpha} \zeta_{2\gamma} -\zeta_2^{\dot\alpha} \zeta_{1\gamma}) + \Omega_\pm^{\dot\alpha\dot\gamma} (\zeta_1^{\alpha} \zeta_{2\dot\gamma} - \zeta_2^{\alpha} \zeta_{1\dot\gamma}) \\ && + \frac{a_3}{6} f_\pm^{\alpha\gamma\dot\alpha\dot\beta} (\zeta_{1\dot\beta} \zeta_{2\gamma} - \zeta_{2\dot\beta}\zeta_{1\gamma}) -\frac{a_0}{4} A_\pm (\zeta_1^\alpha \zeta_2^{\dot\alpha} - \zeta_2^\alpha \zeta_1^{\dot\alpha}) \\ && +\lambda[ f_\pm^{\gamma\dot\alpha} (\zeta_1^{\alpha}\zeta_{2\gamma} - \zeta_2^{\alpha} \zeta_{1\gamma}) + f_\pm^{\alpha\dot\gamma} (\zeta_1^{\dot\alpha} \zeta_{2\dot\gamma} - \zeta_2^{\dot\alpha} \zeta_{1\dot\gamma})] \\ \frac{1}{i\rho_0{}^2}{[\delta_1,\delta_2]} A_\pm &=& - 2 [e_{\beta\dot\beta} B_\pm^{\alpha\beta} (\zeta_1^{\dot\beta}\zeta_{2\alpha} - \zeta_2^{\dot\beta} \zeta_{1\alpha}) + e_{\beta\dot\beta} B_\pm^{\dot\alpha\dot\beta} (\zeta_1^{\beta}\zeta_{2\dot\alpha} - \zeta_2^{\beta} \zeta_{1\dot\alpha})] \\ && -\frac{a_0}{2} f_\pm^{\alpha\dot\beta} (\zeta_{1\dot\beta}\zeta_{2\alpha} - \zeta_{2\dot\beta} \zeta_{1\alpha}) \\ \frac{1}{i\rho_0{}^2}{[\delta_1,\delta_2]} \varphi_\pm &=& \pi_\pm^{\alpha\dot\alpha} (\zeta_{1\dot\alpha} \zeta_{2\alpha} - \zeta_{2\dot\alpha} \zeta_{1\alpha})\end{aligned}$$ where $a_k$ is determined by (\[boson\_date\]) for spin $s$ and $$\rho_0{}^2=\frac{(2s+1)M_1}{2}\rho_1{}^2$$ Summary ======= In this paper we have developed the component Lagrangian description of massive on-shell $N=1$ supermultiplets with arbitrary (half)integer superspin in four dimensional Anti de Sitter space ($AdS_4$). The derivation is based on supersymmetric generalization of frame-like gauge invariant formulation of massive higher spin fields where massive supermultiplets are described by an appropriate set of massless ones. We show that $N=1$ massive supermultiplets can be constructed as a combination of four massive superblocks, each containing one massive boson and one massive fermion. As a result, we have derived both the supertransformations for the components of the one-shell supermultiplets and the corresponding invariant Lagrangians. Thus, the component Lagrangian formulation of the $N=1$ supersymmetric free massive higher spin field theory on the $AdS_4$ space can be considered complete. Let us briefly discuss the possible further generalizations of the results obtained. As we already pointed out, the problem of off-shell supersymmetric massive higher spin theory remains open in general. Only a few examples of such a theory with concrete superspins [@BG1], [@BG2], [@BG3] in flat space have been developed. There are no known examples in the $AdS$ space. There are two possible approaches to study this general problem. One option is to start with on-shell theory and try to find the necessary auxiliary fields closing the superalgebra on the base of Noether’s procedure. Another approach can be based on the use of the superfield techniques from the very beginning. At present, realization of both these approaches seems unclear and will require the development of new methods. Besides, the interesting generalizations of the results obtained can be constructing the partially massless $N=1$ supermultiplets and finding at least on-shell component Lagrangian description for $N$-extended massive supermultiplets in flat and $AdS$ spaces. We hope to attack these problems in the forthcoming works. Acknowledgments {#acknowledgments .unnumbered} =============== The authors are grateful to S.M. Kuzenko for useful comments. I.L.B and T.V.S are thankful to the RFBR grant, project No. 18-02-00153-a for partial support. Their research was also supported in parts by Russian Ministry of Science and High Education, project No. 3.1386.2017. Identities ========== In this appendix we present explicit expression of identities for massive higher spin superblocks. Auxiliary bosonic fields ------------------------ Identities that correspond to (\[SchemIden1\]) $$\begin{aligned} 0 &=& {\cal F}_{\alpha(k-1)\beta\dot\alpha(k-1)} e^\beta{}_{\dot\beta} \Omega^{\alpha(k-2)\dot\alpha(k-1)\dot\beta} \zeta^{\alpha} + \Phi_{\alpha(k-2)\beta\gamma\dot\alpha(k-1)} e^\beta{}_{\dot\beta} {\cal R}^{\alpha(k-2)\dot\alpha(k-1)\dot\beta} \zeta^{\gamma} \\ && - 2{d_k}[(k+1) E^{\dot\beta}{}_{\dot\gamma} \Psi^{\alpha(k-2)\gamma\dot\alpha(k-1)\dot\gamma} \Omega_{\alpha(k-2)\dot\alpha(k-1)\dot\beta} \zeta_{\gamma} \\ && - (k-2) E_\beta{}^\alpha \Psi^{\alpha(k-3)\beta\gamma\dot\alpha(k)} \Omega_{\alpha(k-2)\dot\alpha(k)} \zeta_{\gamma} - E_\beta{}^\gamma \Psi^{\alpha(k-2)\beta\dot\alpha(k)} \Omega_{\alpha(k-2)\dot\alpha(k)} \zeta_{\gamma} ] \\ && - \frac{b_k}{2k} [(k+1)E^\beta{}_\delta \Phi_{\alpha(k-2)\beta\gamma\dot\alpha(k-1)} f^{\alpha(k-2)\delta\dot\alpha(k-1)} - (k-1) E_{\dot\beta}{}^{\dot\alpha} \Phi_{\alpha(k-1)\gamma\dot\alpha(k-1)}f^{\alpha(k-1)\dot\alpha(k-2)\dot\beta} ]\zeta^{\gamma} \\ && + \lambda [E^{\alpha\beta} \Phi_{\alpha(k-1)\beta\dot\alpha(k-1)} \Omega^{\alpha(k-2)\dot\alpha(k-1)\dot\beta} \zeta_{\dot\beta}] \\ && - c_{k+1}[-E_{\beta\gamma} \Phi^{\alpha(k-2)\delta\beta\gamma\dot\alpha(k)} \Omega_{\alpha(k-2)\dot\alpha(k)} \zeta_{\delta}] \\ && - \frac{(k-1)c_k}{(k-1)(k+1)} [kE^{\dot\beta\dot\alpha}\Phi^{\alpha(k-2)\delta\dot\alpha(k-2)} \Omega_{\alpha(k-2)\dot\alpha(k-1)\dot\beta} \zeta_{\delta} + E^{\dot\beta\dot\alpha} \Phi^{\alpha(k-2)\beta\dot\alpha(k-2)} \Omega_{\alpha(k-2)\dot\alpha(k-1)\dot\beta} \zeta_{\beta}] \\ && - \frac{a_{k+1}}{2} [- E_{\dot\beta\dot\gamma} \Phi_{\alpha(k-1)\delta\dot\alpha(k-1)} \Omega^{\alpha(k-1)\dot\alpha(k-1)\dot\beta\dot\gamma} ]\zeta^{\delta}\\ && - \frac{(k-2)a_k}{2k(k+1)} [(k+1)E^{\beta\alpha} \Phi_{\alpha(k-2)\beta\gamma\dot\alpha(k-1)} \Omega^{\alpha(k-3)\dot\alpha(k-1)}] \zeta^{\gamma}\end{aligned}$$ $$\begin{aligned} 0 &=& {\cal F}_{\alpha(k-2)\gamma\dot\alpha(k-2)} e^\gamma{}_{\dot\gamma} \Omega^{\alpha(k-2)\dot\alpha(k-2)\dot\gamma\dot\beta} \zeta_{\dot\beta} + \Phi_{\alpha(k-2)\gamma\dot\alpha(k-2)} e^\gamma{}_{\dot\gamma} {\cal R}^{\alpha(k-2)\dot\alpha(k-2)\dot\gamma\dot\beta} \zeta_{\dot\beta} \\ && + 2{d_{k-1}}[k E^{\dot\gamma}{}_{\dot\delta} \Psi^{\alpha(k-2)\dot\alpha(k-2)\dot\delta} \Omega_{\alpha(k-2)\dot\alpha(k-2)\dot\gamma\dot\beta} \zeta^{\dot\beta} \\ && - (k-2)E_\gamma{}^\beta \Psi^{\alpha(k-3)\gamma\dot\alpha(k-1)} \Omega_{\alpha(k-3)\beta\dot\alpha(k-1)\dot\beta} \zeta^{\dot\beta}] \\ && - \frac{b_{k}}{2k} [- (k-2)E_{\dot\gamma}{}^{\dot\alpha}\Phi_{\alpha(k-1)\dot\alpha(k-2)} f^{\alpha(k-1)\dot\alpha(k-3)\dot\beta\dot\gamma} + (k+1) E^\gamma{}_\beta \Phi_{\alpha(k-2)\gamma\dot\alpha(k-2)} f^{\alpha(k-2)\beta\dot\alpha(k-2)\dot\beta}) \\ && - E_{\dot\gamma}{}^{\dot\beta} \Phi_{\alpha(k-1)\dot\alpha(k-2)} f^{\alpha(k-1)\dot\alpha(k-2)\dot\gamma}] \zeta_{\dot\beta} \\ && + \lambda [E_{\dot\beta\dot\gamma} \Phi_{\alpha(k-2)\gamma\dot\alpha(k-2)} \Omega^{\alpha(k-2)\dot\alpha(k-2)\dot\gamma\dot\beta} \zeta^\gamma] \\ && + c_{k}[-E_{\gamma\delta} \Phi^{\alpha(k-2)\gamma\delta\dot\alpha(k-1)} \Omega_{\alpha(k-2)\dot\alpha(k-1)\dot\beta} \zeta^{\dot\beta}] \\ && + \frac{(k-2)c_{k-1}}{k(k-2)} [kE^{\dot\gamma\dot\alpha}\Phi^{\alpha(k-2)\dot\alpha(k-3)} \Omega_{\alpha(k-2)\dot\alpha(k-2)\dot\gamma\dot\beta} \zeta^{\dot\beta}] \\ && - \frac{a_{k+1}}{2} [- E_{\dot\gamma\dot\delta} \Phi_{\alpha(k-1)\dot\alpha(k-2)} \Omega^{\alpha(k-1)\dot\alpha(k-2)\dot\gamma\dot\beta\dot\delta}] \zeta_{\dot\beta} \\ && - \frac{(k-2)a_k}{2k(k+1)} [(k+1)E^{\gamma\alpha} \Phi_{\alpha(k-2)\gamma\dot\alpha(k-2)} \Omega^{\alpha(k-3)\dot\alpha(k-2)\dot\beta}] \zeta_{\dot\beta}\end{aligned}$$ $$\begin{aligned} 0 &=& {\cal F}_{\alpha\beta\dot\alpha}e^\beta{}_{\dot\beta} \Omega^{\dot\alpha\dot\beta}\zeta^{\alpha} + \Phi_{\beta\gamma\dot\alpha} e^\beta{}_{\dot\beta} {\cal R}^{\dot\alpha\dot\beta}\zeta^{\gamma} \\ && - 2d_2[3E^{\dot\beta}{}_{\dot\gamma} \Psi^{\gamma\dot\alpha\dot\gamma} \Omega_{\dot\alpha\dot\beta} \zeta_{\gamma} - E_\beta{}^\gamma \Psi^{\beta\dot\alpha(2)} \Omega_{\dot\alpha(2)} \zeta_{\gamma} ] \\ && - \frac{b_2}{4} [3E^\beta{}_\delta\Phi_{\beta\gamma\dot\alpha}f^{\delta\dot\alpha} - E_{\dot\beta}{}^{\dot\alpha} \Phi_{\alpha\gamma\dot\alpha} f^{\alpha\dot\beta}]\zeta^{\gamma} +\lambda [E^{\alpha\beta} \Phi_{\alpha\beta\dot\alpha} \Omega^{\dot\alpha\dot\beta} \zeta_{\dot\beta}] \\ && - c_3[-E_{\beta\gamma} \Phi^{\delta\beta\gamma\dot\alpha(2)} \Omega_{\dot\alpha(2)}\zeta_{\delta}] -\ frac{c_2}{3} [2E^{\dot\beta\dot\alpha} \Phi^{\delta} \Omega_{\dot\alpha\dot\beta}\zeta_{\delta} + E^{\dot\beta\dot\alpha} \Phi^{\beta} \Omega_{\dot\alpha\dot\beta} \zeta_{\beta}] \\ && - \frac{a_{3}}{2} [- E_{\dot\beta\dot\gamma} \Phi_{\alpha\delta\dot\alpha} \Omega^{\alpha\dot\alpha\dot\beta\dot\gamma}] \zeta^{\delta} - [\frac{a_0}{4} (4E^\beta{}_{\dot\gamma} \Phi_{\alpha\beta\dot\alpha} B^{\dot\alpha\dot\gamma}) - \frac{a_0\tilde{a}_0}{8} E^{\beta\dot\alpha} \Phi_{\alpha\beta\dot\alpha}\varphi] \zeta^{\alpha} \\ 0 &=& {\cal F}_{\alpha}e^\alpha{}_{\dot\alpha} \Omega^{\dot\alpha\dot\beta} \zeta_{\dot\beta} + \Phi_{\alpha} e^\alpha{}_{\dot\alpha} {\cal R}^{\dot\alpha\dot\beta} \zeta_{\dot\beta} \\ && + [4{d_{1}}E_{\dot\alpha\dot\beta} \Phi^{\dot\beta} \Omega^{\dot\alpha\dot\gamma} \zeta_{\dot\gamma} - c_{2} E_{\alpha\beta} \Phi^{\alpha\beta\dot\beta} \Omega_{\dot\beta\dot\gamma} \zeta^{\dot\gamma} + c_0 E_{\beta\dot\alpha} \phi^\beta \Omega^{\dot\alpha\dot\gamma} \zeta_{\dot\gamma}] \\ && - [-\frac{a_3}{2} E_{\dot\alpha\dot\beta} \Phi_{\beta} \Omega^{\beta\dot\alpha\dot\gamma\dot\beta} + \frac{b_2}{4} (3E^\alpha{}_\beta \Phi_{\alpha} f^{\beta\dot\gamma} - E_{\dot\alpha}{}^{\dot\gamma} \Phi_{\beta} f^{\beta\dot\alpha}) \\ && + {a_0}E^\alpha{}_{\dot\beta} \Phi_{\alpha} B^{\dot\gamma\dot\beta} - \frac{a_0\tilde{a}_0}{8} E^{\alpha\dot\gamma} \Phi_{\alpha}\varphi] \zeta_{\dot\gamma} + \lambda E_{\dot\beta\dot\alpha} \Phi_{\alpha} \Omega^{\dot\alpha\dot\beta} \zeta^\alpha\end{aligned}$$ $$\begin{aligned} 0 &=& E_{\dot\alpha\dot\beta} {\cal F}_{\alpha} B^{\dot\alpha\dot\beta} \zeta^{\alpha} - E_{\dot\alpha\dot\beta}\Phi_{\alpha} {\cal C}^{\dot\alpha\dot\beta} \zeta^{\alpha} + [4{d_{1}} E^\alpha{}_{\dot\gamma} \Phi_{\dot\alpha} B^{\dot\alpha\dot\gamma} \zeta_{\alpha} - 2c_2E_\beta{}_{\dot\gamma}\Phi^{\alpha\beta\dot\beta} B_{\dot\beta}{}^{\dot\gamma}\zeta_{\alpha}] \\ && + [-\frac{a_0}{2} E_{\dot\alpha\dot\beta} \Phi_{\alpha}\Omega^{\dot\alpha\dot\beta} - \frac{\tilde{a}_0}{4} E_{\beta\dot\beta}\Phi_{\alpha} \pi^{\beta\dot\beta}]\zeta^{\alpha} - 2\lambda E^\alpha{}_{\dot\beta} \Phi_{\alpha}B^{\dot\alpha\dot\beta} \zeta_{\dot\alpha} \\ 0 &=& {\cal C}_\alpha E^\alpha{}_{\dot\alpha} B^{\dot\alpha\dot\beta}\zeta_{\dot\beta} - \phi_\alpha E^\alpha{}_{\dot\alpha} {\cal C}^{\dot\alpha\dot\beta} \zeta_{\dot\beta} \\ && + [c_0E_\alpha{}_{\dot\alpha} \Phi^{\alpha} B^{\dot\alpha\dot\gamma} \zeta_{\dot\gamma} + 4d_1E\phi^{\dot\alpha} B^{\dot\alpha\dot\gamma} \zeta_{\dot\gamma}] +[-\frac{a_0}{2} E^\alpha{}_{\dot\alpha} \phi_\alpha \Omega^{\dot\alpha\dot\beta} + \frac{\tilde{a}_0}{8} E\phi_\beta \pi^{\beta\dot\beta}] \zeta_{\dot\beta} \\ 0 &=& - E^\alpha{}_{\dot\alpha} {\cal C}_\alpha \pi^{\beta\dot\alpha} \zeta_{\beta} + \frac12 E_\gamma{}_{\dot\alpha}\phi^\beta {\cal C}^{\gamma\dot\alpha}\zeta_{\beta} -[-c_0E_\alpha{}_{\dot\alpha} \Phi^{\alpha} \pi^{\beta\dot\alpha}\zeta_{\beta} -4d_1E\phi_{\dot\alpha} \pi^{\beta\dot\alpha}\zeta_{\beta}] \\ && + [\frac{a_0\tilde{a}_0}{24} E^\alpha{}_{\dot\alpha} \phi_\alpha f^{\beta\dot\alpha} - \frac{\tilde{a}_0}{12} (-2E\phi_\gamma B^{\beta\gamma}) - \frac{a_0{}^2}{4} \phi^\beta \varphi] \zeta_{\beta} + \lambda E\phi_\beta \pi^{\beta\dot\alpha} \zeta_{\dot\alpha}\end{aligned}$$ Physical bosonic fields ----------------------- Identities that correspond to (\[SchemIden2\]) $$\begin{aligned} 0 &=& {\cal F}_{\alpha(k-1)\beta\dot\alpha(k-1)} e^\beta{}_{\dot\beta} f^{\alpha(k-1)\dot\alpha(k-1)} \zeta^{\dot\beta} \\ && - 2{d_k}[(k+1)E^{\dot\beta}{}_{\dot\gamma} \Psi^{\alpha(k-1)\dot\alpha(k-1)\dot\gamma} f_{\alpha(k-1)\dot\alpha(k-1)} \zeta_{\dot\beta} \\ && - (k-1)E_\beta{}^\alpha \Psi^{\alpha(k-2)\beta\dot\alpha(k-1)\dot\beta} f_{\alpha(k-1)\dot\alpha(k-1)}\zeta_{\dot\beta} ] \\ && - [ (k-1)(- E_{\dot\beta}{}^{\dot\alpha} \Phi_{\alpha(k)\dot\alpha(k-1)} \Omega^{\alpha(k)\dot\alpha(k-2)} + E^\beta{}_\gamma \Phi_{\alpha(k-1)\beta\dot\alpha(k-2)\dot\beta} \Omega^{\alpha(k-1)\gamma\dot\alpha(k-2)}) \zeta^{\dot\beta} \\ && + (k-1)E^{\beta\alpha} \Phi_{\alpha(k-1)\beta\dot\alpha(k-1)}\Omega^{\alpha(k-2)\dot\alpha(k-1)\dot\gamma} \zeta_{\dot\gamma}] \\ && - 2\lambda E_\gamma{}^\beta \Phi_{\alpha(k-1)\beta\dot\alpha(k-1)}f^{\alpha(k-1)\dot\alpha(k-1)} \zeta^{\gamma} \\ && - c_{k+1}[-E_{\beta\gamma} \Phi^{\alpha(k-1)\beta\gamma\dot\alpha(k-1)\dot\beta} f_{\alpha(k-1)\dot\alpha(k-1)} \zeta_{\dot\beta}] \\ && - \frac{(k-1)c_k}{(k-1)(k+1)} [(k+1)E^{\dot\beta\dot\alpha}\Phi^{\alpha(k-1)\dot\alpha(k-2)} f_{\alpha(k-1)\dot\alpha(k-1)}\zeta_{\dot\beta} \\ && - (k-1)E_\beta{}^\alpha \Phi^{\alpha(k-2)\beta\dot\alpha(k-2)} f_{\alpha(k-1)\dot\alpha(k-2)\dot\beta} \zeta^{\dot\beta}] \\ && - \frac{(k-1)a_{k+1}}{2(k+1)}[- E_{\dot\beta\dot\gamma}\Phi_{\alpha(k)\dot\alpha(k-1)} f^{\alpha(k)\dot\alpha(k-1)\dot\gamma}\zeta^{\dot\beta} + E^\beta{}_\gamma \Phi_{\alpha(k-1)\beta\dot\alpha(k-1)} f^{\alpha(k-1)\gamma\dot\alpha(k-1)\dot\gamma} \zeta_{\dot\gamma} ] \\ && - \frac{(k-1)^2a_k}{2k(k-1)}[E^{\beta\alpha} \Phi_{\alpha(k-1)\beta\dot\alpha(k-2)\dot\beta} f^{\alpha(k-2)\dot\alpha(k-2)}] \zeta^{\dot\beta}\end{aligned}$$ $$\begin{aligned} 0 &=& {\cal F}_{\alpha(k-1)\beta\dot\alpha(k-2)\dot\gamma} e^\beta{}_{\dot\beta} f^{\alpha(k-1)\dot\alpha(k-2)\dot\beta}\zeta^{\dot\gamma} \\ && - 2{d_k}[(k+1)E^{\dot\beta}{}_{\dot\delta} \Psi^{\alpha(k-1)\dot\alpha(k-2)\dot\gamma\dot\delta} f_{\alpha(k-1)\dot\alpha(k-2)\dot\beta} \zeta_{\dot\gamma} \\ && - (k-1)E_\beta{}^\alpha \Psi^{\alpha(k-2)\beta\dot\alpha(k-1)\dot\gamma} f_{\alpha(k-1)\dot\alpha(k-1)} \zeta_{\dot\gamma}] \\ && - [kE^\beta{}_\delta \Phi_{\alpha(k-1)\beta\dot\alpha(k-2)\dot\gamma} \Omega^{\alpha(k-1)\delta\dot\alpha(k-2)} -(k-2) E_{\dot\beta}{}^{\dot\alpha} \Phi_{\alpha(k)\dot\alpha(k-2)\dot\gamma}\Omega^{\alpha(k)\dot\alpha(k-3)\dot\beta} ]\zeta^{\dot\gamma} \\ && + \lambda[ E^{\dot\gamma}{}_{\dot\beta} \Phi_{\alpha(k-1)\gamma\dot\alpha(k-2)\dot\gamma} f^{\alpha(k-1)\dot\alpha(k-2)\dot\beta} \zeta^{\gamma} - E_\gamma{}^\beta \Phi_{\alpha(k-1)\beta\dot\alpha(k-1)} f^{\alpha(k-1)\dot\alpha(k-1)} \zeta^{\gamma}] \\ && - c_{k+1}[-E_{\beta\delta} \Phi^{\alpha(k-1)\beta\delta\dot\alpha(k-1)\dot\gamma} f_{\alpha(k-1)\dot\alpha(k-1)} \zeta_{\dot\gamma}] \\ && - \frac{c_k}{(k-1)(k+1)}[(k-2)(k+1) E^{\dot\beta\dot\alpha}\Phi^{\alpha(k-1)\dot\alpha(k-3)\dot\gamma} f_{\alpha(k-1)\dot\alpha(k-2)\dot\beta} \zeta_{\dot\gamma} \\ && + (k+1) E^{\dot\beta\dot\gamma} \Phi^{\alpha(k-1)\dot\alpha(k-2)} f_{\alpha(k-1)\dot\alpha(k-2)\dot\beta} \zeta_{\dot\gamma} + (k-1) E_\beta{}^\alpha \Phi^{\alpha(k-2)\beta\dot\alpha(k-2)} f_{\alpha(k-1)\dot\alpha(k-2)\dot\beta} \zeta^{\dot\beta}] \\ && - \frac{(k-1)a_{k+1}}{2(k+1)} [- E_{\dot\beta\dot\gamma}\Phi_{\alpha(k)\dot\alpha(k-2)\dot\delta} f^{\alpha(k)\dot\alpha(k-2)\dot\beta\dot\gamma} ]\zeta^{\dot\delta} \\ && - \frac{(k-1)a_k}{2k(k-1)} [kE^{\beta\alpha} \Phi_{\alpha(k-1)\beta\dot\alpha(k-2)\dot\gamma} f^{\alpha(k-2)\dot\alpha(k-2)}] \zeta^{\dot\gamma}\end{aligned}$$ $$\begin{aligned} 0 &=& {\cal F}_{\alpha(k-2)\gamma\dot\alpha(k-2)} e^\gamma{}_{\dot\gamma} f^{\alpha(k-2)\beta\dot\alpha(k-2)\dot\gamma}\zeta_\beta \\ && + 2{d_{k-1}}[k E^{\dot\gamma}{}_{\dot\beta} \Psi^{\alpha(k-2)\dot\alpha(k-2)\dot\beta} f_{\alpha(k-2)\beta\dot\alpha(k-2)\dot\gamma} \zeta^\beta \\ && - (k-2)E_\gamma{}^\alpha \Psi^{\alpha(k-3)\gamma\dot\alpha(k-1)} f_{\alpha(k-2)\beta\dot\alpha(k-1)} \zeta^\beta] \\ && - [-(k-2) E_{\dot\gamma}{}^{\dot\alpha} \Phi_{\alpha(k-1)\dot\alpha(k-2)} \Omega^{\alpha(k-1)\beta\dot\alpha(k-3)\dot\gamma} \zeta_\beta \\ && + kE^\gamma{}_\delta \Phi_{\alpha(k-2)\gamma\dot\alpha(k-2)}\Omega^{\alpha(k-2)\beta\delta\dot\alpha(k-2)} \zeta_\beta + E_{\dot\gamma\dot\beta} \Phi_{\alpha(k-2)\gamma\dot\alpha(k-2)} \Omega^{\alpha(k-2)\dot\alpha(k-2)\dot\gamma\dot\beta} \zeta^\gamma] \\ && + \lambda[ E_{\dot\beta\dot\gamma} \Phi_{\alpha(k-1)\dot\alpha(k-2)} f^{\alpha(k-1)\dot\alpha(k-2)\dot\gamma} \zeta^{\dot\beta} + E_\beta{}^\gamma \Phi_{\alpha(k-2)\gamma\dot\alpha(k-2)} f^{\alpha(k-2)\beta\dot\alpha(k-2)\dot\gamma} \zeta_{\dot\gamma} ] \\ && + c_{k}[-E_{\gamma\delta} \Phi^{\alpha(k-2)\gamma\delta\dot\alpha(k-1)} f_{\alpha(k-2)\beta\dot\alpha(k-1)} \zeta^\beta] \\ && + \frac{(k-2)c_{k-1}}{k(k-2)}[kE^{\dot\gamma\dot\alpha}\Phi^{\alpha(k-2)\dot\alpha(k-3)} f_{\alpha(k-2)\beta\dot\alpha(k-2)\dot\gamma} \zeta^\beta] \\ && - \frac{(k-1)a_{k+1}}{2(k+1)} [- E_{\dot\gamma\dot\delta}\Phi_{\alpha(k-1)\dot\alpha(k-2)} f^{\alpha(k-1)\beta\dot\alpha(k-2)\dot\gamma\dot\delta}] \zeta_\beta \\ && - \frac{a_k}{2k(k-1)} [k(k-2)E^{\gamma\alpha} \Phi_{\alpha(k-2)\gamma\dot\alpha(k-2)} f^{\alpha(k-3)\beta\dot\alpha(k-2)} \zeta_\beta \\ && + (k-2)E_{\dot\gamma}{}^{\dot\alpha} \Phi_{\alpha(k-2)\gamma\dot\alpha(k-2)} f^{\alpha(k-2)\dot\alpha(k-3)\dot\gamma} \zeta^\gamma + kE^{\gamma\beta} \Phi_{\alpha(k-2)\gamma\dot\alpha(k-2)} f^{\alpha(k-2)\dot\alpha(k-2)} \zeta_\beta]\end{aligned}$$ $$\begin{aligned} 0 &=& {\cal F}_{\alpha\beta\dot\alpha} e^\beta{}_{\dot\beta} f^{\alpha\dot\alpha} \zeta^{\dot\beta} \\ && - 2{d_2}[3E^{\dot\beta}{}_{\dot\gamma} \Psi^{\alpha\dot\alpha\dot\gamma} f_{\alpha\dot\alpha} \zeta_{\dot\beta} - E_\beta{}^\alpha \Psi^{\beta\dot\alpha\dot\beta} f_{\alpha\dot\alpha} \zeta_{\dot\beta} ] \\ && - [ (- E_{\dot\beta}{}^{\dot\alpha} \Phi_{\alpha(2)\dot\alpha} \Omega^{\alpha(2)} + E^\beta{}_\gamma \Phi_{\alpha\beta\dot\beta} \Omega^{\alpha\gamma}) \zeta^{\dot\beta} + E^{\beta\alpha} \Phi_{\alpha\beta\dot\alpha} \Omega^{\dot\alpha\dot\gamma} \zeta_{\dot\gamma}] \\ && - 2\lambda E_\gamma{}^\beta \Phi_{\alpha\beta\dot\alpha} f^{\alpha\dot\alpha} \zeta^{\gamma} \\ && - c_{3}[-E_{\beta\gamma} \Phi^{\alpha\beta\gamma\dot\alpha\dot\beta} f_{\alpha\dot\alpha} \zeta_{\dot\beta}] - \frac{c_2}{3} [3E^{\dot\beta\dot\alpha} \Phi^{\alpha} f_{\alpha\dot\alpha}\zeta_{\dot\beta} - E_\beta{}^\alpha \Phi^{\beta} f_{\alpha\dot\beta} \zeta^{\dot\beta}] \\ && - \frac{a_3}{6}[- E_{\dot\beta\dot\gamma} \Phi_{\alpha(2)\dot\alpha} f^{\alpha(2)\dot\alpha\dot\gamma}\zeta^{\dot\beta} + E^\beta{}_\gamma \Phi_{\alpha\beta\dot\alpha} f^{\alpha\gamma\dot\alpha\dot\gamma} \zeta_{\dot\gamma} ]+ \frac{a_0}{4} [E^{\beta\alpha} \Phi_{\alpha\beta\dot\beta} A] \zeta^{\dot\beta} \\ 0 &=& {\cal F}_{\alpha\beta\dot\gamma} e^\beta{}_{\dot\beta} f^{\alpha\dot\beta} \zeta^{\dot\gamma} \\ && - 2d_2[3E^{\dot\beta}{}_{\dot\delta} \Psi^{\alpha\dot\gamma\dot\delta} f_{\alpha\dot\beta} \zeta_{\dot\gamma} - E_\beta{}^\alpha \Psi^{\beta\dot\alpha\dot\gamma} f_{\alpha\dot\alpha} \zeta_{\dot\gamma}] \\ && - [2E^\beta{}_\delta \Phi_{\alpha\beta\dot\gamma} \Omega^{\alpha\delta} ]\zeta^{\dot\gamma} + \lambda[ E^{\dot\gamma}{}_{\dot\beta} \Phi_{\alpha\gamma\dot\gamma} f^{\alpha\dot\beta} \zeta^{\gamma} - E_\gamma{}^\beta \Phi_{\alpha\beta\dot\alpha} f^{\alpha\dot\alpha} \zeta^{\gamma}] \\ && - c_{3}[-E_{\beta\delta} \Phi^{\alpha\beta\delta\dot\alpha\dot\gamma} f_{\alpha\dot\alpha}\zeta_{\dot\gamma}] - \frac{c_2}{3}[3 E^{\dot\beta\dot\gamma}\Phi^{\alpha} f_{\alpha\dot\beta} \zeta_{\dot\gamma} + E_\beta{}^\alpha \Phi^{\beta} f_{\alpha\dot\beta} \zeta^{\dot\beta}] \\ && - \frac{a_{3}}{6} [- E_{\dot\beta\dot\gamma} \Phi_{\alpha(2)\dot\delta} f^{\alpha(2)\dot\beta\dot\gamma} ]\zeta^{\dot\delta} + \frac{a_0}{4} [2E^{\beta\alpha} \Phi_{\alpha\beta\dot\gamma} A] \zeta^{\dot\gamma} \\ 0 &=& {\cal F}_{\alpha}e^\alpha{}_{\dot\alpha} f^{\beta\dot\alpha}\zeta_\beta + [4{d_{1}} E_{\dot\alpha\dot\beta} \Phi^{\dot\beta}f^{\gamma\dot\alpha} \zeta_\gamma -c_2E_{\alpha\beta} \Phi^{\alpha\beta\dot\beta} f_{\gamma\dot\beta} \zeta^\gamma + c_0E_{\beta\dot\alpha} \phi^\beta f^{\gamma\dot\alpha} \zeta_\gamma]\\ && - [2E^\alpha{}_\beta \Phi_{\alpha} \Omega^{\gamma\beta} - E_{\dot\alpha\dot\beta} \Phi^{\gamma} \Omega^{\dot\alpha\dot\beta} - \frac{a_{3}}{6} E_{\dot\alpha\dot\beta} \Phi_{\beta} f^{\gamma\beta\dot\alpha\dot\beta} -\frac{a_0}{2} E^{\alpha\gamma}\Phi_{\alpha}A] \zeta_\gamma \\ && + \lambda( E_{\dot\beta\dot\alpha} \Phi_{\beta} f^{\beta\dot\alpha} \zeta^{\dot\beta} + E_\beta{}^\alpha \Phi_\alpha f^{\beta\dot\alpha} \zeta_{\dot\alpha})\end{aligned}$$ $$\begin{aligned} 0 &=& - e^\alpha{}_{\dot\alpha} {\cal F}_{\alpha} A \zeta^{\dot\alpha} - [4{d_{1}} E_{\dot\alpha\dot\beta} \Phi^{\dot\beta}A \zeta^{\dot\alpha} + c_2 E_{\alpha\beta} \Phi^{\alpha\beta\dot\beta} A \zeta_{\dot\beta} + c_0 E_{\beta\dot\alpha} \phi^\beta A \zeta^{\dot\alpha}] \\ && - [ 4(E_{\alpha\dot\beta} \Phi_{\alpha}B^{\alpha(2)} \zeta^{\dot\beta} + E^\beta{}_{\dot\alpha} \Phi_{\beta} B^{\dot\alpha(2)} \zeta_{\dot\alpha}) -\frac{a_0}{2} ( E_{\dot\alpha\dot\beta} \Phi_{\alpha} f^{\alpha\dot\alpha} \zeta^{\dot\beta} - E_\alpha{}^\beta \Phi_{\beta} f^{\alpha\dot\alpha} \zeta_{\dot\alpha})] \\ && + 2\lambda E^\alpha{}_\beta \Phi_{\alpha} A \zeta^\beta \\ 0 &=& - E^\alpha{}_{\dot\alpha} {\cal C}_\alpha \varphi \zeta^{\dot\alpha} - [-c_0E_\alpha{}_{\dot\alpha} \Phi^{\alpha} \varphi \zeta^{\dot\alpha} - 4d_1E \phi_{\dot\alpha} \varphi\zeta^{\dot\alpha}] \\ && + [-E\phi_\alpha \pi^{\alpha\dot\alpha} \zeta_{\dot\alpha} - \frac{\tilde{a}_0}{12} E^\beta{}_{\dot\beta} \phi_\beta A \zeta^{\dot\beta}] - 2\lambda E\phi_\alpha \varphi \zeta^\alpha\end{aligned}$$ [10]{} M. A. Vasiliev, [*Higher Spin Gauge Theories in Various Dimensions,*]{} Fortsch.Phys. [**52**]{} (2004) 702 \[arXiv:hep-th/0401177\]. , X. Bekaert, S. Cnockaert, C. Iazeolla, M. A. Vasiliev, [*Nonlinear higher spin theories in various dimensions,*]{} in Proceedings of 1st Solvay Workshop on Higher Spin Gauge Theories, Brussels, Belgium, 12-14 May, 2004 \[arXiv:hep-th/0503128\]. X. Bekaert, N. Boulanger and P. Sundell, [*How higher spin gravity surpasses the spin-two barrier: no-go theorems versus yes-go examples,*]{} Rev. Mod. Phys. [**84**]{} (2012) 987 \[arXiv:1007.0435\]. E. S. Fradkin, M. A. Vasiliev, [*On the gravitational interaction of massless higher spin fields,*]{} Phys. Lett. [**B189**]{} (1987) 89. T. Curtright, [*Massless Field Supermultiplets with Arbitrary Spin,*]{} Phys. Lett. [**B85**]{} (1979) 214. M.A. Vasiliev, [*Gauge form of description of massless fields with arbitrary spin,*]{} Sov. J. Nucl. Phys. 32 (1980) 439. I. L. Buchbinder, S. M. Kuzenko, [*Ideas and Methods of Supersymmetry and Supergravity,*]{} IOP Publishing, Bristol and Philadelphia, 1998. S. M. Kuzenko, A. G. Sibiryakov and V. V.Postnikov, [*Massless gauge superfields of higher integer superspin,*]{} JETP Lett. [**57**]{} (1993) 534. S. M. Kuzenko and A. G. Sibiryakov, [*Massless gauge superfields of higher integer superspin,*]{} JETP Lett. [**57**]{} (1993) 539. S. M. Kuzenko and A. G. Sibiryakov, [*Free massless higher superspin superfields on the anti-de-Sitter superspace,*]{} Phys. Atom. Nucl. [**57**]{} (1994) 1257 \[arXiv:1112.4612\]. S. J. Gates, S. M. Kuzenko, A. G. Sibiiryakov, [*Towards a unfied theory of masselss superfields of all superspins,*]{} Phys. Lett. [**B 394**]{} (1997) 343 \[arXiv:hep-th/9611193\]. I. L. Buchbinder, S. M. Kuzenko and A. G. Sibiryakov, [*Quantization of higher spin superfields in the anti-De Sitter superspace,*]{} Phys. Lett. [**B352**]{} (1995) 29 \[hep-th/9502148\]. J. Hutomo, S. M. Kuzenko, [*Non-conformal higher spin supercurrents,*]{} Phys. Lett. [**B778**]{} (2018) 242 JHEP [**02**]{} (2018) 137 \[arXiv:1710.10837\] J. Hutomo, S. M. Kuzenko, [*The massless integer superspin multiplets revisited,*]{} JHEP [**02**]{} (2018) 137 \[arXiv:1711.11364\]. E. I. Buchbinder, J. Hutomo, S. M. Kuzenko, [*Higher spin supercurrents in anti-de Sitter space,*]{} JHEP [**09**]{} (2018) 027 \[arXiv:1711.11364\]. I. L. Buchbinder, S. J. Gates, K. Koutrolikos, [*Higher Spin Superfield interactions with the Chiral Supermultiplet: Conserved Supercurrents and Cubic Vertices,*]{} Universe [**4**]{} (2018) no.1, 6 \[arXiv:1708.06262\]. I. L. Buchbinder, S. J. Gates, K. Koutrolikos, [*Interaction of supersymmetric nonlinear sigma models with external higher spin superfields via higher spin supercurrents,*]{} JHEP [**05**]{} (2018) 204 \[arXiv:1804.08539\]. I. L. Buchbinder, S. J. Gates, K. Koutrolikos, [*Conserved higher spin supercurrents for arbitrary spin massless supermultiplets and higher spin superfield cubic interactions,*]{} JHEP [**08**]{} (2018) 055 \[arXiv:1805.04413\]. K. Koutrolikos, P.  Koci, R. von Unge, [*Higher Spin Superfield interactions with Complex linear Supermultiplet: Conserved Supercurrents and Cubic Vertices,*]{} JHEP [**03**]{} (2018) 119 \[arXiv:1712.05150\]. C. Fronsdal [*Massless fields with integer spin,*]{} Phys. Rev. [**D18**]{} 3624 (1978). J. Fang, C. Fronsdal [*Massless fields with half-integral spin,*]{} Phys. Rev. [**D18**]{} 3630 (1978). S. J. Gates Jr, S. M. Kuzenko and A. G. Sibiryakov [*$N=2$ supersymmetry of higher spin massless theories,*]{} Phys. Lett. [**B412**]{} (1997) 59 \[arXiv:hep-th/9609141\]. Yu. M. Zinoviev [*Massive N=1 supermultiplets with arbitrary superspins,*]{} Nucl. Phys. [**B785**]{} (2007) 98 \[arXiv:0704.1535\]. Yu. M. Zinoviev [*Massive spin-2 supermultiplets,*]{} arXiv:hep-th/0206209. Yu. M. Zinoviev [*“Massive supermultiplets with spin 3/2”,*]{} JHEP [**0705**]{} (2007) 092, \[arXiv:hep-th/0703118\]. Nicholas A. Ondo, Andrew J. Tolley [*“Deconstructing Supergravity, I: Massive Supermultiplets”,*]{} \[arXiv:1612.08752\]. S. M. Klishevich, Yu. M. Zinoviev [*“On electromagnetic interaction of massive spin-2 particle”,*]{} Phys. Atom. Nucl. [**61**]{} (1998) 1527, \[arXiv:hep-th/9708150\]. Yu. M. Zinoviev [*“On Massive High Spin Particles in (A)dS”,*]{} \[arXiv:hep-th/0108192\]. R. R. Metsaev [*“Gauge invariant formulation of massive totally symmetric fermionic fields in (A)dS space”,*]{} Phys. Lett. [**B643**]{} (2006) 205-212, \[arXiv:hep-th/0609029\]. I. L.Buchbinder, V. A. Krykhtin, [*Gauge invariant Lagrangian construction for massive bosonic higher spin fields in D dimensions,*]{} Nucl. Phys. [**B727**]{} (2005) 537 \[arXiv:hep-th/0505092\]. I. L.Buchbinder, V. A. Krykhtin, P. M. Lavrov, [*Gauge invariant Lagrangian formulation of higher spin massive bosonic field theory in AdS space,*]{} Nucl. Phys. [**B762**]{} (2007) 344 \[arXiv:hep-th/0608005\]. I. L.Buchbinder, V. A. Krykhtin, L. L. Ryskina, H. Takata, [*Gauge invariant Lagrangian construction for massive higher spin fermionic fields,*]{} Phys. Lett. [**B641**]{} (2006) 386 \[arXiv:hep-th/0603212\]. I. L.Buchbinder, V. A. Krykhtin, A. A.  Reshetnyak, [*BRST approach to Lagrangian construction for fermionic higher spin fields in (A)dS space,*]{} Nucl. Phys. [**B787**]{} (2007) 211 \[arXiv:hep-th/0703049\]. I. L. Buchbinder, S. J. Gates, W. D. Linch, J. Phillips, [*New $4D, {\cal N}=1$ superfield theory: Model of free massive superspin 3/2 multiplet,*]{} Phys.Lett. [**B535**]{} (2002) 280 \[arXiv:hep-th/0201096\]. I. L. Buchbinder, S. J. Gates, W. D. Linch, J. Phillips, [*Dynamical superfield theory of free massive superspin-1 multiplet,*]{} Phys.Lett. [**B549**]{} (2002) 229-236 \[arXiv:hep-th/0207243\]. I. L. Buchbinder, S. J. Gates, S. M. Kuzenko, J. Phillips, [*Massive $4D, {\cal N}=1$ superspin $1 \& 3/2$ multiplets and dualities,*]{} JHEP [**0502**]{} (2005) 056 \[arXiv:hep-th/0501199\]. S. J. Gates, S. M. Kuzenko, G. Tartaglino-Mazzucchelli, [*New massive supergravity multiplets,*]{} JHEP [**0702**]{} (2007) 052 \[arXiv:hep-th/0610333\]. Yu. M. Zinoviev [*“Frame-like gauge invariant formulation for massive high spin particles”,*]{} Nucl. Phys. [**B808**]{} (2009) 185, \[arXiv:0808.1778\]. D. S. Ponomarev, M. A. Vasiliev [*“Frame-Like Action and Unfolded Formulation for Massive Higher-Spin Fields”,*]{} Nucl. Phys. [**B839**]{} (2010) 466, arXiv:1001.0062. I. L. Buchbinder, T. V. Snegirev, Yu. M. Zinoviev [*“Lagrangian description of massive higher spin supermultiplets in $AdS_3$ space”,*]{} JHEP [**08**]{} (2017) 021, \[arXiv:1705.06163\]. [^1]: joseph@tspu.edu.ru [^2]: maksim.khabarov@ihep.ru [^3]: snegirev@tspu.edu.ru [^4]: Yurii.Zinoviev@ihep.ru [^5]: Application of this formulation for quantization of the $N=1$ higher spin superfield model in $AdS_4$ space was considered in [@BKS]. The superfield approach was recently applied for construction of the higher spin supercurrents [@HK-1], [@HK-2], [@BHK]. [@BGK-1], [@BGK-2], [@BGK-3], [@K]. [^6]: Another gauge invariant approach to Lagrangian formulation of massive higher spin fields is given on the base of BRST construction [@BKr05], [@BKrL], [@BKRT], [@BKrR]. [^7]: The attempts to developed the off-shell superfield Lagrangian formulation of the massive higher spin supermultiplets were realized only for some examples in [@BG1], [@BG2], [@BG3], [@GKT]. General superfield formulation is still undeveloped.

--- abstract: 'In this note we give a precise formulation of “resistance to arbitrary side information” and show that several relaxations of differential privacy imply it. The formulation follows the ideas originally due to Dwork and McSherry, stated implicitly in [@Dw06]. This is, to our knowledge, the first place such a formulation appears explicitly. The proof that relaxed definitions (and hence the schemes of [@DKMMN06; @NRS07; @MKAGV08]) satisfy the Bayesian formulation is new.' author: - | Shiva Prasad KasiviswanathanAdam Smith\ Department of Computer Science and Engineering\ Pennsylvania State University\ e-mail: [$\{$kasivisw,asmith$\}$@cse.psu.edu]{}\ bibliography: - '../bibfiles/master.bib' title: | A Note on Differential Privacy:\ Defining Resistance to Arbitrary Side Information --- Introduction ============ Privacy is an increasingly important aspect of data publishing. Reasoning about privacy, however, is fraught with pitfalls. One of the most significant is the auxiliary information (also called external knowledge, background knowledge, or side information) that an adversary gleans from other channels such as the web, public records, or domain knowledge. Schemes that retain privacy guarantees in the presence of independent releases are said to [*compose securely*]{}. The terminology, borrowed from cryptography (which borrowed, in turn, from software engineering), stems from the fact that schemes which compose securely can be designed in a stand-alone fashion without explicitly taking other releases into account. Thus, understanding independent releases is essential for enabling modular design. In fact, one would like schemes that compose securely not only with independent instances of themselves, but with [*arbitrary external knowledge*]{}. Certain randomization-based notions of privacy (such as differential privacy [@DMNS06]) are believed to compose securely even in the presence of arbitrary side information. In this note we give a precise formulation of this statement. First, we provide a Bayesian formulation of differential privacy which makes its resistance to arbitrary side information explicit. Second, we prove that the relaxed definitions of [@DKMMN06; @MKAGV08] still imply the Bayesian formulation. The proof is non-trivial, and relies on the “continuity” of Bayes’ rule with respect to certain distance measures on probability distributions. Our result means that the recent techniques mentioned above [@DKMMN06; @CM06; @NRS07; @MKAGV08] can be used modularly with the same sort of assurances as in the case of strictly differentially-private algorithms. Differential Privacy -------------------- Databases are assumed to be vectors in $\mathcal{D}^n$ for some domain $\mathcal{D}$. The Hamming distance ${{d}}({{\mathrm{x}}},{{\mathrm{y}}})$ on $\mathcal{D}^n$ is the number of positions in which the vectors ${{\mathrm{x}}},{{\mathrm{y}}}$ differ. We let $\Pr[\cdot]$ and ${{\mathbb{E}}}[\cdot]$ denote probability and expectation, respectively. Given a randomized algorithm $\mathcal{A}$, we let $\mathcal{A}({{\mathrm{x}}})$ be the random variable (or, probability distribution on outputs) corresponding to input ${{\mathrm{x}}}$. If $\erert$ and $\Q$ are probability measure on a discrete space $D$, the [*statistical difference*]{} (a.k.a. [*total variation distance*]{}) between $\erert$ and $\Q$ is defined as: $${\mathbf{SD}{\left( {{\erert,\Q}} \right)}}= \max_{S \subset D}|\erert[S]-\Q[S)|.$$ \[def:ind\] A randomized algorithm ${\mathcal{A}}$ is said to be ${\epsilon}$-differentialy private if for all databases ${{\mathrm{x}}},{{\mathrm{y}}}\in \mathcal{D}^n$ at Hamming distance at most 1, and for all subsets $S$ of outputs $$\begin{aligned} \Pr[{\mathcal{A}}({{\mathrm{x}}})\in S] \leq e^{{\epsilon}} \Pr[{\mathcal{A}}({{\mathrm{y}}})\in S].\end{aligned}$$ This definition states that changing a single individual’s data in the database leads to a small change in the [*distribution*]{} on outputs. Unlike more standard measures of distance such as total variation (also called statistical difference) or Kullback-Leibler divergence, the metric here is multiplicative and so even very unlikely events must have approximately the same probability under the distributions ${\mathcal{A}}({{\mathrm{x}}})$ and ${\mathcal{A}}({{\mathrm{y}}})$. This condition was relaxed somewhat in other papers [@DiNi03; @DwNi04; @BDMN05; @DKMMN06; @CM06; @NRS07; @MKAGV08]. The schemes in all those papers, however, satisfy the following relaxation [@DKMMN06]: \[def:indd\] A randomized algorithm ${\mathcal{A}}$ is $({\epsilon},\delta)$-differentially private if for all databases ${{\mathrm{x}}},{{\mathrm{y}}}\in {{\mathcal{D}}}^n$ that differ in one entry, and for all subsets $S$ of outputs, $\Pr[{\mathcal{A}}({{\mathrm{x}}})\in S] \leq e^{{\epsilon}} \Pr[{\mathcal{A}}({{\mathrm{y}}})\in S]+\delta\,.$ The relaxations used in [@DwNi04; @BDMN05; @MKAGV08] were in fact stronger (i.e., less relaxed) than [Definition \[def:ind\]]{}. One consequence of the results below is that all the definitions are equivalent up to polynomial changes in the parameters, and so given the space constraints we work only with the simplest notion.[^1] Semantics of Differential Privacy {#sec:bayes} ================================= There is a crisp, semantically-flavored interpretation of differential privacy, due to Dwork and McSherry, and explained in [@Dw06]: [*Regardless of external knowledge, an adversary with access to the sanitized database draws the same conclusions whether or not my data is included in the original data.*]{} (the use of the term “semantic” for such definitions dates back to semantic security of encryption [@GM84]). In this section, we develop a formalization of this interpretation and show that the definition of differential privacy used in the line of work this paper follows ([@DiNi03; @DwNi04; @BDMN05; @DMNS06]) is essential in order to satisfy the intuition. We require a mathematical formulation of “arbitrary external knowledge”, and of “drawing conclusions”. The first is captured via a [*prior*]{} probability distribution $b$ on ${{\mathcal{D}}}^n$ ($b$ is a mnemonic for “beliefs”). Conclusions are modeled by the corresponding posterior distribution: given a transcript $t$, the adversary updates his belief about the database ${{\mathrm{x}}}$ using Bayes’ rule to obtain a posterior $\bar{b}$: $$\begin{aligned} \label{eqn:bel} \bar{b}[{{\mathrm{x}}}| t] = \frac{\Pr[{\mathcal{A}}({{\mathrm{x}}})=t] b[{{\mathrm{x}}}]}{\sum_{{\mathrm{y}}}\Pr[{\mathcal{A}}({{\mathrm{y}}})=t]b[{{\mathrm{y}}}]}\ . \end{aligned}$$ Note that in an interactive scheme, the definition of ${\mathcal{A}}$ depends on the adversary’s choices; for legibility we omit the dependence on the adversary in the notation. Also, for simplicity, we discuss only discrete probability distributions. Our results extend directly to the interactive, continuous case. For a database ${{\mathrm{x}}}$, define ${{\mathrm{x}}}_{-i}$ to be the same vector where position $i$ has been replaced by some fixed, default value in $D$. Any valid value in $D$ will do for the default value. We can then imagine $n+1$ related games, numbered 0 through $n$. In Game 0, the adversary interacts with ${\mathcal{A}}({{\mathrm{x}}})$. This is the interaction that actually takes place between the adversary and the randomized algorithm ${\mathcal{A}}$. In Game $i$ (for $1\leq i \leq n$), the adversary interacts with ${\mathcal{A}}({{\mathrm{x}}}_{-i})$. Game $i$ describes the hypothetical scenario where person $i$’s data is not included. For a particular belief distribution $b$ and transcript $t$, we can then define $n+1$ [*a posteriori*]{} distributions $\bar{b}_0,\dots,\bar{b}_n$, where the $\bar{b}_0$ is the same as $\bar{b}$ (defined in \[eqn:bel\]) and, for larger $i$, the $i$-th belief distribution is defined with respect to Game $i$: $$\bar{b}_i[{{\mathrm{x}}}| t] = \frac{\Pr[{\mathcal{A}}({{\mathrm{x}}}_{-i})=t] b[{{\mathrm{x}}}]}{\sum_{{\mathrm{y}}}\Pr[{\mathcal{A}}({{\mathrm{y}}}_{-i})=t]b[{{\mathrm{y}}}]}.$$ Given a particular transcript $t$, the privacy has been breached if the adversary would draw different conclusions about the world and, in particular, about a person $i$ depending on whether or not $i$’s data was used. It turns out that the exact measure of “different” here does not matter much. We chose the weakest notion that applies, namely statistical difference. We say there is a problem for transcript $t$ if the distributions $\bar{b}_0[\cdot|t]$ and $\bar{b}_i[\cdot|t]$ are far apart in statistical difference. We would like to avoid this happening for any potential participant. This is captured by the following definition. \[def:sem\] A randomized algorithm ${\mathcal{A}}$ is said to be ${\epsilon}$-semantically private if for all belief distributions $b$ on $\mathcal{D}^n$, for all databases ${{\mathrm{x}}}\in \mathcal{D}^n$, for all possible transcripts $t$, and for all $i = 1,\ldots,n$$\mathrm{:}$ $${\mathbf{SD}{\left( {{\bar{b}_0[{{\mathrm{x}}}|t]\ ,\ \bar{b}_i[{{\mathrm{x}}}|t]\ }} \right)}} \leq {\epsilon}.$$ Dwork and McSherry proposed the notion of semantic privacy, informally, and observed that it is equivalent to differential privacy. We now formally show that the notions of ${\epsilon}$-differential privacy (Definition \[def:ind\]) and ${\epsilon}$-semantic privacy (Definition \[def:sem\]) are very closely related. (Dwork-McSherry) \[thm:eind\] ${\epsilon}$-differential privacy implies $\bar{\epsilon}$-semantic privacy, where $\bar{\epsilon}=e^{{\epsilon}}-1$. $\bar{\epsilon}/2$-semantic privacy implies $2{\epsilon}$-differential privacy. We extend the previous Bayesian formulation to capture situations where bad events can occur with some negligible probability (say, $\delta)$. We relax ${\epsilon}$-semantic privacy to $({\epsilon},\delta)$-semantic privacy and show that it is closely related to $({\epsilon},\delta)$-differential privacy. A randomized algorithm is $({\epsilon},\delta)$-semantically private if for all belief distributions $b$ on $\mathcal{D}^n$, with probability at least $1-\delta$ over pairs $({{\mathrm{x}}},t)$, where the database ${{\mathrm{x}}}$ is drawn according to $b$, and transcript $t$ is drawn according to ${\mathcal{A}}({{\mathrm{x}}})$, and for all $i =1,\dots,n$: $${\mathbf{SD}{\left( {{\bar{b}_0[{{\mathrm{x}}}|t]\ ,\ \bar{b}_i[{{\mathrm{x}}}|t]\ }} \right)}} \leq {\epsilon}.$$ This definition is only interesting when ${\epsilon}>\delta$; otherwise just use statistical difference $2\delta$ and leave ${\epsilon}=0$. Below, we assume ${\epsilon}>\delta$. In fact, in many of the proofs we will be assuming that $\delta$ is a negligible function (of $O(1/n^2)$). In Appendix A, we provide another related definition of $({\epsilon},\delta)$-semantic privacy. \[thm:ind2sdp\] (${\epsilon},\delta$)-differential privacy implies $({\epsilon}',\delta')$-semantic privacy for arbitrary (not necessarily informed) beliefs with ${\epsilon}'=e^{3{\epsilon}}-1+2\sqrt{\delta}$ and $\delta' = O(n\sqrt{\delta})$. $(\bar{\epsilon}/2,\delta)$-semantic privacy implies $(2{\epsilon},2\delta)$-differential privacy with $\bar{\epsilon}=e^{{\epsilon}}-1$. Some Properties of $({\epsilon},\delta)$-Differential Privacy ============================================================= We now describe some properties of $({\epsilon},\delta)$-differential privacy that would be useful later on. This section could be of independent interest. Instead of restricting ourselves to outputs of randomized algorithms, we consider a more general definition of $({\epsilon},\delta)$-differential privacy. \[$({\epsilon},\delta)$-indistinguishability\] Two random variables $X,Y$ taking values in a set $D$ are $({\epsilon},\delta)$-indistinguishable if for all sets $S\subseteq D$, $$\begin{aligned} \Pr[X\in S] \leq e^{\displaystyle {\epsilon}} \Pr[Y\in S] + \delta \ \ \ \ \mbox{ and } \ \ \ \ \Pr[Y\in S] \leq e^{\displaystyle {\epsilon}} \Pr[X\in S] + \delta.\end{aligned}$$ We will also be using a simpler variant of $({\epsilon},\delta)$-indistinguishability, which we call [*point-wise*]{} $({\epsilon},\delta)$-indistinguishability. Claim \[lem:proof\] (Parts \[it:pw2ind\] and \[it:ind2pw\]) shows that $({\epsilon},\delta)$-indistinguishability and point-wise $({\epsilon},\delta)$-indistinguishability are almost equivalent. \[Point-wise $({\epsilon},\delta)$-indistinguishability\] Two random variables $X$ and $Y$ are point-wise $({\epsilon},\delta)$-indistinguishable if with probability at least $1-\delta$ over $a$ drawn from either $X$ or $Y$, we have: $$e^{-{\epsilon}}\Pr[Y=a] \leq \Pr[X=a] \leq e^{{\epsilon}} \Pr[Y=a].$$ The following are useful facts about indistinguishability.[^2] \[lem:proof\] \[it:pw2ind\] If $X,Y$ are point-wise $({\epsilon},\delta)$-indistinguishable then they are $({\epsilon},\delta)$-indistinguishable. \[it:ind2pw\] If $X,Y$ are $({\epsilon},\delta)$-indistinguishable then they are point-wise $(2{\epsilon},\frac{2\delta}{e^{{\epsilon}}{\epsilon}})$-indistinguishable. \[it:ajoint\] Let $X$ be a random variable on $D$. Suppose that for every $a \in D$, ${\mathcal{A}}(a)$ and ${\mathcal{A}}'(a)$ are $({\epsilon},\delta)$-indistinguishable (for some randomized algorithms ${\mathcal{A}}$ and ${\mathcal{A}}'$). Then the pairs $(X,{\mathcal{A}}(X))$ and $(X,{\mathcal{A}}'(X))$ are $({\epsilon},\delta)$-indistinguishable. \[it:joint\] Let $X$ be a random variable. Suppose with probability at least $1-\delta$ over $a {\leftarrow}X$ $(a\,$ drawn from $X)$, ${\mathcal{A}}(a)$ and ${\mathcal{A}}'(a)$ are $({\epsilon},\delta)$-indistinguishable (for some randomized algorithms ${\mathcal{A}}$ and ${\mathcal{A}}'$). Then the pairs $(X,\ {\mathcal{A}}(X))$ and $(X,\ {\mathcal{A}}'(X))$ are $({\epsilon},2\delta)$-indistinguishable. \[it:prob\] If $X,Y$ are $({\epsilon},\delta)$-indistinguishable and ${{\cal G}}$ is some randomized algorithm, then ${{\cal G}}(X)$ and ${{\cal G}}(Y)$ are $({\epsilon},\delta)$-indistinguishable. \[it:sd\] If $X,Y$ are $({\epsilon},\delta)$-indistinguishable, then ${\mathbf{SD}{\left( {{X,Y}} \right)}} \leq \bar{\epsilon}+\delta$, where $\bar{\epsilon}=e^{{\epsilon}}-1$. [*Proof of Part \[it:pw2ind\].*]{} Let $Bad$ be the set of [*bad*]{} values of $a$, that is $$Bad = \{a \,:\, \Pr[X=a] < e^{-{\epsilon}} \Pr[Y=a] \mbox{ or } \Pr[X=a]> e^{{\epsilon}} \Pr[Y=a] \}.$$ By definition, $\Pr[X\in Bad]\leq \delta$. Now consider any set $S$ of outcomes. $$\Pr[X\in S] \leq \Pr[X \in S \setminus Bad] + \Pr[X\in Bad].$$ The first term is at most $e^{\epsilon}\Pr[Y\in S \setminus Bad]\leq e^{\epsilon}\Pr[Y\in S]$. Hence, $\Pr[X \in S] \leq e^{\epsilon}\Pr[Y\in S] + \delta$, as required. The case of $\Pr[Y\in S]$ is symmetric. Therefore, $X$ and $Y$ are $({\epsilon},\delta)$-indistinguishable.\ [*Proof of Part \[it:ind2pw\].*]{} Let $S= \{a \,:\, \Pr[X=a] > e^{2{\epsilon}} \Pr[Y=a]\}$. Then, $$\Pr[X \in S] > e^{2{\epsilon}} \Pr[Y \in S] > e^{{\epsilon}}(1+{\epsilon})\Pr[Y \in S] \Rightarrow \Pr[X \in S] - e^{{\epsilon}}\Pr[Y \in S] > {\epsilon}e^{{\epsilon}} \Pr[Y \in S].$$ Since, $\Pr[X \in S] - e^{{\epsilon}}\Pr[Y \in S] \leq \delta$, we mush have ${\epsilon}e^{{\epsilon}} \Pr[Y \in S] < \delta$. A similar argument when considering the set $S'=\{a \,:\, \Pr[X=a] < e^{-2{\epsilon}} \Pr[Y=a]\}$ shows that ${\epsilon}e^{{\epsilon}} \Pr[Y \in S'] < \delta$. Putting both arguments together, $\Pr[Y \in S \cup S'] \leq 2\delta/({\epsilon}e^{{\epsilon}})$. Therefore, with probability at least $1-2\delta/(e^{{\epsilon}}{\epsilon})$ for any $a$ drawn from either $X$ or $Y$ we have: $e^{-2{\epsilon}}\Pr[Y=a] \leq \Pr[X=a] \leq e^{2{\epsilon}} \Pr[Y=a]$.\ [*Proof of Part \[it:ajoint\].*]{} Let $(X,{{\cal A}}(X))$ and $(X,{{\cal A}}'(X))$ be random variables on $D \times E$. Let $S$ be an arbitrary subset of $D \times E$ and, for every $a \in D$, define $S_a =\{b \in E\,:\, (a,b) \in S\}$. $$\begin{aligned} \Pr[(X,{{\cal A}}(X)) \in S] &\leq& \sum_{a \in D}\Pr[{{\cal A}}(X) \in S_a \,:\, X=a]\Pr[X=a] \\ & < & \sum_{a \in D}(e^{{\epsilon}}\Pr[{{\cal A}}'(X) \in S_a \,:\, X=a]+\delta)\Pr[X=a] \\ & < & \delta + e^{{\epsilon}}\Pr[(X,{{\cal A}}'(X)) \in S].\end{aligned}$$ By symmetry, we also have $\Pr[(X,{{\cal A}}'(X)) \in S] < \delta+\Pr[(X,{{\cal A}}(X)) \in S]$. Since $S$ was arbitrary, $(X,{{\cal A}}(X))$ and $(X,{{\cal A}}'(X))$ are $({\epsilon},\delta)$-indistinguishable.\ [*Proof of Part \[it:joint\].*]{} Let $(X,{{\cal A}}(X))$ and $(X,{{\cal A}}'(X))$ be random variables on $D \times E$. Let $T \subset D$ be the set of $a$’s for which ${{\cal A}}(a) \leq e^{{\epsilon}} {{\cal A}}'(a)$. Now, let $S$ be an arbitrary subset of $D \times E$ and, for every $a \in D$, define $S_a =\{b \in E\,:\, (a,b) \in S\}$. $$\begin{aligned} \Pr[(X,{{\cal A}}(X)) \in S] &\leq& \Pr[X \notin T] + \sum_{a \in T}\Pr[{{\cal A}}(X) \in S_a \,:\, X=a]\Pr[X=a] \\ & < & \delta + \sum_{a \in T}(e^{{\epsilon}}\Pr[{{\cal A}}'(X) \in S_a \,:\, X=a]+\delta)\Pr[X=a] \\ & < & 2\delta + e^{{\epsilon}}\Pr[(X,{{\cal A}}'(X)) \in S].\end{aligned}$$ By symmetry, we also have $\Pr[(X,{{\cal A}}'(X)) \in S] < 2\delta+\Pr[(X,{{\cal A}}(X)) \in S]$. Since $S$ was arbitrary, $(X,{{\cal A}}(X))$ and $(X,{{\cal A}}'(X))$ are $({\epsilon},2\delta)$-indistinguishable.\ [*Proof of Part \[it:prob\].*]{} Let $D$ be some domain. A randomized procedure ${{\cal G}}$ is a pair ${{\cal G}}=(g,R)$, where $R$ is a random variable on some set $E$ and $g$ is a function from $D \times E$ to any set $F$. If $X$ is a random variable on $D$, then ${{\cal G}}(X)$ denotes the random variable on $F$ obtained by sampling $X \otimes R$ and applying $g$ to the result, where the symbol $\otimes$ denotes the tensor product. Now for any set $S \subset F$, $$\begin{aligned} \lefteqn{\Pr[{{\cal G}}(X) \in S] - e^{{\epsilon}}\Pr[{{\cal G}}(Y) \in S]}\\ &=& \Pr[g(X \otimes R) \in S] - e^{{\epsilon}} \Pr[g(Y \otimes R) \in S] \\ &=&\Pr[X\otimes R \in g^{-1}(S)] -e^{{\epsilon}} \Pr[Y\otimes R \in g^{-1}(S)] \\ & \leq & \sum_{r \in E} \Pr[X \in S_r \,:\, R=r]\Pr[R=r] - e^{{\epsilon}} \sum_{r \in E} \Pr[Y \in S_r \,:\, R=r]\Pr[R=r] \\ & =& \sum_{r \in E}(\Pr[X \in S_r \,:\, R=r]-e^{{\epsilon}}\Pr[Y \in S_r \,:\, R=r])\Pr[R=r] \\ &\leq &\sum_{r \in E}\delta \Pr[R=r] =\delta.\end{aligned}$$ By symmetry, we also have $\Pr[{{\cal G}}(Y) \in S] - e^{{\epsilon}}\Pr[{{\cal G}}(X) \in S] \leq \delta$. Since $S$ was arbitrary, ${{\cal G}}(X)$ and ${{\cal G}}(Y)$ are $({\epsilon},\delta)$-indistinguishable.\ [*Proof of Part \[it:sd\].*]{} Let $X$ and $Y$ be random variables on $D$. By definition ${\mathbf{SD}{\left( {{X,Y}} \right)}}=\max_{S \subset D}|\Pr[X \in S] - \Pr[Y \in S]|$. For any set $S \subset D$, $$\begin{aligned} \lefteqn{2|\Pr[X \in S]-\Pr[Y \in S]|} \\ &=& \left |\Pr[X \in S]-\Pr[Y \in S]\right | + \left |\Pr[X \notin S]-\Pr[Y \notin S] \right | \\ &=& \left |\sum_{c \in S}(\Pr[X =c]-\Pr[Y =c])\right | + \left |\sum_{c \notin S}(\Pr[X =c]-\Pr[Y =c])\right | \\ &\leq& \sum_{c \in S}\left |\Pr[X =c]-\Pr[Y =c] \right | + \sum_{c \notin S}\left |\Pr[X =c]-\Pr[Y =c]\right | \\ &=& \sum_{c \in D } \left |\Pr[X =c]-\Pr[Y =c] \right | \\ &\leq& \sum_{c \in D}(e^{{\epsilon}}\Pr[Y =c] + \delta - \Pr[Y =c]) + \sum_{c \in D}(e^{{\epsilon}}\Pr[X =c] +\delta - \Pr[X =c]) \\ &=& 2\delta + (e^{{\epsilon}}-1) \sum_{c \in D}\Pr[Y =c] + (e^{{\epsilon}}-1) \sum_{c \in D}\Pr[X =c] \\ &=& 2(e^{{\epsilon}}-1)+2\delta=2\bar{{\epsilon}}+ 2\delta.\end{aligned}$$ This implies that $|\Pr[X \in S]-\Pr[Y \in S]| \leq \bar{\epsilon}+ \delta$. Since the above inequality holds for every $S \subset D$, it immediately follows that the statistical difference between $X$ and $Y$ is at most $\bar{{\epsilon}}+\delta$. Proofs of Theorems \[thm:eind\] and \[thm:ind2sdp\] =================================================== This section is devoted to proving Theorems \[thm:eind\] and \[thm:ind2sdp\]. For convenience we restate the theorem statements. \[Dwork-McSherry\] ${\epsilon}$-differential privacy implies $\bar{\epsilon}$-semantic privacy, where $\bar{\epsilon}=e^{{\epsilon}}-1$. $\bar{\epsilon}/2$-semantic privacy implies $2{\epsilon}$-differential privacy. Consider any database ${{\mathrm{x}}}$. Consider belief distributions $\bar{b}_0[{{\mathrm{x}}}|t]$ and $\bar{b}_i[{{\mathrm{x}}}|t]$. differential privacy implies that the ratio of $\bar{b}_0[{{\mathrm{x}}}|t]$ and $\bar{b}_i[{{\mathrm{x}}}|t]$ is within $e^{\pm {\epsilon}}$ on every point, i.e., for every $i$ and for every possible transcript $t$: $$e^{-{\epsilon}}\bar{b}_i[{{\mathrm{x}}}|t] \leq \bar{b}_0[{{\mathrm{x}}}|t] \leq e^{{\epsilon}}\bar{b}_i[{{\mathrm{x}}}|t].$$ In the remainder of the proof we fix $i$ and $t$. Substituting $\delta=0$ in Claim \[lem:proof\] (part \[it:sd\]), implies that ${\mathbf{SD}{\left( {{\bar{b}_0[{{\mathrm{x}}}|t],\bar{b}_i[{{\mathrm{x}}}|t]}} \right)}}=\bar{\epsilon}$. To see that $\bar{{\epsilon}}$-semantic privacy implies $2{\epsilon}$-differential privacy, consider a belief distribution $b$ which is uniform over two databases ${{\mathrm{x}}},{{\mathrm{y}}}$ which are at Hamming distance of one. Let $i$ be the position in which ${{\mathrm{x}}}$ and ${{\mathrm{y}}}$ differ. The distribution $\bar{b}_i[\cdot|t]$ will be uniform over ${{\mathrm{x}}}$ and ${{\mathrm{y}}}$ since they induce the same distribution on transcripts in Game $i$. This means that $\bar{b}_0[\cdot|t]$ will assign probabilities $1/2 \pm \bar{\epsilon}/2$ to each of the two databases (follows from ${\epsilon}$-semantic privacy definition). Working through Bayes’ rule shows that $$\frac{\Pr[{\mathcal{A}}({{\mathrm{x}}})=t]}{\Pr[{\mathcal{A}}({{\mathrm{y}}})=t]}= \frac{\Pr[\bar{b}_0[{{\mathrm{x}}}|t]={{\mathrm{x}}}]}{\Pr[\bar{b}_0[{{\mathrm{y}}}|t]={{\mathrm{x}}}]} \leq \frac{{\frac{1}{2}}(1+\bar{\epsilon})}{{\frac{1}{2}}(1-\bar{\epsilon})} \leq e^{2{\epsilon}}.$$ This implies that ${\mathcal{A}}$ is point-wise $2{\epsilon}$-differentialy private. Using Claim \[lem:proof\] (part \[it:pw2ind\]), implies that ${\mathcal{A}}$ is $2{\epsilon}$-differentialy private. We will use the following lemma to establish connections between $({\epsilon},\delta)$-differential privacy and $({\epsilon},\delta)$-semantic privacy. Let $B|_{A=a}$ denote the conditional distribution of $B$ given that $A=a$ for jointly distributed random variables $A$ and $B$. \[lem:bayes\] Suppose two pairs of random variables $(X,{\mathcal{A}}(X))$ and $(Y,{\mathcal{A}}'(Y))$ are $({\epsilon},\delta)$-differentialy private (for some randomized algorithms ${\mathcal{A}}$ and ${\mathcal{A}}'$). Then with probability at least $1-\delta''$ over $t{\leftarrow}{\mathcal{A}}(X)$ (equivalently $t {\leftarrow}{{\cal A}}'(Y)$), the random variables $X|_{{\mathcal{A}}(X) =t}$ and $Y|_{{\mathcal{A}}'(Y)=t}$ are $(\hat{\epsilon},\hat\delta)$-differentialy private with $\hat{\epsilon}=3{\epsilon}$, $\hat\delta=2\sqrt{\delta}$, and $\delta''=\sqrt{\delta}+\frac{2\delta}{{\epsilon}e^{{\epsilon}}}=O(\sqrt{\delta})$. Let $(X,{{\cal A}}(X))$ and $(Y,{{\cal A}}'(Y))$ be random variables on $D \times E$. The first observation is that ${{\cal A}}(X)$ and ${{\cal A}}(Y)$ are $({\epsilon},\delta)$-differentialy private. To prove that consider any set $P \in E$, $$\begin{aligned} \Pr[{{\cal A}}(X) \in P] & =& \Pr[(X,{{\cal A}}(X)) \in D \times P] \leq e^{{\epsilon}}\Pr[(Y,{{\cal A}}'(Y)) \in D \times P] + \delta \\ &=& e^{{\epsilon}}\Pr[{{\cal A}}'(Y) \in P] + \delta.\end{aligned}$$ Since $P$ was arbitrary, ${{\cal A}}(X)$ and ${{\cal A}}'(Y)$ are $({\epsilon},\delta)$-differentialy private. In the remainder of the proof, we will use the notation $X|_t$ for $X|_{{{\cal A}}(X)=t}$ and $Y|_t$ for $Y|_{{{\cal A}}'(Y)=t}$. Define, $$\begin{aligned} &Bad_0 = \{a\,:\, e^{-2{\epsilon}}\Pr[{{\cal A}}'[Y]=a] > \Pr[{{\cal A}}(X)=a] > e^{2{\epsilon}} \Pr[{{\cal A}}'[Y]=a] \}& \\ &Bad_1=\{a \, : \, \exists S \subset D \mbox{ such that } \Pr[X|_a \in S] > e^{\hat{\epsilon}}\Pr[Y|_a \in S] + \hat\delta\}& \\ &Bad_2=\{a \, : \, \exists S \subset D \mbox{ such that } \Pr[Y|_a \in S] > e^{\hat{\epsilon}}\Pr[X|_a \in S] + \hat\delta\}.& \end{aligned}$$ We need an upper bound for the probabilities $\Pr[{{\cal A}}(X) \in Bad_1 \cup Bad_2]$ and $\Pr[{{\cal A}}'(Y) \in Bad_1 \cup Bad_2]$. We know from Claim \[lem:proof\] (part  \[it:ind2pw\]), that $$\Pr[{{\cal A}}(X) \in Bad_0] \leq \frac{2\delta}{{\epsilon}e^{{\epsilon}}} \ \ \mbox{ and } \ \ \Pr[{{\cal A}}'(Y) \in Bad_0] \leq \frac{2\delta}{{\epsilon}e^{{\epsilon}}}.$$ Note that from the initial observation ${{\cal A}}(X)$ and ${{\cal A}}'(Y)$ are $({\epsilon},\delta)$-differentialy private, therefore the condition required for applying Claim \[lem:proof\] (part \[it:ind2pw\]) holds. Now define, $$Bad_1'=Bad_1 \setminus Bad_0 \ \ \mbox{ and } \ \ Bad_2'=Bad_2 \setminus Bad_0.$$ For each $a \in Bad_1'$ and $T \subset D \times E$, define $S_a=\{b \in D \,:\, (b,a) \in T\}$. Define $T_1 = S_a \times \bigcup_{a \in Bad_1'} \{a\}$. $$\begin{aligned} \Pr[(X,{{\cal A}}(X))\in T_1] &=& \sum_{a \in Bad_1'} \Pr[X \in S_a \,:\, {{\cal A}}(X)=a] \Pr[{{\cal A}}(X)=a] \\ &>& \sum_{a \in Bad_1'} (e^{\hat{\epsilon}}\Pr[Y \in S_a \,:\, {{\cal A}}'(Y)=a]+\hat\delta)\Pr[{{\cal A}}(X)=a] \\ &=&\sum_{a \in Bad_1'} e^{\hat{\epsilon}}\Pr[Y \in S_a \,:\, {{\cal A}}'(Y)=a]\Pr[{{\cal A}}(X)=a] + \hat\delta\sum_{a \in Bad_1'} \Pr[{{\cal A}}(X)=a] \\ &=& \sum_{a \in Bad_1'} e^{3{\epsilon}}\Pr[Y \in S_a \,:\, {{\cal A}}'(Y)=a]e^{-2{\epsilon}} \Pr[{{\cal A}}'(Y)=a] + \hat\delta\Pr[{{\cal A}}(X) \in Bad_1'] \\ &=& e^{{\epsilon}}\Pr[(Y,{{\cal A}}'(Y)) \in T_1] + \hat\delta\Pr[{{\cal A}}(X) \in Bad_1'].\end{aligned}$$ The inequality follows because of the definition of $Bad_1'$. By $({\epsilon},\delta)$-differential privacy, $\Pr[(X,{{\cal A}}(X))\in T_1] \leq e^{{\epsilon}}\Pr[(Y,{{\cal A}}(X)) \in T_1] + \delta$. Therefore, $$\hat\delta\Pr[{{\cal A}}(X) \in Bad_1'] \leq \delta \Rightarrow \Pr[{{\cal A}}(X) \in Bad_1'] \leq \delta/\hat\delta.$$ Similarly, $\Pr[{{\cal A}}(X) \in Bad_2'] \leq \delta/\hat\delta.$ Finally, $$\begin{aligned} \Pr[{{\cal A}}(X) \in Bad_1 \cup Bad_2] &\leq& \Pr[{{\cal A}}(X) \in Bad_0] +\Pr[{{\cal A}}(X) \in Bad_1']+\Pr[{{\cal A}}(X) \in Bad_2'] \\ &=& \frac{2\delta}{{\epsilon}e^{{\epsilon}}}+\frac{\delta}{\hat\delta}+\frac{\delta}{\hat\delta} = \frac{2\delta}{{\epsilon}e^{{\epsilon}}}+\sqrt{\delta}.\end{aligned}$$ By symmetry, we also have $\Pr[{{\cal A}}'(Y) \in Bad_1 \cup Bad_2] \leq \frac{2\delta}{{\epsilon}e^{{\epsilon}}}+\sqrt{\delta}$. Therefore, with probability at least $1-\delta''$, $X|_{t}$ and $Y|_{t}$ are $(\hat{\epsilon},\hat\delta)$-differentialy private. The following corollary follows by using the above proposition (with $Y=X$) in conjunction with Claim \[lem:proof\] (part \[it:sd\]). \[cor:bayes\] Let $(X,{\mathcal{A}}(X))$ and $(X,{\mathcal{A}}'(X))$ be $({\epsilon},\delta)$-differentialy private. Then, with probability at least $1-\delta''$ over $t{\leftarrow}{\mathcal{A}}(X)$ $($equivalently $t {\leftarrow}{{\cal A}}'(X)$$)$, the statistical difference between $X|_{{\mathcal{A}}(X) =t}$ and $X|_{{\mathcal{A}}'(X)=t}$ is at most $e^{\hat{\epsilon}}-1+\hat\delta$ with $\hat{\epsilon}=3{\epsilon}$, $\hat\delta=2\sqrt{\delta}$, and $\delta''=O(\sqrt{\delta})$. (${\epsilon},\delta$)-differential privacy implies $({\epsilon}',\delta')$-semantic privacy for arbitrary (not necessarily informed) beliefs with ${\epsilon}'=e^{3{\epsilon}}-1+2\sqrt{\delta}$ and $\delta' = O(n\sqrt{\delta})$. $(\bar{\epsilon}/2,\delta)$-semantic privacy implies $(2{\epsilon},2\delta)$-differential privacy with $\bar{\epsilon}=e^{{\epsilon}}-1$. Let ${\mathcal{A}}$ be a $({\epsilon},\delta)$-differentialy private algorithm. Let $b$ be any belief distribution. From Claim \[lem:proof\] (part \[it:ajoint\]), we know that $(b,{\mathcal{A}}(b))$ and $(b,{\mathcal{A}}_i(b))$ are $({\epsilon},\delta)$-differentialy private. Let $\delta''=O(\sqrt{\delta})$. From Corollary \[cor:bayes\], we get that with probability at least $1-\delta''$ over $t {\leftarrow}{\mathcal{A}}(b)$, the statistical difference between $b|_{{\mathcal{A}}(b) =t}$ and $b|_{{\mathcal{A}}_i(b)=t}$ is at most ${\epsilon}'$. Therefore, for any ${{\mathrm{x}}}{\leftarrow}b$, with probability at least $(1-\delta'')$ over $t {\leftarrow}{\mathcal{A}}({{\mathrm{x}}})$, ${\mathbf{SD}{\left( {{b|_{{\mathcal{A}}({{\mathrm{x}}}) =t},b|_{{\mathcal{A}}_i({{\mathrm{x}}}) =t}}} \right)}} \leq {\epsilon}'$. Taking union bound over all coordinates $i$, implies that for any ${{\mathrm{x}}}{\leftarrow}b$ with probability at least $1-n\delta''$ over $t {\leftarrow}{\mathcal{A}}(b)$, for all $i=1,\dots,n$, we have ${\mathbf{SD}{\left( {{b|_{{\mathcal{A}}({{\mathrm{x}}}) =t},b|_{{\mathcal{A}}_i({{\mathrm{x}}}) =t}}} \right)}} \leq {\epsilon}'$. Therefore, ${\mathcal{A}}$ satisfies $({\epsilon}',\delta')$-semantic privacy for $b$. Since $b$ was arbitrary, we get that (${\epsilon},\delta$)-differential privacy implies $({\epsilon}',\delta')$-semantic privacy. To see that $(\bar{\epsilon}/2,\delta)$-semantic privacy implies $(2{\epsilon},2\delta)$-differential privacy, consider a belief distribution $b$ which is uniform over two databases ${{\mathrm{x}}},{{\mathrm{y}}}$ which are at Hamming distance of one. The proof idea is same as in Theorem \[thm:eind\]. Let $i$ be the position in which ${{\mathrm{x}}}$ and ${{\mathrm{y}}}$ differ. Let $\bar{\mathcal{A}}$ be an algorithm that with probability $1/2$ draws an output from ${\mathcal{A}}({{\mathrm{x}}})$ and with probability $1/2$ draws an output from ${\mathcal{A}}({{\mathrm{y}}})$. Consider a transcript $t$ drawn from $\bar{\mathcal{A}}$. The distribution $\bar{b}_i[\cdot|t]$ will be uniform over ${{\mathrm{x}}}$ and ${{\mathrm{y}}}$ since they induce the same distribution on transcripts in Game $i$. This means that with probability at least $1-\delta$ over $t {\leftarrow}\bar{\mathcal{A}}$, $\bar{b}_0[\cdot |t]$ will assign probabilities $1/2\pm \bar{\epsilon}/2$ to each of the two databases. Working through Bayes’ rule as in Theorem \[thm:eind\] shows that $\bar{\mathcal{A}}$ is point-wise $(2{\epsilon},\delta)$-differentialy private (with probability at least at least $1-2\delta$ of $t {\leftarrow}{\mathcal{A}}({{\mathrm{x}}})$, $e^{-2{\epsilon}}\Pr[{\mathcal{A}}({{\mathrm{y}}})=t] \leq \Pr[{\mathcal{A}}({{\mathrm{x}}})=t] \leq e^{2{\epsilon}} \Pr[{\mathcal{A}}({{\mathrm{y}}})=t]$). Therefore, with probability at least $1-\delta$ of $t {\leftarrow}\bar{\mathcal{A}}$, $e^{-2{\epsilon}}\Pr[{\mathcal{A}}({{\mathrm{y}}})=t] \leq \Pr[{\mathcal{A}}({{\mathrm{x}}})=t] \leq e^{2{\epsilon}} \Pr[{\mathcal{A}}({{\mathrm{y}}})=t]$. Similarly, for $t {\leftarrow}{\mathcal{A}}({{\mathrm{y}}})$. This implies that ${\mathcal{A}}$ is point-wise $(2{\epsilon},2\delta)$-differentialy private. Using Claim \[lem:proof\] (part \[it:pw2ind\]), implies that ${\mathcal{A}}$ is $(2{\epsilon},2\delta)$-differentialy private. Discussion and Consequences =========================== [Theorem \[thm:ind2sdp\]]{} states that the relaxations notions of differential privacy used in some previous work still imply privacy in the face of arbitrary side information. This is [*not*]{} the case for [*all*]{} possible relaxations, even very natural ones. For example, if one replaced the multiplicative notion of distance used in differential privacy with total variation distance, then the following “sanitizer” would be deemed private: choose an index $i\in\{1,\dots,n\}$ uniformly at random and publish the entire record of individual $i$ together with his or her identity (example 2 in [@DMNS06]). Such a “sanitizer” would not be meaningful at all, regardless of side information. Theorems \[thm:ind2sdp\] and A.3 give some qualitative improvements over existing security statements. Theorem A.3 implies that the claims of [@DiNi03; @DwNi04; @BDMN05] can be strengthened to hold for *all* predicates of the input simultaneously (a switch in the order of quantifiers). The strengthening does come at some loss in parameters since $\delta$ is increased. This incurs a factor of 2 in ${\log{\left( {\tfrac{1}{\delta}} \right)}}$, or a factor of $\sqrt{2}$ in the standard deviation. More significantly, [Theorem \[thm:ind2sdp\]]{} shows that noise processes with negligible probability of bad events have nice differential privacy guarantees even for adversaries who are not necessarily informed. There is a hitch however only adversaries whose beliefs somehow represent reality, i.e. for whom the real database is somehow “representative" of the adversary’s view can be said to learn nothing. Finally, the techniques used to prove [Theorem \[thm:ind2sdp\]]{} can also be used to analyze schemes which do not provide privacy for [*all*]{} pairs of neighboring databases ${{\mathrm{x}}}$ and ${{\mathrm{y}}}$, but rather only for [*most*]{} such pairs (remember that neighboring databases are the ones that differ in one entry). Specifically, it is sufficient that those databases where the “differential privacy” condition fails occur only with small probability. \[thm:dsemantic\] Let ${\mathcal{A}}$ be a randomized algorithm. Let $$\mathcal{E} = \{{{\mathrm{x}}}: \forall \mbox{ neighbors }{{\mathrm{y}}}\mbox{ of } {{\mathrm{x}}}, {\mathcal{A}}({{\mathrm{x}}}) \mbox{ and } {\mathcal{A}}({{\mathrm{y}}}) \mbox{ are } ({\epsilon},\delta)\mbox{-differentialy private}\}.$$ Then ${\mathcal{A}}$ satisfies $({\epsilon}',\delta')$-semantic privacy for any belief distribution $b$ such that $b[\mathcal{E}] = \Pr_{{{\mathrm{x}}}{\leftarrow}b}[{{\mathrm{x}}}\in \mathcal{E}] \geq 1-\delta$ with ${\epsilon}'=e^{3{\epsilon}}-1+2\sqrt{\delta}$ and $\delta'=O(n\sqrt{\delta})$. Let $b$ be a belief distribution with $b[\mathcal{E}] \geq 1-\delta$. Let $\delta''=O(\sqrt{\delta})$. From Claim \[lem:proof\] (part \[it:joint\]), we know that $(b,{\mathcal{A}}(b))$ and $(b,{\mathcal{A}}_i(b))$ are $({\epsilon},2\delta)$-differentialy private. From Corollary \[cor:bayes\], we get that with probability at least $1-\delta''$ over $t {\leftarrow}{\mathcal{A}}(b)$, the statistical difference between $b|_{{\mathcal{A}}(b) =t}$ and $b|_{{\mathcal{A}}_i(b)=t}$ is at most ${\epsilon}'$. Therefore, with probability at least $(1-\delta'')$ over pairs $({{\mathrm{x}}},t)$ where ${{\mathrm{x}}}{\leftarrow}b$ and $t {\leftarrow}{\mathcal{A}}({{\mathrm{x}}})$, ${\mathbf{SD}{\left( {{b|_{{\mathcal{A}}({{\mathrm{x}}}) =t},b|_{{\mathcal{A}}_i({{\mathrm{x}}}) =t}}} \right)}} \leq {\epsilon}'$. Taking union bound over all coordinates $i$, implies that with probability at least $1-n\delta''$ over pairs $({{\mathrm{x}}},t)$ where ${{\mathrm{x}}}{\leftarrow}b$ and $t {\leftarrow}{\mathcal{A}}({{\mathrm{x}}})$, for all $i=1,\dots,n$, we have ${\mathbf{SD}{\left( {{b|_{{\mathcal{A}}({{\mathrm{x}}}) =t},b|_{{\mathcal{A}}_i({{\mathrm{x}}}) =t}}} \right)}} \leq {\epsilon}'$. Therefore, ${\mathcal{A}}$ satisfies $({\epsilon}',\delta')$-semantic privacy for belief distribution $b$. Let $LS_f(\cdot)$ denote the local sensitivity of function $f$ (defined in [@NRS07]). Let $Lap(\lambda)$ denote the Laplacian distribution. This distribution has density function $h(y) \propto \exp(-|y|/\lambda)$, mean $0$, and standard deviation $\lambda$. Using the Laplacian noise addition procedure of [@DMNS06; @NRS07], along with Theorem \[thm:dsemantic\] we get, Let $\mathcal{E} =\{ {{\mathrm{x}}}\,:\, LS_f({{\mathrm{x}}}) \leq s \}$. Let ${\mathcal{A}}({{\mathrm{x}}}) = f({{\mathrm{x}}}) + \text{Lap}\left(\frac{s}{{\epsilon}}\right )$. Let $b$ be a belief distribution such that $b[\mathcal{E}] = \Pr_{{{\mathrm{x}}}{\leftarrow}b}[{{\mathrm{x}}}\in \mathcal{E}] \geq 1-\delta$. Then ${\mathcal{A}}$ satisfies $({\epsilon}',\delta')$-semantic privacy for the belief distribution $b$ with ${\epsilon}'=e^{3{\epsilon}}-1+2\sqrt{\delta}$ and $\delta'=O(n\sqrt{\delta})$. Acknowledgements {#acknowledgements .unnumbered} ================ We are grateful for helpful discussions with Cynthia Dwork, Frank McSherry, Moni Naor, Kobbi Nissim, and Sofya Raskhodnikova. Appendix A: Another View of Semantic Privacy {#appendix-a-another-view-of-semantic-privacy .unnumbered} ============================================ In this section, we discuss another possible definition of $({\epsilon},\delta)$-semantic privacy. Even though this definition seems to be the more desirable one, it also seems hard to achieve. \[reality-oblivious (${\epsilon},\delta$)-semantic privacy\] A randomized algorithm is reality-oblivious $({\epsilon},\delta)$-semantically private if for all belief distributions $b$ on $\mathcal{D}^n$, for all databases ${{\mathrm{x}}}\in \mathcal{D}^n$, with probability at least $1-\delta$ over transcripts $t$ drawn from ${{\cal A}}({{\mathrm{x}}})$, and for all $i=1,\dots,n$: $${\mathbf{SD}{\left( {{\bar{b}_0[{{\mathrm{x}}}|t]\ ,\ \bar{b}_i[{{\mathrm{x}}}|t]\ }} \right)}} \leq {\epsilon}.$$ We first prove if the adversary has arbitrary beliefs, then $({\epsilon},\delta)$-differential privacy doesn’t provide any reasonable reality-oblivious $({\epsilon}',\delta')$-semantic privacy guarantee. \[lem:noind2sdp\][^3] (${\epsilon},\delta$)-differential privacy does not imply reality-oblivious $({\epsilon}',\delta')$-semantic privacy for any reasonable values of ${\epsilon}'$ and $\delta'$. This counterexample is due to Dwork and McSherry: suppose that the belief distribution is uniform over $\{(0^n),(1,0^{n-1})\}$, but that real database is $(1^{n})$. Let the database ${{\mathrm{x}}}=(x_1,\dots,x_n)$. Say we want to reveal $f({{\mathrm{x}}})=\sum_ix_i$. Adding Gaussian noise with variance $\sigma^2={\log{\left( {\tfrac{1}{\delta}} \right)}}/{\epsilon}^2$ satisfies $({\epsilon},\delta)$-differential privacy (refer [@DMNS06; @NRS07] for details). However, with overwhelming probability the output will be close to $n$, and this will in turn induce a very non-uniform distribution over $\{(0^n),(1,0^{n-1})\}$ since $(1,0^{n-1})$ is exponentially (in $n$) more likely to generate a value near $n$ than $(0^n)$. More precisely, due to the Gaussian noise added, $$\frac{\Pr[{\mathcal{A}}({{\mathrm{x}}})=n \, | \, {{\mathrm{x}}}=(0^n)]}{\Pr[{\mathcal{A}}({{\mathrm{x}}})= n \, | \, {{\mathrm{x}}}=(1,0^{n-1})]} = \frac{\exp\left (\frac{-n^2}{2\sigma} \right)}{\exp \left(\frac{-(n-1)^2}{2\sigma}\right )} = \exp\left (\frac{-2n+1}{2\sigma} \right).$$ Therefore, given that the output is close to $n$, the posterior distribution of the adversary would be exponentially more biased toward $(1,0^{n-1})$ than $(0^n)$. Hence, it is exponentially far away from the prior distribution which was uniform. On the other hand, if the adversary believes he is seeing ${\mathcal{A}}({{\mathrm{x}}}_{-1})$, then no update will occur and the posterior distribution will remain uniform. Since the posterior distributions in these two situations are exponentially far apart (one exponentially far from uniform, other uniform), it shows that (${\epsilon},\delta$)-differential privacy does not imply any reasonable guarantee on reality-oblivious semantic privacy. However, $({\epsilon},\delta)$-differential privacy does provide a strong reality-oblivious $({\epsilon}',\delta')$-semantic privacy guarantee for [*informed*]{} belief distributions. Using terminology from [@BDMN05; @DMNS06], we say that a belief distribution $b$ is informed if $b$ is constant on $n-1$ coordinates and agrees with the database in those coordinates. This corresponds to the adversary knowing some set of $n-1$ entries in the database before interacting with the algorithm, and then trying to learn the remaining one entry from the interaction. Let ${\mathcal{A}}_i$ be a randomized algorithm such that for all databases ${{\mathrm{x}}}$, ${\mathcal{A}}_i({{\mathrm{x}}})={\mathcal{A}}({{\mathrm{x}}}_{-i})$. \[thm:ind2sdp-inf\] (${\epsilon},\delta$)-differential privacy implies reality-oblivious $({\epsilon}',\delta')$-semantic privacy for informed beliefs with ${\epsilon}'=e^{3{\epsilon}}-1+2\sqrt{\delta}$ and $\delta' = O(n\sqrt{\delta})$.[^4] Let ${\mathcal{A}}$ be a $({\epsilon},\delta)$-differentialy private algorithm. Let ${{\mathrm{x}}}$ be any database. Let $b$ be any informed belief distribution. This means that $b$ is constant on all $n-1$ coordinates, and agrees with ${{\mathrm{x}}}$ in those $n-1$ coordinates. Let $i$ be the coordinate which is not yet fixed in $b$. From Claim \[lem:proof\] (part \[it:ajoint\]), we know that $(b,{\mathcal{A}}(b))$ and $(b,{\mathcal{A}}_i(b))$ are $({\epsilon},\delta)$-differentialy private. Therefore, we can apply Lemma \[lem:bayes\]. Let $\delta''=O(\sqrt{\delta})$. From Corollary \[cor:bayes\], we get that with probability at least $1-\delta''$ over $t {\leftarrow}{\mathcal{A}}(b)$, the statistical difference between $b|_{{\mathcal{A}}(b) =t}$ and $b|_{{\mathcal{A}}_i(b)=t}$ is at most ${\epsilon}'$. Therefore, for ${{\mathrm{x}}}$, with probability at least $(1-\delta'')$ over $t {\leftarrow}{\mathcal{A}}({{\mathrm{x}}})$, ${\mathbf{SD}{\left( {{b|_{{\mathcal{A}}({{\mathrm{x}}}) =t},b|_{{\mathcal{A}}_i({{\mathrm{x}}}) =t}}} \right)}} \leq {\epsilon}'$. Taking union bound over all coordinates $i$, implies that with probability at least $1-n\delta''$ over $t {\leftarrow}{\mathcal{A}}({{\mathrm{x}}})$, for all $i=1,\dots,n$, we have ${\mathbf{SD}{\left( {{b|_{{\mathcal{A}}({{\mathrm{x}}}) =t},b|_{{\mathcal{A}}_i({{\mathrm{x}}}) =t}}} \right)}} \leq {\epsilon}'$. Therefore, ${\mathcal{A}}$ satisfies reality-oblivious $({\epsilon}',\delta')$-semantic privacy for $b$. Since ${{\mathrm{x}}}$ was arbitrary, we get that (${\epsilon},\delta$)-differential privacy implies reality-oblivious $({\epsilon}',\delta')$-semantic privacy for informed beliefs. [^1]: That said, some of the other relaxations, such as probabilistic differential privacy from  [@MKAGV08], might lead to better parameters in [Theorem \[thm:ind2sdp\]]{}. [^2]: A few similar properties relating to statistical difference were shown in [@SV99]. Note that $({\epsilon},\delta)$-indistinguishability is not a metric, unlike statistical difference. But it does inherit some nice metric like properties. [^3]: Note that adversaries whose belief distribution is very different from the real database (as in the counterexample of Theorem A.2 may think they have learned a lot. But does such “learning" represent a breach of privacy? We do not think so, but leave the final decision to the reader. [^4]: Reality-oblivious $(\bar{\epsilon}/2,\delta)$-semantic privacy implies $(2{\epsilon},2\delta)$-differential privacy with $\bar{\epsilon}=e^{{\epsilon}}-1$. For details see the proof of Theorem \[thm:ind2sdp\].

--- abstract: 'Recent high resolution spectroscopic analysis of nearby FGK stars suggests that a high C/O ratio of greater than 0.8, or even 1.0, is relatively common. Two published catalogs find C/O$>0.8$ in 25-30% of systems, and C/O$>1.0$ in $\sim$ 6-10%. It has been suggested that in protoplanetary disks with C/O$>0.8$ that the condensation pathways to refractory solids will differ from what occurred in our solar system, where C/O$=0.55$. The carbon-rich disks are calculated to make carbon-dominated rocky planets, rather than oxygen-dominated ones. Here we suggest that the derived stellar C/O ratios are overestimated. One constraint on the frequency of high C/O is the relative paucity of carbon dwarfs stars ($10^{-3}-10^{-5}$) found in large samples of low mass stars. We suggest reasons for this overestimation, including a high C/O ratio for the solar atmosphere model used for differential abundance analysis, the treatment of a Ni blend that affects the O abundance, and limitations of one-dimensional LTE stellar atmosphere models. Furthermore, from the estimated errors on the measured stellar C/O ratios, we find that the significance of the high C/O tail is weakened, with a true measured fraction of C/O$>0.8$ in 10-15% of stars, and C/O$>1.0$ in 1-5%, although these are still likely overestimates. We suggest that infrared T-dwarf spectra could show how common high C/O is in the stellar neighborhood, as the chemistry and spectra of such objects would differ compared to those with solar-like abundances. While possible at C/O$>0.8$, we expect that carbon-dominated rocky planets are rarer than others have suggested.' author: - 'Jonathan J. Fortney' title: 'On the Carbon-to-Oxygen Ratio Measurement in Nearby Sunlike Stars: Implications for Planet Formation and the Determination of Stellar Abundances' --- Introduction ============ The Composition of Stars and Planets ------------------------------------ The determination of the abundances of atoms in the atmospheres of stars is an essential element of modern astronomy. Recently, tremendous work has occurred on understanding the relationship between planets and the abundances of planet-hosting and non-planet-hosting stars. Since the pioneering work of [Gonzalez97]{}, many investigators have worked to understand connections between stellar abundances and the observed frequency [Santos04,Fischer05,Johnson10]{} and composition [Guillot06,Burrows07,Miller11]{} of planets. Our solar system is one realization of the complex planet formation process. The raw materials that made up the Sun and solar nebula, through a process of condensation, grain growth, and accumulation, gave rise to four rocky planets in our inner solar system that are predominantly composed of Mg-Si-O-bearing rocks and Fe-Ni metals. In other solar systems, with parent star disks with other abundances, a different selection of refractory materials, or in different relative abundances, surely occur. For instance, if a nebula’s carbon-to-oxygen (C/O) ratio is $\gtrsim$0.8, condensation pathways can change dramatically, leading to carbon-dominated rocky planets, as recently discussed in detail by [Bond10]{}. There had been prior intermittent interested in carbon-dominated planets in the past decade, from [Gaidos00]{}, [Lodders04]{}, and [Kuchner05]{}, to name three examples. In particular, [Gaidos00]{} discussed different formation scenarios for giant planet cores and rocky planets in disks with varied C/O ratios, as well as how the chemical evolution of the galaxy generally can lead to enhanced C/O through time. [Lodders04]{} suggested that the planetesimals that make up Jupiter’s heavy element enrichment were carbon-rich, and that Jupiter initially formed at the “tar line” rather than the “ice line.” This is one possible explanation for the low water abundance measured by the *Galileo Entry Probe* [Wong04]{}. [Kuchner05]{}, similar to [Gaidos00]{}, were interested in giant planets and terrestrial planets that could form in environments where the local (or entire disk’s) C/O$>1$, leading to carbon-dominated (rather than oxygen-dominated) silicates. More recently [Bond10]{} coupled protoplanetary disk abundances derived from stellar spectra to a model of disk chemistry, which yields the condensation sequence of solids. Their work further coupled the formation of solids to an N-body model of planet formation [OBrien06]{}. For particular planetary systems, with measured C/O and Mg/Si ratios of the host star, they calculated the equilibrium disk chemistry and solid composition for the initial planetesimal distribution. [Bond10]{} furthermore kept track of the contribution of particular planetesimals as they add their mass to growing protoplanets, and in the end find the relative contributions of C, O, Mg, Si, etc. to the masses of formed planets. Within the context of giant planets, [Madhu11]{} suggested that day-side photometry of the transiting planet WASP-12b indicates an atmosphere with C/O$>1.0$. More recently [Madhu11b]{} and [Oberg11]{} have investigated the accumulation of gas and icy planetesimals in disks with a range of C/O ratios to understand possible pathways to forming “carbon-rich” gas giants. C/O Ratio in Stars ------------------ Composition-dependent planet formation models depend on the stellar abundances of C and O for the initial conditions of disk chemistry. The stellar C/O ratios in [Bond10]{} were taken from determinations of C from [Ecuvillon04]{} and of O from [Ecuvillon06]{}. Motivated by [Bond10]{}, larger tabulations of C and O abundances were recently made by [Delgado10]{}, for 370 FGK stars from the HARPS planet-search sample, and [Petigura11]{}, for 457 F and G stars from the California Planet Survey sample. These two studies are relatively similar, as they cover large samples that include planet-hosting stars and those not found to host planets. The two studies do have some differences in the lines of C and O chosen. For the carbon abundance, the [Delgado10]{} work used CI lines at 5380.3 and 5052.2 Å, with only 5380.3 Å used for stars with [$T_{\rm eff}$]{}$< 5100$ K. For oxygen, the forbidden lines of \[OI\] at 6300 and 6363 Å were used. The derivation of the abundances was done with a combination of the code MOOG, for the generation of synthetic spectra \[\]\[as updated in 2002\][Sneden73]{}, the Kurucz ATLAS9 atmosphere grid with overshooting [Kurucz93]{}, and the equivalent widths were measured using the ARES program [Sousa07]{}. The [Petigura11]{} study used a CI line for carbon at 6587 Å, and the \[OI\] line for oxygen at 6300 Å. The derivation of the abundances was performed with the Spectroscopy Made Easy (SME) code [Valenti96]{} with Kurucz stellar atmospheres. Tabulations from [Bond10]{} (who quote Ecuvillon et al. values), [Delgado10]{}, and [Petigura11]{} are shown in . Both of the large studies found a somewhat similar shape. They found a maximum at C/O ratios modestly higher that that of the Sun \[0.55, from\]\[\][Asplund09]{}, with a noticeably enhanced peak in the distribution found by [Delgado10]{}, shown most clearly in *b*. Of particular interest to all of these authors, and our *Letter*, is the tail off to higher C/O ratios $>0.8$ (dotted line) and even further onto $>1.0$ (short dashed line). [Delgado10]{} find C/O$>0.8$ for 24% of their stars, and C/O$>1.0$ for 6%. For the [Petigura11]{} sample, they find C/O$>0.8$ for 29% of their stars, and C/O$>1.0$ for 10%. The numbers quoted are for the mixed sample of planet-hosting and non-planet hosting stars. Taken at face value, and the condensation chemistry in [Bond10]{}, this potentially implies carbon planets (formed when C/O$>0.8$) in $\sim$25% of planetary systems. However, one must take care when estimating the fraction of stars with high C/O ratios, given observational error bars. Of interest is the positive tail at high C/O, and a tail such as this is expected since the error is approximately constant in the logarithmic abundance ratio, \[C/O\]. This leads to an error distribution that is log-normal in the C/O ratio, and is seen in, for instance, Table 6 in [Petigura11]{}. Their average error at C/O$=1$ is 0.23, which is 1.61$\sigma$ above the solar C/O ratio they use, of 0.63. From this 1.61$\sigma$, 5.4% of the stellar sample is thus expected to be found with C/O$>1$, just due to observational errors. This would yield a “true” fraction of stars with C/O$>1$ of $\sim$5%, rather than 10%. [Delgado10]{} do not provide individual errors on their C/O determinations, but assuming similar errors, their entire sample of C/O$>1$ stars (6%) can be explained by errors. At C/O$=0.8$, in the [Petigura11]{} sample, the average error is 0.16, a difference of 0.17 above their solar value, or 1.06$\sigma$. This yields an expected fraction of 14.5%, which would move the 29% with C/O$>0.8$ down to 15%. For [Delgado10]{}, this expected fraction moves their 24% at C/O$>0.8$ down to 10%. The upshot is a much reduced tail of stars with high C/O ratios. ![*Top*: Histogram of the C/O ratios from the papers of [Ecuvillon04,Ecuvillon06]{} (as tabulated in [@Bond10]), [Delgado10]{}, and [Petigura11]{}, in blue, green, and orange, respectively. C/O ratios of 0.55 (the solar value from Asplund et al. 2009), 0.8, and 1.0 are shown as long dashed, dotted, and short dashed lines, respectively. The adopted solar C/O ratios from [Delgado10]{} and [Petigura11]{} are shown as thick green and orange ticks at the top. *Bottom*: The data sets are normalized, yielding fractions of each sample, instead of number. The large samples of [Delgado10]{} and [Petigura11]{} find 24-29% of FGK stars have C/O$>0.8$, and 6-10% have C/O$>1$.[]{data-label="histo1"}](f1a.eps "fig:"){width="4.5in"} ![*Top*: Histogram of the C/O ratios from the papers of [Ecuvillon04,Ecuvillon06]{} (as tabulated in [@Bond10]), [Delgado10]{}, and [Petigura11]{}, in blue, green, and orange, respectively. C/O ratios of 0.55 (the solar value from Asplund et al. 2009), 0.8, and 1.0 are shown as long dashed, dotted, and short dashed lines, respectively. The adopted solar C/O ratios from [Delgado10]{} and [Petigura11]{} are shown as thick green and orange ticks at the top. *Bottom*: The data sets are normalized, yielding fractions of each sample, instead of number. The large samples of [Delgado10]{} and [Petigura11]{} find 24-29% of FGK stars have C/O$>0.8$, and 6-10% have C/O$>1$.[]{data-label="histo1"}](f1b.eps "fig:"){width="4.5in"} Motivation ========== Deriving the abundances of C and O in stellar atmospheres is a demanding task. Even for the Sun, this has been an especially difficult over the past decade \[e.g.,\]\[\][Allende01,Allende02,Asplund09]{}. The most recent state-of-the-art work includes intense efforts to find unblended lines, calculated NLTE corrections where applicable, and solar atmosphere pressure-temperature profiles that come directly from three-dimensional simulations [Asplund09,Caffau11]{}. While deriving the abundances of atoms and molecules in low-mass M and K dwarf stellar atmospheres is more difficult that than of Sunlike stars, they do have one natural asset. Cooler dwarf atmospheres have a natural “flip” in the chemistry at C/O$=1$. At C/O$<1$, most O goes into the CO molecule, but additional O is left over and partitions into H$_2$O and, for M stars, TiO/VO gases. TiO/VO absorption bands dominate the optical spectra of M stars. However, at C/O$>1$, the CO molecules use up nearly all O in cool stars, leaving excess C, which goes into molecules such as C$_2$ and CN. In such a cool star, the optical region is dominated by C$_2$ and CN. We suggest that if C/O$>1$ in FGK stars is a relatively common phenomenon then cool carbon stars showing C$_2$ and CN bands should be relatively common in the solar neighborhood. This is something that can be investigated, and was even briefly noted by [Petigura11]{}. True Fraction of Carbon Dwarfs? =============================== Carbon dwarfs, which appear to have C/O$>1$ in their atmosphere, are rare stars. The best current understanding of their formation is that they are *not* made of primordial C/O$>1$ gas, but instead have high C abundances due to accretion of AGB dredge-up material from a companion \[e.g.,\]\[\][deKool95,Steinhardt05]{}. Carbon dwarfs have been found by many authors, with most being found by large surveys, such as the SDSS [Margon02,Downes04]{}. Their numbers are predominantly spread among what would otherwise be G, K, and M spectral types. For our purposes here, instead of forming these carbon-rich stars by accretion, we can take the extremely optimistic view that these are in fact primordial C/O$>1$ systems. Then the fraction of dwarfs that are carbon dwarfs would be the upper limit on the fraction of stars with primordial C/O$>1$. [deKool95]{} have previously estimated the frequency of carbon dwarfs, based on detections before SDSS. They estimate a space density in the disk of $\sim 1 \times 10^{-6}$ pc${-3}$, but noted this may be an overestimate. This can be compared to the space density of stars with mass $<0.7$ [$M_{\odot}$]{}, $6.5 \times 10^{-2}$ pc${-3}$, from [Bochanski10]{}. This is a 4-5 order of magnitude difference. There are other ways to estimate the carbon dwarf frequency. In particular, [Covey08]{} have analyzed SDSS spectra and 2MASS photometry of 25,000 sources, in an effort to take a census of low mass stars out to $J$=16.2, over a mass range from 0.1 - 0.7 [$M_{\odot}$]{}. For these cool stars, if C/O$>1$ did occur, it should be clear, given the abundance of carbon-rich molecules (C$_2$, CN) that would occur in such cool atmospheres. Of their sample of 9,649 low mass stars, they note 24 “exotic contaminants” including 4 carbon stars. This implies a C/O$>1$ in only 0.04% of cases, although based on color cuts $\sim$1/2 of the carbon dwarfs may have been missed (K.Covey, personal communication). This work, and the fact that [Downes04]{} found only $\sim$100 carbon dwarfs from a large SDSS search at $15.6<r<20.8$ to specifically find such stars, certainly strengthens the point of [deKool95]{} that carbon dwarfs are quite rare. The relative frequency is likely $10^{-3}-10^{-5}$. We can think of no reason that Sunlike stars in the solar neighborhood would be uniformly more enriched in carbon (or oxygen deficient) than the KM stars. One potential way out could be if the later-type stars are systematically older than the earlier-type stars, so that some amount of galactic chemical evolution could have taken place. In this case the younger (FG) stars could have higher C/O \[e.g.\]\[\][Gaidos00]{}. However, the HARPS/California samples are not young stars, so constructing a credible explanation through this path seems unlikely. Diagnosing high C/O from Brown Dwarfs? ====================================== The K & M dwarfs from the SDSS give us leverage into the fraction of dwarfs with C/O$>1$. Are there are any stellar populations that could be used to understand the fraction of stars with $0.8<$C/O$<1$? One possibility are the L and T dwarfs. L-type dwarfs, from [$T_{\rm eff}$]{}s of $\sim$ 2400-1400 K, have atmospheres whose infrared opacity is dominated by molecular bands of H$_2$O, CO, and cloud layers made up of refractory condensates like corundum (Al$_2$O$_3$), enstatite (MgSiO$_3$), forsterite (Mg$_2$SiO$_4$), and iron (Fe) [AM01]{}. Corundum, enstatite, and forsterite need abundant oxygen to form. If C/O$>0.8$, or C/O$>1$, are common in L-dwarfs, then perhaps we could see a different kind of cloudy L-dwarf, with condensates dominated by SiC instead of the Mg-Si-O forms. Certainly the spectra of such objects would be highly abnormal at C/O$>1$, as they would strongly favor CO at the expense of H$_2$O, but perhaps even at C/O ratios closer to 0.8, the cloud condensation pathway would be different than normal L dwarfs. In the cooler objects, the T-dwarfs, the silicate clouds appear to reside below most of the visible atmosphere. Quite importantly, these refractory clouds remove 21% of oxygen from the atmosphere \[e.g.\]\[and references therein\][Visscher10]{}. This removal of oxygen from the gas raises the C/O ratio in the visible atmosphere. [Lodders10]{} have investigated chemistry as a function of C/O ratio at 1100 K and 0.01 bar, which is after silicate condensation. Lodders finds a dramatic change in chemistry even at C/O$=0.8$. At C/O$>0.8$, methane becomes more abundant than water (by two orders of magnitude at C/O$=0.9$) and HCN becomes nearly as abundant as water. Since CO, CH$_4$, H$_2$O (and HCN) all have prominent opacity in the infrared, C/O$>0.8$ brown dwarfs could appear distinctly different than those with a C/O ratio closer to that of the Sun. Since most brown dwarfs are found from color-color diagrams in surveys such as SDSS and 2MASS, it is quite possible that high C/O brown dwarfs, should they exist, could have previously eluded detection. The cuts from SDSS use $i-z$ color, which could in principle be less sensitive to the C/O chemistry since the optical spectra of brown dwarfs are shaped by pressure-broadened lines of Na and K [bms]{}. But 2MASS uses $JHK_{\rm s}$ cuts, which could lead to high C/O dwarfs being missed, since in some cases H$_2$O would not be the dominant opacity source, leading to different near-infrared colors. As far as we are aware, there are no compelling outliers for high C/O brown dwarfs, but a search for them could be valuable. A full exploration of C/O ratio in brown dwarfs, and its affects on clouds, is beyond the scope of this work. However, there is an additional point that we can motivate. In T dwarfs, essentially all infrared opacity is due to water, methane, and carbon dioxide, which tie up nearly all C and O. The relatively cloud-free spectra of T dwarfs may allow for accurate determinations of C/O in the solar neighborhood. In Figure \[ratio\] we show model spectra of two 900 K dwarfs, one with C/O$=0.55$ (the solar value) and another at C/0$=0.7$. These models were computed using the atmosphere code of M. Marley and J. Fortney and collaborators, described elsewhere [Marley02,Fortney07b]{}. The opacity and chemistry databases are taken from a previous tabulation [Fortney05]{}. The differences between the two spectra can be prominent, in particular as a ratio. One can see differences of 10% in the JHK peaks, and 20-30% in the water bands. With future improvements in the accuracy of opacities, in particular a high-temperature database for CH$_4$, we suggest it may be possible to derive relative C and O abundances, from H$_2$O, CH$_4$, and CO, from infrared T-dwarf spectra. High spectral resolution observations of brown dwarfs are now being achieved as well [Rice10]{}, which could allow further progress in abundance analysis. The utility of such efforts would be a better understanding of the C/O ratio of stars in the solar neighborhood. Recently, some work by [Tsuji11]{} on a small number of brown dwarfs with AKARI observations has moved in this direction. Discussion ========== We suggest that recent high-resolution spectroscopic analyses have overestimated the fraction of FGK stars with C/O$>1.0$, and, by extension, perhaps with C/O$>0.8$ as well. [Gaidos00]{} and [Bond10]{} suggested that the pathway to form “rocky” planets in C/O$>0.8$ systems will differ in these carbon-rich system, making carbon-rich planets. While this logic seems secure, we find it is less likely than some have anticipated. This is because the true fraction of carbon-rich parent stars is quite low, $10^{-3}-10^{-5}$. However, our work should not be interpreted as claiming that carbon-rich terrestrial or giant planets cannot form. The C/O ratio surely varies in the ISM and the region of phase space between $0.8<\mathrm{C/O}<1$ is less constrained by our work. However, given the overestimation at C/O$>1$ it appears probable that this region is less-populated than has been recently suggested [Bond10,Delgado10,Petigura11]{}, although not empty. Furthermore we showed in §1.2 that the quoted observational error bars from these groups imply smaller measured fractions of high C/O stars, which eliminated 15% from C/O$>0.8$ and 5% from C/O$>1.0$. These authors of course put in great effort to understand the sources of their errors, and to correct for them. The choice of lines used could be re-examined. The choice of low-excitation forbidden \[OI\] lines along with high excitation permitted CI lines could introduce systemic errors. A revised (although difficult) study could examine \[CI\]/\[OI\] or CI/OI, since these lines would behave similarly at a given stellar temperature and in deviations from LTE (M. Asplund, personal communication). An additional point is that both the large [Delgado10]{} and [Petigura11]{} works tuned their abundance retrieval methods to match Sun’s C/O ratio, as a standard. [Asplund09]{}, in a recent review, find C/O$=0.55$. However, in [Delgado10]{} the particular abundances used for the Sun lead to C/O$=0.66$, while in [Petigura11]{}, it is 0.63. (See the thick tick marks at the top of Figure 1). So there clearly could be an offset of $\sim$0.1, towards higher C/O, in these works. An issue with the Ni blend of the \[OI\] line used by [Petigura11]{} may potentially skew their results towards higher C/O ratios. These authors adopt the abundances of O and Ni from 3D abundance analyses, and then use them in their 1D analysis. To obtain a good fit to the solar spectrum, they adopted $\mathrm{log}(gf)=-1.98$ for the NiI line, which is 35% higher than the laboratory measurement of measured by [Johansson03]{}. Consequently, [Petigura11]{} in their Figure 3, show a NiI line that is 40% stronger than found in, for example, [Allende01]{}. This could explain the upward trend in C/O with increasing \[Fe/H\] in their Figure 16, as follows: a known trend is that \[Ni/Fe\] increases slightly \[e.g.\]\[\][Robinson06]{} and \[O/Fe\] decreases as metallicity (\[Fe/H\]) increases \[e.g.\]\[\][Delgado10]{}. A Ni blend that is too strong in the solar fit will increase in importance as \[Fe/H\] increases, leading to an overestimation of the Ni blend, an underestimation of the O abundance, and a C/O ratio that is too high. In the [Petigura11]{} sample, this could explain why most of the C/O$>1$ stars occur when \[Fe/H\]$>0.2$. We note that [Delgado10]{} and [Petigura11]{} both relied on the stellar atmosphere models of Kurucz. As a comparison, one could alternatively use the PHOENIX [Hauschildt99]{} or MARCS [Gustafsson08]{} stellar atmospheres. [Gustafsson08]{} have compared their MARCS models to published models from Kurucz and PHOENIX, and the agreement does indeed vary with [$T_{\rm eff}$]{} and with plane parallel vs. spherical symmetry. There are still other pathways towards understanding extrasolar abundances. As FG stars evolve onto the red giant branch and cool to [$T_{\rm eff}$]{} values of $\sim$ 4000 K, it may be easier to constrain their C/O ratios in the same qualitative manner we suggest for M dwarfs. The composition of extrasolar planetesimals can be determined by studying externally polluted white dwarf atmospheres, and some early evidence has been found for carbon-poor planetesimals [Jura06]{}. While we have focused exclusively on stellar abundances, it is important to recall that condensation of solids within a disk itself can potentially lead to non-stellar C/O ratios in nebula gas. For instance, condensation of water in a protoplanetary disk can leave the surrounding nebula gas relatively carbon-rich (through large abundance of gaseous CO), which can change the relative ratios compared to those of the parent star [Stevenson88,Lodders10,Oberg11]{}. This may be a pathway to forming giant planets with relatively high C/O ratios in their envelopes. A giant planet atmosphere with C/O$>1$ was recently suggested for planet WASP-12b, based on fits to 7 photometric points of day-side planet emission [Madhu11]{}. Their finding could be made more robust with the inclusion into their model of the absorption bands of molecules that are expected in the high C/O regime, such as HCN and C$_2$H$_2$ [Lodders10]{}, which were not considered in the [Madhu11]{} study. Carbon-dominated rocky planets would be extremely interesting objects. Their prevalence around stars of Sunlike abundances, and those stars with enhanced C/O ratios, would certainly tell us much about nebular condensation chemistry and the planet formation process. While these planets may be inherently rare, we look forward to additional advances in the future. JJF acknowledges support from the Alfred P. Sloan Foundation. JJF thanks the anonymous referee for a important suggestions regarding statistics and nickel blends. JJF thanks Debra Fischer, Martin Asplund, Bruce Margon, Katharina Lodders, Channon Visscher, Mark Marley, Mike Irwin, Kevin Covey, Mike Jura, Graeme Smith, Greg Laughlin, Geoff Marcy, Travis Barman, Sean Raymond, Andrew Howard, Jade Bond, and David O’Brien for many stimulating conversations throughout the course of this project. JJF thanks Jacob Bruns for compiling the C/O data. [49]{} natexlab\#1[\#1]{} , A. S., & [Marley]{}, M. S. 2001, , 556, 872 , C., [Lambert]{}, D. L., & [Asplund]{}, M. 2001, , 556, L63 —. 2002, , 573, L137 , M., [Grevesse]{}, N., [Sauval]{}, A. J., & [Scott]{}, P. 2009, , 47, 481 , J. J., [Hawley]{}, S. L., [Covey]{}, K. R., [West]{}, A. A., [Reid]{}, I. N., [Golimowski]{}, D. A., & [Ivezi[ć]{}]{}, [Ž]{}. 2010, , 139, 2679 , J. C., [O’Brien]{}, D. P., & [Lauretta]{}, D. S. 2010, , 715, 1050 , A., [Hubeny]{}, I., [Budaj]{}, J., & [Hubbard]{}, W. B. 2007, , 661, 502 , A., [Marley]{}, M. S., & [Sharp]{}, C. M. 2000, , 531, 438 , E., [Ludwig]{}, H.-G., [Steffen]{}, M., [Freytag]{}, B., & [Bonifacio]{}, P. 2011, , 268, 255 , K. R., [et al.]{} 2008, , 136, 1778 , M., & [Green]{}, P. J. 1995, , 449, 236 , E., [Israelian]{}, G., [Gonz[á]{}lez Hern[á]{}ndez]{}, J. I., [Bond]{}, J. C., [Santos]{}, N. C., [Udry]{}, S., & [Mayor]{}, M. 2010, , 725, 2349 , R. A., [et al.]{} 2004, , 127, 2838 , A., [Israelian]{}, G., [Santos]{}, N. C., [Mayor]{}, M., [Villar]{}, V., & [Bihain]{}, G. 2004, , 426, 619 , A., [Israelian]{}, G., [Santos]{}, N. C., [Shchukina]{}, N. G., [Mayor]{}, M., & [Rebolo]{}, R. 2006, , 445, 633 , D. A., & [Valenti]{}, J. 2005, , 622, 1102 , J. J., & [Marley]{}, M. S. 2007, , 666, L45 , J. J., [Marley]{}, M. S., [Lodders]{}, K., [Saumon]{}, D., & [Freedman]{}, R. 2005, , 627, L69 , E. J. 2000, , 145, 637 , G. 1997, , 285, 403 , T., [Santos]{}, N. C., [Pont]{}, F., [Iro]{}, N., [Melo]{}, C., & [Ribas]{}, I. 2006, , 453, L21 , B., [Edvardsson]{}, B., [Eriksson]{}, K., [J[ø]{}rgensen]{}, U. G., [Nordlund]{}, [Å]{}., & [Plez]{}, B. 2008, , 486, 951 , P. H., [Allard]{}, F., [Ferguson]{}, J., [Baron]{}, E., & [Alexander]{}, D. R. 1999, , 525, 871 , S., [Litz[é]{}n]{}, U., [Lundberg]{}, H., & [Zhang]{}, Z. 2003, , 584, L107 , J. A., [Aller]{}, K. M., [Howard]{}, A. W., & [Crepp]{}, J. R. 2010, , 122, 905 , M. 2006, , 653, 613 , M. J., & [Seager]{}, S. 2005, ArXiv:0504214 , R. 1993, CD-ROM 13, ATLAS9 Stellar Atmosphere Programs and 2 km/s Grid (Cambridge:SAO) , K. 2004, , 611, 587 —. 2010, [in Formation and Evolution of Exoplanets, ed. R. Barnes]{} (Wiley), 157–186, arXiv:0910.0811 , N., [Mousis]{}, O., [Johnson]{}, T. V., & [Lunine]{}, J. I. 2011, , 743, 191 , N., [et al.]{} 2011, , 469, 64 , B., [et al.]{} 2002, , 124, 1651 , M. S., [Seager]{}, S., [Saumon]{}, D., [Lodders]{}, K., [Ackerman]{}, A. S., [Freedman]{}, R. S., & [Fan]{}, X. 2002, , 568, 335 , N., & [Fortney]{}, J. J. 2011, , 736, L29 , K. I., [Murray-Clay]{}, R., & [Bergin]{}, E. A. 2011, , 743, L16 , D. P., [Morbidelli]{}, A., & [Levison]{}, H. F. 2006, , 184, 39 , E. A., & [Marcy]{}, G. W. 2011, , 735, 41 , E. L., [Barman]{}, T., [Mclean]{}, I. S., [Prato]{}, L., & [Kirkpatrick]{}, J. D. 2010, , 186, 63 , S. E., [Laughlin]{}, G., [Bodenheimer]{}, P., & [Fischer]{}, D. 2006, , 643, 484 , N. C., [Israelian]{}, G., & [Mayor]{}, M. 2004, , 415, 1153 , C. A. 1973, PhD thesis, THE UNIVERSITY OF TEXAS AT AUSTIN. , S. G., [Santos]{}, N. C., [Israelian]{}, G., [Mayor]{}, M., & [Monteiro]{}, M. J. P. F. G. 2007, , 469, 783 , C. L., & [Sasselov]{}, D. D. 2005, ArXiv:0502152 , D. J., & [Lunine]{}, J. I. 1988, , 75, 146 , T., [Yamamura]{}, I., & [Sorahana]{}, S. 2011, , 734, 73 , J. A., & [Piskunov]{}, N. 1996, , 118, 595 , C., [Lodders]{}, K., & [Fegley]{}, Jr., B. 2010, , 716, 1060 , M. H., [Mahaffy]{}, P. R., [Atreya]{}, S. K., [Niemann]{}, H. B., & [Owen]{}, T. C. 2004, Icarus, 171, 153

--- abstract: 'A search is performed for collimated muon pairs displaced from the primary vertex produced in the decay of long-lived neutral particles in proton-proton collisions at $\sqrt{s}$ = 7   centre-of-mass energy, with the ATLAS detector at the LHC. In a [1.9 fb$^{-1}$]{}event sample collected during 2011, the observed data are consistent with the Standard Model background expectations. Limits on the product of the production cross section and the branching ratio of a Higgs boson decaying to hidden-sector neutral long-lived particles are derived as a function of the particles’ mean lifetime.' author: - The ATLAS Collaboration title: '[Search for displaced muonic lepton jets from light Higgs boson decay in proton-proton collisions at [$\sqrt{s}$]{}  = 7    with the ATLAS detector]{}' --- A search is presented for long-lived neutral particles decaying to final states containing collimated muon pairs in proton-proton collisions at $\sqrt{s}$ = 7   centre-of-mass energy. The event sample, collected during 2011 at the LHC with the ATLAS detector, corresponds to an integrated luminosity of [1.9 fb$^{-1}$]{}. The model considered in this analysis consists of a Higgs boson decaying to a new hidden sector of particles which finally produce two sets of collimated muon pairs, but the search described is equally valid for other, distinct models such as heavier Higgs boson doublets, singlet scalars or a $Z^\prime$ that decay to a hidden sector and eventually produce collimated muon pairs.\ Recently, evidence for the production of a boson with a mass of about 126  has been published by ATLAS [@higgsatl] and CMS [@higgscms]. The observation is compatible with the expected production and decay of the Standard Model (SM) Higgs boson [@HIGGS1; @HIGGS2; @HIGGS3] at this mass. Testing the SM Higgs hypothesis is currently of utmost importance. To this end two effects may be considered: (i) additional resonances which arise in an extended Higgs sector found in many extensions of the SM, or (ii) rare Higgs boson decays which may deviate from those predicted by the SM. In this Letter we search for a scalar that decays to a light hidden sector, focusing on the 100 GeV to 140 GeV mass range. In doing so, we cover both of the above aspects, deriving constraints on additional Higgs-like bosons, as well as placing bounds on the branching ratio of the discovered 126 GeV resonance into a hidden sector of the kind described below.\ The phenomenology of light hidden sectors has been studied extensively over the past few years [@b1; @b2; @b3; @b4; @b5]. Possible characteristic topological signatures of such extensions of the SM are “lepton jets". A lepton jet is a cluster of highly collimated particles: electrons, muons and possibly pions [@b2; @b6; @b7; @b8]. These arise if light unstable particles with masses in the  to  range (for example dark photons, [$\gamma_{d}$]{}) reside in the hidden sector and decay predominantly to SM particles. At the LHC, hidden-sector particles may be produced with large boosts, causing the visible decay products to form jet-like structures. Hidden-sector particles such as [$\gamma_{d}$]{}may be long-lived, resulting in decay lengths comparable to, or larger than, the detector dimensions. The production of lepton jets can occur through various channels. For instance, in supersymmetric models, the lightest visible superpartner may decay into the hidden sector. Alternatively, a scalar particle that couples to the visible sector may also couple to the hidden sector through Yukawa couplings or the scalar potential. This analysis is focused on the case where the Higgs boson decays to the hidden sector [@b9; @b10]. The SM Higgs boson has a narrow width into SM final states if $m_{H} < 2 m_W$. Consequently, any new (non-SM) coupling to additional states, which reside in a hidden sector, may contribute significantly to the Higgs boson decay branching ratios. Even with new couplings, the total Higgs boson width is typically small, well below the order of one GeV. If a SM-like Higgs boson is confirmed, it will remain important to constrain possible rare decays, e.g. into lepton jets.\ Neutral particles with large decay lengths and collimated final states represent, from an experimental point of view, a challenge both for the trigger and for the reconstruction capabilities of the detector. Collimated particles in the final state can be hard to disentangle due to the finite granularity of the detectors; moreover, in the absence of inner tracking detector information and a primary vertex constraint, it is difficult to reconstruct charged-particle tracks from decay vertices far from the interaction point (IP). The ATLAS detector [@ATLASTDR] is equipped with a muon spectrometer (MS) with high-granularity tracking detectors that allow charged-particle tracks to be reconstructed in a standalone configuration using only the muon detector information (MS-only). This is a crucial feature for detecting muons not originating from the primary interaction vertex.\ The search presented in this Letter focuses on neutral particles decaying to the simplest type of muon jets (MJs), containing only two muons; prompt MJ searches have been performed both at the Tevatron [@tevatron1; @tevatron2] and at the LHC [@CMS]. Other searches for displaced decays of a light Higgs boson to heavy fermion pairs have also been performed at the LHC [@Hiddenv].\ The benchmark model used for this analysis is a simplified scenario where the Higgs boson decays to a pair of neutral hidden fermions ($f_{d2}$) each of which decays to one long-lived [$\gamma_{d}$]{}and one stable neutral hidden fermion ($f_{d1}$) that escapes the detector unnoticed, resulting in two lepton jets from the [$\gamma_{d}$]{}decays in the final state (see Fig. \[fig:model\]). The mass of the [$\gamma_{d}$]{}(0.4 ) is chosen to provide a sizeable branching ratio to muons [@b9]. ![Schematic picture of the Higgs boson decay chain, H$\rightarrow$2($f_{d2}\rightarrow f_{d1}$[$\gamma_{d}$]{}). The Higgs boson decays to two hidden fermions ($f_{d2}$). Each hidden fermion decays to a [$\gamma_{d}$]{}and to a stable hidden fermion ($f_{d1}$), resulting in two muon jets from the [$\gamma_{d}$]{}decays in the final state.[]{data-label="fig:model"}](fig_01a.pdf){width="55mm"} ATLAS is a multi-purpose detector [@ATLASTDR] at the LHC, consisting of an inner tracking system (ID) embedded in a superconducting solenoid, which provides a 2 T magnetic field parallel to the beam direction, electromagnetic and hadronic calorimeters and a muon spectrometer using three air-core toroidal magnet systems[^1]. The trigger system has three levels [@L1TRIG] called Level-1 (L1), Level-2 (L2) and Event Filter (EF). L1 is a hardware-based system using information from the calorimeter and muon spectrometer, and defines one or more Regions of Interest (ROIs), geometrical regions of the detector, identified by ($\eta$, $\phi$) coordinates, containing interesting physics objects. L2 and the EF (globally called the High Level Trigger, HLT) are software-based systems and can access information from all sub-detectors. The ID, consisting of silicon pixel and micro-strip detectors and a straw-tube tracker, provides precision tracking of charged particles for . The electromagnetic and hadronic calorimeter system covers and, at , has a total depth of 9.7 interaction lengths (22 radiation lengths in the electromagnetic part). The MS provides trigger information () and momentum measurements () for charged particles entering the spectrometer. It consists of one barrel and two endcap parts, each with 16 sectors in $\phi$, equipped with precision tracking chambers and fast detectors for triggering. Monitored drift tubes are used for precision tracking in the region and cathode strip chambers are used for 2.0 $\leq$ . The MS detectors are arranged in three stations of increasing distance from the IP: inner, middle and outer. The air core toroidal magnetic field allows an accurate charged particle reconstruction independent of the ID information. The three planes of trigger chambers (resistive plate chambers in the barrel and the thin gap chambers in the endcaps) are located in middle and outer (only in the barrel) stations. The L1 muon trigger requires hits in the middle stations to create a low tranverse momentum () muon ROI or hits in both the middle and outer stations for a high  ROI. The muon ROIs have a spatial extent of () in the barrel and of in the endcap. L1 ROI information seeds, at HLT level, the reconstruction of muon momenta using the precision chamber information. In this way sharp trigger thresholds up to 40  can be obtained. The set of parameters used to generate the signal Monte Carlo samples is listed in Table \[tab:param\]. The Higgs boson is generated through the gluon-gluon fusion production mechanism which is the dominant process for a low mass Higgs boson. The gluon-gluon fusion Higgs boson production cross section in [*pp*]{} collisions at [$\sqrt{s}$]{}= 7 , estimated at the next-to-next-to-leading order (NNLO) [@HiggsCrossS], is $\sigma_{\textrm{\small SM}} = $ 24.0 pb for $m_{H}=$ 100  and $\sigma_{\textrm{\small SM}} = $ 12.1 pb for $m_{H}=$ 140 . The [[PYTHIA]{}]{} generator [@PYTHIA] is used, linked together with [[MadGraph]{}]{}4.4.2 [@b12] and [[BRIDGE]{}]{} [@BRIDGE], for gluon-gluon fusion production of the Higgs boson and the subsequent decay to hidden-sector particles.\ As discussed in the introduction, the signal is chosen to enable a study of rare, non-SM, Higgs boson decays in the (possibly extended) Higgs sector. To do so we choose two points which envelope a mass range covering the 126 GeV resonance. The lower mass point, $m_H=$ 100 GeV, is chosen to be compatible with the decay-mode-independent search by OPAL at LEP [@opal]. The higher mass point, $m_H=$ 140 GeV, is chosen well below the $WW$ threshold, where a sizeable branching ratio into a hidden sector may be naturally achieved. The masses of $f_{d2}$ and $f_{d1}$ are chosen to be light relative to the Higgs boson mass, and far from the kinematic threshold at $m_{f_{d1}} + m_{\gamma_{d}} = m_{f_{d2}}$. For the chosen dark photon mass (0.4 ), the [$\gamma_{d}$]{}decay branching ratios are expected to be [@b9]: 45$\%~ \ee$, 45$\%~\mu^+\mu^-$, 10$\%~\pi^+\pi^-$. Thus $20\%$ of the Higgs [[*H*]{}$\rightarrow \gamma_{d}~\gamma_{d}+ X$]{}decays are expected to have the required four-muon final state.\ The mean lifetime $\tau$ of the [$\gamma_{d}$]{}(expressed throughout this Letter as $\tau$ times the speed of light $c$) is a free parameter of the model. In the generated samples [c$ \tau$]{}is chosen so that a large fraction of the decays occur inside the sensitive ATLAS detector volume, i.e. up to 7 m in radius and 13 m along the $z$-axis, where the trigger chambers of the middle stations are located. The detection efficiency can then be estimated for a range of [$\gamma_{d}$]{}mean lifetime through re-weighting of the generated samples.\ ------------ -------------- -------------- ---------------------- -------------- Higgs mass $m_{f_{d2}}$ $m_{f_{d1}}$ [$\gamma_{d}$]{}mass [c$ \tau$]{} $[\gev]$ $[\gev]$ $[\gev]$ $[\gev]$ \[mm\] 100 5.0 2.0 0.4 47 140 5.0 2.0 0.4 36 ------------ -------------- -------------- ---------------------- -------------- : Parameters used for the Monte Carlo simulation. The last column is the [$\gamma_{d}$]{}mean lifetime $\tau$ multiplied by the speed of light $c$, expressed in mm.[]{data-label="tab:param"} Potential backgrounds include all the processes which lead to real prompt muons in the final state such as the SM processes [*W*]{}+jets, [*Z*]{}+jets, , [*WW*]{}, [*WZ*]{}, and [*ZZ*]{}. However, the main contribution to the background is expected from processes giving a high production rate of secondary muons which do not point to the primary vertex, such as decays in flight of $K/\pi$ and heavy flavour decays in multi-jet processes, or muons due to cosmic rays. The prompt lepton background samples are generated using [[PYTHIA]{}]{} ([*W*]{}+jets, and [*Z*]{}+jets) and [[MC@NLO]{}]{} [@mcatlno] (, [*WW*]{}, [*WZ*]{}, and [*ZZ*]{}). The generated Monte Carlo events are processed through the full ATLAS simulation chain based on [[GEANT4]{}]{} [@GEANT4; @ATLSIM]. Additional [*pp*]{} interactions in the same and nearby bunch crossings (pile-up) are included in the simulation. All Monte Carlo samples are re-weighted to reproduce the observed distribution of the number of interactions per bunch crossing in the data. For the multi-jet background evaluation a data-driven method is used. The cosmic-ray background is also evaluated from data. The main kinematic characteristics of the signal sample are: - [The [$\gamma_{d}$]{}pair are emitted approximately back-to-back in $\phi$, with an angular spread of the distribution due to the emission of the $f_{d1}$. ]{} - [The average  of the $\gamma_{d}$ in the laboratory frame is about 20  for [$m_{H}=~$100 GeV]{}and 30  for [$m_{H}=~$140 GeV]{}; due to the small mass of the [$\gamma_{d}$]{}, large boost factors in the decay length should be expected.]{} - [ Fig. \[fig:drmu\] shows the distribution of $\Delta R = \sqrt{(\Delta\eta)^2 + (\Delta\phi)^2}$ between the two muons from the [$\gamma_{d}$]{}decay. The $\Delta R$ is computed at the decay vertex of the [$\gamma_{d}$]{}from the vector momenta of the two muons. Due to the small mass of the [$\gamma_{d}$]{}the $\Delta R$ is almost always below 0.1.]{} Since the two $f_{d1}$ are, like the two [$\gamma_{d}$]{}, emitted back-to-back in $\phi$, the observed missing transverse momentum , computed at the event-generator level, is small and cannot be used as a discriminating variable against the background. ![$\Delta R$ distribution between the two muons from the $\gamma_{d}$ decay for the signal Monte Carlo samples with $m_{H}$ = 100  and $m_{H}$ = 140 .[]{data-label="fig:drmu"}](fig_02a.pdf){width="70mm"} The dataset used for this analysis was collected at a centre-of-mass energy of 7  during the first part of 2011, where a low level of pile-up events in the same bunch-crossing was present (an average of $\approx 6$ interactions per crossing). Only periods in which all ATLAS subdetectors were operational are used. The total integrated luminosity used is 1.94 $\pm$ $0.07$ fb$^{-1}$ [@LUMI1; @LUMI2]. All events are required to have at least one reconstructed vertex along the beam line with at least three associated tracks, each with  $\geq$ 0.4 . The primary interaction vertex is defined to be the vertex whose constituent tracks have the largest $\Sigma$[$p_{\mathrm{T}}^{\mathrm{2}}$]{}. This analysis deals with displaced [$\gamma_{d}$]{}decays with final states containing only muons. Signal events are therefore characterized by a four-muon final state with the four muons coming from two displaced decay vertices. Due to the relatively low  of the muons and to the displaced decay vertex, a low- multi-muon trigger with muons reconstructed only in the MS is needed. In order to have an acceptably low trigger rate at a low  threshold, a multiplicity of at least three muons is required. Candidate events are collected using an unprescaled HLT trigger with three reconstructed muons of  $ \geq $ 6 , seeded by a L1-accept with three different muon ROIs. These muons are reconstructed only in the MS, since muons originating from a neutral particle decaying outside the pixel detector will not have a matching track in the ID tracking system. The trigger efficiency for the Monte Carlo signal samples, defined as the fraction of events passing the trigger requirement with respect to the events satisfying the analysis selection criteria (described in Section 6) is 0.32$\pm 0.01_{\textrm{\small stat}}$ for [$m_{H}=~$100 GeV]{}and 0.31$\pm 0.01_{\textrm{\small stat}}$ for [$m_{H}=~$140 GeV]{}.\ The main reason for the relatively low trigger efficiency is the small opening $\Delta R$ between the two muons of the [$\gamma_{d}$]{}decay ($\Delta R \leq 0.1$) shown in Fig. \[fig:drmu\]. These values of $\Delta R$ are often smaller than the L1 trigger granularity; in this case the L1 produces only one ROI. The trigger only fires if at least one of the [$\gamma_{d}$]{}produces two distinct L1 ROIs. The single [$\gamma_{d}$]{}ROI efficiency, $\varepsilon_{\textrm{\footnotesize 2ROI}}$ ($\varepsilon_{\textrm{\footnotesize 1ROI}}$), defined as the fraction of [$\gamma_{d}$]{}passing the offline selection that give two (one) trigger ROIs is 0.296 $\pm~0.004_{\textrm{\small stat}}$ (0.626 $\pm~0.004_{\textrm{\small stat}}$) for [$m_{H}=~$100 GeV]{}and 0.269 $\pm~0.003_{\textrm{\small stat}}$ (0.653 $\pm~0.003_{\textrm{\small stat}}$) for [$m_{H}=~$140 GeV]{}. Fig. \[fig:treggeffVSetadr2\] shows the $\varepsilon_{\textrm{\footnotesize 2ROI}}$ as a function of the dark photon $\eta$ and of the [$\Delta R$]{}of the two muons from the [$\gamma_{d}$]{}decay. The increased trigger granularity in the endcap and the efficiency decrease at small values of [$\Delta R$]{}are clearly visible.\ The systematic uncertainty on the trigger efficiency is estimated with a sample of  from collision data and a corresponding sample of Monte Carlo events, using the tag-and-probe (TP) method. A cut on $\Delta R \leq 0.1$ between the two muons is used to reproduce the small track-to-track spatial separation in the MS of the signal. The tag is a (MS+ID) combined muon, defined as a MS-reconstructed muon that is associated with a trigger object and combined with a matching “good ID track". Good ID tracks must have at least one hit in the pixel detector, at least six hits in the silicon micro-strip detectors and at least six hits in the straw-tube tracker. The probe is a good ID track which, when combined with the tag track, gives an invariant mass inside a 100  window around the  mass. A muon ROI that matches the probe in $ \eta$ and $\phi$, and is different from the ROI associated with the tag, is searched for. The number of probes with a matched ROI divided by the number of probes without a matched ROI gives the $\varepsilon$$_{\textrm{\footnotesize 2ROI}}^{\textrm{\footnotesize TP}}$/$\varepsilon$$_{\textrm{\footnotesize 1ROI}}^{\textrm{\footnotesize TP}}$ ratio. Values of $\varepsilon$$_{\textrm{\footnotesize 2ROI}}^{\textrm{\footnotesize TP}}$/$\varepsilon$$_{\textrm{\footnotesize 1ROI}}^{\textrm{\footnotesize TP}}$ = 0.42$\pm 0.05_{\textrm{\small stat}}$ for the  data and $\varepsilon$$_{\textrm{\footnotesize 2ROI}}^{\textrm{\footnotesize TP}}$/$\varepsilon$$_{\textrm{\footnotesize 1ROI}}^{\textrm{\footnotesize TP}}$ = 0.39$\pm 0.05_{\textrm{\small stat}}$ for the corresponding Monte Carlo sample are obtained. The relative statistical uncertainty on the difference between these two estimates is 17$\%$ and this is taken conservatively to be the systematic uncertainty on the trigger efficiency. ![$\varepsilon_{\textrm{\tiny 2ROI}}$ as a function (a) of the $\eta$ of the [$\gamma_{d}$]{}and (b) of the [$\Delta R$]{}of the muon pair for the Monte Carlo samples with Higgs boson masses of 100  and 140 . The errors are statistical only.[]{data-label="fig:treggeffVSetadr2"}](fig_03a.pdf "fig:"){width="70mm"} ![$\varepsilon_{\textrm{\tiny 2ROI}}$ as a function (a) of the $\eta$ of the [$\gamma_{d}$]{}and (b) of the [$\Delta R$]{}of the muon pair for the Monte Carlo samples with Higgs boson masses of 100  and 140 . The errors are statistical only.[]{data-label="fig:treggeffVSetadr2"}](fig_03b.pdf "fig:"){width="70mm"} MJs from displaced [$\gamma_{d}$]{}decays are characterized by a pair of muons in a narrow cone, produced away from the primary vertex of the event. Consequently tracks reconstructed in the MS with a good quality track fit [@reco] are used. MJs are identified using a simple clustering algorithm that associates all the muons in cones of $\Delta R = 0.2$, starting with the muon with highest . The size of the cone takes into account the multiple scattering of the muons in the calorimeters. All the muons found in the cone are associated with a MJ. After this procedure, if any muons are unassociated with a MJ the search is repeated for this remainder, starting again with the highest  muon. This continues until all possible MJs are formed. The MJ direction and momentum are obtained from the vector sum over all muons in the MJ. Only MJs with two reconstructed muons are accepted and only events with two MJs are kept for the subsequent analysis.\ The possible contribution to the background of SM processes which lead to real prompt muon pairs in the final state is evaluated using simulated samples. After the trigger and the requirement of having two MJs in the event, their contributions have been found to be negligible. The only significant background sources are expected to be from processes giving a high production rate of secondary muons which do not point to the primary vertex, such as decays in flight of $K/\pi$ and heavy flavour decays in multi-jet production, or cosmic-ray muons not pointing to the primary vertex.\ In order to separate the signal from the background, a number of discriminating variables have been studied. The multi-jet background can be significantly reduced by using calorimeter isolation requirements around the MJ direction. The calorimetric isolation variable [$E_{\mathrm{T}}^{\mathrm{isol}}$]{}is defined as the difference between the transverse calorimetric energy  in a cone of $\Delta R = 0.4$ around the highest  muon of the MJ and the  in a cone of $\Delta R = 0.2$; a cut [$E_{\mathrm{T}}^{\mathrm{isol}}$]{}$ \leq$ 5$~ \gev~$ keeps almost all the signal. The isolation modelling is validated for real isolated muons with a sample of muons coming from $Z \rightarrow \mu\mu$ decays. To further improve the signal-to-background ratio, two additional discriminating variables are used: [$\Delta \phi$]{}  between the two MJs and [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}for the MJ, defined as the scalar sum of the transverse momentum of the tracks, measured in the ID, inside a cone $\Delta R = 0.4$ around the direction of the MJ. The muon tracks of the MJ in the ID, if any, are not removed from the isolation sum, so that prompt muons, which give a reconstructed track in both the ID and MS, will contribute to the [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}. As a consequence a cut on [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}of a few  will remove prompt MJs or MJs with very short decay length. {width="60mm"} {width="60mm"} {width="60mm"} For the background coming from cosmic-ray muons (mainly pairs of almost parallel cosmic-ray muons crossing the detector) a cut on the impact parameters of the muon tracks with respect to the primary interaction vertex is used.\ The final set of selection criteria used is the following: - Topology cut: events are required to have exactly two MJs, $N_{\textrm{\footnotesize MJ}}$ = 2. - MJ isolation: require MJ isolation with [$E_{\mathrm{T}}^{\mathrm{isol}}$]{}$ \leq$ 5$~ \gev~$ for both MJs in the event. - Require $|\Delta \phi|$ $\geq$ 2 between the two MJs. - Require opposite charges for the two muons in a MJ (Q$_{\textrm{\footnotesize MJ}} = 0$). - Require a cut on the transverse and longitudinal impact parameters of the muons with respect to the primary vertex: $|d_0| <$ 200 mm and $|z_0|< $ 270 mm. - Require [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}$< 3 \GeV$ for both MJs. The distributions of the relevant variables at the different steps of the cut flow are shown in Fig. \[fig:sumpt\_scan1\]. The results are summarized in Table \[tab:cutflowABCD\]. No events survive the selection in the data sample whereas the expected signals from Monte Carlo simulation, assuming 100$\%$ branching ratio for [[*H*]{}$\rightarrow \gamma_{d}~\gamma_{d}+ X$]{}and the parameters given in Table \[tab:param\], are 75 or 48 events for Higgs boson masses of 100  and 140  respectively. The method used to estimate the cosmic-ray and multi-jet background yields, quoted in Table \[tab:cutflowABCD\], is discussed in Section 7. cut cosmic-rays multi-jet total background [$m_{H}=~$100 GeV]{} [$m_{H}=~$140 GeV]{} data ------------------------------------------------------ ------------------- ---------------------------------- ------------------------------------- ------------------------ ---------------------- ------ $N_{\textrm{\scriptsize MJ}}= 2 $ $3.0\pm 2.1 $ N/A N/A 135$\pm11_{-21}^{+29}$ 90$\pm9_{-13}^{+17}$ 871 [$E_{\mathrm{T}}^{\mathrm{isol}}$]{}$ \leq$ 5$~\gev$ $3.0\pm 2.1 $ N/A N/A 132$\pm11_{-21}^{+28}$ 88$\pm9_{-13}^{+17}$ 219 $|\Delta \phi|\geq$ 2 $1.5\pm 1.5 $ 153 $\pm$ 18 $\pm$ 9 155 $\pm$ 18 $\pm$ 9 123$\pm11_{-19}^{+26}$ 81$\pm9_{-12}^{+15}$ 104 Q$_{\textrm{\scriptsize MJ}}$ = 0 $1.5 \pm 1.5$ 57 $\pm$15$\pm$22 59 $\pm$ 15 $\pm$ 22 121$\pm11_{-19}^{+26}$ 79$\pm8_{-12}^{+15}$ 80 $|d_0|$, $|z_0|$ $0_{-0} ^{+1.64}$ 111$\pm$39$\pm$63 111$\pm$39$\pm$63 105$\pm10_{-16}^{+22}$ 66$\pm8_{-10}^{+12}$ 70 [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}$< 3~\GeV$ $0_{-0} ^{+1.64}$ 0.06$\pm 0.02 ^{+0.66} _{-0.06}$ $0.06^{+1.64 +0.66} _{-0.02 -0.06}$ 75$\pm9_{-12}^{+16}$ 48$\pm7_{-7}^{+9}$ 0 The resulting single [$\gamma_{d}$]{}reconstruction efficiency for the mean lifetimes given in Table \[tab:param\] is shown in Fig. \[fig:Efficiencies\] as a function of $\eta$, the [$\Delta R$]{}separation of the two muons from the [$\gamma_{d}$]{}decay and the decay length in the transverse plane, , of the [$\gamma_{d}$]{}. The efficiency is defined as the number of [$\gamma_{d}$]{}passing the offline selection divided by the number of [$\gamma_{d}$]{}in the spectrometer acceptance () with both muons having  $\geq $ 6 . The low reconstruction efficiency at very short is a consequence of the [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}cut. ![[$\gamma_{d}$]{}reconstruction efficiency $\varepsilon_{\textrm{\footnotesize rec}}$ as a function (a) of $\eta$, (b) of [$\Delta R$]{}and (c) of the transverse decay length of the [$\gamma_{d}$]{}for [$m_{H}=~$100 GeV]{}and [$m_{H}=~$140 GeV]{}and for the mean lifetimes given in Table \[tab:param\]. The reconstruction efficiency is defined as the number of [$\gamma_{d}$]{}passing the offline selection divided by the number of [$\gamma_{d}$]{}in the spectrometer acceptance () with both muons having  $\geq $ 6 . The uncertainties are statistical only.[]{data-label="fig:Efficiencies"}](fig_05a.pdf "fig:"){width="70mm"} ![[$\gamma_{d}$]{}reconstruction efficiency $\varepsilon_{\textrm{\footnotesize rec}}$ as a function (a) of $\eta$, (b) of [$\Delta R$]{}and (c) of the transverse decay length of the [$\gamma_{d}$]{}for [$m_{H}=~$100 GeV]{}and [$m_{H}=~$140 GeV]{}and for the mean lifetimes given in Table \[tab:param\]. The reconstruction efficiency is defined as the number of [$\gamma_{d}$]{}passing the offline selection divided by the number of [$\gamma_{d}$]{}in the spectrometer acceptance () with both muons having  $\geq $ 6 . The uncertainties are statistical only.[]{data-label="fig:Efficiencies"}](fig_05b.pdf "fig:"){width="70mm"} ![[$\gamma_{d}$]{}reconstruction efficiency $\varepsilon_{\textrm{\footnotesize rec}}$ as a function (a) of $\eta$, (b) of [$\Delta R$]{}and (c) of the transverse decay length of the [$\gamma_{d}$]{}for [$m_{H}=~$100 GeV]{}and [$m_{H}=~$140 GeV]{}and for the mean lifetimes given in Table \[tab:param\]. The reconstruction efficiency is defined as the number of [$\gamma_{d}$]{}passing the offline selection divided by the number of [$\gamma_{d}$]{}in the spectrometer acceptance () with both muons having  $\geq $ 6 . The uncertainties are statistical only.[]{data-label="fig:Efficiencies"}](fig_05c.pdf "fig:"){width="70mm"} The systematic uncertainty on the reconstruction efficiency is evaluated using a tag-and-probe method by comparing the reconstruction efficiency $\varepsilon$$_{\textrm{\footnotesize rec}}^{\textrm{\footnotesize TP}}$ for  samples from collision data and  Monte Carlo simulation. The tag-and-probe definitions and the cut on $\Delta R \leq 0.1$ between the two muons are the same as in Section 5. To measure the reconstruction efficiency the ID probe track is associated with a MS-only muon track, different from the one associated with the tag. The result is shown in Fig. \[fig:effi-DR-Reco\].\ The relative difference between the result obtained from the  data and the   Monte Carlo sample in the same range of $\Delta R \leq 0.1$, as for the signal, is taken as the systematic uncertainty on the reconstruction efficiency and amounts to 13$\%$. ![Tag-and-probe reconstruction efficiency $\varepsilon$$_{\textrm{\scriptsize rec}}^{\textrm{\scriptsize TP}}$ as a function of the [$\Delta R$]{}between the two muons, evaluated on a sample of  from collision data and a corresponding sample of Monte Carlo events. The $\varepsilon$$_{\textrm{\scriptsize rec}}^{\textrm{\scriptsize TP}}$ for the signal Monte Carlo, evaluated with a similar tag-and-probe method, is also shown. The uncertainties are statistical only.[]{data-label="fig:effi-DR-Reco"}](fig_06a.pdf){width="70mm"} To estimate the multi-jet background contamination in the signal region we use a data-driven ABCD method slightly modified to cope with the problem of the very low number of events in the control regions. The ABCD method assumes that two variables can be identified, which are relatively uncorrelated, and which can each be used to separate signal and background. It is assumed that the multi-jet background distribution can be factorized in the MJ [$E_{\mathrm{T}}^{\mathrm{isol}}$]{} – $|\Delta \phi|$ plane. The region A is defined by [$E_{\mathrm{T}}^{\mathrm{isol}}$]{}$\leq$ 5   and $|\Delta \phi|<$ 2; the region B, defined by [$E_{\mathrm{T}}^{\mathrm{isol}}$]{}$\leq$ 5   and $|\Delta \phi|\geq$ 2, is the signal region. The regions C and D are the anti-isolated regions ([$E_{\mathrm{T}}^{\mathrm{isol}}$]{}$>$ 5 ) and they are defined by $|\Delta \phi|<$ 2 and $|\Delta \phi|\geq$ 2, respectively. Neglecting the signal contamination in regions A, C and D ([$E_{\mathrm{T}}^{\mathrm{isol}}$]{}$>$ 5   or $|\Delta \phi|< $ 2) the number of multi-jet background events in the signal region can be evaluated as $N_B = N_D \times N_A/ N_C$. Due to the very low number of events in the control regions the values of $N_A$, $N_C$ and $N_D$ as a function of the cut on the final discriminant variable [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}are extracted by modelling the [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}distributions with bifurcated Gaussian templates, with parameters fitted from the data in the corresponding regions, and by integrating the fitted function in the range 0 $<$ [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}$<$ 3 . The low statistics in the four regions at each step of the cut flow give rise to large fluctuations in the multi-jet background estimate. The extracted yields are $N_A = (7.1 \pm 1.5_{\textrm{\small stat}})\cdot 10^{-3}$, $N_C = (1.81 \pm 1.0_{\textrm{\small stat}})\cdot 10^{-2}$ and $N_D = (1.51 \pm 0.07_{\textrm{\small stat}})\cdot 10^{-1}$ and the estimated number of multi-jet background events in the signal region is $N_B = 0.06 \pm 0.02_{\textrm{\small stat}}$. Possible sources of systematic uncertainty related to the background estimation method are also evaluated. The functional form is changed and the procedure to estimate the number of multi-jet background events in the signal region is repeated. The difference in $N_B$ is taken as the systematic uncertainty in the modelling of the multi-jet background shape and it amounts to $^{+0.66} _{-0.06}$. The effect of possible signal leakage in the background regions is also considered and is found to be negligible.\ The background induced by muons from cosmic-ray showers is evaluated using events collected by the trigger active when there are no collisions (empty bunch crossings). The number of triggered events is rescaled by the collision to empty bunch crossing ratio and for the active time (since the trigger in the empty bunch crossing was not active in all the runs). No events survived the requirements on the impact parameters with respect to the primary vertex ($|d_0| <$ 200 mm and $|z_0|<$ 270 mm), resulting in a cosmic-ray contamination estimate of $0^{+1.64} _{-0}$. The final yields for the different background sources are summarized in Table \[tab:cutflowABCD\]. The following effects are considered as possible sources of systematic uncertainty: - [**Luminosity**]{}\ The overall normalisation uncertainty of the integrated luminosity is $3.7\%$ [@LUMI1; @LUMI2]. - [**Muon momentum resolution**]{}\ The systematic uncertainty on the muon momentum resolution for MS-only muons has been evaluated by smearing and shifting the momenta of the muons by scale factors derived from  data-Monte Carlo comparison, and by observing the effect of this shift on the signal efficiency. The overall effect of the muon momentum resolution uncertainty is negligible. - [**Trigger**]{}\ The systematic uncertainty on the single [$\gamma_{d}$]{}trigger efficiency, evaluated using a tag-and-probe method is $17\%$ (see Section 5). - [**Reconstruction efficiency**]{}\ The systematic uncertainty on the reconstruction efficiency, evaluated using a tag-and-probe method for the single [$\gamma_{d}$]{}reconstruction efficiency, is $13\%$ (see Section 6). - [**Effect of pile-up**]{}\ The systematic uncertainty on the signal efficiency related to the effect of pile-up is evaluated by comparing the number of signal events after imposing all the selection criteria on the signal Monte Carlo sample increasing the average number of interactions per crossing from $\approx 6$ to $\approx 16$. This systematic uncertainty is negligible. - [**Effect of [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}cut**]{}\ Since the [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}cut could affect the minimum [c$ \tau$]{}value that can be excluded, the effect of this cut on the signal Monte Carlo has been studied. A variation of $10\%$ on the [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}cut results in a relative variation of $<$1$\%$ on the signal, which can therefore be neglected. - [**Background evaluation**]{}\ The systematic uncertainties that can affect the background estimation are related to the data-driven method used. The functional model used to fit the [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}distribution is varied to evaluate the systematic uncertainty in the modelling of its shape, which also includes the effect of the [$ \Sigma p_{\mathrm{T}}^{\mathrm{ID}}$]{}cut on the background estimation. This systematic uncertainty amounts to $^{+0.66} _{-0.06}$ events. The effect of signal leakage is also negligible. The efficiency of the selection criteria described above is evaluated for the simulated signal samples (see Table \[tab:param\]) as a function of the mean lifetime of the [$\gamma_{d}$]{}. Using pseudo-experiments with [c$ \tau$]{}ranging from 0 to 700 mm the number of [$\gamma_{d}$]{}that decay in each region of the detector is weighted by the corresponding total efficiency for that region. In this way the number of expected signal events is predicted as a function of the [$\gamma_{d}$]{}mean lifetime. These numbers, together with the expected number of background events (multi-jet and cosmic rays) and taking into account the zero data events surviving the selection criteria in [1.9 fb$^{-1}$]{}, are used as input to obtain limits at the 95$\%$ confidence level ([*CL*]{}). The [*CLs*]{} method [@CLspaper] is used to set 95$\%$ [*CL*]{} upper limits on the cross section times branching ratio ([$\sigma\times$BR ]{}) for the process [[*H*]{}$\rightarrow \gamma_{d}~\gamma_{d}+ X$]{}. Here the branching ratio of [[$\gamma_{d}$]{}$\rightarrow \mu~\mu$]{}is set to $45\%$ with the [$\gamma_{d}$]{}mass set to 0.4 , as previously discussed. The [$\sigma\times$BR ]{}is given as a function of the [$\gamma_{d}$]{}mean lifetime, expressed as [c$ \tau$]{}for [$m_{H}=~$100 GeV]{}and [$m_{H}=~$140 GeV]{}. These limits are shown on Fig. \[fig:CLboth\]. Table \[tab:limits\] shows the ranges in which the $\gamma_d$ [c$ \tau$]{}is excluded at the $95\%$ [*CL*]{} for [[*H*]{}$\rightarrow \gamma_{d}~\gamma_{d}+ X$]{}branching ratios of $100\%$ and $10\%$. ------------------ ----------------------------- ----------------------------- Higgs boson mass excluded c$\tau$ $[mm]$ excluded c$\tau$ $[mm]$ $[\gev]$ BR($100\%$) BR($10\%$) 100 1 $\leq$ c$\tau$ $\leq$ 670 5 $\leq$ c$\tau$ $\leq$ 159 140 1 $\leq$ c$\tau$ $\leq$ 430 7 $\leq$ c$\tau$ $\leq$ 82 ------------------ ----------------------------- ----------------------------- : Ranges in which [$\gamma_{d}$]{}[c$ \tau$]{}is excluded at $95\%$ [*CL*]{} for [$m_{H}=~$100 GeV]{}and [$m_{H}=~$140 GeV]{}, assuming 100$\%$ and 10$\%$ branching ratio of [[*H*]{}$\rightarrow \gamma_{d}~\gamma_{d}+ X$]{}.[]{data-label="tab:limits"} ![ The 95$\%$ upper limits on the [$\sigma\times$BR ]{}for the process [[*H*]{}$\rightarrow \gamma_{d}~\gamma_{d}+ X$]{}as a function of the dark photon [c$ \tau$]{}for the benchmark sample with (a) [$m_{H}=~$100 GeV]{}and with (b) [$m_{H}=~$140 GeV]{}. The expected limit is shown as the dashed curve and the solid curve shows the observed limit. The horizontal lines correspond to the Higgs boson SM cross sections at the two mass values.[]{data-label="fig:CLboth"}](fig_07a.pdf "fig:"){width="70mm"} ![ The 95$\%$ upper limits on the [$\sigma\times$BR ]{}for the process [[*H*]{}$\rightarrow \gamma_{d}~\gamma_{d}+ X$]{}as a function of the dark photon [c$ \tau$]{}for the benchmark sample with (a) [$m_{H}=~$100 GeV]{}and with (b) [$m_{H}=~$140 GeV]{}. The expected limit is shown as the dashed curve and the solid curve shows the observed limit. The horizontal lines correspond to the Higgs boson SM cross sections at the two mass values.[]{data-label="fig:CLboth"}](fig_07b.pdf "fig:"){width="70mm"} The ATLAS detector at the LHC was used to search for a light Higgs boson decaying into a pair of hidden fermions ($f_{d2}$), each of which decays to a [$\gamma_{d}$]{}and to a stable hidden fermion ($f_{d1}$), resulting in two muon jets from the [$\gamma_{d}$]{}decay in the final state. In a [1.9 fb$^{-1}$]{}sample of $\sqrt{s} =7$ TeV proton-proton collisions no events consistent with this Higgs boson decay mode are observed. The observed data are consistent with the Standard Model background expectations.\ Limits are set on the [$\sigma\times$BR ]{}to [[*H*]{}$\rightarrow \gamma_{d}~\gamma_{d}+ X$]{}as a function of the long-lived particle mean lifetime for $m_{H}=$ 100  and 140  with the chosen [$\gamma_{d}$]{}mass that gives a decay branching ratio of 45% for [[$\gamma_{d}$]{}$\rightarrow \mu~\mu$]{}. Assuming the SM production rate for a 140  Higgs boson, its branching ratio to two hidden-sector photons is found to be below 10%, at $95\%$ [*CL*]{}, for hidden photon [c$ \tau$]{}in the range 7 mm $\leq$ c$\tau$ $\leq$ 82 mm. Bounds on the [$\sigma\times$BR ]{}of a 126  Higgs boson may be conservatively extracted using the corresponding 140  exclusion curve. We thank CERN for the very successful operation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be operated efficiently. We acknowledge the support of ANPCyT, Argentina; YerPhI, Armenia; ARC, Australia; BMWF and FWF, Austria; ANAS, Azerbaijan; SSTC, Belarus; CNPq and FAPESP, Brazil; NSERC, NRC and CFI, Canada; CERN; CONICYT, Chile; CAS, MOST and NSFC, China; COLCIENCIAS, Colombia; MSMT CR, MPO CR and VSC CR, Czech Republic; DNRF, DNSRC and Lundbeck Foundation, Denmark; EPLANET and ERC, European Union; IN2P3-CNRS, CEA-DSM/IRFU, France; GNSF, Georgia; BMBF, DFG, HGF, MPG and AvH Foundation, Germany; GSRT, Greece; ISF, MINERVA, GIF, DIP and Benoziyo Center, Israel; INFN, Italy; MEXT and JSPS, Japan; CNRST, Morocco; FOM and NWO, Netherlands; BRF and RCN, Norway; MNiSW, Poland; GRICES and FCT, Portugal; MERYS (MECTS), Romania; MES of Russia and ROSATOM, Russian Federation; JINR; MSTD, Serbia; MSSR, Slovakia; ARRS and MVZT, Slovenia; DST/NRF, South Africa; MICINN, Spain; SRC and Wallenberg Foundation, Sweden; SER, SNSF and Cantons of Bern and Geneva, Switzerland; NSC, Taiwan; TAEK, Turkey; STFC, the Royal Society and Leverhulme Trust, United Kingdom; DOE and NSF, United States of America. The crucial computing support from all WLCG partners is acknowledged gratefully, in particular from CERN and the ATLAS Tier-1 facilities at TRIUMF (Canada), NDGF (Denmark, Norway, Sweden), CC-IN2P3 (France), KIT/GridKA (Germany), INFN-CNAF (Italy), NL-T1 (Netherlands), PIC (Spain), ASGC (Taiwan), RAL (UK) and BNL (USA) and in the Tier-2 facilities worldwide. The ATLAS Collaboration, *Observation of a New Particle in the Search for the Standard Model Higgs boson with the ATLAS detector at the LHC*, Phys. Lett. B 716 (2012) 1-29 \[arXiv:1207.7214\]. The CMS Collaboration, *Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC*, Phys. Lett. B 716 (2012) 30-61 \[arXiv:1207.7235\]. F. Englert and R. Brout, *Broken Symmetry and the Mass of Gauge Vector Mesons*, Phys. Rev. Lett. 13 (1964) 321. P.W. Higgs, *Broken symmetries, massless particles and gauge fields*, Phys. Lett. 12 (1964) 132. G.S. Guralnik, C.R. Hagen and T.W.B. Kibble, *Global Conservation Laws and Massless Particles*, Phys. Rev. Lett. 13 (1964) 585. M. J. Strassler and K. M. Zurek, *Echoes of a Hidden Valley at Hadron Colliders*, Phys. Lett. B 651 (2007) 374 \[arXiv:0604261\]. N. Arkani-Hamed and N. Weiner, *LHC Signals for a Superunified Theory of Dark Matter*, JHEP 12 (2008) 104 \[arXiv:0810.0714\]. T. Han, Z. Si, K. M. Zurek, and M. J. Strassler, *Phenomenology of Hidden Valleys at Hadron Colliders*, JHEP 07 (2008) 008. S. Gopalakrishna, S. Jung, and J. D. Wells, *Higgs Boson Decays to Four Fermions Through an Abelian Hidden Sector*, Phys. Rev. D 78 (2008) 055002 \[arXiv:0801.3456\]. M. J. Strassler and K. M. Zurek, *Discovering the Higgs Through Highly-Displaced Vertices*, Phys. Lett. B 661 (2008) 263-267 \[arXiv:0605193\]. M. Baumgart, C. Cheung, J. T. Ruderman, L. T. Wang, and I. Yavin, *Non-Abelian Dark Sectors and Their Collider Signatures*, JHEP 04 (2009) 014 \[arXiv:0901.0283\]. C. Cheung, J. T. Ruderman, L. T. Wang, and I. Yavin, *Lepton Jets in (Supersymmetric) Electroweak Processes*, JHEP 04 (2010) 116 \[arXiv:0909.0290\]. Y. Bai and Z. Han, *Measuring the Dark Force at the LHC*, Phys. Rev. Lett. 103 (2009) 051801 \[arXiv:0902.0006\]. A. Falkowski, J. T. Ruderman, T. Volansky, and J. Zupan, *Hidden Higgs Decaying to Lepton Jets*, JHEP 05 (2010) 077. A. Falkowski, J. T. Ruderman, T. Volansky, and J. Zupan, *Discovering Higgs Decays to Lepton Jets at Hadron Colliders*, Phys. Rev. Lett. 105 (2010) 241801 \[arXiv:1007.3496\]. , *The ATLAS Experiment at the CERN Large Hadron Collider*, JINST 3 (2008) S08003. V.M. Abazov [*et al.*]{} \[D0 Collaboration\], *Search for Dark Photons from Supersymmetric Hidden Valleys*, Phys. Rev. Lett. 103 (2009) 081802. V.M. Abazov [*et al.*]{} \[D0 Collaboration\], *Search for Events with Leptonic Jets and Missing Transverse Energy in pp Collisions at [$\sqrt{s}$]{}= 1.96 [TeV]{}*, Phys. Rev. Lett. 105 (2010) 211802. The CMS Collaboration, *Search for Light Resonances Decaying into Pairs of Muons as a Signal of New Physics*, JHEP 07 (2011) 098 \[arXiv:1106.2375\]. The ATLAS Collaboration, *Search for a light Higgs boson decaying to long-lived weakly-interacting particles in proton-proton collisions at $\sqrt{s}$ = 7  with the ATLAS detector*, Phys. Rev. Lett. 108 (2012) 251801. , *Performance of the ATLAS Trigger System in 2010*, Eur. Phys. J. C 72 (2012) 1849 \[arXiv:1110.1530\]. LHC Higgs Cross Section Working Group, S. Dittmaier, C. Mariotti, G. Passarino, and R. Tanaka (Eds.), *Handbook of LHC Higgs Cross Sections: 1. Inclusive Observables*, CERN-2011-002 (CERN, Geneva, 2011) \[arXiv:1101.0593\]. S. Mrenna, T. Sjöstrand and P. Z. Skands, *PYTHIA 6.4 physics and manual*, JHEP 05 (2006) 026 \[arXiv:0603175\]. J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer and T. Stelzer, *MadGraph 5 : Going Beyond*, JHEP 1106 (2011) 128 \[arXiv:1106.0522\]. P. Meade and M. Reece, *Bridge: Branching Ratio Inquiry / Decay Generated Events* \[arXiv:0703031\]. G. Abbiendi [*et al.*]{} \[OPAL Collaboration\], *Decay mode independent searches for new scalar bosons with the OPAL detector at LEP*, Eur. Phys. J. C 27 (2003) 311 \[arXiv:0206022\]. S. Frixione and B.R. Webber, *Matching NLO QCD computations and parton shower simulations*, JHEP 0206 (2002) 029 \[arXiv:0204244\]. S. Agostinelli et al., *GEANT4 - a simulation toolkit*, Nucl. Instrum. Meth. A 506 (2003) 250. , *The ATLAS Simulation Infrastructure*, Eur. Phys. J. C 70 (2010) 823 \[arXiv:1005.4568\]. , *Luminosity Determination in pp Collisions at $\sqrt{s}$ = 7  using the ATLAS Detector at the LHC*, Eur. Phys. J. C 71 (2011) 1630 \[arXiv:1101.2185\]. , *Luminosity Determination in pp Collisions at $\sqrt{s}$ = 7  using the ATLAS Detector in 2011*, ATLAS-CONF-2011-116. http://cdsweb.cern.ch/record/1376384/files/ATLAS-CONF-2011-116.pdf , *Expected Performance of the ATLAS Experiment - Detector, Trigger and Physics*, 2009 \[arXiv:0901.0512\]. A. L. Read, *Presentation of search results: The CL(s) technique*, J. Phys. G 28 (2002) 2693-2704. [^1]: ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the $z$-axis coinciding with the beam pipe axis. The $x$-axis points from the IP to the centre of the LHC ring, and the $y$-axis points upward. Cylindrical coordinates ($r$,$\phi$) are used in the transverse plane, $\phi$ being the azimuthal angle around the beam pipe. The pseudorapidity is defined in terms of the polar angle $\theta$ as .

--- abstract: 'In the search engine of Google, the PageRank algorithm plays a crucial role in ranking the search results. The algorithm quantifies the importance of each web page based on the link structure of the web. We first provide an overview of the original problem setup. Then, we propose several distributed randomized schemes for the computation of the PageRank, where the pages can locally update their values by communicating to those connected by links. The main objective of the paper is to show that these schemes asymptotically converge in the mean-square sense to the true PageRank values. A detailed discussion on the close relations to the multi-agent consensus problems is also given.' author: - | Hideaki Ishii\ Department of Computational Intelligence and Systems Science\ Tokyo Institute of Technology\ 4259 Nagatsuta-cho, Midori-ku, Yokohama 226-8502, Japan\ E-mail: ishii@dis.titech.ac.jp\ Roberto Tempo\ CNR-IEIIT, Politecnico di Torino\ Corso Duca degli Abruzzi 24, 10129 Torino, Italy\ E-mail: roberto.tempo@polito.it title: | Distributed Randomized Algorithms for\ the PageRank Computation[^1] --- Introduction {#sec:intro} ============ In the last decade, search engines have become widely used indispensable tools for searching the web. For such engines, it is essential that the search results not only consist of web pages related to the query terms, but also rank the pages properly so that the users quickly have access to the desired information. The PageRank algorithm at Google is one of the successful algorithms that quantify and rank the importance of each web page. This algorithm was initially proposed in [@BriPag:98], and an overview can be found in, e.g., [@LanMey:06; @BryLei:06]. One of the main features of the PageRank algorithm is that it is based solely on the link structure inherent in the web. The underlying key idea is that links from important pages make a page more important. More concretely, each page is considered to be voting the pages to which it is linked. Then, in the ranking of a page, the total number of votes as well as the importance of the voters are reflected. This problem is mathematically formulated as finding the eigenvector corresponding to the largest eigenvalue of a certain stochastic matrix associated with the web structure. For the PageRank computation, a critical aspect is the size of the web. The web is said to be composed of over 8 billion pages, and its size is still growing. Currently, the computation is performed centrally at Google, where the data on the whole web structure is collected by crawlers automatically browsing the web. In practice, the class of algorithms that can be applied is limited. In fact, the basic power method is employed, but it is reported that this computation takes about a week [@LanMey:06]. This clearly necessitates more efficient computational methods. In this regard, several approaches have recently been proposed. In [@KamHavGol:04], an adaptive computation method is developed, which classifies web pages into groups based on the speed of convergence to the PageRank values and allocates computational resources accordingly. Another line of research is based on distributed approaches, where the computation is performed on multiple servers communicating to each other. For example, Monte Carlo methods are used in [@AvrLitNem:07], while the work in [@ZhuYeLi:05] utilizes the block structure of the web to apply techniques from the Markov chain literature. In [@deJBra:07; @KolGalSzy:06], methods based on the so-called asynchronous iterations [@BerTsi:89] in numerical analysis are discussed. In this paper, we follow the distributed approach and, in particular, develop a randomized algorithm for the PageRank computation; for recent advances on probabilistic approaches in systems and control, see [@TemCalDab_book]. This algorithm is fully distributed and has three main features as follows: First, in principle, each page can compute its own PageRank value locally by communicating with the pages that are connected by direct links. That is, each page exchanges its value with the pages that it is linked to and those linked to it. Second, the pages make the decision to initiate this communication at random times which are independent from page to page. This means that, in its implementation, there is neither a fixed order among the pages nor a centralized agent in the web that determines the pages to update their values. Third, the computation required for each page is very mild. The main result of the paper shows that the algorithm converges to the true PageRank values in the mean-square sense. This is achieved by computing the time average at each page. From a technical viewpoint, an important characteristic of the approach is that the stochasticity of the matrix in the original problem is preserved and exploited. We first propose a basic distributed update scheme for the pages and then extend this into two directions to enhance its performance and flexibility for implementation. It is further noted that in [@IshTem_acc:09; @IshTemBaiDab:09], this approach has been generalized to incorporate failures in the communication as well as aggregation of the web structure. In [@IshTem_sice:09], a related result on finding the variations in the PageRank values when the web data may contain errors is given. We emphasize that the approach proposed here is particularly motivated by the recent research on distributed consensus, agreement, and flocking problems in the systems and control community; see, e.g., [@JadLinMor:03; @BerTsi:07; @Wu:06; @HatMes:05; @BoyGhoPra:06; @QuWanHul:08; @TahJad:08; @TemIsh:07; @CarFagFoc:05; @KasBasSri:07; @YuHenFid:07; @MarBroFra:04; @RenBea:05; @Moreau:05; @LinFraMag:05]. For additional details, we refer to [@AntBai:07; @csm:07; @BerTsi:89]. Among such problems, our approach to PageRank computation is especially related to the consensus, where multiple agents exchange their values with neighboring agents so that they obtain consensus, i.e., all agents reach the same value. The objective is clearly different from that of the PageRank problem, which is to find a specific eigenvector of a stochastic matrix via the power method. However, the recursion appearing in the consensus algorithm is exactly in the same form as the one for our distributed PageRank computation except that the class of stochastic matrices is slightly different. These issues will be discussed further. The organization of this paper is as follows: In Section \[sec:pagerank\], we present an overview of the PageRank problem. The distributed approach is introduced in Section \[sec:dist1\], where we propose a basic scheme and prove its convergence. Its relation with multi-agent consensus problems is discussed in Section \[sec:consensus\]. We then develop two extensions of the basic distributed algorithm: One in Section \[sec:simul\] is to improve the rate of convergence by allowing multiple pages to simultaneously update and the other in Section \[sec:approx\] to reduce the communication load among the pages. The proposed algorithm is compared with an approach known as asynchronous iteration from the field of numerical analysis in Section \[sec:asynch\]. Numerical examples are given in Section \[sec:example\] to show the effectiveness of the proposed schemes. We conclude the paper in Section \[sec:concl\]. Part of the material of this paper has appeared in a preliminary form in [@IshTem_cdc:08]. [*Notation*]{}:  For vectors and matrices, inequalities are used to denote entry-wise inequalities: For $X,Y\in\R^{n\times m}$, $X\leq Y$ implies $x_{ij}\leq y_{ij}$ for $i=1,\ldots,n$ and $j=1,\ldots,m$; in particular, we say that the matrix $X$ is nonnegative if $X\geq 0$ and positive if $X> 0$. A probability vector is a nonnegative vector $v\in\R^n$ such that $\sum_{i=1}^n v_i = 1$. Unless otherwise specified, by a stochastic matrix, we refer to a column-stochastic matrix, i.e., a nonnegative matrix $X\in\R^{n\times n}$ with the property that $\sum_{i=1}^n x_{ij}=1$ for $j=1,\ldots,n$. Let $\one\in\R^n$ be the vector with all entries equal to $1$ as $\one:=[1\;\cdots\;1]^T$. Similarly, $S\in\R^{n\times n}$ is the matrix with all entries being $1$. For $x\in\R^n$, we denote by $\abs{x}$ the vector containing the absolute values of the corresponding entries of $x$. The norm $\norm{\cdot}$ for vectors is the Euclidean norm. The spectral radius of the matrix $X\in\R^{n\times n}$ is denoted by $\rho(X)$. We use $I$ for the identity matrix. The PageRank problem {#sec:pagerank} ==================== In this section, we provide a brief introductory description of the PageRank problem; this material can be found in, e.g., [@LanMey:06; @BryLei:06; @BriPag:98]. Consider a network of $n$ web pages indexed by integers from 1 to $n$. This network is represented by the directed graph $\Gcal=(\Vcal,\Ecal)$. Here, $\Vcal:=\{1,2,\ldots,n\}$ is the set of vertices corresponding to the web page indices while $\Ecal\subset\Vcal\times\Vcal$ is the set of edges representing the links among the pages. The vertex $i$ is connected to the vertex $j$ by an edge, i.e., $(i,j)\in\Ecal$, if page $i$ has an outgoing link to page $j$, or in other words, page $j$ has an incoming link from page $i$. To avoid trivial situations, we assume $n\geq 2$. The objective of the PageRank algorithm is to provide some measure of importance to each web page. The PageRank value, or simply the value, of page $i\in\Vcal$ is a real number in $[0,1]$; we denote this by $x_i^*$. The values are ordered such that $x_i^*>x_j^*$ implies that page $i$ is more important than page $j$. The basic idea in ranking the pages in terms of the values is that a page having links from important pages is also important. This is realized by determining the value of one page as a sum of the contributions from all pages that have links to it. In particular, the value $x_i^*$ of page $i$ is defined as $$x_i^* = \sum_{j\in \Lcal_i} \frac{x_j^*}{n_j},$$ where $\Lcal_i:=\{j:\; (j,i)\in\Ecal\}$, i.e., this is the index set of pages linked to page $i$, and $n_j$ is the number of outgoing links of page $j$. It is customary to normalize the total of all values so that $\sum_{i=1}^{n} x^*_i = 1$. Let the values be in the vector form as $x^*\in[0,1]^n$. Then, from what we described so far, the PageRank problem can be restated as $$x^* = A x^*,~~x^*\in[0,1]^n,~~\sum_{i=1}^n x^*_i = 1, \label{eqn:xA:pr}$$ where the matrix $A=(a_{ij})\in\R^{n\times n}$, called the link matrix, is given by $$a_{ij} := \begin{cases} \frac{1}{n_j} & \text{if $j\in \Lcal_i$},\\ 0 & \text{otherwise}. \end{cases} \label{eqn:A}$$ Note that the value vector $x^*$ is a nonnegative unit eigenvector corresponding to the eigenvalue 1 of the nonnegative matrix $A$. In general, for this eigenvector to exist and to be unique, it is sufficient that the web as a graph is strongly connected [@HorJoh:85][^2]. However, the web is known not to be strongly connected. Thus, the problem is slightly modified as follows[^3]. First, note that in the real web, the so-called dangling nodes, which are pages having no links to others, are abundant. Such pages can be found, e.g., in the form of PDF document files having no outgoing links. These pages introduce zero columns into the link matrix. To simplify the discussion, we redefine the graph and thus the matrix $A$ by bringing in artificial links for all dangling nodes (e.g., links back to the pages having links to a dangling node). As a result, the link matrix $A$ becomes a stochastic matrix, that is, $\sum_{i=1}^n a_{ij}=1$ for each $j$. This implies that there exists at least one eigenvalue equal to 1. To emphasize the changes in the links that we have just made, we state the following as an assumption. The link matrix $A$ given in is a stochastic matrix. Next, to guarantee the uniqueness of the eigenvalue 1, a modified version of the values has been introduced in [@BriPag:98] as follows: Let $m$ be a parameter such that $m\in(0,1)$, and let the modified link matrix $M\in\R^{n\times n}$ be defined by $$M := (1-m)A + \frac{m}{n}S. \label{eqn:M}$$ In the original algorithm in [@BriPag:98], a typical value for $m$ is chosen as $m=0.15$; we use this value throughout this paper[^4]. Notice that $M$ is a positive stochastic matrix. By the Perron theorem [@HorJoh:85], this matrix is primitive[^5]; in particular, the eigenvalue 1 is of multiplicity 1 and is the unique eigenvalue of maximum modulus (i.e., with the maximum absolute value). Furthermore, the corresponding eigenvector is positive. Hence, we redefine the value vector $x^*$ by using $M$ in place of $A$ in as follows. The PageRank value vector $x^*$ is given by $$x^* = M x^*,~~x^*\in[0,1]^n,~~\sum_{i=1}^n x^*_i = 1. \label{eqn:prvec}$$ As mentioned in the Introduction, due to the large dimension of the link matrix $M$, the computation of the eigenvector corresponding to the eigenvalue 1 is difficult. The solution that has been employed in practice is based on the power method. That is, the value vector $x^*$ is computed through the recursion $$\begin{aligned} x(k+1) &= M x(k) = (1-m)A x(k) + \frac{m}{n}\one, \label{eqn:xM0} \end{aligned}$$ where $x(k)\in\R^n$ and the initial vector $x(0)\in\R^n$ is a probability vector. The second equality above follows from the fact $Sx(k)=\one$, $k\in\Z_+$. For implementation, the form on the far right-hand side is important, exhibiting that multiplication is required using only the sparse matrix $A$, and not the dense matrix $M$. Based on this method, we can asymptotically find the value vector as shown below; see, e.g., [@HorJoh:85]. In the update scheme , for any initial state $x(0)$ that is a probability vector, it holds that $x(k)\rightarrow x^*$ as $k\rightarrow\infty$. We now comment on the convergence rate of this scheme. Denote by $\lambda_1(M)$ and $\lambda_2(M)$ the largest and the second largest eigenvalues of $M$ in magnitude. Then, for the power method applied to $M$, the asymptotic rate of convergence is exponential and depends on the ratio $\abs{\lambda_2(M)/\lambda_1(M)}$. Since $M$ is a positive stochastic matrix, we have $\lambda_1(M)=1$ and $\abs{\lambda_2(M)}<1$. Furthermore, it is shown in [@LanMey:06] that the structure of the link matrix $M$ leads us to the bound $$\abs{\lambda_2(M)} \leq 1-m. \label{eqn:lambda2}$$ We next provide a simple example for illustration. \ \[ex:1\]Consider the web with four pages shown in Fig. \[fig:4nodes\]. As a graph, this web is strongly connected, and there are no dangling nodes. The link matrix $A$ and the modified link matrix $M$ can easily be constructed by and , respectively, as $$\begin{aligned} A &= \begin{bmatrix} 0 & 0 & 0 & \frac{1}{3}\\ 1 & 0 & \frac{1}{2} & \frac{1}{3}\\ 0 & \frac{1}{2} & 0 & \frac{1}{3}\\ 0 & \frac{1}{2} & \frac{1}{2} & 0 \end{bmatrix},~~ M = \begin{bmatrix} 0.0375 & 0.0375 & 0.0375 & 0.3208\\ 0.8875 & 0.0375 & 0.4625 & 0.3208\\ 0.0375 & 0.4625 & 0.0375 & 0.3208\\ 0.0375 & 0.4625 & 0.4625 & 0.0375 \end{bmatrix},\end{aligned}$$ where we used the value $m=0.15$ from [@BriPag:98]. Then, the value vector $x^*$ can be computed as $x^* = \bigl[ 0.119 ~ 0.331 ~ 0.260 ~ 0.289 \bigr]^T$. Notice that page 2 has the largest value since it is linked from three pages while page 1, which has only one link to it, has the smallest value. On the other hand, pages 3 and 4 have the same number of incoming links, but page 4 has a larger value. This is because page 4 has more outgoing links, and thus it receives more contribution from page 3 than what it gives back. A distributed randomized approach {#sec:dist1} ================================= In this section, we propose a distributed approach to compute the value vector $x^*$. Consider the web from the previous section. The basic protocol of the scheme is as follows: At time $k$, page $i$ initiates its PageRank value update (i) by sending the value of page $i$ to the pages that are linked and (ii) by requesting the values from the pages that are linked to page $i$. All pages involved in this process renew their values based on the latest available information. To implement the scheme in a distributed manner, we assume that the pages taking the update action are determined in a random manner. This is specified by the random process $\theta(k)\in\Vcal$, $k\in\Z_+$. If at time $k$, $\theta(k)=i$, then page $i$ initiates an update action by communicating and exchanging the values with the pages connected by incoming and outgoing links. Specifically, $\theta(k)$ is assumed to be i.i.d., and its probability distribution is given by $$\Prob\{\theta(k)=i\} = \frac{1}{n},~~\forall k\in\Z_+. \label{eqn:theta}$$ This means that each page takes the update action with equal probability. In principle, this scheme may be implemented without requiring a centralized decision maker or any fixed order among the pages. In particular, consider the distributed update scheme in the following form: $$x(k+1) = (1-\hat{m}) A_{\theta(k)} x(k) + \frac{\hat{m}}{n}\one, \label{eqn:xMi1}$$ where $x(k)\in\R^n$ is the state whose initial state $x(0)$ is a probability vector, $\theta(k)\in\{1,\ldots,n\}$ is the mode of the system, and $A_i$, $i=1,\ldots,n$, are called the distributed link matrices and are to be determined; $\hat{m}\in(0,1)$ is a parameter replacing $m$ in the centralized scheme . The objective here is to design this distributed update scheme by finding the appropriate link matrices $A_i$ and the parameter $\hat{m}$ so that the PageRank values are computed through the time average of the state $x$. Let $y(k)$ be the average of the sample path $x(0),\ldots,x(k)$ as $$y(k) := \frac{1}{k+1}\sum_{\ell=0}^{k} x(\ell). \label{eqn:yk}$$ We say that, for the distributed update scheme, the PageRank value $x^*$ is obtained through the time average $y$ if, for each initial state $x(0)$ that is a probability vector, $y(k)$ converges to $x^*$ in the mean-square sense as follows: $$E\left[ \bigl\| y(k) - x^* \bigr\|^2 \right] \rightarrow 0,~~~k\rightarrow\infty. \label{eqn:thm:erg}$$ This type of convergence is called ergodicity for stochastic processes [@PapPil:02]. In what follows, we develop the distributed update scheme of . The main result is presented as Theorem \[thm:erg\] showing the convergence of the scheme. The design consists of two steps, one for the link matrices $A_i$ and then the parameter $\hat{m}$. In later sections, this approach will be extended to improve the convergence rate and the necessary communication load. Distributed link matrices and their average ------------------------------------------- The first step in the development is to introduce the distributed link matrices. For each $i$, the matrix $A_i\in\R^{n\times n}$ is obtained as follows: (i) The $i$th row and column coincide with those of $A$; (ii) the remaining diagonal entries are equal to $1-a_{i\ell}$, $\ell=1,\ldots,n$, $\ell\neq i$; and (iii) all the remaining entries are zero. More formally, we have $$\begin{aligned} (A_i)_{j\ell} &:= \begin{cases} a_{j\ell} & \text{if $j=i$ or $\ell=i$},\\ 1-a_{i\ell} & \text{if $j=\ell\neq i$},\\ 0 & \text{otherwise}, \end{cases} ~~~~~i=1,\ldots,n. \label{eqn:Ai}\end{aligned}$$ It follows that these matrices are stochastic because the original link matrix $A$ possesses this property. As we shall see later, this property indeed is critical for the convergence of the scheme. \[ex:3\]We continue with the 4-page web in Example \[ex:1\]. The link matrices $A_i$ are given by $$\begin{aligned} A_1 &= \begin{bmatrix} 0 & 0 & 0 & \frac{1}{3}\\ 1 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & \frac{2}{3} \end{bmatrix},~~ A_2 = \begin{bmatrix} 0 & 0 & 0 & 0\\ 1 & 0 & \frac{1}{2} & \frac{1}{3}\\ 0 & \frac{1}{2} & \frac{1}{2} & 0\\ 0 & \frac{1}{2} & 0 & \frac{2}{3} \end{bmatrix},~~ A_3 = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & \frac{1}{2} & \frac{1}{2} & 0\\ 0 & \frac{1}{2} & 0 & \frac{1}{3}\\ 0 & 0 & \frac{1}{2} & \frac{2}{3} \end{bmatrix},~~ A_4 = \begin{bmatrix} 1 & 0 & 0 & \frac{1}{3}\\ 0 & \frac{1}{2} & 0 & \frac{1}{3}\\ 0 & 0 & \frac{1}{2} & \frac{1}{3}\\ 0 & \frac{1}{2} & \frac{1}{2} & 0 \end{bmatrix}. \tag*{\text{\End}} $$ To clarify the properties of the link matrices $A_i$ just introduced, we consider the simpler update scheme $$x(k+1) = A_{\theta(k)} x(k), \label{eqn:xAi}$$ where $x(k)$ is the state with $x(0)$ being a probability vector, and the mode $\theta(k)$ is specified in . In particular, we focus on its average dynamics. The mean $\overline{x}(k):=E[x(k)]$ of the state $x(k)$ follows the recursion $\overline{x}(k+1)=\overline{A} \overline{x}(k)$, where $\overline{A}:= E[A_{\theta(k)}]$ is the average matrix and $\E[\,\cdot\,]$ is the expectation with respect to the random process $\theta(k)$. Hence, we now inspect this matrix $\overline{A}$. Due to the probability distribution of $\theta(k)$ in , we have $$\overline{A} = \frac{1}{n} \sum_{i=1}^n A_i. \label{eqn:Abar0}$$ It is obvious that $\overline{A}$ is a stochastic matrix since all $A_i$ are stochastic. The following lemma shows some properties of this matrix $\overline{A}$ that will be useful later. \[lem:Abar\]For the average matrix $\overline{A}$ given in , we have the following: 1. $\overline{A} = \frac{2}{n} A + \frac{n-2}{n} I$. 2. There exists a vector $z_0\in\R_+^n$ which is an eigenvector corresponding to the eigenvalue 1 for both matrices $A$ and $\overline{A}$. [[*Proof:*]{}]{}(i)  By definition of $\overline{A}$, we have $$\begin{aligned} (\overline{A})_{j\ell} &= \frac{1}{n} \sum_{i=1}^{n} (A_i)_{j\ell} = \begin{cases} \frac{1}{n} \bigl[ a_{jj} + \sum_{i=1,\ i\neq j}^n (1- a_{ij}) \bigr] \\ \hspace*{1.5cm} \text{if $j=\ell$},\\ \frac{2}{n} a_{j\ell}~~~~\text{if $j\neq \ell$}.\\ \end{cases}\end{aligned}$$ By definition of $A$, we have $a_{jj}=0$ and $\sum_{i=1,\ i\neq j}^n a_{ij} = 1$. Thus, the expression for $\overline{A}$ follows. (ii)  From (i), we have $\overline{A}-I = 2/n(A-I)$. This implies that every eigenvector $z_0$ of the link matrix $A$ associated with the eigenvalue 1 is also an eigenvector of the average matrix $\overline{A}$ for the same eigenvalue. The lemma above provides some justification for the proposed distributed approach. That is, even though the matrices $A$ and $\overline{A}$ have different structures, they share an eigenvector for the eigenvalue 1, which corresponds to the PageRank vector. Mean-square convergence of the distributed update scheme -------------------------------------------------------- As in the case with the original link matrix $A$, for the average matrix $\overline{A}$, the eigenvector corresponding to the eigenvalue 1 may not be unique. We follow an argument similar to that in Section \[sec:pagerank\] and introduce the modified versions of the distributed link matrices. Since the link matrices $A_i$ are stochastic, we can rewrite the distributed update scheme in as $$x(k+1) = M_{\theta(k)} x(k), \label{eqn:xM}$$ where the modified distributed link matrices are given by $$M_i := (1-\hat{m}) A_i + \frac{\hat{m}}{n} S,~~~i=1,\ldots,n. \label{eqn:Mi}$$ This expression is derived by the relation $Sx(k)=\one$ because $x(k)$ in is a probability vector for each $k$. Note that $M_i$ are positive stochastic matrices. Similarly to the argument on the link matrices $A_i$ above, the problem at this second step is as follows: We shall find the modified link matrices $M_i$ by choosing the parameter $\hat{m}$ such that their average and the link matrix $M$ from share an eigenvector for the eigenvalue 1. Since such an eigenvector is unique for $M$, it is necessarily equal to the value vector $x^*$ (see ). Let $\overline{x}(k):=\E[x(k)]$ be the mean of the state $x(k)$ of the system . Its dynamics is expressed as $$\overline{x}(k+1) = \overline{M} \overline{x}(k), \label{eqn:xMbar}$$ where $\overline{x}(0)=x(0)$ and the average matrix $\overline{M}$ is given by $\overline{M} := \E[M_{\theta(k)}]$. A simple way of defining $M_i$ would be to let $\hat{m}=m$ as in the case with $M$; however, it can be shown that there is no clear relation between the original matrix $M$ and the average matrix $\overline{M}$ such as that between $A$ and $\overline{A}$ as we have seen in Lemma \[lem:Abar\]. Instead, we take the parameter $\hat{m}$ as $$\hat{m} = \frac{2m}{n - m(n-2)}. \label{eqn:mhat}$$ For the value $m=0.15$ used in this paper, we have $\hat{m} = 0.3/(0.85n+0.3)$. For this choice of $\hat{m}$, the next result holds. \[lem:Mbar\]For the parameter $\hat{m}$ given in , we have the following: 1. $\hat{m}\in(0,1)$ and $\hat{m}<m$. 2. $\overline{M} = \frac{\hat{m}}{m} M + \left(1 - \frac{\hat{m}}{m}\right)I$. 3. For the average matrix $\overline{M}$, the eigenvalue 1 is simple and is the unique eigenvalue of maximum modulus. The value vector $x^*$ is the corresponding eigenvector. [[*Proof:*]{}]{}(i)  By the assumptions $m\in(0,1)$ and $n\geq 2$, $\hat{m}$ in is positive. Also, we have $1-\hat{m} = n(1-m)/[n - m(n-2)]$. Hence, $1-\hat{m}> 1-m$, that is, $\hat{m}< m< 1$. (ii)  This can be shown by direct calculation as follows: $$\begin{aligned} \overline{M} &= (1-\hat{m}) \overline{A} + \frac{\hat{m}}{n} S~~~ \text{(by the definition of $\overline{M}$, \eqref{eqn:Mi}, and then \eqref{eqn:Abar0})}\\ &= (1-\hat{m}) \biggl[ \frac{2}{n} A + \frac{n-2}{n} I \biggr] + \frac{\hat{m}}{n} S~~~ \text{(by Lemma~\ref{lem:Abar}\;(i))}\\ &= \frac{\hat{m}}{m} M + \biggl( 1 - \frac{\hat{m}}{m} \biggr)I~~~ \text{(by $\hat{m}$ in \eqref{eqn:mhat} and the definition of $M$ in \eqref{eqn:M})}.\end{aligned}$$ (iii)  From (ii), we have $\overline{M}-I = \hat{m}/m (M-I)$. Hence, $\overline{M}$ and $M$ share an eigenvector for the eigenvalue 1. However, both $\overline{M}$ and $M$ are positive stochastic matrices. Therefore, by the Perron theorem [@HorJoh:85], for these matrices, the eigenvalue 1 is of multiplicity 1 and is the unique one having the maximum magnitude. Moreover, by , the corresponding eigenvector coincides with $x^*$. From (iii) in the lemma above, it follows that the value vector $x^*$ can be obtained by the power method, i.e., by the average system in as $\overline{x}(k) \rightarrow x^*$, $k\rightarrow\infty$. Hence, in an average sense, the distributed update scheme asymptotically provides the correct values. It is interesting to observe that this can be achieved though the original link matrix $A$ does not explicitly appear in the scheme. In fact, an eigenvector of the matrix $M$ is computed through randomly switching among the distributed link matrices $M_i$. However, this property turns out not to be sufficient to guarantee convergence of $x(k)$ to the true value $x^*$. From Lemma \[lem:tauM\](ii) in the Appendix, we can show that for any sequence $\{\theta(k)\}$, there exists a sequence of probability vectors $\{v(k)\}$ such that, for any $x(0)$, it holds that $x(k)-v(k) \one^T x(0)=x(k)-v(k)\rightarrow 0$ as $k\rightarrow\infty$. The vector $v$ and hence the state $x$ in general do not converge. This can be seen in Example \[ex:3\] when, e.g., page 1 initiates an update ($\theta(k)=1$); the update for page 4 is given by $x_4(k+1) = 2(1-m)/3 x_4(k) + m/4$, showing that $x_4$ cannot stay at its equilibrium value. Therefore, in the distributed approach, we resort to computing the time average $y(k)$ of the states. The following theorem is the main result of this section. It shows that the time average indeed converges to the value vector in the mean-square sense. \[thm:erg\]In the distributed update scheme in , the PageRank value $x^*$ is obtained through the time average $y$ in as $E\bigl[ \bigl\| y(k) - x^* \bigr\|^2 \bigr] \rightarrow 0$, $k\rightarrow\infty$. The theorem highlights an ergodic property in the proposed update scheme. It can be shown by general Markov process results in, e.g., [@Cogburn:86]. For completeness, however, a proof more specific to the current setup is provided in Appendix \[sec:app:B\]; it employs tools for stochastic matrices and moreover is useful for an extension given in Section \[sec:approx\]. Regarding the convergence of this algorithm, we see from in the proof that it is of order $1/k$ and moreover depends on the size of $n$ linearly through the parameter $\hat{m}$ in . Several remarks are in order. In practice, each page needs to communicate with the pages that are directly connected by incoming or outgoing links. We emphasize that the recursion to be used is and not the equivalent expression of ; in the latter case, the link matrices $M_i$ are positive, which can imply that the values of all pages are required for an update of a page. Nevertheless, as can be seen in , the link matrices $A_i$ are sparse. Thus, at time $k$, communication is required only among the pages corresponding to the nonzero entries in $A_{\theta(k)}$. Each page then performs weighted addition of its own value, the values just received, and the extra term $\hat{m}/n$. Consequently, the amount of computation required at each page is limited at any time. Implementation issues such as how web pages can exactly make local computations are outside the scope of this paper. However, it is clear that certain regulations may be necessary so that page owners cooperate with the search engine and the PageRank values computed by them can be trusted[^6]. Another issue concerning reliability of the ranking is that of link spam, i.e., links added to enhance the PageRank of certain pages on purpose; a method to detect such spamming is studied in, e.g., [@AndBorHop:07; @LanMey:06]. Relations to consensus problems {#sec:consensus} =============================== In this section, we discuss the relation between the two problems of PageRank and consensus. First, we describe a stochastic version of the consensus problem. Such problems have been studied in, e.g., [@BoyGhoPra:06; @Wu:06; @HatMes:05; @TahJad:08]; see also [@TemIsh:07]. Consider a set $\Vcal=\{1,2,\ldots,n\}$ of agents having scalar values. The network of agents is represented by the directed graph $\Gcal=(\Vcal,\Ecal)$. The vertex $i$ is connected to the vertex $j$ by an edge $(i,j)\in\Ecal$ if agent $i$ can communicate its value to agent $j$. Assume that the graph is strongly connected[^7]. The objective is that all agents reach a common value by communicating to each other, where the pattern in the communication is randomly determined at each time. Let $x_i(k)$ be the value of agent $i$ held at time $k$, and let $x(k):=[x_1(k)\cdots x_n(k)]^T\in\R^n$. The values are updated via the recursion $$x(k+1) = A_{\theta(k)} x(k), \label{eqn:consensus:x}$$ where $\theta(k)\in\{1,\ldots,d\}$ is the mode specifying the communication pattern among the agents and $d$ is the number of such patterns. The communication patterns are given as follows: Each $i\in\{1,\ldots,d\}$ corresponds to the subset $\Ecal_i\subset\Ecal$ of the edge set. Then, the matrix $A_i$ has $(A_i)_{j\ell}>0$ if and only if $(\ell,j)\in\Ecal_i$. We assume that (i) $(j,j)\in\Ecal_i$ for all $j$, (ii) $\bigcup_{i=1}^d\Ecal_i=\Ecal$, and (iii) the matrix $A_{i}$ is a row-stochastic matrix. The communication pattern is random, and in particular, $\theta(k)$ is an i.i.d. random process. Its probability distribution is given by $\Prob\{\theta(k)=i\} = 1/d$ for $k\in\Z_+$. We say that consensus is obtained if for any initial vector $x(0)\in\R^n$, it holds that $$\abs{x_i(k)-x_j(k)}\rightarrow 0,~~k\rightarrow\infty \label{eqn:consensus}$$ with probability one for all $i,j\in\Vcal$. A well-known approach is to update the value of each agent by taking the average of the values received at that time. In this case, the matrix $A_i$ is constructed as $$(A_{i})_{j\ell} := \begin{cases} \frac{1}{n_{ij}} & \text{if $(\ell,j)\in\Ecal_{i}$},\\ 0 & \text{otherwise}, \end{cases}$$ where $n_{ij}$ is the number of agents $\ell$ with $(\ell,j)\in\Ecal_{i}$, i.e., those that transmit their values to agent $j$. \[ex:consensus\]Consider the graph in Example \[ex:1\] with four agents. We introduce four communication patterns arising from the protocol in the distributed PageRank algorithm: The edge subset $\Ecal_i$ contains all $(i,j)$ and $(j,i)$ in the original edge set $\Ecal$ including $(i,i)$ that corresponds to a self-loop for $i,j=1,2,3,4$. The matrices $A_i$ can be written as $$\begin{aligned} A_1 &= \begin{bmatrix} \frac{1}{2} & 0 & 0 & \frac{1}{2}\\ \frac{1}{2} & \frac{1}{2} & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{bmatrix},~~ A_2 = \begin{bmatrix} 1 & 0 & 0 & 0\\ \frac{1}{4} & \frac{1}{4} & \frac{1}{4} & \frac{1}{4}\\ 0 & \frac{1}{2} & \frac{1}{2} & 0\\ 0 & \frac{1}{2} & 0 & \frac{1}{2} \end{bmatrix},~~ A_3 = \begin{bmatrix} 1 & 0 & 0 & 0\\ 0 & \frac{1}{2} & \frac{1}{2} & 0\\ 0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3}\\ 0 & 0 & \frac{1}{2} & \frac{1}{2} \end{bmatrix},~~ A_4 = \begin{bmatrix} \frac{1}{2} & 0 & 0 & \frac{1}{2}\\ 0 & \frac{1}{2} & 0 & \frac{1}{2}\\ 0 & 0 & \frac{1}{2} & \frac{1}{2}\\ 0 & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \end{bmatrix}. \tag*{\text{\End}}\end{aligned}$$ We now present the convergence result for consensus. Assume that the graph is strongly connected. Then under the scheme of , where the communication pattern is chosen randomly, consensus is obtained in the sense of . [*Outline of the proof:*]{}  Let $\overline{A}:=E[A_{\theta(k)}]$ be the average matrix. This matrix is stochastic and irreducible. This is because the original graph is strongly connected, and hence under the probability distribution of $\theta(k)$, we have $(\overline{A})_{j\ell}>0$ for each $(\ell,j)\in\Ecal$. Furthermore, by definition, the diagonal entries are positive, and thus $\overline{A}$ is a primitive matrix, implying that it has the unique eigenvalue 1 of maximum modulus [@HorJoh:85]. Thus, by [@TahJad:08], it follows that consensus is obtained. In comparison with the distributed PageRank problem, the consensus problem has the features below: 1. The graph is assumed to be strongly connected. 2. The goal is that all values $x_i(k)$ become equal, and moreover there is no restriction on its size. 3. Convergence with probability one can be attained for the values $x_i(k)$ directly; there is no need to consider their time average (as in $y_i(k)$ in ). 4. The matrices $A_i$ are row stochastic and the diagonal entries are all positive. In contrast, in our distributed PageRank computation scheme, the link matrices are column stochastic. However, the coefficient of ergodicity, which is the tool employed for proving Theorem \[thm:erg\], is useful also for this problem; see, e.g., [@TahJad:08]. It is clear that many similarities exist between the algorithms for consensus and PageRank. We emphasize that the distributed PageRank approach in this paper has been particularly motivated by the recent advances in the consensus literature. We highlight two points that provide helpful insights into the PageRank problem as follows: 1. At the conceptual level, it is natural to view the web as a network of agents that can make its own computation as well as communication with their neighboring agents. 2. At the technical level, it is important to impose stochasticity on the distributed link matrices. For the distributed PageRank computation, very few works exploit this viewpoint. Extensions to simultaneous updates {#sec:simul} ================================== So far, we have discussed the update scheme where only one page initiates an update at each time instant. In the web with billions of pages, however, this approach may not be practical. In this section, we extend the distributed algorithm by allowing multiple pages to simultaneously initiate updates. Consider the web with $n$ pages from Sections \[sec:pagerank\] and \[sec:dist1\]. As before, at each time $k$, the page $i$ initiates its PageRank value update (i) by sending its value to the pages that it is linked to and (ii) by requesting the pages that link to it for their values. All pages involved here update their values based on the new information. The difference from the simpler scheme before is that there may be pages that are requested for their values by multiple pages at the same time. The current scheme handles such situations. These updates can take place in a fully distributed and randomized manner. The decision to make an update is a random variable. In particular, this is determined under a given probability $\alpha\in(0,1]$ at each time $k$, and hence, the decision can be made locally at each page. Note that the probability $\alpha$ is a global parameter in that all pages share the same $\alpha$. Formally, the proposed distributed update scheme is described as follows. Let $\eta_i(k)\in\{0,1\}$, $i=1,\ldots,n$, $k\in\Z_+$, be Bernoulli processes given by $$\eta_i(k) = \begin{cases} 1 & \text{if page $i$ initiates an update at time $k$},\\ 0 & \text{otherwise}, \end{cases}$$ where their probability distributions are specified as $$\alpha := \Prob\bigl\{ \eta_i(k)=1 \bigr\}. \label{eqn:alpha}$$ The process $\eta_i(k)$ is generated at the corresponding page $i$. Similarly to the argument in Section \[sec:dist1\], we start with the update law as in : $$x(k+1) = (1-\hat{m}) A_{\eta_1(k),\ldots,\eta_n(k)} x(k) + \frac{\hat{m}}{n} \one, \label{eqn:xAeta}$$ where $x(k)\in\R^n$, the initial state $x(0)$ is a probability vector, $\hat{m}\in(0,1)$ is the parameter used instead of $m$ in the centralized case, and $A_{\eta_1(k),\ldots,\eta_n(k)}$ are the distributed link matrices. The problem of distributed PageRank computation is formulated as follows: Find the distributed link matrices $A_{p_1,\ldots,p_n}$ and the parameter $\hat{m}$ such that, for the corresponding distributed update scheme , the PageRank value $x^*$ is obtained through the time average. This problem is a generalization of that in Section \[sec:dist1\], where only one page initiates an update at a time. The current approach is called the distributed scheme with simultaneous updates. Its analysis is more involved as we shall see. Distributed link matrices and their average ------------------------------------------- We introduce the distributed link matrices. Let the matrices $A_{p_1,\ldots,p_n}$ be given by $$\begin{aligned} &\bigl( A_{p_1,\ldots,p_n} \bigr)_{ij} := \begin{cases} a_{ij} & \text{if $p_i = 1$ or $p_j = 1$},\\ 1 - \sum_{h:~p_h=1} a_{hj} & \text{if $p_i = 0$ and $i=j$},\\ 0 & \text{if $p_i = p_j=0$ and $i\neq j$} \end{cases} \label{eqn:Aeta}\end{aligned}$$ for $p_r\in\{0,1\}$, $r\in\{1,\ldots,n\}$, and $i,j\in\{1,\ldots,n\}$. Clearly, there are $2^n$ matrices. They have the property that (i) if $p_i=1$, then the $i$th column and the $i$th row are the same as those in the original link matrix $A$, (ii) if $p_i=0$, then the $i$th diagonal entry is chosen so that the entries of the $i$th column add up to 1, and (iii) all other entries are 0. Hence, these are constructed as stochastic matrices. Notice that the link matrix $A_{p_1,\ldots,p_n}$ coincides with $A_i$ in when $p_i=1$ and $p_j=0$ for all $j\neq i$. We next analyze the average dynamics of the distributed update scheme in . For simplicity, as in Section \[sec:dist1\], we use the same notation $\overline{A}$ for the average link matrix given by $$\overline{A}:= E\bigl[A_{\eta_1(k),\ldots,\eta_n(k)}\bigr], \label{eqn:Abar}$$ where $E[\,\cdot\,]$ is the expectation with respect to $\eta_i(k)$, $i\in\Vcal$. This matrix $\overline{A}$ is stochastic. The following result shows that the average link matrix $\overline{A}$ has a clear relation to the original link matrix $A$. In particular, it implies that the two matrices share the eigenvector for eigenvalue 1. \[prop:simul:Aave\]For the average link matrix $\overline{A}$ given in , we have the following: 1. $\overline{A} = \bigl[1-(1-\alpha)^2\bigr] A + (1-\alpha)^2 I$. 2. There exists a vector $z_0\in\R_+^n$ which is an eigenvector corresponding to the eigenvalue 1 for both $A$ and $\overline{A}$. The proof of this proposition is preceded by a preliminary result. Observe that $\overline{A}$ can be written as $$\overline{A} = \sum_{\ell=0}^{n} \alpha^{\ell} (1-\alpha)^{n-\ell} \hat{A}_{\ell}, \label{eqn:simul:Aave}$$ where the matrices $\hat{A}_{\ell}$, $\ell=0,1,\ldots,n$, are given by $$\hat{A}_{\ell} := \sum_{\substack{p_r\in\{0,1\},~r=1,\ldots,n:\;\\ \sum_{r=1}^{n}p_r = \ell}} A_{p_1,\ldots,p_n}. \label{eqn:Ahatl}$$ The matrix $\hat{A}_{\ell}$ is the sum of matrices for the cases where $\ell$ pages simultaneously initiate updates. These matrices can be explicitly written in terms of the original link matrix $A$. Here, we use the binomial coefficient given by $\comb{r}{k}:=r!/[(r-k)!\;k!]$. Note that $\comb{r}{0}=1$ for any $r\in\Z_+$. \[lem:Ahatl\]The matrices $\hat{A}_{\ell}$, $\ell=0,1,\ldots,n$, in can be expressed as follows: $$\hat{A}_{\ell} = \begin{cases} A & \text{if $\ell=n$},\\ nA & \text{if $\ell=n-1$},\\ \bigl(\comb{n}{\ell} - \comb{n-2}{\ell}\bigr) A + \comb{n-2}{\ell} I & \text{if $\ell=0,\ldots,n-2$}. \end{cases} \label{eqn:lem:Ahatl}$$ [[*Proof:*]{}]{}We consider four cases as follows. (1) $\ell=n$:  This is the case when all pages initiate updates, and thus by definition $\hat{A}_n = A_{1,\ldots,1} = A$. (2) $\ell=n-1$:  When all but one page initiate updates, it is obvious from the definition that $A_{0,1,\ldots,1} = A_{1,0,1,\ldots,1} = \cdots = A_{1,\ldots,1,0} = A$. Since there are $n$ such cases, their sum is $\hat{A}_{n-1} = nA$. (3) $\ell=0$:  In the case when none of the pages initiates an update, by definition, the matrix $A_{0,\ldots,0}$ reduces to the identity matrix as $\hat{A}_0=A_{0,\ldots,0}=I$. Noting that $\comb{n}{0}=\comb{n-2}{0}=1$, we have . (4) $\ell=1,\ldots,n-2$:  To prove the expression of $\hat{A}_{\ell}$ for these cases, we must show for each entry that $$\begin{aligned} \bigl( \hat{A}_{\ell} \bigr)_{ij} &= \begin{cases} \bigl( \comb{n}{\ell} - \comb{n-2}{\ell} \bigr) a_{ij} & \text{if $i\neq j$},\\ \comb{n-2}{\ell} & \text{if $i= j$}, \end{cases}\notag\\ &\hspace*{2cm}i,j\in\{1,\ldots,n\}. \label{eqn:Ahatl:4}\end{aligned}$$ In the following, the proof is divided into two steps for the cases of $i\neq j$ and $i=j$. (i) $i\neq j$:  By the definition of $A_{p_1,\ldots,p_n}$ in , its $(i,j)$ entry reduces to $$\bigl( A_{p_1,\ldots,p_n} \bigr)_{ij} = \begin{cases} a_{ij} & \text{if $p_i = 1$ or $p_j = 1$},\\ 0 & \text{otherwise}. \end{cases}$$ Thus, from $$\begin{aligned} \bigl( \hat{A}_{\ell} \bigr)_{ij} &= \sum_{\substack{p_r\in\{0,1\},~r=1,\ldots,n:\\ \sum_{r=1}^{n}p_r = \ell}} \bigl( A_{p_1,\ldots,p_n} \bigr)_{ij} \notag\\ &= \sum_{\substack{p_r\in\{0,1\},~r=1,\ldots,n:\\ \sum_{r=1}^{n}p_r = \ell~\text{and}\\ (p_i = 1~\text{or}~p_j = 1)}} a_{ij} \notag\\ &= \bigl( 2 \comb{n-1}{\ell-1} - \comb{n-2}{\ell-2} \bigr) a_{ij}, \label{eqn:lem:Ahatl:4a}\end{aligned}$$ where the last equality is established by counting the number of cases where $p_i=1$ or $p_j=1$ holds among all possible combinations of $p_1,\ldots,p_n\in\{0,1\}$ such that $\ell$ of them are equal to 1. Using the formula for binomial coefficients $$\comb{r}{k}=\comb{r-1}{k} + \comb{r-1}{k-1},~~~r,k\in\Z_+, \label{eqn:binomial}$$ we can show that $2 \comb{n-1}{\ell-1} - \comb{n-2}{\ell-2} = \comb{n}{\ell} - \comb{n-2}{\ell}$. Hence, from , we arrive at for $i\neq j$. (ii) $i=j$:  Since $a_{ii}=0$ in the link matrix $A$ in and by , the $(i,i)$ entry of $A_{p_1,\ldots,p_n}$ is $$\bigl( A_{p_1,\ldots,p_n} \bigr)_{ii} = \begin{cases} 1 - \sum_{h:~p_h=1} a_{hi} & \text{if $p_i = 0$},\\ 0 & \text{if $p_i = 1$}. \end{cases}$$ Hence, $$\begin{aligned} \bigl( \hat{A}_{\ell} \bigr)_{ii} &= \sum_{\substack{p_r\in\{0,1\},~r=1,\ldots,n:\\ \sum_{r=1}^{n}p_r = \ell}} \bigl( A_{p_1,\ldots,p_n} \bigr)_{ii} = \sum_{\substack{p_r\in\{0,1\},~r=1,\ldots,n:\\ \sum_{r=1}^{n}p_r = \ell,~p_i = 0}} \biggl( 1 - \sum_{h:\;p_h=1} a_{hi} \biggr)\\ &= \comb{n-1}{\ell} - \sum_{\substack{h=1\\ h\neq i}}^{n} \sum_{\substack{p_r\in\{0,1\},~r=1,\ldots,n:~\\ \sum_{r=1}^{n}p_r = \ell,\;p_i = 0,\;p_h=1}} a_{hi},\end{aligned}$$ where the first term is obtained by counting the number of possible combinations of $p_1,\dots,p_n$ such that their sum equals $\ell$ and $p_i=0$; the second term is a consequence of switching the order of two summations. By a combinatorial argument again, we have $\bigl( \hat{A}_{\ell} \bigr)_{ii} = \comb{n-1}{\ell} - \comb{n-2}{\ell-1} \sum_{h=1, h\neq i}^n a_{hi}$. The original link matrix $A$ in is stochastic with diagonal entries being 0. This fact together with yields $\bigl( \hat{A}_{\ell} \bigr)_{ii} = \comb{n-1}{\ell} - \comb{n-2}{\ell-1} = \comb{n-2}{\ell}$. Therefore, is attained for the case $i=j$.   (i)  By and Lemma \[lem:Ahatl\], the matrix $\overline{A}$ can be computed directly as $$\begin{aligned} \overline{A} &= \sum_{\ell=0}^n \alpha^{\ell} (1-\alpha)^{n-\ell} \hat{A}_{\ell}\notag\\ &= \sum_{\ell=0}^{n-2} \alpha^{\ell} (1-\alpha)^{n-\ell} \bigl[ \bigl( \comb{n}{\ell} - \comb{n-2}{\ell} \bigr) A + \comb{n-2}{\ell} I \bigr] + \alpha^{n-1} (1-\alpha) n A + \alpha^{n} A \notag\\ &= \sum_{\ell=0}^{n} \alpha^{\ell} (1-\alpha)^{n-\ell} \comb{n}{\ell}A + \sum_{\ell=0}^{n-2} \alpha^{\ell} (1-\alpha)^{n-\ell} \comb{n-2}{\ell} \bigl( I - A \bigr). \label{eqn:prop:simul:1} \end{aligned}$$ In the first term above, we have by the binomial identity $\sum_{\ell=0}^{n} \alpha^{\ell} (1-\alpha)^{n-\ell} \comb{n}{\ell} = [ \alpha + (1-\alpha)]^n = 1$. Similarly, for the second term, $\sum_{\ell=0}^{n-2} \alpha^{\ell} (1-\alpha)^{n-\ell} \comb{n-2}{\ell} = (1-\alpha)^2$. Substituting these relations into , we have $\overline{A} = A + (1-\alpha)^2 (I - A)$, which is the desired expression of $\overline{A}$. (ii)  The equality in (i) implies that any eigenvector $z_0$ of the link matrix $A$ associated with the eigenvalue 1 is also an eigenvector of the average matrix $\overline{A}$ for this eigenvalue. Mean-square convergence of the distributed update scheme -------------------------------------------------------- To guarantee that the distributed scheme yields the PageRank value, we now examine the modified versions of the distributed link matrices. We first express the distributed update scheme of in its equivalent form as $$\begin{aligned} x(k+1) &= M_{\eta_1(k),\ldots,\eta_n(k)} x(k), \label{eqn:xMmod}\end{aligned}$$ where the modified distribution link matrices are given by $$\begin{aligned} M_{p_1,\ldots,p_n} &:= (1-\hat{m}) A_{p_1,\ldots,p_n} + \frac{\hat{m}}{n}S, ~~~~p_1,\ldots,p_n\in\{0,1\}. \label{eqn:Meta}\end{aligned}$$ This form can be obtained by using the facts that the link matrices $A_{p_1,\ldots,p_n}$ are stochastic and that $Sx(k)=\one$. Clearly, these matrices $M_{p_1,\ldots,p_n}$ are positive and stochastic. The objective here is to find the modified link matrices $M_{p_1,\ldots,p_n}$, by selecting the parameter $\hat{m}$, so that their average and the link matrix $M$ from share an eigenvector corresponding to the eigenvalue 1. Since such an eigenvector is unique for $M$, it is necessarily equal to the value vector $x^*$. As in the earlier case in Section \[sec:dist1\], we take the parameter $\hat{m}$ to be different from the original $m$. In particular, let $$\begin{aligned} \hat{m} = \frac{m[1 - (1 - \alpha)^2]}{1 - m (1 - \alpha)^2}. \label{eqn:mhat2}\end{aligned}$$ For the value $m=0.15$ used in this paper, we have $\hat{m} = 0.15[1 - (1 - \alpha)^2]/[1 - 0.15 (1 - \alpha)^2]$. Then, let the average link matrix be $\overline{M}:= \E[M_{\eta_1(k),\ldots,\eta_n(k)}]$. Here, the distributed link matrices are positive stochastic matrices, which implies that the average matrix $\overline{M}$ enjoys the same property. The next lemma is the key to establish the desired relation between the distributed link matrices and their average. It is stated without proof; it follows similarly to that for Lemma \[lem:Mbar\]. \[lem:Mbar2\]The scalar $\hat{m}$ in and the link matrices $M_{p_1,\ldots,p_n}$ in have the following properties: 1. $\hat{m}\in(0,1)$ and $\hat{m}\leq m$. 2. $\overline{M} = \frac{\hat{m}}{m} M + \left(1 - \frac{\hat{m}}{m}\right)I$. 3. For the average matrix $\overline{M}$, the eigenvalue 1 is simple and is the unique eigenvalue of maximum modulus. The value vector $x^*$ is the corresponding eigenvector. We can show by (iii) in the lemma that, in an average sense, the distributed update scheme asymptotically obtains the correct values. More precisely, we have $E[x(k)]=\overline{M}^{k} x(0)\rightarrow x^*$ as $k\rightarrow\infty$. Further, as discussed in Section \[sec:pagerank\], the asymptotic rate of convergence is dominated by the second largest eigenvalue $\lambda_2(\overline{M})$ in magnitude. By and (ii) in the lemma, this eigenvalue can be bounded as $$\begin{aligned} \abs{\lambda_2(\overline{M})} &= \frac{\hat{m}}{m}\abs{\lambda_2(M)} + 1 - \frac{\hat{m}}{m} \leq \frac{1 - m}{1 - m (1 - \alpha)^2}.\end{aligned}$$ It is clear that this is a monotonically decreasing function of $\alpha$ and $m$. That is, higher probability in updates and/or larger $m$ results in faster convergence in average. We are now ready to state the main result of this section. \[thm:erg2\]Consider the distributed scheme with simultaneous updates in . For any update probability $\alpha\in(0,1]$, the PageRank value $x^*$ is obtained through the time average $y$ in as $E\bigl[ \bigl\| y(k) - x^* \bigr\|^2 \bigr] \rightarrow 0$, $k\rightarrow\infty$. The proof follows along similar lines as that in Theorem \[thm:erg\]. More specifically, we can prove either by the general Markov chain results of, e.g., [@Cogburn:86] or by Appendix \[sec:app:B\] where we replace the expression of $\overline{M}$ there with the one in Lemma \[lem:Mbar2\]. Hence, the convergence is of order $1/k$, as in the algorithm of Section \[sec:dist1\]; it also depends on the update probability $\alpha$ but is independent of $n$. We remark that this scheme is fully decentralized when $\alpha<1$. It is parameterized by $\alpha$, which determines the frequency in the updates, communication load among the pages, and the rate of convergence in the mean as we have seen above. In practice, the recursion in must be implemented in the equivalent form . It is clear that communication is required only over the links corresponding to the nonzero entries in the link matrices there. Each page then computes weighted additions of its own value, the values received from others, and the constant $\hat{m}/n$. On the other hand, when $\alpha=1$, the scheme reduces to the original centralized one in Section \[sec:pagerank\]. In this case, the distributed link matrix is $M_{1,\ldots,1}$ and coincides with the original $M$ because $\hat{m}=m$ and $A_{1,\ldots,1}=A$ from Lemma \[lem:Ahatl\]. Update termination in PageRank computation {#sec:approx} ========================================== In this section, we further develop the distributed algorithm for calculating the PageRank. We relax the objective and aim at obtaining approximate values of the PageRank. The key feature here is to allow the pages to terminate their updates at the point when the values have converged to a certain level. The benefit is that such values need to be transmitted only once to the linked pages; hence, the computation and communication load can be reduced. In a centralized setting, the idea of update termination for the PageRank computation has been introduced by [@KamHavGol:04]. We extend this idea to the distributed update scheme of Section \[sec:simul\]. First, we consider a simple case to attain a convergence result. Then, we provide the details of the proposed algorithm. Convergence properties for the distributed scheme ------------------------------------------------- Consider the distributed update scheme with simultaneous updates in for computing the values $x(k)$ together with their time average $y(k)$. Within this subsection, we fix the sample paths $\{\eta_j(k)\}_{k=0}^{k_0-1}$, $j=1,\ldots,n$, up to time $k_0-1$ of the processes specifying the updates in the pages. Suppose that some of the time averages $y_i(k_0)$ have, in an approximate sense, converged. This is measured by finding those that have varied only within sufficiently small ranges for a certain number of time steps. We introduce two parameters: Let $\delta\in(0,1)$ be the relative error level, and let $N_s$ be the number of steps. Using the history of its own time average $y_i$, each page $i$ then determines at time $k_0$ whether the following condition holds: $$\abs{y_i(k_0) - y_i(k_0-\ell)} \leq \delta y_i(k_0),~~\ell=1,2,\ldots,N_s. \label{eqn:yconv}$$ If so, then (i) the page $i$ will terminate its update and fix its estimate at $y_i(k_0)$, and then (ii) this value $y_i(k_0)$ is transmitted to the pages connected to $i$ by direct links. After this point, these values will be used at the pages performing further updates. The question of interest is whether the pages that continue with their updates after time $k_0$ will reach a good estimate of their true values. In what follows, we show that the answer is positive and the approximation level achievable in the estimate will be as good as that for the pages that have terminated their updates. Note that the analysis is based on the given sample paths $\{\eta_j(k)\}_{k=0}^{k_0-1}$, and hence the state $x$ and its average $y$ up to time $k_0$ are fixed; we study their stochastic behaviors after this time. Let $\Ccal(k_0)$ be the set of page indices that have reached good estimates at time $k_0$ as $$\begin{aligned} \Ccal(k_0) &:= \bigl\{ i\in\Vcal:~ \abs{y_i(k_0) - y_i(k_0-\ell)} \leq \delta y_i(k_0), ~~\ell=1,2,\ldots,N_s \bigr\}.\end{aligned}$$ The cardinality of this set is denoted by $n_1(k_0)$. We assume $n_1(k_0)\geq 1$. Also, let $\Ncal(k_0) := \Vcal \setminus \Ccal(k_0)$. Based on these sets, we introduce a coordinate transformation for the state $x(k)$ and partition it as $$x(k) = \begin{bmatrix} x_{\Ccal}(k)\\ x_{\Ncal}(k) \end{bmatrix},~~~k\geq k_0,$$ where $x_{\Ccal}(k)\in\R^{n_1(k_0)}$ contains the values of the pages in $\Ccal(k_0)$ and $x_{\Ncal}(k)\in\R^{n-n_1(k_0)}$ contains those of the pages in $\Ncal(k_0)$. With slight abuse of notation, we write the transformed state by $x(k)$. Also, we use the shorthand notation $A_{p}$ for $A_{p_1,\ldots,p_n}$, $p_i\in\{0,1\}$, $i\in\Vcal$. Then, the distributed link matrices $A_{p}$ in and the average link matrix $\overline{A}$ in are partitioned accordingly as $$A_{p} = \begin{bmatrix} A_{p,\Ccal\Ccal} & A_{p,\Ccal\Ncal}\\ A_{p,\Ncal\Ccal} & A_{p,\Ncal\Ncal} \end{bmatrix},~~~ \overline{A} = \begin{bmatrix} \overline{A}_{\Ccal\Ccal} & \overline{A}_{\Ccal\Ncal}\\ \overline{A}_{\Ncal\Ccal} & \overline{A}_{\Ncal\Ncal} \end{bmatrix}. \label{eqn:Abar:part}$$ Since the time average $y_{\Ccal}$ has converged sufficiently by time $k_0$, the proposed approach employs the value $y_{\Ccal}(k_0)$ as $x_{\Ccal}(k)$ for all $k\geq k_0$. Hence, the value at time $k_0$ is reset as $$x(k_0) = \begin{bmatrix} y_{\Ccal}(k_0)\\ x_{\Ncal}(k_0) \end{bmatrix}.$$ The updates are carried out through the distributed algorithm given by $$x(k+1) = \widetilde{A}_{\eta(k)} x(k) + \frac{\hat{m}}{n} \tilde{s}, ~~~~k\geq k_0, \label{eqn:approx:dist:x}$$ where $$\begin{aligned} \widetilde{A}_{\eta(k)} &= \begin{bmatrix} I & 0\\ \widetilde{A}_{\eta(k), \Ncal\Ccal} & \widetilde{A}_{\eta(k),\Ncal\Ncal} \end{bmatrix} := \begin{bmatrix} I & 0\\ (1-\hat{m})A_{\eta(k), \Ncal\Ccal} & (1-\hat{m})$ $A_{\eta(k),\Ncal\Ncal} \end{bmatrix}, \tilde{s} := \begin{bmatrix} 0\\ \one \end{bmatrix}, \label{eqn:approx:dist:Aeta}\end{aligned}$$ with $\one\in\R^{n-n_1(k_0)}$. We note that the matrices $\widetilde{A}_{p}$ are nonnegative, but are no longer stochastic; the sums of the entries of the first $n_1(k_0)$ columns are larger than 1 while those of the other columns are smaller than 1. Hence, though $x(k)\geq 0$ still holds, the state $x(k)$ may not be a probability vector. In addition, this scheme is in the distributed form of , and not the one in based on the modified link matrices. The time average $y(k)$ is also modified by fixing the entries for $i\in\Ccal(k_0)$ as $$y(k) = \begin{bmatrix} y_{\Ccal}(k_0)\\ y_{\Ncal}(k) \end{bmatrix},~~~~k\geq k_0,$$ where $y_{\Ncal}(k)$ is determined through the original formula . For the approximate update scheme , its average state $\overline{x}(k):=E[x(k)]$ follows the recursion $$\overline{x}(k+1) = \widehat{A}\; \overline{x}(k) + \frac{\hat{m}}{n}\tilde{s}, ~~~k\geq k_0, \label{eqn:approx:xave}$$ where the average link matrix $\widehat{A}:= E[\widetilde{A}_{\eta(k)}]$ is given by $$\widehat{A} = \begin{bmatrix} I & 0\\ \widehat{A}_{\Ncal\Ccal} & \widehat{A}_{\Ncal\Ncal} \end{bmatrix} := \begin{bmatrix} I & 0\\ (1-\hat{m})\overline{A}_{\Ncal\Ccal} & (1-\hat{m})\overline{A}_{\Ncal\Ncal} \end{bmatrix}. \label{eqn:Mtilbar}$$ Regarding this average link matrix, the following result will be useful in the subsequent development. \[lem:Ann\]The submatrix $\widehat{A}_{\Ncal\Ncal}$ of the average link matrix $\widehat{A}$ as given in satisfies the following: 1. $\rho(\widehat{A}_{\Ncal\Ncal})\in[0,1-\hat{m}]$ and, in particular, $\widehat{A}_{\Ncal\Ncal}$ is a stable matrix. 2. $(I-\widehat{A}_{\Ncal\Ncal})^{-1}\geq 0$. [[*Proof:*]{}]{}(i) Since the original average link matrix $\overline{A}$ in is a stochastic matrix, the block diagonal matrix $\diag(0,\overline{A}_{\Ncal\Ncal})$ containing the submatrix $\overline{A}_{\Ncal\Ncal}$ satisfies $\overline{A}\geq \diag(0,\overline{A}_{\Ncal\Ncal})\geq 0$. By the property of nonnegative matrices [@HorJoh:85], it follows that $1=\rho(\overline{A})\geq \rho(\overline{A}_{\Ncal\Ncal})\geq 0$. Therefore, $\rho(\widehat{A}_{\Ncal\Ncal})= \rho((1-\hat{m})\overline{A}_{\Ncal\Ncal})\in[0,1-\hat{m}]$. By Lemma \[lem:Mbar2\] (i), $\hat{m}\in(0,1)$ and hence $\rho(\widehat{A}_{\Ncal\Ncal})<1$. (ii) Let $\lambda:=1-\rho(\widehat{A}_{\Ncal\Ncal})$. This is the eigenvalue of $I-\widehat{A}_{\Ncal\Ncal}$ with the smallest real part. By (i), we have $\lambda>0$. Thus, $I-\widehat{A}_{\Ncal\Ncal}$ is an $M$-matrix[^8], so it has an inverse that is nonnegative [@HorJoh:91]. We remark that in (i) in the lemma, the level of stability is affected by the parameter $\alpha$ as it determines the size of $\hat{m}$. It is clear that the value vector $x^*$ (with the coordinate change) is an equilibrium of the system . We partition it as $$x^* = \begin{bmatrix} x_{\Ccal}^*\\ x_{\Ncal}^* \end{bmatrix}. \label{eqn:approx:xast}$$ It is also simple to show that the vector $\widetilde{x}(k_0)$ given by $$\begin{aligned} \widetilde{x}(k_0) &= \begin{bmatrix} \widetilde{x}_{\Ccal}(k_0)\\ \widetilde{x}_{\Ncal}(k_0) \end{bmatrix} := \begin{bmatrix} y_{\Ccal}(k_0)\\ \bigl( I - \widehat{A}_{\Ncal\Ncal} \bigr)^{-1} \bigl( \widehat{A}_{\Ncal\Ccal}\; y_{\Ccal}(k_0) + \frac{\hat{m}}{n}s \bigr) \end{bmatrix} \label{eqn:approx:xdash2}\end{aligned}$$ is an equilibrium of the system . This vector always exists by (ii) in the lemma above. After the pages in $\Ccal(k_0)$ have terminated their updates, the dynamics of the scheme can be characterized as follows. \[lem:approx:xdash2\]For the distributed update scheme and its average system , the following statements hold. 1. The average state $\overline{x}(k)$ converges to $\widetilde{x}(k_0)$ and, in particular, $\overline{x}_{\Ncal}(k)\rightarrow \widetilde{x}_{\Ncal}(k_0)$ as $k\rightarrow\infty$. 2. If $\abs{\widetilde{x}_{\Ccal}(k_0)-x_{\Ccal}^*}\leq\delta x_{\Ccal}^*$, then $\abs{\widetilde{x}_{\Ncal}(k_0)-x_{\Ncal}^*}\leq \delta x_{\Ncal}^*$. [[*Proof:*]{}]{}(i) Since $\widetilde{x}(k_0)$ is an equilibrium of the average system , it follows that $$\begin{aligned} \overline{x}(k+1) - \widetilde{x}(k_0) &= \begin{bmatrix} I & 0\\ \widehat{A}_{\Ncal\Ccal} & \widehat{A}_{\Ncal\Ncal} \end{bmatrix} \bigl( \overline{x}(k)-\widetilde{x}(k_0) \bigr).\end{aligned}$$ Here, we have $\overline{x}_{\Ccal}(k) - \widetilde{x}_{\Ccal}(k_0)=0$, $k\geq k_0$. Thus, $$\begin{aligned} \overline{x}_{\Ncal}(k+1) - \widetilde{x}_{\Ncal}(k_0) &= \widehat{A}_{\Ncal\Ncal} \left( \overline{x}_{\Ncal}(k) - \widetilde{x}_{\Ncal}(k_0) \right). $$ By Lemma \[lem:Ann\](i), $\widehat{A}_{\Ncal\Ncal}$ is a stable matrix. Hence, we have $\overline{x}_{\Ncal}(k) - \widetilde{x}_{\Ncal}(k_0) \rightarrow 0$ as $k\rightarrow\infty$. (ii) For the average system , $\widetilde{x}(k_0)$ and $x^*$ are both equilibria, and thus $$\widetilde{x}_{\Ncal}(k_0) - x_{\Ncal}^* = \bigl( I - \widehat{A}_{\Ncal\Ncal} \bigr)^{-1} \widehat{A}_{\Ncal\Ccal} \bigl( \widetilde{x}_{\Ccal}(k_0) - x^*_{\Ccal} \bigr).$$ By Lemma \[lem:Ann\](ii), $(I - \widehat{A}_{\Ncal\Ncal})^{-1}\geq 0$. Moreover, by construction, $\widehat{A}_{\Ncal\Ccal}\geq 0$. Thus, using the assumption, we have $$\begin{aligned} \left| \widetilde{x}_{\Ncal}(k_0) - x_{\Ncal}^* \right| &\leq \bigl( I - \widehat{A}_{\Ncal\Ncal} \bigr)^{-1} \widehat{A}_{\Ncal\Ccal} \; \left| \widetilde{x}_{\Ccal}(k_0) - x^*_{\Ccal} \right| \\ &\leq \bigl( I - \widehat{A}_{\Ncal\Ncal} \bigr)^{-1} \widehat{A}_{\Ncal\Ccal} \; \delta x^*_{\Ccal}\\ &\leq \delta \bigl( I - \widehat{A}_{\Ncal\Ncal} \bigr)^{-1} \biggl( \widehat{A}_{\Ncal\Ccal} \; x^*_{\Ccal} + \frac{\hat{m}}{n}\one \biggr) = \delta x^*_{\Ncal},\end{aligned}$$ where the last equality follows because $x^*$ is an equilibrium of . The lemma shows that if the values in $y_{\Ccal}(k_0)=\widetilde{x}_{\Ccal}(k_0)$ are actually close to the true values $x^*_{\Ccal}$, then via the recursion in , we can still obtain an approximate value $\widetilde{x}_{\Ncal}(k_0)$ in the average sense for all other states; the approximation level is the same as that for $\widetilde{x}_{\Ccal}(k_0)$, represented by the parameter $\delta$. The following is the main convergence result for the scheme described in this section. \[thm:approx:erg\]Consider the distributed scheme in , where under the given sample paths $\{\eta_i(k)\}_{k=0}^{k_0-1}$, $i=1,\ldots,n$, at time $k_0$, the updates at $n_1(k_0)$ pages have terminated. The time average $y_{\Ncal}(k)$, $k\geq k_0$, converges to the equilibrium $\widetilde{x}_{\Ncal}(k_0)$ in the mean-square sense as $E\bigl[ \bigl\| y_{\Ncal}(k) - \widetilde{x}_{\Ncal}(k_0) \bigr\|^2 \bigr] \rightarrow 0$, $k\rightarrow\infty$. The proof is presented in Appendix \[sec:app:C\]. It is based on that for Theorem \[thm:erg\]. However, unlike the setup there, the distributed link matrices in the current scheme are not stochastic. This means that we cannot employ general Markov process results of, e.g., [@Cogburn:86]. In contrast, the proof relies on the stability of the submatrix $\widehat{A}_{\Ncal\Ncal}$ as shown in Lemma \[lem:Ann\](i). Also, for this reason, the analysis does not involve the modified link matrices such as $M_{\eta_1(k),\ldots,\eta_n(k)}$ that appeared in the previous section. Distributed algorithm with update termination --------------------------------------------- We present the distributed algorithm with update termination based on the results from this section. \[alg:dist\] For $i\in\Vcal$, page $i$ executes the following. 1. Initialize the parameters $n$, $\alpha$, $x_i(0)$, $N_s$, and $\delta$. Set $k=0$, $\Ccal(0)=\emptyset$, and $n_1(0)=0$. 2. At time $k$, generate $\eta_i(k)\in\{0,1\}$ under the probability $\alpha$. If $\eta_i(k)=1$, then send the value $x_i(k)$ to pages $j\nin\Ccal(k)$ that it is linked to, and request pages $j\nin\Ccal(k)$ having links to it for their values. 3. Update the value $x_i(k)$ and its time average by $$\begin{split} x_i(k+1) &= \sum_{j=1}^n \bigl( \widetilde{A}_{\eta(k)} \bigr)_{ij}\; x_j(k) + \frac{\hat{m}}{n},\\ y_i(k) &= \frac{1}{k+1} \sum_{\ell=0}^{k}x_i(\ell), \end{split} \label{eqn:alg:dist:x}$$ where $\widetilde{A}_{\eta(k)}$ is constructed by using $\Ccal(k)$. 4. Check if $y_i(k)$ has sufficiently converged based on . If so, then (i) add $i$ to the set $\Ccal(k)$, (ii) send $y_i(k)$ to the pages having direct links to page $i$, and (iii) fix $x_i(\ell)=y_i(\ell)=y_i(k)$ for $\ell\geq k$. 5. If $\Ccal(k)=\Vcal$, then terminate the algorithm. Otherwise, set $\Ccal(k+1)=\Ccal(k)$ and $k=k+1$, and then go to Step 1. We remark that, in this scheme, the choice of the parameters $\delta$ and $N_s$ affects the accuracy in the values when the pages terminate their updates as well as the time when the pages decide to do so. Taking $\delta$ smaller and/or $N_s$ larger will improve the value estimates, but will require longer time before the updates terminate; this in turn will keep the computation and communication load higher. Discussion on asynchronous iteration methods {#sec:asynch} ============================================ In this section, we discuss the application of a numerical analysis method known as asynchronous iteration [@BerTsi:89] to the distributed computation of PageRank values. Deterministic algorithms for the PageRank problem under this approach have been discussed in, e.g., [@deJBra:07; @KolGalSzy:06]. We present a randomization-based algorithm and clarify its relation to the schemes of this paper. Consider the original update scheme in based on the power method. Let $\eta_i(k)\in\{0,1\}$ for $i\in\Vcal$, $k\in\Z_+$, be the i.i.d. random processes whose distributions are as in . Similarly to the scheme with simultaneous updates in Section \[sec:simul\], when $\eta_i(k)=1$ at time $k$, then page $i$ initiates an update; such an event occurs with probability $\alpha$. However, the difference is that this update is performed as in the power method, and moreover pages whose corresponding processes $\eta_i(k)$ are zero do not make any updates. The distributed update recursion is given as follows: $$\check{x}(k+1) = \check{M}_{\eta_1(k),\ldots,\eta_n(k)} \check{x}(k), \label{eqn:xasynch}$$ where the initial state $\check{x}(0)$ is a probability vector and the link matrices are given by $$\begin{aligned} \bigl( \check{M}_{p_1,\ldots,p_n} \bigr)_{ij} := \begin{cases} (1-m)a_{ij}+ \frac{m}{n} & \text{if $p_i = 1$},\\ 1 & \text{if $p_i = 0$ and $i=j$},\\ 0 & \text{otherwise}. \end{cases}\end{aligned}$$ It is clear that these matrices keep the rows of the original link matrix $M$ in for the pages that initiate updates. Other pages just keep their previous values. Thus, these matrices are not stochastic. The following result shows that through this algorithm, we can compute the PageRank values. \[lem:asynch\]Under the distributed update scheme of , for every initial state $x(0)$ that is a probability vector, the PageRank value $x^*$ is obtained as $\check{x}(k)\rightarrow x^*$, $k\rightarrow\infty$, with probability one. The distributed update scheme in is a randomized version of the one in [@BerTsi:89 Section 6.2], and the proof can be extended in a straightforward way. Specifically, it relies on the property $\rho(M)=1$ that the link matrix $M$ has. The algorithm is based on general asynchronous iteration algorithms for distributed computation of fixed points in the field of numerical analysis. It is interesting to note that the proof of the result above employs an argument similar to that of Lyapunov functions. We also point out that the convergence rate is exponential and the general scheme can handle delays in the communication. In comparison, the algorithms proposed in this paper have the following characteristic features. First, the link matrices in the update schemes and are stochastic, and this property is exploited in the convergence analysis. It moreover provides the relation to consensus type problems as discussed in Section \[sec:consensus\]. Second, there is a difference with regard to the type of links over which communication takes place. In particular, it is shown in a subsequent paper [@IshTemBaiDab:09] that the present approach can be extended in such a way that each page communicates only with those connected by outgoing links; the information of such links are by default available locally. By contrast, in the asynchronous iteration algorithm, pages must utilize the incoming links. This means that popular pages linked from many pages need extra storage to keep the data of such links. Numerical example {#sec:example} ================= We present an example with 1,000 web pages ($n=1,000$). The links among the pages were randomly generated. The first ten pages are designed to have high rankings and are given links from over 90% of the pages. Others have between 2 and 333 links per page. We ran Algorithm \[alg:dist\] in Section \[sec:approx\] where each page initiates an update at a fixed probability $\alpha=0.01$ and terminates the updates when an approximate value is obtained. In the distributed scheme , we generated sample paths of the processes $\eta_i(k)$, $i\in\Vcal$, which determine the pages initiating updates, and computed the state $x(k)$. The initial state $x(0)$ was taken as a random probability vector. The parameters for the update termination were chosen as follows: The number $N_s$ of steps before stopping the update was $N_s=800$ and the parameter $\delta$ determining the level of approximation was $\delta=0.01$. We chose these values so that the characteristics of this scheme are visible in the plots. The responses of the time average $y_i$ for $i=21,\ldots,30$ are shown in Fig. \[fig:y\_term\]. The time when the corresponding pages terminated their updates are marked by $\bigcirc$. We observe that the convergence is fairly fast, and the updates stop by time $k=4,500$ for these pages. Let the errors in the estimates be $e(k):=y(k)-x^*$. In Fig. \[fig:err\_term\], these are shown for two cases: $\norm{e(k)}_{1}$ under the $\ell_1$ norm and $\norm{e(k)}_{\infty}$ under the $\ell_{\infty}$ norm. The plot shows that the error in the individual values of $y_i$ (measured by the $\ell_{\infty}$ norm) rapidly decreases and remains small while the total error (in the $\ell_1$ norm) also decreases but at a slower rate. In Fig. \[fig:pr\_term\], the final values of $y_i$ for the first twenty pages are plotted as $\bigcirc$ together with the acceptable ranges of error, that is, $[(1-\delta)x^*_i,(1+\delta)x^*_i]$ by two lines connected in the middle. As we mentioned in Section \[sec:approx\], the time average $y$ is no longer normalized in this case. However, the sum of all $y_i$ at $k=8,000$ turned out to be about $0.989$, which is very close to the desired value 1. Conclusion {#sec:concl} ========== In this paper, we first gave an overview of the PageRank computation problem, which is critical in making accurate search results with Google. We introduced a randomization-based distributed approach for the computation of PageRank values and showed the mean-square convergence of the proposed schemes. It was demonstrated that the approach has a clear relation to consensus type problems. The algorithms were generalized in the recent papers [@IshTem_acc:09; @IshTemBaiDab:09], where random link failures and computations based on aggregating groups of pages are addressed and more discussions on the advantages of this approach can be found. Future research will deal with the effects of communication delays and the improvement of convergence rate, and also study issues related to implementation of the proposed distributed algorithms. [*Acknowledgement*]{}: We are thankful to Er-Wei Bai, B. Ross Barmish, Tamer Başar, Fabrizio Dabbene, Soura Dasgupta, Shinji Hara, Zhihua Qu, M. Vidyasagar, and Yutaka Yamamoto for their helpful comments and discussions on this work. [10]{} Special section on complex networked control systems. , 27(4), 2007. R. Andersen, C. Borgs, J. Chayes, J. Hopcroft, V.S. Mirrokni, and S.-H. Teng. Local computation of [P]{}age[R]{}ank contributions. In [*Algorithms and Models for the Web-Graph*]{}, volume 4863 of [ *Lect. Notes Comp. Sci.*]{}, pages 150–165. Springer, Berlin, 2007. P.J. Antsaklis and J. Baillieul[, Guest Editors]{}. Special [I]{}ssue on the [T]{}echnology of [N]{}etworked [C]{}ontrol [S]{}ystems. , 95(1), 2007. K. Avrachenkov, N. Litvak, D. Nemirovsky, and N. Osipova. Monte [C]{}arlo methods in [P]{}age[R]{}ank computation: [W]{}hen one iteration is sufficient. , 45:890–904, 2007. D.P. Bertsekas and J.N. Tsitsiklis. . Prentice-Hall, Englewood Cliffs, NJ, 1989. D.P. Bertsekas and J.N. Tsitsiklis. Comments on “[C]{}oordination of groups of mobile autonomous agents using nearest neighbor rules”. , 52:968–969, 2007. S. Boyd, A. Ghosh, B. Prabhakar, and D. Shah. Randomized gossip algorithms. , 52:2508–2530, 2006. S. Brin and L. Page. The anatomy of a large-scale hypertextual [Web]{} search engine. , 30:107–117, 1998. K. Bryan and T. Leise. The \$25,000,000,000 eigenvector: [T]{}he linear algebra behind [G]{}oogle. , 48:569–581, 2006. R. Carli, F. Fagnani, A. Speranzon, and S. Zampieri. Symmetries in the coordinated consensus problem. In P. J. Antsaklis and P. Tabuada, editors, [*Networked Embedded Sensing and Control Workshop*]{}, volume 331 of [*Lect. Notes Contr. Info. Sci.*]{}, pages 25–51. Springer, Berlin, 2005. R. Cogburn. On products of random stochastic matrices. , 50:199–213, 1986. D.V. de Jager and J.T. Bradley. Asynchronous iterative solution for state-based performance metrics. In [*Proc. ACM SIGMETRICS*]{}, pages 373–374, 2007. D.J. Hartfiel. , volume 1695 of [*Lect. Notes Math.*]{} Springer, Berlin, 1998. Y. Hatano and M. Mesbahi. Agreement over random networks. , 50:1867–72, 2005. R.A. Horn and C.R. Johnson. . Cambridge Univ. Press, 1985. R.A. Horn and C.R. Johnson. . Cambridge Univ. Press, 1991. H. Ishii and R. Tempo. A distributed randomized approach for the [P]{}age[R]{}ank computation: Parts 1 and 2. In [*Proc. [47]{}th IEEE Conf. on Decision and Control*]{}, pages 3523–3528 and 3529–3534, 2008. H. Ishii and R. Tempo. Computing the [P]{}age[R]{}ank variation for fragile web data. , 2:1–9, 2009. H. Ishii and R. Tempo. Distributed [P]{}age[R]{}ank computation with link failures. In [*Proc. American Control Conf.*]{}, pages 1976–1981, 2009. H. Ishii, R. Tempo, E.-W. Bai, and F. Dabbene. Distributed randomized [P]{}age[R]{}ank computation based on web aggregation. In [*Proc. [48]{}th IEEE Conf. on Decision and Control and Chinese Control Conference*]{}, pages 3026–3031, 2009. A. Jadbabaie, J. Lin, and A.S. Morse. Coordination of groups of mobile autonomous agents using nearest neighbor rules. , 48:988–1001, 2003. S. Kamvar, T. Haveliwala, and G. Golub. Adaptive methods for the computation of [P]{}age[R]{}ank. , 386:51–65, 2004. A. Kashyap, T. Başar, and R. Srikant. Quantized consensus. , 43:1192–1203, 2007. G. Kollias, E. Gallopoulos, and D.B. Szyld. Asynchronous iterative computations with web information retrieval structures: [T]{}he [P]{}age[R]{}ank case. In G.R. Joubert [*et al.*]{}, editor, [*Parallel Computing: Current and Future Issues of High-End Computing*]{}, volume 33 of [*NIC Series*]{}, pages 309–316. John von Neumann-Institut for Computing, Julich, Germany, 2006. A.N. Langville and C.D. Meyer. . Princeton Univ. Press, 2006. Z. Lin, B.A. Francis, and M. Maggiore. Necessary and sufficient graphical conditions for formation control of unicycles. , 50:121–127, 2005. J.A. Marshall, M.E. Broucke, and B.A. Francis. Formations of vehicles in cyclic pursuit. , 49:1963–1974, 2004. L. Moreau. Stability of multiagent systems with time-dependent communication links. , 50:169–182, 2005. A. Papoulis and S.U. Pillai. . McGraw Hill, New York, 2002. Z. Qu, J. Wang, and R.A. Hull. Cooperative control of dynamical systems with application to autonomous vehicles. , 53:894–911, 2008. W. Ren and R.W. Beard. Information consensus in multivehicle cooperative control. , 50:655–661, 2005. E. Seneta. . Springer, New York, 1981. A. Tahbaz-Salehi and A. Jadbabaie. A necessary and sufficient condition for consensus over random networks. , 53:791–795, 2008. R. Tempo, G. Calafiore, and F. Dabbene. . Springer, London, 2005. R. Tempo and H. Ishii. Monte [C]{}arlo and [L]{}as [V]{}egas randomized algorithms for systems and control: [A]{}n introduction. , 13:189–203, 2007. C.W. Wu. Synchronization and convergence of linear dynamics in random directed networks. , 51:1207–1210, 2006. C. Yu, J.M. Hendrickx, B. Fidan, B.D.O. Anderson, and V.D. Blondel. Three and higher dimensional autonomous formations: [R]{}igidity, persistence and structural persistence. , 43:387–402, 2007. Y. Zhu, S. Ye, and X. Li. Distributed [P]{}age[R]{}ank computation based on iterative aggregation-disaggregation methods. In [*Proc. [14]{}th ACM Conf. on Information and Knowledge Management*]{}, pages 578–585, 2005. Appendix ======== In this appendix, following some preliminary material, the proofs of Theorems \[thm:erg\] and \[thm:approx:erg\] are given. Preliminaries ------------- We present some results related to infinite products of stochastic matrices from [@Seneta:81; @Hartfiel:98]. These are required for the proof of Theorem \[thm:erg\] given in the next subsection. First, the notion of weak ergodicity is introduced. Given a sequence of stochastic matrices $\{P(k)\}_{k=0}^{\infty}\subset\R^{n\times n}$, let their (backward) product be $T(k):=P(k)\cdots P(0)$. The sequence $\{P(k)\}_{k=0}^{\infty}$ is said to be weakly ergodic if $$t_{ri}(k) - t_{rj}(k) \rightarrow 0,~~ k\rightarrow \infty,~~\forall i,j,r\in\{1,\ldots,n\}. \label{eqn:ergodic}$$ In , $t_{ri}(k)$ for all $i$ tend to be equal as $k\rightarrow\infty$, that is, all columns of the product matrix $T(k)$ coincide in the limit. However, in general, the columns do not converge to a single vector. To characterize matrix sequences that are weakly ergodic, we employ the tool known as the coefficient of ergodicity. Let $\tau(\cdot)$ be the scalar function for stochastic matrices in $\R^{n\times n}$ given by $$\tau(P) := \frac{1}{2}\max_{i,j}\sum_{r=1}^{n}\abs{p_{ri}-p_{rj}}. \label{eqn:tau}$$ This function $\tau(\cdot)$ has the following properties. \[lem:coef\] 1. $\tau(P)\in[0,1]$ and, moreover, $\tau(P)=0$ if and only if there exists a probability vector $v\in\R^n$ such that $P=v \one^T$, where $\one=[1\;\cdots\;1]^T\in\R^n$. 2. $\tau(P) = \max\{ \norm{Px}_1:x\in\R^n,\norm{x}_1=1,\sum_{i}x_i=0 \}$. 3. $\tau(PQ)\leq \tau(P)\tau(Q)$ for stochastic matrices $P,Q$. Weak ergodicity can be characterized by the following lemma. \[lem:weakerg\] For a sequence of stochastic matrices $\{P(k)\}_{k=0}^{\infty}$, their product $P(k)\cdots P(0)$ is weakly ergodic if $\tau(P(k))\leq \tau_0$ for all $k$, where $\tau_0\in(0,1)$ is a scalar. The distributed update scheme {#sec:app:B} ----------------------------- We now analyze the proposed algorithm in Section \[sec:dist1\]. \[lem:tauM\]For the distributed update scheme and its average system , the following hold. 1. The matrices $M_i$, $i\in\{1,\ldots,n\}$, and $\overline{M}$ satisfy $\tau(M_i) < 1-\hat{m}$ and $\tau(\overline{M}) < 1-\hat{m}$. 2. For any mode sequence $\{\theta(k)\}$, the matrix sequence $\{M_{\theta(k)}\}$ is weakly ergodic. [[*Proof:*]{}]{}(i)  We show only for $\overline{M}$ since the case of $M_i$ is similar. Recall that by and , the average matrix $\overline{M}$ can be expressed as $\overline{M}= (1-\hat{m}) \overline{A} + \frac{\hat{m}}{n} S$. Thus, we have $$\begin{aligned} \tau(\overline{M}) &= \frac{1}{2}\max_{i,j}\sum_{r=1}^{n} \left| (\overline{M})_{ri} - (\overline{M})_{rj} \right|\\ &= (1-\hat{m})\frac{1}{2}\max_{i,j}\sum_{r=1}^{n} \left| (\overline{A})_{ri} - (\overline{A})_{rj} \right| = (1-\hat{m})\tau(\overline{A}).\end{aligned}$$ The matrix $\overline{A}$ is stochastic, and hence, by Lemma \[lem:coef\] (i), it holds that $\tau(\overline{A})\leq 1$. Consequently, we arrive at the inequality $\tau(\overline{M})\leq 1-\hat{m}$. (ii)  This follows from (i) and Lemma \[lem:weakerg\] since $\hat{m}\in(0,1)$ by Lemma \[lem:Mbar\] (i).    Let the error from the average be $e(k) := x(k) - x^*$. Note that $e(k)$ satisfies $\sum_{i=1}^n e_i(k)=0$. This is because in the systems and , by assumption, the initial states are probability vectors, and furthermore, $M_i$, $i=1,\ldots,n$, and $\overline{M}$ are stochastic matrices; hence, both $x(k)$ and $x^*$ are nonnegative vectors whose entries add up to 1. Observe that $$\begin{aligned} y(k) - x^* &= \frac{1}{k+1}\sum_{\ell=0}^{k} (x(\ell) - x^*) = \frac{1}{k+1}\sum_{\ell=0}^{k} e(\ell).\end{aligned}$$ Thus, $$\begin{aligned} E[\norm{y(k) - x^*}^2] &= E\biggl[ \biggl\| \frac{1}{k+1}\sum_{\ell=0}^{k} e(\ell) \biggr\|^2 \biggr] = \frac{1}{(k+1)^2} E\biggl[ \sum_{\ell=0}^{k} e(\ell)^T e(\ell) + 2 \sum_{\ell=0}^{k-1} \sum_{r=1}^{k-\ell} e(\ell)^T e(\ell+r) \biggr]\notag\\ &= \frac{1}{(k+1)^2} \biggl\{ \sum_{\ell=0}^{k} E[e(\ell)^T e(\ell)] + 2 \sum_{\ell=0}^{k-1} \sum_{r=1}^{k-\ell} E[e(\ell)^T e(\ell+r)] \biggr\}. \label{eqn:thm:erg:2}\end{aligned}$$ We use the norm relation $\norm{z}\leq\norm{z}_1$ for $z\in\R^n$ [@HorJoh:85] and the property $\norm{x(\ell)}_1=\norm{x^*}_1=1$ to obtain the bound $\norm{e(\ell)}\leq 2$. Then, in the first summation term in , we have $$\sum_{\ell=0}^{k} E[e(\ell)^T e(\ell)] \leq 4(k+1). \label{eqn:thm:erg:2a}$$ In the second summation term, we see that the summands can be written as $$\begin{aligned} E[e(\ell)^T e(\ell+r)] &= E[ e(\ell)^T (x(\ell+r) - x^*)] \notag\\ &= E\bigl[ e(\ell)^T \bigl( M_{\theta(\ell+r-1)}\cdots M_{\theta(\ell)} x(\ell) - x^* \bigr) \bigr]. \label{eqn:thm:erg:3}\end{aligned}$$ Here, by taking the expectation of the matrix product $M_{\theta(\ell+r-1)}\cdots M_{\theta(\ell)}$ with respect to the random variables $\theta(\ell+r-1),\ldots,\theta(\ell)$, $$\begin{aligned} E[e(\ell)^T e(\ell+r)] &= E\bigl[ e(\ell)^T \bigl( E[M_{\theta(\ell+r-1)}\cdots M_{\theta(\ell)}] x(\ell) - x^* \bigr) \bigr] \notag\\ &= E\bigl[ e(\ell)^T \bigl( E[M_{\theta(\ell+r-1)}]\cdots E[M_{\theta(\ell)}] x(\ell) - x^* \bigr) \bigr] \notag\\ &= E\bigl[ e(\ell)^T ( \overline{M}^r x(\ell) - x^* ) \bigr], \label{eqn:thm:erg:3b}\end{aligned}$$ where the second and third equalities follow from the independence of $\theta(\ell+r-1),\ldots,\theta(\ell)$ and the definition of the average matrix $\overline{M}$, respectively. Since, by Lemma \[lem:Mbar\] (iii), $x^*$ is the eigenvector of $\overline{M}$ for the eigenvalue 1, it follows that $\overline{M}^r x(\ell) - x^*=\overline{M}^r (x(\ell) - x^*)$. Further, we have $x(\ell)-x^*=e(\ell)$ and again apply the fact $\norm{z}\leq\norm{z}_1$, $z\in\R^n$, to derive from that $$\begin{aligned} E[e(\ell)^T e(\ell+r)] &= E\bigl[ e(\ell)^T \overline{M}^r e(\ell) \bigr] \leq E\bigl[ \norm{e(\ell)} \norm{\overline{M}^r e(\ell)} \bigr] \notag\\ &\leq 2\; E\bigl[ \norm{\overline{M}^r e(\ell)}_1 \bigr], \label{eqn:thm:erg:3a}\end{aligned}$$ where in the last inequality, we also used $\norm{e(\ell)}\leq 2$. As we have mentioned above, it holds that $\sum_{i=1}^n e_i(\ell)=0$. Thus, apply Lemma \[lem:coef\] (ii) and (iii) to and obtain $$\begin{aligned} E[e(\ell)^T e(\ell+r)] &\leq 2\; \tau(\overline{M}^r) E\bigl[ \norm{e(\ell)}_1 \bigr]\notag\\ &\leq 2\;\tau(\overline{M})^r E\bigl[ \norm{e(\ell)}_1 \bigr] \leq 4\;\tau(\overline{M})^r, \label{eqn:thm:erg:6}\end{aligned}$$ where the last inequality is due to $\norm{e(\ell)}_1\leq 2$. Note that by Lemma \[lem:tauM\], $\tau(\overline{M})<1$. Finally, by substituting and into , we have $$\begin{aligned} E[\norm{y(k) - x^*}^2] &\leq \frac{1}{(k+1)^2} \biggl\{ 4(k+1) + 2\sum_{\ell=0}^{k-1} \sum_{r=0}^{k-\ell} 4\, \tau(\overline{M})^r \biggr\}\notag\\ &\leq \frac{4}{k+1} \biggl( 1 + \frac{2}{1-\tau(\overline{M})} \biggr), \end{aligned}$$ and hence using the bound on $\tau(\overline{M})$ in Lemma \[lem:tauM\] (i), we obtain $$\begin{aligned} E[\norm{y(k) - x^*}^2] &\leq \frac{4(2 + \hat{m})}{\hat{m}(k+1)} \rightarrow 0,~~~k\rightarrow\infty. \label{eqn:thm:erg:conv}\end{aligned}$$ Thus, the PageRank value $x^*$ is obtained through the time average $y$. Proof of Theorem \[thm:approx:erg\] {#sec:app:C} ----------------------------------- For simplicity, let the initial time of the update scheme to be $k_0+1 =0$. Further, we write $\tilde{x}_{\Ncal}$ for $\tilde{x}_{\Ncal}(k_0)$. Denote the error between the state and the average by $e(k) := x_{\Ncal}(k) - \widetilde{x}_{\Ncal}$. Then, $$\begin{aligned} y_{\Ncal}(k) - \widetilde{x}_{\Ncal} &= \frac{1}{k+1}\sum_{\ell=0}^{k} (x_{\Ncal}(\ell) - \widetilde{x}_{\Ncal}) = \frac{1}{k+1}\sum_{\ell=0}^{k} e(\ell).\end{aligned}$$ Thus, $$\begin{aligned} E[\norm{y_{\Ncal}(k) - \widetilde{x}_{\Ncal}}^2] &= \frac{1}{(k+1)^2} \biggl\{ \sum_{\ell=0}^{k} E[e(\ell)^T e(\ell)] + 2 \sum_{\ell=0}^{k-1} \sum_{r=1}^{k-l} E[e(\ell)^T e(\ell+r)] \biggr\}. \label{eqn:prop:approx:erg:2}\end{aligned}$$ In what follows, we must evaluate the two summation terms on the right-hand side. First, we claim that $e(k)$ is uniformly bounded and in particular, for each $k\geq 0$, $$\norm{e(k)}_1 \leq \norm{x_{\Ncal}(0)}_1 + \frac{\varepsilon}{\hat{m}} + \norm{\widetilde{x}_{\Ncal}}_1 =: \gamma, \label{eqn:gamma}$$ where $\varepsilon := \max_{p\in\{0,1\}^n} \norm{ \widetilde{A}_{p, \Ncal\Ccal} \widetilde{x}_{\Ccal} + \hat{m}/n\,\one }_1$. Notice that $$\norm{e(k)}_1 = \norm{x_{\Ncal}(k) - \widetilde{x}_{\Ncal}}_1 \leq \norm{x_{\Ncal}(k)}_1 + \norm{\widetilde{x}_{\Ncal}}_1. \label{eqn:normek}$$ From the distributed update law in , it easily follows that $$x_{\Ncal}(k+1) = \widetilde{A}_{\eta(k),\Ncal\Ncal} x_{\Ncal}(k) + \widetilde{A}_{\eta(k),\Ncal\Ccal} \widetilde{x}_{\Ccal} + \frac{\hat{m}}{n} \one.$$ By definition, $\widetilde{A}_{\eta(k),\Ncal\Ncal} = (1-\hat{m})A_{\eta(k),\Ncal\Ncal}$ and $A_{\eta(k),\Ncal\Ncal}$ is a submatrix of the stochastic matrix $A_{\eta(k)}$. Consequently, we have $\norm{\widetilde{A}_{\eta(k),\Ncal\Ncal}}_1 \leq 1 - \hat{m}$. Using this bound, we obtain $$\begin{aligned} \norm{x_{\Ncal}(k+1)}_1 &\leq \bigl\| \widetilde{A}_{\eta(k),\Ncal\Ncal} \bigr\|_1\cdot \norm{x_{\Ncal}(k)}_1 + \biggl\| \widetilde{A}_{\eta(k),\Ncal\Ccal}\widetilde{x}_{\Ccal} + \frac{\hat{m}}{n}\one \biggr\|_1\\ &\leq (1 - \hat{m})\norm{x_{\Ncal}(k)}_1 + \varepsilon.\end{aligned}$$ Thus, $$\begin{aligned} \norm{x_{\Ncal}(k)}_1 &\leq (1 - \hat{m})^{k} \norm{x_{\Ncal}(0)}_1 + \varepsilon \sum_{\ell=0}^{k-1}(1 - \hat{m})^{\ell}\\ &\leq \norm{x_{\Ncal}(0)}_1 + \frac{\varepsilon}{\hat{m}}.\end{aligned}$$ Therefore, substituting this into , we have shown . Now, with the bound , the first summation term in can be upper bounded as $$\sum_{\ell=0}^{k} E[e(\ell)^T e(\ell)] \leq (k+1)\gamma^2. \label{eqn:prop:approx:erg:2a}$$ We next look at the the second summation term of . The summands can be written as $$\begin{aligned} &E[e(\ell)^T e(\ell+r)] = E[ e(\ell)^T [0~I] (x(\ell+r) - \widetilde{x})] \notag\\ &~~= E\biggl[ e(\ell)^T [0~I] \biggl( \widetilde{A}_{\eta(\ell+r-1)}\cdots \widetilde{A}_{\eta(\ell)} x(\ell) + \sum_{j=\ell}^{\ell+r-1} \widetilde{A}_{\eta(\ell+r-1)}\cdots \widetilde{A}_{\eta(j+1)} \frac{\hat{m}}{n}\tilde{s} - \widetilde{x} \biggr) \biggr].\end{aligned}$$ Here, by taking the expectation of the matrix products $\widetilde{A}_{\eta(\ell+r-1)}\cdots \widetilde{A}_{\eta(\ell+j)}$, $j=0,1,\ldots,r-1$, with respect to the random variables $\eta(\ell+r-1),\ldots,\eta(\ell+j)$, we have $$\begin{aligned} E[e(\ell)^T e(\ell+r)] &= E\biggl[ e(\ell)^T [0~I] \biggl(\hspace*{-0.4mm} E\bigl[ \widetilde{A}_{\eta(\ell+r-1)}\cdots \widetilde{A}_{\eta(\ell)} \bigr] x(l) \\ &\hspace*{2.5cm}\mbox{} + \hspace*{-0.5mm} \sum_{j=\ell}^{\ell+r-1} \hspace*{-0.5mm} E\bigl[ \widetilde{A}_{\eta(\ell+r-1)} \cdots \widetilde{A}_{\eta(j+1)} \bigr] \frac{\hat{m}}{n}\tilde{s} - \widetilde{x} \biggr)\hspace*{-0.3mm} \biggr] \notag\\ &= E\biggl[ e(\ell)^T [0~I] \biggl(\hspace*{-0.4mm} E\bigl[ \widetilde{A}_{\eta(\ell+r-1)}]\cdots E[\widetilde{A}_{\eta(\ell)} \bigr] x(\ell) \\ &\hspace*{2.5cm}\mbox{} + \hspace*{-0.4mm} \sum_{j=\ell}^{\ell+r-1} \hspace*{-0.4mm} E\bigl[ \widetilde{A}_{\eta(\ell+r-1)}] \cdots E[\widetilde{A}_{\eta(j+1)} \bigr] \frac{\hat{m}}{n}\tilde{s} - \widetilde{x} \biggr)\hspace*{-0.3mm} \biggr]\\ &= E\biggl[ e(\ell)^T [0~I] \biggl( \widehat{A}^r x(\ell) + \hspace*{-0.4mm} \sum_{j=\ell}^{\ell+r-1} \hspace*{-0.4mm} \widehat{A}^{\ell+r-1-j}\frac{\hat{m}}{n}\tilde{s} - \widetilde{x} \biggr)\hspace*{-0.3mm} \biggr],\end{aligned}$$ where the second and third equalities, respectively, follow from the independence of $\eta(\ell+r-1),\ldots,\eta(\ell+j)$ and the definition of $\widehat{A}$ in . As $\widetilde{x}$ is an equilibrium of the average system in , it can be shown that $E[e(\ell)^T e(\ell+r)] = E\bigl[ e(\ell)^T [0~I] \widehat{A}^r (x(\ell) - \widetilde{x}) \bigr]$. Here, we have $x(\ell) - \widetilde{x}=[0^T~e(\ell)^T]^T$ and use the norm relation $\norm{z}\leq \norm{z}_1$ for all $z\in\R^n$ [@HorJoh:85] to derive $$\begin{aligned} E[e(\ell)^T e(\ell+r)] &= E\biggl[ e(\ell)^T \begin{bmatrix}0 & I\end{bmatrix} \widehat{A}^r \begin{bmatrix} 0\\ e(\ell) \end{bmatrix} \biggr]\notag\\ &= E\bigl[ e(\ell)^T \widehat{A}_{\Ncal\Ncal}^r e(\ell) \bigr] \leq E\bigl[ \norm{e(\ell)}\cdot \bigl\| \widehat{A}_{\Ncal\Ncal}^r e(\ell) \bigr\| \bigr]\notag\\ &\leq E\bigl[ \norm{e(\ell)}_1 \cdot \bigl\| \widehat{A}_{\Ncal\Ncal}^r e(\ell) \bigr\|_1 \bigr] \leq E\bigl[ \norm{e(\ell)}_1^2\cdot \bigl\| \widehat{A}_{\Ncal\Ncal}^r \bigr\|_1 \bigr] \notag\\ &\leq \gamma^2 \bigl\| \widehat{A}_{\Ncal\Ncal} \bigr\|^r_1, \label{eqn:prop:approx:erg:3a}\end{aligned}$$ where in the last inequality, we used $\norm{e(\ell)}_1\leq \gamma$ from . The average matrix $\widehat{A}_{\Ncal\Ncal}$ can be bounded as $\bigl\|\widehat{A}_{\Ncal\Ncal}\bigr\|_1 \leq 1 - \hat{m}$ because $\widehat{A}_{\Ncal\Ncal}=(1-\hat{m})\overline{M}_{\Ncal\Ncal}$, and $\overline{M}_{\Ncal\Ncal}$ is a submatrix of a stochastic matrix. Thus, we arrive at $$\begin{aligned} E[e(\ell)^T e(\ell+r)] &\leq \gamma^2 \left( 1 - \hat{m} \right)^r. \label{eqn:prop:approx:erg:6}\end{aligned}$$ Finally, by substituting and into and by $\hat{m}\in(0,1)$ from Lemma \[lem:Mbar2\] (i), $$\begin{aligned} &E[\norm{y_{\Ncal}(k) - \widetilde{x}_{\Ncal}}^2]\notag\\ &~~\leq \frac{1}{(k+1)^2} \biggl\{ (k+1)\gamma^2 + 2\sum_{\ell=0}^{k-1} \sum_{r=1}^{k-\ell} \gamma^2 \left( 1 - \hat{m} \right)^r \biggr\} \notag\\ &~~\leq \frac{\gamma^2}{k+1} \biggl( 1 + 2 \frac{1-\hat{m}}{\hat{m}} \biggr) $$ The far right-hand side converges to zero as $k\rightarrow \infty$, which completes the proof. [^1]: This paper is a preliminary version of the article that appeared in the IEEE Transactions on Automatic Control, 55: 1987-2002, 2010. This work was supported in part by the Ministry of Education, Culture, Sports, Science and Technology, Japan, under KAKENHI Grant No. 21760323. [^2]: A directed graph is said to be strongly connected if for any two vertices $i,j\in\Vcal$, there is a sequence of edges which connects $i$ to $j$. In terms of the link matrix $A$, strong connectivity of the graph is equivalent to $A$ being irreducible. [^3]: In fact, in the consensus literature, it is known that the eigenvalue 1 of a row-stochastic matrix is simple if and only if the underlying graph has at least one globally reachable node; this means that there is a node from which each node in the graph can be reached via a sequence of edges (see, e.g., [@LinFraMag:05; @RenBea:05; @Moreau:05]). For our purpose, it is indeed possible to provide the column-stochastic counterpart of global reachable nodes, but the real web does not possess this property either. [^4]: In [@BriPag:98], no specific reason is given for this choice of $m$. As shown later in , however, large $m$ has the effect of faster convergence in the computation while it can also average out the PageRank values. [^5]: A nonnegative matrix $X\in\R^{n\times n}$ is said to be primitive if it is irreducible and has only one eigenvalue of maximum modulus. [^6]: In the consensus literature, problems involving cheating have been studied. An example is the Byzantine agreement problem, where among the agents there are malicious ones who send confusing information so that other agents cannot achieve consensus (see, e.g., [@TemIsh:07]). [^7]: As discussed in Section \[sec:pagerank\], this assumption can be replaced with a weaker one that a globally reachable node exists. [^8]: A matrix $X\in\R^{n\times n}$ is said to be an $M$-matrix if its off-diagonal entries are nonpositive and all eigenvalues have positive real parts.

--- abstract: 'An exact diagonalization study reveals that a matter-wave bright soliton and the Goldstone mode are simultaneously created in a quasi-one-dimensional attractive Bose-Einstein condensate by superpositions of quasi-degenerate low-lying many-body states. Upon formation of the soliton the maximum eigenvalue of the single-particle density matrix increases dramatically, indicating that a fragmented condensate converts into a single condensate as a consequence of the breaking of translation symmetry.' author: - Rina Kanamoto - Hiroki Saito - Masahito Ueda title: | Symmetry Breaking and Enhanced Condensate Fraction\ in a Matter-Wave Bright Soliton --- Fragmentation of a Bose-Einstein condensate (BEC), which occurs as a consequence of a certain exact symmetry of the system, has recently been discussed in a number of articles [@NS; @fr; @UL]. In contrast to the conventional BEC, characterized by a unique macroscopic eigenvalue in the single-particle density matrix [@PO], the fragmented BEC is characterized by more than one macroscopic eigenvalue [@NS]. If the system has an exact symmetry and if the many-body theory predicts fragmentation of the ground state, the Gross-Pitaevskii (GP) mean-field theory does not predict a fragmented condensate but approximates it with a single condensate whose symmetry is spontaneously broken. For example, a quasi-one-dimensional (1D) BEC with attractive interaction forms bright solitons [@BS-EX], which are well described by the GP theory [@BS-MF]. Efforts to elucidate how such symmetry-broken states emerge from exact many-body states have been made in diverse systems [@symmetry]. In this Letter, we show that the formation of a broken-symmetry soliton and an enhancement of the condensate fraction are caused by superpositions of the low-lying states of the symmetry-preserving many-body Hamiltonian. We find that the many-body spectrum exhibits a number of quasi-degenerate states in the regime where the exact ground state is a fragmented condensate. Superposition of these quasi-degenerate levels simultaneously generates the broken-symmetry bright soliton and the Goldstone mode, accompanied by a significant increase in the condensate fraction. By introducing a small symmetry-breaking perturbation or by considering the action of a quantum measurement, we explicitly show that the fragmented condensate is very fragile against the soliton formation. Also elucidated in the language of the many-body theory is the mechanism underlying a partial breaking of the quantized circulation in the presence of a rotating drive. We consider a system of $N$ attractive bosons with mass $m$ on a 1D ring with circumference $2\pi R$. Length and energy are measured in units of $R$ and $\hbar^2/(2mR^2)$, respectively. The Hamiltonian for our system is given by $$\begin{aligned} \label{hamiltonian} \hat{H}=\!\int_0^{2\pi}\!\!d\theta \left[-\hat{\psi}^{\dagger}(\theta)\frac{\partial^2}{\partial\theta^2}\hat{\psi}(\theta)-\frac{\pi g}{2} \hat{\psi}^{\dagger 2}(\theta){\hat{\psi}}^2(\theta)\right],\end{aligned}$$ where $\hat{\psi}(\theta)$ is the field operator, which annihilates an atom at position $\theta$, and $g$ $(>0)$ denotes the strength of attractive interaction. According to the GP mean-field approximation for the Hamiltonian (\[hamiltonian\]), the ground state is either a uniform condensate or a broken-symmetry bright soliton, depending on whether the parameter $gN$ is below or above the critical value, $gN= 1$. In contrast, all eigenstates of the original Hamiltonian are translation invariant, and many-body theory predicts that the ground state is either a [*single*]{} ($gN\lesssim 1$) or [*fragmented*]{} ($gN\gtrsim 1$) condensate [@QPT]. ![ (a) Excitation spectrum obtained by exact diagonalization of Hamiltonian (1) for $N=200$. The inset shows the corresponding result obtained with truncation $l_{\rm c}=2$ near the critical point. (b) Bogoliubov spectrum corresponding to (a), where branch $A'$ represents the Goldstone mode, $B'$ the breathing mode of a bright soliton, and $C'$ the second harmonic of $B'$. (c) Energy gap $\Delta E$ between the ground and the first excited states in the many-body spectrum versus the total number of atoms $N$ with $gN=1.4$ held fixed. Triangles and circles denote results obtained with $l_{\rm c}=1$ and $l_{\rm c}=2$, respectively. []{data-label="fig1"}](fig1.eps) Figure \[fig1\] (a) shows the low-lying spectrum obtained by exact diagonalization of the Hamiltonian (\[hamiltonian\]). The dramatic change in the landscape of the energy spectrum around $gN \simeq 1$ is a consequence of the quantum phase transition between a single condensate and a fragmented one. Figure \[fig1\] (b) presents the Bogoliubov spectrum obtained from the Bogoliubov-de Gennes equations. By comparing Figs. 1(a) and (b), we find that the Bogoliubov spectrum has a one-to-one correspondence with the many-body spectrum for $gN \lesssim 1$. For $gN\gtrsim 1$, however, the many-body spectrum becomes much more intricate than the Bogoliubov one. In the Bogoliubov spectrum for $gN \gtrsim 1$, there appears a Goldstone mode $A'$ (the translation mode of the soliton) associated with the symmetry breaking of the ground state, the breathing mode $B'$, and the second harmonic of the breathing mode $C'$. In Fig. \[fig1\] (a) for $gN\gtrsim 1$, in contrast, a number of quasi-degenerate levels appear with the density of states peaking around the Bogoliubov levels; we denote the corresponding groups as $A, B$, and $C$, respectively. The basis states for the diagonalization are restricted to the angular-momentum states $l=0,\pm 1$ ($l_{\rm c}=1$) unless otherwise stated, and the field operator is given by ${\hat \psi}(\theta)=({\hat c}_0+e^{i\theta}{\hat c}_1+e^{-i\theta}{\hat c}_{-1})/\sqrt{2\pi}$, where ${\hat c}_l$ is the annihilation operator of a boson with angular momentum $\hbar l$. The validity of this cutoff has been confirmed by the inclusion of higher angular-momentum states as shown in the inset of Fig. \[fig1\] (a) with $l_{\rm c}=2$ [@NP], where the energy landscape and the degree of degeneracy are unchanged from those of $l_{\rm c}=1$. We denote the eigenstates of the Hamiltonian as $|{\cal L}\rangle_{\sigma}$, where ${\cal L}$ is the total angular momentum. The index $\sigma=A,B,\cdots$ labels the eigenstate in the ascending order of energy ($E_{|{\cal L}\rangle_A} < E_{|{\cal L}\rangle_B} < \cdots$) for each ${\cal L}$. The states $|{\cal L}\rangle_{\sigma}$ and $|-{\cal L}\rangle_{\sigma}$ are degenerate in the absence of a rotating drive. At $gN=0$, the eigenstates are the Fock states $|{\cal L}\rangle_{\sigma}=|n_1,n_{-1}\rangle$, where $n_{\pm 1}$ denote the numbers of atoms with angular momenta $l=\pm 1$ and ${\cal L}=n_1-n_{-1}$. The energies of the states $|n_1,n_{-1}\rangle$ are given by $n_1+n_{-1}\equiv j$, and these states are thus $(j+1)$-fold degenerate. The index $\sigma=A,B, \cdots$ corresponds to the number of $l=\pm 1$ pairs being given by $(j-|{\cal L}|)/2=0,1,\cdots$. For $0< gN \lesssim 1$, the energy branches are characterized by $j$, and the $(j+1)$-fold degeneracy is almost maintained. As $gN$ approaches 1, each branch begins to ramify, and the energy landscape for $gN \gtrsim 1$ is characterized by the index $\sigma$ and ${\cal L}$ as $E_{|0\rangle_{\sigma}}\lesssim E_{|\pm 1\rangle_{\sigma}} \lesssim E_{|\pm 2\rangle_{\sigma}}\lesssim\cdots$. There is no Goldstone mode because the ground state possesses the translation symmetry, and the lowest excited states $|\pm 1\rangle_A$ have a finite energy gap $\Delta E\equiv E_{|\pm 1\rangle_A}-E_{|0\rangle_A}$, since the system is finite. However, the density of states above the ground state becomes higher for larger $N$, and the gap $\Delta E$ collapses as $1/N$ \[Fig. \[fig1\] (c)\]. The ground state is therefore unstable against excitations of the quasi-degenerate low-lying states. We construct the many-body counterparts of the bright soliton $|\Psi_{\theta}\rangle$ and the Goldstone mode $|\Phi_{\theta}\rangle$, such that $\langle\Psi_{\theta}|\Phi_{\theta}\rangle=0$ by superpositions of the ground and quasi-degenerate states: $$\begin{aligned} |\Psi_{\theta}\rangle \!\!\!&\equiv&\!\!\! e^{-i{\hat L}\theta} \left[\beta_0|0\rangle_A+\sum_{{\cal L}>0} \beta_{\cal L}\left( |{\cal L}\rangle_A + |-{\cal L}\rangle_A\right)\right],\label{soliton}\\ |\Phi_{\theta} \rangle \!\!\!&\equiv&\!\!\! \frac{d}{d\theta}|\Psi_{\theta}\rangle \!\!=\!\!-ie^{-i{\hat L}\theta}\sum_{{\cal L}>0}{\cal L} \beta_{\cal L}\left(|{\cal L}\rangle_A-|-{\cal L}\rangle_A\right)\label{Goldstone}, \end{aligned}$$ where $\hat{L}\equiv \int d\theta \hat{\psi}^{\dagger}(\theta)(-i\partial_{\theta})\hat{\psi}(\theta)$ is the angular momentum operator, and $\beta_{\cal L}$’s satisfy $\sum_{\cal L} |\beta_{\cal L}|^2=1$. The energy cost associated with the superposition (\[soliton\]), $E_{|\Psi_{\theta}\rangle}-E_{|0\rangle_A} =\sum_{{\cal L}}|\beta_{\cal L}|^2\! \left(E_{|{\cal L}\rangle_A}-E_{|0\rangle_A}\right)$, is on the order of $1/N$. This indicates that the symmetry breaking from the exact ground state $|0\rangle_A$ to the bright soliton $|\Psi_{\theta}\rangle$ costs little energy, and the superposition thus occurs by an “infinitesimal” perturbation on the order of $1/N$. The emergence of quasi-degenerate levels also plays a crucial role in breaking the quantized circulation. In the presence of a rotating drive with angular frequency $2\Omega$, the many-body ground state is either a single or fragmented condensate, depending on whether $f(g,N,\Omega)\equiv (1-gN)/2-2(\Omega-[\Omega+1/2])^2$ is positive or negative [@rotation], where $[\Omega+1/2]$ denotes the maximum integer that does not exceed $\Omega+1/2$. Figure \[fig2\] shows the total angular momentum of the ground state ${\cal L}_{\rm g}$ and low-lying eigenvalues of the Hamiltonian ${\hat H}-2\Omega\hat{L}+\Omega^2$ in the rotating frame. There clearly appear two distinct regimes where the density of states of excitations is low ($f>0$) and high ($f<0$) in the spectrum, and the circulation $h{\cal L}_g/m$ is quantized only when $f>0$. When the density of states above the ground state is sparse, the ground state cannot make a transition to states with higher angular momenta even if $\Omega$ is increased, since the term $-2\Omega{\hat L}$ is not large enough to make up for the excitation energy due to a large energy gap. The total angular momentum ${\cal L}_{\rm g}$ therefore does not increase with $\Omega$, and is quantized at integral multiples of $N$ for $f>0$. However, a number of branches with higher angular momenta decrease in energy as $f$ becomes negative; then, as $\Omega$ is increased, one branch after another takes the place of the ground level upon intersection \[Fig. \[fig2\] (b)\]. The angular momentum of the ground state ${\cal L}_{\rm g}$ thus increases stepwise each time the excitation energy collapses. The interval between collapses of the excitation energy, i.e., the width of the steps of ${\cal L}_{\rm g}$, becomes narrower as $N$ becomes larger, eventually resulting in the breaking of the quantized circulation, as indicated by the slopes in the thick line in Fig. \[fig2\] (a). The emergence of quasi-degenerate levels also induces symmetry breaking in a manner similar to Eq. (\[soliton\]). In fact, the ground state in the GP theory is a localized soliton for $f<0$ [@rotation]. ![ Low-lying spectrum (thin lines, left scale) and expectation value of the total angular momentum (thick line, right scale) of the ground state versus the angular frequency of the rotating drive for $g=2\times 10^{-3}$ and $N=100$. (b) is an enlargement of (a). []{data-label="fig2"}](fig2.eps) We next investigate the effect of a symmetry-breaking perturbation by diagonalizing the Hamiltonian $\hat{H}+\varepsilon \hat{V}$ where $\hat{H}$ is given by Eq. (\[hamiltonian\]) and $\hat{V}=\int d\theta \hat{\psi}^{\dagger}(\theta)\cos\theta\hat{\psi}(\theta)$. As the perturbation is switched on, the ground state begins to localize by superposing the quasi-degenerate states. The largest eigenvalue $\lambda_{\rm M}^{\rm (\varepsilon)}$ of the single-particle density matrix is plotted as a function of $\varepsilon N^2$ in Fig. \[fig3\]. The fragmented condensate approaches the single condensate as $\varepsilon N^2$ increases. We note that $\lambda_{\rm M}^{(\varepsilon)}$ depends only on $\varepsilon N^2$, which indicates that a single condensate is realized for $\varepsilon \gtrsim N^{-2}$. The fragmented condensate with $N\gg 1$ is hence very fragile against the symmetry-breaking perturbation. The condensate fraction $\lambda_{\rm M}^{(\varepsilon)}$ and the distribution of $\beta_{\cal L}$ in Eq. (\[soliton\]) can be derived analytically with the Bogoliubov approximation [@derivation]. Since the frequency of the translation mode of the broken-symmetry soliton is proportional to $\varepsilon^{1/2}$ and small, the number of excited atoms in this mode is much larger than in the other modes. We thus consider only the translation mode, and obtain the depletion as $(2N)^{-1} \varepsilon^{-1/2} [(2F)^{-1/2} - \varepsilon^{1/2} + O(\varepsilon)]\equiv 1-\lambda_{\rm B}^{(\varepsilon)}$, where $F \equiv 7 (3gN+4)^{1/2} / [(5gN+2)(2gN-2)^{1/2}]$ is a decreasing function for $gN > 1$. The behavior of $\lambda_{\rm B}^{(\varepsilon)}$ is in excellent agreement with the numerical result as shown in Fig. \[fig3\]. Assuming that $\beta_{\cal L} |{\cal L}\rangle_A$ in Eq. (\[soliton\]) corresponds to the projection of the Bogoliubov ground state on the angular momentum ${\cal L}$, i.e., $\beta_{\cal L}|{\cal L}\rangle_A \simeq\int\frac{d\theta}{2\pi} e^{i ({\cal L} - \hat{L}) \theta} |\Psi_{\rm B}\rangle$, we obtain the distribution of $\beta_{\cal L}$ as $$\begin{aligned} |\beta_{\cal L}|^2\simeq\frac{1}{\sqrt{2 \pi d^2}}\exp\left[-\frac{{\cal L}^2}{2 d^2}\right], \qquad\qquad\qquad\label{Gaussian}\\ d^2=N \varepsilon^{1/2} \frac{2(gN-1)}{7gN} \left[(2F)^{1/2} + F \varepsilon^{1/2} + O(\varepsilon)\right]\label{widthP}, \end{aligned}$$ which is in excellent agreement with the numerical results as shown in the inset of Fig. \[fig3\]. The behavior of the width $d\propto N^{1/2} \varepsilon^{1/4}$ indicates that the center-of-mass fluctuation is proportional to $N^{-1/2} \varepsilon^{-1/4}$. ![ The largest eigenvalue of the reduced single-particle density matrix obtained by the exact diagonalization ($\lambda_{\rm M}^{(\varepsilon)}$) and Bogoliubov theory ($\lambda_{\rm B}^{(\varepsilon)}$) versus $\varepsilon N^2$ for $gN=1.4$. Inset: Distribution of $|\beta_{\cal L}|^2=|{}_A\langle {\cal L} | \Psi(\varepsilon)\rangle|^2$ with $\varepsilon=1\times 10^{-4}$. The solid curves depict Eq. (\[Gaussian\]) with Eq. (\[widthP\]). []{data-label="fig3"}](fig3.eps) ![ Ensemble-averaged largest eigenvalue $\bar{\lambda}_{\rm M}^{(j)}$ of the reduced single-particle density matrix versus the number of quantum measurements $j$ for $gN_{\rm init}=1.6$. The solid curve denotes the analytical result (\[anaM\]). Inset: Ensemble-averaged value of $|\beta_{\cal L}|^2=|{}_A\langle {\cal L} | \Psi^{(j)}\rangle|^2$ after $j=5,20$ and $50$-time measurements. The solid curves depict Eq. (\[Gaussian\]) with $d^2 = 4 j (gN_{\rm init}-1)/(7gN_{\rm init})$. []{data-label="fig4"}](fig4.eps) Finally, we investigate what happens to the exact ground state under the action of a quantum measurement [@PD]. The ground state $|0\rangle_A$ is prepared as an initial state with the number of atoms $N_{\rm init}$ so that $gN_{\rm init} > 1$. Suppose that one atom is detected at position $\theta_j$ in the $j$-th measurement. The postmeasurement state $|\Psi^{(j)} \rangle$ is related to the premeasurement one $|\Psi^{(j-1)} \rangle$ by $|\Psi^{(j)}\rangle = \hat{\psi}(\theta _j)|\Psi^{(j-1)}\rangle/\sqrt{\langle \Psi^{(j-1)}|\hat{\psi}^{\dagger}(\theta_j)\hat{\psi}(\theta_j)|\Psi^{(j-1)}\rangle}$. The normalized single-particle density is given by $n^{(j)}(\theta)=\langle\Psi^{(j)}|\hat{\psi}^{\dagger}(\theta) \hat{\psi}(\theta)|\Psi^{(j)}\rangle/N_j$, and the position of the $(j+1)$-th measurement $\theta_{j+1}$ is probabilistically determined after the $j$-th measurement according to the probability distribution $n^{(j)}(\theta)$. Since each run is a stochastic process, we numerically perform sequential runs of independent simulations and take the ensemble average for each $j$. We find that the ensemble-averaged value of the condensate fraction $\bar{\lambda}_{\rm M}^{(j)}$ is independent of $N_{\rm init}$ for a fixed $gN_{\rm init}$, and that it monotonically increases as shown in Fig. \[fig4\]. Therefore, when $N_{\rm init}\gg 1$, $\bar{\lambda}_{\rm M}^{(j)}$ reaches the order of one for $j/N_{\rm init} \ll 1$, indicating that the fragmented condensate rapidly becomes a single condensate by the action of quantum measurements. The single-particle density $n^{(j)}(\theta)$ localizes and the state $|\Psi_{\theta}^{(j)}\rangle$ reaches the form of Eq. (\[soliton\]). Suppose that the initial state is a uniform superposition of the soliton state $\int d \theta_0 A(\theta_0) \phi_{\rm sol}^{\rm GP}(\theta-\theta_0)$, where $\theta_0$ is the center of mass and $A(\theta_0)$ is a constant for $j=0$. We can show [@derivation] that the probability distribution is changed by the $j$-time measurements as $A^2_j(\theta_0) \propto \exp (-4 c j \theta_0^2)$ with $c\equiv 2(gN_{\rm init}-1)/(7gN_{\rm init})$, and hence the distribution of $|\beta_{\cal L}|^2$ reduces to the Gaussian form (\[Gaussian\]) with $d^2 = 2 c j$, i.e., $d\propto \sqrt{j}$ for $j \lesssim \sqrt{N_{\rm init}}$. The center-of-mass fluctuation thus reduces to $\Delta \theta_0 \propto j^{-1/2}$ by the $j$-time measurements. The condensate fraction can also be derived analytically [@derivation] as $$\begin{aligned} \label{anaM} \lambda_{\rm M}^{(j)}&=&\frac{1}{2} \left[1-c+c e^{-2\alpha}\right.\nonumber\\ &+&\!\!\!\!\!\left.\sqrt{(1-c+c e^{-2\alpha})^2-4c(1-2c)(1-e^{-\alpha})^2}\right], \end{aligned}$$ where $\alpha=1/(8cj)$. These results are in excellent agreement with the numerical ones as shown in Fig. \[fig4\]. In conclusion, we employed the exact diagonalization method to investigate the simultaneous emergence of a bright soliton and the Goldstone mode in a 1D attractive BEC. We found that the existence of a number of quasi-degenerate states above a critical strength of attractive interaction makes the ground state fragile against symmetry-breaking perturbations. By introducing a symmetry-breaking perturbation or quantum measurements, we showed that a localized state with an enhanced condensate fraction is generated by a superposition of quasi-degenerate many-body states, and that the resulting state closely resembles the Bogoliubov ground state. The mechanism underlying a partial breaking of the quantized circulation is also elucidated in terms of the many-body spectrum. This work was supported by a 21st Century COE program at Tokyo Tech “Nanometer-Scale Quantum Physics”, Special Coordination Funds for Promoting Science and Technology, and a Grant-in-Aid for Scientific Research (Grant No. 15340129) by the Ministry of Education, Culture, Sports, Science and Technology. [30]{} P. Nozières and D. Saint James, J. Phys. (France) [**43**]{}, 1133 (1982). N.K. Wilkin, J.M.F. Gunn, and R.A. Smith, Phys. Rev. Lett. [**80**]{}, 2265 (1998); D.S. Rokhsar, e-print cond-mat/9812260; C.K. Law [*et al*]{}., Phys. Rev. Lett. [**81**]{}, 5257 (1998); E.J. Mueller, T.L. Ho, G. Baym, and M. Ueda, unpublished; M. Koashi and M. Ueda, Phys. Rev. Lett. [**84**]{}, 1066 (2000); T.-L. Ho and S.-K. Yip, Phys. Rev. Lett. [**84**]{}, 4031 (2000); S. Ashhab and A.J. Leggett, Phys. Rev. A [**68**]{}, 063612 (2003). M. Ueda and A.J. Leggett, Phys. Rev. Lett. [**83**]{}, 1489 (1999). O. Penrose and L. Onsager, Phys. Rev. [**104**]{}, 576 (1956). K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, Nature (London) [**417**]{}, 150 (2002); L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, C. Salomon, Science [**296**]{}, 1290 (2002). U. Al Khawaja, H.T.C. Stoof, R.G. Hulet, K.E. Strecker, and G.B. Partridge, Phys. Rev. Lett. [**89**]{}, 200404 (2002); L.D. Carr and J. Brand, Phys. Rev. Lett. [**92**]{}, 040401 (2004). V.E. Zakharov and A.B. Shabat, Zh. Eksp. Teor. Fiz. [**61**]{}, 118 (1971) \[Sov. Phys. JETP [**34**]{}, 62 (1972)\]. B. Bernu, C. Lhuillier, and L. Pierre, Phys. Rev. Lett. [**69**]{}, 17 (1992); E.H. Rezayi, F.D.M. Haldane, and K. Yang, Phys. Rev. Lett. [**83**]{}, 1219 (1999); Y. Castin, e-print cond-mat/0012040; N.R. Cooper, N.K. Wilkin, and J.M.F. Gunn, Phys. Rev. Lett. [**87**]{}, 120405 (2001); T. Nakajima and M. Ueda, Phys. Rev. Lett. [**91**]{}, 140401 (2003). R. Kanamoto, H. Saito, and M. Ueda, Phys. Rev. A [**67**]{}, 013608 (2003). We employed the Lanczos algorithm in TITPACK Ver. 2 developed by H. Nishimori. R. Kanamoto, H. Saito, and M. Ueda, Phys. Rev. A [**68**]{}, 043619 (2003); P.F. Kartsev, Phys. Rev. A [**68**]{}, 063613 (2003). Details of the derivations will be reported elsewhere. J. Javanainen and S.M. Yoo, Phys. Rev. Lett. [**76**]{}, 161 (1996); Y. Castin and J. Dalibard, Phys. Rev. A [**55**]{}, 4330 (1997).

--- abstract: 'We study the harvesting of entanglement and mutual information by Unruh-DeWitt particle detectors from thermal and squeezed coherent field states. We prove (for arbitrary spatial dimensions, switching profiles and detector smearings) that while the entanglement harvesting ability of detectors decreases monotonically with the field temperature $T$, harvested mutual information grows linearly with $T$. We also show that entanglement harvesting from a general squeezed coherent state is independent of the coherent amplitude, but depends strongly on the squeezing amplitude. Moreover, we find that highly squeezed states i) allow for detectors to harvest much more entanglement than from the vacuum, and ii) ensure that the entanglement harvested does not decay with their spatial separation. Finally we analyze the spatial inhomogeneity of squeezed states and its influence on harvesting, and investigate how much entanglement one can actually extract from squeezed states when the squeezing is bandlimited.' author: - Petar Simidzija - 'Eduardo Martín-Martínez' bibliography: - 'references.bib' title: Harvesting correlations from thermal and squeezed coherent states --- Introduction {#sec:intro} ============ The entanglement structure of a quantum field has been an important area of research over the last few decades. Besides being an interesting focus of study in its own right, the presence of entanglement between local degrees of freedom in general field states (and in particular the vacuum [@Summers1985; @Summers1987]) has been used as a means to better understand important fundamental questions, from the black hole information loss problem [@Hawking1975; @Hawking1976; @Susskind1993; @Almheiri2013; @Braunstein2013; @Hawking2016], to the dynamics of quantum phase transitions in statistical mechanics [@Vidal2002; @Calabrese2004]. Moreover, operational approaches which harness this entanglement to perform useful tasks have also been studied, leading to, for example, the development of protocols for *quantum energy teleportation* [@Hotta2008; @Hotta2009; @Frey2014]. Another widely studied protocol making use of the entanglement present in a quantum field is concerned with the extraction of field entanglement onto a pair of initially uncorrelated first-quantized systems (detectors). These so called *entanglement harvesting* protocols were initially studied in the 90s by Valentini [@Valentini1991], then later by Reznik *et al.* [@Reznik2003; @Reznik2005], and have in the last decade or so experienced a great deal of attention from many different perspectives [@Steeg2009; @Brown2013a; @Martinez2013a; @Brown2014; @Drago2014; @Pozas2015; @Salton2015; @Pozas2016; @Martinez2016a; @Nambu; @Sachs2017; @Guillaume2017; @Sachs2018; @Trevison2018]. Many of these recent lines of research into entanglement harvesting are related to the fact that the amount of harvestable entanglement is generally sensitive to the many variable parameters of the setup. For instance, the sensitivity of entanglement harvesting on the position and motion of the detectors has resulted in harvesting-based proposals in metrology — from rangefiding [@Salton2015] to precise vibration detection [@Brown2014] — while, on the more fundamental side, it has also been shown that entanglement harvesting is sensitive to the geometry [@Steeg2009] and topology [@Martinez2016a] of the background spacetime. Furthermore, while most of these entanglement harvesting studies have focused on conventional linear Unruh-DeWitt (UDW) particle detectors [@DeWitt1979] coupled to real scalar fields [@Pozas2015], there have also been several interesting results coming from other variations of the setup. Some examples include: hydrogenoid atomic detectors coupled to the full electromagnetic field [@Pozas2016], non-linear couplings of UDW detectors to neutral [@Sachs2017] and charged [@Sachs2018] scalar fields, tripartite entanglement in flat spacetime [@Drago2014], and multiple detector harvesting in curved spacetimes [@Nambu]. Entanglement harvesting using infinite dimensional harmonic oscillator detectors has been looked at in several works as well. An example which is very relevant to this paper is an article by Brown where the issue of harvesting from thermal states is considered [@Brown2013a]. While some of the above mentioned parameters affecting entanglement harvesting are difficult to control in a lab setting (such as the geometry and topology of spacetime), other parameters, such as the energy gap of the detectors or the state of the field, are more easily tunable. A major motivation for studying the sensitivity of entanglement harvesting to these types of parameters is that it may lead to experimental realizations of entanglement harvesting protocols. This would not only be an important achievement from a fundamental perspective, but it could also potentially be a method of obtaining entanglement that could then be used for quantum information purposes [@Martinez2013a]. With this ultimate motivation in mind, it has been shown that a non-zero detector energy gap is crucial in protecting an entanglement harvesting UDW pair against fluctuation induced, entanglement harming, local noise [@Pozas2017; @Simidzija2018]. Furthermore, for harmonic oscillator detectors, this noise has been found to increase with field temperature, leading to detrimental effects on the amount of entanglement harvested [@Brown2013a] by oscillator pairs. Meanwhile, and perhaps surprisingly, for UDW detectors interacting with coherent states of the field, the presence of leading order local noise does not end up affecting the amount of entanglement that can be harvested from the field [@Simidzija2017b; @Simidzija2017c]. In this paper, we fill in significant gaps in the study of entanglement harvesting sensitivity on thermal and general squeezed coherent field states. While, to our knowledge, this is the first study of squeezed state entanglement harvesting, we would also like to point out that our study of thermal state harvesting differs in several crucial regards to the previous work in [@Brown2013a]. In [@Brown2013a] it was shown that for a pair of pointlike oscillator detectors interacting with a massless field in a one-dimensional cavity, the amount of entanglement extracted decays rapidly with the temperature. In contrast, i) we consider spatially smeared qubit detectors interacting with a field of any mass in a spacetime of any dimensionality, rather than pointlike oscillator detectors interacting with a massless field in (1+1)-dimensions, ii) we look at the continuum free space case rather than being in a cavity, and hence we are not forced to introduce any UV cutoffs to handle numerical sums, and iii) we directly compute the evolved detectors’ density matrix from the field’s one and two-point functions, rather than using the significantly different formalism of Gaussian quantum mechanics (see, e.g. [@Adesso2007]). Despite these significant differences between our approach and that in [@Brown2013a], we will find that, for thermal states, our results are in qualitative agreement with their general conclusions, i.e. that temperature is detrimental to entanglement harvesting. However, since we obtain analytical expressions for entanglement measures, rather than being restricted to numerical calculations, we are able to provide an explicit proof that the amount of entanglement that (qubit) detectors can harvest from the field rapidly decays with its temperature. In particular, we will show that the optimal thermal state for harvesting entanglement from the field is the vacuum. On the other hand, we will see that this is not the case for the harvesting of mutual information, which is a measure of the total (quantum and classical) correlations of the detector pair. In fact we will see that for high field temperatures $T$ (while still in the perturbative regime) the mutual information harvested by the detectors *increases* proportionally with $T$. We will then consider the case of squeezed coherent states [@Loudon1987], where, to the authors’ knowledge, no previous literature exists. We will first prove that the statement “entanglement harvesting is independent of the field’s coherent amplitude" is true not only for non-squeezed coherent states, as was shown in [@Simidzija2017b], but also for arbitrarily squeezed coherent states. On the other hand we will show that, unlike the coherent amplitude, the choice of field’s squeezing amplitude $\zeta(\bm k)$ does in fact affect the ability of UDW detectors to become entangled, and moreover the Fourier transform of $\zeta(\bm k)$ directly gives the locations in space near which entanglement harvesting is optimal. Perhaps surprisingly, we will also find that for highly and uniformly squeezed field states, the amount of entanglement that the detectors can harvest is independent of their spatial separation, and is often much higher than the amount obtainable from the vacuum. We will also analyze whether this advantage carries over to more experimentally attainable field configurations where states are squeezed across a narrow frequency range of field modes. This paper is structured as follows: We begin in Sec. \[sec:setup\] by reviewing the setup of entanglement harvesting by UDW detectors from arbitrary states of a scalar field. In Sec. \[sec:thermal\] we particularize to the case of thermal field states, and study the harvesting of entanglement and mutual information in this setting. Then, in Sec. \[sec:squeezed\] we look at entanglement harvesting from squeezed field states, both those with uniform and bandlimited squeezing amplitudes. Finally, Sec. \[sec:conclusions\] is left for the conclusions. Units of $\hbar=c=k_\textsc{b}=1$ are used throughout. Correlation harvesting setup {#sec:setup} ============================ Before studying the harvesting of correlations from thermal and squeezed coherent field states, let us review the general correlation harvesting setup that can be found in extensive literature (see, e.g. [@Martinez2015] and references therein) and that is applicable to any field state. We start with a free Klein-Gordon field $\hat\phi$ in $(n+1)$-dimensional Minkowski spacetime, which can be expressed in a basis of plane wave modes as $$\label{eq:field} \hat{\phi}(\bm{x},t) = \int\frac{{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm{k}\, }}{\sqrt{2(2\pi)^n {\omega_{\bm{k}}}}}\left[{\hat{a}_{\bm{k}}^\dagger} e^{{\mathrm{i}}({\omega_{\bm{k}}} t-\bm{k}\cdot\bm{x})}+\text{H.c.}\right],$$ where ${\omega_{\bm{k}}}:=\sqrt{|\bm k|^2+m^2}$, and the creation and annihilation operators, ${\hat{a}_{\bm{k}}^\dagger}$ and ${\hat{a}_{\bm{k}}}$, satisfy the canonical commutation relations $$\label{eq:CCR} [{\hat{a}_{\bm{k}}},{\hat{a}_{\bm{k'}}}]= [{\hat{a}_{\bm{k}}^\dagger},{\hat{a}_{\bm{k'}}^\dagger}]=0, \quad [{\hat{a}_{\bm{k}}},{\hat{a}_{\bm{k'}}^\dagger}]=\delta^{(n)}(\bm{k}-\bm{k'}).$$ We denote by ${| {0} \rangle}$ the ground state of the field, by which we mean the state annihilated by all the ${\hat{a}_{\bm{k}}}$ operators. For now, let us suppose that the field is in an arbitrary (potentially mixed) state ${\hat{\rho}_\phi}$. We will later particularize to the case of thermal and squeezed coherent states. Next we consider the pair of first-quantized particle detectors that couple to the field with the aim of extracting (i.e. *harvesting*) entanglement. We will model the detectors (labeled $\nu\in\{\text{A},\text{B}\}$) as two-level quantum systems, with ground states ${| {g_\nu} \rangle}$, excited states ${| {g_\nu} \rangle}$, and proper energy gaps $\Omega_\nu$. We assume that the detectors are at rest at positions $\bm x_\nu$, that they have spatial profiles given by the *smearing functions* $F_\nu(\bm x)$, and that they are initially (i.e. prior to interacting with the field) in the separable state ${\hat{\rho}_\textsc{a}}\otimes{\hat{\rho}_\textsc{b}}$. Then, we describe the interaction of the detectors and the field using the Unruh-DeWitt (UDW) model [@DeWitt1979], which is a successful model of the light-matter interaction when angular momentum exchange can be neglected [@Pozas2016; @Pablo]. In this model the coupling of detectors to field is given by the interaction picture interaction Hamiltonian, ${\hat H_\textsc{i}}(t) = \hat H_\textsc{i,a}(t)+\hat H_\textsc{i,b}(t)$, where $$\label{eq:H_nu} \hat{H}_{\textsc{i},\nu}(t) := \lambda_\nu \chi_\nu(t) \hat{\mu}_\nu(t) \int { \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm{x}\, } F_\nu(\bm{x}-\bm{x}_\nu) \hat{\phi}(\bm{x},t).$$ Here, $\lambda_\nu$ is the coupling strength of detector $\nu$ to the field, $\chi_\nu(t)$ is the time-dependent *switching function* which models the duration of the interaction and how the detector $\nu$ is turned on and off, and the $\hat \mu_\nu(t)$ are operators on the two detector Hilbert space given by $ \hat{\mu}_\textsc{a}(t) := \hat{m}_\textsc{a}(t)\otimes\mathds{1}_\textsc{b}$, and $ \hat{\mu}_\textsc{b}(t) := \mathds{1}_\textsc{a}\otimes\hat{m}_\textsc{b}(t)$, where $\hat{m}_\nu(t)$ is the interaction picture monopole moment of detector $\nu$: $$\label{eq:m_nu} \hat{m}_\nu(t)= {| {e_\nu} \rangle}{\langle {g_\nu} |} e^{{\mathrm{i}}\Omega_\nu t}+ {| {g_\nu} \rangle}{\langle {e_\nu} |} e^{-{\mathrm{i}}\Omega_\nu t}.$$ To determine how entangled (if at all) the detectors are following their interactions with the field, we calculate the time-evolved two-detector state ${\hat{\rho}_\textsc{ab}}$ as $$\label{eq:rhoab1} {\hat{\rho}_\textsc{ab}}:={\text{Tr}}_\phi\left[\hat U\left({\hat{\rho}_\textsc{a}}\otimes{\hat{\rho}_\textsc{b}}\otimes{\hat{\rho}_\phi}\right)\hat U^\dagger\right],$$ where the time-evolution unitary $\hat U$ is formally given by $$\label{eq:U} \hat{U} = \mathcal{T}\exp\left[{-{\mathrm{i}}\int_{-\infty}^{\infty}\!\!\!\dif t\, {\hat H_\textsc{i}}(t)}\right],$$ with $\mathcal T$ denoting the time-ordering operation. By assuming that the detector-field coupling constants $\lambda_\nu$ — which have units of $(\text{length})^{(n-3)/2}$ in $(n+1)$-dimensional spacetime — are small compared to other scales with the same units in the setup, we can expand $\hat U$ in powers of $\lambda_\nu$, obtaining $$\label{eq:Dyson} \hat{U}={\mathds{1}}\! \underbrace{-{\mathrm{i}}\!\int_{-\infty}^\infty\!\!\!\!\!\dif t {\hat H_\textsc{i}}(t)} _{\hat{U}^{(1)}} \underbrace{-\!\!\int_{-\infty}^{\infty}\!\!\!\!\!\dif t\! \int_{-\infty}^t\!\!\!\!\!\dif t' {\hat H_\textsc{i}}(t)\hat{H}_\textsc{i,a}(t')}_{\hat{U}^{(2)}} +\mathcal{O}(\lambda_\nu^3).$$ Then, the final two-detector state ${\hat{\rho}_\textsc{ab}}$ in Eq.  can be perturbatively expressed as $$\label{eq:rhoab_general} {\hat{\rho}_\textsc{ab}}= \hat{\rho}_\textsc{ab}^{(0)}+ \hat{\rho}_\textsc{ab}^{(1)}+ \hat{\rho}_\textsc{ab}^{(2)}+ \mathcal{O}(\lambda_\nu^3),$$ where $$\begin{aligned} \hat{\rho}_\textsc{ab}^{(0)}&:= {\hat{\rho}_\textsc{a}}\otimes{\hat{\rho}_\textsc{b}}\otimes{\hat{\rho}_\phi}, \\ \hat{\rho}_\textsc{ab}^{(1)}&:= {\text{Tr}}_\phi\left(\hat{U}^{(1)}\hat{\rho}_0 +\hat{\rho}_0 \hat{U}^{(1)\dagger}\right), \\ \hat{\rho}_\textsc{ab}^{(2)}&:= {\text{Tr}}_\phi\left(\hat{U}^{(2)}\hat{\rho}_0 +\hat{U}^{(1)} \hat{\rho}_0 \hat{U}^{(1)\dagger} +\hat{\rho}_0 \hat{U}^{(2)\dagger}\right).\end{aligned}$$ By using the definitions of $\hat U^{(1)}$ and $\hat U^{(2)}$ in Eq.  and the expression for ${\hat H_\textsc{i}}$ given by Eq. , it is straightforward to show that $\hat{\rho}_\textsc{ab}^{(1)}$ and $\hat{\rho}_\textsc{ab}^{(2)}$ take the forms $$\begin{aligned} \label{eq:rhoab^1} \hat{\rho}_{\textsc{ab}}^{(1)}&= {\mathrm{i}}\!\!\!\!\sum_{\nu\in\{\text{A,B}\}}\!\!\!\!\lambda_\nu \int_{-\infty}^{\infty}\!\!\!\!\!\dif t\chi_\nu(t) [\hat{\rho}_{\textsc{ab}}^{(0)},\hat{\mu}_\nu(t)]V(\bm{x}_\nu,t), \\ \label{eq:rhoab^2} \hat{\rho}_{\textsc{ab}}^{(2)}&= \!\!\!\!\sum_{\nu,\eta\in\{\text{A,B}\}}\!\!\!\! \lambda_\nu\lambda_\eta \Bigg[\int_{-\infty}^{\infty}\!\!\!\!\!\dif t \int_{-\infty}^{\infty}\!\!\!\!\!\dif t' \chi_\nu(t')\chi_\eta(t) \notag\\ &\phantom{{}= \!\!\!\!\sum_{\nu,\eta\in\{\textsc{a,b}\}}\!\!\!\! \lambda_\nu} \times\hat{\mu}_\nu(t') \hat{\rho}_{\textsc{ab}}^{(0)}\hat{\mu}_\eta(t) W(\bm{x}_\eta,t,\bm{x}_\nu,t')\notag\\ &\phantom{{}=\,\,} -\int_{-\infty}^{\infty}\!\!\!\!\!\dif t \int_{-\infty}^{t}\!\!\!\!\!\dif t' \chi_\nu(t)\chi_\eta(t') \notag\\ &\phantom{{}= \!\!\!\!\sum_{\nu,\eta\in\{\textsc{a,b}\}}\!\!\!\! \lambda_\nu} \times\hat{\mu}_\nu(t)\hat{\mu}_\eta(t') \hat{\rho}_{\textsc{ab}}^{(0)} W(\bm{x}_\nu,t,\bm{x}_\eta,t')\notag\\ &\phantom{{}=\,\,} -\int_{-\infty}^{\infty}\!\!\!\!\!\dif t \int_{-\infty}^{t}\!\!\!\!\!\dif t' \chi_\nu(t)\chi_\eta(t') \notag\\ &\phantom{{}= \!\!\!\!\sum_{\nu,\eta\in\{\textsc{a,b}\}}\!\!\!\! \lambda_\nu} \times \hat{\rho}_{\textsc{ab}}^{(0)}\hat{\mu}_\eta(t')\hat{\mu}_\nu(t) W(\bm{x}_\eta,t',\bm{x}_\nu,t) \Bigg].\end{aligned}$$ Here, $V(\bm{x}_\nu,t)$ and $W(\bm{x}_\eta,t,\bm{x}_\nu,t')$ are given by $$\begin{aligned} \label{eq:V} V(\bm{x}_\nu,t) &\coloneqq \int{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm{x}\, } F_\nu(\bm{x}-\bm{x}_\nu) v(\bm{x},t),\\ \label{eq:W} W(\bm{x}_\eta,t,\bm{x}_\nu,t') &\coloneqq \int\!\!{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm{x}\, }\!\!\!\int\!\!{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm{x'}\, } F_\eta(\bm{x}-\bm{x}_\eta) F_\nu(\bm{x'}-\bm{x}_\nu) \notag\\ &\phantom{{}=}\times w(\bm{x},t,\bm{x'},t'),\end{aligned}$$ while the one- and two-point correlation functions, $v(\bm x,t)$ and $w(\bm x,t,\bm x',t')$, of the field in the state ${\hat{\rho}_\phi}$, are defined as $$\begin{aligned} \label{eq:v} v(\bm{x},t) &:= \operatorname{Tr}_\phi \left[\hat{\phi}(\bm{x},t){\hat{\rho}_\phi}\right], \\ \label{eq:w} w(\bm{x},t,\bm{x'},t') &:= \operatorname{Tr}_\phi \left[\hat{\phi}(\bm{x},t) \hat{\phi}(\bm{x'},t') {\hat{\rho}_\phi}\right].\end{aligned}$$ After computing the evolved two-detector state ${\hat{\rho}_\textsc{ab}}$ using Eq. , we can use it to compute the amount of correlations present between the detectors A and B following their interactions with the field. In this paper we will focus on two types of correlations: entanglement and mutual information. More precisely, we will quantify the entanglement that the detectors A and B harvest from the field by computing the negativity $\mathcal N$, which, for a state ${\hat{\rho}_\textsc{ab}}$ on the Hilbert space $\mathcal H_\textsc{a}\otimes\mathcal H_\textsc{b}$, is defined as [@Vidal2002] $$\label{eq:neg} \mathcal{N}\left[{\hat{\rho}_\textsc{ab}}\right] \coloneqq \sum_i \max \left(0,-E_{\textsc{ab},i}^{{\text{\textbf{t}}}_\textsc{a}} \right),$$ where the $E_{\textsc{ab},i}^{{\text{\textbf{t}}}_\textsc{a}}$ are the eigenvalues of the partially transposed matrix ${\hat{\rho}_\textsc{ab}^{{\text{\textbf{t}}}_\textsc{a}}}$. It is well known that the negativity of a two-qubit system is an entanglement monotone that vanishes if and only if the two-qubit state is separable [@Peres1996; @Horodecki1996]. Hence the negativity is often used as a measure of entanglement in harvesting scenarios, and it is the measure that we will use. It is also possible for Alice and Bob to be classically correlated via their interactions with the field. We will quantify the total amount of correlations (quantum and classical) between them by computing the mutual information, $I$, which is defined as $$\begin{aligned} \label{eq:mut_info} I[{\hat{\rho}_\textsc{ab}}]:= S[{\hat{\rho}_\textsc{a}}]+S[{\hat{\rho}_\textsc{b}}]-S[{\hat{\rho}_\textsc{ab}}],\end{aligned}$$ where $S[\hat\rho]:=-{\text{Tr}}(\hat\rho\log\hat\rho)$ is the von Neumann entropy of the state $\hat\rho$, while ${\hat{\rho}_\textsc{a}}:={\text{Tr}}_\textsc{b}({\hat{\rho}_\textsc{ab}})$ and ${\hat{\rho}_\textsc{b}}:={\text{Tr}}_\textsc{a}({\hat{\rho}_\textsc{ab}})$ are the reduced states of detectors A and B following the detector-field interactions. In particular, if entanglement is zero and the mutual information is not, the correlations have to be either classical correlations or discord [@Ollivier2001; @Henderson2001]. Thermal field state {#sec:thermal} =================== Let us suppose now that the two Unruh-DeWitt detectors are initially in their ground states, $\hat\rho_\nu={| {g_\nu} \rangle}{\langle {g_\nu} |}$, and that the field is in a thermal state ${\hat\rho_{_\beta}}$ of inverse temperature $\beta$. It will be sufficient for our purposes to formally define ${\hat\rho_{_\beta}}$ as a Gibbs state in the usual way. Namely we write $$\label{eq:rb} {\hat\rho_{_\beta}}:=\frac{\exp(-\beta\hat H_\phi)}{Z},$$ where $Z:={\text{Tr}}[\exp(-\beta\hat H_\phi)]$ is the partition function of the free field. Here $\hat H_\phi$ is the Shrödinger picture free field Hamiltonian, which, after subtracting off an infinite zero-point energy (which does not affect any observable dynamics), takes the form $$\hat H_\phi = \int { \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm k\, } {\omega_{\bm{k}}} {\hat{a}_{\bm{k}}^\dagger}{\hat{a}_{\bm{k}}}.$$ We would like to emphasize that, strictly speaking, the Gibbs definition of ${\hat\rho_{_\beta}}$ in Eq.  is not well defined when $\hat H_\phi$ is the Hamiltonian of a field in free space, since then $\hat H_\phi$ is an operator acting on a Hilbert space of uncountably many dimensions, and certain technical issues arise in with performing its exponentiation and trace. We could proceed rigorously by instead considering our field to be in a large box of length $L$, such that its Hilbert space is of countable dimension, and then in the end taking the limit $L\rightarrow\infty$. Alternatively we could formalize our treatment by making use of the Kubo-Martin-Schwinger (KMS) definition of a thermal state, which is rigorously defined even for continuous variable systems [@Kubo1957; @Martin1959]. In this case the definition of ${\hat\rho_{_\beta}}$ would correspond to a KMS state of KMS parameter $\beta$ with respect to the time $t$ proper to both detectors. However we will shortly see that, for our limited purposes, these more rigorous definitions of ${\hat\rho_{_\beta}}$ are unnecessary in the sense that formal calculations using the Gibbs definition in Eq.  yield the same results. This can be checked by comparing the results we will obtain with, e.g., [@Strocchi2008]. To see this concretely, from the definition  of ${\hat\rho_{_\beta}}$ and the canonical commutation relations (CCRs) in Eq. , we can straightforwardly calculate the one- and two-point correlation functions defined in  and . Because the field is composed of a linear superposition of ${\hat{a}_{\bm{k}}}$ and ${\hat{a}_{\bm{k}}^\dagger}$ operators, we first compute the following useful expression: $$\begin{aligned} \label{eq:expectation_a} {\text{Tr}}_\phi\left({\hat\rho_{_\beta}}{\hat{a}_{\bm{k}}}\right) &= \frac{1}{Z}{\text{Tr}}_\phi\left(e^{-\beta{\hat H_\phi}}{\hat{a}_{\bm{k}}}\right)\\ &= \frac{1}{Z}{\text{Tr}}_\phi\left(e^{-\beta{\hat H_\phi}}{\hat{a}_{\bm{k}}} e^{\beta{\hat H_\phi}}e^{-\beta{\hat H_\phi}}\right)\notag\\ &= \frac{e^{\beta{\omega_{\bm{k}}}}}{Z}{\text{Tr}}_\phi\left({\hat{a}_{\bm{k}}} e^{-\beta{\hat H_\phi}}\right)\notag\\ &= e^{\beta{\omega_{\bm{k}}}}{\text{Tr}}_\phi\left({\hat\rho_{_\beta}}{\hat{a}_{\bm{k}}}\right)\notag,\end{aligned}$$ where in the third line we made use of the identity $e^{-\beta{\hat H_\phi}}{\hat{a}_{\bm{k}}} e^{\beta{\hat H_\phi}}=e^{\beta{\omega_{\bm{k}}}}{\hat{a}_{\bm{k}}}$, which can be easily proved using the Zassenhaus formula and the CCRs. Then, comparing the first and last lines of Eq. , we conclude that ${\text{Tr}}_\phi\left({\hat\rho_{_\beta}}{\hat{a}_{\bm{k}}}\right)=0$. Hence ${\text{Tr}}_\phi({\hat\rho_{_\beta}}{\hat{a}_{\bm{k}}^\dagger})=0$, and therefore the one-point function $v(\bm x,t)=0$. Then, from Eqs.  and , we conclude that the first order contribution $\hat\rho_\textsc{ab}^{(1)}$ to ${\hat{\rho}_\textsc{ab}}$ is identically zero for a thermal field state. To calculate the two-point function $w(\bm x,t,\bm x',t')$ we first compute: $$\begin{aligned} \label{eq:expectation_a_ad} {\text{Tr}}_\phi\left({\hat\rho_{_\beta}}{\hat{a}_{\bm{k}}}{\hat{a}_{\bm{k'}}^\dagger}\right) &= \frac{1}{Z}{\text{Tr}}_\phi\left(e^{-\beta{\hat H_\phi}}{\hat{a}_{\bm{k}}}{\hat{a}_{\bm{k'}}^\dagger}\right)\\ &= \frac{1}{Z}{\text{Tr}}_\phi\left(e^{-\beta{\hat H_\phi}}{\hat{a}_{\bm{k}}} e^{\beta{\hat H_\phi}}e^{-\beta{\hat H_\phi}{\hat{a}_{\bm{k'}}^\dagger}}\right)\notag\\ &= \frac{e^{\beta{\omega_{\bm{k}}}}}{Z}{\text{Tr}}_\phi\left({\hat{a}_{\bm{k}}} e^{-\beta{\hat H_\phi}{\hat{a}_{\bm{k'}}^\dagger}}\right)\notag\\ &= e^{\beta{\omega_{\bm{k}}}}{\text{Tr}}_\phi\left({\hat\rho_{_\beta}}{\hat{a}_{\bm{k'}}^\dagger}{\hat{a}_{\bm{k}}}\right)\notag\\ &=e^{\beta{\omega_{\bm{k}}}}\left[{\text{Tr}}_\phi\left({\hat\rho_{_\beta}}{\hat{a}_{\bm{k}}}{\hat{a}_{\bm{k'}}^\dagger}\right)+\delta(\bm k-\bm k')\right]\notag,\end{aligned}$$ where in the last step we again made use of the CCRs. Comparing the first and last lines of this expression gives the result $$\label{eq:exp_a_ad} {\text{Tr}}({\hat\rho_{_\beta}}{\hat{a}_{\bm{k}}}{\hat{a}_{\bm{k'}}^\dagger})= \frac{e^{\beta{\omega_{\bm{k}}}}}{e^{\beta{\omega_{\bm{k}}}}-1}\delta^3(\bm k-\bm k').$$ Similarly we obtain the identities $$\begin{aligned} {\text{Tr}}({\hat\rho_{_\beta}}{\hat{a}_{\bm{k}}^\dagger}{\hat{a}_{\bm{k'}}}) &= \frac{1}{e^{\beta{\omega_{\bm{k}}}}-1}\delta^3(\bm k-\bm k'),\\ {\text{Tr}}({\hat\rho_{_\beta}}{\hat{a}_{\bm{k}}}{\hat{a}_{\bm{k'}}}) &=0,\\ {\text{Tr}}({\hat\rho_{_\beta}}{\hat{a}_{\bm{k}}^\dagger}{\hat{a}_{\bm{k'}}^\dagger}) &=0. \label{eq:exp_ad_ad}\end{aligned}$$ Notice that, as alluded to above, the calculations in Eqs.  and would turn out the same if we rigorously considered the field in a box and then took the $L\rightarrow\infty$ limit in the end. In particular the only difference would be that the CCRs contain a Kronecker delta, which in the limit of free space becomes a Dirac delta, thus recovering our results in a more rigorous fashion. Furthermore, our final expressions in Eqs. - are equal to those obtained using the KMS definition of ${\hat\rho_{_\beta}}$ (see equation 14.3 in [@Strocchi2008]). Hence our formal use of the Gibbs definition of ${\hat\rho_{_\beta}}$ in Eq.  is justified. We can now use the identities in Eqs. - to write the two-point function of the field, defined by $w(\bm x,t,\bm x',t'):={\text{Tr}}[{\hat\rho_{_\beta}}\hat\phi(\bm x,t)\hat\phi(\bm x',t')]$, as $$\begin{aligned} \label{eq:thermal_wightman} w(\bm x,t,\bm x',t') = w^\text{vac}(\bm x,t,\bm x',t') +w^\text{th}_\beta(\bm x,t,\bm x',t').\end{aligned}$$ Here $w^\text{vac}(\bm x,t,\bm x',t')$ and $w^\text{th}_\beta(\bm x,t,\bm x',t')$ are the vacuum ($\beta$-independent) two-point function and the thermal ($\beta$-dependent) contribution, respectively, and are explicitly given by $$\begin{aligned} \label{eq:wvac} w^\text{vac}(\bm x,t,\bm x',t')&=\! \int \!\frac{{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm k\, }}{2(2\pi)^n{\omega_{\bm{k}}}} e^{-{\mathrm{i}}{\omega_{\bm{k}}}(t-t')} e^{{\mathrm{i}}\bm k\cdot(\bm x-\bm x')}, \\ \label{eq:wbeta} w^\text{th}_\beta(\bm x,t,\bm x',t')&= \int \frac{{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm k\, }\left[ e^{{\mathrm{i}}{\omega_{\bm{k}}}(t-t')} e^{-{\mathrm{i}}\bm k\cdot(\bm x-\bm x')} +\text{c.c} \right]}{2(2\pi)^n{\omega_{\bm{k}}}\left(e^{\beta{\omega_{\bm{k}}}}-1\right)} .\end{aligned}$$ Before we proceed to use the two-point function to calculate the time-evolved two-detector density matrix ${\hat{\rho}_\textsc{ab}}$, it should be noted that in the literature one often finds a very different looking expression for the two-point function of a thermal field state. For instance, in [@Weldon2000], the thermal two-point function for a massless field in $(3+1)$-dimensions is shown to be $$\begin{aligned} \label{eq:thermal_wightman_alt} w(\bm x, t, 0, 0)=& \frac{1}{8\pi r\beta} \!\left[\coth\!\left(\frac{\pi(r+t)}{\beta}\right)\!+\coth\!\left(\frac{\pi(r-t)}{\beta}\right)\! \right]\notag\\ &+ \frac{{\mathrm{i}}}{8\pi r} \left[\delta^{(3)}(r+t)-\delta^{(3)}(r-t)] \right],\end{aligned}$$ where $r:=|\bm x|$. The advantage of this expression over the one in Eq.  is that there are no integrals over momentum space that have to be evaluated. The disadvantage is that it is restrictive to the massless $(3+1)$-dimensional case. Furthermore the method used in [@Weldon2000] to obtain Eq.  is much less direct than the method we employed in obtaining Eq. . In any case, as a consistency check in Appendix \[app:thermal\_wightman\] we show that the expression in Eq.  is indeed a specific case of Eq.  when $m=0$, $n=3$, and $\bm x' = t' = 0$. We now come back to our main objective: use the two-point function $w(\bm x,t,\bm x',t')$ in Eq.  to compute the density matrix ${\hat{\rho}_\textsc{ab}}$ in . Substituting into we obtain $$\label{eq:rhoab_thermal} {\hat{\rho}_\textsc{ab}}= \begin{pmatrix} 1-\mathcal{L}_\textsc{aa}(\beta)-\mathcal{L}_\textsc{bb}(\beta)& 0 & 0 & \mathcal{M}^*(\beta) \\ 0 & \mathcal{L}_\textsc{bb}(\beta) & \mathcal{L}_\textsc{ab}^*(\beta) & 0 \\ 0 & \mathcal{L}_\textsc{ab}(\beta) & \mathcal{L}_\textsc{aa}(\beta) & 0 \\ \mathcal{M}(\beta) & 0 & 0 & 0 \end{pmatrix},$$ to second order in the coupling strength $\lambda$, and where we work in the basis $\{ {| {g_\textsc{a}} \rangle}{| {g_\textsc{b}} \rangle}, {| {g_\textsc{a}} \rangle}{| {e_\textsc{b}} \rangle}, {| {e_\textsc{a}} \rangle}{| {g_\textsc{b}} \rangle}, {| {e_\textsc{a}} \rangle}{| {e_\textsc{b}} \rangle}\}$. The terms $\mathcal{L}_{\nu\eta}(\beta)$ and $\mathcal M(\beta)$ are defined to be $$\begin{aligned} \label{eq:L_thermal} \mathcal{L}_{\nu\eta}(\beta)&= \mathcal{L}_{\nu\eta}^\text{vac} +2\pi\lambda_\nu\lambda_\eta \int\!\! \frac{{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm k\, }\bar F_\nu^*(\bm k) \bar F_\eta(\bm k) e^{{\mathrm{i}}\bm k\cdot(\bm x_\eta-\bm x_\nu)}}{2{\omega_{\bm{k}}}\left(e^{\beta{\omega_{\bm{k}}}}-1\right)}\notag\\ &\hspace{1.5cm}\times\Big[ \bar \chi_\nu^*({\omega_{\bm{k}}}-\Omega_\nu) \bar \chi_\eta({\omega_{\bm{k}}}-\Omega_\eta) \notag\\ &\hspace{2cm}+ \bar \chi_\nu({\omega_{\bm{k}}}+\Omega_\nu) \bar \chi_\eta^*({\omega_{\bm{k}}}+\Omega_\eta) \Big], \\ \label{eq:M_thermal} \mathcal M(\beta)&= \mathcal M^\text{vac} -2\pi\lambda_\textsc{a}\lambda_\textsc{b} \int\!\! \frac{{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm k\, }\bar F_\textsc{a}(\bm k)\bar F_\textsc{b}^*(\bm k) e^{{\mathrm{i}}\bm k\cdot(\bm x_\textsc{a}-\bm x_\textsc{b})}}{2{\omega_{\bm{k}}}\left(e^{\beta{\omega_{\bm{k}}}}-1\right)}\notag\\ &\hspace{1.5cm}\times\Big[ \bar \chi_\textsc{a}^*({\omega_{\bm{k}}}-\Omega_\textsc{a}) \bar \chi_\textsc{b}({\omega_{\bm{k}}}+\Omega_\textsc{b}) \notag\\ &\hspace{2cm}+\bar \chi_\textsc{a}({\omega_{\bm{k}}}+\Omega_\textsc{a}) \bar \chi_\textsc{b}^*({\omega_{\bm{k}}}-\Omega_\textsc{b}) \Big].\end{aligned}$$ Here we define the Fourier transform $\bar g:\mathbb R^m\rightarrow\mathbb C$ of a function $g:\mathbb R^m\rightarrow\mathbb R$ as $$\label{eq:FT} \bar{g}(\bm{k}) := \frac{1}{\sqrt{(2\pi)^m}}\int{ \mathrm{d} \ifx\relaxm\relax\else \rule{-0.02em}{1.5ex}^{m}\rule{0.08em}{0ex}\! \fi \bm{x}\, } g(\bm{x})e^{{\mathrm{i}}\bm{k}\cdot\bm{x}},$$ and as always we use the superscript “vac" to denote quantities that do not depend on the inverse temperature $\beta$, i.e. those terms which arise from the “vacuum" part $w^\text{vac}$ of the two-point function. The vacuum terms $\mathcal{L}_{\nu\eta}^\text{vac}$ and $\mathcal M^\text{vac}$ are explicitly given by $$\begin{aligned} \label{eq:Lvac} \mathcal L_{\nu\eta}^\text{vac} &= 2\pi\lambda_\nu\lambda_\eta \int\frac{{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm k\, }}{2{\omega_{\bm{k}}}} \bar F_\nu^*(\bm k)\bar F_\eta(\bm k) e^{-{\mathrm{i}}\bm k\cdot(\bm x_\nu-\bm x_\eta)} \\ &\hspace{2cm}\times \bar \chi_\nu({\omega_{\bm{k}}}+\Omega_\nu) \bar \chi_\eta^*({\omega_{\bm{k}}}+\Omega_\eta), \notag\\ \label{eq:Mvac} \mathcal M^\text{vac} &= -\lambda_\textsc{a}\lambda_\textsc{b} \int\frac{{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm k\, }}{2{\omega_{\bm{k}}}} \int_{-\infty}^\infty\dif t \int_{-\infty}^t\dif t' e^{-{\mathrm{i}}{\omega_{\bm{k}}}(t-t')} \\ &\hspace{1cm}\times \Big[ \bar F_\textsc{a}(\bm k) \bar F_\textsc{b}^*(\bm k) e^{{\mathrm{i}}\bm k\cdot(\bm x_\textsc{a}-\bm x_\textsc{b})} \bar \chi_\textsc{a}(t) \bar \chi_\textsc{b}(t') \notag\\ &\hspace{1.5cm}\times e^{{\mathrm{i}}(\Omega_\textsc{a}t+\Omega_\textsc{b}t')} + (\text{A}\leftrightarrow\text{B}) \Big]. \notag\end{aligned}$$ Harvesting entanglement {#eq:thermal:harvesting_entanglement} ----------------------- Having computed the time-evolved density matrix ${\hat{\rho}_\textsc{ab}}$ of the Unruh-DeWitt detector pair, we can now compute the negativity of this state and thus quantify the amount of entanglement the detectors harvest from the thermal field state. Using the expression for ${\hat{\rho}_\textsc{ab}}$, we find that in the same computational basis, to $\mathcal O(\lambda^2)$ the partially transposed matrix ${\hat{\rho}_\textsc{ab}^{{\text{\textbf{t}}}_\textsc{a}}}$ takes the form $$\label{eq:rhoab_thermal_pt} {\hat{\rho}_\textsc{ab}^{{\text{\textbf{t}}}_\textsc{a}}}= \begin{pmatrix} 1-\mathcal{L}_\textsc{aa}(\beta)-\mathcal{L}_\textsc{bb}(\beta)& 0 & 0 & \mathcal{L}_\textsc{ab}^*(\beta) \\ 0 & \mathcal{L}_\textsc{bb}(\beta) & \mathcal{M}^*(\beta) & 0 \\ 0 & \mathcal{M}(\beta) & \mathcal{L}_\textsc{aa}(\beta) & 0 \\ \mathcal{L}_\textsc{ab}(\beta) & 0 & 0 & 0 \end{pmatrix}.$$ As discussed in [@Pozas2015], at $\mathcal O(\lambda^2)$ a matrix of this form has only one potentially negative eigenvalue: $$\begin{aligned} \label{eq:neg2} E_{\textsc{ab},1}^{{\text{\textbf{t}}}_\textsc{a}} = \frac{1}{2}\Big(& \mathcal{L}_\textsc{aa}(\beta)+\mathcal{L}_\textsc{bb}(\beta)\\ &-\sqrt{(\mathcal{L}_\textsc{aa}(\beta)-\mathcal{L}_\textsc{bb}(\beta))^2+ 4|\mathcal{M}(\beta)|^2}\Big).\notag\end{aligned}$$ Hence we find that the negativity $\mathcal N$, defined in Eq. , can be written as $$\mathcal N[{\hat{\rho}_\textsc{ab}}]=\max\left(0,-E_{\textsc{ab},1}^{{\text{\textbf{t}}}_\textsc{a}}\right).$$ Now suppose that the detectors A and B are identical. That is, they have the same shapes $F(\bm x)=F_\nu(\bm x)$, the same proper energy gaps $\Omega=\Omega_\nu$, the same coupling constants $\lambda=\lambda_\nu$, and the same switching profiles $\chi(t-t_\nu)=\chi_\nu(t)$. Note that we are still allowing for the detectors to couple to the field at potentially different spacetime locations $(t_\textsc{a},\bm x_\textsc{a})$ and $(t_\textsc{b},\bm x_\textsc{b})$. However, since the local terms $\mathcal{L}_{\nu\nu}$ are translationally invariant, we find that $\mathcal L_\textsc{aa}(\beta)=\mathcal L_\textsc{bb}(\beta)$, and the negativity can be written more simply as $$\label{eq:neg_thermal} \mathcal N=\max\left[0,|\mathcal M(\beta)|-\mathcal L_{\nu\nu}(\beta)\right].$$ As acknowledged in [@Pozas2015], this form for the negativity makes evident the competition between the non-local term $|\mathcal M(\beta)|$, which increases the negativity, and the local term $\mathcal L_{\nu\nu}(\beta)$, which decreases it. We note however, that although this interpretation of Eq.  is pleasantly consistent with the intuition that entanglement is a non-local phenomenon, it should not be taken too literally. For instance, in [@Simidzija2017b; @Simidzija2017c] it was shown that a detector pair interacting with a coherent field state extracts the exact same amount of entanglement as it would from a vacuum state, despite the fact that inherently local terms of $\mathcal O(\lambda)$ appear in ${\hat{\rho}_\textsc{ab}}$ for the former but not the latter case. Having obtained an expression in for the negativity $\mathcal N$ of two identical Unruh-DeWitt detectors following their interactions with a thermal field state, we would now like to determine the temperature dependence of $\mathcal N$. In other words, we want to answer the question, “what is the optimal field temperature for Unruh-DeWitt detectors to harvest entanglement?" To answer this question, let us first particularize the terms $\mathcal L_{\nu\eta}(\beta)$ and $\mathcal M(\beta)$ in Eqs.  and for identical detectors. We obtain $$\begin{aligned} \label{eq:L_thermal_identical} \mathcal{L}_{\nu\eta}(\beta)&= \mathcal{L}_{\nu\eta}^\text{vac} +\pi\lambda^2 \int\!\! \frac{{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm k\, }|\bar F(\bm k)|^2}{{\omega_{\bm{k}}}\left(e^{\beta{\omega_{\bm{k}}}}-1\right)} e^{{\mathrm{i}}\bm k\cdot(\bm x_\eta-\bm x_\nu)} \notag \\ &\hspace{0.3cm}\times \Big( |\bar \chi({\omega_{\bm{k}}}-\Omega)|^2 e^{{\mathrm{i}}({\omega_{\bm{k}}}-\Omega)t_\eta} e^{-{\mathrm{i}}({\omega_{\bm{k}}}-\Omega)t_\nu} \notag\\ &\hspace{0.7cm} + |\bar \chi({\omega_{\bm{k}}}+\Omega)|^2 e^{-{\mathrm{i}}({\omega_{\bm{k}}}+\Omega)t_\eta} e^{{\mathrm{i}}({\omega_{\bm{k}}}+\Omega)t_\nu} \Big), \\ \label{eq:M_thermal_identical} \mathcal M(\beta)&= \mathcal M^\text{vac} -2\pi\lambda^2\!\! \int\!\! \frac{{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm k\, }|\bar F(\bm k)|^2 e^{{\mathrm{i}}\Omega(t_\textsc{a}+t_\textsc{b})} e^{{\mathrm{i}}\bm k\cdot(\bm x_\textsc{a}-\bm x_\textsc{b})}}{{\omega_{\bm{k}}}\left(e^{\beta{\omega_{\bm{k}}}}-1\right)}\notag \\ &\hspace{0.3cm} \times \bar \chi^*({\omega_{\bm{k}}}-\Omega) \bar \chi({\omega_{\bm{k}}}+\Omega) \cos[{\omega_{\bm{k}}}(t_\textsc{a}-t_\textsc{b})].\end{aligned}$$ Now, let us consider two temperatures, $\beta_1^{-1}<\beta_2^{-1}$. Then, defining $$\label{eq:h} h(\bm k):= \frac{1}{e^{\beta_2{\omega_{\bm{k}}}}-1} - \frac{1}{e^{\beta_1{\omega_{\bm{k}}}}-1},$$ which is strictly greater than zero, we can rewrite $\mathcal L_{\nu\nu}(\beta)$ and $\mathcal M(\beta)$ to read $$\begin{aligned} \label{eq:L_thermal_identical_2} \mathcal{L}_{\nu\nu}(\beta_2)&= \mathcal{L}_{\nu\nu}(\beta_1) +\pi\lambda^2 \int\!\! \frac{{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm k\, }h(\bm k)|\bar F(\bm k)|^2}{{\omega_{\bm{k}}}} \notag \\ &\hspace{1cm}\times \Big( |\bar \chi({\omega_{\bm{k}}}-\Omega)|^2 + |\bar \chi({\omega_{\bm{k}}}+\Omega)|^2 \Big), \\ \label{eq:M_thermal_identical_2} \mathcal M(\beta_2)&= \mathcal M(\beta_1) -2\pi\lambda^2\!\! \int\!\! \frac{{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm k\, }h(\bm k)|\bar F(\bm k)|^2 }{{\omega_{\bm{k}}}}\notag \\ &\hspace{1cm} \times e^{{\mathrm{i}}\Omega(t_\textsc{a}+t_\textsc{b})} e^{{\mathrm{i}}\bm k\cdot(\bm x_\textsc{a}-\bm x_\textsc{b})} \cos[{\omega_{\bm{k}}}(t_\textsc{a}-t_\textsc{b})]\notag \\ &\hspace{1cm} \times \bar \chi^*({\omega_{\bm{k}}}-\Omega) \bar \chi({\omega_{\bm{k}}}+\Omega).\end{aligned}$$ Taking the magnitude of the latter expression we obtain $$\begin{aligned} \label{eq:M_thermal_norm} |\mathcal M(\beta_2)| &\le |\mathcal M(\beta_1)| +2\pi\lambda^2 \Bigg| \int\!\! \frac{{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm k\, }h(\bm k)|\bar F(\bm k)|^2 }{{\omega_{\bm{k}}}}\notag \\ &\hspace{1cm} \times e^{{\mathrm{i}}\Omega(t_\textsc{a}+t_\textsc{b})} e^{{\mathrm{i}}\bm k\cdot(\bm x_\textsc{a}-\bm x_\textsc{b})} \cos[{\omega_{\bm{k}}}(t_\textsc{a}-t_\textsc{b})]\notag \\ &\hspace{1cm} \times \bar \chi^*({\omega_{\bm{k}}}-\Omega) \bar \chi({\omega_{\bm{k}}}+\Omega) \Bigg| \notag \\ &\le |\mathcal M(\beta_1)| +2\pi\lambda^2 \int\!\! \frac{{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm k\, }h(\bm k)|\bar F(\bm k)|^2 }{{\omega_{\bm{k}}}}\notag \\ &\hspace{1cm} \times |\bar \chi^*({\omega_{\bm{k}}}-\Omega)| |\bar \chi({\omega_{\bm{k}}}+\Omega)|\end{aligned}$$ Finally, combining Eqs.  and we find $$\begin{aligned} \label{eq:thermal_ineq} &\,\, |\mathcal M(\beta_2)|-\mathcal L_{\nu\nu}(\beta_2)\notag \\ \le&\,\, |\mathcal M(\beta_1)|-\mathcal L_{\nu\nu}(\beta_1) -\pi\lambda^2\!\! \int\!\! \frac{{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm k\, }h(\bm k)D(\bm k)}{{\omega_{\bm{k}}}} \notag\\ \le&\,\, |\mathcal M(\beta_1)|-\mathcal L_{\nu\nu}(\beta_1),\end{aligned}$$ where $D(\bm k):=|\bar F(\bm k)|^2( |\bar \chi({\omega_{\bm{k}}}-\Omega)|\!-\! |\bar \chi({\omega_{\bm{k}}}+\Omega)| )^2$ is a non-negative function characterized by the switching, smearing, and energy gap of the detectors. Hence, using the definition of the negativity, Eq.  proves our first result: the amount of entanglement that two identical UDW detectors can harvest from a thermal field state decreases with the temperature $\beta^{-1}$. This is true regardless of the dimensionality of spacetime, the mass of the field, and the properties (spatial smearing, temporal switching, energy gap) of the detectors. In fact, we can obtain a somewhat stronger statement about the negativity of a pair of detectors interacting with a thermal field state. First, notice from Eq.  that for given values of $\beta_1$ and $\bm k$, the value of the function $h(\bm k)$ can be increased arbitrarily by choosing a small enough value of $\beta_2$. Therefore, from Eq. , as long as $D(\bm k)$ is not identically equal to zero, we find that the value of $|\mathcal M(\beta_2)|-\mathcal L_{\nu\nu}(\beta_2)$ can be made negative by taking a large enough temperature $\beta_2^{-1}$. Hence, not only does the amount of entanglement harvested by a UDW detector pair decreases monotonically with the temperature, but also by increasing the temperature of the field to a high enough value we can always (as long as $D(\bm k)$ is not identically zero) ensure that the thermal noise prevents the detectors from becoming entangled at all. This is true regardless the mass of the field, spacetime dimensionality and the detector properties. Knowing that the negativity $\mathcal N$ of a detector pair decreases with the temperature of the field, we can ask what is the rate of this decrease. We can straightforwardly obtain a bound on $\dif \mathcal N/\dif \beta$ from Eq. . First, writing $E_{\textsc{ab},1}^{{\text{\textbf{t}}}_\textsc{a}}(\beta)=\mathcal L_{\nu\nu}(\beta)-|\mathcal M(\beta)|$ for identical detectors, the second line of Eq.  can be expressed as $$\begin{aligned} E_{\textsc{ab},1}^{{\text{\textbf{t}}}_\textsc{a}}(\beta_1) - E_{\textsc{ab},1}^{{\text{\textbf{t}}}_\textsc{a}}(\beta_2) \le -\pi\lambda^2\!\! \int\!\! \frac{{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm k\, }h(\bm k)D(\bm k)}{{\omega_{\bm{k}}}}.\end{aligned}$$ Dividing both sides of this expression by $\beta_1-\beta_2$, taking the limit $\beta_1\rightarrow\beta_2$, and using the fact that $$\lim_{\beta_1\rightarrow\beta_2} \frac{h(\bm k)}{\beta_1-\beta_2} = -\frac{\dif}{\dif\beta_1} \left( \frac{1}{e^{\beta_1{\omega_{\bm{k}}}}-1} \right) = \frac{{\omega_{\bm{k}}} e^{\beta_1{\omega_{\bm{k}}}}}{\left(e^{\beta_1{\omega_{\bm{k}}}}-1\right)^2},$$ we find the rate of change of the eigenvalue $E_{\textsc{ab},1}^{{\text{\textbf{t}}}_\textsc{a}}(\beta)$ with respect to the inverse temperature $\beta$ to be bounded from below according to $$\begin{aligned} \frac{\dif}{\dif\beta} E_{\textsc{ab},1}^{{\text{\textbf{t}}}_\textsc{a}}(\beta) &\le -\pi\lambda^2\!\! \int { \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm k\, }D(\bm k) \frac{e^{\beta{\omega_{\bm{k}}}}}{\left(e^{\beta{\omega_{\bm{k}}}}-1\right)^2} .\end{aligned}$$ Therefore in regions where the negativity $\mathcal N(\beta)$ is non-zero, we have that $$\frac{\dif\mathcal N}{\dif\beta} \ge \pi\lambda^2\!\! \int { \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm k\, }D(\bm k) \frac{e^{\beta{\omega_{\bm{k}}}}}{\left(e^{\beta{\omega_{\bm{k}}}}-1\right)^2}.$$ This puts a lower bound on how fast $\mathcal N$ must grow with the inverse temperature $\beta$, in regions where $\mathcal N$ is non-zero. Of course if $\mathcal N$ is zero, then increasing $\beta$ will only result in $\mathcal N$ remaining zero. Having proven the general result that temperature is always detrimental to entanglement harvesting (at least for identical detectors), let us now consider some particular parameters for the detectors A and B, so that we may explicitly see the manifestation of this phenomenon. To that end, let us suppose that the two detectors are located in $(3+1)$ dimensional spacetime, that they have Gaussian spatial profiles of width $\sigma$, $$\begin{aligned} \label{eq:smearing_gaussian} F(\bm x)=\frac{1}{(\sqrt{\pi}\sigma)^3} e^{-\frac{|\bm x|^2}{\sigma^2}},\end{aligned}$$ and that their temporal switching functions are also Gaussians (of width $\uptau$), $$\begin{aligned} \label{eq:switching_gaussian} \chi(t)=e^{-\frac{t^2}{\uptau^2}}.\end{aligned}$$ Then it is straightforward to show that the terms $\mathcal M^\text{th}$, $\mathcal L_\textsc{ab}^\text{th}$ and $L_{\nu\nu}^\text{th}$, which make up the thermal contributions to the density matrix ${\hat{\rho}_\textsc{ab}}$, evaluate to $$\begin{aligned} \mathcal M^\text{th}&= -\frac{\tilde\lambda^2e^{-\frac{1}{2}\tilde\Omega^2}e^{{\mathrm{i}}\tilde\Omega\tilde\Delta^+}}{4\pi\tilde d} \int_0^\infty\dif \tilde k \frac{e^{-\frac{1}{2}\tilde k^2(1+\tilde\sigma^2)}}{e^{\tilde\beta\tilde k}-1} \\ \notag &\hspace{3cm}\times \sin(\tilde d\tilde k)\cos(\tilde\Delta^-\tilde k), \\ \mathcal L_\textsc{ab}^\text{th}&= \frac{\tilde\lambda^2e^{-\frac{1}{2}\tilde\Omega^2}e^{-{\mathrm{i}}\tilde\Omega\tilde\Delta^-}}{2\pi\tilde d} \int_0^\infty\dif \tilde k \frac{e^{-\frac{1}{2}\tilde k^2(1+\tilde\sigma^2)}}{e^{\tilde\beta\tilde k}-1} \\ \notag &\hspace{3cm}\times \sin(\tilde d\tilde k)\cosh[(\tilde\Omega+{\mathrm{i}}\tilde\Delta^-)\tilde k], \\ \mathcal L_{\nu\nu}^\text{th}&= \frac{\tilde\lambda^2e^{-\frac{1}{2}\tilde\Omega^2}}{2\pi} \int_0^\infty\dif \tilde k \frac{\tilde k e^{-\frac{1}{2}\tilde k^2(1+\tilde\sigma^2)}}{e^{\tilde\beta\tilde k}-1} \cosh(\tilde\Omega\tilde k).\end{aligned}$$ Here, every quantity with a tilde is a dimensionless expression of the scales of the problem in units of $\uptau$ (e.g. $\tilde\Omega:=\Omega \uptau$, $\tilde\beta:=\beta/\uptau$), and we have defined $\tilde d:=|\bm{ x}_\textsc{a}-\bm{ x}_\textsc{b}|/\uptau$ and $\tilde \Delta^\pm:= (t_\textsc{b}\pm t_\textsc{a})/\uptau$. Meanwhile the terms $\mathcal M^\text{vac}$ and $\mathcal L_{\nu\eta}^\text{vac}$, which give the vacuum ($\beta$ independent) contributions to ${\hat{\rho}_\textsc{ab}}$, can be found in equations 29-31 in [@Pozas2015]. ![Negativity of identical detectors as a function of field temperature, for different spatial separations $d$ of their centers of mass. The detectors are coupled to the field at the same time according to a Gaussian switching function of width $\uptau$, their spatial profiles are Gaussians of width $\sigma=\uptau$, and their energy gap is $\Omega=3/\uptau$.[]{data-label="fig:NegVsT_thermal"}](NegVsT_thermal.pdf){width="0.9\linewidth"} Assuming these detector spatial profiles and switching functions, in Fig. \[fig:NegVsT\_thermal\] we show the dependence of the negativity of the detector pair on the temperature $T=\beta^{-1}$ of the field. We see that, in accordance with our general discussion above, the negativity is a monotonically decreasing function of $T$, and that it is identically zero after a certain finite temperature. These findings are qualitatively the same as what was found in [@Brown2013a], namely that harmonic oscillator detectors in a (1+1)D cavity harvest less entanglement as the field temperature increases. This is, of course, all in agreement with our intuition that “thermal noise" is detrimental to the detectors obtaining non-local correlations. We will soon see however, that this seemingly reasonable intuition does not apply when we quantify the correlations using the mutual information rather than the negativity. In particular we will show that the mutual information between the detector pair can increase with the field temperature. ![Negativity of identical detectors as a function of their energy gap, for different field temperatures $T$. The detectors are coupled to the field at the same time according to a Gaussian switching function of width $\uptau$, and they have Gaussian spatial profiles of width $\sigma=\uptau$, the centers of which are separated in space by $d=2\uptau$.[]{data-label="fig:NegVsOmega_thermal"}](NegVsOmega_thermal.pdf){width="0.9\linewidth"} To conclude this section, let us briefly investigate how the negativity of the detectors varies with their energy gap $\Omega$. These results are summarized in Fig. \[fig:NegVsOmega\_thermal\]. Notice that, for a given field temperature $T$, the detectors cannot become entangled if their energy gap is below some finite value $\Omega_\text{min}(T)$. We also notice that $\Omega_\text{min}(T)$ is a monotonically increasing function of temperature. This tells us that if we have a way to control the energy gap of the detectors, then by measuring the amount of entanglement that this detector pair harvests from the field we have, in principle, a quantum thermometer capable of measuring the field temperature. Harvesting mutual information {#eq:thermal:harvesting_mut_info} ----------------------------- Having shown that the amount of entanglement harvested by two Unruh-DeWitt detectors decreases with the temperature of the field with which they interact, we can ask what happens to other types of correlations. As mentioned above, the mutual information $I[{\hat{\rho}_\textsc{ab}}]$, defined in Eq. , quantifies the total correlations (quantum and classical) present between the two detectors. Using the time-evolved density matrix ${\hat{\rho}_\textsc{ab}}$ in Eq.  for the two detectors, we find that $I[{\hat{\rho}_\textsc{ab}}]$ takes the form $$\begin{aligned} \label{eq:mut_info_2} I[{\hat{\rho}_\textsc{ab}}]=& \mathcal L_+\log(\mathcal L_+)+ \mathcal L_-\log(\mathcal L_-) \\ &-\mathcal L_\textsc{aa}\log(\mathcal L_\textsc{aa}) -\mathcal L_\textsc{bb}\log(\mathcal L_\textsc{bb}) +\mathcal O(\lambda^4) \notag,\end{aligned}$$ where $\mathcal L_\pm$ is defined as $$\begin{aligned} \mathcal L_\pm = \frac{1}{2} \left( \mathcal L_\textsc{aa}+\mathcal L_\textsc{bb} \pm \sqrt{(\mathcal L_\textsc{aa}-\mathcal L_\textsc{bb})^2+4|\mathcal L_\textsc{ab}|^2} \right).\end{aligned}$$ Although the general dependence of $I[{\hat{\rho}_\textsc{ab}}]$ on the temperature $\beta^{-1}$ is highly non-trivial, from Eq.  it is straightforward to derive the asymptotic behaviour as $\beta^{-1}\rightarrow\infty$. Defining $\mathscr L_{\pm}:=\beta\mathcal L_{\pm}$ and $\mathscr L_{\nu\eta}:=\beta\mathcal{L}_{\nu\eta}$, we notice from Eq.  that $\mathscr L_{\pm}$ and $\mathscr L_{\nu\eta}$ are independent of $\beta$ in the limit $\beta^{-1}\rightarrow\infty$. Then from Eq.  it is straightforward to show that in the $\beta^{-1}\rightarrow\infty$ limit the mutual information goes as $$\begin{aligned} I[{\hat{\rho}_\textsc{ab}}]\sim \frac{1}{\beta} \big(& \mathscr L_{+}\log\mathscr L_{+} + \mathscr L_{-}\log\mathscr L_{-} \notag\\ &- \mathscr L_{\textsc{aa}}\log\mathscr L_{\textsc{aa}} - \mathscr L_{\textsc{bb}}\log\mathscr L_{\textsc{bb}} \big).\end{aligned}$$ Combining this with the fact that the mutual information is always non-negative, we conclude that in the large temperature limit (of course with a coupling constant small enough so that we are still within the perturbative regime) the total correlations that the detectors harvest from the field grow proportionally to the temperature $\beta^{-1}$. ![Mutual information of identical detectors as a function of field temperature, for different spatial separations $d$ of their centers of mass. The detectors are coupled to the field at the same time according to a Gaussian switching function of width $\uptau$, their spatial profiles are Gaussians of width $\sigma=\uptau$, and their energy gap is $\Omega=3/\uptau$.[]{data-label="fig:MutInfoVsT_thermal"}](MutInfoVsT_thermal.pdf){width="0.9\linewidth"} To see explicitly the dependence of $I[{\hat{\rho}_\textsc{ab}}]$ on the temperature, let us once again particularize to the case of identical detectors with Gaussian spatial smearings and Gaussian switching functions . These results are plotted in Fig. \[fig:MutInfoVsT\_thermal\]. We see that for low $T=\beta^{-1}$ the mutual information approaches a constant finite value, which corresponds to the correlations that the detectors would obtain if they interacted with the field vacuum. For intermediate field temperatures, we find that the mutual information has a non-trivial dependence on $T$, and in fact, unlike the negativity, $I[{\hat{\rho}_\textsc{ab}}]$ does not always increase with $T$. However, as we showed for the case of arbitrary detectors above, in the asymptotic limit $T\rightarrow\infty$ the mutual information is proportional to $T$. It should be emphasized that in a full, non-perturbative calculation, this upwards trend of $I[{\hat{\rho}_\textsc{ab}}]$ with temperature would not continue indefinitely for the simple reason that for a two qubit system the mutual information is bounded from above by $2\log 2$. Nevertheless it is interesting that, at least in the perturbative regime (i.e. if for a given temperature we consider a small enough coupling strength), the amount of entanglement harvested from the field by an Unruh-DeWitt detector pair is hindered by high field temperatures, whereas the total correlations in fact grow with $T$. Squeezed coherent field state {#sec:squeezed} ============================= Again let us suppose that each Unruh-DeWitt detector is in its ground state, and that now the field is in an arbitrary, multimode, squeezed coherent state. The physical relevance of squeezed coherent states is that they are the most general set of states that saturate the Heisenberg uncertainty principle. The most general multimode squeezed coherent state is given by ${| {\alpha(\bm k),\zeta(\bm k,\bm k')} \rangle}={\hat D_\alpha}{\hat S_\zeta}{| {0} \rangle}$, where the *displacement operato* ${\hat D_\alpha}$ and the *squeezing operator* ${\hat S_\zeta}$ are unitary operators defined by [@Loudon1987] $$\begin{aligned} \label{eq:displacement_operator} {\hat D_\alpha}&:=\exp\left[\int{ \mathrm{d} \ifx\relax3\relax\else \rule{-0.02em}{1.5ex}^{3}\rule{0.08em}{0ex}\! \fi \bm k\, }\left(\alpha(\bm k){\hat{a}_{\bm{k}}^\dagger}-\text{H.c.}\right)\right] , \\ {\hat S_\zeta}&:=\exp\left[ \frac{1}{2}\int\!{ \mathrm{d} \ifx\relax3\relax\else \rule{-0.02em}{1.5ex}^{3}\rule{0.08em}{0ex}\! \fi \bm k\, }\!\!\int\!{ \mathrm{d} \ifx\relax3\relax\else \rule{-0.02em}{1.5ex}^{3}\rule{0.08em}{0ex}\! \fi \bm k'\, }\left(\zeta^*(\bm k,\bm k'){\hat{a}_{\bm{\bm k}}}{\hat{a}_{\bm{\bm k'}}}-\text{H.c.}\right) \right]\!,\end{aligned}$$ We call the complex valued distributions $\alpha(\bm k)$ and $\zeta(\bm k,\bm k')$ respectively the *coherent amplitude* and *squeezing amplitude* of the state ${| {\alpha(\bm k),\zeta(\bm k,\bm k')} \rangle}$. Through the integrals in the definitions of ${\hat D_\alpha}$ and ${\hat S_\zeta}$, these distributions generalize the familiar notion of a squeezed coherent state of a single harmonic oscillator to the case where we have an uncountably infinite number of field mode oscillators that can be pairwise two-mode squeezed with each other. In order to calculate the one and two-point functions of the field in a squeezed coherent state, we will make use of the identities governing the action of ${\hat D_\alpha}$ and ${\hat S_\zeta}$ on the creation and annihilation operators. Namely, by using the canonical commutation relations and the Baker-Campbell-Hausdorff lemma it is straightforward to show that $$\begin{aligned} {\hat D_\alpha}^\dagger{\hat{a}_{\bm{k}}}{\hat D_\alpha}&= {\hat{a}_{\bm{k}}}+\alpha(\bm k)\openone.\end{aligned}$$ On the other hand, we are not aware of a similarly convenient closed-form expression for ${\hat S_\zeta}^\dagger{\hat{a}_{\bm{k}}}{\hat S_\zeta}$ in the case of an arbitrary, continuous, multimode squeezing. However, since ${\hat S_\zeta}$ is the exponential of terms quadratic in ${\hat{a}_{\bm{k}}}$ and ${\hat{a}_{\bm{k}}^\dagger}$, by expanding out the exponentials in ${\hat S_\zeta}^\dagger{\hat{a}_{\bm{k}}}{\hat S_\zeta}$ it is not difficult to prove that this expression takes the form of a linear superposition of ${\hat{a}_{\bm{k}}}$ and ${\hat{a}_{\bm{k}}^\dagger}$ operators, i.e. $$\label{eq:SaS} {\hat S_\zeta}^\dagger{\hat{a}_{\bm{k}}}{\hat S_\zeta}= \int{ \mathrm{d} \ifx\relax3\relax\else \rule{-0.02em}{1.5ex}^{3}\rule{0.08em}{0ex}\! \fi \bm k'\, } \left[K_1(\bm k,\bm k'){\hat{a}_{\bm{k'}}} + K_2(\bm k,\bm k'){\hat{a}_{\bm{k'}}^\dagger} \right],$$ for some bi-distributions $K_1$ and $K_2$. In particular this implies that $$\begin{aligned} &{\langle {\alpha(\bm k),\zeta(\bm k,\bm k')} |}{\hat{a}_{\bm{k''}}}{| {\alpha(\bm k),\zeta(\bm k,\bm k')} \rangle}\notag\\ = \,&{\langle {0} |}{\hat S_\zeta}^\dagger[{\hat{a}_{\bm{k''}}}+\alpha(\bm k'')]{\hat S_\zeta}{| {0} \rangle}\notag\\ = \,&\alpha(\bm k''),\end{aligned}$$ and hence, using the mode expansion  of the field operator, the one-point function  of the field in the state ${| {\alpha(\bm k),\zeta(\bm k,\bm k')} \rangle}$ is $$\begin{aligned} \label{eq:v_sqeezed_coherent} v(\bm x,t)= \int\frac{{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm k\, }}{\sqrt{2(2\pi)^n{\omega_{\bm{k}}}}} \left( \alpha(\bm k)e^{-{\mathrm{i}}({\omega_{\bm{k}}} t-\bm k\cdot \bm x)} + \text{c.c} \right).\end{aligned}$$ Thus we see that the one-point function is independent of the squeezing amplitude $\zeta(\bm k,\bm k')$. Similarly we can show that the two-point function  in the state ${| {\alpha(\bm k),\zeta(\bm k,\bm k')} \rangle}$ is of the form $$\begin{aligned} \label{eq:w_general_sqeezed_coherent} w(\bm x, t,\bm x',t') = w^\text{ind}(\bm x, t,\bm x',t') + w^\text{coh}(\bm x, t,\bm x',t'),\end{aligned}$$ where $w^\text{ind}$ is independent of the coherent amplitude $\alpha(\bm k)$, where $w^\text{coh}$ is given by a product of one-point functions: $$\label{eq:wcoh} w^\text{coh}(\bm x,t,\bm x',t') = v(\bm x,t)v(\bm x',t'),$$ and vanishes if $\alpha(\bm k)= 0$ for all $\bm k$. Even without calculating the $\alpha(\bm k)$-independent contribution $w^\text{ind}$ to the two-point function, we can see that it is the product of two one-point functions. In [@Simidzija2017c] it was shown that when this is the case, then the $\alpha(\bm k)$-dependent contributions of ${\hat{\rho}_\textsc{ab}}$ arising from the one-point function exactly cancel the contributions from the two-point function, so that the eigenvalues of ${\hat{\rho}_\textsc{ab}}$ and ${\hat{\rho}_\textsc{ab}^{{\text{\textbf{t}}}_\textsc{a}}}$ — and therefore the negativity $\mathcal N[{\hat{\rho}_\textsc{ab}}]$ as well — are completely independent of $\alpha(\bm k)$. This result was used in [@Simidzija2017c] to prove that the entanglement harvested by an Unruh-DeWitt detector pair is independent of the coherent amplitude of a (non-squeezed) coherent state. Since this is a general consequence of the special relationship between the $\alpha(\bm k)$-dependent parts of the one and two-point functions, we conclude that this result is true even in the presence of squeezing. Namely, to $\mathcal O(\lambda^2)$, the negativity of a detector pair interacting with a general squeezed coherent state ${| {\alpha(\bm k),\zeta(\bm k,\bm k')} \rangle}$ is independent of the coherent amplitude distribution $\alpha(\bm k)$. In other words, entanglement harvesting from a squeezed coherent state is insensitive to the coherent amplitude. Therefore, since we are interested in studying the entanglement harvested by the detector pair from a general squeezed coherent state, we can, without loss of generality, restrict our attention only to squeezed vacuum states (i.e. we can make $\alpha(\bm k)$ identically zero). Additionally, for mathematical simplicity—i.e. in order to obtain an explicit expression for ${\hat S_\zeta}^\dagger{\hat{a}_{\bm{k}}}{\hat S_\zeta}$ in Eq. —from here on we will consider only squeezed coherent states in which the squeezing is not “mixed" between modes, i.e. such that the squeezing amplitude is of the form $\zeta(\bm k,\bm k')=\zeta(\bm k)\delta(\bm k-\bm k')$. In this case we find that ${\hat S_\zeta}$ simplifies to $${\hat S_\zeta}=\exp\left[ \frac{1}{2}\int{ \mathrm{d} \ifx\relax3\relax\else \rule{-0.02em}{1.5ex}^{3}\rule{0.08em}{0ex}\! \fi \bm k\, }\left(\zeta^*(\bm k)\hat a_{\bm k}^2-\text{H.c.}\right) \right],$$ and that ${\hat S_\zeta}^\dagger{\hat{a}_{\bm{k}}}{\hat S_\zeta}$ can be conveniently expressed as $$\begin{aligned} {\hat S_\zeta}^\dagger{\hat{a}_{\bm{k}}}{\hat S_\zeta}&= \cosh[r(\bm k)]{\hat{a}_{\bm{k}}}-e^{{\mathrm{i}}\theta(\bm k)}\sinh[r(\bm k)]{\hat{a}_{\bm{k}}^\dagger},\end{aligned}$$ where we have written $\zeta(\bm k)=r(\bm k)e^{{\mathrm{i}}\theta(\bm k)}$ in polar form. The two-point function  of the state ${\hat S_\zeta}{| {0} \rangle}$, with ${\hat S_\zeta}$ in the above form, can be written as $$\begin{aligned} \label{eq:w_squeezed} w(\bm x, t,\bm x',t') = w^\text{vac}(\bm x, t,\bm x',t') + w^\text{sq}(\bm x, t,\bm x',t'),\end{aligned}$$ where $w^\text{vac}$ is the vacuum two-point function given in Eq. , while $w^\text{sq}$ is the contribution that depends on $\zeta(\bm k)$ and vanishes if $\zeta(\bm k)=0$ for all $\bm k$. Explicitly $w^\text{sq}(\bm x,t,\bm x',t')$ is given by $$\begin{aligned} \label{eq:wsq} &w^\text{sq}(\bm x,t,\bm x', t') = \int\frac{{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm k\, }}{2(2\pi)^n{\omega_{\bm{k}}}} \sinh[r(\bm k)] \\ &\hspace{0.5cm}\times \Big( \!\!-e^{{\mathrm{i}}\theta(\bm k)}\cosh[r(\bm k)] e^{-{\mathrm{i}}{\omega_{\bm{k}}}(t+t')}e^{{\mathrm{i}}\bm k\cdot(\bm x+\bm x'} \notag\\ &\hspace{1cm} +\sinh[r(\bm k)] e^{-{\mathrm{i}}{\omega_{\bm{k}}}(t-t')}e^{{\mathrm{i}}\bm k\cdot(\bm x-\bm x'} \Big) +\text{c.c},\end{aligned}$$ Notice that, unlike Eq.  for a thermal field state, the two-point function for a squeezed coherent state is not invariant with respect to spacetime translations. As we will see, a physical consequence of this is that the negativity harvested by a pair of UDW detectors from a squeezed coherent state depends not only on the spacetime interval between the detectors, but also on where in the spacetime they are centered. With the expression  for the two-point function of a squeezed vacuum field state, and with the vanishing one-point function , we can proceed to calculate the evolved state ${\hat{\rho}_\textsc{ab}}$ of the two UDW detectors following their interactions with this field. From  we obtain $$\label{eq:rhoab_squeezed} {\hat{\rho}_\textsc{ab}}= \begin{pmatrix} 1-\mathcal{L}_\textsc{aa}[\zeta]-\mathcal{L}_\textsc{bb}[\zeta]& 0 & 0 & \mathcal{M}^*[\zeta] \\ 0 & \mathcal{L}_\textsc{bb}[\zeta] & \mathcal{L}_\textsc{ab}^*[\zeta] & 0 \\ 0 & \mathcal{L}_\textsc{ab}[\zeta] & \mathcal{L}_\textsc{aa}[\zeta] & 0 \\ \mathcal{M}[\zeta] & 0 & 0 & 0 \end{pmatrix},$$ to second order in the coupling strength $\lambda$, and where we work in the basis $\{ {| {g_\textsc{a}} \rangle}{| {g_\textsc{b}} \rangle}, {| {g_\textsc{a}} \rangle}{| {e_\textsc{b}} \rangle}, {| {e_\textsc{a}} \rangle}{| {g_\textsc{b}} \rangle}, {| {e_\textsc{a}} \rangle}{| {e_\textsc{b}} \rangle}\}$. The matrix terms $\mathcal{L}_{\nu\eta}[\zeta]$ and $\mathcal M[\zeta]$ are now functionals of the squeezing distribution $\zeta(\bm k)$, and they take the forms $$\begin{aligned} \mathcal L_{\nu\eta}[\zeta]&= \mathcal L_{\nu\eta}^\text{vac} + \mathcal L_{\nu\eta}^\text{sq}[\zeta], \\ \mathcal M[\zeta]&=\mathcal M^\text{vac}+\mathcal M^\text{sq}[\zeta].\end{aligned}$$ As before, the vacuum terms $\mathcal L_{\nu\eta}^\text{vac}$ and $\mathcal M^\text{vac}$ are given by Eqs.  and , while the $\zeta(\bm k)$ dependent terms read $$\begin{aligned} \label{eq:L_squeezed_general} \mathcal L_{\nu\eta}^\text{sq}[\zeta] &= \pi\lambda_\nu\lambda_\eta \int\frac{{ \mathrm{d} \ifx\relax3\relax\else \rule{-0.02em}{1.5ex}^{3}\rule{0.08em}{0ex}\! \fi \bm k\, }}{{\omega_{\bm{k}}}} \Big( \sinh^2[r(\bm k)] \bar F_\nu(\bm k) \bar F_\eta^*(\bm k) \bar \chi_\nu^*({\omega_{\bm{k}}}-\Omega_\nu) \bar \chi_\eta({\omega_{\bm{k}}}-\Omega_\eta) e^{{\mathrm{i}}\bm k\cdot(\bm x_\nu-\bm x_\eta)} \\ &\hspace{3cm} + \sinh^2[r(\bm k)] \bar F_\nu^*(\bm k) \bar F_\eta(\bm k) \bar \chi_\nu({\omega_{\bm{k}}}+\Omega_\nu) \bar \chi_\eta^*({\omega_{\bm{k}}}+\Omega_\eta) e^{-{\mathrm{i}}\bm k\cdot(\bm x_\nu-\bm x_\eta)} \notag \\ &\hspace{3cm} - e^{-{\mathrm{i}}\theta(\bm k)} \sinh[r(\bm k)]\cosh[r(\bm k)] \bar F_\nu^*(\bm k) \bar F_\eta^*(\bm k) \bar \chi_\nu({\omega_{\bm{k}}}+\Omega_\nu) \bar \chi_\eta({\omega_{\bm{k}}}-\Omega_\eta) e^{-{\mathrm{i}}\bm k\cdot(\bm x_\nu+\bm x_\eta)} \notag \\ &\hspace{3cm} - e^{{\mathrm{i}}\theta(\bm k)} \sinh[r(\bm k)]\cosh[r(\bm k)] \bar F_\nu(\bm k) \bar F_\eta(\bm k) \bar \chi_\nu^*({\omega_{\bm{k}}}-\Omega_\nu) \bar \chi_\eta^*({\omega_{\bm{k}}}+\Omega_\eta) e^{{\mathrm{i}}\bm k\cdot(\bm x_\nu+\bm x_\eta)} \Big), \notag \\ \label{eq:M_squeezed_general} \mathcal M^\text{sq}[\zeta] &= 2\pi\lambda_\textsc{a}\lambda_\textsc{b} \int\frac{{ \mathrm{d} \ifx\relax3\relax\else \rule{-0.02em}{1.5ex}^{3}\rule{0.08em}{0ex}\! \fi \bm k\, }}{{\omega_{\bm{k}}}} \Big( e^{-{\mathrm{i}}\theta(\bm k)} \sinh[r(\bm k)]\cosh[r(\bm k)] \bar F_\textsc{a}^*(\bm k) \bar F_\textsc{b}^*(\bm k) \bar \chi_\textsc{a}({\omega_{\bm{k}}}+\Omega_\textsc{a}) \bar \chi_\textsc{b}({\omega_{\bm{k}}}+\Omega_\textsc{b}) e^{-{\mathrm{i}}\bm k\cdot(\bm x_\textsc{a}+\bm x_\textsc{b})} \\ &\hspace{3cm} + e^{{\mathrm{i}}\theta(\bm k)} \sinh[r(\bm k)]\cosh[r(\bm k)] \bar F_\textsc{a}(\bm k) \bar F_\textsc{b}(\bm k) \bar \chi_\textsc{a}^*({\omega_{\bm{k}}}-\Omega_\textsc{a}) \bar \chi_\textsc{b}^*({\omega_{\bm{k}}}-\Omega_\textsc{b}) e^{{\mathrm{i}}\bm k\cdot(\bm x_\textsc{a}+\bm x_\textsc{b})} \notag \\ &\hspace{3cm} - \sinh^2[r(\bm k)] \bar F_\textsc{a}^*(\bm k) \bar F_\textsc{b}(\bm k) \bar \chi_\textsc{a}({\omega_{\bm{k}}}+\Omega_\textsc{a}) \bar \chi_\textsc{b}^*({\omega_{\bm{k}}}-\Omega_\textsc{b}) e^{-{\mathrm{i}}\bm k\cdot(\bm x_\textsc{a}-\bm x_\textsc{b})} \notag \\ &\hspace{3cm} - \sinh^2[r(\bm k)] \bar F_\textsc{a}(\bm k) \bar F_\textsc{b}^*(\bm k) \bar \chi_\textsc{a}^*({\omega_{\bm{k}}}-\Omega_\textsc{a}) \bar \chi_\textsc{b}({\omega_{\bm{k}}}+\Omega_\textsc{b}) e^{{\mathrm{i}}\bm k\cdot(\bm x_\textsc{a}-\bm x_\textsc{b})} \Big). \notag\end{aligned}$$ Harvesting entanglement {#harvesting-entanglement} ----------------------- In order to study the dependence of field squeezing on the ability of detectors to harvest entanglement, let us once again particularize to the case of a massless field and identical UDW detectors with Gaussian spatial profiles of width $\sigma$, given by Eq. , and Gaussian temporal switching functions of width $\uptau$, as in Eq. . Then the matrix elements $\mathcal L_{\nu\eta}^\text{sq}[\zeta]$ and $\mathcal M^\text{sq}[\zeta]$ given by Eqs.  and  become $$\begin{aligned} \label{eq:L_squeezed_gaussian} \mathcal L_{\nu\eta}^\text{sq}[\zeta] &= \frac{\tilde\lambda^2 e^{-\frac{1}{2}\tilde\Omega^2}}{16\pi^2} \int\frac{{ \mathrm{d} \ifx\relax3\relax\else \rule{-0.02em}{1.5ex}^{3}\rule{0.08em}{0ex}\! \fi \tilde{\bm k}\, }}{|\tilde{\bm k}|} e^{-\frac{1}{2}|\tilde{\bm k}|^2(1+\tilde\sigma^2)} \Big( \sinh^2[r(\bm k)] e^{|\tilde{\bm k}|\tilde\Omega} e^{-{\mathrm{i}}|\tilde{\bm k}|(\tilde t_\nu-\tilde t_\eta)} e^{{\mathrm{i}}\tilde{\bm k}\cdot(\tilde{\bm x}_\nu-\tilde{\bm x}_\eta)} \\ &\hspace{5cm} + \sinh^2[r(\bm k)] e^{-|\tilde{\bm k}|\tilde\Omega} e^{{\mathrm{i}}|\tilde{\bm k}|(\tilde t_\nu-\tilde t_\eta)} e^{-{\mathrm{i}}\tilde{\bm k}\cdot(\tilde{\bm x}_\nu-\tilde{\bm x}_\eta)} \notag \\ &\hspace{5cm} - e^{-{\mathrm{i}}\theta(\bm k)} \sinh[r(\bm k)]\cosh[r(\bm k)] e^{{\mathrm{i}}|\tilde{\bm k}|(\tilde t_\nu+\tilde t_\eta)} e^{-{\mathrm{i}}\tilde{\bm k}\cdot(\tilde{\bm x}_\nu+\tilde{\bm x}_\eta)} \notag \\ &\hspace{5cm} - e^{{\mathrm{i}}\theta(\bm k)} \sinh[r(\bm k)]\cosh[r(\bm k)] e^{-{\mathrm{i}}|\tilde{\bm k}|(\tilde t_\nu+\tilde t_\eta)} e^{{\mathrm{i}}\tilde{\bm k}\cdot(\tilde{\bm x}_\nu+\tilde{\bm x}_\eta)} \notag \Big), \notag \\ \label{eq:M_squeezed_gaussian} \mathcal M^\text{sq}[\zeta] &= -\frac{\tilde\lambda^2 e^{-\frac{1}{2}\tilde\Omega^2}}{16\pi^2} \int\frac{{ \mathrm{d} \ifx\relax3\relax\else \rule{-0.02em}{1.5ex}^{3}\rule{0.08em}{0ex}\! \fi \tilde{\bm k}\, }}{|\tilde{\bm k}|} e^{-\frac{1}{2}|\tilde{\bm k}|^2(1+\tilde\sigma^2)} \Big( \sinh^2[r(\bm k)] e^{-{\mathrm{i}}|\tilde{\bm k}|(\tilde t_\textsc{a}-\tilde t_\textsc{b})} e^{{\mathrm{i}}\tilde{\bm k}\cdot(\tilde{\bm x}_\textsc{a}-\tilde{\bm x}_\textsc{b})} \\ &\hspace{5cm} + \sinh^2[r(\bm k)] e^{{\mathrm{i}}|\tilde{\bm k}|(\tilde t_\textsc{a}-\tilde t_\textsc{b})} e^{-{\mathrm{i}}\tilde{\bm k}\cdot(\tilde{\bm x}_\textsc{a}-\tilde{\bm x}_\textsc{b})} \notag \\ &\hspace{5cm} - e^{-{\mathrm{i}}\theta(\bm k)} \sinh[r(\bm k)]\cosh[r(\bm k)] e^{-|\tilde{\bm k}|\tilde\Omega} e^{{\mathrm{i}}|\tilde{\bm k}|(\tilde t_\textsc{a}+\tilde t_\textsc{b})} e^{-{\mathrm{i}}\tilde{\bm k}\cdot(\tilde{\bm x}_\textsc{a}+\tilde{\bm x}_\textsc{b})} \notag \\ &\hspace{5cm} - e^{{\mathrm{i}}\theta(\bm k)} \sinh[r(\bm k)]\cosh[r(\bm k)] e^{|\tilde{\bm k}|\tilde\Omega} e^{-{\mathrm{i}}|\tilde{\bm k}|(\tilde t_\textsc{a}+\tilde t_\textsc{b})} e^{{\mathrm{i}}\tilde{\bm k}\cdot(\tilde{\bm x}_\textsc{a}+\tilde{\bm x}_\textsc{b})} \notag \Big), \notag\end{aligned}$$ where, as before, we denote by a tilde any quantity referred to the scale $\uptau$ (e.g., $\tilde\Omega=\Omega\uptau$, $\tilde \sigma =\sigma/\uptau$, etc.). With these explicit expressions for the matrix elements of ${\hat{\rho}_\textsc{ab}}$ at hand, we can now readily compute the negativity $\mathcal N=\max\left(0,|\mathcal M[\zeta]|-\mathcal L_{\nu\nu}[\zeta]\right)$, and thus quantify the amount of entanglement that the two detectors harvest from the field. ### Uniform squeezing {#sec:uniform_squeezing} Let us begin by considering the simplest possible type of squeezing: that in which all field modes are squeezed equally. To that end we take $\zeta(\bm k)=r$, where we also assume that $r$ is real and positive. (We will shortly see what the effect is of $r$ having a complex phase.) ![Negativity of identical detectors as a function of their center of mass position, for different values of the squeezing parameter $r=|\zeta(\bm k)|$. Here the squeezing is uniform across all field modes. The detectors are coupled to the field through Gaussian switching functions of width $\uptau$ centered at $t=0$, and their energy gaps are $\Omega=\uptau^{-1}$. The detectors are centered at $(x_\textsc{com}\pm \uptau,0,0)$ and have Gaussian spatial profiles of width $\sigma=\uptau$.[]{data-label="fig:neg_vs_xcom"}](neg_vs_xcom.pdf){width="0.9\linewidth"} In Fig. \[fig:neg\_vs\_xcom\], for different values of $r$, we plot the negativity of the detectors following their interactions with the field as a function of their joint center of mass. We see that—as we anticipated already from the two-point function—a squeezed field state is in general not translationally invariant, and as such the entanglement harvesting ability of a pair of detectors from such a state is not translationally invariant either. In particular we find that if the detectors’ center of mass is near the spatial origin of the coordinate system, then the detectors can harvest more entanglement from a uniformly squeezed field state than from the vacuum. On the other hand if the detectors are far enough away from the origin, then, regardless of the amount of squeezing, they are unable to extract entanglement. The proximity to the origin that is necessary for squeezing to be beneficial for entanglement harvesting is dictated by the amount of squeezing $r$: for a highly squeezed field state the detectors can harvest a lot more entanglement, but they have to be highly centered near the origin; for a less squeezed state the improvement in harvesting is not as noticeable, but the detectors do not need to be so precisely centered. Let us now attempt to better understand the non-translation-invariance of squeezed field states in general, and in particular the consequences of this for entanglement harvesting from these states. Concretely, with regards to the plots in Fig. \[fig:neg\_vs\_xcom\], it is natural to ask why is the spatial origin of our chosen coordinate system the preferred location of UDW detectors that hope to harvest entanglement? First, let us note once again that, as can be seen in Fig \[fig:neg\_vs\_xcom\], in the absence of squeezing the translation-invariance of entanglement harvesting is restored. Therefore, the picking out of a preferred point in space near which entanglement harvesting is maximized (in this case the origin of the coordinate system) must be a direct consequence of the squeezing amplitude $\zeta(\bm k)$ that we choose for the field. In fact, we notice that the Fourier transform of the uniform amplitude $\zeta(\bm k)=r$ is proportional to $\delta(\bm x)$, and therefore the origin $\bm x=0$ is clearly a special point in this case. As we will now show, this relationship between the Fourier transform of the squeezing amplitude and the preferred location of detectors trying to harvest entanglement is valid in general. To that end, let us consider an arbitrary squeezing amplitude $\zeta(\bm k)$. With this choice of squeezing, there will be some preferred points in space near which it is easier for detectors to harvest entanglement, and others near which it is more difficult. Suppose now that we change the squeezing by a local phase $\zeta(\bm k) \rightarrow \zeta'(\bm k)=e^{{\mathrm{i}}\bm k\cdot\bm x_0} \zeta(\bm k)$. How do the positions of the preferred points change? To answer this question, let us recall from Eq.  that the state ${\hat{\rho}_\textsc{ab}}$ of the two detectors following their interactions with a squeezed field state with amplitude $\zeta'$ is given by $${\hat{\rho}_\textsc{ab}}={\text{Tr}}_\phi\left[\hat U'\left({\hat{\rho}_\textsc{a}}\otimes{\hat{\rho}_\textsc{b}}\otimes\hat S_{\zeta'}^\dagger{| {0} \rangle}{\langle {0} |}\hat S_{\zeta'}\right)\hat U'^\dagger\right],$$ where $\hat U'$ is the time-evolution unitary $$\begin{aligned} \hat{U'} = \mathcal{T}\exp \Big[ &-{\mathrm{i}}\!\int\!\dif t \sum_\nu \lambda_\nu \chi_\nu(t) \hat{\mu}_\nu(t) \\ &\times \int \!{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm{x}\, } F_\nu(\bm{x}-\bm{x}_\nu) \hat{\phi}(\bm{x},t) \Big]. \notag\end{aligned}$$ Now let us define the field *momentum operator* to be $\hat{\bm P}:=\int{ \mathrm{d} \ifx\relax3\relax\else \rule{-0.02em}{1.5ex}^{3}\rule{0.08em}{0ex}\! \fi \bm k\, } \bm k{\hat{a}_{\bm{k}}^\dagger}{\hat{a}_{\bm{k}}}$. Then, using the fact that $$\label{eq:translation_of_a} e^{{\mathrm{i}}\hat{\bm P}\cdot \bm x_0}{\hat{a}_{\bm{k}}} e^{-{\mathrm{i}}\hat{\bm P}\cdot \bm x_0}={\hat{a}_{\bm{k}}} e^{{\mathrm{i}}\bm k\cdot\bm x_0},$$ we find that we can write $\hat S_{\zeta'}=e^{-{\mathrm{i}}\hat{\bm P}\cdot \bm x_0/2}\hat S_{\zeta}e^{{\mathrm{i}}\hat{\bm P}\cdot \bm x_0/2}$. Making use of the cyclicity of the partial trace with respect to the subsystem being traced over, we find that ${\hat{\rho}_\textsc{ab}}$ can be expressed as $${\hat{\rho}_\textsc{ab}}={\text{Tr}}_\phi\left[\hat U\left({\hat{\rho}_\textsc{a}}\otimes{\hat{\rho}_\textsc{b}}\otimes\hat S_{\zeta}^\dagger{| {0} \rangle}{\langle {0} |}\hat S_{\zeta}\right)\hat U^\dagger\right],$$ where $\hat U:=e^{{\mathrm{i}}\hat{\bm P}\cdot \bm x_0/2}\hat U'e^{-{\mathrm{i}}\hat{\bm P}\cdot \bm x_0/2}$. Using  we readily obtain $$\begin{aligned} \hat{U} = \mathcal{T}\exp \Big[ &-{\mathrm{i}}\!\int\!\!\dif t\! \sum_\nu \lambda_\nu \chi_\nu(t) \hat{\mu}_\nu(t) \\ &\times \int \!{ \mathrm{d} \ifx\relaxn\relax\else \rule{-0.02em}{1.5ex}^{n}\rule{0.08em}{0ex}\! \fi \bm{x}\, } F_\nu\left(\bm{x}-\bm{x}_\nu-\frac{\bm x_0}{2}\right) \hat{\phi}(\bm{x},t) \Big]. \notag\end{aligned}$$ Hence changing the field’s squeezing amplitude by a local phase $\zeta \rightarrow e^{{\mathrm{i}}\bm k\cdot\bm x_0} \zeta$ is equivalent to shifting the detectors in space by an amount $\bm x_0/2$. In other words, a local phase change of the squeezing amplitude effects a translation of the points in space near which it is easier for the detectors to harvest entanglement. However, such a local phase change of $\zeta$ also effects a translation of its Fourier transform: namely $\bar\zeta(\bm x)\rightarrow \bar \zeta(\bm x-\bm x_0)$. Note that the discrepancy by a factor of 2 between the amount that the preferred points are translated ($\bm x_0/2$) and the amount that the Fourier transform $\bar\zeta$ is shifted by ($\bm x_0$) can be removed by choosing a different convention for the exponent in the definition  of a Fourier transform. Therefore we conclude that (up to a potential re-scaling) the Fourier transform of the field’s squeezing amplitude $\zeta$ directly tells us where in space the UDW detectors should be centered if they want to harvest more entanglement from the squeezed field state. These preferred locations are commensurate with where the fluctuations of the field amplitude, and the stress energy density, are localized in space. Having expounded the dependence of the detectors’ center of mass on their ability to harvest entanglement from a squeezed field state, and having related this to the local phase of the squeezing amplitude, let us now turn to the question of how the *magnitude* of the squeezing amplitude affects the detector’s abilities to harvest entanglement. In Fig. \[fig:neg\_vs\_r\] we plot the negativity of a UDW detector pair as a function of $\zeta(\bm k)=r$, which we once again assume to be uniform across all field modes. We notice several interesting features from these plots. Interestingly, high squeezing can remove the dependence of entanglement harvesting on the distance between the detectors. Indeed, we find that while at low squeezing amplitude the amount of entanglement that the detectors can harvest depends on their spatial separation $d:=|\bm x_\textsc{a}-\bm x_\textsc{b}|$, at high squeezing this is not the case. In other words, in the limit of large uniform squeezing of the field, a detector pair separated by a large spatial distance will harvest the same amount of entanglement as if they were at the same location in space. A similar effect of removal of the distance scale in a setup where vacuum entanglement is relevant was seen in [@Hotta2014] where quantum energy teleportation could be made independent of separation between sender and receiver if one uses squeezed field states. Furthermore, from Fig. \[fig:neg\_vs\_r\], we find that the amount of entanglement that the detectors harvest is also independent of the squeezing parameter $\zeta(\bm k)=r$ in the limit as $r\rightarrow\infty$. Hence although squeezing the field modes often increases the amount of harvestable entanglement from that allowed by the field vacuum, this trend of increasing negativity does not continue indefinitely, but rather plateaus to a constant asymptotic value at large $r$. ### Bandlimited squeezing {#sec:bandlimited_squeezing} To an experimentalist looking to make an entanglement harvesting measurement in the lab, perhaps the most interesting results of the previous section are that i) the amount of entanglement harvested by a pair of UDW detectors from a highly (uniformly) squeezed field state is *independent* of the spatial separation of the detectors, and ii) if the detectors are centered near the “preferred" locations in space (as determined by the Fourier transform of the squeezing function $\zeta(\bm k)$), then the amount of entanglement that they harvest could be much higher than in the case of a vacuum field state. However such an experimentalist would be quick to note that there is an obvious difficulty with attempting to translate the theoretical results of the previous section into an actual experiment in the lab. Namely, in the previous section we assumed the field to be uniformly squeezed across all field modes, while squeezed states in experimental quantum optics [@Bachor2004] and superconducting setups [@Zagoskin2008] are generally bandlimited to a very narrow range of field modes. We expect that in this case, where only a narrow frequency range of modes are squeezed, the field state will behave more similarly to the vacuum state, in which case squeezing might not give much of an advantage in terms of entanglement harvesting. The key question is then: what range of field modes must be squeezed in order to produce a significant entanglement harvesting advantage over the vacuum state? To answer this question, let us now assume that only the field modes *near* some momentum $\bm k$ are uniformly squeezed, while all other modes are in their vacuum states. More precisely, we set $$\label{eq:bandlimited_squeezing} \zeta(\bm k') = \begin{cases} r & \text{if }|k'_i-k_i|<\frac{\epsilon}{2} \text{ for } i\in\{x,y,z\}\\ 0 & \text{otherwise} \end{cases},$$ where $\bm k' = (k'_x,k'_y,k'_z)$, $\bm k = (k_{x},k_{y},k_{z})$, and $\epsilon$ parametrizes the bandwidth of the squeezing. With this choice of squeezing amplitude, and assuming again that the spatial and temporal profiles of the detectors are Gaussians given by Eqs.  and , the matrix elements $\mathcal L_{\nu\eta}^\text{sq}[\zeta]$ and $\mathcal M^\text{sq}[\zeta]$ of the evolved two detector density matrix ${\hat{\rho}_\textsc{ab}}$ are again given by the expressions in Eqs.  and , except that now the limits of momentum space integration are such that $|k'_i-k_{i}|<\epsilon/2$. With the use of these expressions we can compute the negativity $\mathcal N=\max\left(0,|\mathcal M[\zeta]|-\mathcal L_{\nu\nu}[\zeta]\right)$, and thus observe how the amount of entanglement that the detectors can harvest depends on the bandwidth $\epsilon$ of the field’s squeezing amplitude. However before showing plots of $\mathcal N$ versus $\epsilon$, since we are in this section trying to upgrade our theoretical findings to the realm of what is experimentally feasible, it is important that we also discuss what values of squeezing amplitude $r$ we can expect to obtain in our bandlimited frequency range. As far as we are aware, the highest experimentally attained squeezed state of the electromagnetic field resulted in a squeezed quadrature noise reduction of 15 dB below the vacuum level [@Henning2016]. Using the conversion formula [@Lvovsky2015] $$\Delta\text{Noise} \text{ (in dB)}= 10 \log_{10}\left(2\langle\Delta \hat X^2\rangle\right),$$ between the reduction in noise of the squeezed quadrature $\hat X$ and the variance $\langle\Delta \hat X^2\rangle:=\langle\hat X^2\rangle-\langle \hat X\rangle^2$ of that quadrature in the squeezed state ${| {\zeta(\bm k)} \rangle}$, as well as the expression $$\langle\Delta \hat X^2\rangle = \frac{1}{2}e^{-2r},$$ between $\langle\Delta \hat X^2\rangle$ and $r$, we find that $$\Delta\text{Noise} \text{ (in dB)} = -20\log_{10}(e)r.$$ Hence a noise reduction of 15 dB corresponds to a squeezing amplitude of $r\approx 1.7$. To be on the safe side with respect to experimental feasibility, we will for the below discussion set $r=1$ (corresponding to $\sim 8.7$ dB). In Fig. \[fig:neg\_vs\_epsilon\] we plot the dependence of the negativity that two UDW detectors can harvest from the field, as a function of the bandwidth $\epsilon$ of field modes that are squeezed (we assume the squeezed modes to be centered around some wavevector $\bm k$). In the top plot of this figure, we suppose that the detectors are near enough in space such that they are able to harvest entanglement from the field vacuum ($\epsilon=0$). Perhaps unintuitively, we find that as we start squeezing around the mode $\bm k$ (i.e. we increase $\epsilon$), the negativity of the detectors initially begins to decrease. That is, for a small bandwidth $\epsilon$ of field squeezing, regardless of the mode $\bm k$ around which the squeezing is being performed, the amount of entanglement that the detectors can harvest from the field is actually less than what they could harvest from the vacuum. Eventually however, as the bandwidth is increased further, the amount of entanglement that the detectors can harvest from the field becomes higher than in the vacuum case. Meanwhile, detectors with a large spatial separation (bottom plot of Fig. \[fig:neg\_vs\_epsilon\]) are unable to harvest entanglement from the vacuum ($\epsilon=0$), as was already shown in Ref. [@Pozas2015]. In this case increasing the squeezing bandwidth allows the detectors to harvest some entanglement, but this only occurs for $\epsilon$ larger than some critical value $\epsilon_c$. Hence, regardless of separation, the ability of a pair of UDW detectors to harvest more entanglement from a squeezed field state than from the vacuum is dependent on whether a large enough frequency interval of field modes is squeezed, i.e. if the bandwidth $\epsilon$ is larger than some critical value $\epsilon_c$. We notice from the plots in Fig. \[fig:neg\_vs\_epsilon\] that the critical bandwidth $\epsilon_c$ necessary to achieve an improvement in entanglement harvesting over the vacuum is at least of the order $|\bm k|$, where $\bm k$ is the wavevector of the mode around which we squeeze. Hence for instance if we wanted to use a 300 THz squeezed laser source to entangle a pair of atomic detectors, we would need to squeeze all the modes up to 600 THz with wavevectors pointing in the direction of the laser, as well a wide range of field modes pointing in other directions. As far as we are aware, current experimental setups featuring squeezed electromagnetic field states do not squeeze such large bandwidths of field modes. Hence, in order to make use of the benefits of squeezed field states with respect to entanglement harvesting, it may be necessary to increase the experimentally achievable squeezing bandwidth. Alternatively, it might still be possible to obtain high levels of harvestable entanglement with narrowly bandlimited squeezed states, but for which the squeezing amplitude $\zeta(\bm k)$ is non-uniform in the bandlimited range. This remains to be investigated in future work. Conclusions {#sec:conclusions} =========== We studied the ability of a pair of Unruh DeWitt particle detectors to harvest quantum and classical correlations from thermal and squeezed states of a scalar field with which they interact. We find several interesting results: First, we prove that the amount of entanglement that a pair of identical detectors (with arbitrary spatial profiles and time-dependent switching functions) can harvest from a thermal state of the field decreases monotonically with temperature. Additionally, we obtain a lower bound on this rate of decrease, and hence show that for temperatures higher than a certain threshold the detectors are unable to harvest any entanglement from the field. With these findings we also extend the main results in [@Brown2013a], where it was numerically shown (using the very different formalism of Gaussian quantum mechanics) that temperature is detrimental to entanglement harvesting by harmonic oscillator detectors from a massless field in 1+1 dimensional spacetime. Indeed, we prove that this is also the case for qubit detectors of arbitrary shape and switching interacting with a field of any mass in any dimensionality of spacetime. On the other hand, we find that unlike the negativity, the mutual information — which is a measure of the total (quantum and classical) correlations — that the detectors harvest from the field actually increases linearly with the field temperature (again extending the numerical findings of [@Brown2013a] to qubit detectors). Hence, while thermal noise hinders the ability of UDW detectors to harvest entanglement, it is beneficial in the harvesting of non-entanglement correlations. Moving on to squeezed field states, we start by proving that, at least to leading perturbative order, the amount of entanglement that a UDW detector pair can harvest from a squeezed coherent state is independent of its coherent amplitude. This greatly generalizes the result of Ref. [@Simidzija2017b], which considered only unsqueezed coherent states, to hold for all general squeezed coherent states. We also show that, unlike the coherent amplitude, the field’s squeezing amplitude $\zeta(\bm k)$ *does* affect the amount of entanglement that the detectors can harvest from the field. In particular, we find that the amount of entanglement that detectors centered at a spatial point $\bm x_0$ can harvest is directly related to the amplitude of the Fourier transform of $\zeta(\bm k)$ evaluated at $\bm x_0$. Hence, contrary to vacuum [@Pozas2015], coherent [@Simidzija2017b], and thermal states, harvesting entanglement from general squeezed states is generally *not* a translationally invariant process. However, and perhaps surprisingly, we find that for detectors centered at a particular location $\bm x_0$, the amount of entanglement harvested from a highly and uniformly squeezed state is independent of the spatial separation of the detectors. Moreover, this amount of entanglement is often much larger than detectors at the same separation would be able to harvest from the vacuum, raising the idea of the possibility of using squeezed states to experimentally test entanglement harvesting. This result is commensurate with the finding that squeezed states can remove the distance decay of protocols that rely on field entanglement such as quantum energy teleportation [@Hotta2014]. Finally, we have also studied how entanglement harvesting is modified when we allow for squeezing only in a finite frequency bandwidth of field modes. We find that if we restrict the modes of the field that are squeezed to a narrow bandwidth (namely, when the bandwidth is below the order of the frequency being squeezed), then squeezing states give no noticeable advantage over vacuum entanglement harvesting, at least for uniform squeezing. It remains to be seen whether a more general squeezing amplitude (e.g. with continuously varying magnitude and phase) can provide the necessary advantages in entanglement harvesting that we have found here for uniform squeezing, while at the same time being implementable in a lab setting. This is an important direction for future research, since such a squeezed field state could overcome the main experimental limitation of entanglement harvesting: the fast decay with detector separation. P.S. gratefully acknowledges the support of the NSERC CGS-M and Ontario Graduate Scholarships. E.M.-M. acknowledges the funding from the NSERC Discovery program and his Ontario Early Research Award. Thermal two-point function {#app:thermal_wightman} ========================== We will show that our expression for the thermal two-point function in Eq.  reduces to the special case in Eq.  when $m=0$, $n=3$, and $\bm x' = t' = 0$. Let us first evaluate the second term in Eq. , $w_\beta(\bm x,t,0,0)$, which is given in Eq. . Working in polar coordinates, with $k:=|\bm k|$ and $r:=|\bm x|$, we straightforwardly obtain $$\begin{aligned} \label{eq:wbeta2} w_\beta(\bm x,t,0,0) &= \frac{1}{2\pi^2 r}\int_0^\infty \frac{\dif k}{e^{\beta k}-1} \sin(kr)\cos(kt)\notag\\ &= \mathcal P\Bigg( -\frac{1}{4\pi^2(r^2-t^2)} \\ & +\!\frac{1}{8\pi r\beta}\! \!\left[\coth\!\!\left(\frac{\pi(r+t)}{\beta}\!\right)\!\!+\coth\!\!\left(\frac{\pi(r-t)}{\beta}\!\right)\!\! \right]\!\! \Bigg)\!, \notag\end{aligned}$$ where $\mathcal P$ denotes the principal value of the integral (this expression only has meaning as a distribution). Interestingly, notice that the last term does not depend on the temperature. We can similarly calculate the first term in Eq. , $w_0(\bm x,t,0,0)$, which is given in Eq. . We obtain $$\begin{aligned} w_0(\bm x,t,0,0) &= \frac{1}{8\pi^2{\mathrm{i}}r} \int_0^\infty\!\!\dif k \left( e^{-{\mathrm{i}}k(t-r)}-e^{-{\mathrm{i}}k(t+r)} \right)\notag\\ &= \frac{1}{8\pi^2{\mathrm{i}}r} \lim_{s\rightarrow\infty} \int_0^s\!\!\!\dif k \left(\! e^{-{\mathrm{i}}k(t-r)}-e^{-{\mathrm{i}}k(t+r)} \!\right)\notag\\ &= \mathcal P\left(\frac{1}{4\pi^2(r^2-t^2)}\right) +\lim_{s\rightarrow\infty} \frac{1}{8\pi^2 r} \label{eq:w0_intermediate}\\ &\phantom{=}\times \Bigg[\frac{{\mathrm{i}}\sin\left(s(r+t)\right)}{r+t}-\frac{{\mathrm{i}}\sin\left(s(r-t)\right)}{r-t} \notag\\ &\phantom{===}- \frac{\cos\left(s(r+t)\right)}{r+t}-\frac{\cos\left(s(r-t)\right)}{r-t} \Bigg].\notag\end{aligned}$$ Notice that although these limits do not converge as real functions, they do converge as distributions on test functions. Namely we have $$\begin{aligned} \lim_{s\rightarrow\infty} \frac{\sin(s x)}{\pi x}&= \delta(x), \\ \lim_{s\rightarrow\infty} \frac{\cos(s x)}{\pi x}&= 0 =\text{the zero distribution}.\end{aligned}$$ Hence Eq.  simplifies to $$\begin{aligned} \label{eq:w0_2} w_0(\bm x,t,0,0) =& \mathcal P\left(\frac{1}{4\pi^2(r^2-t^2)}\right) \notag\\ &+ \frac{{\mathrm{i}}}{8\pi r} \left[\delta^{(3)}(r+t)-\delta^{(3)}(r-t)] \right],\end{aligned}$$ where it should again be emphasized that the principal value and the delta functions only make sense as distributions. Finally, combining Eqs.  and , we find that for a massless field in $(3+1)$-dimensions our expression for the two-point function, Eq. , reduces to the distribution in Eq. , which was obtained in [@Weldon2000] by a completely different method.

--- author: - | Shuchi Chawla\ UW-Madison\ [shuchi@cs.wisc.edu]{} - | Evangelia Gergatsouli\ UW-Madison\ [gergatsouli@wisc.edu]{} - | Yifeng Teng\ UW-Madison\ [yifengt@cs.wisc.edu]{} - | Christos Tzamos\ UW-Madison\ [tzamos@wisc.edu]{} - | Ruimin Zhang\ UW-Madison\ [rzhang274@wisc.edu]{} bibliography: - 'allrefs.bib' title: Learning Optimal Search Algorithms from Data --- Extension to other feasibility constraints {#sec:extensions} ========================================== In this section we extend the problem in cases where there is a feasibility constraint $\mathcal{F}$, that limits what or how many boxes we can choose. We consider the cases where we are required to select $k$ distinct boxes, and $k$ independent boxes from a matroid. In both cases we design SPA strategies that can be converted to PA. These two variants are described in more detail in subsections \[subsec:generalK\] and \[subsec:matroids\] that follow.

--- abstract: 'An Archimedean copula is characterised by its generator. This is a real function whose inverse behaves as a survival function. We propose a semiparametric generator based on a quadratic spline. This is achieved by modelling the first derivative of a hazard rate function, in a survival analysis context, as a piecewise constant function. Convexity of our semiparametric generator is obtained by imposing some simple constraints. The induced semiparametric Archimedean copula produces Kendall’s tau association measure that covers the whole range $(-1,1)$. Inference on the model is done under a Bayesian approach and for some prior specifications we are able to perform an independence test. Properties of the model are illustrated with a simulation study as well as with a real dataset.' author: - | [Ricardo Hoyos & Luis Nieto-Barajas]{}\ [*Department of Statistics, ITAM, Mexico*]{}\ \ title: '**A Bayesian semiparametric Archimedean copula**' --- [*Keywords*]{}: Archimedean copula, Bayes nonparametrics, piecewise constant, survival analysis, quadratic spline. [*AMS Classification*]{}: 60E05 $\cdot$ 62G05 $\cdot$ 62N86. Introduction {#sec:intro} ============ Let $\varphi(\cdot)$ be a continuous, strictly decreasing function from $[0,1]$ to $[0,\infty)$ such that $\varphi(1)=0$. Let $\varphi^{-1}(\cdot)$ be the inverse or the pseudo-inverse of $\varphi$, where the latter is defined as zero for $t>\varphi(0)$. If $\varphi(0)=\infty$ the generator is called strict. An Archimedean copula $C(u,v)$ with generator $\varphi$ is a function from $[0,1]^2$ to $[0,1]$ defined as $$\label{eq:arqc} C(u,v)=\varphi^{-1}\left(\varphi(u)+\varphi(v)\right).$$ A further requirement for to be well defined is that $\varphi$ must be convex [e.g. @nelsen:06]. There are many properties that characterize Archimedean copulas, for instance, they are symmetric, associative and their diagonal section $C(u,u)$ is always less than $u$ for all $u\in(0,1)$. Generators $\varphi(\cdot)$ are usually parametric families defined by a single parameter. Most of them are summarised in @nelsen:06 [Table 4.1]. Association measures induced by Archimedean copulas are a function of the generator. For instance, Kendall’s tau becomes $$\label{eq:ktau} \kappa_\tau=1+4\int_0^1\frac{\varphi(t)}{\varphi^{\prime}(t^+)}\d t,$$ where $\varphi^{\prime}(t^+)$ denotes right derivative of $\varphi$ at $t$. In this work we propose a Bayesian semiparametric generator defined through a quadratic spline. Within a survival analysis context, we model the first derivative of a hazard rate function with a piecewise constant function. The hazard rate and the cumulative hazard functions become linear and quadratic continue functions, respectively. The induced survival function is used as an inverse generator for an Archimedean copula. Convexity constrains are properly addressed and inference on the model is done under a Bayesian approach. Other studies on semiparametric generators for Archimedean copulas can be found in where their model is based on an empirical Kendall’s process. A new approach and extensions of this latter methodology can be found in . In the model arises from the one-to-one correspondence between an Archimedean generator and a distribution function of a nonnegative random variable. In particular they use a mixture of Pólya trees as a prior for the corresponding distribution function under a Bayesian nonparametric approach. In a work more related to ours, use the relationship between quantile functions and Archimedean generators to define a semiparametric generator by supplementing a parametric generator with $n+1$ dependence parameters. Differing to their work, our model is not based on any parametric generator and the Kendall’s tau can take values on the whole interval $(-1,1)$. The contents of the rest of the paper is as follows. In Section \[sec:model\] we present our proposal and characterise its properties. In Section \[sec:post\] we provide details of how to make posterior inference under a Bayesian approach. In Section \[sec:illust\] we illustrate the performance of our model with a simulation study as well as with a real data set. We conclude with some remarks in Section \[sec:concl\]. Model {#sec:model} ===== To define our proposal we realise that $\varphi^{-1}$ is a decreasing function from $[0,\infty)$ to $[0,1]$, so it behaves as a survival function, in a failure time data analysis context . The idea is to propose a semi/non parametric form for the inverse generator $\varphi^{-1}$ by using survival analysis ideas. For that we recall some basic definitions. Let $h(t)$ be a nonnegative function with domain in $[0,\infty)$ such that $H(t)=\int_0^t h(s)\d s\to\infty$ as $t\to\infty$. Then $S(t)=\exp\{-H(t)\}$ is a decreasing function from $[0,\infty)$ to $[0,1]$, so it behaves like an inverse generator $\varphi^{-1}(t)$. In a survival analysis context, functions $h(\cdot)$, $H(\cdot)$ and $S(\cdot)$ are the hazard rate, cumulative hazard and survival functions, respectively. In particular, if $h(t)=\theta$, i.e. constant for all $t$, then $S(t)=e^{-\theta t}$. If we take $\varphi(t)^{-1}=e^{-\theta t}$, then $\varphi(t)=-(\log t)/\theta$. Using we obtain that the resulting copula $C(u,v)=uv$ is the independence copula, and what is interesting, is that it does not depend on $\theta$. Using these ideas we construct a semiparametric generator in the following way. We first consider a partition of size $K$ of the positive real line, with interval limits given by $0=\tau_0<\tau_1<\cdots<\tau_K=\infty$. Then, we define the first derivative of the hazard rate, as a piecewise constant function of the form $$\label{eq:hp} h'(t)=\sum_{k=1}^K \theta_k I(\tau_{k-1}<t\leq\tau_k),$$ where $\theta_K\equiv 0$. We recover the hazard rate function as $h(t)=\int_0^t h'(s)\d s+\theta_0$, where $h(0)=\theta_0>0$ is an initial condition. Using , the hazard rate becomes a piecewise linear function of the form $$\label{eq:h} h(t)=\sum_{k=1}^K \left(A_k+\theta_k t\right) I(\tau_{k-1}<t\leq\tau_k),$$ where $A_1=\theta_0$ and $A_k=\theta_0+\sum_{j=1}^{k-1}(\theta_j-\theta_{j+1})\tau_j$, for $k=2,\ldots,K$. Integrating now the hazard function , the cumulative hazard is a piecewise quadratic function given by $$\label{eq:H} H(t)=\sum_{k=1}^K \left(B_k+A_k t+\frac{\theta_k}{2} t^2\right) I(\tau_{k-1}<t\leq\tau_k),$$ where $B_1=0$ and $B_k=\sum_{j=2}^k(\theta_j-\theta_{j-1})\tau_{j-1}^2/2$, for $k=2,\ldots,K$. We therefore define a semiparametric inverse generator as the induced survival function, which can be written as $$\label{eq:phiinv} \varphi^{-1}(t)=\exp\{-H(t)\},$$ where $H(t)$ is given in . After doing some algebra, we can invert this function to obtain an expression for the generator $$\begin{aligned} \nonumber \varphi(t)=\sum_{k=1}^{K}&\left(\left[\operatorname{sgn}(\theta_k)\left\{\frac{2}{\theta_k}\left(\frac{A_k^2}{2\theta_k}-B_k-\log(t)\right)\right\}^{1/2}-\frac{A_k}{\theta_k}\right]I(\theta_k\neq 0)\right.\\ \nonumber &\left.\hspace{5mm}-\frac{B_k+\log(t)}{A_k}I(\theta_k=0)\right) I\left(\varphi^{-1}(\tau_k)\leq t< \varphi^{-1}(\tau_{k-1})\right). \\ \label{eq:phi}\end{aligned}$$ The value $K$ controls the flexibility of the generator, and thus of the copula. If $K=1$, the induced Archimedean copula is the independent copula, whereas for larger $K$, the generator, and the induced copula, become more nonparametric. Potentially $K$ could be infinite. We now discuss some properties of our semiparametric generator. \[prop:1\] Consider the semiparametric inverse generator $\varphi^{-1}(t)$, given in , and the corresponding generator $\varphi(t)$, given in , and assume that $\{\theta_k\}$ are such that $\theta_0>0$, $\theta_K=0$ and satisfy conditions (C1) and (C2) given by 1. $A_k+\theta_k t\geq 0$, for $t\in(\tau_{k-1},\tau_k]$ and for all $k=1,\ldots,K$. 2. $(A_k + \theta_k t)^2 > \theta_k$, for $t\in(\tau_{k-1},\tau_k]$ and for all $k=1,\ldots,K$. Then, 1. $\varphi^{-1}(t)$ and $\varphi(t)$ are continuous functions of $t$, 2. $\varphi^{-1}(t)$ is a convex function, 3. $\varphi(t)$ a strict generator. For (i) we know that $h'(t)$, as in , is a piecewise constant discontinuous function, however, function $h(t)$, as in , is continuous. To see this, for each $k=1,\ldots,K$, the limit from the left is $\lim_{t\to\tau_k^{-}} h(t) = \lim_{t \to \tau_k^{-}} A_k + \theta_k t = A_k + \theta_k \tau_k$, and the limit from the right becomes $\lim_{t \to \tau_k^{+}} h(t) = \lim_{t \to \tau_k^{+}} A_{k+1} + \theta_{k+1} t = A_{k+1} + \theta_{k+1} \tau_k$. Since $A_{k+1} = A_{k} + (\theta_{k} - \theta_{k+1})\tau_{k}$, then both limits coincide. For (ii) we take the second derivative of $\varphi^{-1}(t)$ which becomes $\varphi^{-1(\prime \prime)}(t) = \{h(t)\}^2 \exp\{-H(t)\} - h^{\prime}(t)\exp\{-H(t)\}$, this is positive if and only if $\{h(t)\}^2-h'(t)>0$. For this to happen we require condition $(C2)$. For (iii), $\varphi^{-1}(t)$ must be a proper survival function, that is, $h(t)$ must be nonnegative, which is achieved by imposing condition $(C1)$. Furthermore, we need $\lim_{t\to\infty}\varphi^{-1}(t)=0$, which is equivalent to prove that $\lim_{t\to\infty}H(t)=\lim_{t\to\infty}\left(B_K+A_K t+\theta_K t^2/2\right)=\infty$. This is true since $B_K$ is a finite constant, $A_K>0$ and $\theta_K=0$, so the linear part goes to infinity when $t\to\infty$. To see the kind of association induced by our proposal, we computed the Kendall’s tau using expression with generator . This is given in the following result. \[prop:kt\] The Kendall’s tau obtained by the Archimedean copula with semiparametric generator is given by $$\kappa_\tau = -1+2\sum_{k=1}^K A_k \int_{\tau_{k-1}}^{\tau_k}\exp\left(-2B_k-2A_k t-\theta_k t^2\right)\d t.$$ Moreover, this $\kappa_\tau\in(-1,1)$. Rewriting expression in terms of the inversed generator we obtain $\kappa_\tau=1-4\int_0^\infty t\{\varphi^{-1(\prime)}(t)\}^2\d t$. Computing the derivative we get $\varphi^{-1(\prime)}(t)=-\sum_{k=1}^K(A_k+\theta_k t)\times$ $\exp\{-(B_k + A_k t+\theta_k t/2)\}I(\tau_{k-1}<t\leq\tau_k)$. Doing the integral we obtain the expression. To obtain the range of possible values of $\kappa_\tau$ it is easier to re-write $\kappa_\tau$ in terms of $h(t)$ and $H(t)$. This becomes $\kappa_{\tau} = -2 \int_0^{\infty} t h^{\prime}(t)\exp\{-H(t)\}\,dt$. Here it is straightforward to see that the sign of $\kappa_\tau$ is determined by the sign of $h'(t)$, therefore $h'(t)>0$ for all $t$ implies $-1 < \kappa_{\tau} < 0$ and $h'(t)\le 0$ implies $0 \le\kappa_{\tau}<1$. The expression for $\kappa_\tau$ tells us that the concordance induced by our semiparametric copula is a function of both, the parameters $\{\theta_k\}$, as well as of the partition limits $\{\tau_k\}$. It depends on a definite integral and can be evaluated numerically. What is more important is that $\kappa_\tau$ covers the whole range from $-1$ to $1$, showing that our proposal is very flexible. To illustrate the flexibility of our model we define a partition of the positive real line of size $K=10$, such that $\tau_k=-\log(1-k/10)$ for $k=0,1,\ldots,10$. We consider two scenarios for the values of the parameters $\{\theta_k\}$. The first scenario is defined by $\theta_k<0$ for all $k$, whereas the second scenario contains $\theta_k>0$ for all $k$. Conditions $(C1)$ and $(C2)$ were satisfied in both cases. Figure \[fig:ilust1\] contains functions $h'(t)$, $H(t)$ and $\varphi^{-1}(t)$ for two different scenarios, the solid (blue) line corresponds to the first scenario and the dotted (red) line to the second scenario. In the first case the corresponding hazard function (middle panel) is decreasing, whereas for the second case the hazard function is increasing. The induced concordance values are $\kappa_\tau=0.368$ and $\kappa_\tau=-0.202$, respectively. As a second example, we consider a partition of size $K=50$, such that $\tau_k = -\log(1 - k/50)$ for $k=0,1,\ldots,50$. We consider three different scenarios for the parameters $\{\theta^{(i)}_k\}$ with $i=1,2,3$, respectively. In the first scenario we assume $\theta_1^{(1)}\sim \un(-1,1)$, in the second $\theta_1^{(2)}\sim \un(-50,0)$ and in the third $\theta_1^{(3)}\sim \un(0,1)$. Posteriorly, we define sequentially $\theta_k^{(i)}\sim \un(a_k^{(i)},b_k^{(i)})$ with $a_k^{(i)}$ and $b_k^{(i)}$ constants such that constrains $(C1)$ and $(C2)$ are satisfied, for $k=2,\ldots,K-1$ and $i=1,2,3$. We repeated sampling from these distributions a total of 5,000 times, and for each repetition we computed $\kappa_\tau$. The induced histogram densities for the three scenarios are presented in Figure \[fig:simkt\]. For the first scenario, the values of $\kappa_\tau$ range from $-0.3$ to $0.4$, showing that our model can capture both negative and positive concordance measures. For the second scenario, the values of $\kappa_\tau$ are all positive and the distribution is right skewed, and for the third scenario the values of $\kappa_\tau$ are all negative showing a left skewed distribution. According to [@nelsen:06], new generators can be defined if we apply a scale transformation of the form $\phi^{-1}(t)=\varphi^{-1}(\alpha t)$ iff $\phi(t)=\varphi(t)/\alpha$, for $\alpha>0$, where $\phi(t)$ becomes a new Archimedean copula generator. More recently, realised that the new generator $\phi(t)$ induces exactly the same copula as that obtained with $\varphi(t)$. To see this we have that $C_\phi(u,v)=\phi^{-1}(\phi(u)+\phi(v))=\varphi^{-1}\left(\alpha\left\{\frac{1}{\alpha}\varphi(u)+\frac{1}{\alpha}\varphi(v)\right\}\right)=C_\varphi(u,v)$. In other words, an Archimedean copula generator is not unique. Moreover, in terms of the hazard rate functions, $h_\phi(t)$ and $h_\varphi(t)$, induced by generators $\phi$ and $\varphi$, respectively, the relationship becomes $h_\phi(t)=\alpha h_\varphi(\alpha t)$. In order to make our semiparametric generator identifiable, without loss of generality, we impose the new constraint 1. $\theta_0=1$. This constraint is equivalent to impose $h(0)=1$ in definition . Posterior inference {#sec:post} =================== The copula density $f_C(u,v)$, of an Archimedean copula, can be obtained by taking the second crossed derivatives with respect to $u$ and $v$ in expression . In terms of the generator and its inverse this density becomes $$\label{eq:cdensity} f_C(u,v)=\varphi^{-1(\prime\prime)}\left(\varphi(u)+\varphi(v)\right) \varphi^{(\prime)}(u)\varphi^{(\prime)}(v),$$ where the single and double primes denote first and second derivatives, respectively, and are given by $$\varphi^{-1(\prime\prime)}(t)=\sum_{k=1}^K \left\{(A_k+\theta_k t)^2-\theta_k\right\}\exp\left\{-\left(B_k+A_k t+\frac{\theta_k}{2}t^2\right)\right\}I(\tau_{k-1}<t\leq\tau_K)$$ and $$\varphi^{(\prime)}(t)=-\sum_{k=1}^{K}\frac{1}{t}\left(-2\theta_k B_k+A_k^2-2\theta_k\log(t)\right)^{-1/2}I\left(\varphi^{-1}(\tau_k)\leq t< \varphi^{-1}(\tau_{k-1})\right).$$ Let $(U_i,V_i)$, $i=1,\ldots,n$ be a bivariate sample of size $n$ from $f_C(u,v)$ defined in . With this we can construct the likelihood for ${\boldsymbol{\theta}}=(\theta_0,\theta_1,\ldots,\theta_K)$ as $\mbox{lik}({\boldsymbol{\theta}}\mid\bu,\bv)=\prod_{i=1}^n f_C(u_i,v_i\mid{\boldsymbol{\theta}})$, where we have made explicit the dependence on ${\boldsymbol{\theta}}$ in the notation of the copula density. Recall that the parameters must satisfy several conditions, $(C1)$ and $(C2)$ given in Proposition \[prop:1\], $(C3)$ to make our generator unique, and $\theta_K=0$. We assume a prior distribution for the $\theta_k$’s of the form $$\label{eq:prior} f(\theta_k)=\pi_0 I(\theta_k=0)+(1-\pi_0)\mbox{N}(\theta_k\mid \mu_0,\sigma^2_0),$$ independently for $k=1,\ldots,K-1$. Note that we explicitly allow the $\theta_k$’s, for $k=1,\ldots,K-1$ to be zero with positive probability $\pi_0$. This prior choice is useful to define an independence test. Specifically, the hypothesis $H_0:U \mbox{ and } V$ independent is equivalent to $H_0:\theta_1=\cdots=\theta_{K-1}=0$. To perform the test we can compute the posterior probability of $H_0$ and its complement and make the decision, say via Bayes factors . The posterior distribution of ${\boldsymbol{\theta}}$ is simply given by the product of expressions and , up to a proportionality constant. It is somehow easier to characterize the posterior distribution by implementing a Gibbs sampler and sampling from the conditional posterior distributions $$\label{eq:postc} f(\theta_k\mid {\boldsymbol{\theta}}_{-k},\data)\propto\mbox{lik}({\boldsymbol{\theta}}\mid\bu,\bv)f(\theta_k),$$ for $k=1,\ldots,K-1$. However, sampling from conditional distributions is not trivial, we therefore propose a Metropolis-Hastings step [@tierney:94] by sampling $\theta_k^*$ at iteration $(r+1)$ from a random walk proposal distribution $$q(\theta_k\mid{\boldsymbol{\theta}}_{-k},\theta_k^{(r)})=\pi_1 I(\theta_k=0)+(1-\pi_1)\un(\theta_k\mid \max\{a_k,\theta_k^{(r)}-\delta c_k\},\min\{b_k,\theta_k^{(r)}+\delta c_k\})$$ where the interval $(a_k,b_k)$ represents the conditional support of $\theta_k$, $c_k=b_k-a_k$ is its length, with $a_k = \max_{k \le j \le K-1} \left\{\left(\sqrt{\theta_{j+1}}I(\theta_{j+1}\geq 0) -\theta_0 - \sum_{i=1,i \ne k}^{j}(\tau_i - \tau_{i-1})\theta_i\right)/(\tau_{k} - \tau_{k-1})\right\}$, for $k=1,\ldots,K-1$, $b_k = \left(\theta_0 + \sum_{j=1}^{k-1} (\tau_j - \tau_{j-1})\theta_j \right)^2$, for $k=2,\ldots,K-1$, and $b_1 = 1$. The justification of these bounds obeys the inclusion of constraints $(C1)$ and $(C2)$ and their derivations are given in Appendix \[sec:appendix\]. The parameters $\pi_1$ and $\delta$ are tuning parameters that control de acceptance rate. Therefore, at iteration $r+1$ we accept $\theta_k^*$ with probability $$\alpha\left(\theta_k^*,\theta_k^{(r)}\right)=\min\left\{1\,,\;\frac{f(\theta_k^*\mid{\boldsymbol{\theta}}_{-k},\data)\,q(\theta_k^{(r)}\mid{\boldsymbol{\theta}}_{-k},\theta_k^{*})}{f(\theta_k^{(r)}\mid{\boldsymbol{\theta}}_{-k},\data)\,q(\theta_k^*\mid{\boldsymbol{\theta}}_{-k},\theta_k^{(r)})}\right\}.$$ Numerical studies {#sec:illust} ================= We illustrate the performance of our model in two ways, through a simulation study, and with a real data set. To define the partition $\{\tau_k\}$ of the positive real line we consider a Log-$\alpha$ partition defined by $\tau_k = - \alpha \log(1-k/K)$ for $k = 0,\ldots,K-1$, with $\alpha>0$. This partition is the result of transforming a uniform partition in the interval $[0,1]$ via a convex function. In particular we inspired ourselves in the generator of the product copula. Larger values of $\alpha$ increase the spread of the partition along the positive real line. Simulation study ---------------- We generated simulated data from four parametric Archimedean copulas, namely the product, Clayton, Ali-Mikhail-Haq (AMH) and Gumbel copulas. Their features are summarised in Table \[tab:parcopulas\], where we include the parameter space, the generator, the inverse generator, an indicator whether the copula is strict or not and the induced $h(t)$ function obtained through inversion of relationship . For each parametric copula we took a sample of size $n=200$. To specify the copulas we took $\theta\in\{-0.8,1\}$ for the Clayton copula, $\theta\in\{-0.7,0.7\}$ for the AMH copula, and $\theta=1.4$ for the Gumbel copula. For the partition size we compared $K\in\{10,20\}$ and tried values $\alpha \in \{0.3,0.5,0.9,1,2,\ldots,10\}$. For the prior distributions we took $\pi_0=0, \mu_0=-1$ and $\sigma_0^2=10$. We implemented a MH step with-in the Gibbs sampler where the proposal distributions were specified by $\pi_1=0$ and $\delta=0.25$. The acceptance rate attained with these specifications are around 30%, which according to are optimal for random walks. Finally, the chains were ran for 20,000 iterations with a burn-in of 2,000 and keeping one of every 5$^{th}$ iteration to produce posterior estimates. To assess goodness of fit (GOF) we computed several statistics. The logarithm of the pseudo marginal likelihood (LPML), originally suggested by , to assess the fitting of the model to the data. The supremum norm, defined by $\sup_{t}|\varphi^{-1}(t)-\widehat{\varphi}^{-1}(t)|$ to assess the discrepancy between our posterior estimate (posterior mean) $\widehat{\varphi}^{-1}(t)$ from the true inverse generator $\varphi^{-1}(t)$. We also computed the Kendall’s tau coefficient and compare the posterior point and 95% interval estimates with the true value. These values are shown in Tables \[tab:prod\] to \[tab:gumbel14\]. Although we fitted our model with all values of $\alpha$ mentioned above, we only show results for some of them in the tables. Note that, due to the nonunicity of an Archimedean generator, an equivalent constraint to $(C3)$ has to be imposed to the parametric generators that we are comparing to. That is we set $h(0)=1$ for the product, Clayton and AMH copulas, and $h(\epsilon)=1$ for the Gumbel copula, for say $\epsilon=0.01$. The difference in the latter case is because, for a Gumbel copula, $h(t)\to\infty$ when $t\to 0$. These conditions are already included in the parametrisation used in Table \[tab:parcopulas\]. For the product copula the GOF measures are presented in Table \[tab:prod\]. With exception of the partition Log-$3$ for $K=30$, for all settings considered, the true $\kappa_\tau$ lies inside the 95% credible intervals. The LPML chooses the model with Log-$1$ partitions of size $K=10$, and corresponds to the second smallest value of the supremum norm. Posterior estimates of functions $h(t)$ and $\varphi^{-1}(t)$ are shown in Figure \[fig:prod\]. In both cases the true function lies inside the 95% credible intervals. For the Clayton copula we have two choices of $\theta$, $-0.8$ and $1$. The first choice, $\theta=-0.8$, corresponds to a generator that is not strict, that is, $\varphi^{-1}(t)>0$ for $t\in[0,5/4]$, and $\varphi^{-1}(t)=0$ for $t>5/4$. This is an interesting challenge because our model defined only strict generators. The settings with smallest supremum norm, Log-$0.5$ with $K=10$, produces the 95% credible interval for $\kappa_\tau$ closest to the true value, however it does not achieve the largest LPML. The inconsistency of the GOF measures might be due to the non strictness feature of the true generator. Moreover, if we look at the graphs of the posterior estimates of $h(t)$ and $\varphi^{-1}(t)$ (Figure \[fig:clay\_08\]), for larger values of $t$ the true functions lie outside of our posterior estimates. For $\theta=1$, the best model is obtained with a Log-$6$ partition of size $10$. In this case, posterior estimates of functions $h(t)$ and $\varphi^{-1}(t)$ with the best fitting (Figure \[fig:clay1\]), contain the true functions. For the AMH copula we have two values of $\theta$, $-0.7$ and $0.7$. The best fitting consistently chosen by the three GOF criteria is obtained with a Log-$6$ and Log-$1$ partitions of size $K=10$, respectively for the two values of $\theta$. Posterior estimates of functions $h(t)$ and $\varphi^{-1}(t)$ with the best fitting are shown in Figures \[fig:amh\_07\] and \[fig:amh07\], respectively. In all cases the true functions lie within the 95% credible intervals. For the Gumbel copula with $\theta=1.4$ we have an interesting behaviour. The true $h(t)$ function has the feature that $h(0)=\infty$. This represents a challenge for our model since we have imposed the constrain $(C3)$ which is equivalent to $h(0)=1$. The highest LPML value is obtained with a Log-$7$ partition of size $K=10$, however the posterior 95% credible interval for $\kappa_\tau$ does not contain the true value. On the other hand, the second best value of LPML is obtained with an Log-$3$ partition of size $10$, and in this case the 95% credible interval for $\kappa_\tau$ does contain the true value. We select this latter as the best fitting. Posterior estimates of functions $h(t)$ and $\varphi^{-1}(t)$ are shown in Figure \[fig:gumbel14\]. Recalling that the true hazard function goes asymptotically to infinity when $t\to 0$, therefore, for values close to zero the true $h(t)$ lies outside our posterior credible intervals, something similar happens in the estimates of the inverse generator. Apart from this, our posterior estimates are very good for $t>\epsilon$. An important learning from the previous examples is that increasing the partition size doe not necessarily implies a better fitting. Real data analysis ------------------ In public health it is important to study the factors that determine the birth weight of a child. Low birth weight is associated with high perinatal mortality and morbility . We study the dependence structure between the age of a mother ($X$) and the weight of her child ($Y$), and concentrated on mothers of 35 years old and above. The dataset was obtained from the General Hospital of Mexico through the opendata platform that can be accessed at https://datos.gob.mx/busca/dataset/perfiles-metabolicos-neonatales/resource/4ab603eb-b73a-498f-8c56-0dc6d21930e8. It contains $n = 208$ records of the neonatal metabolic profile of male babies registered in the year 2017 in Mexico City. The marginal distributions for variables $U$ and $V$, induced by copula , are uniform. In practice, copulas are used to model the dependence for any pair of random variables regardless of their marginal distributions. Let $X$ and $Y$ be two random variables with marginal cumulative distributions $F(x)$ and $G(y)$ respectively. Then the joint cumulative distribution function for $(X,Y)$ is obtained as [@sklar:59], $H(x,y)=C(F^{-1}(x),G^{-1}(y))$, where $C$ is given in . Since we are just interested in modelling the dependence between $X$ and $Y$, it is common in practice to transform the original data, $(X_i,Y_i)$, $i=1,\ldots,n$, to the unit interval via a modified rank transformation [@Deheuvels:79] in the following way. Let $\bX'=(X_1,\ldots,X_n)$ and $\bY'=(Y_1,\ldots,Y_n)$ then $U_i=\mbox{rank}(i,\bX)/n$ and $V_i=\mbox{rank}(i,\bY)/n$ are the transformed data, where $\mbox{rank}(i,\bX)=k$ iff $X_i=X_{(k)}$ for $i,k=1,\ldots,n$. This is based on the probability integral transform using the empirical cumulative distribution function of each coordinate. In Figure \[fig:realdata\] we show a dispersion diagram of the original data (left panel) and the rank transformed data (right panel). To avoid problems due to ties in the original data, we fist include a perturbation to the data by adding a uniform random variable $\un(0,0.01)$ to each coordinate. The sample Kendall’s tau value for the transformed data is $\tilde{\kappa}_{\tau} = -0.1122$. We fitted our model to the transformed data with the following specifications. To define the partitions we took values $\alpha \in \{0.3,0.5,0.9,1,2,\ldots,10\}$ with sizes $K\in\{10,20\}$. For the prior we took $\pi_0=0$, $\mu_0=-1$ and $\sigma_0^2=10$. The MCMC specifications were the same as those used for the simulated data. The GOF measures computed were the LPML and the posterior estimates (point and 95% credible interval) of $\kappa_\tau$. The results are reported in Table \[tab:realdata\]. The best fitting model according to LPML is that obtained with a partition of size $K=10$ and Log-$10$. The sample concordance $\tilde{\kappa}_\tau$ is included in our posterior 95% credible interval estimate $\kappa_\tau\in(-0.213,-0.098)$. The estimated hazard rate function $h(t)$ and the inverse generator $\varphi^{-1}(t)$, with the best fitting model, are included in Figure \[fig:realdata\]. The solid thick line corresponds to the point estimates and the solid thin lines to the 95% credible intervals. For a visual comparison, the blue dotted line corresponds to the function of the independence (product) copula. This confirms that there is a negative (weak) dependence between the age of the mother and the birth weight of the child. The older the mother, the less weight of the child. This finding could potentially help the policy makers to focus campaigns to help the awareness of future mothers. As mentioned in Section \[sec:model\], we can use our model to undertake an independence test. For that we choose the prior distribution for the $\theta_k$’s, as in , such that the prior probability of $H_0:\theta_1=\cdots=\theta_{K-1}=0$ is around $1/2$, in other words, we want $\P(H_0)=\pi_0^{K-1}=1/2$. Particularly, for a partition of size $K=10$ we need to specify $\pi_0=0.9258$. In order to get a point of mass proposal in the Metropolis-Hasting step we consider $\pi_1 = 0.3$. We re-ran our model using these values with the other specifications left unchanged. The posterior probability of $H_0$ becomes $\P(H_0\mid\data)= 0.53$, which leads to an inconclusive test. This might be explained by the fact that although the association is negative, it is very close to zero. To calibrate our independence test, we consider the simulated data from previous section of two copulas, the product copula and the Clayton copula with $\theta=-0.8$. In these cases the best fitting was obtained with a partition of size $K=10$, so we chose the same prior values as for the real data to perform an independence test. Posterior probabilities of $H_0$ are $0.81$ and $0$, respectively, showing that for independent data the posterior probability is a lot larger that 0.5, whereas for clearly dependent data the posterior probability of independence is zero. Concluding remarks {#sec:concl} ================== We have proposed a semiparametric Archimedean copula that is flexible enough to capture the behaviour of several families of parametric arquimedean copulas. Our model is capable of modelling positive and negative dependence. The number of parameters in the model to produce a good estimation of the dependence in the data should not be extremely high. For most of the examples considered here 10 parameters are enough. Defining an appropriate partition to analyse real data sets is not trivial. We suggest to try different values of $\alpha$ in a wide range and compare using a GOF criteria like the LPML we used here. In the exposition and in examples considered here, we concentrated on bivariate copulas, however extensions to more than two dimensions is also possible, say $C(u_1,\ldots,u_m)=\varphi^{-1}\left(\varphi(u_1)+\cdots+\varphi(u_m)\right)$. Performance of our semiparametric copula in this multivariate setting is worth studying. Our model is motivated by semiparametric proposals for survival analysis functions and appropriately modified to satisfy the properties of an Archimedean generator. The semiparametric generator presented here turned out to be based on quadratic splines, however, alternative proposals are possible. Instead of starting with a piecewise function for the derivative of a hazard rate, we could start by defining a piecewise constant function for the hazard rate itself. That is $h(t)=\sum_{k=1}^K\theta_k I(\tau_{k-1}<t\leq\tau_k)$ with $\theta_k>0$, and $\{\tau_k\}$ a partition of the positive real line. In this case the cumulative hazard function becomes $H(t)=\sum_{j=1}^{k}\theta_j\Delta_j+\theta_{k}(t-\tau_{k-1})$, for $t\in(\tau_{k-1},\tau_{k}]$, with $\Delta_j=\tau_{j}-\tau_{j-1}$. The inverse generator is then a linear spline of the form $$\nonumber \varphi^{-1}(t)=\exp\left\{-\sum_{k=1}^K\theta_k w_k(t)\right\},$$ with $$w_k(t)=\left\{\begin{array}{ll} \Delta_k, & t>\tau_k \\ t-\tau_{k-1} & t\in(\tau_{k-1},\tau_k] \\ 0 & \mbox{otherwise} \end{array}\right.$$ and the corresponding Archimedean generator has the form $$\nonumber \varphi(t)=\sum_{k=1}^K \left\{\tau_{k-1}-\frac{1}{\theta_{k}}(\log t+\vartheta_{k-1})\right\}I(\vartheta_{k-1}<-\log t\leq\vartheta_k),$$ with $\vartheta_k=\sum_{j=1}^{k}\theta_j\Delta_j$. To ensure convexity of the generator we further require $\theta_1\geq\theta_2\geq\cdots\geq\theta_K$. Furthermore, the Kendall’s tau has a simpler expression $$\nonumber \kappa_\tau=1+\sum_{k=1}^K\left\{e^{-2\vartheta_k}(1+2\theta_k\tau_k)-e^{-2\vartheta_{k-1}}(1+2\theta_k\tau_{k-1})\right\}.$$ However, it can be shown that this expression for the Kendall’s tau only allows positive values, constraining the possible associations captured by the model. Acknowledgements {#acknowledgements .unnumbered} ================ This research was supported by *Asociación Mexicana de Cultura, A.C.* Derivation of posterior conditional support of $\theta_k$ {#sec:appendix} ========================================================= In order to satisfy constraint $(C1)$, we consider first the case $\theta_k\leq 0$. Therefore\ $\min_{t \in (\tau_{k-1},\tau_k]}A_k + \theta_k t = A_k + \theta_k \tau_{k}$. This implies the following constraint for $\theta_k$, $$\theta_k \geq \max_{k \le j \le K-1} \left\{- \left( \theta_0 + \sum_{i=1,i \ne k}^{j}(\tau_i - \tau_{i-1})\theta_i\right)/(\tau_{k} - \tau_{k-1})\right\},$$ for $k = 1,\ldots,K-1$, where we define the empty sum as zero. On the order hand, if $\theta_k > 0$ we have $\min_{t \in (\tau_{k-1},\tau_k]}A_k + \theta_k t = A_k + \theta_k \tau_{k-1},$ and we get, from condition $(C2)$, the following restriction $$\theta_k < \left(\theta_0 + \sum_{i=1}^{k-1} (\tau_i - \tau_{i-1})\theta_i \right)^2.$$ This defines the upper bound $b_k$, for $k = 2,\ldots,K-1$, and $b_1=1$. Because the term $\theta_k$ appears on the right side of the previous inequality for $j=k+1,\ldots,K-1$, we need to consider the following restriction $$\theta_k > \left.\left( \sqrt{\theta_j} - \theta_0 - \sum_{i=1,i \ne k}^{j-1}(\tau_i - \tau_{i-1})\theta_i\right)\right/(\tau_{k} - \tau_{k-1})$$ if $\theta_j \geq 0$. Combining this with the constraint when $\theta_k\leq 0$ above, we get the lower bound $a_k$ for $k=1,\ldots,K-1$. [99]{} Deheuvels, P. (1979). La fonction de dépendance empirique et ses proprietés. Un test non paramétrique d’indépendance. [*Roy. Belg. Bull. Cl. Sci.*]{}, [**65**]{}, (5), 274–292. Di Bernardino, E. and Rullière, D. (2013). On certain transformation of Archimedean copulas: Application to non-parametric estimation of their generators. *Dependence Modeling* [**1**]{}, 1–36. Geisser, S. and Eddy, W. F. (1979). A predictive approach to model selection. *Journal of the American Statistical Association* **74**, 153–160. Genest, C., Nešlehová J. and Ziegel J. (2011). Inference in multivariate Archimedean copula models. *TEST* **20**: 223. https://doi.org/10.1007/s11749-011-0250-6. Genest, C. and Rivest, L. (1993). Statistical inference procedures for bivariate Archimedean copulas . *Journal of the American Statistical Association*, **88**, 1034-1043. Guillote, S. and Perron, J. (2015). Inference on Archimedean copulas using mixtures of Pólya trees. *Journal of Statistical Planning and Inference*, **166**, 2-13. Kass, R.E. and Raftery, A.E. (1995). Bayes Factors. *Journal of the American Statistical Association* **90**, 773–795. Klein, J.P. and Moeschberger, M.L. (2003). [*Survival analysis*]{}. Springer, New York. Nelsen, R.B. (2006). [*An introduction to copulas*]{}. Springer, New York. Neuhaus, G. (1971). On weak convergence of stochastic processes with multidimensional time parameter. *The Annals of Mathematical Statistics* , **42** (4), 1285-1295. Nieto-Barajas, L.E. and Walker, S.G. (2002). Markov beta and gamma processes for modeling hazard rates. *Scandinavian Journal of Statistics* **29**, 413–424. Robert, C. P. and Casella, G. (2010). [*Introducing Monte Carlo methods with R*]{}. Springer, New York. Sklar, M. (1959). Fonctions de répartition á $n$ dimensions et leurs marges. Université Paris 8. Smith, A. and Roberts, G. (1993). Bayesian computations via the Gibbs sampler and related Markov chain Monte Carlo methods. [*Journal of the Royal Statistical Society, Series B*]{} [**55**]{}, 3-–23. Stevens-Simon, C. and Orleans, M. (2001). Low-birthweight prevention programs: The enigma of failure. *Birth* **26**, 184–191. Tierney, L. (1994). Markov chains for exploring posterior distributions. [*Annals of Statistics*]{} [**22**]{}, 1701-–1722. Vanderhende, F. and Lampert, P. (2005). Local dependence estimation using semiparametric Archimedean copulas. [*The Canadian Journal of Statistics*]{} [**33**]{}, 377–388. ![Functions $h'(t)$ (first panel), $h(t)$ (second panel) and $\varphi^{-1}(t)$ (third panel) for two scenarios of $\{\theta_k\}$. All negative values (solid line), and all positive values (dotted line).[]{data-label="fig:ilust1"}](Grafica_Derivadah.pdf "fig:"){width="5.5cm" height="5.cm"} ![Functions $h'(t)$ (first panel), $h(t)$ (second panel) and $\varphi^{-1}(t)$ (third panel) for two scenarios of $\{\theta_k\}$. All negative values (solid line), and all positive values (dotted line).[]{data-label="fig:ilust1"}](Grafica_h.pdf "fig:"){width="5.5cm" height="5.cm"} ![Functions $h'(t)$ (first panel), $h(t)$ (second panel) and $\varphi^{-1}(t)$ (third panel) for two scenarios of $\{\theta_k\}$. All negative values (solid line), and all positive values (dotted line).[]{data-label="fig:ilust1"}](Grafica_GeneradorInverso.pdf "fig:"){width="5.5cm" height="5.cm"} \[fig:2\] ![Prior distributions of Kendall’s tau under three different scenarios.[]{data-label="fig:simkt"}](HistTauKendall_v1.pdf "fig:"){width="5.5cm" height="5.cm"} ![Prior distributions of Kendall’s tau under three different scenarios.[]{data-label="fig:simkt"}](HistTauKendall_thetanegative.pdf "fig:"){width="5.5cm" height="5.cm"} ![Prior distributions of Kendall’s tau under three different scenarios.[]{data-label="fig:simkt"}](HistTauKendall_thetapositive.pdf "fig:"){width="5.5cm" height="5.cm"} ![ \[fig:prod\]](Grafh_Producto.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:prod\]](GrafInvGen_Producto.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:clay\_08\]](Grafh_Clayton-08.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:clay\_08\]](GrafInvGen_Clayton-08.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:clay1\]](Grafh_Clayton.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:clay1\]](GrafInvGen_Clayton.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:amh07\]](Grafh_AMH07.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:amh07\]](GrafInvGen_AMH07.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:amh\_07\]](Grafh_AMH-07.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:amh\_07\]](GrafInvGen_AMH-07.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:gumbel14\]](Grafh_Gumbel14.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:gumbel14\]](GrafInvGen_Gumbel14.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:realdata\]](MuestraOriginal.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:realdata\]](MuestraTransformada.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:perealdata\]](Grafh_Datos.pdf "fig:"){width="6.5cm" height="6.0cm"} ![ \[fig:perealdata\]](GrafInvGen_Datos.pdf "fig:"){width="6.5cm" height="6.0cm"} Copula $\Theta$ $\varphi(t)$ $\varphi^{-1}(t)$ Estrict? h(t) --------- --------------- ------------------------------------------------------------------------- -------------------------------------------------------------------------------- -------------------- -------------------------------------------------------------- Product - $ - \log(t)$ $e^{-t}$ Yes $1$ Clayton $[-1,\infty)$ $\frac{1}{\theta}(t^{- \theta}-1)$ $ (1+ \theta t)^{-1/\theta}$ If $\theta \geq 0$ $\frac{1}{1 + \theta t}$ AMH $[-1,1)$ $\frac{1}{1-\theta}\log \left( \frac{1 - \theta + \theta t}{t} \right)$ $ \frac{1- \theta}{e^{(1-\theta)t} - \theta}$ Yes $\frac{(1-\theta)e^{(1-\theta)t}}{e^{(1-\theta)t} - \theta}$ Gumbel $[1,\infty)$ $ \epsilon\left(\frac{-\log(t)}{\epsilon\theta}\right)^{\theta}$ $\exp\left\{-\epsilon\theta\left(\frac{t}{\epsilon}\right)^{1/\theta}\right\}$ Yes $\left(\frac{\epsilon}{t}\right)^{1- 1/\theta}$ : Summary of some parametric Archimedean copulas parametrised such that $h(0)=1$, for the first three copulas, and $h(\epsilon)=1$, for the Gumbel copula.[]{data-label="tab:parcopulas"}

--- abstract: 'The variability inherent in solar wind composition has implications for the variability of the physical conditions in its coronal source regions, providing constraints on models of coronal heating and solar wind generation. We present a generalized prescription for constructing a wavelet power significance measure (confidence level) for the purpose of characterizing the effects of missing data in high cadence solar wind ionic composition measurements. We describe the data gaps present in the 12-minute ACE/SWICS observations of ${\rm O}^{7+}/{\rm O}^{6+}$ during 2008. The decomposition of the in-situ observations into a ‘good measurement’ and a ‘no measurement’ signal allows us to evaluate the performance of a filler signal, i.e., various prescriptions for filling the data gaps. We construct Monte Carlo simulations of synthetic ${\rm O}^{7+}/{\rm O}^{6+}$ composition data and impose the actual data gaps that exist in the observations in order to investigate two different filler signals: one, a linear interpolation between neighboring good data points, and two, the constant mean value of the measured data. Applied to these synthetic data plus filler signal combinations, we quantify the ability of the power spectra significance level procedure to reproduce the ensemble-averaged time-integrated wavelet power per scale of an ideal case, i.e. the synthetic data without imposed data gaps. Finally, we present the wavelet power spectra for the ${\rm O}^{7+}/{\rm O}^{6+}$ data using the confidence levels derived from both the Mean Value and Linear Interpolation data gap filling signals and discuss the results.' author: - 'J. K. Edmondson, B. J. Lynch, S. T. Lepri, and T. H. Zurbuchen' title: 'Analysis of High Cadence In-Situ Solar Wind Ionic Composition Data Using Wavelet Power Spectra Confidence Levels' --- Introduction ============ Decades of in-situ plasma observations have revealed a rich picture of the solar wind [@Zurbuchen07 and references therein], whose overall structure and magnetic topology follows the solar magnetic activity cycle. Heliospheric solar wind observations reflect the structure of their coronal source regions: a relatively cool, fast solar wind with relatively homogeneous ionic composition and elemental abundances originating from coronal holes [@Geiss95; @McComas02], and a relatively hot, slow solar wind that exhibits considerably more variability in ionic composition and elemental abundances, originating either directly from within the vicinity of coronal streamers [@Gosling97; @Zurbuchen02]. In-situ observations of ionic charge state composition, especially of carbon (${\rm C}^{6+}/{\rm C}^{4+}$) and oxygen (${\rm O}^{7+}/{\rm O}^{6+}$) offer insight into coronal dynamics at temperatures of order one million degrees [e.g., @vonSteiger1997; @Zhao09; @Landi12; @Gilbert12]. Identifiable temporal scales from within the compositional variability may provide insights into the nature of the source regions of the solar wind. Wavelet transforms are used to identify transient structure coherency as well as global periodicities in time series data [see e.g., @Daubechies1992; @TorrenceCompo1998; @Liu2007]. Wavelet analyses have an advantage over traditional spectral methods by being able to isolate both large timescale and small timescale periodic behavior that occur over only a subset of the time series. Thus, we are able to analyze the frequency decomposition as a function of time. This is extremely useful if we expect the time series to originate from either a time varying source region or, equivalently, to be consecutively sampling many different source regions with varying physical properties, such as in the solar wind. Recently, @Katsavrias12 used wavelets to examine four solar cycles worth of solar wind plasma, interplanetary magnetic field, and geomagnetic indices to verify intermittent periodicities on timescales shorter than the solar cycle. Common solar timescales ranging from a decade down to hours have been characterized, and timescales of the order of a Carrington Rotation period (approx. 27 days) and shorter (e.g., 14, 9, and 7 days) have been consistently identified in various heliospheric and geomagnetic data [e.g., @Bolzan05; @Fenimore78; @Gonzalez87; @Gonzalez93; @Mursula98; @Nayar01; @Nayar02; @Svalgaard75]. @Temmer07 linked the 9 day timescale to coronal hole variability in the declining phase of solar cycle 23 and @Neugebauer97 used wavelet analyses of [*Ulysses*]{} solar wind speed data to investigate polar microstreams occurring on timescales of 16 hours. Wavelet power spectra are a powerful tool to identify and characterize structures with specific transient timescales and global periodicities, but all commonly used algorithms require fully populated data-sets. That is inconsistent with solar wind composition data – as well as almost all in situ data-sets – because data gaps occur for a number of reasons. The experiment may undergo maintenance and data may not be available, or the signal to noise of the instrument at a given time may have prevented a valid and accurate measurement. Thus, care must be taken to account for spurious results caused by such data gaps. Thus, to identify characteristic timescales smaller than the largest gap duration, one must either break-up the full data set into disjoint segments of continuous data measurements, or quantify the spurious information introduced into the data set by filling-in the no-measurement times. It is with the latter solution that the methodology described in this paper is concerned. Our purpose here is to describe a generalized procedure for the construction of wavelet power significance levels that quantify the relative influence of a filler signal of generally arbitrary form interleaved within a measured data signal. The decomposition of the time series allows for a similar decomposition of the total wavelet power spectrum, and thereby quantifying the power spectra associated with the filler signal and nonlinear interference, for comparison against the measured data signal power. Using the decomposition of the signal power spectra, we identify a statistical confidence level against the null hypothesis that a given feature in the total wavelet power spectrum is due to the filler signal and/or interference effects; in other words, we construct a significance measure for the the total wavelet power spectrum that identifies power spectral features resulting from the measured signal. The structure of the paper is as follows. In Section \[S:WaveletCharacteristics\] we briefly review the wavelet transform, power spectrum, and methods for identifying global periodicities (akin to Fourier modes) as well as transient coherency characteristics. In Section \[S:Data\] we discuss the solar wind ionic composition data obtained by ACE/SWICS during the quiet solar conditions of 2008, and the origin and characteristics of no-measurement data gaps in the context of wavelet analysis. In Section \[S:DataReductionScheme\] we derive the wavelet power statistical confidence level to characterize the effects, and quantify the influence of no-measurement gaps in the data. In Section \[S:MonteCarlo\] we evaluate the performance of two filler signal forms (Linear Interpolation and constant Mean Value) using ensemble-averaged Monte Carlo simulations of a statistical ${\rm O}^{7+}/{\rm O}^{6+}$ ratio model random (1$^{st}$-order Markov) process. In Section \[S:WaveletO7O6\] we examine the wavelet power spectra of actual 12 minute ${\rm O}^{7+}/{\rm O}^{6+}$ data from 2008 with the Linear Interpolation filler signal for the high cadence data gaps, and present our conclusions in Section \[S:Conclusions\]. Rectified Wavelet Power Spectrum Analysis {#S:WaveletCharacteristics} ========================================= The wavelet transform of a time series $T(t)$ is given by $$\label{E:WaveletTransform} W_{\rm T}( t , s ) = \int {\rm T} ( t' ) \ \psi^* ( t' , t , s ) \ dt'.$$ In our calculations, the wavelet bases are generated from the Morlet family, though we note all following analysis is valid for any wavelet basis family. The Morlet family is a time-shifted, time-scaled, complex exponential modulated by a Gaussian envelope, $$\psi \left( t' , t , s \right) = \frac{\pi^{1/4}}{\vert s \vert^{1/2}} {\rm exp}\left[ i \omega_{0} \left( \frac{t' - t}{s} \right) \right] {\rm exp}\left[ - \frac{1}{2} \left( \frac{t' - t}{s} \right)^2 \right]$$ where $ \left( t' , t \right) \in I_{T} \times I_{T} \subset \mathbb{R} \times \mathbb{R}$ is the time and time-translation center, respectively, and $s \in I_{S} \subset \mathbb{R}$ is the timescale over which the Gaussian envelope is substantially different from zero. The $\omega_{0} \in \mathbb{R}$ is a non-dimensional frequency parameter defining the number of oscillations of the complex exponential within the Gaussian envelope; we set $\omega_{0} = 6$, yielding approximately three oscillations within the envelope. The wavelet power spectrum is given by, $\vert W_{\rm T}( t , s ) \vert^2$, for $\psi , T \in L^2 \left( \mathbb{R} \right)$. @TorrenceCompo1998 identify a bias in favor of large timescale features in the canonical power spectrum, which they attribute to the width of the wavelet filter in frequency-space; at large timescales the function is highly compressed yielding sharper peaks of higher amplitude. Equivalently, high frequency peaks tend to be underestimated because the wavelet filter is broad at small timescales. @Liu2007 showed this effect is the difference between the energy and the integration of the energy with respect to time, and thus may be rectified, in practice, by multiplying the wavelet power spectra by the corresponding frequency. Thus, throughout this paper we use the rectified power spectrum given by $$\mathcal{P}_{T} \left( t , s \right) = \vert s \vert^{-1} \vert W_{T} \left( t , s \right) \vert^2.$$ The (rectified) wavelet power spectrum for a general time series and wavelet basis can be highly structured and complex. For solar wind composition data, the time series will likely include characteristic global solar oscillation frequencies, such as the approximate 27-day solar rotation period. In addition to any characteristic global oscillations, the time series will likely be full of transient (non-stationary) ‘coherent structures’ in which variations in the coronal source parameters lead to local variations in the composition ratio data. Thus, we define a localized ‘coherent structure’ as data points that become locally elevated with respect the surrounding measurements, which we note, may spread over a variety of timescales. Global periodicities feature as horizontal bands of (relatively) strong power in the wavelet power spectrum. We characterize the global periodic behavior by integrating the wavelet power spectrum along each correlation timescale to calculate the energy contained in all wavelet coefficients at that timescale; this is known as the global wavelet power spectrum [see e.g., @LeWang03; @Bolzan05], and for the Morlet family is akin to the Fourier modes. The global periodic frequencies within the time series (e.g., Fourier modes) are identified with the local maxima in the integrated power per timescale. Transient (non-stationary) ‘coherent structures’ feature as localized 2D maxima in the wavelet power spectrum. The timescale corresponding to such local 2D maxima demarcates the coherency size of the transient feature. [*ACE*]{}/SWICS Measurements of ${\rm O}^{7+}/{\rm O}^{6+}$ Solar Wind Composition {#S:Data} ================================================================================== The [*Advanced Composition Explorer*]{} spacecraft (ACE) is currently in orbit about the L1 point, $\sim$1.5 million km sunward of Earth [@Stone1998]. Here we analyze data obtained with the Solar Wind Ionic Composition Spectrometer [SWICS; @Gloeckler98]. SWICS is a time-of-flight (TOF) mass spectrometer paired with energy-resolving solid-state detectors (SSDs) and an electrostatic analyzer (ESA) that measures the ionic composition of the solar wind. Ions with the appropriate energy per charge are selected in the ESA. Ion speed is determined in the TOF telescope and the residual energy measured by the SSDs enables particle identification. These measurements allow the independent determination of mass, $M$, charge, $Q$, and energy, $E$, and are virtually free of background contamination [e.g., see @vonSteiger2000]. In order that the wavelet transform defined by equation (\[E:WaveletTransform\]) to be well-defined, and the analysis capable of identifying coherent structure and global oscillation frequencies to the highest time resolution, the (full year) input time series of charge state ratio values must be fully populated at the given cadence. Typically the composition measurements in 1- and 2-hour averages have superb counting statistics, however in the highest time resolution data (12 minute cadence) the flux levels are occasionally too low for a valid measurement to be recorded. For an ionic composition ratio, the presence of a valid data point is subject to the relatively restrictive condition that the ACE/SWICS instrument must have made a measurement for both numerator and denominator with enough counting statistics such that the data reduction algorithm derives a value at the given time resolution, and the denominator must be non-zero. In other words, under low-flux conditions, which occurred throughout 2008, the charge state ratios under consideration occasionally could not be constructed. The top panel of Figure \[datafig\] shows 12-minute average ${\rm O}^{7+}/{\rm O}^{6+}$ ACE/SWICS measurements for the full 2008 year. Qualitatively, there are no discernible gaps in the time series over the full year. However, upon closer viewing for example of individual Carrington Rotations (bottom panels of Figure \[datafig\]), the ubiquity of such no-measurement data gaps becomes clear. Note the sporadic nature of the gaps between 42 and 49 days within CR2066, as well as the day 121 in CR2069. The frequency of the data gap durations is quantified in Figure \[MissingDataPDF\] which plots the probability distribution function (PDF) of the missing data time durations. From this, we find the vast majority, $90.4\%$, of the no-measurement durations occur on timescales less than 0.1 days (2.4 hours). In addition, $9.4\%$ of the gap durations occur on timescales between 0.1 and 1 days. Only $0.2\%$ of the data gaps have durations greater than 1 day. The single maximum no-measurement duration is 2.5 days (occurring at 214 days into the year). Any analysis of the wavelet power spectrum of a time series that includes data gaps is only valid to the timescales of the largest data gap (in this case, 2.4 days). Below this timescale, one must break the full year data time series into subsets of continuous measurement durations, and perform similar power spectrum analyses on the individual subset time series. Such a procedure, while valid, leads to a host of issues. For example, any physical global oscillations on timescales below the largest data gap are lost. In addition, the boundary effects associated with the cone of influence within the individual wavelet power spectra [see e.g., @TorrenceCompo1998], become amplified as the size of the data set decreases. In this paper, we take a different approach. We retain as much physical information below the largest data gap timescale as possible by filling the data gaps with a particular signal form and quantifying the propagation of new information introduced into the system throughout the analysis. Constructing Wavelet Power Confidence Levels to Characterize the Effects of Data Gap {#S:DataReductionScheme} ==================================================================================== In order to attempt to keep any physical information of global oscillation frequencies and coherent structures below the timescale of the largest data gap, we require a fully-populated time series for the full time interval under scrutiny. Therefore, we introduce a particular signal form model to fill the data gaps, and quantify the new information introduced to the system by constructing a statistical confidence level as a measure of the influence of the filler signal on the total wavelet power spectrum. Wavelet Power Spectrum from a Superposition of Signals ------------------------------------------------------ From a qualitative standpoint, the wavelet power at any given timescale is determined by several factors, the strength of the measured signal, the strength of the filler signal, and interference effects between the data signal and filler signal. To quantify this decomposition of the wavelet power spectrum, we first note that the full time interval, $t \in I_{T} \subset \ \mathbb{R}$, may be decomposed into (discontinuous) interleaved subsets of measurement time, $t \in I_{D} \subset I_{T} \subset \ \mathbb{R}$, and no-measurement time, $t \in I_{F} \subset I_{T} \subset \ \mathbb{R}$. Note, $I_{T} = I_{D} + I_{F}$. With this decomposition of the time interval, we may then decompose the full time series, $T(t)$, into a linear superposition of two signals over the full duration consisting of the measurement data signal, $D(t)$, such that the values within the no-measurement intervals are set equal to zero; and the no-measurement filler signal, $F(t)$, in which non-zero values fall within the no-measurement intervals. $$\begin{aligned} D \left( t \right) = & \left\{ \begin{array}{ccr} D \left( t \right) & , & t \in I_{D} \\ 0 & , & t \in I_{F} \end{array} \right. \\ F \left( t \right) = & \left\{ \begin{array}{ccr} F \left( t \right) & , & t \in I_{F} \\ 0 & , & t \in I_{D} \end{array} \right. \label{E:SignalDecomposition}\end{aligned}$$ The full time series is, therefore, $T(t) = D(t) + F(t) \ \forall \ t \in I_{T}$. We note, the full time series may contain zero values, though only where zero measurements were in fact made. On the other hand, the no-measurement intervals are filled by a model of a known functional form. To demonstrate the procedure, we construct the following example time series shown in Figure \[O7O6model\]. The data signal, $D(t)$, shown in black in the top panel of Figure \[O7O6model\] is a synthetic 1-year time series of ${\rm O}^{7+}/{\rm O}^{6+}$ data (described in further detail in Section \[SS:ModelData\]), into which we introduce the observed 2008 data gaps. The filler signal, $F(t)$, shown in red in the middle panel of Figure \[O7O6model\], is a simple linear interpolation across the data gaps. The filler signal form over an $N$-point gap between good data points ${\rm D}_1$ and ${\rm D}_2$ is given by ${\rm F}_n = {\rm D}_1 + ({\rm D}_2 - {\rm D}_1)(n/(N+1))$ for $n = 1$ to $N$. The bottom panels of Figure \[O7O6model\] plot the composite time series across two Carrington rotations equivalent to those shown in Figure \[datafig\]. The wavelet integral transform is linear in the input signals. For a linear superposition of input signals, $T(t) = D(t) + F(t)$, the wavelet transform in a given basis, $\psi \left( t , t' , s \right)$, of the total signal is simply the linear superposition of the wavelet integral transforms of the component signals. $$\begin{aligned} \label{E:LinearWaveletIntegral} \displaystyle W_\text{T} \left( t , s \right) & = & \int_{I_{T}} \text{T} \left( t' \right) \psi \left( t , t' , s \right) dt' \nonumber \\ \displaystyle W_\text{T} \left( t , s \right) & = & \int_{I_{T}} \left( \text{D} \left( t' \right) + \text{F} \left( t' \right) \right) \psi \left( t , t' , s \right) dt' \nonumber \\ \displaystyle W_\text{T} \left( t , s \right) & = & W_\text{D} \left( t , s \right) + W_\text{F} \left( t , s \right) $$ Where the integration is taken over the entire set, $I_{T} \subset \mathbb{R}$. In the most general case, the wavelet transform given by equation (\[E:LinearWaveletIntegral\]) is a complex number, $W_{T} : I_{T} \times I_{S} \rightarrow \ \mathbb{C}$, where $t \in I_{F} \subset \ \mathbb{R}$ is the full time interval, and $s \in I_{S} \subset \ \mathbb{R}$ is the timescale interval, and is given by, $$\label{E:ComplexWavelet} W_{T} \left( t , s \right) = \mathfrak{Re} \lbrace W_{T} \left( t , s \right) \rbrace + i\ \mathfrak{Im} \lbrace W_{T} \left( t , s \right) \rbrace$$ The (rectified) wavelet power signal for the same time and timescale intervals is the square of the amplitude of the wavelet transform, $\mathcal{P}_{T} : I_{T} \times I_{S} \rightarrow \ \mathbb{R}$. $$\label{E:WaveletPower} \mathcal{P}_{T} \left( t , s \right) = \vert s \vert^{-1} \ \vert W_{T} \left( t , s \right) \vert^2 = \vert s \vert^{-1} \ \left[ \mathfrak{Re}^2 \lbrace W_{T} \left( t , s \right) \rbrace + \mathfrak{Im}^2 \lbrace W_{T} \left( t , s \right) \rbrace \right] \\ $$ In general, the real and imaginary components may take on positive, zero, and negative values, and the square of the real and imaginary components ensures the (rectified) total wavelet power spectrum is positive, semi-definite (i.e., non-negative), for all $\left( t , s \right) \in I_{T} \times I_{S}$. Substituting the signal decomposition of equations (\[E:LinearWaveletIntegral\]), the amplitude of the wavelet power constructed from a superposition of signals necessarily involves not only the power amplitudes of the individual component signals, but also interference effects between the signals, $$\label{E:NonLinearWaveletPower} \mathcal{P}_{T} \left( t , s \right) = \mathcal{P}_{D} \left( t , s \right) + \mathcal{P}_{F} \left( t , s \right) + \mathcal{P}_{I} \left( t , s \right)$$ Where we have defined the data signal power, filler signal power, and interference power by, $$\label{E:PowerDecomposition1} \mathcal{P}_{D} \left( t , s \right) \equiv \ \vert s \vert^{-1} \ \vert W_{D} \left( t , s \right) \vert^2 = \vert s \vert^{-1} \left[ \mathfrak{Re}^2 \lbrace W_{D} \left( t , s \right) \rbrace + \mathfrak{Im}^2 \lbrace W_{D} \left( t , s \right) \rbrace \right]$$ $$\label{E:PowerDecomposition2} \mathcal{P}_{F} \left( t , s \right) \equiv \ \vert s \vert^{-1} \ \vert W_{F} \left( t , s \right) \vert^2 = \vert s \vert^{-1} \left[ \mathfrak{Re}^2 \lbrace W_{F} \left( t , s \right) \rbrace + \mathfrak{Im}^2 \lbrace W_{F} \left( t , s \right) \rbrace \right]$$ $$\label{E:PowerDecomposition3} \begin{split} \mathcal{P}_{I} \left( t , s \right) & \equiv 2 \ \vert s \vert^{-1} \left( \ \mathfrak{Re} \lbrace W_{D} \left( t , s \right) \rbrace \ \mathfrak{Re} \lbrace W_{F} \left( t , s \right) \rbrace \right) \\ & + 2 \ \vert s \vert^{-1} \left( \ \mathfrak{Im} \lbrace W_{D} \left( t , s \right) \rbrace \ \mathfrak{Im} \lbrace W_{F} \left( t , s \right) \rbrace \right) \end{split}$$ Figure \[WaveletPowerDecomposition1\] plots the wavelet power spectra decomposition for the time series used in Figure \[O7O6model\]: for the data signal $\mathcal{P}_{D}$ (top panel), the filler signal $\mathcal{P}_{F}$ (middle panel), and the interference signal $\mathcal{P}_{I}$ (bottom panel). From equations (\[E:WaveletPower\]), (\[E:PowerDecomposition1\]) and (\[E:PowerDecomposition2\]), the sets of values realized by the total signal power, the data signal power, and the filler signal power spectrograms are all bounded and non-negative, $\mathcal{P}_{T} \left( t , s \right) \geq 0$, $\mathcal{P}_{D} \left( t , s \right) \geq 0$, and $\mathcal{P}_{F} \left( t , s \right) \geq 0$, for all $\left( t , s \right) \in I_{T} \times I_{S}$ (note, the equality holding if and only if the real and imaginary components of the wavelet transform of the particular time series are simultaneously zero). However, for a given $\left( t , s \right) \in I_{T} \times I_{S}$, the real and imaginary components of the respective data and filler transforms may not be of a similar sign, and thus the respective cross terms may be negative. Therefore, in general, the interference power, $\mathcal{P}_{I} \left( t , s \right)$, of equation (\[E:PowerDecomposition3\]) may realize all real values (positive, zero, and negative). The negative values of the interference power are interpreted simply as destructive interference, reducing the strictly constructive sum of the individual data and filler signal power spectra such that the total wavelet power spectrum remains a physically meaningful non-negative value. To prove this assertion, we note the decomposition of the time series into measured data and no-measurement filler signals leads to the decomposition of the total wavelet power spectrum given by equation (\[E:NonLinearWaveletPower\]). By equation (\[E:WaveletPower\]), the total wavelet power is positive, semi-definite, $\mathcal{P}_{T} \left( t , s \right) \geq 0$, for all $\left( t , s \right) \in I_{T} \times I_{S}$, thus the decomposition of equation (\[E:NonLinearWaveletPower\]) must be positive, semi-definite for all $\left( t , s \right) \in I_{T} \times I_{S}$, $$\label{E:PositiveSemiDefiniteDecomposition_1} \mathcal{P}_{D} \left( t , s \right) + \mathcal{P}_{F} \left( t , s \right) + \mathcal{P}_{I} \left( t , s \right) \geq 0 \\$$ It is sufficient to show condition (\[E:PositiveSemiDefiniteDecomposition\_1\]) holds for all $\left( t , s \right) \in I_{T} \times I_{S}$. For any fixed $\left( t_{0} , s_{0} \right) \in I_{T} \times I_{S}$, the values realized by the data and filler power spectra are, by equations (\[E:PowerDecomposition1\]) and (\[E:PowerDecomposition2\]) respectively, $\mathcal{P}_{D} \left( t_{0} , s_{0} \right) = M \geq 0$ and $\mathcal{P}_{F} \left( t_{0} , s_{0} \right) = N \geq 0$. Additionally, their sum is positive, semi-definite, $\mathcal{P}_{D} \left( t_{0} , s_{0} \right) + \mathcal{P}_{F} \left( t_{0} , s_{0} \right) = M + N \geq 0$ (the equality holds if and only if both $M = 0$ and $N = 0$). In the case $\left( t_{0} , s_{0} \right)$ correspond to a positive or zero interference power value, $\mathcal{P}_{I} \left( t_{0} , s_{0} \right) = P \geq 0$, condition (\[E:PositiveSemiDefiniteDecomposition\_1\]) is trivially satisfied. In the case $\left( t_{0} , s_{0} \right)$ correspond to a negative interference power, $\mathcal{P}_{I} \left( t_{0} , s_{0} \right) = P < 0$, condition (\[E:PositiveSemiDefiniteDecomposition\_1\]) may be written, $$\label{E:PositiveSemiDefiniteDecomposition_2} \begin{array}{c} \vert M \vert + \vert N \vert - \vert P \vert \geq 0 \\ \vert M \vert + \vert N \vert \geq \vert P \vert \\ \end{array}$$ Since the choice of fixed $\left( t_{0} , s_{0} \right) \in I_{T} \times I_{S}$ is arbitrary, the assertion is proved for all $\left( t , s \right) \in I_{T} \times I_{S}$. We note, the power decomposition of equation (\[E:NonLinearWaveletPower\]) constrains the form of the filler signal power, and subsequently the interference power, to be comparable with that of the data signal power. For a general signal, the wavelet power amplitude distribution at a given timescale depends on the relative magnitude of the range of values over which the signal is distributed. If a particular filler signal model extends the total signal range too far, then the total wavelet power spectrum will be dominated by filler signal and interference effects, completely saturating the measured signal. This constraint requires the range of values of the filler signal model to be at least of similar order as those of the measured signal (examples include, the mean or RMS values of the measured signal, a bounded linear or spline interpolation between measured data points.). In this paper, we compare Linear Interpolation filler signal and a constant Mean Value filler signal. Comparison Power Spectrum and Confidence Levels ----------------------------------------------- We seek to quantify the new information introduced into the total wavelet power spectrum with the choice of filler signal, by constructing a cofidence level against the null-hypothesis that a given feature in the total wavelet power spectrum is the result of the filler signal and/or nonlinear interference effects. In other words, by filling the no-measurement gaps with a filler signal of arbitrary form we are introducing new information into the system. We aim to quantify the influence of the new information in overall the power spectrum, and thereby elucidate the physical information contained in the (incomplete) measured signal to the highest possible cadence. @TorrenceCompo1998 discuss stationary significance tests for both red-noise and white-noise by equating a weighted local wavelet power spectrum distribution to an assumed (normal) probability distribution, and then calculating the confidence level according to the particular assumed distribution. @Lachowicz09 offered a prescription to construct a significance level for wavelet power spectra against a time series that is the realization of some physical process that generates a signal with an intrinsic power law, $f^{-\alpha}$, variability. The main underlying assumption is that the Fourier power spectra of the comparison signal approximates that for the given signal. In the case of solar wind composition data, we have no *a priori* reason to suspect that a particular ion (or ions in the case of a composition ratio) are generated by a process with an intrinsic power law variability. Thus, we are not interested in comparing against some physical process governed by (say) an intrinsic red-noise power law, but rather simply looking to quantify the effects of both the (arbitrary) filler signal and its associated interference in the total wavelet power spectrum. Thus, the assumption of a comparison of the standard Fourier power spectra between the two signals is no longer physically relevant. We construct a statistical confidence level, based on the prescription of @Lachowicz09, against the null hypothesis that a particular feature in the total power spectrum is due to either the filler signal, a nonlinear interference effect, or a combination of both; equivalently, that a particular feature in the total signal power spectrum is significant as the result of coherent structures in the measured data signal. Thus, we seek a quantitative comparison measure of the structures of the total signal power against the power spectrum consisting of both the filler signal power and interference power. From equation (\[E:NonLinearWaveletPower\]), we define the comparison power to be, $$\mathcal{P}_{C} \left( t , s \right) \equiv \mathcal{P}_{F} \left( t , s \right) + \mathcal{P}_{I} \left( t , s \right) \label{E:ComparisonPower}$$ Recall, that while the power of the total power signal is strictly non-negative, $\mathcal{P}_{T} \left( t , s \right) \geq 0$, the range of values of the comparison power spectrum, $\mathcal{P}_{C} \left( t , s \right)$, will, in general, cover some bounded interval that includes zero in the interior, the bounding values of which depend on the relative values of $\mathcal{P}_{F} \left( t , s \right)$ and $\mathcal{P}_{I} \left( t , s \right)$. In other words, the comparison signal includes destructive interference terms of a larger magnitude that the positive filler signal power. That the comparison power may realize negative values requires us to consider the situation in which at a given timescale the comparison power may be so dominated by destructive interference that the resulting confidence level will also realize a negative value. Such an operation is meaningless, since in some sense, it is a comparison between ‘coherent structures’ in the data signal with the process of destructive interference between the data and filler signals. To rectify this, we use the fact that the comparison power spectrum is bounded from below, $M \equiv \text{inf} \lbrace \mathcal{P}_{C} \left( t , s \right) \rbrace \ \text{for all} \ \left( t , s \right) \in I_{T} \times I_{S}$. We note, in general, $M < 0$, thus, we construct an adjusted total power structure by adding the absolute value of this constant to both sides of equation (\[E:NonLinearWaveletPower\]). $$\mathcal{P}_{T} \left( t , s \right) + \vert M \vert = \mathcal{P}_{D} \left( t , s \right) + \mathcal{P}_{C} \left( t , s \right) + \vert M \vert \label{E:AdjustedPower}$$ Strictly speaking, we are now constructing a confidence level against the null hypothesis that structures in the adjusted total power, $\mathcal{P}_{T} \left( t , s \right) + \vert M \vert$, are the result of structures in the power spectrum of the adjusted comparison signal, $\mathcal{P}_{C} \left( t , s \right) + \vert M \vert$. We ascribe no physical interpretation to the addition of this constant power value across all timescales. It is required to make the confidence level physically consistent across all possible situations; the idea of comparing physical structures with physical structures by “translating" the process of destructive interference into physically coherent structures. Mathematically, the addition of a constant does not change the relative structure sizes within the power spectra, and thus a significance level constructed on the adjusted spectra retains the physically meaningful information. For continuous wavelet basis families there is information overlap between timescales (e.g., the basis family is in general not orthonormal), thus we must construct the $p^{th}$ quantile information as a function of timescale. At each fixed timescale, $s_{0} \in I_{S}$, we assume the adjusted comparison power spectrum, $\mathcal{P}_{C} \left( t , s_{0} \right) + \vert M \vert$, is distributed in time as a bounded continuous random variable and construct a probability distribution function, $\rho \left( P ; s_{0} \right)$, from the histogram of the adjusted comparison power values over the full time interval, $t \in I_{T}$; for notational clarity we include the dependence on the given fixed timescale $s_{0}$. There is some ambiguity as to the proper power histogram bin resolution. Under the continuous variable assumption, the bin resolution, $dP$, must be such that all the structures in the adjusted power spectrum are well resolved at the given timescale $s_{0}$. In practice this can be a very small value and therefore computationally expensive. For this study, $dP$ is on the order of 10$^{-5}$. Physically, the probability distribution function, $\rho \left( P ; s_{0} \right)$, is a measure of the relative influence of the (adjusted) comparison power in the (adjusted) total power spectrum at the given timescale $s_{0}$. The $p^{th}$ quantile significant power level at each timescale is given by the power value, $X_{p} \left( s_{0} \right)$, such that $\text{Prob}\left( \; \rho \left( P ; s_{0} \right) \leq X_{p} \left( s_{0} \right) \; \right)$. Formally, $$\text{Prob}\left( \; \rho \left( P ; s_{0} \right) \leq X_{p} \left( s_{0} \right) \; \right) = \int_{0}^{X_{p} \left( s_{0} \right)} \rho \left( P ; s_{0} \right) \ dP \label{E:PDF}$$ Note, for each fixed timescale, $s_{0} \in I_{S}$, $X_{p} \left( s_{0} \right)$ is a constant. Therefore, at a given fixed timescale, $s_{0} \in I_{S}$, where the adjusted total power is greater than the power level of the $p^{th}$ quantile, $X_{p} \left( s_{0} \right)$, $$\mathcal{P}_{T} \left( t , s_{0} \right) + \vert M \vert \geq X_{p} \left( s_{0} \right) \label{E:SignificantCondition}$$ we can say with $p^{th} \%$ confidence that the particular power structure is not due to the filler signal, nor an interference effect between the data signal and the filler signal. There are often timescales in which condition (\[E:SignificantCondition\]) is not satisfied, and thus no (adjusted) total power features are significant with respect to the (adjusted) comparison (filler plus interference) power. For example, an $80\%$ significance level at each timescale, $s \in I_{S}$, is constructed by (numerically) integrating equation (\[E:PDF\]) until the integral value of exceeds 0.8. The corresponding upper-integration limit, $X_{p} \left( s \right)$, at which this condition is met is the 80$\%$ significant power level at that timescale. Condition (\[E:SignificantCondition\]) then denotes whether the adjusted total power is significant relative to the adjusted comparison power at the coordinates $\left( t , s \right)$. To illustrate, we choose a fixed timescale, $s_{0} = 2.133$ days, with nice overall variability in the adjusted total power, $\mathcal{P}_{T} \left( t , s_{0} \right)$. Figure \[WaveletPowerDecomposition2\] top panel plots the adjusted comparison wavelet power spectrum, $\mathcal{P}_{C} \left( t , s \right) + |M|$, with a horizontal dashed black line demarcating the (fixed) timescale $s_{0} = 2.133$ days. The bottom panel plots the corresponding PDF, $\rho \left( P ; s_{0} \right)$, of the comparison power signal at the (fixed) timescale $s_{0} = 2.133$ days with the 80%, 90%, and 95% significance power levels, $X_{\lbrace 0.8 , 0.9 , 0.95 \rbrace} \left( s_{0} \right)$, demarcated as vertical red lines. Figure \[WaveletPowerDecomposition3\] top panel shows the adjusted total power, $\mathcal{P}_{T} \left( t , s \right) + |M|$, with (fixed) timescale $s_{0} = 2.133$ day marker. The bottom panel plots the adjusted total power signal at the $s=2.133$ day timescale with the 80%, 90%, and 95% significance power levels, $X_{\lbrace 0.8 , 0.9 , 0.95 \rbrace} \left( s_{0} \right)$, marked respectively with horizontal red lines, corresponding to the power levels calculated from the adjusted comparison signal PDF. For every adjusted total power value greater than the chosen significance level, we can say with 80%, respectively, 90% and 95%, confidence that the power associated with that feature is not due to filler signal or interference effects. We note, similar effects are seen in the case of the same synthetic time series with the same introduced data gaps, and a constant Mean Value filler signal form (see Appendix \[S:Appendix1\]). Qualitatively, despite the differences between the Linear Interpolation filler signal and constant Mean Value filler signal, the adjusted comparison power spectra share many $0^{th}$-order features (cf. Figure \[WaveletPowerDecomposition2\] and Figure \[fA2\]). Thus, it is the locations and durations of the data gaps, and therefore the interference power $\mathcal{P}_{I} \left( t , s \right)$ that dominates the (adjusted) comparison power spectra, $\mathcal{P}_{C} \left( t , s \right) + |M|$; as opposed to the particular filler power spectrum, $\mathcal{P}_{F} \left( t , s \right)$ associated with a particular form of the $F (t)$ signal. Evaluating Filler Signal Performance with Monte Carlo Ensemble Modeling {#S:MonteCarlo} ======================================================================= We have repeated the procedure described in Section \[S:DataReductionScheme\] for an ensemble of 100 different realizations of synthetic ${\rm O}^{7+}/{\rm O}^{6+}$ time series that have the observed 2008 data gaps imposed on each realization. In this section we compare results obtained for the Linear Interpolation filler signal (e.g., Figure \[O7O6model\]) described previously and a constant Mean Value filler signal (e.g., Figure \[fA1\]) where every missing data point is set to the average value of the measurement data points. From these three ensemble sets (the ideal data gap-free model, and the two cases with imposed gaps filled with Linear Interpolation and Mean Value filler forms), we compute the wavelet power spectra for every realization, as well as the 80% confidence level for both filler signal cases (see Appendix \[S:Appendix1\] for representative Figures \[fA1\], \[fA2\], and \[fA3\] corresponding to construction of the power spectra confidence levels for the Mean Value filler signal). From the individual wavelet power spectra for each of the three ensemble sets, we calculate the mean time-integrated power spectra across all (fixed) timescales, as well as the standard deviation. This ensemble-averaged time-integrated power per scale of the ideal set (the synthetic data without the imposed data gaps) is used to compare with the results of the ensemble-averaged time-integrated power per scale *above the 80% significance level* computed for each of the synthetic data sets with gaps and their respective filler signal. Our application of Monte Carlo modeling can be thought of as a mechanism for investigating the particular frequency response or transfer function of some unknown “black box" in the traditional signal processing sense. The ideal (gap-free) synthetic data corresponds to a set of input waveforms that yield a certain ensemble-averaged, global time-integrated power per scale spectrum. The presence of data gaps, our choice of values to fill those gaps, and our power spectra confidence level threshold condition result in a set of output waveforms which have a well-defined, quantified significance and their own (potentially very different) ensemble-averaged, global time-integrated power per scale spectrum. Understanding and characterizing the influence of missing data on features and properties of the wavelet power spectra is an important and necessary step towards linking those features and properties with the underlying physical processes of their origin. Modeling Synthetic ${\rm O}^{7+}/{\rm O}^{6+}$ Time Series {#SS:ModelData} ---------------------------------------------------------- In order to use Monte Carlo techniques to evaluate the performance of the different filler signals used to replace missing data, we must have a procedure for generating model time series. Obviously, the model time series should be constructed to have statistical properties as similar to the observations as possible, and in our case here, the ${\rm O}^{7+}/{\rm O}^{6+}$ composition ratio data. Due to its intrinsic variability, a number of authors have suggested that solar wind ionic composition measurements can be reasonably approximated by a first-order Markov process [e.g., @Zurbuchen00; @Hefti2000]. Therefore, we construct a random process with the following recursion $$Z_n = Z_{n-1}{\rm exp}\left[ -\Delta t/\tau_{1/e} \right] + G_n$$ where $\Delta t$ in the exponential decay term is the resolution of the data (12 min) and $G_n$ is a random number drawn from a normalized Guassian distribution. For the @Zurbuchen00 $e$-folding time of $\tau_{1/e} = 0.42$ days ($\sim$10 hours), the exponential decay term describing how much memory the process retains is close to unity for the 12-minute data ($e^{-0.02} \sim 0.98$) and slightly less if we were to model the 2-hour averages ($e^{-0.20} \sim 0.82$). The model composition time series is then computed as, $$\label{E:THZModel} Y_n = {\rm exp}[ \sigma_\ell \hat{Z_n} + \mu_\ell ],$$ where $\hat{Z_n}$ is $Z_n$ normalized to unit variance and $\mu_\ell$, $\sigma_\ell$ are the mean and standard deviation of the natural logarithm of the measured ionic composition ratio. @Zurbuchen00 showed that the ${\rm O}^{7+}/{\rm O}^{6+}$ data had a log-normal distribution with ($\mu_\ell$, $\sigma_\ell$)=($-$1.32, 0.45) and we use those values here. Our model time series reproduces the 10-hour $e$-folding time of the autocorrelation function and has an FFT power spectra that falls off between $f^{-1}$ and $f^{-2}$, consistent with the @Zurbuchen00 analysis. We note that in the @Edmondson2013 companion paper, we present the results of this modeling tuned to the ${\rm C}^{6+}/{\rm C}^{4+}$ ionic charge state ratio. There we show that, while this type of Markov process modeling produces the log-normal distribution of the in-situ measurements (by construction), the global time-integrated power per scale spectra of the observations contains real information about the physical structure and dynamics of their source region, including properties of the plasma and coronal magnetic field, that are not and cannot be accounted for by a purely random process. Ensemble-Averaged Integrated Power per Scale {#SS:AvgIPPS} -------------------------------------------- Figure \[FillerPerformance\] plots the Monte Carlo simulation results for our three ensemble set averages. The ideal case (with no missing data) is shown as the black line; all of the wavelet power of each realization is deemed significant because there are no data gaps. Thus, the ideal integrated power per scale represents the ensemble-average of the global periodicities (Fourier modes) of the ‘input waveforms’. The integrated power per scale for the ensemble-average Linear Interpolation (red asterisks) and ensemble-average Mean Value (blue triangles) cases are the ‘output waveforms’ that result from taking the ideal set of Monte Carlo realizations, adding the 2008 data gaps and a particular filler signal, and applying the 80% power spectra confidence level threshold condition. In other words, the ensemble-average global periodicities (Fourier modes) above the 80% significance level. The error bars in each color represent the statistical uncertainty of one standard deviation in each timescale bin for each of the ensemble sets. We see that for the Linear Interpolation case, the shape of the ideal cases’ integrated power per scale is well preserved for $s \gtrsim 1$ day but shows increasing departure from the the ideal case at increasingly smaller time scales. The overall relative shape of the global periodicities are qualitatively similar, but the Linear Interpolation case is increasingly attenuating the ‘input waveform’ power for $s < 1$ day. The Mean Value filler ensemble-averaged results however show a very different spectrum response. While there is much more attenuation across all scales, the behavior for scales $s \gtrsim 1$ day retains the shape of both the ideal and Linear Interpolation cases. However, for scales below $\sim$0.1 days (2.4 hrs), the Mean Value ensemble-averaged integrated power per scale rebounds from a local minimum and increases in magnitude through to the Nyquist frequency of the time series. The Mean Value filler case’s spectral response for $s \lesssim 0.1$ days (2.4 hrs) is primarily the signature of the occurrence frequency of the data gap durations. This small scale (high frequency) amplification is a direct spurious effect following from the interleaving of the constant filler signal within gap durations with relatively high frequency of occurrence, and the measured data signal. The smallest duration data gaps occur ubiquitously throughout the measured data signal, and filling these gaps with any constant value has the effect of creating spurious small scale, pulse-like structures in the full (data plus filler) signal as the wavelet transform passes through the small scales. At wavelet transform scales larger than the majority of gap durations, this effect is mitigated as the gap durations become much smaller than the wavelet filter band pass, hence the integrated power per scale shape reflects that of the ideal gap-free case. In this example, the distribution of 2008 gap durations in the 12 minute ${\rm O}^{7+}/{\rm O}^{6+}$ measurements (Figure \[MissingDataPDF\]) indicates the largest gap is $\sim$2.5 days, the time scale above which the integrated power per scale for all three cases reflect similar trends. Additionally, the vast majority of gap durations, $90.4$%, occur at timescales below 0.1 days in duration, at which point the spurious high-frequency effect dominates. Finally, there is a transition zone between $\sim$0.1 and $\sim$2.5 days in which the slope is much shallower than the ideal gap-free case. On the other hand, with the Linear Interpolation filler signal, we are able to maintain the relative shape of the ideal data gap-free ensemble-averaged integrated power per scale spectrum over a broader range of scales but with increasing attenuation at progressively smaller scales $s \lesssim 1$ day. This is essentially due to the fact that, in any given data gap, the difference between filler signal values and neighboring synthetic model values are, by construction, much closer (as opposed to the synthetic model values and Mean Value filler). Figures \[WaveletPowerDecomposition2\] and \[WaveletPowerDecomposition3\], illustrate the construction of significance levels at 80%, 90%, and 95% comparison power. Using this procedure, we calculated the ensemble-averaged integrated power per scale for the Linear Interpolation filler at the 80% (shown as red asterisks in Figure \[FillerPerformance\]), as well as the 90% and 95% significance levels to examine the attenuation due to the significance level threshold conditions. Figure \[HybridFillerPerformance\] plots these results normalized to the ideal gap-free average integrated power per scale spectrum. The ideal case is shown as the black line at unity and the 80%, 90%, and 95% Linear Interpolation cases are shown as red asterisks, green squares, and blue crosses, respectively. Here the scale-dependence of the attenuation with respect to the ideal gap-free ensemble spectrum is readily visible showing a drop from roughly 0.70–0.80 of the ideal average power for $s > 1$ days down to $\sim$0.30 of the ideal power at $s \sim 0.02$ days. For our particular set of model time series and data gap structure, the ensemble-averaged power per scale curves for the different significance levels show very little separation with respect to each other. This could be expected from examination of Figure \[WaveletPowerDecomposition3\] where the time-integrated power for the $s=2.133$ day cut shows only minor differences in the total area under the $\mathcal{P}_T+|M|$ curve and above the various significance level thresholds. Therefore, our selection of the 80% significance level appears reasonable, at least for the time series and data gap properties analyzed here. Mean Value filler comparisons across 80%, 90%, and 95% significance levels exhibit similar trends, albeit with stronger relative attenuation. Wavelet Analysis of ${\rm O}^{7+}/{\rm O}^{6+}$ 12-Min Data {#S:WaveletO7O6} =========================================================== We have applied the analysis procedure outlined in Section \[S:DataReductionScheme\] to the ACE/SWICS ${\rm O}^{7+}/{\rm O}^{6+}$ data shown in Figure \[datafig\] using both the Linear Interpolation and constant Mean Value prescriptions for filling the data gaps. The wavelet power spectra for full data plus both filler signal models were calculated, as well as the wavelet power spectra decomposition from the two filler signals and their respective nonlinear interference components. The adjusted comparison power was then used to construct the 80% significance levels for each timescale. The results are shown in Figures \[figO7O6wavelet\], \[figO7O6wavelet2\], and \[figO7O6wavelet3\]. In Figure \[figO7O6wavelet\], the total wavelet power spectra for the ${\rm O}^{7+}/{\rm O}^{6+}$ data with a Linear Interpolation filler and the constant Mean Value filler are shown in the top and bottom panels, respectively. Figure \[figO7O6wavelet2\] plots the corresponding wavelet power that exceeds the 80% confidence level thresholds. Figure \[figO7O6wavelet3\] plots on a linear scale, the normalized time-integrated power per scale for both the overall total power spectra (top row) and 80% significant power (bottom row), for the Linear Interpolation filler signal (right column) and constant Mean Value filler signal (left column). For the Morlet wavelet family, the time-integrated power per timescale (also known as the global wavelet power) of Figure \[figO7O6wavelet3\] is akin to Fourier mode decomposition. The integrated wavelet power per scales of the ${\rm O}^{7+}/{\rm O}^{6+}$ for both filler signal forms, in both the total and significant power, exhibit a number of well defined peaks corresponding to relatively well defined Fourier modes (globally periodic timescales) in similar timescale neighborhoods. In the Linear Interpolation filler signal case (left column), there are three strong Fourier modes (peaks) occurring at approximately $\lbrace$ 3, 8–10, 18–28 $\rbrace$ days, in both total and significant power cases. Below $\sim$1 day timescales, it becomes difficult to discern Fourier modes (peaks) from the power law shape. As explained above, the large high-frequency effect (timescales $\lesssim$ 0.1 days) in the Mean Value case (right column) reflects the nature of the gap durations. Outside of this effect, there are well-defined Fourier modes in both the total power and significant power that occur at approximately, $\lbrace$ 2.5, 8–9, 13–17, 25–30, 45–50 $\rbrace$ day timescales. The smallest identifiable Fourier modes, at 2.5 and 3 days, respectively, may be an artifact of the largest data gap duration in the measured signal. However, @Zhao09 showed that the slow solar wind, as determined by ${\rm O}^{7+}/{\rm O}^{6+} \ge 0.145$, has a mean width centered on the heliospheric current sheet of approximately 20$^\circ$ (40$^\circ$) during solar minimum (maximum). We note that our other significant integrated power peak at the 3–4 day correlation timescale corresponds to an $\sim$45$^\circ$ width given the 13$^\circ$ day$^{-1}$ solar rotation rate. This may reflect crossing the slow solar wind region surrounding the helmet streamer belt in a highly inclined configuration and would be consistent with the width of the slow solar wind distribution observed in the Ulysses fast latitude scan [@McComas00]. @Temmer07 identified 9-day periodicities in ACE solar wind parameters over the 1998-2006 period and showed these likely arose due to the distribution of coronal holes via time series of coronal hole area. @Katsavrias12 likewise identified both the 9-day and 13.5 day peaks in solar wind speed, proton temperature, density, and components of the magnetic field over a four solar cycle interval (1966–2010). The 18–28 day, and 25–30 day timescales are clearly associated with Carrington rotation effects. Interestingly, the Carrington rotation periodicity is absent from the significant Linear Interpolation filler power spectra. This is largely due to the unusual solar minimum conditions during 2008. In the 12-minute ${\rm O}^{7+}/{\rm O}^{6+}$ data shown in Figure \[datafig\], one may identify a *qualitative* recurrent $\sim$5 day enhancement repeating with a 27-day periodicity for three consecutive Carrington Rotations at the beginning of the year (CR2066–CR2068). However, this enhancement is absent (potentially due to a data gap) in the fourth Carrington Rotation (CR2069) and virtually indistinguishable during the remainder of the year. In fact, the wavelet power in the top panel of Figure \[figO7O6wavelet\] also shows this as a reasonably strong intensity stripe at the 27-day timescale that falls outside of the 80% significance contours for the first 100 days and a more modest intensity signal at that timescale up until 200 days; thus, the signal is present in the total wavelet power, but does not exceed the 80% confidence level threshold derived from the Linear Interpolation filler signal and its interference effects. On the other hand, while the 27-day periodicity is absent, its first harmonic at approximately 13.5 days is present above the 80% significance level for the entire data set duration. The preference for this periodicity seems likely due to the large-scale coronal magnetic field structure and consequently, the solar wind structure in the heliosphere. During the solar cycle 23 solar minimum, the polar fields were substantially weaker than usual resulting in a more highly warped helmet streamer belt, more pseudostreamers, and more complexity in the mapping of the solar wind source regions to smaller, low latitude coronal holes [e.g., @Lee09; @Riley12]. @MursulaZieger1996 have argued the 13.5 day periodicity could be explained by a two slow-fast stream structure per Carrington Rotation that may result from a highly warped helmet streamer belt, but it is not clear this is universally applicable [e.g., see discussion by @Temmer07]. Finally, the 45–50 day Fourier mode that shows up in the large timescale tail of the Mean Value filler case, is likely a harmonic of the Carrington rotation periodicity. Above this scale, for a single year data set (2008), the total integrated power per scale is primarily bounded by the wavelet cone-of-influence (see Figure \[figO7O6wavelet\]). Conclusions {#S:Conclusions} =========== We have presented a generalized procedure for constructing a wavelet power spectrum significance level measure that quantifies the relative influence of two interleaved signals. In this investigation, the total signal is an interleaved combination of measured data and some (general, arbitrary) signal imposed to fill the ‘no measurement’ gaps at the given cadence. We constructed a statistical confidence level against the null-hypothesis that a given feature in the total wavelet power spectrum is strictly due to filler signal and the nonlinear interference effects between the filler signal and measured data signal. We apply this power spectra confidence procedure on Monte Carlo simulations of synthetic ${\rm O}^{7+}/{\rm O}^{6+}$ ionic composition data to evaluate the performance of the Mean Value and Linear Interpolation filler signals. Using the performance criteria of reproducing the ideal, data gap-free ensemble-averaged time-integrated power per scale, we show that the Linear Interpolation filler signal does a better job than the Mean Value signal across all but the smallest temporal scales and effectively acts as a low-pass filter suppressing the inherent high frequency (small timescale) power that arise from the frequency of the missing data and duration of the data gaps. We show that for our sparsely populated data set, the 80%, 90%, and 95% confidence levels yield almost identical results for the synthetic data ensemble. We calculated the ${\rm O}^{7+}/{\rm O}^{6+}$ wavelet power during the quiet-sun solar minimum of 2008 using both the Mean Value and Linear Interpolation filler signals, show the structure of their derived 80% power confidence levels, and present the total and $\ge$80% significant time-integrated power per scale spectra. Our analysis using the Linear Interpolation data gap filler signal yields strong Fourier mode harmonics in both the total and significant integrated power per scale spectrum at $\lbrace$ 3, 8–10, 18–28 $\rbrace$ days. Each of these peaks are also visible in the total and significant integrated power per scale when using the Mean Value filler, but the relative magnitude of the spectrum for scales $\gtrsim$0.10 days is dwarfed by the (spurious) power associated with very-high frequencies ($s < 0.10$ days). In a companion publication [@Edmondson2013], we have applied the power spectra confidence analysis presented here to the ${\rm C}^{6+}/{\rm C}^{4+}$ ionic composition ratio and discuss the implications of the coherent structure and variability of ionic composition ratios for current theories of solar wind generation. The authors would like to thank the reviewer for their comments and suggestions which substantially improved the paper. J.K.E., S.T.L., and T.H.Z. acknowledge support from NASA LWS NNX10AQ61G and NNX07AB99G. B.J.L acknowledges support from AFOSR YIP FA9550-11-1-0048 and NASA HTP NNX11AJ65G. Power Spectra and Confidence Levels for the Mean Value Filler Signal {#S:Appendix1} ==================================================================== In Section \[S:DataReductionScheme\] we used one realization of the Zurbuchen et al. inspired Markov process modeling to illustrate the wavelet power spectra confidence procedure. First, we generated a synthetic time series time series from Equations (19) and (20), then imposed data gaps corresponding to the missing data intervals in the 2008 12 minute ${\rm O}^{7+}/{\rm O}^{6+}$ data. The analysis of Section \[S:DataReductionScheme\] used the Linear Interpolation filler signal to populate the missing data intervals and calculate the various wavelet power spectra associated with the ‘good measurement’ data, the filler signal, and their nonlinear interference. A reference comparison power was constructed from the filler signal and interference power contributions and used to quantify the significance levels associated with features in the total wavelet power spectra. In section \[S:MonteCarlo\], we presented the ensemble-averaged results from performing this procedure on a set of 100 realizations of the Markov process using both the Linear Interpolation filler signal and a constant Mean Value filler signal. Here we present details of our power spectra confidence level construction for the model realization example of section \[S:DataReductionScheme\] with the Mean Value filler signal. Figure \[fA1\] shows time series in two zoomed in views for same synthetic data $D(t)$ shown in Figure \[O7O6model\] but for the constant Mean Value filler signal $F(t)$. Following the decomposition of the time series and calculation the power of the wavelet transforms of the constituent components, the top panel of Figure \[fA2\] plots the comparison power plus offset $\mathcal{P}_{C} + |M|$. The bottom panel of Figure \[fA2\] plots the PDF of the comparison power at the $s = 2.133$ day scale and the 80%, 90%, and 95% levels of the distribution. The similarities and differences in the comparison power between the Mean Value filler signal and the Linear Interpolation case are readily apparent when comparing Figures \[fA2\] and \[WaveletPowerDecomposition2\]. First, we see that the comparison power wavelet has both a similar range in magnitude and qualitative large scale structure in $(t, s)$. For example, the regions of relatively low comparison power levels (saturated as white in the color scale) at the $s\sim10$ day scale features are quite similar in shape and location, the overall trend of comparison power levels at scales $0.1 \lesssim s \lesssim 1.0$ days being elevated with respect to $s \gtrsim 1.0$ days, and the largest power levels (saturated with magenta in the color scale) corresponding to many fine-scale linear striations for $s \lesssim 0.1$ days. From the relative amount of color scale saturation at the smallest scales, the Mean Value comparison power has a broader temporal extent indicating more high frequency interference power throughout the time series. The lower panels of Figures \[fA2\] and \[WaveletPowerDecomposition2\] however, have a very different PDFs for their respective comparison power levels (although the overall range of values are comparable). While the Linear Interpolation PDF is symmetric and centered around a mid-point value of $\sim$44, the Mean Value PDF has more of an exponential fall off from a maximum at $\sim$41.3. Thus, we can see the relative contributions to the comparison power from: (1) the form and values of the $F(t)$ filler signal in shape of the PDF at a given scale, and (2) in the location, duration, and distribution of the data gaps (i.e., where $F(t) \ne 0$) and the resulting interference power in the overall, large scale properties of the comparison wavelet power. Figure \[fA3\] plots the total power plus offset $\mathcal{P}_{T} + |M|$ in the top panel and the $s = 2.133$ day cut in the lower panel for the Mean Value filler in the same format as and for direct comparison with Linear Interpolation results in Figure \[WaveletPowerDecomposition3\]. Again, there are both important similarities in the overall qualitative properties of the wavelet power and important differences arising from the different filler signals. In the wavelet power, the $s \gtrsim 5.0$ days features are less prominent in the Mean Value case, whereas the smallest scale structures at $s \lesssim 0.1$ days are much more prominent. The lower panel of Figures \[fA3\] shows that the Mean Value case has no total power at the $s=2.133$ day scale that falls above the 95% significance level, and only two temporal locations that exceed the 80% level. The most prominent Mean Value total power feature at $t\sim$DOY 145 is obviously present in the Linear Interpolation wavelet power, but only slightly exceeds the Linear Interpolation 80% significance level at this scale. The overall, qualitative scale-dependent influence of the Mean Value and Linear Interpolation filler signals to the total wavelet power in the single representative synthetic data example from Section \[S:DataReductionScheme\] are reproduced in the properties of the ensemble-averaged behavior obtained from the Monte Carlo simulations in Section \[S:MonteCarlo\] and in the application to the actual ${\rm O}^{7+}/{\rm O}^{6+}$ measurements presented in Section \[S:WaveletO7O6\]. Bolzan, M., Rosa, R.R., Ramos, F.M., Fagundes, P.R., & Sahai, Y. 2005, J. Atmos. Solar-Terr. Phys. 67, 1843 Daubechies, I. 1992, *Ten Lectures on Wavelets*, Society for Industrial and Applied Mathematics. Edmondson, J. K., Lynch, B. J., Lepri, S. T., & Zurbuchen, T. H. 2013, “Coherent Structure in Solar Wind ${\rm C}^{6+}/{\rm C}^{4+}$ Ionic Composition Data During the Quiet-Sun Conditions of 2008," , in press Fenimore, E. E., Asbridge, J. R., Bame, S. J., Feldman, W. C., & Gosling, J. T. 1978, , 83, 4353 Geiss, J., Gloeckler, G., von Steiger, R., Balsiger, H., Fisk, L. A., Galvin, A. B., Ipavich, F. M., Livi, S., McKenzie, J. F., Ogilvie, K. W., & Wilken, B. 1995, Science, 268, 1033 Gilbert, J. A., Lepri, S. T., Landi, E., & Zurbuchen, T. H. 2012, , 751, 20 Gloeckler, G., Cain, J., Ipavich, F. M., Tums, E. O., Bedini, P., Fisk, L. A., Zurbuchen, T. H., Bochsler, P., Fischer, J., Wimmer-Schweingruber, R. F., Geiss, J., & Kallenbach, R. 1998, , 86, 497 Gonzalez A. L. C., & Gonzalez, W. D. 1987, , 92, 4357. Gonzalez A. L. C., Gonzalez, W. D., Dutra, S. L., & Tsurutani, B. T. 1993, , 98, 9215 Gosling, J. T. 1997, AIP Conf. Ser., 385, 17 Hefti, S., Grünwaldt, H., Bochsler, P., & Aellig, M. R. 2000, , 108, A5, 10527 Katsavrias, Ch., Preka-Papadema, P., & Moussas, X. 2012, , 280, 623 Lachowicz, P. 2009, IEEE Transactions on Signal Processing, arXiv:0906.4176v1 \[astro-ph.HE\] Landi, E., Alexander, R. L., Gruesbeck, J. R., Gilbert, J. A., Lepri, S. T., Manchester, W. B., & Zurbuchen, T. H. 2012, , 744, 100 Le, G.-M., & Wang, J.-L. 2003, Chinese Journal of Astronomical and Astrophysical, 3, 391Ð394 Lee, C. O., Luhmann, J. G., Zhao, X. P., Liu, Y., Riley, P., Arge, C. N., Russell, C. T., & de Pater, I. 2009, , 256, 345 Liu, Y., San Liang, X., & Weisberg, R. H. 2007, J. Atmos. Oceanic Technol., 24, 2093 McComas, D. J., Barraclough, B. L., Funsten, H. O., et al. 2000, , 105, A5, 10419 McComas, D. J., Elliott, H. A., & von Steiger, R. 2002, , 29, 9, 28-1 Mursula K. & Zieger, B., 1996, , 101, 27077 Mursula K. & Zieger, B., 1998, , 25, 1851 Neugebauer, M., Ruzmaikin, A., & McComas, D. J. 1997, AIP Conf. Proc., 385, 41 Prabhakaran Nayar, S. R., Sanalkumaran Nair, V., Radhika, V. N., & Revathy, K. 2001, , 201, 405 Prabhakaran Nayar, S. R., Radhika, V. N., Revathy, K., & Ramadas, V. 2002, , 208, 359 Riley, P., & Luhmann, J. G. 2012, , 277, 355 Stone, E. C., Frandsen, A. M., Mewaldt, R. A., Christian, E. R., Margolies, D., Ormes, J. F., & Snow, F. 1998, , 86, 1 Svalgaard, L. & Wilcox, J. M. 1975, , 41, 461 Temmer, M., Vrsnak, B., & Veronig, A.M. 2007, , 241, 371 Torrence, C., & Compo, G. P. 1998, “A practical guide to wavelet analysis", Bull. Am. Meteorol. Soc., 61-78 von Steiger, R., Schwadron, N. A., Fisk, L. A., et al. 2000, , 105, 27217 von Steiger, R., Geiss, J., & Gloeckler, G. 1997, Cosmic Winds and the Heliosphere, Space Sci. Ser., edited by J. R. Jokipii, C. P. Sonett, and M. S. Giampapa, Univ. of Ariz. Press, Tucson. Zhao, L., T. H. Zurbuchen, & L. A. Fisk 2009, , 36, L14104 Zurbuchen, T. H., Hefti, S., Fisk, L. A., Gloeckler, G., & Schwadron, N. A. 2000, , 105, A8, 18327 Zurbuchen, T. H., Fisk, L. A., Gloeckler, G., & von Steiger, R. 2002, ., 29, 9, 66-1 Zurbuchen, T. H. 2007, , 45, 297

--- abstract: 'We report long-slit spectroscopic observations of the quasar SDSS J082303.22+052907.6 ($z_{\rm CIV}$$\sim$3.1875), whose Broad Line Region (BLR) is partly eclipsed by a strong damped Lyman-$\alpha$ (DLA; log$N$(HI)=21.7) cloud. This allows us to study the Narrow Line Region (NLR) of the quasar and the Lyman-$\alpha$ emission from the host galaxy. Using [cloudy]{} models that explain the presence of strong NV and PV absorption together with the detection of SiII$^*$ and OI$^{**}$ absorption in the DLA, we show that the density and the distance of the cloud to the quasar are in the ranges 180 $<$ $n_{\rm H}$ $<$ 710 cm$^{-3}$ and 580 $>$ $r_0$ $>$230 pc, respectively. Sizes of the neutral($\sim$2-9pc) and highly ionized phases ($\sim$3-80pc) are consistent with the partial coverage of the CIV broad line region by the CIV absorption from the DLA (covering factor of $\sim$0.85). We show that the residuals are consistent with emission from the NLR with CIV/Lyman-$\alpha$ ratios varying from 0 to 0.29 through the profile. Remarkably, we detect extended Lyman-$\alpha$ emission up to 25kpc to the North and West directions and 15kpc to the South and East. We interpret the emission as the superposition of strong emission in the plane of the galaxy up to 10kpc with emission in a wind of projected velocity $\sim$500km s$^{-1}$ which is seen up to 25kpc. The low metallicity of the DLA (0.27 solar) argues for at least part of this gas being in-falling towards the AGN and possibly being located where accretion from cold streams ends up.' date: 'Accepted ....... Received ....... ' title: ' A coronagraphic absorbing cloud reveals the narrow-line region and extended Lyman-$\alpha$ emission of QSO J0823$+$0529 [^1]' --- \[firstpage\] galaxies: evolution – intergalactic medium – quasars: absorption lines – quasars: individual: SDSS J082303.22$+$052907.6 Introduction ============ Luminous high-redshift quasars consist of supermassive black holes residing at the center of massive galaxies and growing through mass accretion of gas in an accretion disk. Bright quasars play an important role in shaping their host galaxies through the emission of ionizing flux but also through launching powerful and high-velocity outflows of gas. These outflows inject energy and material to the disk of the galaxy and may be influencing the physical state up to larger distances. It has remained unclear however what are the mechanisms that drive energy from the very center of the active galactic nuclei (AGN) to the outskirts of the galaxy. Observational evidence for outflows and winds in AGNs is seen as prominent radio-jets in radio-loud sources, broad absorption lines observed in broad absorption line (BAL) quasars, or through the photoionized warm absorber which is frequently observed in the soft X-rays (e.g. Crenshaw et al. 2003). Gravitational micro-lensing studies have shown that the primary X-ray emission region in AGN is of the order of a few tens of gravitational radii in size (e.g. Dai et al. 2010) and X-ray spectroscopy shows that highly ionized outflows launched from this region are seen in high-$z$ quasars (Chartas et al. 2009) and in at least 40% of them with velocities up to 0.1 c (Gofford et al. 2013). Outflows are observed also on large scales. Mullaney et al. (2013) used the SDSS spectroscopic data base to study the one-dimensional kinematic properties of \[OIII\]$\lambda$5007 by performing multicomponent fitting to the optical emission-line profiles of about 24000, $z<0.4$ optically selected AGNs. They showed that approximately 17 percent of the AGNs have emission-line profiles that indicate their ionized gas kinematics are dominated by outflows and a considerably larger fraction are likely to host ionized outflows at lower levels. Harrison et al. (2014) find high-velocity ionized gas (velocity widths of about 600-1500 km s$^{-1}$) with observed spatial extents of (6-16) kpc in a representative sample of $z<0.2$, luminous (i.e. $L$\[O [iii]{}\]$>$10$^{41.7}$ erg s$^{-1}$) type 2 AGNs. Therefore galaxy-wide energetic outflows are not confined to the most extreme star-forming galaxies or radio-luminous AGNs. If outflows are observed both on small and large scales, how the small scale outflows are transported at larger distances remains unclear (Faucher-Giguère et al. 2012, Ishibashi & Fabian 2015, King & Pounds 2015). This is however a crucial question as these outflows are massive and energetic enough to significantly influence star formation in the host galaxy and provide significant metal enrichment to the interstellar and intergalactic media (e.g. Dubois et al. 2013). At high redshift where quasars are more luminous, the consequences of such outflows are of first importance for galaxy formation. One way to study the interplay between the quasar and its surrounding is to search for Lyman-$\alpha$ emission around quasars (e.g. Hu & Cowie 1987, Hu et al. 1996, Petitjean et al. 1996, Bunker et al. 2003, Christensen et al. 2006). These observations reveal gas infalling onto the galaxy (Weidinger et al. 2005), positive feed-back from the AGN (Rauch et al. 2013) or a correlation between the luminosity of the extended emission and the ionizing flux from the quasar (North et al. 2012). Very recently, we searched quasar spectra from the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS; Dawson et al. 2014) for the rare occurrences where a strong damped Lyman-$\alpha$ absorber (DLA) blocks the Broad Line Region emission (BLR) from the quasar and acts as a natural coronagraph to reveal narrow Lyman-$\alpha$ emission from the host galaxy (Finley et al. 2013; see also Hennawi et al. 2009). This constitutes a new way to have direct access to the quasar host galaxy and possibly, when the size of the DLA is small enough, to the very center of the AGN. Out of a total of more than 70,000 spectra of $z>2$ quasars (Pâris et al. 2012), we gathered a sample of 57 such quasars and followed-up six of them with the slit spectrograph Magellan-MagE to search for the very special cases where the DLA coronagraph reveals the very center of the host galaxy and extended Lyman-$\alpha$ emission. In the course of this follow-up program, we found one object SDSS J0823+0529 where the DLA does not cover the Lyman-$\alpha$ broad line region entirely and reveals the emission of the Lyman-$\alpha$ and C [iv]{} narrow line regions. We show here that this is a unique opportunity to study the link between the properties of the central regions of the AGN to that of the gas in the halo of the quasar. The paper is organized as follows. In Section 2 we describe the observations and data reduction. We derive properties of the gas associated with the DLA (metallicity, ionization state, density, distance to the quasar, typical size) in Section 3. We discuss the properties of the quasar narrow line region and of the extended Lyman-$\alpha$ emission in Sections 4 and 5, respectively, We then finally, present our conclusions in Section 6. In this work, we use a standard CDM cosmology with $\Omega_{\Lambda}$ = 0.73, $\Omega_{m}$ = 0.27, and H$_0$ =70 km s$^{-1}$ Mpc$^{-1}$ (Komatsu et al. 2011). Therefore 1 arcsec corresponds to about 7.1 kpc at the redshift of the quasar ($z$ = 3.1875 see below). In the following we will use solar metallicities from Asplund et al. (2009). [c]{} [c]{} Observations and data reduction =============================== The spectrum of the quasar SDSS J0823$+$0529 was observed with the Magellan Echellete spectrograph (MagE; Marshall et al. 2008) mounted on the 6.5 m Clay telescope located at Las Campanas Observatory. MagE is a medium-resolution long-slit spectrograph that covers the full wavelength range of the visible spectrum (3200 $\textup{\AA}$ $-$ 1 $\mu$ m). Its 10 arcsec long slit and 0.30$^{"}$ per pixel sampling in the spatial direction, are ideal for observing high-redshift extended astrophysical objects. The spectrograph was designed to have high throughput in the blue, with the peak throughput reaching 22 $\%$ at 5600 $\textup{\AA}$ including the telescope. The quasar was observed in December 2012 with an one arcsec width slit aligned along two different position angles (PAs) for 1 hour each. One position was South-North (PA = 0) and the other position was East-West (PA = 90). Another 1-hour exposure with PA = 90  was taken in February 2013 but the resultant spectrum has a lower SNR. Following each exposure, the spectrum of a standard star was also recorded allowing us to precisely flux-calibrate the quasar spectra. The seeing, measured on the extracted trace of the quasar is 1.06 arcsec for PA = 0 and 0.93 arcsec for PA = 90. We reduced the spectrum using the Mage\_Reduce pipeline written in the Interactive Data Language (IDL) by George Becker[^2]. In addition to the 1-dimensional (1D) spectrum, the pipeline provides 2-dimensional (2D) sky-subtracted science frames as well as the corresponding 2D wavelength solution corrected for the vacuum heliocentric velocity shifts. These 2D images will later be used to rectify the curved spectral orders (see below). Here, we note that since there is an extended Lyman-$\alpha$ emission in order 9 of our 2D spectra, we preferentially used a wider extraction window to extract this order. Two-dimensional spectra when imaged onto a detector are often curved with respect to the natural (x,y)-coordinate system of the detector defined by the CCD columns (Kelson 2003). To rectify the curvature of the orders we first rebin the wavelength area (given by the 2D wavelength solution) and position area (given by the 2D slitgrid array provided by the pipeline) increasing the number of pixel by a factor of hundred. We define a new 2D array, with one dimension representing the wavelength ($\lambda$) and the other the position on the slit ($s$). For each pixel on the 2D spectrum of the quasar, we take its corresponding wavelength and its position on the slit directly in the rebinned 2D areas. Therefore, the (x,y) coordinates of each pixel (defined by the CCD columns) can now be transformed to an ($s$, $\lambda$) coordinate defined by the new 2D array introduced above. In this new ($s$, $\lambda$)-coordinate system, the curvature of the orders and the tilting of the spectral lines are all rectified. In this study, we use these rectified 2D images when we discuss the spatial extension of the Lyman-$\alpha$ emission line in the quasar spectrum. Note that the rebinning process has very little effect on the Lyman-$\alpha$ emission because it is conveniently placed in the middle of the corresponding order where curvature is at minimum. Finally, the extracted 1D spectrum of each individual order is corrected for the relative spectral response of the instrument and flux-calibrated using the spectrum of a standard star (HR 1544) observed during the same night. The spectrum of the standard star can be found in the ESO standard star catalogue webpage[^3]. We emphasize that our standard star spectrum was obtained immediately after observing the quasar spectrum. We also flux calibrated in the same way the 2D spectra in the Lyman-$\alpha$ range. These flux-calibrated spectra are then combined weighting each pixel by the inverse of its variance. The resulting spectrum (after binning with a 3 $\times$ 3 box) has $\sim$ 27 km s$^{-1}$ per pixel and 3 pixels per resolution element, and therefore FWHM $\sim$ 80 km s$^{-1}$. We have fitted the quasar C [iv]{} emission line with two Gaussian functions (to mimic the two lines of the doublet) to estimate the redshift of the quasar. We derive $z_{\rm em}$ = 3.1875. This is $\sim$330 km s$^{-1}$ smaller than the redshift of the DLA ($z_{\rm DLA}$ = 3.1910 derived from the fit of Si [ii]{} and Fe [ii]{} absorptions). However, it is well known that redshifts from C [iv]{} are smaller by up to 600 km s$^{-1}$ compared to redshifts from narrow forbidden lines (e.g. Hewett & Wild 2010). Therefore we cannot be certain that the DLA is redshifted compared to the quasar. We would need to detect \[O[iii]{}\] lines redshifted to the infra-red to have a better estimate of $z_{\rm em}$. [c]{} A DLA acting as a coronagraph ============================= Figure \[QSO\_whole\_spect\] shows the co-added spectrum of SDSS J0823+0529 using all the exposures taken at different PAs. It is apparent that a strong DLA is located close to the redshift of the quasar. However, strong Lyman-$\alpha$ emission is present in the center of the DLA trough. This is possible if the absorbing cloud has a transverse extension which is smaller than the central narrow line region (NLR) of the quasar. Note that the strength of the Lyman-$\alpha$ emission together with the size of the cloud derived below makes it very improbable that the Lyman-$\alpha$ emission could be a consequence of star formation in the DLA itself. The redshift of the DLA ($z_{\rm DLA}$ = 3.1910) was determined by conducting a simultaneous Voigt profile fit of the Fe [ii]{}, Si [ii]{}$^{*}$, and O [i]{}$^{**}$ absorption profiles. Elemental abundances -------------------- A Voigt profile fit was conducted on the damped Lyman-$\alpha$ absorption line of this system, resulting in a neutral hydrogen column density of log $N_{\rm HI}$ = 21.70 $\pm$ 0.10. Note that the placement of the quasar continuum is very uncertain as the DLA covers the Lyman-$\alpha$ and the N [v]{} broad emission lines. However, the core and especially the red wing of the DLA profile allowed us to satisfactorily constrain the H [i]{} column density. We will come back to this in Section 3.4. We detect absorption lines from O [i]{}, O [i]{}$^{**}$, Si [ii]{}, Si [ii]{}$^*$, Fe [ii]{}, Al [ii]{}, Al [iii]{}, Ar [i]{}, C [ii]{}, C [ii]{}$^{*}$, C [iv]{}, Si [iv]{}, P [v]{} and N [v]{}. Absorption profiles are shown in Fig. \[j08231D\] and results from fitting these lines are given in Tables \[lowion\] $\&$ \[hiion\]. Techniques used here are similar to those in Fathivavsari et al (2013). It must be noted that most of the lines are saturated, preventing us from deriving accurate column densities especially at the resolution of our data (R$\sim$4000). The profiles are dominated by a main strong component clearly seen in particular in Fe [ii]{}$\lambda$1608 and Si [ii]{}$^*$$\lambda$1533. To constrain the Doppler parameter of this component, we take advantage of the fact that Fe [ii]{}$\lambda$1608 is not saturated and well defined while Fe [ii]{}$\lambda$1611 is not detected (see Fig \[dopplerb\]). We start by fitting together Si [ii]{}$\lambda$1808 and 1526, imposing the presence of the main component with a fixed Doppler parameter. We then use the resulting decomposition to fit the Fe [ii]{} lines. Results are shown in Fig \[dopplerb\] for $b$ = 10 and 20 km/s. It is apparent that components narrower than $b$ $\sim$ 20 km/s are not favored and $b$ = 10 km/s is definitely rejected. Therefore, we are confident that imposing $b$ $\sim$ 20 km/s for the main component in the DLA will give us a good estimate of column densities. For some of the species, several lines with very different oscillator strengths (either in doublets or multiplets) are present so that we can derive robust estimates of the column densities. This is the case for Si [ii]{}, Si [ii]{}$^{*}$, and Fe [ii]{}. From this, we derive metallicities relative to solar, \[Si/H\] = $-$0.79 and \[Fe/H\] = $-$1.87. We also derive an upper limit on $N$(Fe [iii]{}). We also fit the high-ionization species. We did not try to tie the components in different profiles because this high-ionization phase could be highly perturbed. In Table \[hiion\], except for the first two components of the C [iv]{} absorption profiles, all reported column densities are upper limits because the profiles are strongly saturated. It can be seen however that the decompositions in sub-components are fairly consistent between the different species. We detect absorptions from O [i]{}$^{**}$ and Si [ii]{}$^{*}$ which are rarely detected in DLAs (see Noterdaeme et al. 2015, Neeleman et al. 2015) and are more commonly seen in DLAs associated to GRB afterglows (Vreeswijk et al. 2004; Chen et al. 2005; Fynbo et al. 2006). Absorption from the fine structure state of Si [ii]{} will be used to constrain the density of the absorbing cloud (see below). The high measured depletion of iron relative to silicon (i.e. \[Si/Fe\] = +1.08) in this DLA suggests the presence of dust. Consequently, extinction due to dust might be significant along this line of sight. Indeed, the median g $-$ r color for a sample of 697 non-BAL quasars with redshift within $\Delta$$z$ = $\pm$0.02 around $z_{\rm em}$ = 3.1910 is ${\left< g-r \right>}$ = 0.30 when $g-r$ = 1.1 for QSO J0823+0529. The reddening for this line of sight is E(B-V) = 0.09, measured with an SMC reddening law template, which places it among the most reddened of the sight lines in the Finley et al. (2013) statistical sample. We will take this reddening into account in the following while discussing the properties of the quasar. Physical conditions in the DLA gas ---------------------------------- In this section we use the photo-ionization code [cloudy]{} to constrain the ionization state of the absorber and its distance to the quasar central engine. Observed ionic ratios of Si [ii]{}, Si [iv]{}, Al [iii]{}, and Ar [i]{} are used to constrain the plane parallel models constructed for a range of ionization parameters. The calculations were stopped when a neutral hydrogen column density of log $N$(H [i]{}) = 21.70 is reached. Solar relative abundances are assumed and the metallicity is taken to be Z = 0.16Z$_{\odot}$ from the Silicon abundance of the DLA derived in Section 3.1. [c c c c]{} Redshift & Ion & log(N)$^a$ & log(N)\ $ $ & $ $ & $\rm Observed$ & $\rm Model$\ 3.190974 & Si [ii]{} & 16.42$\pm$0.10 & 16.53\ & Si [ii]{}$^{*}$ & 15.49$\pm$0.30 & 15.49\ & Fe [ii]{} & 15.33$\pm$0.20 & 16.50\ & Fe [iii]{} & $\le$ 15.00 & 15.09\ & Ar [i]{} & 14.97$\pm$0.20 & 15.04\ & Al [iii]{} & 14.80$\pm$0.10 & 14.64\ & O [i]{}$^{**}$ & 16.45$\pm$0.45 & 15.70\ & O [i]{} & 17.08$\pm$0.50 & 17.62\ \ \[lowion\] [c c c c c]{} Redshift & Ion & b & log(N) & log(N)\ $ $ & $ $ & \[km s$^{-1}$\] & $\rm Observed$ & $\rm Model$$^a$\ 3.185037 & C [iv]{} & 23 & 13.60$\pm$0.10 & $ $\ 3.186553 & C [iv]{} & 32 & 13.90$\pm$0.10 & $ $\ 3.189333 & C [iv]{} & 86$\pm$10 & $\ge$15.40$\pm$0.10 & $ $\ 3.190953 & C [iv]{} & 38$\pm$14 & $\ge$14.65$\pm$0.30 & $ $\ 3.192634 & C [iv]{} & 39$\pm$6 & $\ge$15.40$\pm$0.50 & $ $\ Total(C [iv]{}) & & & $\ge$15.75 & 18.43\ 3.189333 & Si [iv]{} & 56$\pm$10 & $\ge$14.30$\pm$0.10 & $ $\ 3.190953 & Si [iv]{} & 25$\pm$14 & $\ge$14.60$\pm$0.60 & $ $\ 3.192634 & Si [iv]{} & 25$\pm$6 & $\ge$15.72$\pm$0.90 & $ $\ Total(Si [iv]{}) & & & $\ge$15.75 & 16.14\ 3.189821 & N [v]{} & 99$\pm$39 & $\ge$14.80$\pm$0.30 & $ $\ 3.191157 & N [v]{} & 34$\pm$29 & $\ge$14.89$\pm$0.20 & $ $\ 3.192807 & N [v]{} & 74$\pm$19 & $\ge$14.87$\pm$0.10 & $ $\ Total(N [v]{}) & & & $\ge$15.33 & 16.55\ 3.189788 & P [v]{} & 136 & $\ge$14.20$\pm$0.20 & $ $\ 3.191109 & P [v]{} & 97 & $\ge$13.95$\pm$0.30 & $ $\ 3.192757 & P [v]{} & 13 & $\ge$13.80$\pm$0.60 & $ $\ Total(P [v]{}) & & & $\ge$14.50 & 15.30\ \ \[hiion\] [c]{} -------- --------------- ----------- -------------- --------------- log U $n$$_{\rm H}$ $r$$_{0}$ $l$(H [i]{}) $l$(C [iv]{}) $ $ \[cm$^{-3}$\] \[pc\] \[pc\] \[pc\] $-$1.1 710 579 2.3 3.0 $-$0.7 500 435 3.2 5.8 $-$0.3 355 326 4.6 14.6 +0.0 250 275 6.5 33.7 +0.3 180 229 9.1 80.0 -------- --------------- ----------- -------------- --------------- : Hydrogen density ($n_{\rm H}$) in the cloud, its distance to the central AGN ($r$$_{0}$), size of the DLA ($l_{\rm HI}$) and transverse size of the C [iv]{} phase for different values of the ionization parameter (U). \[DLAsize\] The ionizing spectrum incident on the cloud is taken as the combination of the standard AGN spectrum of Mathews & Ferland (1987), Haardt-Madau extragalactic spectrum (Haardt & Madau 2005, HM05) and the CMB radiation both at $z$ = 3.1910. Fig. \[Model\] summarizes the results of these calculations. Bottom panel gives resulting ionic ratios and top panel shows column densities. In both panels measurements and upper limits are indicated by dashed lines. For Al [ii]{}, we scale the Si [ii]{} column density assuming solar metallicity ratios. The column density ratio log$N$(Al [ii]{})/$N$(Al [iii]{}) yields ionization parameter ranging from $-$1.1 to 0.3. Both log$N$(Si[ii]{})/$N$(Ar[i]{}) ratio and the limit on log$N$(Si[ii]{})/$N$(Si[iv]{}) are consistent with this range of ionization parameters. Our preferred value is log $U$ = $-0.3$ and we indicate the column densities for this model in Tables 1 and 2. Note that we detect a trough at the position of P[v]{}$\lambda$1117 with an absorption profile which is consistent with that of other high-ionization species (see Fig. 2). The second weaker line of the doublet, P[v]{}$\lambda$1128 is affected by noise. The fit to these two lines gives a column density which is consistent with the results of the preferred model. The same is true for N [v]{}. This strongly supports the fact that the cloud is highly ionized. To determine the hydrogen density of this cloud (i.e. $n_{\rm H}$), a series of [cloudy]{} models with fixed ionization parameter (varying from log U = $-$1.1 to 0.3) and varying $n_{\rm H}$ were constructed. Note that in this series of models, the metallicity and incident radiation are the same as those considered above. By increasing the hydrogen density we are indeed trying to collisionally populate the excited states of the Si [ii]{} ground state to explain the observed Si [ii]{}$^*$ column density. Knowing the ionization parameter and the density, we can derive the distance of the cloud to the quasar by estimating the number of ionizing photons emitted by the quasar. Since the quasar is reddened, we first estimate the flux at 20370Å  (corresponding to H$\beta$ at the redshift of the quasar) by extrapolating with a power-law the continuum observed at 6125 and 8165 Å. Following Srianand & Petitjean (2000), we then assume that the de-reddened quasar continuum is a power law ($f_{\lambda}$ $\sim$ $\lambda^{\alpha_{\lambda}}$; with $\alpha_{\lambda}$ = $-$1.5) and we consider that the flux at $\sim$ 20370 $\textup{\AA}$ is not affected by the reddening. We thus estimate the flux at the Lyman limit in the rest frame of the absorber to be $f_{912{\tiny\textup{\AA}}}$ = 3.40$\times$10$^{-17}$ erg s$^{-1}$cm$^{-2}$$\textup{\AA}^{-1}$. This flux corresponds to a luminosity of $L_{912{\tiny\textup{\AA}}}$ = 3.25$\times$10$^{42}$ erg s$^{-1}$$\textup{\AA}$$^{-1}$ at the Lyman limit. We can now estimate the rate at which hydrogen ionizing photons are impinging upon the face of the cloud by integrating $L_{\nu}$/$h\nu$ over the energy range 1 to 20 Ryd. If we assume a flat spectrum for the quasar over this energy range, we get $Q$ = 6.78$\times$10$^{55}$ photons per second. From the definition of the ionization parameter U, $$U = \frac{Q}{4 \pi r_{0}^{2} n_{\rm H} c}$$ one can see that for given values of $n_{\rm H}$, U, and $Q$, one can uniquely estimate the distance, $r_{0}$, from the quasar to the absorbing cloud ($c$ is the speed of light). We can then derive the size of the cloud along the line of sight. For the neutral part, the size along the line of sight is $l$ $\sim$ $N_{\rm HI}$/$n_{\rm H}$. Results are summarized in Table \[DLAsize\] which gives the hydrogen density, distance to the quasar and size of the neutral phase of the cloud for different values of log $U$. The longitudinal size range from 2.3 to 9.1 pc for a distance of, respectively, 579 to 229 pc from the quasar and a hydrogen density of 710 to 180 cm$^{-3}$. Using the structure given by the model and assuming that the cloud is spherical we can derive the transverse size of the C [iv]{} phase. It is given in the last column of Table \[DLAsize\] and range from 3 to 80 pc. These estimates for the transverse size of the high-ionization zone are only rough estimates because our model is very simple (only one density) and we assume spherical geometry. Residual flux in the bottom of the DLA trough --------------------------------------------- We observe residual flux in the bottom of the DLA trough extending in velocity well beyond the narrow emission line. This can be seen in Fig. \[Lya\_residual\_2PA\] where we overplot zooms of the Lyman-$\alpha$ regions observed along the two PAs. It is apparent that the flux is never at zero in the trough. We checked in the Lyman-$\alpha$ forest that the bottom of saturated lines have on an average zero flux. The residual flux is consistent in these two spectra as is demonstrated in the bottom panel of the figure which gives the difference between the two spectra. This means that the DLA cloud does not cover the background source completely. Since the source of the quasar continuum is much smaller than the broad line region (e.g. Hainline et al. 2013), this excess is due either to the cloud not covering the BLR completely or to a second continuum source. We investigate here if this excess can be due to an additional continuum source. If this excess is due to the continuum of the host galaxy then we would expect some residual at the bottom of metal lines. Indeed C [ii]{}$\lambda$1334 is saturated and seems to show some weak residual (see Fig. 2). We thus have scaled the LBG mean continuum as given by Kornei et al. (2010) to the bottom of C [ii]{}$\lambda$1334. We find that this could explain 20 to 50% of the DLA residual. The corresponding magnitude of the galaxy would be 23.22 which would be very bright. In any case this possibility cannot explain all the residual. [c]{} It is probable that part of the residual flux in the bottom of the DLA trough is due to the fact that part of the Lyman-$\alpha$ emission from the BLR is not covered by the DLA. The size of the broad line region (BLR) can be inferred from time delay measurements between variations in the continuum and in the broad lines. Recent investigations of low-redshift AGNs show a tight relation between this size and the luminosity of the AGN, $R=A\times (L/10^{43})^B$, where $R$ is the radius of the BLR, $A$ is a typical distance in light-days and $L$ is the luminosity either in an emission line (H$\beta$ or C [iv]{}) or in the continuum. The index is found to have a value close to B$\sim$0.6-0.7 when the typical distance A is in the range 20-80 light-days for local AGNs (Wu et al. 2004; Kaspi et al. 2005). More recently Bentz et al. (2009) find log $(R_{\rm BLR})=K+\alpha {\rm log}(\lambda L_{\lambda}$(5100Å)) with $\alpha$ = 0.519$^{+0.063}_{-0.066}$ and K = $-$21.3$^{+2.9}_{-2.8}$. The slope suggests that brighter AGNs have to a first approximation the same structure as fainter AGNs with only larger dimensions. Therefore, extending the Kaspi et al. (2005) relation to higher luminosities yields a typical radius of the order of 1 pc for the BLR of bright high-z quasars. In the present case, luminosity is $L_{\rm CIV}~\sim~9~\times~10^{44}$ erg s$^{-1}$ which gives a size of the BLR of $\le$ 1.1 pc. We have estimated in the previous Section that the longitudinal size of the DLA is of order 2-9 pc. This is larger than the size of the BLR. It therefore would mean either that the cloud is much smaller in the transverse direction, corresponding possibly to a filamentary structure or that some holes are present in the cloud or that the cloud is not centered on the quasar. QSO BLR Lyman-$\alpha$ emission and covering factor --------------------------------------------------- To determine the H [i]{} column density accurately, one needs to reconstruct the shape of the quasar spectrum at the position of the DLA profile. We can apply principal component (PCA) reconstruction of the quasar flux using the red part of the spectrum to estimate the shape of the Lyman-$\alpha$$-$N [v]{} emission (Pâris et al. 2011, 2013). To do so, we first subtract the residual flux seen at the bottom of the DLA (i.e. residuals from BEL and NEL regions) so that we get zero flux in the DLA trough. We then de-redden this spectrum and add again the residuals subtracted above. We now have the complete de-reddened observed spectrum. Applying the Principle Component Analysis (PCA) method on this spectrum, we derive the PCA spectrum shown as the dotted red curve in the upper panel of Fig. \[simulateHI\]. Then, we subtract the continuum from the PCA spectrum (green dashed curve) and scale it so that it is consistent with the residual flux seen at the bottom of the DLA trough (solid blue curve). This residual flux is only $\sim$7 % of that of the Lyman-$\alpha$ broad emission, indicating that $\sim$93 % of the Lyman-$\alpha$ BLR is covered by the cloud. To obtain the flux seen by the DLA, we subtract from the dashed green curve the residual flux seen at the bottom of the DLA (i.e. the solid blue curve) and re-add the continuum. We then fit the DLA to obtain the solid red curve in the lower panel. We also checked that we get the same result if, before fitting the DLA, we first re-redden the PCA continuum. As can be seen in Fig. \[simulateHI\], the fit is slightly high near the N [v]{} emission line although within errors. This is possibly because the N [v]{} emission line in this quasar may be weaker than what is predicted by the PCA method. It should be reminded that the PCA reconstruction is an estimate of the quasar spectrum which stays close to the median spectrum in the overall quasar population (see Pâris et al. 2011). The result (log $N$(H [i]{}) = 21.7$\pm$0.1) is consistent with what was derived previously. [c]{} [c]{} Figure \[simulateQSO\] shows the quasar spectrum as it would be observed if no DLA were present. In this figure, we have added the Lyman-$\alpha$ narrow emission component seen in the DLA trough to the PCA spectrum. The final spectrum is degraded to the SNR $\sim$ 20 by adding Gaussian noise to it. It is apparent from this spectrum that the emission from the narrow line region is quite strong in this quasar. The quasar narrow line region ============================= We call here the narrow line region (NLR), the region of the quasar host galaxy that is located within the PSF of the observations. This corresponds to about 1 arcsec or 7.1 kpc at the redshift of the quasar (or a distance of 3.55 kpc on both sides of the quasar). The emission seen beyond this will be called extended emission. We have shown that the DLA is of much smaller dimension so that most of the Lyman-$\alpha$ NLR is not covered and is detected as Lyman-$\alpha$ emission in the bottom of the DLA trough. We can extract this emission which is shown in the top panel of Fig. \[CIV\_NEL\_II\]. It can be seen that the emission is spread over more than 1200 km s$^{-1}$ with FWHM $\sim$ 900 km s$^{-1}$. CIV partial coverage -------------------- The large rest equivalent width of the C [iv]{} and Si [iv]{} absorption doublets and the flat-bottomed structure of their profiles suggest that these lines are saturated (see Fig. \[CIV\_NEL\_II\]). However, the flux at the bottom of the C [iv]{} doublet absorption lines apparently does not reach the zero flux level, indicating that the absorbing cloud is only partially covering the background emission-line region. When partial coverage occurs, the residual intensity seen at the bottom of absorption lines can be written as at each wavelength: $$I(\lambda)~=~I_{0}(\lambda)(1~-~C_{\rm f})~+~C_{\rm f}~I_{0}(\lambda)~exp~[-\tau(\lambda)]$$ where $I_{0}(\lambda)$ is the incident (unabsorbed) intensity, $\tau(\lambda)$ is the optical depth of the cloud at the considered wavelength, and $C_{\rm f}$ is the fraction of the background emitting region that is covered by the absorbing cloud (i.e. the covering factor). In the case of doublets, and assuming the covering factor is the same for each component of the doublet, we can write: $$C_{\rm f}~=~\frac{1~+~R_{2}^{2}~-~2R_{2}}{1~+~R_{1}~-~2R_{2}}$$ where $R_{1}$ and $R_{2}$ are the normalized residual intensities in the two absorption lines of the doublet (Petitjean & Srianand 1999; Srianand & Shankaranarayanan 1999). In Fig. \[CIV\_Cf\], we show the covering factor of the C [iv]{}, Si [iv]{}, and N [v]{} doublets. Here, the red and blue histograms are the profiles of the weaker and stronger transitions of each doublet, respectively. The green solid line is the covering factor at each point of the profile calculated from Eq.(3). The green vertical dotted lines mark the regions used to calculate the mean values of $C_{\rm f}$, avoiding the wings of the profiles. The green horizontal dotted lines along with the green filled squares show these mean values. Error bars are calculated as the standard deviation of $C_{\rm f}$ in different velocity bins. The measured values of $C_{\rm f}$ for C [iv]{} and Si [iv]{} are $\sim$ 0.85 and 0.90 respectively suggesting that the size of the corresponding BEL region may be similar for the two high-ionization species. Note also that N [v]{} on the contrary seems fully covered. The above numbers are however only indicative because the resolution of our spectrum is not high ($R\sim 4000$). In addition there is a possible FeII$\lambda$2586 line at $z_{\rm abs}~=~1.5140$ blended with the red wing of C [iv]{}$\lambda$1550 (cyan curve in the C [iv]{}$\lambda$1550 panel in Fig \[j08231D\]). In Fig. \[CIV\_NEL\_II\], we demonstrate the effect of partial coverage on the observed C [iv]{} doublet profiles taking into account the resolution of the spectrum and the presence of the FeII line. The contribution of the Fe [ii]{}$\lambda$2586 absorption to this fit is robustly determined by fitting together Fe [ii]{}$\lambda$$\lambda$2344,2600 in the same system. In panel (b) of Fig. \[CIV\_NEL\_II\] the red curve is the fit conducted with VPFIT[^4] on the original data. Here, we can see that there is no way to fit most of the profile without invoking partial coverage. Reconstructing C [iv]{} narrow emission line (NEL) profile ---------------------------------------------------------- As mentioned earlier, the flux at the bottom of the C [iv]{} doublet absorption lines does not reach the zero level. This residual flux could be due to the partial coverage of either the C [iv]{} BLR or NLR. However, the radius of the C [iv]{} phase associated with the DLA is larger and sometimes much larger than the size of the BLR (see Section 3.2) which strongly suggests that the residual flux is due to the C [iv]{} NLR. We thus tried to reproduce the residual by C [iv]{} emission, including [both]{} lines of the doublet. Doing this we would like to ask the question whether we can associate the emission with the Lyman-$\alpha$ emission. Indeed, this could bring important clues on the origin of the Lyman-$\alpha$ emission. In case the Lyman-$\alpha$ emission [*cannot*]{} be associated with C [iv]{} emission then this would strongly suggest that the Lyman-$\alpha$ emission corresponds to scattered light after radiative transfer. We thus first try to simply scale the Lyman-$\alpha$ profile. This is bound to fail as it is apparent that the residuals do not follow the shape of the Lyman-$\alpha$ emission. We therefore decomposed the Lyman-$\alpha$ profile in three Gaussian functions (see top panel of Fig. \[CIV\_NEL\_II\]) as suggested by the profile itself. This allows us to scale different parts of the Lyman-$\alpha$ profile differently. For each pixel we define the ratio $R$ = C [iv]{}/Lyman-$\alpha$ by combining the three Gaussian functions using the following equation: $$R~=~\frac{R_{1}\times G_{1}~+~R_{2}\times G_{2}~+~R_{3}\times G_{3}}{G_{1}~+~G_{2}~+~G_{3}} \label{eq_CIV_NEL}$$ where $R_{1}$, $R_{2}$, and $R_{3}$ are the weights for each component directly related to the C [iv]{}/Lyman-$\alpha$ ratio in this component. By assigning different values to $R_{1}$, $R_{2}$, and $R_{3}$, we can scale each Gaussian individually. For instance, if $R_{1}$ = $R_{2}$ = $R_{3}$ = 1.0, this results in the red profile shown in panel (a) of Fig. \[CIV\_NEL\_II\]. which is simply the combination of the three Gaussian functions. One can now use the factor $R$ = C [iv]{}/Lyman-$\alpha$ defined for each wavelength to properly scale the Lyman-$\alpha$ NEL profile. We thus have to adjust the $R_{1}$, $R_{2}$, and $R_{3}$ parameters until the C [iv]{} emission is consistent with the residual seen at the bottom of the C [iv]{} doublet absorption lines. We find that the residual can be reproduced (see panel (c) of Fig. \[CIV\_NEL\_II\]) with $R_{1}$ = 0.29, $R_{2}$ = 0.045, and $R_{3}$ = 0.18. The variation of $R$ through the profile is given as a dashed pink line in panels (c) and (e) of Fig. \[CIV\_NEL\_II\]. The corresponding fit of C [iv]{} after removing the residuals is given in panels (d) and (f). Since the weight of the second emission component is much smaller than the two other ones, we ask the question whether it would be possible that the second component has no C [iv]{} associated. To test this, we impose $R_{2}$ = 0.001. The result of the fit is given in panels (e) and (f). It is apparent that we can find a solution with no C [iv]{} associated with the second component. We thus conclude from all this that (i) the C [iv]{} emission is strongest around $v$ $\sim$ $-$200 km s$^{-1}$; this position could indicate the redshift of the quasar, implying that most of the Lyman-$\alpha$ emission and the DLA are redshifted; (ii) at least part of the NLR Lyman-$\alpha$ emission has no C [iv]{} emission associated (predominantly around $v=+100$ km s$^{-1}$) which means that the Lyman-$\alpha$ emission in this component is due to scattered light or that the emitted gas is located within a distance from the quasar smaller than the transverse size of the C [iv]{} phase associated with the cloud so that the corresponding C [iv]{} emission is absorbed by the high-ionization phase of the DLA cloud. [c]{} Extended Lyman-$\alpha$ emission ================================ In the middle and bottom panels of Fig. \[2D\_both\_PAs\] we show the 2D spectra of the Lyman-$\alpha$ emission detected in the DLA trough for the two PAs. It is apparent that the Lyman-$\alpha$ emission is extended and slightly displaced relative to the quasar trace. To quantify this we integrate the whole Lyman-$\alpha$ profile in the spectral direction and compare the result to the spatial PSF derived from the integration of the quasar spectrum over the rest of the order beyond 6520 Å. This is shown in the right panels in Fig. \[2D\_both\_PAs\]. It is apparent that the emission is extended well beyond the PSF for both PAs and mostly in one direction implying that the total emission is shifted towards this direction. We investigate whether the extension of the emission varies with the velocity position. For this, we split the velocity range over which the emission is seen in several regions, following the profile, and integrate the spatial emission profile over these regions. We then fit the profile by a Gaussian function. The results show that along PA = NS the spatial extent of the emission is larger than 5 arcsec over about 2000 km s$^{-1}$ and that the shift is about 0.2 arcsec towards the North direction in the same region. For PA = EW, extension is about the same but the shift is consistent with zero meaning that the extended emission is more symmetric around the trace. The Lyman-$\alpha$ emission is detected up to more than 25 kpc from the quasar and there is a strong excess emission along PA = NS to the North. To better visualize the extended emission, we will subtract the emission associated with the central PSF e.g. the emission located on the quasar trace. To do so, the 2D spectral order outside the Lyman-$\alpha$ region of each PA is split into several chunks, and in each chunk the counts are integrated along the spectral axis. A Gaussian is then fitted on the profiles to get the central pixel values of the spatial profiles. Finally, we fit a straight line on these values to determine the center of the trace for each spectral pixel. Once we know the position of the trace exactly, we extract the spatial profile in each wavelength pixel and we fit a Gaussian with width equal to the continuum PSF width. We then subtract this Gaussian from the profile. Results are given in Fig \[Lya\_vs\_trace\_extension\]. Top panels show the resulting 2D spectrum of the extension. Bottom panels show the Lyman-$\alpha$ profiles integrated along the spectral direction: in red the emission which was fitted on top of the trace and in black is the extension. [c]{} Along PA = NS, most of the extended emission is seen to the North up to projected distances of 3.5 arcsec or $\sim$25 kpc from the center. The striking observation here is that the emission is extended over more than 1000 km s$^{-1}$ at all distances from the quasar. Along PA = EW, extension is more apparent in one direction as well, to the West. However the velocity extension gets smaller when the distance to the quasar increases. Fig \[Distance\] shows the variation of the integrated flux with distance. In this Figure, the red and blue curves indicate variations of flux with distance following the relation $F$ = $F_{0}$($r$/$r_{0}$)$^{-2}$ relation. The data roughly follow such a law. In case this emission is due to recombination and since the ionization flux is decreasing with distance as $r^{-2}$ this would mean that the number of ionized hydrogen atoms intersected by the slit is the same at all distances. This could be the case if the gas is clumpy so that changes in density has little effect. The velocity profiles of the extended Lyman-$\alpha$ emission at different distances from the AGN are shown in Fig \[4\_Extession\_profiles\] and Table \[flux\_luminosity\] summarizes the integrated flux and luminosity of the trace and the extension. Discussion and conclusions ========================== We have performed slit spectroscopic observations of QSO J0823+0529 with the MagE spectrograph mounted on the Magellan telescope along two PAs, in the North-South and East-West directions. The quasar is unique because a DLA is located at a redshift very similar to that of the quasar ($z_{\rm DLA}$ = 3.1910 and $z_{\rm CIV}$ = 3.1875, i.e. $\Delta v$ $\sim$330 km s$^{-1}$) and acts as a coronagraph blocking most of the flux from the central regions of the AGN. In the present case, the DLA cloud is small enough so that it covers only approximately 90% of the Lyman-$\alpha$ BLR. This puts us in a unique position to be able to directly observe the quasar narrow line region and the extended emission line region. Indeed, Lyman-$\alpha$ emission is detected up to more than 25 kpc from the quasar along both PAs. The quasar NLR -------------- Finley et al. (2013) have gathered a sample of 57 DLAs acting as a coronagraph in front of quasars; the DLAs have redshifts within 1500 km s$^{-1}$ from the quasar redshift. Their statistical sample of 31 quasars shows an excess of such DLAs compared to what is expected from the distribution of intervening DLAs. This can be explained if most of these DLAs are part of galaxies clustering around the quasar. However 25% of such DLAs reveal Lyman-$\alpha$ emission probably from the quasar host galaxy implying that these DLAs have sizes smaller than the quasar emission region. The Lyman-$\alpha$ luminosities are consistent with those of Lyman-$\alpha$ emitters in 75% of the cases and 25% have much higher luminosities. The later systems probably reveal the central NLR of the quasar. QSO J0823+0529 is part of this later class of coronagraph DLAs. We have shown that the size of the neutral phase is of order 2-9 pc which means that the gas does not cover the Lyman-$\alpha$ NLR. Three main components of Lyman-$\alpha$ emission from the NLR are clearly detected at $\Delta V$ = $-$300, +100 and +400 km s$^{-1}$ relative to the DLA redshift (see Fig. 9). The corresponding C [iv]{} emission seems to be absent in the second strongest component. Either the Lyman-$\alpha$ emission is scattered or the associated C [iv]{} emission is absorbed by the high-ionization phase of the DLA. To be covered, the corresponding gas must be located within $\sim$3-80 pc from the quasar which, we have seen, corresponds to the extension of the high-ionization phase of the cloud assuming spherical geometry. At least part of the two other components could therefore be located beyond this distance from the quasar, although part of the third component could also be hidden, as the C [iv]{}/H [i]{} ratio is larger by a factor of about two in the first component compared to the third component (see Fig. 9). Note that the relative velocities of components 1 and 3 can be interpreted as the result of a conical outflow with mean projected velocities of about $\pm$300 km s$^{-1}$. [c c]{} Lyman-$\alpha$ emission from the center to the outskirts of the host galaxy --------------------------------------------------------------------------- Extended Lyman-$\alpha$ emission has been observed around high-redshift radio galaxies (Heckman et al. 1991; van Ojik et al. 1996), as well as around radio-quiet quasars (Bunker et al. 2003; Christensen et al. 2006). For radio-loud quasars or radio-galaxies, the Lyman-$\alpha$ flux of the nebula is an order of magnitude higher (Christensen et al. 2006), presumably because the emission of radio-loud quasar gaseous envelopes is enhanced by interactions with the radio jets. More recently, North et al. (2012) have used a careful treatment of the spectral PSF to extract quasar traces. This revealed four detections of extended emission out of six radio-quiet quasars at $z\sim 4.5$ with extensions of diameter $16<d<64$ kpc down to a luminosity of 2$\times$10$^{-17}$ erg s$^{-1}$ cm$^{-2}$ arcsec$^{-2}$. The emission has 900 $<$ FWHM $<$ 2200 km s$^{-1}$. Our observations are in line with these numbers. The extended emission we detect in QSO J0823+0529 has a diameter of $\sim$50 kpc and FWHM $\sim$ 900 km s$^{-1}$. However, QSO J0823+0529 does not seem to follow the $L$(Lyman-$\alpha$) vs $L$(BLR) relation indicated by these authors. Indeed the Lyman-$\alpha$ luminosity in QSO J0823+0529 is more than an order of magnitude larger than what would be expected from this relation even after correcting for dust attenuation. It is still possible that dust is present closer to the quasar and further attenuates the BLR Lyman-$\alpha$ emission. It is also possible that the decomposition between the broad and narrow line region emissions was ambiguous in previous studies so that the NLR emission could have been underestimated. ---------------------- ---------------------------- ------------------------- -- $ $ Integrated flux Luminosity $ $ \[erg s$^{-1}$ cm$^{-2}$\] \[erg s$^{-1}$\] Trace(PA=NS) 3.59 $\times$ 10$^{-16}$ 3.43 $\times$ 10$^{43}$ Extension(PA=NS) 3.97 $\times$ 10$^{-16}$ 3.79 $\times$ 10$^{43}$ Trace(PA=EW) 3.36 $\times$ 10$^{-16}$ 3.21 $\times$ 10$^{43}$ Extension(PA=EW) 2.68 $\times$ 10$^{-16}$ 2.56 $\times$ 10$^{43}$ Lyman-$\alpha$(BELR) 7.10 $\times$ 10$^{-15}$ 6.78 $\times$ 10$^{44}$ ---------------------- ---------------------------- ------------------------- -- : Lyman-$\alpha$ integrated fluxes and luminosities in the PSF centered on the quasar trace and in the extension. The last row gives the integrated flux and luminosity of the broad Lyman-$\alpha$ emission calculated using the simulated spectrum shown in Fig \[simulateQSO\]. \[flux\_luminosity\] The emission is more extended to the North-West of the object (see Figs. 11, 13 and 15). There are two notable features in the spatial and velocity structure of the nebula. First, in the North-West direction, the kinematics are strikingly similar along the trace and 10 kpc away from the center (see Fig. 13) with a velocity spread of more than 1000 km s$^{-1}$. The emission is quite strong in this region. Secondly, there is a clear gradient of about 1000 km s$^{-1}$ between 15 kpc to the East and 20 kpc to the West. It is tempting to interpret these features as the superposition of the emission of gas in the disk of the galaxy, where the density is higher and turbulent kinematics prevent gas clouds to be well organized with emission from gas flowing out of the disk with velocities of the order of 500 km s$^{-1}$. This gas is best seen up to 20 kpc to the West and 10 kpc to the East. Such winds can be reproduced by recent models taking into account the effects of radiation trapping (Ishibashi & Fabian 2015). Nature of the DLA ----------------- It is well demonstrated that bright quasars are surrounded by large amounts of gas both in extended ionized halos up to 10 Mpc from the quasar (e.g. Rollinde et al. 2005) and extended (300 kpc) reservoirs of neutral gas in the halo of the host galaxy (Prochaska et al. 2013, 2014; Johnson et al. 2015). However, along the line of sight to quasars, the incidence of neutral gas is less (e.g. Shen & Ménard 2012) indicating that the ionizing emission from quasars is highly anisotropic. The DLA we discuss here is part of the so-called proximate DLAs in the sense that the absorption redshift is similar to that of the quasar ($z_{\rm abs}$ $\sim$ z$_{\rm em}$, see e.g. Ellison et al. 2010). However, this is the first time it can be demonstrated that the gas is located very close to the quasar ($<$ 400 pc) and probably associated with the central part of the host galaxy. We have modelled the physical state of the gas in the DLA using photo-ionization models. The ionization parameter is found to be in the range $-1.1$ $<$ log$U$ $<$ $+0.3$ which means that the cloud is mostly highly ionized. A density of $n_{\rm H}$ $\sim$ 710-180 cm$^{-3}$ is needed to explain the absorption lines from O [i]{} and Si [ii]{} ground state excited levels implying that the neutral phase should have dimensions of the order of 2 to 9 pc and be embedded in an ionized cloud of size $\sim$3-80 pc. From this, we could derive that the cloud is located between 230 and 580 pc from the quasar. The metallicity of the cloud, $Z$ = 0.16 $Z_{\odot}$, is typical of the metallicity of standard intervening DLAs (Rafelski et al. 2012). [c]{} It is intriguing to note that the high-ionization phase of the cloud we see has characteristics similar to those of some of the warm absorbers seen in many AGNs. Tombesi et al. (2013) argued that warm absorbers (WA) and ultra-fast outflows (UFO) could represent parts of a single large-scale stratified outflow observed at different locations from the black hole. The UFOs are likely launched from the inner accretion disc and the WAs at larger distances, such as the outer disc and/or torus. There are still significant uncertainties on the exact location of this material, which ranges from a few pc up to kpc scales, (e.g. Krolik & Kriss 2001; Blustin et al. 2005). The absorption lines are systematically blue-shifted, indicating outflow velocities of the WAs in the range 100-1000 km s$^{-1}$. King & Pounds (2013) have argued that the dense gas which surrounds the AGN when it starts shining is swept out by the fast winds powered by the accretion luminosity. The wind is halted by collisions near the radius where radiation pressure drops. The shocked gas must rapidly cool and mix with the swept-up ISM. Distance from the AGN and properties of the gas are similar to what is observed for warm-absorbers, as in our case. Therefore, in QSO J0843+0529, the DLA could be located in the galactic disk at the terminal position of the wind, at the limit of the interstellar medium of the host galaxy. The presence of dust in the gas can be considered as supporting this view as it is expected in the dense environment of AGNs (see e.g. Leighly et al. 2015). [c c]{} The only caveat with this idea is that the metallicity of the DLA is typical of intervening DLAs, when we would expect the gas in the ISM of the quasar host galaxy to have larger metallicity. This, together with the low outflow velocity (the DLA is centered on the Lyman-$\alpha$ emission), argues for another explanation for the origin of this gas. Given the distance to the central AGN (230 $<$ $r_0$ $<$ 580 pc), one is tempted to conjecture that it could be the final fate of infalling gas. Note that in that case, the presence of dust is not a problem as the observed amount is not unusual compared to what is seen in typical high column density DLAs. Indeed, accretion is believed to happen through cold flows which are expected to be of lower metallicity compared to the environment of the quasar (see Bouché et al. 2013). Little is known about how quasars at high redshift interact with cold infalling streams of gas, and in particular whether these collimated structures can survive the energy released by the AGN. Therefore the DLA cloud could be in a transitory phase before it is completely destroyed by the AGN ionizing radiation field (Namekata et al. 2014). The two alternatives are actually not incompatible, since some of the warm absorbers could be part of the galactic ISM swept away from the center by the AGN and placed at large distances from the AGN (e.g. King & Pounds 2013). These outflowing warm absorbers may then intersect with accreted cold streams. Concluding remark ----------------- There is little doubt that DLAs acting as coronagraphs are important targets to be observed and analyzed. Besides revealing interesting characteristics of the quasar host galaxy they are, like in QSO J0823+0529, potentially part of the machinery that power the AGN. Finley et al. (2013) found 57 such systems out of about a third of the BOSS targets. With the on-going eBOSS-SDSSIV survey and the forseen DESI survey (Schlegel et al. 2011), the number of such systems will increase by a large factor in the future. It will thus be important to gather a large sample of these objects to be able to study their characteristics statistically. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank the anonymous referee for their constructive comments, which helped us to improve the paper. We also thank George Becker for advices on MagE data reduction and Hadi Rahmani for useful discussion. H.F. was supported by the Agence Nationale pour la Recherche under program ANR-10-BLAN-0510-01-02. SL has been supported by FONDECYT grant number 1140838 and partially by PFB-06 CATA. Asplund, M., Grevesse, N., Sauval, A. J., Scott, P., 2009, ARA&A, 47, 481 Bentz M. C., Peterson B. M., Netzer H., Pogge R. W., Vestergaard M., 2009, ApJ, 697, 160 Bentz, M. C., Walsh, J. L., Barth, A. J., Baliber, N., Bennert, V. N., et al., 2009, ApJ, 705, 199 Blustin, A. J., Page, M. J., Fuerst, S. V., Branduardi-Raymont, G., Ashton, C. E. 2005, A&A, 431, 111 Bouché, N., Murphy, M. T., Kacprzak, G. G., Péroux, C., Contini, T., Martin, C. L., Dessauges-Zavadsky, M. 2013, Science, 341, 50 Bunker, A., Smith, J., Spinrad, H., Stern, D., Warren, S., 2003, Ap&SS, 284, 357 Chartas, G., Saez, C., Brandt, W. N., Giustini, M., Garmire, G. P., 2009, ApJ, 706, 644 Chen, H.-W., Prochaska, J. X., Bloom, J. S., Thompson, I. B., 2005, ApJ, 634, L25 Christensen, L., Jahnke, K., Wisotzki, L., Sánchez, S. F., 2006, A&A, 459, 717 Crenshaw D. M., Kraemer S. B., George I. M., 2003, ARA&A, 41, 117 Dai X., Kochanek C. S., Chartas G., Kozłowski S., Morgan C. W., Garmire G., Agol E., 2010, ApJ, 709, 278 De Cia, A., Ledoux, C., Fox, A. J., et al., 2012, A&A, 545, 64 Dubois, Y., Pichon, C., Devriendt, J., Silk, J., Haehnelt, M., Kimm, Taysun, Slyz, A., 2013, MNRAS, 428, 2885 Ellison, Sara L., Prochaska, J. X., Hennawi, J., Lopez, S., Usher, C., Wolfe, A. M., Russell, D. M., Benn, C. R., 2010, MNRAS, 406,1435 Fathivavsari, H., Petitjean, P., Ledoux, C., Noterdaeme, P., Srianand, R., Rahmani, H., Ajabshirizadeh, A., 2013, MNRAS, 435, 1727 Faucher-Giguère, C.-A., Quataert, E., 2012, MNRAS, 425, 605 Finley, H., Petitjean, P., Pâris, I., Noterdaeme, P., Brinkmann, J. et al., 2013, A&A, 558, 111 Fynbo, J. P. U., Starling, R. L. C., Ledoux, C., et al., 2006, A&A, 451, L47 Gofford, J., Reeves, J. N., Tombesi, F., et al., 2013, MNRAS, 430, 60 Hainline, L. J., Morgan, C. W., MacLeod, C. L., Landaal, Z. D., Kochanek, C. S., et al., 2013, ApJ, 774, 69 Harrison, C. M., Alexander, D. M., Mullaney, J. R., Swinbank, A. M., 2014, MNRAS, 441, 3306 Heckman, T. M., Lehnert, M. D., Miley, G. K., van Breugel, W., 1991, ApJ, 381, 373 Hennawi, J. F., Prochaska, J. X., Kollmeier, J., Zheng, Zheng 2009, ApJ, 693, L49 Hewett, P. C., Wild, V., 2010, MNRAS, 405, 2302 Hu, E. M., Cowie, L. L. 1987, ApJ, 317, L7 Hu, E. M., McMahon, R. G., Egami, C. 1996, ApJ, 459, L53 Ishibashi, W., Fabian, A. C., 2015, arXiv:1504.07393 Johnson, S. D., Hsiao-Wen C., Mulchaey, J. S., 2015, arXiv:1505.07838 Kaspi S., Maoz D., Netzer H., Peterson B. M., Vestergaard M., Jannuzi B. T., 2005, ApJ, 629, 61 King, A., Pounds, K., 2014, MNRAS, 437, 81 King, A., Pounds, K. 2015, arXiv:1503.05206 Komatsu, E., Smith, K. M., Dunkley, J., et al., 2011, ApJS, 192, 18 Krolik, J. H., Kriss, G. A., 2001, ApJ, 561, 684 Leighly, K. M., Cooper, E., Grupe, D., Terndrup, D. M., Komossa, S. 2015, arXiv:1508.01257 Liu G., Zakamska N. L., Greene J. E., Nesvadba N. P. H., Liu X., 2013, MNRAS, 436, 2576 Mullaney J. R., Alexander D. M., Fine S., Goulding A. D., Harrison C. M., Hickox R. C., 2013, MNRAS, 433, 622 Namekata, D., Umemura, M., Hasegawa, K., 2014, MNRAS, 443, 2018 Neeleman, M., Prochaska, J. X., Wolfe, A. M., 2015, ApJ, 800, 7 Nesvadba N. P. H., Polletta M., Lehnert M. D., Bergeron J., De Breuck C., Lagache G., Omont A., 2011, MNRAS, 415, 2359 North, P. L., Courbin, F., Eigenbrod, A., Chelouche, D., 2012, A&A, 542, 91 Noterdaeme, P., Srianand, R., Rahmani, H., Petitjean, P., Pâris, I. et al., 2015, A&A, 577, 24 Osterbrock D. E., Ferland G. J., 2006, Astrophysics of Gaseous Nebulae and Active Galactic Nuclei. Univ. Science Books, Mill Valley, CA Pâris, I., Petitjean, P., Aubourg, E., et al., 2011, A&A, 530, 50 Pâris, I., Petitjean, P., Aubourg, E., et al., 2012, A&A, 548, 66 Pâris, I., Petitjean, P., Aubourg, E., et al., 2014, A&A, 563, 54 Petitjean, P., Pécontal, E., Valls-Gabaud, D., Charlot, S. 1996, Nature, 380, 411 Prochaska J. X., Hennawi J. F., Simcoe R. A., 2013, ApJL, 762, L19 Prochaska J. X., Lau M. W., Hennawi J. F., 2014, Ap, 796, 140 Rauch, M., Becker, G. D., Haehnelt, M. G., Carswell, R. F., Gauthier, J.-R. 2013, MNRAS, 431, 68 Rafelski, M., Wolfe, A. M., Prochaska, J. X., Neeleman, M., Mendez, A. J. 2012, ApJ, 755, 89 Rollinde, E., Srianand, R., Theuns, T., Petitjean, P., Chand, Hum 2005, MNRAS, 361, 1015 Schlegel, D., Abdalla, F., Abraham, T., 2011, arXiv1106.1706 Shen Y., Ménard B., 2012, ApJ, 748, 131 Silva, A. I., Viegas, S. M., 2002, MNRAS, 329, 135 Srianand, R., Petitjean, P., 2000, A&A, 357, 414 van Ojik, R., Roettgering, H. J. A., Carilli, C. L., et al., 1996, A&A, 313, 25 Vreeswijk, P. M., Ellison, S. L., Ledoux, C., et al., 2004, A&A, 419, 927 Wang J.-G., et al., 2009, ApJ, 707, 1334 Weidinger, M., M oller, P., Fynbo, J. P. U., Thomsen, B. 2005, A&A, 436, 825 Wu X.-B., Wang R., Kong M. Z., Liu F. K., Han, J. L., 2004, A&A, 424, 793 [^1]: Based on data obtained with MagE at the Clay telescope of the Las Campanas Observatory (CNTAC Prgm. ID CN2012B-51 and CN2013A-121) [^2]: <ftp://ftp.ast.cam.ac.uk/pub/gdb/mage_reduce/> [^3]: <https://www.eso.org/sci/observing/tools/standards.html> [^4]: <http://www.ast.cam.ac.uk/~rfc/vpfit.html>

--- abstract: 'We review different computation methods for the renormalised energy momentum tensor of a quantised scalar field in an Einstein Static Universe. For the extensively studied conformally coupled case we check their equivalence; for different couplings we discuss violation of different energy conditions. In particular, there is a family of masses and couplings which violate the weak and strong energy conditions but do not lead to spacelike propagation. Amongst these cases is that of a minimally coupled massless scalar field with no potential. We also point out a particular coupling for which a massless scalar field has vanishing renormalised energy momentum tensor. We discuss the backreaction problem and in particular the possibility that this Casimir energy could both source a short inflationary epoch and avoid the big bang singularity through a bounce.' author: - | [Carlos A. R. Herdeiro and Marco Sampaio]{}\ \ [*Departamento de Física e Centro de Física do Porto*]{}\ [*Faculdade de Ciências da Universidade do Porto*]{}\ [*Rua do Campo Alegre, 687, 4169-007 Porto, Portugal*]{} date: October 2005 title: | [**[Casimir energy and a\ cosmological bounce]{}**]{} --- Introduction ============ The Casimir effect [@Casimir] is a manifestation of the vacuum fluctuations of a quantum field. It was first considered in systems with boundaries, and it is known that the Casimir force is highly sensitive to the size, geometry and topology of such boundaries. In particular it may change from attractive to repulsive depending on such shape [@review]. But the Casimir force is also present in systems with no boundaries and a compact topology, since the latter imposes periodicity conditions which resemble boundary conditions. If our universe is either open or flat, with non-trivial topology, or closed, every quantum field living on it should generate a Casimir type force, which led many authors to study the Casimir effect in FRW models (see [@review] and references therein for a review). In the case of a spherical universe, most computations of the Casimir energy, or more generically, of the renormalised stress energy tensor, have focused on conformally coupled scalar fields. For instance, a massless conformally coupled scalar field, the electromagnetic field and the massless Dirac field on an Einstein static universe, have been considered in [@Ford1; @Ford2]; their Casimir energies have been shown to be of the form $\alpha/R^4$, with $\alpha$, respectively, $1/(480\pi^2)$, $11/(240\pi^2)$ and $17/(1920\pi^2)$. Note that all of these are positive. Since these are all conformally coupled fields, they have equation of state $p=\rho/3$, and obey the strong energy condition. This means the Casimir force is attractive. But this is not always so in the cosmological context. Zel’dovich and Starobinskii [@zeldovich] have indeed verified long ago that the Casimir energy of a scalar field could drive inflation in a flat universe with toroidal topology. The purpose of this letter is to exhibit a family of quantum scalar fields which originate a repulsive Casimir force in a closed universe, since they violate the strong energy condition. Interestingly, this family includes the simplest case one could consider: a minimally coupled massless scalar field with no potential. Our computation will be performed in the Einstein Static Universe (ESU), which can be faced as an approximation to a dynamical FRW model in sufficiently small time intervals, and avoids having to deal with the complexities of quantum field theory in a time dependent spacetime, like particle creation.[^1] Quantum scalar field with arbitrary coupling in the ESU ======================================================= We consider the ESU, which is a well known solution of the Einstein equations sourced by a perfect fluid with positive energy density ($\rho>0$) and zero pressure ($p=0$) together with a positive cosmological constant ($\Lambda>0$): $G_{\mu \nu}+\Lambda g_{\mu \nu} =T_{\mu \nu}$, with $T_{\mu \nu}=\rho u_{\mu}u_{\nu}$. The metric is $$ds^2_{ESU}=-dt^2+R^2d\Omega_{S^3}=-dt^2+\frac{R^2}{4}\left((\sigma_R^1)^2+(\sigma_R^2)^2+(\sigma_R^3)^2\right) \ .$$ We have written the metric on the unit 3-sphere, $d\Omega_{S^3}$, in terms of right forms on $SU(2)$. In order to achieve a static solution, the cosmological constant and the energy density are related by $\Lambda=\rho/2=1/R^2$. Another viewpoint is that the ESU is supported by a perfect fluid with $p=-\rho/3=-1/R^2$. It is well known that this universe is unstable against small radial perturbations as was first argued by de Sitter. The reason is that the energy density of the perfect fluid increases/decreases with decreasing/increasing radius, whereas the one of the cosmological constant is kept constant. Since the former gives an attractive contribution and the latter a repulsive one, any displacement from the original equilibrium position will grow, rendering such original position as unstable. But even without such classical perturbations this universe is unstable due to quantum mechanics. These are the instabilities we will focus on, in the spirit discussed at the end of the introduction. Let us consider a free (i.e with no potential) scalar field $\Phi$, with mass $\mu$ and with a coupling (not necessarily conformal) to the Ricci scalar of the background $\mathcal{R}=6/R^2$, governed by $$\left(\Box -\xi \mathcal{R}\right)\Phi=\mu^2\Phi \ .$$ Conformal coupling is obtained in four spacetime dimensions by taking the coefficient $\xi=1/6$ (and the theory is then conformal if $\mu=0$), whereas minimal coupling corresponds to $\xi=0$. The compactness of the spatial sections of the background guarantees a discrete mode spectrum which can be easily obtained using elementary group theory. We take the D’Alembertian in the form $$\Box=-\frac{\partial^2}{\partial t^2}+\frac{4}{R^2}\left(({\bf k}^R_1)^2+({\bf k}^R_2)^2+({\bf k}^R_3)^2\right)=-\frac{\partial^2}{\partial t^2}+\frac{4}{R^2}{\bf k}^2 \ ,$$ where ${\bf k}^R_i$ are the right vector fields dual to $\sigma^i_R$ and ${\bf k}^2$ is one of the two Casimirs in $SO(4)$. Notice that the eigenfunctions of the Klein-Gordon operator $(\Box -\xi \mathcal{R}-\mu^2)$ may be taken in the form \_n= e\^[-it]{}\_j\^[m\_L,m\_R]{}  , \[eigenfunctions\] where the index $n$ represents all quantum numbers $j,m_L,m_R$ and $\mathcal{D}_j^{m_L,m_R}$ represents a Wigner D-function [@Edmonds]. Such function may be thought of as a spherical harmonic on the 3-sphere or as a matrix element of the rotation operator $\langle j, m_L|\hat{R}(\alpha,\beta,\gamma)| j, m_R\rangle$, where $|j,m\rangle$ is the basis of a representation of $SU(2)$ and $(\alpha,\beta,\gamma)$ are Euler angles. It follows straightforwardly that the dispersion relation becomes \_j\^2=+\^2  ,      j=0,,1,,…\[scalaresu\] with the degeneracy of each frequency being $d_j=(2j+1)^2$, in agreement with the spectrum found in [@Ford1]. Note that there are no unstable modes for $\xi\in \mathbb{R}_0^+$, which includes minimal and conformal coupling. This is the range of couplings we will analyse in the following. Canonical quantisation of the scalar field can be performed unambiguously. One finds the mode expansion $$\hat{\Phi}=\sum_n \hat{a}_n^{\dagger}\Psi_n+\hat{a}_n\Psi^*_n \ , \ \ \ \ \ \Psi_n=\sqrt{\frac{2j+1}{2\omega_j V}}\Phi_n \ ,$$ with $V=2\pi^2R^3$ being the volume of the constant $t$ hypersurfaces and with the operators $\hat{a}_n^{\dagger},\hat{a}_n$ obeying the usual commutation relation $[\hat{a}_n,\hat{a}_{n'}^{\dagger}]=\delta_{nn'}$. The classical energy momentum tensor of the scalar field is more conveniently written in the natural tetrad basis ${\bf e}^a=\{dt,R\sigma^1_R/2,R\sigma^2_R/2,R\sigma^3_R/2\}$, T\_[ab]{}=[**k**]{}\_a\_b-\_c\^c-g\_[ab]{}\^2+(G\_[ab]{}-\_a[**k**]{}\_b+g\_[ab]{})\^2  , \[emtensor\] where we have denoted ${\bf k}_a=\{\partial/\partial t, {\bf k}_i^R\}$. The conformal case ($\xi=1/6,\mu=0$) has zero trace; quantum mechanically, however, the renormalised energy momentum tensor for a conformally coupled, free massless scalar field generically develops a trace anomaly, which can be written solely in terms of geometric quantities of the background [@Brown; @Wald]. In four spacetime dimensions such anomaly takes the form T\^a\_[  a]{}\_[ren]{} ={C\_[abcd]{}C\^[abcd]{}-(R\_[abcd]{}R\^[abcd]{}-4R\_[ab]{}R\^[cd]{}+4\^2)} + \^2  . \[anomaly\] The coefficient $\chi$ is renormalisation scheme dependent. This trace is zero for the ESU: the Ricci scalar is constant, the Weyl tensor $C_{abcd}$ vanishes since the geometry is conformally flat and the second Euler density also vanishes, since the Euler characteristic of any odd dimensional sphere is zero. Renormalisation =============== Denoting the non-vanishing components of $\langle T^a_{\ \ b}\rangle$ as $\langle T^0_{\ \ 0}\rangle\equiv-\rho$ and $\langle T^1_{\ \ 1}\rangle=\langle T^2_{\ \ 2}\rangle=\langle T^3_{\ \ 3}\rangle\equiv p$, we find the unrenormalised quantities \_0=\_[n=1]{}\^[+]{}n\^2  ,     p\_0=\_[n=1]{}\^[+]{}n\^2  , \[massive1\] with \_n=  ,     n  . \[wmassive\] In [@Ford2] the particular case with $\xi=1/6$ was studied. These infinite quantities were renormalised by introducing a damping factor $e^{-\beta n}$ in the sums, subtracting the flat space contribution, and taking the parameter $\beta$ to zero. The result is\  , \[rhorenm1\]$$\barr{l} \displaystyle{p_{ren}=\frac{1}{12\pi^2R^4}\left\{\sum_{n=1}^{N-1}\left[\frac{n^3}{\sqrt{1+\left(\frac{\mu R}{n}\right)^2}}-n^3+\frac{(\mu R)^2}{2}n-\frac{3(\mu R)^4}{8n}\right] +\sum_{n=N}^{+\infty}\sum_{k=3}^{+\infty}c_k\frac{(\mu R)^{2k}}{n^{2k-3}}\right.} \spa{0.6cm}\\ \displaystyle{~~~~~~~~~~~~~~~~~~~~~ \left.+\frac{1}{120}+\frac{(\mu R)^2}{24}+\frac{(\mu R)^4}{32}\left(12\ln{\frac{\mu R}{2}}+7+12\gamma\right)\right\}} \earr \ ,$$ where $\gamma$ is Euler’s constant, $N$ is an integer such that $N>\mu R$ and the explicit formulae for the coefficients $b_k$, $c_k$ are $$b_k=\frac{(-1)^{k-1}(2k-3)!!}{2^kk!} \ , \ \ \ \ \ \ c_k=\frac{(-1)^{k}(2k-1)!!}{2^kk!} \ .$$ These expressions are not very enlightening; therefore we plot the renormalised energy density and pressure, for fixed radius, in terms of the mass in figure \[rhomassivefig\]. The point we would like to emphasise is that they are always positive. Therefore we conclude that a conformally coupled scalar field with arbitrary mass does not violate the strong energy condition and hence produces solely an attractive effect in the universe. (0,0)(0,0) At this point let us make a comment which also works as a consistency check. The first law of thermodynamics yields $p=-\partial E/\partial V$, where the total energy $E=\rho V=\rho 2\pi^2R^3$. We can easily check that this holds both for the infinite unrenormalised quantities (\[massive1\]) and the finite renormalised ones (\[rhorenm1\]) $$p_0=-\frac{1}{3R^2}\frac{\partial (R^3\rho_0)}{\partial R} \ , \ \ \ p_{ren}=-\frac{1}{3R^2}\frac{\partial (R^3\rho_{ren})}{\partial R} \ .$$ Thus, we could have only computed the energy density and have deduced the pressure via the first law of thermodynamics, which obviously only states the conservation of energy $T^{\mu \nu}_{\ \ \ ;\mu}=0$. This is exactly what shall be done in the cases to follow. As a check on the result (\[rhorenm1\]) let us consider another renormalisation method, namely zeta function. We can write the regularised expression as |(s)=\_[n=1]{}\^[+]{}  , \[masszeta\] where $\mu_0$ has dimension of mass and $$a^2\equiv \mu^2R^2+6\xi-1 \ ,$$ which, for $a^2\ge 0$, has the appropriate form to be written in terms of Epstein-Hurwitz zeta functions |(s)=(\_[EH]{}(s/2-1,a\^2)-a\^2\_[EH]{}(s/2,a\^2))  .\[massreg\] In order to obtain the renormalised energy density we should now consider the limit of this expression when $s\rightarrow -1$. The Epstein-Hurwitz zeta function is only defined by the infinite sum $\sum_{n=1}^{+\infty}(n^2+a^2)^{-s}$ for $Re(s)>1/2$. Its analytic continuation to other values of $s$, which converges for all $s$ with the exception of an infinite number of poles is given by [@nesterenko] \_[EH]{}(s,a\^2)=-+(a\^2)\^[1/2-s]{}+\_[n=1]{}\^[+]{}n\^[s-1/2]{}K\_[s-1/2]{}(2n )  , \[EHzeta\] where $K_{\nu}$ are modified Bessel functions, and $a^2$ is required to be positive. It is simple to see that the infinite series converges for all $s$, since the modified Bessel functions have asymptotic behaviour $$K_{\nu}(z)\sim \sqrt{\frac{\pi}{2z}}e^{-z} \ , \ \ \ \ \ |z|\rightarrow \infty \ .$$ The poles of $\zeta_{EH}(s,a^2)$ arise at the poles of $\Gamma(s-1/2)$, that is at $s=1/2-n$, $n\in \mathbb{N}_0$. Since $a^2$ must be positive this technique does not apply in the case of a massless minimally coupled scalar field. Let us focus on the case with $a^2>0$. Applying (\[EHzeta\]) to (\[massreg\]) we find\   . \[zeta\] The point $s=-1$ is exactly at one of the poles of the analytic continuation of the Epstein-Hurwitz zeta function. That is, the right hand side of the last formula is an infinite quantity, and it might seem that the zeta function method fails. With the correct interpretation this is not so. Let us consider separately two cases: $\bullet$ : Conformal coupling ($\xi=1/6$) and arbitrary mass: This case is obtained by substituting $a^2=\mu^2R^2$ in (\[zeta\]). The infinite contribution, i.e the term with $\Gamma(-2)$ in the $s=-1$ limit, is exactly the $R$ independent term. Thus, despite its (infinite) contribution to the Casimir energy it will not contribute to the Casimir *force*. This argument would be enough to neglect it, and to suspect this is the flat space contribution. To confirm this is indeed the case compute the flat space result by taking the infinite radius limit of (\[masszeta\]): $$\lim_{R\rightarrow +\infty}\bar{\rho}(s)=\frac{\mu_0^{s+1}}{4\pi^2}\int_0^{+\infty}dk k^2(k^2+\mu^2)^{-s/2}=\frac{\mu_0^{s+1}}{4\pi^2}\frac{\sqrt{\pi}}{4}\frac{\Gamma\left(\frac{s-3}{2}\right)}{\Gamma\left(\frac{s}{2}\right)}(\mu)^{3-s} \ .$$ This is exactly the infinite term obtained after zeta function regularisation. $\bullet$ : Nonconformal coupling: In the generic case, the divergent term cannot be identified with the flat space contribution, since it does depend on $R$. However, it turns out that the correct result is still obtained by simply dropping out the divergent terms, as will be shown below. Doing so, the final answer for the renormalised vacuum energy density of a scalar field with generic coupling is\   . \[rhorenm2\] An expression equivalent to (\[rhorenm2\]) was first derived, for the particular case of $\xi=1/6$, in [@Dowker:1976pr], using a different renormalisation technique. It was argued therein that such renormalised energy momentum tensor could lead to a self consistent ESU solution of the semi-classical Einstein equations. We shall come back to this point in section 4. The advertised check on our calculation of the Casimir energy for a massive, conformally coupled, scalar in the ESU can be performed graphically; plotting the result (\[rhorenm2\]) for $\xi=1/6$, we have checked that it perfectly coincides with the previous one (\[rhorenm1\]), plotted in figure \[rhomassivefig\]. The zeta function method confirmed the damping factor method used originally in [@Ford2]. But is unfortunately inapplicable to the most general situation when $a^2<0$ which includes the interesting case of a massless minimally coupled scalar field. We will now study this case by a variation of the damping factor method. The essential ingredient will be the Abel-Plana formula which we use in the form [@Olver] \_[m=b]{}\^[+]{}G(m)=\_b\^[+]{}G(t)dt++i\_0\^[+]{} dt  . \[abelplana\] We regularise the vacuum energy density in the following way |()=\_[n=1]{}\^[+]{}e\^[-(n/R)]{}  , \[regomega\] where $\Omega$ is a positive function of its argument that grows sufficiently fast with $n$ so as to make the sum converge for any $\beta>0$. A convenient choice obeys $$\frac{d\Omega}{d(\frac{n}{R})}=\frac{n^2}{R^2}\sqrt{\frac{n^2}{R^2}+\frac{\mu^2R^2+6\xi-1}{R^2}} \ ;$$ it can be written explicitly as $$\barr{l} \displaystyle{\Omega\left(\frac{n}{R}\right)=\frac{1}{8}\left\{2\frac{n}{R}\left(\frac{n^2}{R^2}+\frac{\mu^2R^2+6\xi-1}{R^2}\right)^{3/2} - \frac{\mu^2R^2+6\xi-1}{R^2}\frac{n}{R}\left(\frac{n^2}{R^2}+\frac{\mu^2R^2+6\xi-1}{R^2}\right)^{1/2}\right. } \spa{0.4cm}\\ \displaystyle{ \left. ~~~~~~~~~~ - \left(\frac{\mu^2R^2+6\xi-1}{R^2}\right)^2\ln \left(\frac{\frac{n}{R}+\left(\frac{n^2}{R^2}+\frac{\mu^2R^2+6\xi-1}{R^2}\right)^{1/2}}{ \sqrt{\frac{|\mu^2R^2+6\xi-1|}{R^2}}}\right)\right\}} \ . \earr$$ The integration constant has been chosen such that $\Omega \rightarrow 0$ as the frequency vanishes, which means that the damping factor is not altering the long wavelength modes. This seems to be the most physical choice, since these long wavelength modes are the ones responsible for the Casimir energy. We divide the situation with a stable spectrum into two cases which we analyse separately: $\bullet$ [$\mu^2R^2+6\xi-1>0$ (includes conformal coupling with arbitrary mass)]{}. Applying the Abel-Plana formula (\[abelplana\]) with $b=0$ to the regularised expression (\[regomega\]) we find $$\barr{l} \displaystyle{ \bar{\rho}(\beta)=\frac{1}{4\pi^2}\int_0^{+\infty}d\Omega e^{-\beta \Omega}~~~~~ } \spa{0.4cm}\\ \displaystyle{~~~~ -\frac{i}{4\pi^2R^4}\int_0^{+\infty}t^2\left[\frac{\sqrt{\frac{it-ia}{it+ia}}(it+ia)e^{-\beta\Omega(it/R)}-\sqrt{\frac{-it-ia}{-it+ia}}(-it+ia)e^{-\beta\Omega(-it/R)}}{exp(2\pi t)-1}\right]dt}\ . \earr$$ The first integral became exactly the contribution of flat space (which justifies our choice of $\Omega$). Indeed, from (\[regomega\]) \_[R+]{}|()=\_0\^[+]{}dx x\^2e\^[-|(x)]{}=\_0\^[+]{}d| e\^[-|]{}  , \[flats\] where $$\bar{\Omega}(x)=\lim_{R\rightarrow \infty}\Omega(x)\ .$$ The second integral, where $a=\sqrt{\mu^2R^2+6\xi-1}$, converges in the $\beta\rightarrow 0$ limit. Thus $$\rho_{ren}=-\frac{i}{4\pi^2R^4}\int_0^{+\infty}t^2\left[\frac{\sqrt{\frac{it-ia}{it+ia}}(it+ia)-\sqrt{\frac{-it-ia}{-it+ia}}(-it+ia)}{exp(2\pi t)-1}\right]dt\ .$$ The square root representation in the complex plane that has been used, has a branch cut for $t\in [-a,a]$. Splitting the $t$ integral in $\int_0^a+\int_a^{+\infty}$, one can check that only the second one contributes. The final result is \_[ren]{}=\_\^[+]{}dt  . \[rhorenm3\] This expression was first obtained for the special case of conformal coupling $\xi=1/6$ in [@mamayev] (see also [@review]). Specialising for such coupling, the plot of this function exactly coincides with the ones in figure \[rhomassivefig\], again checking the result. It is quite striking that expressions apparently as distinct as (\[rhorenm1\]), (\[rhorenm2\]) and (\[rhorenm3\]) are actually different representations of the same function. For a more general coupling with $\mu^2R^2+6\xi-1>0$, we can also check that the result (\[rhorenm3\]) coincides with the one computed with the zeta function method (\[rhorenm2\]), where we dropped out the divergent term, even though it could not be clearly interpreted as the flat space contribution. Such term should seems to renormalise the gravitational action in the way described in [@Streeruwitz:1975sv]. Since we are mostly interested in the renormalised energy-momentum tensor, which will be determined by the finite part of the result, we will not dwell any longer on this point. (0,0)(0,0) (0,0)(0,0) $\bullet$ [$-1\le\mu^2R^2+6\xi-1<0$ (includes minimal coupling with zero mass)]{}. Applying the Abel-Plana formula (\[abelplana\]) with $b=1$ to the regularised expression (\[regomega\]) we find $$\barr{l} \displaystyle{ \bar{\rho}(\beta)=\int_{\Omega(\frac{1}{R})}^{+\infty}\frac{d\Omega}{4\pi^2} e^{-\beta \Omega}+\frac{\sqrt{\mu^2R^2+6\xi}}{8\pi^2R^4}e^{-\beta\Omega(\frac{1}{R})} +\frac{i}{4\pi^2R^4}\int_0^{+\infty}\frac{f_{\beta}(1+it)-f_{\beta}(1-it)}{exp(2\pi t)-1}dt}\ , \earr$$ where $$f_{\beta}(x)\equiv x^2\sqrt{x^2+\mu^2R^2+6\xi-1}e^{-\beta\Omega(\frac{x}{R})} \ .$$ The flat space contribution is still (\[flats\]), when $\mu^2>0$. For the tachyonic case the flat space contribution is more subtle, so we will restrict our analysis to $-1\le\mu^2R^2+6\xi-1<0$ and $\mu^2>0$. Subtracting the flat space quantity and removing the regulator $\beta$ we find the renormalised energy density\  . Despite the apparent unfriendly expression, this is a real quantity which can be plotted without great difficulty. Using the last formula and (\[rhorenm3\]) we have plotted several cases, with different $\xi$’s, in figure \[minimal\]. The most noticeable feature is that both the renormalised energy density and pressure may become negative for a range of values. Clearly this leads to violations of the strong energy condition, as will be discussed in more detail in the next section. Let us close by remarking that for zero mass, the renormalised energy density and pressure can be written, for arbitrary coupling, in the form \_[ren]{}=  ,         p\_[ren]{}=  , \[masslessprho\] where -+\_0\^[+]{}dt  .The quantity $\alpha$ is plotted as a function of $\xi$ in figure \[secfig\]. In particular, the special value of the coupling where $\alpha$ changes sign corresponds to a theory where the renormalised energy momentum tensor vanishes in the ESU. Its numerical value is $\xi_c\simeq 0.05391$. In figure \[ec\] some energy conditions are displayed. (0,0)(0,0) Discussion ========== The main purpose of this paper was to point out that the Casimir effect in the ESU becomes repulsive for a family of scalar fields with various couplings to the Ricci scalar and masses. This can lead to an inflationary era in the early universe, which generically seems to be too short to solve the usual big bang model problems. More interestingly it can lead to a cosmological bounce. To understand how, let us briefly consider the backreaction problem by taking the semiclassical Einstein equations, $$G_{\mu \nu}+\Lambda g_{\mu \nu}=T^{(matter)}_{\mu \nu}+\langle T^{\phi}_{\mu \nu} \rangle \ .$$ Analysing the simple massless case (\[masslessprho\]), for $\alpha>0$, we conclude that the quantum fluid can support a self consistent Einstein Static Universe with some radius, a fact first noticed in [@Dowker:1976pr]. Considering the Friedmann equation and Raychaudhuri equations (reinserting Newton’s constant) $$\dot{R}^2+k=\frac{8\pi G}{3}\rho R^2 \ , \ \ \ \ \ \ddot{R}=-\frac{4\pi G}{3}(\rho + 3p)R \ ,$$ and taking the quantum fluid and a positive cosmological constant $$\rho=\Lambda+\frac{\alpha}{R^4} \ , \ \ \ p=-\Lambda+\frac{\alpha}{3R^4} \ ,$$ a self consistent solution is obtained with k=+1  ,     =  ,     R=  . Clearly, this solution suffers from fine tuning, and is a universe of the order of the Planck size. For the case with $\alpha<0$, the fluid can, with the help of dust and a positive cosmological constant, produce an inflationary era followed by a decelerating phase and the present accelerating era. Indeed, taking $$\rho=\Lambda-\frac{|\alpha|}{R^4}+\frac{\eta}{R^3} \ , \ \ \ p=-\Lambda-\frac{|\alpha|}{3R^4} \ ,$$ one finds the generic behaviour displayed in figure \[sf\]. The universe has a minimal radius $R=R_c$ where it bounces from a collapsing to an expanding epoch, thus avoiding the Big Bang singularity. For $R>R_c$ there is a short accelerating phase, which is followed by a matter era and then cosmological constant domination. What is most interesting on having this Casimir energy induced bounce is that it does not rely on any particular form of a potential, and the scalar field needs not having classical effects whatsoever. It is its shear existence that originates the effect. Of course this is by no means a complete cosmological model of our universe. One obvious problem is how to include the radiation era in the picture, which has the same $1/R^4$ dependence for the energy density as the Casimir energy momentum tensor. Indeed, nucleosynthesis data constraints the influence that the quantum fluid could have over that period and therefore, since they vary equally with $R$, over any period where they both exist. A way around this problem could be to assume that radiation is only created after the quantum fluid domination epoch. Such mechanism could be similar to the usual reheating at the end of inflation. In any case, this simple model illustrates the point that the Casimir energy could both source an early inflationary epoch and avoid the Big Bang singularity in a closed universe. Note that despite having zero mass $\mu$, the non-minimal coupling $\xi$ works, in the Einstein static universe as an effective mass of order $1/R^2$. Thus, this massless case can have its classical dynamics frozen during inflation due to such effective mass. . It would certainly be interesting to further generalise this analysis to the case of a non-static FRW model where the dynamical Casimir effect takes place. (0,0)(0,0) Acknowledgements {#acknowledgements .unnumbered} ================ We are very grateful to Pedro Avelino, Filipe Paccetti Correia, Malcolm Perry and Gary Gibbons for discussions and suggestions. We would especially like to thank J. S. Dowker for reading the manuscript. C.H. is supported by FCT through the grant SFRH/BPD/5544/2001. M.S. was supported by the Marie Curie research grant MERG-CT-2004-511309. This work was also supported by Fundação Calouste Gulbenkian through *Programa de Estímulo à Investigação* and by the FCT grants POCTI/FNU/38004/2001 and POCTI/FNU/50161/2003. Centro de Física do Porto is partially funded by FCT through POCTI programme. [99]{} H. B. G. Casimir “On the attraction between two perfectly conducting plates,” Proc. Kon. Nederl. Akad. Wet. [**51**]{} (1948) 793. M. Bordag, U. Mohideen and V. M. Mostepanenko, “New developments in the Casimir effect,” Phys. Rept.  [**353**]{} (2001) 1 \[arXiv:quant-ph/0106045\]. Y. B. Zeldovich and A. A. Starobinsky, “Quantum Creation Of A Universe In A Nontrivial Topology,” Sov. Astron. Lett.  [**10**]{} (1984) 135. E. Elizalde and A. C. Tort, “A note on the Casimir energy of a massive scalar field in positive curvature space,” Mod. Phys. Lett. A [**19**]{} (2004) 111 \[arXiv:hep-th/0306049\]. A. R. Edmonds “Angular Momentum in Quantum Mechanics” Princeton University Press, Third Printing, 1974, Chapter 4. L. H. Ford, “Quantum Vacuum Energy In General Relativity,” Phys. Rev. D [**11**]{} (1975) 3370. L. H. Ford, “Quantum Vacuum Energy In A Closed Universe,” Phys. Rev. D [**14**]{} (1976) 3304. L. S. Brown, “Stress Tensor Trace Anomaly In A Gravitational Metric: Scalar Fields,” Phys. Rev. D [**15**]{}, 1469 (1977). R. M. Wald, “Trace Anomaly Of A Conformally Invariant Quantum Field In Curved Space-Time,” Phys. Rev. D [**17**]{} (1978) 1477. S. G. Mamayev, V. M. Mostepanenko, A. A. Starobinsky, Sov.  Phys. - JETP (USA) [**43**]{} (1976) 823. V. V. Nesterenko and I. G. Pirozhenko, “Justification of the zeta function renormalization in rigid string model,” J. Math. Phys.  [**38**]{} (1997) 6265 \[arXiv:hep-th/9703097\]. J. S. Dowker and R. Critchley, “Vacuum Stress Tensor In An Einstein Universe. Finite Temperature Effects,” Phys. Rev. D [**15**]{} (1977) 1484. F. W. J. Olver, “Asymptotics and Special Functions,” Academic Press, New York, 1974. E. Streeruwitz, “Vacuum Fluctuations Of A Quantized Scalar Field In A Robertson-Walker Universe,” Phys. Rev. D [**11**]{} (1975) 3378. [^1]: Recent work on the Casimir effect on the ESU can be found in [@Elizalde:2003ke] and references therein.

--- abstract: 'We study a natural notion of decoherence on quantum random walks over the hypercube. We prove that this model possesses a decoherence threshold beneath which the essential properties of the hypercubic quantum walk, such as linear mixing times, are preserved. Beyond the threshold, we prove that the walks behave like their classical counterparts.' author: - Gorjan Alagic$^1$ - Alexander Russell$^2$ title: Decoherence in quantum walks on the hypercube --- Introduction ============ The notion of a *quantum random walk* has emerged as an important element in the development of efficient quantum algorithms. In particular, it makes a dramatic appearance in the most efficient known algorithm for element distinctness [@A03]. The technique has also provided simple separations between quantum and classical query complexity [@CCD03], improvements in mixing times over classical walks [@NV00; @MR01], and some interesting search algorithms [@CG04; @AA05]. The basic model has two natural variants, the *continuous* model of Childs, et al. [@CFG01], on which we will focus, and the *discrete* model introduced by Aharonov, et al. [@AAKV01]. We refer the reader to Szegedy’s [@S04] article for a more detailed discussion. In the continuous model, a quantum walk on a graph $G$ is determined by the time-evolution of the Schrödinger equation using $kL$ as the Hamiltonian, where $L$ is the Laplacian of the graph and $k$ is a positive scalar to which we refer as the “jumping rate” or “energy”. In addition to being a physically attractive model, it has been successfully applied to some algorithmic problems as indicated above. Such walks have been studied over a variety of graphs with special attention given to Cayley graphs, whose algebraic structure has provided immediate methods for determining the spectral resolution of the linear operators that determine the system’s dynamics. Once it had been discovered that quantum random walks can offer improvement over their classical counterparts with respect to such basic phenomena as mixing and hitting times, it was natural to ask how robust these walks are in the face of decoherence, as this would presumably be an issue of primary importance for any attempt at implementation [@LP03; @SBTK03; @DRKB02]. In this article, we study the effects of a natural notion of decoherence on the hypercubic quantum walk. Our notion of decoherence corresponds, roughly, to independent measurement “accidentally” taking place in each coordinate of the walk at a certain rate $p$. We discover that for values of $p$ beneath a threshold depending on the energy of the system, the walk retains the basic features of the non-decohering walk; these features disappear beyond this threshold, where the behavior of the classical walk is recovered. Moore and Russell [@MR01] analyzed both the discrete and the continuous quantum walk on a hypercube. Kendon and Tregenna [@KT03] performed a numerical analysis of the effect of decoherence in the discrete case. In this article, we extend the continuous case with the model of decoherence described above. In particular, we show that up to a certain rate of decoherence, both linear instantaneous mixing times and linear instantaneous hitting times still occur. Beyond the threshold, however, the walk behaves like the classical walk on the hypercube, exhibiting $\Theta(n \log n)$ mixing times. As the rate of decoherence grows, mixing is retarded by the quantum Zeno effect. Results ------- Consider the continuous quantum walk on the $n$-dimensional hypercube with energy $k$ and decoherence rate $p$, starting from the initial wave function $\Psi_0 = \vert 0 \rangle ^{\otimes n}$, corresponding to the corner with Hamming weight zero. We prove the following theorems about this walk. When $p < 4k$, the walk has instantaneous mixing times at $$t_{mix} = \frac {n (2\pi c - \arccos(p^2/8k^2-1))}{\sqrt{16k^2 - p^2}}$$ for all $c \in \mathbb{Z}$, $c > 0$. At these times, the total variation distance between the walk distribution and the uniform distribution is zero. This result is an extension of the results in [@MR01], and an improvement over the classical random walk mixing time of $\Theta(n \log n)$. Note that the mixing times decay with $p$ and disappear altogether when $p \geq 4k$. Further, for large $p$, we will see that the walk is retarded by the quantum Zeno effect. When $p < 4k$, the walk has approximate instantaneous hitting times to the opposite corner $(1, \dots , 1)$ at times $$t_{hit} = \frac{2 \pi n (2c + 1)}{\sqrt{16k^2 - p^2}}$$ for all $c \in \mathbb{Z}$, $c \geq 0$. However, the probability of measuring an exact hit decays exponentially in $c$; the probability is $$P_{hit} = \left[\frac{1}{2} + \frac{1}{2}e^{-\frac{p \pi (2c + 1)} {\sqrt{16k^2 - p^2}}}\right]^n\enspace.$$ In particular, when no decoherence is present, the walk hits at $t_{hit} = \frac{n \pi(2c+1)}{2k},$ and it does so exactly, i.e. $P_{hit} = 1$. For $p \geq 4k$, no such hitting occurs. This result is a significant improvement over the exponential hitting times of the classical random walk, with the caveat that decoherence has a detrimental effect on the accuracy of repeated hitting times. Finally, we show that under high levels of decoherence, the measurement distribution of the walk actually converges to the uniform distribution in time $\Theta(n \log n)$, just as in the classical case. For a fixed $p \geq 4k$, the walk mixes in time $\Theta(n \log n)$. In the remainder of the introduction, we describe the continuous quantum walk model, and recall the graph product analysis of Moore and Russell [@MR01]. In the second section, we describe our model of decoherence, derive a superoperator that governs the behavior of the decohering walk, and prove that it is decomposable into an $n$-fold tensor product of a small system. We then fully analyze the small system in the third section, and use those results to draw conclusions about the general walk in 3 distinct regimes: $p < 4k$, $p = 4k$, and $p > 4k$. These regimes are roughly analogous to underdamping, critical damping, and overdamping (respectively) of a simple harmonic oscillator with damping rate $p$ and angular frequency $2k$. The continuous quantum walk on the hypercube -------------------------------------------- A continuous quantum walk on a graph $G$ begins at a distinguished vertex $v_0$ of $G$, the initial wave function of the walk being $\Psi_0$, where $\langle \Psi_0 \vert v \rangle = 1$ if $v = v_0$ and $0$ otherwise. The walk then evolves according to the Schrödinger equation. In our case, the graph is the $n$-dimensional hypercube. Concretely, we identify the vertices with $n$-bit strings, with edges connecting those pairs of vertices that differ in exactly one bit. Since the hypercube is a regular graph, we can let the Hamiltonian $H$ be the adjacency matrix instead of the Laplacian [@GW03]; the dynamics are then given by the unitary operator $U_t = e^{iHt}$ and the state of the walk at time $t$ is $\Psi_t = U_t \Psi_0$. The following analysis makes use of the hypercube’s product graph structure; this structure will be useful again later when we consider the effects of decoherence. The analysis below diverges from that of Moore and Russell [@MR01] only in that we allow each qubit to have energy $k/n$ instead of $1/n$. The energy of the entire system is then $k$. Let $$\sigma_x = \left(\begin{matrix} 0 & k/n \\ k/n & 0 \end{matrix} \right),$$ and let $$H = \sum_{j=1}^n \identity \otimes \cdots \otimes \sigma_x \otimes \cdots \otimes \identity\enspace,$$ where the $j$th term in the sum has $\sigma_x$ as the $j$th factor in the tensor product. Then we have $$\begin{aligned} U_t &= e^{iHt} = \prod_{j=1}^n \identity \otimes \cdots \otimes e^{it\sigma_x} \otimes \cdots \otimes \identity = \left[e^{it\sigma_x} \right]^{\otimes n} \\ & = \left[\begin{matrix}\cos(kt/n) & i~\sin(kt/n) \\ i~\sin(kt/n) & \cos(kt/n) \end{matrix}\right]^{\otimes n}\enspace.\end{aligned}$$ Applying $U_t$ to the initial state $\Psi_0 = \vert 0 \rangle ^{\otimes n}$, we have $$U_t \Psi_0 = \left[ \cos\left(\frac{kt}{n}\right) \vert 0 \rangle + i~\sin\left(\frac{kt}{n}\right) \vert 1 \rangle \right]^{\otimes n}$$ which corresponds to a uniform state exactly when $\frac{kt}{n}$ is an odd multiple of $\frac{\pi}{4}$. A derivation of the superoperator ================================= We begin by recalling a model of decoherence commonly used in the discrete model, with the intention of deriving a superoperator $U_t$, acting on density matrices, which mimics these dynamics in our continuous setting. The discrete model, described in [@KT03], couples unitary evolution according to the discrete-time quantum random walk model of Aharonov et al. [@AAKV01] with partial measurement at each step occurring with some fixed probability $p$. Specifically, the evolution of the density matrix can be written as $$\rho_{t+1} = (1-p) U \rho_t U^{\dagger} + p \sum_{i} \mathbf{P_i} U \rho_t U^{\dagger} \mathbf{P_i}$$ where $U$ is the unitary operator of the walk, $i$ runs over the dimensions where the decoherence occurs, and the $\mathbf{P_i}$ project in the usual “computational” basis [@KT03]. In the continuous setting, the unitary operator that governs the non-decohering walk is $U_t = e^{-iHt}$, where $H$ is the normalized adjacency matrix of the hypercube times an energy constant. To extend the above decoherence model to this setting, recall that the superoperator $U_t \otimes U_t^\dagger$ associated with these dynamics has the property that $$\frac{d\, U_t \otimes U_t^\dagger}{dt} = i\left(e^{-iHt} \otimes e^{iHt} \right) \left[ \identity \otimes H - H \otimes \identity\right]\enspace;$$ wishing to augment these dynamics with measurement occurring at some prescribed rate $p$, we desire a superoperator $S_t$ that satisfies $$S_{t+dt} = S_t[e^{-iHdt} \otimes e^{iHdt}][(1 - p\,dt) \identity + pdt(\mathbf{P})]$$ where $\mathbf{P}$ is the operator associated with the decohering measurement. Intuitively, the unitary evolution of the system is punctuated by measurements taking place with rate $p$, analogous to the discrete case. Letting $e^{-iHdt} = \identity - iHdt$, we can expand and simplify: $$\begin{aligned} S_{t+dt} & = S_t[e^{-iHdt} \otimes e^{iHdt}][(1 - pdt) \identity + pdt(\mathbf{P})] \\ & = S_t[(\identity - iHdt) \otimes (\identity + iHdt)][(1 - pdt) \identity + pdt(\mathbf{P})] \\ & = S_t[\identity \otimes \identity + idt(\identity \otimes H - H \otimes \identity) - pdt \identity \otimes \identity + pdt(\mathbf{P})]\enspace.\end{aligned}$$ In terms of a differential equation, $$\begin{aligned} \frac{dS_t}{dt} & = \frac{S_{t+dt} - S_t}{dt}\\ & = \frac{S_t[\identity \otimes \identity + idt(\identity \otimes H - H \otimes \identity) - pdt \left(\identity \otimes \identity + \mathbf{P}\right)] - S_t}{dt}\\ & = S_t[i(\identity \otimes H - H \otimes \identity) - p\identity \otimes \identity + p(\mathbf{P})]\enspace.\end{aligned}$$ The solution is $$\label{superop-soln} S_t = \exp\left([i(\identity \otimes H - H \otimes \identity) - p \identity \otimes \identity + p(\mathbf{P})]t\right)\enspace.$$ We now define the decoherence operator $\mathbf P$. This operator will correspond to choosing a coordinate uniformly at random and measuring it by projecting to the computational basis $\{|0\rangle, |1\rangle\}.$ Let $\Pi_0$ and $\Pi_1$ be the single qubit projectors onto $\vert 0 \rangle$ and $\vert 1 \rangle$, respectively. We define $$\mathbf P = \frac{1}{n} \sum_{1 \leq i \leq n} [\Pi^i_0 \otimes \Pi^i_0 + \Pi^i_1 \otimes \Pi^i_1]$$ where $\Pi^i_0 = \identity \otimes \cdots \otimes \identity \otimes \Pi_0 \otimes \identity \otimes \cdots \otimes \identity$ with the nonidentity projector appearing in the $i$th place. We define $\Pi^i_1$ similarly, so that $\Pi^i_j$ ignores all the qubits except the $i$th one, and projects it onto $\vert j \rangle$ where $j \in \{0,1\}$. Note that $$\Pi^i_j \otimes \Pi^i_j = [\identity \otimes \identity] \otimes \cdots \otimes [\Pi_j \otimes \Pi_j] \otimes \cdots \otimes [\identity \otimes \identity]$$ for $j \in \{0,1\}$. The superoperator as an $n$-fold tensor product ----------------------------------------------- The pure continuous quantum walk on the $n$-dimensional hypercube is easy to analyze, in part, because it is equivalent to a system of $n$ non-interacting qubits. We now show that, with the model of decoherence described above, each dimension still behaves independently. In particular, the superoperator that dictates the behavior of the walk is decomposable into an $n$-fold tensor product. Recall the product formulation of the non-decohering Hamiltonian $$H = \sum_{j=1}^n \identity \otimes \cdots \otimes \sigma_x \otimes \cdots \otimes \identity$$ where $$\sigma_x = \left(\begin{matrix} 0 & k/n \\ k/n & 0 \end{matrix} \right)$$ with $\sigma_x$ appearing in the $j$th place in the tensor product. We have given each single qubit energy $k/n$, resulting in a system with energy $k$. This choice will allow us to precisely describe the behavior of the walk in terms of the relationship between the energy of the system and the rate of decoherence. We can write each of the terms in the exponent of the superoperator from (\[superop-soln\]) as follows: $$\begin{aligned} \identity \otimes H &= \sum_{j=1}^n [\identity \otimes \identity] \otimes \cdots \otimes [\identity \otimes \sigma_x] \otimes \cdots \otimes [\identity \otimes \identity]\enspace, \\ H \otimes \identity &= \sum_{j=1}^n [\identity \otimes \identity] \otimes \cdots \otimes [\sigma_x \otimes \identity ] \otimes \cdots \otimes [\identity \otimes \identity]\enspace.\end{aligned}$$ Our decoherence operator can also be written in this form: $$\begin{aligned} {\mathbf P} &=& \frac{1}{n} \sum_{j=1}^n [\Pi^i_0 \otimes \Pi^i_0 + \Pi^i_1 \otimes \Pi^i_1] \\ &=& \frac{1}{n} \sum_{j=1}^n ([{\identity} \otimes {\identity}] \otimes \cdots \otimes [\Pi_0 \otimes \Pi_0] \otimes \cdots \otimes [{\identity} \otimes {\identity}]\\ &&+ [{\identity} \otimes {\identity}] \otimes \cdots \otimes [\Pi_1 \otimes \Pi_1] \otimes \cdots \otimes [{\identity} \otimes {\identity}])\enspace.\end{aligned}$$ The identity operator has a consistent decomposition: $\identity \otimes \identity = \frac{1}{n}\sum_{j=1}^n [{\identity} \otimes {\identity}] \otimes \cdots \otimes [{\identity} \otimes {\identity}].$ We can now put these pieces together to form the superoperator: $$\begin{aligned} S_t & = \exp\left(it(\identity \otimes H) - it(H \otimes \identity) - pt \identity \otimes \identity + pt\mathbf{P}\right) \\ & = \exp\left(\sum_{j=1}^n [{\identity} \otimes {\identity}] \otimes \cdots \otimes \mathbf{A} \otimes \cdots [{\identity} \otimes {\identity}]\right) \\ & = \prod_{j=1}^n [{\identity} \otimes {\identity}] \otimes \cdots \otimes e^\mathbf{A} \otimes \cdots [{\identity} \otimes {\identity}]\\ & = \left[e^\mathbf{A}\right]^{\otimes n}\end{aligned}$$ where $$\begin{aligned} \mathbf{A} &=& \frac{t}{n}[(\identity \otimes in\sigma_x) - (in\sigma_x \otimes \identity) - p({\identity} \otimes {\identity})\\ &&+ p(\Pi_1 \otimes \Pi_1) + p(\Pi_0 \otimes \Pi_0)] \\ &=& \frac{t}{n}\left( \begin{matrix} 0 & ik & -ik & 0 \\ ik & -p & 0 & -ik \\ -ik & 0 & -p & ik \\ 0 & -ik & ik & 0 \\ \end{matrix} \right).\end{aligned}$$ Notice that for $p = 0$, $\left[e^{\mathbf{A}}\right]^{\otimes n} = \left[e^{-it\sigma_x} \otimes e^{it\sigma_x}\right]^{\otimes n}$, which is exactly the superoperator formulation of the dynamics of the non-decohering walk. Small-system behavior and analysis of the walk ============================================== So far we have shown that the walk with decoherence is still equivalent to $n$ non-interacting single-qubit systems. We now analyze the behavior of a single-qubit system under the superoperator $e^\mathbf{A}$. The structure of this single particle walk will allow us to then immediately draw conclusions about the entire system. The eigenvalues of $\mathbf{A}$ are $0$, $- \frac{pt}{n}$, $\frac{-p t - \alpha t}{2n}$ and $\frac{-p t + \alpha t}{2n}$. Here $\alpha = \sqrt{p^2-16k^2}$ is a complex constant that will later turn out to be important in determining the behavior of the system as a function of the rate of decoherence $p$ and the energy $k$. The matrix exponential of $\mathbf{A}$ in this spectral basis can then be computed by inspection. To see how our superoperator acts on a density matrix $\rho_0$, we may change $\rho_0$ to the spectral basis, apply the diagonal superoperator to yield $\rho_t$, and finally change $\rho_t$ back to the computational basis. At that point we can apply the usual projectors $\Pi_0$ and $\Pi_1$ to determine the probabilities of measuring $0$ or $1$ in terms of time. Let $\Psi_0 = \vert 0 \rangle$ and $\rho_0 = \vert \Psi_0 \rangle \langle \Psi_0 \vert$. In the diagonal basis, $$\rho_0 = \left[\begin{matrix} 1/2 \\ 0 \\ \frac{1}{4}(-1 + \frac{p}{\alpha}) \\ \frac{1}{4}(-1 - \frac{p}{\alpha}) \end{matrix} \right]$$ and thus at time $t$ we have $$\rho_t = e^{\mathbf{A}}\rho_0 = \left[\begin{matrix} 1/2 \\ 0 \\ \frac{1}{4}e^{\frac{-p t - \alpha t}{2n}}(-1 + \frac{p}{\alpha}) \\ \frac{1}{4}e^{\frac{-p t + \alpha t}{2n}}(-1 - \frac{p}{\alpha}) \end{matrix} \right].$$ If we then change back to the computational basis and project by $\Pi_0$ and $\Pi_1$, we may compute the probabilities of measuring $0$ and $1$ at a particular time $t$: $$P[0] = \frac{1}{4}\left[2 + e^{\frac{-p t - \alpha t}{2n}}(1 - p/ \alpha) + e^{\frac{-p t + \alpha t}{2n}}(1 + p/ \alpha)\right]$$ $$P[1] = \frac{1}{4}\left[2 - e^{\frac{-p t - \alpha t}{2n}}(1 - p/ \alpha) - e^{\frac{-p t + \alpha t}{2n}}(1 + p/ \alpha)\right]$$ which can be simplified somewhat to $$P[0] = \frac{1}{2} + \frac{1}{2} e^{-\frac{pt}{2n}} \left[\cos\left(\frac{\beta t}{2n}\right) + \frac{p}{\beta}~\sin\left(\frac{\beta t}{2n}\right) \right]$$ $$P[1] = \frac{1}{2} - \frac{1}{2} e^{-\frac{pt}{2n}} \left[\cos\left(\frac{\beta t}{2n}\right) + \frac{p}{\beta}~\sin\left(\frac{\beta t}{2n}\right) \right].$$ Here we have let $\beta = -i \alpha = \sqrt{16k^2-p^2}$ for simplicity. A quick check shows that when $p = 0,$ $P[0] = \cos^2\left(\frac{kt}{n}\right)$ and $P[1] = \sin^2\left(\frac{kt}{n}\right)$, which are exactly the dynamics of the non-decohering walk. The probabilities for this non-decohering case are shown in Figure \[fig1\]. ![The $p=0$ case - no decoherence: a plot of $P[0]$ and $P[1]$ versus time, for $k =1$, $n=5$, $p = 0$[]{data-label="fig1"}](pequals0){width="3in"} The three regimes mentioned before are immediately apparent. For $p < 4k$, $\beta$ is real. When $p = 4k$, we have $\beta = 0$, which appears to be a serious problem at first glance. Finally, for $p > 4k$, $\beta$ is imaginary. We now address each of these three situations in detail. The case $p < 4k$ : linear mixing and hitting times --------------------------------------------------- ![The $p<4k$ case: a plot of $P[0]$ and $P[1]$ versus time, for $k =1$, $n=5$, $p = 0.5$[]{data-label="fig2"}](pless4k){width="3in"} When $p < 4k$, we recover the perhaps most interesting feature of the non-decohering walk: the instantaneous mixing time is linear in $n$. To exactly determine the mixing times for our decohering walk, we solve $P[0] = P[1] = \frac{1}{2}$; this amounts to determining when $$\gamma = \frac{1}{2} e^{-\frac{pt}{2n}} \left[\cos\left(\frac{\beta t}{2n}\right) + \frac{p}{\beta}~\sin\left(\frac{\beta t}{2n}\right) \right]$$ equals zero. Clearly the exponential decay term results in mixing as $t \to \infty$; our principle concern, however, is with the periodic mixing times analogous to those of the original walk. We thus ignore the exponential term when solving the equality $\gamma = 0$, which yields $$\frac{p^2}{\beta^2} = \frac{1 + \cos(\beta t/n)}{1 - \cos(\beta t/n)}.$$ This equation actually has more solutions than the one we started with, because of the use of half-angle formulas for simplification. The solutions that we want are $$t_{mix} = \frac {n}{\beta} \left[2\pi c - \arccos\left(\frac{p^2}{8k^2}-1\right) \right]$$ where $c$ ranges over the positive integers. Evidently, the mixing times still occur in linear time; an example is shown in Figure \[fig2\]. Note also that if we let $p = 0$, we have $t_{mix} = n\pi(2c - 1)/(4k)$, which are exactly the nice periodic mixing times of the non-decohering walk. In the decohering case, however, these mixing times drift towards infinity, and cease to exist altogether beyond the threshold of $p = 4k$. This proves Theorem 1. We now wish to determine when our small system is as close as possible to $\vert 1 \rangle$. Since our large-system walk begins at $\vert 0 \rangle^{\otimes n}$, this will correspond to approximate hitting times to the opposite corner $\vert 1 \rangle^{\otimes n}$. These times correspond to local maxima of $P[1]$; the solutions are $$t_{hit} = 2n \pi \left( \frac{2c + 1}{\beta} \right)$$ where $c$ ranges over the non-negative integers. At these points in time, the value of $P[1]$ is $$\frac{1}{2} + \frac{1}{2}e^{-(2c+1)\frac{p \pi}{\beta}}$$ which immediately yields Theorem 2. The breakpoint case $p = 4k$ ---------------------------- We first observe that $t_{mix} \to \infty$ as $p \to 4k$. Hence, we do not expect to see any mixing in this case. To analyze the probabilities exactly, we take the limit of $\gamma$ as $p \to 4k$. The solution is $$\label{pequals4k} \lim_{p \to 4k} \gamma = \frac{1}{2}e^{-\frac{2kt}{n}}\left[1 + \frac{2kt}{n} \right]\enspace.$$ Indeed, since $k$, $t$ and $n$ are all positive, $\gamma$ is zero only in the limit as $t \to \infty$. The linear mixing and hitting behavior from the previous section has entirely disappeared. As in the critical damping of simple harmonic motion, a small decrease in the rate $p$ can result in drastically different behavior, in this case a return to linear mixing and hitting. We leave the limiting mixing analysis of this case for the next section, where we develop some relevant tools. The case $p > 4k$ and the limit to the classical walk ----------------------------------------------------- ![The $p>4k$ case: a plot of $P[0]$ and $P[1]$ versus time, for $k=1$, $n=5$, $p = 9$[]{data-label="fig3"}](pgreater4k){width="3in"} The goal of this section is to show two interesting consequences of the presence of substantial decoherence in the quantum walk on the hypercube. First, we will show that for a fixed $p \geq 4k$, the walk behaves much like the classical walk on the hypercube, mixing in time $\Theta(n \log n)$. Second, we show that as $p \to \infty$, the walk suffers from the quantum Zeno effect. Informally stated, the rate of decoherence is so large that the walk is continuously being reset to the initial wave function $|0\rangle^{\otimes n}$ by measurement. ### Recovering classical behavior Consider a single qubit. Let $P$ be the distribution obtained by full measurement at time $t$, and $U$ the uniform distribution: $$P(0) = \frac{1}{2}+ \gamma, \qquad P(1) = \frac{1}{2}- \gamma, \qquad U(0) = U(1) = \frac{1}{2},$$ where $$\gamma = \frac{1}{4}\left[e^{\frac{(-p-\alpha)t}{2n}}(1-p/\alpha) + e^{\frac{(\alpha-p)t}{2n}}(1+p/\alpha)\right].$$ For $x = (x_1, \dots, x_n) \in \mathbb{Z}_2^n$, $$P^n(x) = \prod_{i=1}^n P(x_i) \qquad\text{and}\qquad U^n(x) = 2^{-n}$$ are the analogous product distributions in the $n$-dimensional case. To analyze the limiting mixing behavior of the walk, we will consider the total variation distance $\|P^n - U^n\| = \sum_x |P^n(x) - U^n(x)|$ between these distributions. In order to give bounds for total variation, we will use *Hellinger distance* [@ASYMP], defined as follows: $$H(A, B)^2 = \sum_{x} \left(\sqrt{A(x)} - \sqrt{B(x)}\right)^2 = 1 - \sum_{x}\sqrt{A(x)B(x)}.$$ We will make use of the following two properties of Hellinger distance: $$1 - H(A^n, B^n)^2 = (1-H(A,B)^2)^n\enspace,$$ and $$\|A - B\| \leq 2H(A, B) \leq 2\|A - B\|^{1/2}. \label{helltv}$$ The first property makes it easy to work with product distributions. The second gives a nice relationship between Hellinger distance and total variation distance. In our case, $$\begin{aligned} H(P^n,U^n)^2 & = 1 - (1-H(P,U)^2)^n \\ & = 1 - \left(\frac{1}{2}\sqrt{1+2\gamma} + \frac{1}{2}\sqrt{1-2\gamma}\right)^n\\ & = 1 - \left(1 - \frac{\gamma^2}{2} + O(\gamma^3)\right)^n.\end{aligned}$$ And hence, by (\[helltv\]), $$\|P_n - U_n\|^2 \leq 4 - 4\left(1 - \frac{\gamma^2}{2} + O(\gamma^3)\right)^n.$$ Consider the walk with decoherence rate $p > 4k$. We have $\alpha = \sqrt {p^2 - 16k^2} < p$, where $\alpha$ and $p$ are positive and real. It follows that for a fixed $p > 4k$, $\gamma \to 0$ and $\|P^n - U^n\| \to 0$ as $t \to \infty$. Hence the walk does indeed mix eventually, and the measurement distribution in fact converges to the uniform distribution. Let $t = d \cdot n\log n$ where $d > 0$ is a constant, and rewrite $\gamma$ as follows: $$\gamma = \frac{1}{4}e^{-(p-\alpha)\frac{d \log n}{2}}\left[(1-p/\alpha) + e^{\frac{-\alpha d \log n}{2}}(1+p/\alpha)\right].$$ Suppose we choose $d$ such that $d > (p - \alpha)^{-1}$. Then $\gamma = o(n^{-1/2})$, which implies that $\|P^n - U^n\| = o(1)$. On the other hand, if $d < (p - \alpha)^{-1}$, then $\gamma = \omega(n^{-1/2})$ and there exists a constant $\epsilon$ such that $\|P^n - U^n\| \geq \epsilon > 0$. This shows that the walk mixes in time $\Theta(n \log n)$ when $p > 4k$. Notice that when $p = 4k$, $(p - \alpha)^{-1} = (4k)^{-1}$, so that the same technique easily extends to that case via equation (\[pequals4k\]). This completes the proof of Theorem 3. ### Quantum Zeno effect for large $p$ Recall from the previous section that the time required to mix when $p > 4k$ is $$t \geq \frac{n~ \log n}{p - \alpha}$$ which clearly increases with $p$. Further, for large $p$, $p/\alpha$ tends to $1$, and hence $\gamma$ tends to $1/2.$ Notice that $\gamma = 1/2$ corresponds to remaining at the initial state forever. We conclude that the mixing of the walk is retarded by the quantum Zeno effect, where measurement occurs so often that the system tends to remain in the initial state. Alexander Russell gratefully acknowledges the support of the National Science Foundation, under the grants CAREER CCR-0093065, CCR-0220264, EIA-0218443, and ARO grant W911NF-04-R-0009. The authors are grateful to Viv Kendon for helpful suggestions. [15]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , in ** (, ), pp. , . , , , , , , in ** (, ), pp. , . (), . , in ** (), . , ****, (), . , in ** (, ), pp. , . , , , **** (), . , , , , in ** (, ), pp. , . , in ** (, ), pp. , . , ****, (), . , , , , ****, (), . , , , , ****, (), . , ****, (), . , in ** (), . , .

--- abstract: | We study nine S0–Sb galaxies with (photometric) bulges consisting of two distinct components. The outer component is a flattened, kinematically cool, disklike structure: a “disky pseudobulge”. Embedded inside is a rounder, kinematically hot spheroid: a “classical bulge”. This indicates that pseudobulges and classical bulges are not mutually exclusive: some galaxies have both. The disky pseudobulges almost always have an exponential disk (scale lengths = 125–870 pc, mean $\sim 440$ pc) with disk-related subcomponents: nuclear rings, bars, and/or spiral arms. They constitute 11–59% of the galaxy stellar mass (mean $PB/T = 0.33$), with stellar masses $\sim 7 \times 10^{9}$–$9 \times 10^{10}$ [$\mathrm{M}_{\sun}$]{}. Classical-bulge components have Sérsic indices of 0.9–2.2, effective radii of 25–430 pc and stellar masses of $5 \times 10^{8}$–$3 \times 10^{10}$ [$\mathrm{M}_{\sun}$]{} (usually $< 10$% of the galaxy’s stellar mass; mean $B/T = 0.06$). The classical bulges show rotation, but are kinematically hotter than the disky pseudobulges. Dynamical modeling of three systems indicates that velocity dispersions are isotropic in the classical bulges and equatorially biased in the disky pseudobulges. In the mass–radius and mass–stellar mass density planes, classical-bulge components follow sequences defined by ellipticals and (larger) classical bulges. Disky pseudobulges *also* fall on this sequence; they are more compact than similar-mass large-scale disks. Although some classical bulges are quite compact, they are distinct from nuclear star clusters in both size and mass, and coexist with nuclear clusters in at least two galaxies. Since almost all the galaxies in this study are barred, they probably *also* host boxy/peanut-shaped bulges (vertically thickened inner parts of bars). NGC 3368 shows evidence for such a zone outside its disky pseudobulge, making it a galaxy with all three types of “bulge”. author: - | Peter Erwin$^{1,2,7}$, Roberto P. Saglia$^{1,2}$, Maximilian Fabricius$^{1,2}$, Jens Thomas$^{1,2}$, Nina Nowak$^{3}$, Stephanie Rusli$^{1,2}$, Ralf Bender$^{1,2}$, Juan Carlos Vega Beltr[á]{}n$^{4}$, and John E. Beckman$^{4,5,6}$\ $^{1}$Max-Planck-Insitut für extraterrestrische Physik, Giessenbachstrasse, 85748 Garching, Germany\ $^{2}$Universitäts-Sternwarte München, Scheinerstrasse 1, D-81679 München, Germany\ $^{3}$Stockholm University, Department of Astronomy, Oskar Klein Centre, SE-10691 Stockholm, Sweden\ $^{4}$Instituto de Astrofísica de Canarias, C/ Via Láctea s/n, 38200 La Laguna, Tenerife, Spain\ $^{5}$Departamento de Astrofísica, Universidad de La Laguna, Avda. Astrofísico Fco. Sánchez s/n, 38200, La Laguna, Tenerife, Spain\ $^{6}$Consejo Superior de Investigaciones Científicas, Spain\ $^{7}$Guest investigator of the UK Astronomy Data Centre nocite: - '[@erwin14-smbh]' - '[@comeron10]' - '[@delorenzo-caceres12; @delorenzo-caceres13]' - '[@kormendy04]' title: 'Composite Bulges: The Coexistence of Classical Bulges and Disky Pseudobulges in S0 and Spiral Galaxies' --- \[firstpage\] galaxies: bulges – galaxies: structure – galaxies: elliptical and lenticular, cD – galaxies: spiral – galaxies: kinematics and dynamics – galaxies: evolution. Introduction ============ In the standard picture of galaxy structure, disk galaxies have two main stellar components. The defining component is the disk: a highly flattened structure dominated by rotation, often (but not always) with a radial density profile which is exponential; disks often have significant substructure, particularly bars and spiral arms. The secondary component, present in early and intermediate Hubble types, is the bulge. Traditionally, the bulge has been seen as something very like a small elliptical galaxy embedded within the disk: more spheroidal than the disk, with stellar motions dominated by velocity dispersion rather than rotation, and having a strongly concentrated structure – e.g., having a surface brightness profile similar or identical to the stereotypical $R^{1/4}$ profile of an elliptical). In addition, the stellar populations of bulges were said to resemble those of ellipticals in being older (and possibly more metal-rich and alpha-enhanced) than the majority of stars in the disk. (See, e.g., @wyse97 and @renzini99 for reviews.) Taken all together, this seemed to argue for a bulge formation mechanism similar to that proposed for ellipticals, either via monolithic collapse or by rapid, violent mergers of initial subcomponents at high redshift. The past decade or two has seen the growing realization that this picture is probably *not* true for many bulges, at least when bulges are defined as the excess stellar light in the central regions of the galaxy when compared to the dominant exponential profile of the disk.[^1] Instead, bulges are now seen as falling into two rather different classes: classical bulges (the traditional model) and *pseudobulges* [e.g., @kormendy93; @kormendy04], which are conceived of as something much more like disks than spheroids; i.e., they are flattened and dominated by rotation, with profiles which are close to exponential. Added to this complexity is the existence of so-called “boxy” and “peanut-shaped” bulges, which are now well understood as the vertically thickened inner parts of bars [see @athanassoula05 for a discussion of the distinctions]; confusingly, these structures are also sometimes called pseudobulges. Although some authors are careful to point out the possibility that classical bulges could coexist with pseudobulges [see, e.g., @athanassoula05; @fisher-drory10], it is common to suggest that galaxies have one or the other, but not both. For example, in observational surveys such as those of @fisher-drory08 or @gadotti09, photometrically identified bulges are classified as either classical or pseudobulge. Similarly, in studies of how central supermassive black holes (SMBHs) relate to their host galaxies, disk galaxies are divided into those with classical bulges and those with pseudobulges [e.g., @hu08; @greene10; @kormendy11]. In this paper, we present evidence for the *coexistence* in nine galaxies of both a classical bulge – that is, a round, kinematically hot stellar structure which is significantly larger than a nuclear star cluster – and a disky pseudobulge – that is, a flattened stellar system, distinct from the main disk, whose kinematics are at least partly dominated by rotation and which (usually) hosts nuclear bars, nuclear rings, or other disky morphology. Two of the galaxies discussed here – NGC 3368 and NGC 3489 – were previously discussed, in an abbreviated fashion, in @nowak10, using the term “composite pseudobulges”. Some analysis of the morphological substructure in NGC 3945 and NGC 4371 has been previously presented in @erwin99 and @erwin03-id. The outline of this paper is as follows. After some initial discussion of data sources and reduction (Section \[sec:obs\]), we lay out our terminology in Section \[sec:terms\]. We introduce our methodology for identifying classical bulges by considering in Section \[sec:simple-classical\] two examples of *simple* classical-bulge-plus-disk systems (galaxies with *only* a classical bulge in addition to their disk). With this as a reference, we then consider two galaxies (NGC 3945 and NGC 4371) in some detail in Section \[sec:n3945n4371\], demonstrating first that much of their photometrically defined bulges are *not* classical bulges as previously defined, but something else: (disky) pseudobulges. We then go on to show that inside each pseudobulge is an additional structure which *does* resemble a classical bulge. The evidence for composite bulges in seven other galaxies follows a similar pattern, but is postponed to the Appendix (Section \[sec:others-full\]) so as not to interrupt the flow of the paper. Section \[sec:dynamics\] uses the results of Schwarzschild modeling for three composite-bulge galaxies to investigate the 3D stellar dynamics of the classical-bulge and disky-pseudobulge components. Section \[sec:discussion\] considers these composite-bulge galaxies and their subcomponents, including an analysis of their place in the mass-radius and surface-density–mass diagrams and a demonstration that the classical-bulge components, while generally rather small, are not in the same class as nuclear star clusters. Finally, Section \[sec:summary\] summarizes our findings. Data Sources {#sec:obs} ============ Imaging Data and Surface-Brightness Profiles -------------------------------------------- Imaging data for this study comes from a variety of sources; for each galaxy, we specify the individual images used and their origins. Some of the large-scale, ground-based optical images come from the WIYN Survey [@erwin-sparke03], the INT-WFC observations of @erwin08, or from the Sloan Digital Sky Survey [SDSS; @york00]; for the latter, we use Data Release 7 [DR7; @abazajian09]. Other ground-based image sources include WHT-INGRID $K$-band images from @knapen03, available via NED, and images from the European Southern Observatory (ESO) archive and the Isaac Newton Group (ING) archive. For high-resolution imaging of the central regions of galaxies, we rely on archival images from the *Hubble Space Telescope*, obtained with the Wide Field Planetary Camera 2 (WFPC2), the Wide Field Channel of the Advanced Camera for Surveys (ACS-WFC), or the NICMOS2 and NICMOS3 near-IR imagers. In some cases – i.e., when *HST* imaging data is lacking or when the centre of a galaxy is particularly dusty – we use adaptive-optics $K$-band images derived from our VLT-SINFONI IFU observations by collapsing the datacubes along the wavelength direction. The latter observations are described in more detail in @nowak10, @rusli11 and @erwin14-smbh. The surface-brightness profiles we construct and analyze combine data from multiple images for each galaxy: high-resolution data from *HST* or AO imaging for the smallest radii, where good spatial resolution is critical, and non-AO ground-based imaging for larger radii, since the galaxies extend beyond the high-resolution imaging fields of view. To combine these data, we must match the overall intensity scaling, and we must also account for the fact that the high-resolution images are too small for accurate background subtraction. (The VLT-SINFONI AO observations were taken with offsets to blank sky, but these were spaced too far apart in time for accurate removal of the overall background levels.) The problem is then to find the correct multiplicative scaling $k$ to match the background-corrected “inner” profile (from the high-resolution image) with the “outer” profile (from the low-resolution image, which is large enough for its own background to be properly estimated), while also determining the unknown background level $B_{i}$ in the high-resolution image, so that we can transform the observed inner profile $I_{i}$ to a corrected profile $I_{i}^{\prime}$: $$I_{i}^{\prime} \; = \; k (I_{i} - B_{i}).$$ Fortunately, these two issues can be dealt with in combination, by taking advantage of the fact that the same galaxy was observed in both images. The trick is to identify a radial overlap zone between the two profiles (outside the region where seeing distorts the low-resolution data) and iteratively fit for the values of $k$ and $B_{i}$ which minimize the difference $I_{i}^{\prime}(r) - I_{o}(r)$ for values of $r$ in the overlap zone (with $I_{o}$ being the outer profile). The two profiles are then merged at the best-matching radius in the overlap region. Although the profiles we analyze are usually major-axis cuts, in practice we determine $k$ and $B_{i}$ using profiles from ellipse fits with fixed position angle and ellipticity, to increase the signal-to-noise ratio. This approach is ideal when the high- and low-resolution images were taken with the same filter; in some cases, we are forced to match and combine profiles from images with dissimilar filters (e.g., F814W and $R$, or $K$ and $z$). When decomposing our surface-brightness profiles – which are typically cuts along the major axis of the galaxy – we do not attempt to correct for PSF convolution, though we do exclude the inner 2–3 pixels of the profile from the fit. For several galaxies with *HST* images, we estimated the possible effects of neglecting PSF convolution by also extracting profiles from images which had been deconvolved using the Lucy-Richardson algorithm (via the <span style="font-variant:small-caps;">IRAF</span> task `lucy`) and TinyTim-generated PSFs. (This was not possible for all galaxies, since for some galaxies we rely on SINFONI data for which PSFs cannot be determined with nearly the same precision.) Comparison of fits to both uncorrected and “deconvolved” profiles of the same galaxy showed that parameters for the central (classical) bulges differ by $\la 15$% in the Sérsic index $n$ and $\la 3$% for other parameters. Spectroscopic Data ------------------ Some of the spectroscopic data used for NGC 2859 and NGC 4371 is based on previously unpublished data obtained with the ISIS double spectrograph on the 4.2m William Herschel Telescope. Details of the observations are provided in the Appendix (Section \[sec:data-spec\]). For NGC 3368, NGC 3945 and NGC 4371 we use long-slit data obtained with the Marcario Low Resolution Spectrograph at the Hobby-Eberly Telescope, previously presented in @fabricius12. For NGC 3368 and NGC 4699, we also use IFU data from our SINFONI $K$-band SMBH measurement program [@nowak10]; @erwin14-smbh. ### Other Sources We also make use of various published long-slit and IFU kinematic data; the specific sources are listed in the discussions of each galaxy. We note here some datasets provided directly to us. For NGC 1068, this includes both long-slit data from @shapiro03, provided by Joris Gerssen, and SINFONI data for NGC 1068 [@davies07], provided by Ric Davies. Large-scale SINFONI kinematic data for NGC 3368 [@hicks13] were provided by Erin Hicks. Finally, for NGC 4262 we make use of OASIS IFU data [@mcdermid06], provided by Richard McDermid. Terminology and Definitions: What Do *We* Mean by “Classical Bulge” and “Pseudobulge”? {#sec:terms} ====================================================================================== The terms “pseudobulge” and “classical bulge” are unfortunately rather ambiguous at present. Sometimes they are defined in terms of their presumed formation methods: e.g., pseudobulges are central concentrations of stars formed from bar- or spiral-driven inflows of gas in the disk plane, or even by any process that does not explicitly involve major mergers, while classical bulges are those structures formed by violent relaxation in major mergers (usually at high redshifts). The fundamental problem with such approaches is that there are few if any clear observational predictions for how to distinguish such formation methods in nearby galaxies. For example, the formation of central “bulges” by the merger of massive star-forming clumps in gas-rich, high-$z$ disks can produce thick, dispersion-dominated central structures with $\sim R^{1/4}$ light profiles and $\alpha$-enhanced metallicities [e.g., @immeli04; @elmegreen08], fulfilling most of the traditional criteria for classical bulges. If one insists on major mergers as the formation mechanism, then these are not classical bulges – but we would have little or no way to distinguish these structures at $z \sim 0$ from “proper” classical bulges. Other theoretical studies of this formation mechanism argue that the resulting bulges should be smaller and more exponential-like, with Sérsic indices of $\sim 2$ or even $\sim 1$, and significant rotation [e.g., @hopkins12; @inoue12]. What this means is that a classification of structures in present-day galaxies based on their supposed formation mechanisms, though desirable, is probably still premature. If we turn to the more feasible approach of observationally-based classification, we still find considerable discord, if not an outright cacophony. Some surveys classify the central region of a galaxy as classical or pseudobulge depending on the presence or absence of certain morphological features: e.g., a smooth light distribution means a classical bulge, while the presence of dust lanes, spiral arms, rings, nuclear-scale bars, or so-called boxy/peanut-shaped isophotes means a pseudobulge [@kormendy04; @fisher-drory08 e.g.,]. Other studies make distinctions based on photometric profiles of the “bulge” component that results from a bulge-disk decomposition – e.g., pseudobulges are by definition any central structure with a Sérsic index $< 2$, or even any such structure with $n < 4$ [@laurikainen09] – or some combination of mean surface brightness and size for the bulge component [e.g., @gadotti09]. Because of this confusion, we feel it is important to be clear about our terminology and our methods for identifying and classifying different types of “bulges”. As part of our analysis, we first identify what we call **photometric bulges**. This term refers to the region of a galaxy where the observed stellar surface brightness is brighter than an inward extrapolation of the outer disk component, or is brighter than the inward extrapolation of a previously identified disky pseudobulge. We require that the photometric bulge should be more extended than a simple nuclear star cluster (i.e., the half-light radius should be $\ga 10$ pc). This is, as the name suggests, a purely photometric classification, and is used only as a preliminary tool (and for comparison with the results of purely photometric methodologies). We then analyse the photometric bulges and classify them into two categories: - **Classical bulge:** This is a photometric bulge which is some type of kinematically hot spheroid. That is, it must be clearly *rounder* in a three-dimensional sense than the main galaxy disk (i.e., $(c/a)_{\rm bulge} > (c/a)_{\rm disk}$, where $c$ is the vertical scale length and $a$ is the radial) *and* must have stellar kinematics which are dominated by velocity dispersion rather than rotation. - **Disky pseudobulge:** This is a photometric bulge which is “disklike” in *two* main ways: it has a flattening similar or identical to that of the main galaxy disk, and the stellar kinematics are dominated by rotation rather than velocity dispersion at least some point within the photometric bulge region. We also consider the presence of clear morphological features such as bars, rings and spiral arms to be additional signatures of a disky pseudobulge, but do not rely on them alone. It is important to note that our definition of classical bulge does *not* assume a particular surface-brightness profile shape: we are not assuming that kinematically hot spheroids must have de Vaucouleurs $R^{1/4}$ profiles, nor that they must have Sérsic indices greater than some minimum value. We also note that we are not considering several characteristics which are sometimes, as alluded to previously, discussed as indicative of “pseudobulges” [@kormendy04]. For example, we are mostly not concerned with the presence or absence of dust in the centres of these galaxies, since this can sometimes be due to off-plane or counter-rotating gas (likely the result of accretion), and in other cases may merely indicate that the disk and bulge are co-extensive. Since many of the galaxies we consider are lenticular, we do not require the presence of current or recent star formation as a pseudobulge indicator either. Thus, we are explicitly including what @fisher09 called “inactive pseudobulges” (objects in their sample which had what they considered the morphological and photometric signatures of pseudobulges but which showed little or no evidence for recent star formation; see also @fisher-drory10). Finally, we are, for the most part, explicitly excluding boxy/peanut-shaped bulges from consideration in this study. As @athanassoula05 pointed out, these are the vertically thickened inner parts of bars, the result of a common dynamical instability which appears to accompany the formation of most bars. As such, they are *not* the sort of highly flattened, axisymmetric structures we are most interested in. None the less, we *do* consider the question of their possible co-existence with disky pseudobulges and classical bulges later on in the paper (Section \[sec:boxy\]). In Section \[sec:simple-classical\], we present two cases of S0 galaxies with purely classical bulges, as a way of providing both examples of how we identify classical bulges and some context for the composite bulges we discuss later. In Section \[sec:n3945n4371\] we then go on to analyze two composite-bulge S0 galaxies in detail; we start by identifying disky pseudobulges in each galaxy. While we would ideally like to present an example or two of “pure disky pseudobulge” systems before moving on to the composite bulges, we have encountered difficulty in trying to identify any clear examples in the very nearby (e.g., $D \la 20$ Mpc), early-type disk galaxy population for which the necessary data (particularly stellar kinematics with the right combination of high spatial resolution and radial extent) exist. The problem is not so much identifying candidate disky pseudobulges in other nearby galaxies (as numerous others have done), but rather being able to clearly demonstrate that these are *not also* composite bulge galaxies: i.e., that there are no classical bulges, however small, inside these galaxies. The fact that we do not present any examples of S0–Sb disk galaxies with pure disky pseudobulges should not, however, be taken as a claim that such systems are absent in the local universe. Methods and Applications: Simple Classical Bulges {#sec:simple-classical} ================================================= Basic Methodology ----------------- How does one identify the “bulge” of a galaxy, and how does one determine whether such a structure is more like a classical bulge or a disky pseudobulge? Our basic approach has three steps. First, we identify the photometric bulge via a standard Sérsic + exponential decomposition of the entire galaxy. The outer boundary of the (photometric) “bulge-dominated” region is then identified by finding the radius [$R_{bd}$]{} where the Sérsic and exponential components are equal in brightness: $$\mu_{\rm Ser}({\ensuremath{R_{bd}}}) \; = \; \mu_{\rm exp}({\ensuremath{R_{bd}}})$$ Note that this radius may vary somewhat depending on the wavelengths of the data being used for the fit. Second, we analyse the morphology of the photometric bulge region, focusing especially on the shape of the isophotes. Our working assumption is that the photometric bulge and the outer disk share a common equatorial plane (i.e., they have the same line of nodes and, most importantly, the same inclination to the line of sight). This means that if the bulge is intrinsically rounder (more spheroidal) than the disk, its projected isophotes should appear rounder than those of the outer disk; in the extreme case of a spherical bulge, we would expect to see the elliptical isophotes of the (projected) outer disk give way to circular isophotes at small radii, where the bulge dominates the light. Finally, we analyse the stellar kinematics in the photometric bulge region, trying to determine whether they are more dominated by rotation or velocity dispersion. Traditionally, one way of using stellar kinematics to discriminate between classical bulges and pseudobulges [going back to @kormendy82] has been to note the position of the bulge in question on the $V/\sigma$–$\epsilon$ diagram [@illingworth77], where $V$ and $\sigma$ are the “characteristic” stellar velocity and velocity dispersion and $\epsilon$ is the ellipticity.[^2] One can define a curve in this diagram which corresponds to an “isotropic oblate rotator” (IOR), a simple model for a classical bulge or elliptical with isotropic velocity dispersion and possible flattening due to modest amounts of stellar rotation [e.g., @binney78; @binney05]. If the object clearly lies above the IOR curve, then the argument is that the object is too dominated by rotation to be considered a classical bulge. (Kinematically hot systems with little or no rotation but significant anisotropy will tend to lie below the IOR line.) This is one of the methods by which some of the original “pseudobulges” (avant la lettre) were identified [@kormendy82; @kormendy93]. Recent discussion of this diagram, primarily in the context of elliptical and S0 galaxies and taking advantage of 2D kinematics, include, e.g., @cappellari07, @spolaor10, and @emsellem11. There are, however, some problems with the $V/\sigma$–$\epsilon$ approach. The underlying theoretical arguments for the reference IOR models presuppose simple, coherent stellar systems with unique, unambiguous values for the ellipticity, velocity and velocity dispersion. The original application envisaged was elliptical galaxies, where at least the domain (the entire galaxy) is unambiguous. But in the case of complex systems such as a bulge embedded within a disk containing secondary structures (nuclear rings, bars, etc.), it is not at all clear how one is supposed to define “the” ellipticity; nor is it clear how to define “the” velocity dispersion when the latter can vary significantly with radius. Even the common technique of choosing the *maximum* stellar velocity as “the” velocity runs into trouble if the rotation curve continues to rise throughout the bulge region and on into the disk-dominated part of the galaxy (or if the rotation curve has multiple peaks). Faced with these difficulties, we opt for a different approach: we define a simple *local* measurement of the relative importance of rotation versus disperson, by deprojecting the observed rotation to its in-plane value $V_{\rm dp} = V_{\rm obs} / \sin i$ and then dividing this by the observed velocity dispersion $\sigma$ *at the same radius*. The resulting quantity – [$V_{\rm dp} / \sigma$]{} – is a continually varying function of the radius, not a “universal” value for an entire galaxy (or entire galactic component). We adopt an admittedly crude and ad-hoc limit of ${\ensuremath{V_{\rm dp} / \sigma}}= 1$ as the dividing line between kinematically “cool” and kinematically “hot” systems, so that classical bulges should have ${\ensuremath{V_{\rm dp} / \sigma}}< 1$ within the region where they dominate the galaxy’s light. In the following subsections we provide some partial justification for this criterion by showing that galaxies with simple disk + spheroidal bulge morphologies do seem to have ${\ensuremath{V_{\rm dp} / \sigma}}< 1$ within their bulge-dominated regions. Since we identify the photometric bulge region via decomposition of the major-axis profile, it makes sense to use major-axis value of [$V_{\rm dp} / \sigma$]{}. This lets us use major-axis long-slit spectroscopy, which is in many cases the only available stellar kinematic data – or the only data covering the full radial range of interest – for the galaxies we examine. To show how this approach works in the simple case of disk galaxies *without* pseudobulges, the following subsections apply our methodology to two S0s with classical bulges. Simple Classical Bulge Example: NGC 7457 {#sec:n7457} ---------------------------------------- NGC 7457 is a nearby ($D = 12.9$ Mpc)[^3] low-luminosity S0 galaxy, seen at moderate inclination ($i \approx 58\degr$; @gutierrez11); it has a central velocity dispersion of only $\sim 60$–70 [km s$^{-1}$]{} [e.g., @trager98; @wegner03; @ho09]. Although it has been suggested as a possible pseudobulge host in the past, largely on the basis of supposed deviations from the Faber-Jackson relation [e.g., @kormendy93; @pinkney03], more recent analyses clearly identify it as having a classical bulge [e.g., @fisher-drory08; @fisher-drory10; @kormendy11]. The upper left panel of Figure \[fig:n7457-complete\] shows the $V$-band isophotes for NGC 7457, based on an archival image from the Jacobus Kapteyn Telescope (JKT) of the ING [see @gutierrez11 for details]. The upper right panel shows our B/D decomposition of a major-axis cut which combines data from the $V$-band image with data from an HST WFPC2 F555W image (for $r < 6$ arcsec). The fit excluded both the outer part of the disk [which has an antitruncated profile; @gutierrez11] and the inner nuclear excess at $r < 0.35$ arcsec, which has been attributed to either AGN emission [@gebhardt03] or a nuclear star cluster [@graham09]. The bulge/disk crossover radius is at ${\ensuremath{R_{bd}}}= 6.2$ arcsec; the disk is clearly the dominant component at $r \ga 15$–20 arcsec. Turning to the isophotes shapes, the lower left panel of Figure \[fig:n7457-complete\] shows the results of fitting ellipses to both images. The ellipticity stays roughly constant in to $a \sim 12$ arcsec, and then becomes progressively rounder inside. This is consistent with the influence of a round bulge embedded within a highly elliptical (inclined) disk, so we have evidence that the photometric bulge identified in the B/D decomposition is a rounder (and thus more spheroidal) object than the disk. Finally, the lower right panel of Figure \[fig:n7457-complete\] shows [$V_{\rm dp} / \sigma$]{} as a function of radius along the major axis, using the long-slit kinematic data of @simien97c. Within the photometric bulge region, ${\ensuremath{V_{\rm dp} / \sigma}}$ is consistently $< 1$ (in fact, it never gets above $\sim 0.6$); it increases to larger radii, finally becoming $> 1$ at $r \ga 15$ arcsec. Note that [$V_{\rm dp} / \sigma$]{} continues to increase as we move into the (photometric and morphological) disk region, reaching values $\ga 2$ in the region which is unambiguously disk-dominated. As a crude approximation, then, we can identify “kinematically disklike” regions as having ${\ensuremath{V_{\rm dp} / \sigma}}> 1$. {width="6.8in"} Simple Classical Bulge Example: NGC 1332 {#sec:n1332} ---------------------------------------- With a $K$-band luminosity of $1.56 \times 10^{11}$ [$\mathrm{L}_{\sun}$]{} and a central velocity dispersion of 328 [km s$^{-1}$]{} [@rusli11], NGC 1332 is one of the most massive S0 galaxies in the nearby universe ($D = 22.3$ Mpc, from @tonry01 and @mei05); it also hosts a central supermassive black hole with a mass of $1.45 \times 10^{9}$ [$\mathrm{M}_{\sun}$]{} [@rusli11]. Figure \[fig:n1332-complete\] shows the overall morphology of the galaxy in panel a: a highly elliptical outer disk with a distinctly rounder inner zone. Decomposition of the major-axis profile (panel b, combining data from a ground-based $R$-band image with *HST* WFPC2 F814W data at smaller radii and data from our SINFONI $K$-band datacube at the very smallest radii to reduce the effects of circumnuclear dust extinction; see @rusli11 for details) shows a photometric bulge dominating the light at $r < {\ensuremath{R_{bd}}}= 12$ arcsec. As was the case for NGC 7457, the isophotes in the bulge-dominated region are clearly rounder (panel c) than those of the outer disk. Previous B/D decompositions for this galaxy (both 1D and 2D) are discussed in @rusli11. Of particular note is the fact that their 2D decomposition had a best fit using a Sérsic bulge component with ellipticity $= 0.27$, in contrast to the best-fitting exponential disk ellipticity of 0.73. This is clear support for the idea that the photometric bulge corresponds to a region which is significantly rounder than the disk. The slight twisting and rounding of isophotes between $a \sim 20$ and 40 arcsec may indicate a very weak bar or lens, but otherwise this galaxy is very close to an ideal exponential disk + Sérsic bulge system. Finally, panel d of Figure \[fig:n1332-complete\] shows the radial trend of ${\ensuremath{V_{\rm dp} / \sigma}}$, using data from @kuijken96 as re-reduced by @rusli11. ${\ensuremath{V_{\rm dp} / \sigma}}$ clearly reaches a plateau value ($\sim 0.5$–0.6) within the photometric bulge; as was the case for NGC 7457, the ratio only becomes $> 1$ outside the photometric bulge region. As in the case of NGC 7457, we conclude that the photometric bulge in NGC 1332 is a structure which is clearly rounder than the disk and has stellar kinematics dominated by velocity dispersion: in other words, a classical bulge. {width="6.0in"} Composite Bulges: Detailed Examples {#sec:n3945n4371} =================================== In this section, we turn to a set of galaxies whose photometric bulges show significantly more complex structure than was true for the two S0 galaxies considered in the previous section. Basic parameters for these galaxies are presented in Table \[tab:galaxies\]. We begin by examining two S0 galaxies – NGC 3945 and NGC 4371 – in detail, as paradigms for the analysis we apply to the whole set. Individual details and analysis for the other galaxies are discussed in the Appendix (Section \[sec:others-full\]). ---------- ------------------ ------ -------- ---------- ----- ------- -------------- Galaxy RC3 Type D Source $M_{B}$ PA $i$ [$R_{bd}$]{} Mpc deg deg arcsec (1) (2) (3) (4) (5) (6) (7) (8) NGC 1068 (R)SA(rs)b 14.2 1 $-21.23$ 86 31/40 24 NGC 1543 (R)SB(l)$0^0$ 20.0 2 $-20.12$ — 20? 26 NGC 1553 SA(r)$0^0$ 18.0 2 $-21.08$ 152 48 16 NGC 2859 R)SB(r)$0^{+}$ 24.2 1 $-20.21$ 85 32 52 NGC 3368 SAB(rs)ab 10.5 3 $-20.37$ 172 50 52 NGC 3945 (R)SB(rs)$0^{+}$ 19.8 1 $-19.94$ 158 55 17 NGC 4262 SB(s)$0^-$? 15.4 4 $-18.72$ 155 30 7.2 NGC 4371 SB(r)$0^{+}$ 16.9 4 $-19.49$ 90 58 32 NGC 4699 SAB(rs)b 18.9 1 $-21.33$ 35 37 44 ---------- ------------------ ------ -------- ---------- ----- ------- -------------- Basic characteristics of galaxies with composite bulges. Column 1: Galaxy name. Column 2: Hubble type . Column 3: Distance. Column 4: Source for distance: 1 = Virgocentric-infall-corrected redshift from HyperLeda; 2 = @tonry01, with correction from @mei05; 3 = @freedman01; 4 = @blakeslee09. Column 5: absolute $B$ luminosity (HyperLeda $B_{tc}$ + adopted distance). Column 6: adopted position angle of disk. Column 7: adopted inclination. Column 8: Radius where luminosity of Sérsic component = luminosity of exponential component from initial photometric decomposition (see text). ---------- ---------------------- --------- -------- ---------------------- -------- Galaxy Telescope/Instrument Filter Source Telescope/Instrument Source Imaging Spectroscopy (1) (2) (3) (4) (5) (6) NGC 1068 VLT-SINFONI (AO) $H$ 1 VLT-SINFONI (AO) 1 *HST*-NICMOS3 F200N Gemini-GMOS 2 2MASS $K$ KNPO-4m-RCFS 3 SDSS $i$ … NGC 1543 *HST*-WFPC2 F814W … *Spitzer*-IRAC IRAC2 … NGC 1553 *HST*-WFPC2 F814W CTIO-4m-RCS 4 *HST*-NICMOS2 F160W ESO-1.52m-B&C 5 *Spitzer*-IRAC IRAC1 … NGC 2859 *HST*-ACS/WFC F814W WHT-ISIS 6 WIYN-3.5m $R$ 7 WHT-SAURON 8 SDSS $i$ … NGC 3368 VLT-SINFONI (AO) $K$ 9 VLT-SINFONI (AO) 9 *HST*-NICMOS2 F160W HET-MLRS 10 WHT-INGRID $K$ 11 … SDSS $r$ … NGC 3945 *HST*-WFPC2 F814W *HST*-STIS 12 WIYN-3.5m $R$ 7 HET-MLRS 10 NGC 4262 *HST*-ACS/WFC F850LP WHT-OASIS 13 SDSS $i$ WHT-SAURON 14 NGC 4371 *HST*-ACS/WFC F850LP VLT-SINFONI (AO) 15 INT-WFC $r$ 16 WHT-ISIS 6 NGC 4699 VLT-SINFONI (AO) $K$ 15 VLT-SINFONI (AO) 15 SDSS $i$,$z$ Las Campanas-2.5m-MS 17 ---------- ---------------------- --------- -------- ---------------------- -------- Imaging and spectroscopic data used for the composite-bulge galaxies. For each galaxy, we list the data in order of decreasing spatial resolution (e.g., *HST* or AO data, followed by ground-based data). Column 1: Galaxy name. Column 2: Telescope + instrument or survey for imaging data (“AO” = adaptive optics used). Column 3: Filter used for imaging data. Column 4: Source of imaging data, if not from public telescope archives. Column 5: Telescope + instrument for spectroscopic/kinematic data. Column 6: Source of spectroscopic/kinematic data. References: 1 = @davies07; 2 = @gerssen06; 3 = @shapiro03; 4 = @kormendy84a; 5 = @longo94; 6 = this paper (Appendix \[sec:data-spec\]); 7 = @erwin-sparke03; 8 = @delorenzo-caceres08; 9 = @nowak10; 10 = @fabricius12; 11 = @knapen03; 12 = @gultekin09a; 13 = @mcdermid06; 14 = @emsellem04; 15 = Erwin et al., in prep.; 16 = @erwin08; 17 = @bower93. Composite Bulge Example: NGC 3945 --------------------------------- NGC 3945 is a double-barred S0 galaxy [@erwin99; @erwin04] which was originally singled out as an unusual object by @kormendy82; it had the largest value of [$(V_{\rm max} / \sigma_{0})^{\star}$]{}[^4] of any of the galaxies in his sample [see also Figure 17 of @kormendy04], indicating an unusually high degree of rotational support for a “bulge”. @erwin99 re-examined this galaxy using *HST* imaging. They pointed out that much of the inner region (i.e., the photometric bulge) appeared to be similar to a small exponential disk, complete with a nuclear bar surrounded by a stellar ring, embedded inside the main body of the galaxy, and that this region had a flattening roughly consistent with that of the outer disk. @erwin03-id revisited this analysis by performing B/D decompositions and a further analysis of the original kinematics of @kormendy82; they noted the existence of a separate photometric component within the inner 2 arcsec, with isophotes which were rounder than the main part of the photometric bulge (and the outer disk). Because NGC 3945 is such a paradigmatic case for the phenomenon of composite bulges, we repeat most of their analysis here, with the addition of *HST* STIS stellar kinematics from @gultekin09a which enables us to explore the kinematic status of the classical bulge; we also make use of new, large-scale kinematics obtained with the HET. {width="6.0in"} {width="6.0in"} ### Photometric Bulge as Disky Pseudobulge As we did for the pure-classical-bulge S0s in the previous section, we performed a simple B/D decomposition of the major-axis profile (panel b of Figure \[fig:n3945a\]). This is not entirely successful, since the outer disk is *not* a simple exponential; instead, it is distorted by the presence of a very luminous outer ring, which we exclude from the fit. (On the other hand, this profile largely avoids any contribution from the primary bar, which is oriented close to the galaxy minor axis.) None the less, there *is* a clear inner excess which dominates the light at $r \la 20$ arcsec; from our fit, we find ${\ensuremath{R_{bd}}}= 17$ arcsec. The fit in Figure \[fig:n3945a\] is somewhat different from the global $B/D$ decomposition presented in @erwin03-id, which used an ellipse-fit profile rather than a major-axis cut, and tried to fit the outer ring beyond 80 arcsec with the same exponential component. That fit increased the role of the Sérsic component (in part because it included light from the bar, which our cut largely avoids because the bar is oriented almost perpendicular to the major axis), moving [$R_{bd}$]{} out to $\sim 30$ arcsec; a similar ${\ensuremath{R_{bd}}}$ value can be seen in the ellipse-fit-based B/D decomposition of this galaxy in @fisher-drory10. If we use either of those fits instead, the photometric bulge region becomes larger, but the conclusions of our analysis remain unchanged. Thus far, our analysis of this galaxy has not shown any clear deviation from the simple classical-bulge systems discussed in Section \[sec:simple-classical\]. Something rather different does emerge, however, when we look at the isophote shapes in the photometric bulge region (panels c and e of Figure \[fig:n3945a\]). The ellipticity of this region is quite large: it reaches a maximum value of 0.36 at $a \approx 10$ arcsec, which is close to that of the outer disk. Given that these isophotes may still be distorted by light from the (primary) bar outside, it is possible that the underlying shape is actually the *same* as the outer disk; in any case, it suggests that much of the photometric bulge region is nearly as flat as the outer disk. Unsharp masking (panel d of Figure \[fig:n3945a\]) shows the signature of a stellar nuclear ring in this region (with $a \sim 6.5$ arcsec); additional unsharp masking also reveals the presence of a nuclear bar inside the ring [see Figure 2 of @erwin99]. This inner bar actually produces a *minimum* in the ellipticity at $a \approx 2.6$ arcsec because it is oriented close to the minor axis of the galaxy. The presence of these structures reinforces the idea that the photometric bulge of NGC 3945 is predominantly a disklike structure. Finally, panel f of Figure \[fig:n3945a\] shows [$V_{\rm dp} / \sigma$]{}, using our HET kinematics. In contrast to the simple classical-bulge galaxies studied above, [$V_{\rm dp} / \sigma$]{} reaches a peak value $> 2$ within the photometric bulge region. This is clear, dramatic evidence that the photometric bulge region is *not* a simple, kinematically hot structure; instead, the local stellar motions are sufficiently dominated by rotation as to resemble the *disk* regions of NGC 1332 and NGC 7457. Combined with the previous morphological evidence, this makes NGC 3945 one of the clearest cases of an S0 galaxy with a disky pseudobulge. ### Inner Photometric Bulge as Classical Bulge All of the foregoing is strong evidence that the photometric bulge of NGC 3945 is mostly, if not entirely, a disky pseudobulge. However, there is also good evidence that the innermost regions of the galaxy are dominated by a separate component, distinct from the pseudobulge. The inner part ($r < 20$ arcsec) of the major-axis surface-brightness profile (panel b of Figure \[fig:n3945b\]) has two interesting characteristics: first, most of the profile is very nearly a perfect exponential; second, the inner $r \la 2$ arcsec show a steep central excess. @erwin03-id pointed out that this profile appeared eerily similar to that of an exponential plus an inner Sérsic component, and proceeded to treat it in just that fashion. Panel b of Figure \[fig:n3945b\] produces a revised version of that fit, this time including the contribution from the lens/outer-disk (dashed green line in the bottom left corner of the upper panel). Although the details of the fit are slightly different from @erwin03-id, the result is still an excellent match to the profile; the small deviations at $r < 5$ arcsec are due to the presence of the inner bar and its surrounding nuclear ring. Given this *inner* decomposition, we can identify a new, much smaller photometric bulge region, with ${\ensuremath{R_{bd,i}}}= 1.1$ arcsec (we use [$R_{bd,i}$]{} to indicate an *inner* bulge = disk radius, in contrast to the [$R_{bd}$]{} value from the global decomposition of the previous subsection). The inner ellipse fits – particularly those of the HST image – show that the region inside ${\ensuremath{R_{bd,i}}}= 1.1$ arcsec is distinctly *rounder* than the main part of the disky pseudobulge (and rounder than the outer disk), with an ellipticity $\approx 0.21$ (panel c of Figure \[fig:n3945b\]). Photometrically and morphologically, then, we have evidence for a distinct central component, rounder than the outer disk *and* the disky pseudobulge. What about the stellar kinematics? Here, we make use of the HST-STIS kinematics published by @gultekin09a. Panel d of Figure \[fig:n3945b\] shows that [$V_{\rm dp} / \sigma$]{} reaches a plateau value of $\sim 0.5$ in the region $\la 1.1$ arcsec. Since the stellar kinematics of this central structure appear dominated by random motions, much like the larger classical bulges of NGC 1332 and NGC 7457 (above), we conclude that this inner structure is a (small) classical bulge. Composite Bulge Example: NGC 4371 --------------------------------- Like NGC 3945, NGC 4371 was noted by @kormendy82 for its rather large value of [$(V_{\rm max} / \sigma_{0})^{\star}$]{} – second only to NGC 3945 in his sample. The variation in ellipticity of the inner isophotes in this barred S0 galaxies led @wozniak95 to suggest that the galaxy might have a total of *three* bars: the obvious large outer bar and two more in the photometric bulge region. Using *HST* images, @erwin99 were able to show that the apparent signature of the inner two “bars” was actually the result of a bright, stellar nuclear ring distorting the isophotes [see also @erwin01-nr]. ### Photometric Bulge as Disky Pseudobulge Our B/D decomposition of the major-axis profile is shown in panel b of Figure \[fig:n4371a\]. The fit shows significant residuals, due in part to the presence of the nuclear ring at $a \sim 10$ arcsec; however, it does allow us to define the photometric bulge boundary as ${\ensuremath{R_{bd}}}= 25$ arcsec. (This is smaller than the bulge region defined by the decomposition of @fisher-drory10, presumably because their ellipse-fit-derived profile includes light from the bar, which our major-axis cut largely excludes.) As was the case for NGC 3945, a close-up of the photometric bulge region shows very elliptical isophotes interior to those defining the bar (panels c and e of Figure \[fig:n4371a\]); the peak ellipticity of $\sim 0.40$ is close to that of the outer disk. (Note that the plotted ellipticity in panel e peaks at 0.54 at $a \sim 140$ arcsec due to the presence of an outer ring; the true outer-disk ellipticity of 0.45 is determined from isophotes further out; see, e.g., Fig 3b of @erwin05.) Unsharp masking (panel d) shows the nuclear ring, which is a purely stellar phenomenon with no signs of gas, dust, or ongoing star formation, though @comeron10 did find that the ring has a slightly bluer colour than the surrounding light in their *HST* colour map. Unlike the case of NGC 3945, there is no evidence for a nuclear bar in this galaxy. The peculiarly box-shaped isophotes interior to the nuclear ring, which are visible at $a \sim 5$–7arcsec and which produce the local ellipticity minimum in the ellipse fits, can be explained as the side effect of adding the isophotes of an elliptical ring to those of a rounder structure inside [see @erwin01-nr]. Finally, analysis of our major-axis WHT-ISIS spectroscopy shows that the [$V_{\rm dp} / \sigma$]{} profile (panel f) has a peak of $\sim 1.5$ within the photometric bulge region – in fact, the peak is more or less at the radius of the nuclear ring. Once again, we have good evidence that the photometric bulge region is kinematically more like a disk than a classical spheroid, in addition to the clear morphological evidence for a disky pseudobulge. {width="6.0in"} ### Inner Photometric Bulge as Classical Bulge {#sec:n4371b} Although the inner profile is not as clean and simple as that of NGC 3945, we can still identify a clear central excess at $r \la 5$ arcsec. Panel b of Figure \[fig:n4371b\] shows a plausible decomposition, where we treat the inner disk + nuclear ring as a single component with a broken-exponential profile [@erwin08]. (See Erwin et al. 2014 for a more complex, 2D decomposition which yields similar results in terms of the classical bulge.) As we did for NGC 3945, we can define an inner value of ${\ensuremath{R_{bd,i}}}= 5$ arcsec, where the Sérsic component is brighter than the sum of the outer exponential plus the nuclear-ring/inner-disk component. The ellipticity of the *HST* isophotes interior to this radius (panel c of Figure \[fig:n4371b\]) is consistently $\approx 0.30$ (the variation at $a \approx 0.3$–0.6 arcsec is due to a circumnuclear dust ring, noted by Comerón et al. 2010). Moreover, the stellar kinematics for this region (panel d of the same figure) shows that [$V_{\rm dp} / \sigma$]{} reaches a plateau of $\sim 0.65$ within ${\ensuremath{R_{bd,i}}}= 5$ arcsec, so the kinematics of this region are dominated by velocity dispersion instead of rotation. In other words, the inner $r < 5$ arcsec of this galaxy appears to be dominated by a classical bulge. {width="6.0in"} Other Composite Bulges {#sec:others} ====================== Detailed discussion and analysis of the other seven composite-bulge galaxies can be found in Appendix \[sec:others-full\]. We note here that not all these galaxies are as clear-cut as the two just discussed (NGC 3945 and NGC 4371). For example, we lack kinematic data with high enough spatial resolution to properly resolve the interior of the classical-bulge region for some of those galaxies. In most such cases, however, [$V_{\rm dp} / \sigma$]{} is significantly $< 1$ in the (resolved) region just *outside* the classical bulge, so it is unlikely that ${\ensuremath{V_{\rm dp} / \sigma}}$ is actually $> 1$ inside the classical bulge. For some galaxies, we use profiles from position angles other than the galaxy major axis, especially when strong nuclear bars are present. This includes profiles perpendicular to nuclear bars (when present), to minimize its contribution, as well as alternate cases where we use a profile along the nuclear-bar major axis, including an extra component to account for the bar contribution. The results of these analyses, combined with the previous ones for NGC 3945 and NGC 4371, are presented in Tables \[tab:disk-features\]–\[tab:BtoT\]. ---------- --------------- ------------------------------------------- ----------------- --------------------- ---------- --------- --------------------------- Galaxy Exp. Profile? Max$({\ensuremath{V_{\rm dp} / \sigma}})$ Disky Features $\mu_{0}$ $h$ $\log M_{\star}$ (mag arcsec$^{-2}$) (arcsec) (pc) ([$\mathrm{M}_{\sun}$]{}) (1) (2) (3) (4) (5) (6) (7) (8) NGC 1068 Y 2.80 bar, spirals 13.12 ($K$) 6.80 468 10.89 NGC 1543 Y —$^{1}$ bar, NR 17.15 ($I$) 7.03 665 9.84 NGC 1553 Y? 1.38 bar, NR/spirals 15.31 ($I$) 5.33 466 10.57 NGC 2859 Y 1.56 bar, NR/spirals 16.88 ($i$) 4.45 522 10.49 NGC 3368 Y 1.36 bar, spirals 12.71 ($K$) 3.05 149 9.85 NGC 3945 Y 2.18 bar, NR 15.69 ($I$) 5.11 491 10.43 NGC 4262 Y 1.58 NR 14.45 ($i$) 1.66 124 10.09 NGC 4371 Y\[\*\] 1.50 NR —$^{2}$ —$^{2}$ —$^{2}$ 9.88 NGC 4699 Y 1.36$^{3}$ bar, spirals 15.83 ($z$) 7.61 697 10.97 ---------- --------------- ------------------------------------------- ----------------- --------------------- ---------- --------- --------------------------- Column 1: Galaxy name. Column 2: Indicates whether surface-brightness profile is exponential. Column 3: Maximum value of [$V_{\rm dp} / \sigma$]{} in the disky pseudobulge. Column 4: “Disky” features found in the pseudobulge region (NR = nuclear ring); note that for NGC 1068 and NGC 3368, the bars are only seen clearly in the near-IR. Column 5: Central surface brightness of exponential component of disky pseudobulge; band is listed in parentheses. Columns 6–7: Exponential scale length of same, listed in both angular and linear sizes. Column 8: Logarithm of estimated stellar mass of disky pseudobulge (including nuclear bar components; see text). Notes: 1 = galaxy too close to face-on to reliably deproject velocities; 2 = In NGC 4371, the disky pseudobulge profile is distorted by the strong nuclear ring, and is not modeled as a simple exponential (see Section \[sec:n4371b\]); 3 = kinematics do not extend to edge of pseudobulge, so this may be only a *lower* limit on Max$({\ensuremath{V_{\rm dp} / \sigma}})$. ---------- ------ ---------- ----------- --------------------- ------------------ --------------------------- Name $n$ $\mu_{e}$ $\epsilon$ $\log M_{\star}$ (arcsec) (pc) (mag arcsec$^{-2}$) ([$\mathrm{M}_{\sun}$]{}) (1) (2) (3) (4) (5) (6) (7) NGC 1068 0.98 0.50 34 11.05 ($K$) 0.14 9.47 NGC 1543 1.50 2.73 258 17.02 ($I$) 0.05 9.49 NGC 1553 1.66 1.48 129 15.73 ($I$) 0.1 9.32 NGC 2859 1.68 1.06 124 17.00 ($i$) 0.15 9.47 NGC 3368 1.34 0.63 31 13.38 ($K$) 0.00 8.73 NGC 3945 2.02 1.24 119 16.20 ($I$) 0.20 9.63 NGC 4262 0.89 0.30 23 15.02 ($i$) 0.05 8.70 NGC 4371 2.18 5.20 427 18.70 ($z$) 0.30 9.64 NGC 4699 1.43 2.15 247 15.42 ($z$) 0.11 10.46 ---------- ------ ---------- ----------- --------------------- ------------------ --------------------------- Characteristics the of the classical bulges in our composite-bulge galaxies. Column 1: Galaxy name. Columns 2–5: Parameters of S[é]{}rsic fit; effective radius is given in both angular and linear sizes, and the band of the $\mu_{e}$ value is in parenthesis (these are usually based on the reddest available images, which vary from galaxy to galaxy). Column 6: Adopted mean isophotal ellipticity of classical bulge. Column 7: Logarithm of estimated stellar mass (see text). ---------- ---------------------------- ------------------------------ ------------------ ---------- Name $B/T_{\star, \mathrm{cl}}$ $B/T_{\star, \mathrm{phot}}$ $B/T_{L}$ (lit.) Source (1) (2) (3) (4) (5) NGC 1068 0.021 0.40 0.11 L10 NGC 1543 0.10 0.00 0.34 L10 NGC 1553 0.031 0.41 0.23 L10 NGC 2859 0.070 0.38 0.29, 0.47 L10, F12 NGC 3368 0.024 0.54 0.26, 0.41 F11, F12 NGC 3945 0.045 0.37 0.39, 0.36 L10, F12 NGC 4262 0.045 0.65 0.55 L10 NGC 4371 0.093 0.27 0.22, 0.38 L10, F12 NGC 4699 0.089 0.71 0.34 L04 ---------- ---------------------------- ------------------------------ ------------------ ---------- : $B/T$ Values[]{data-label="tab:BtoT"} Bulge-to-total values for composite-bulge galaxies. Column 1: Galaxy name. Column 2: Ratio of classical bulge stellar mass to total galaxy stellar mass. Column 3: Ratio of photometric bulge stellar mass to total galaxy stellar mass, from simple 1-D decomposition. Column 4: Near-IR $B/T$ luminosity values from the literature. Column 5: Sources for values in Column 4 – L04 = 2D $K$-band decompositions of @laurikainen04b; L10 = 2D $K$-band decompositions of @laurikainen10; F11 = “near-IR” 1D decompositions of @fisher-drory11; F12 = $H$-band 1D decompositions of @fabricius12. Stellar Dynamics of Classical Bulges and Disky Pseudobulges {#sec:dynamics} =========================================================== {width="6.0in"} Three of the composite-bulge galaxies, along with one of the example pure-classical-bulge galaxies (NGC 1332), have been observed with the SINFONI IFU during a project measuring supermassive black hole masses in galaxy centres [@nowak07; @nowak08; @nowak10; @rusli11; @rusli13a; @erwin14-smbh]. As part of the analysis, we perform Schwarzschild orbit-superposition modeling in order to reproduce both the stellar light distribution and the observed stellar kinematics; see @thomas04, @nowak10 and @rusli11 for details. Briefly stated, this involves constructing a galaxy potential from the combination of a central SMBH and one or more deprojected, 3D luminosity-density distributions (e.g., a components for the classical bulge and one for the disky-pseudobulge+outer-disk), converted to 3D stellar mass density distributions via a stellar mass-to-light ($M/L$) ratio. Trial values of the SMBH mass and the stellar $M/L$ ratios are assigned, and then several tens of thousands of sample orbits are integrated in the potential. The resulting orbit library is weighted to produce the best match to the observed light distribution and the observed stellar kinematics, and the process is then iterated with new values of $M/L$ and SMBH mass to map out the $\chi^{2}$ landscape. The end result, in addition to best-fit values for the stellar $M/L$ ratios and the SMBH mass, is a library of stellar orbits and corresponding weights which can be used to investigate the phase-space distribution of the stars and to look for things such as radial trends in 3D stellar dynamical quantities. We have previously made use of best-fit orbit libraries to study radial anisotropy trends in core and non-core elliptical galaxies [@thomas14]; here, we use the best-fit models for the S0 NGC 1332 (which has only a classical bulge) and the composite-bulge galaxies NGC 3368, NGC 4371 and NGC 4699 to explore how the models might shed light on the internal 3D kinematics of composite bulges. Data and modeling for NGC 1332 and NGC 3368 have already been presented in @rusli11 and @nowak10, respectively; full data and modeling results, including SMBH measurements, for the other galaxies will be presented elsewhere [@erwin14-smbh]. Once the best-fitting model is determined, stellar-dynamical quantities can be extracted using the weighted means of orbits in different radial and angular bins. Working in cylindrical coordinates $(R,\varphi,z)$, we compute the velocity anisotropy by comparing the vertical dispersion $\sigma_{z}^{2}$ with the mean velocity dispersion in the equatorial plane $\sigma_{e}^{2}$; the latter is defined as $$\sigma_{e}^2 = (\sigma_{R}^2 + \sigma_{\varphi}^2)/2 \, .$$ All of these values are averages of the orbits over angular bins running from $\theta = -23\degr$ to $\theta = +23\degr$ with respect to the equatorial plane at each radius, using the orbit weights from the best-fitting solution. We then define the anisotropy $\beta_{\rm eq}$ as $$\beta_{\rm eq} \; = \; 1 \: - \: \sigma_{e}^{2} / \sigma_{z}^{2} \, ;$$ this is $\sim 0$ for the isotropic case and $< 0$ for planar-biased anisotropy. For comparison, $\beta_{\rm eq} \approx -1.0$ for the Galactic disk in the Solar neighborhood [using $z = 0$ values from Table 1 of @bond10], or $-1.2$ for the Galactic thick disk [using dispersions from Table 5 of @carollo10]. Figure \[fig:aniso\_figure\] shows the radial trend of the anisotropy for the simple classical-bulge S0 NGC 1332 and for three composite-bulge galaxies. We also plot the radial trends of $V_{\varphi} / \langle \sigma \rangle$, where $V_{\varphi}$ is the intrinsic mean rotation velocity measured in the two bins closest to the equatorial plane and $\langle \sigma \rangle$ is the total velocity dispersion: $$\langle \sigma \rangle = \sqrt{(\sigma_{R}^2 + \sigma_{\varphi}^2 + \sigma_{z}^2)/3} \, .$$ For all galaxies, the anisotropy parameter $\beta_{\rm eq}$ is $\sim 0$ in the classical-bulge region, and decreases outside, indicating a shift from isotropic velocity dispersion to a dispersion which is dominated by planar motions. The latter is what we expect for flattened, disklike structures, and supports the idea that the disky pseudobulges are indeed dynamically distinct from the classical bulges. In most cases, the [$V_{\varphi} / \langle\sigma\rangle$]{} profiles show a trend similar to what we have seen in the major-axis [$V_{\rm dp} / \sigma$]{} profiles: dispersion-dominated kinematics within the classical-bulge region and rotation-dominated kinematics in the disky pseudobulge (or main disk in the case of NGC 1332) outside. The exception is NGC 4371, where there is also an *inner* peak in [$V_{\varphi} / \langle\sigma\rangle$]{} at $r \sim 0.4$ arcsec, deep within the classical-bulge region. Curiously, the radius where [$V_{\varphi} / \langle\sigma\rangle$]{} reaches its local maximum is also where the isophotal ellipticity has a local maximum (panel c of Figure \[fig:n4371b\]), though *HST* colour maps indicate that this is also a region marked by strong circumnuclear dust [see @comeron10]. This might represent the existence of an additional disky component with $r \sim 30$ pc deep inside the classical bulge. Discussion {#sec:discussion} ========== In the preceding sections of this paper (and in the Appendix) we have provided a kind of existence proof demonstrating that at least some lenticular and early-type spiral galaxies can host *both* disky pseudobulges *and* compact classical bulges, with the latter nestled inside the former. This shows that classical bulges and pseudobulges are not always exclusive phenomena. The disky pseudobulges can almost always be described with exponential profiles,[^5] with the addition of various disk-like features: nuclear bars, stellar rings, or spiral arms. Our major-axis decompositions yield exponential scale lengths of 125–870 pc, with a mean of 440 pc. These are relatively massive structures, with stellar masses ranging from $7.1 \times 10^{9}$ to $9.4 \times 10^{10}$ [$\mathrm{M}_{\sun}$]{} (mean = $3.3 \times 10^{10}$ [$\mathrm{M}_{\sun}$]{}), or anywhere from 11 to 59 per cent of the total galaxy stellar mass (mean fraction = 33 per cent). Thus in many cases a significant fraction of the galaxy’s stars are part of the disky pseudobulge. (See Section \[sec:pb-plots\] for details on the computation of the disky-pseudobulge stellar masses.) The classical bulges, in contrast, are relatively compact, low-mass structures. Our fits yield Sérsic indices of 0.89–2.18 (mean index = 1.52), half-light radii between 23 and 426 pc (mean $R_{e} = 143$ pc), and stellar masses ranging from $5.0 \times 10^{8}$ to $2.9 \times 10^{10}$ [$\mathrm{M}_{\sun}$]{}. (See Section \[sec:cb-plots\] for details of the stellar-mass estimation.) In only two galaxies is the stellar mass of the classical bulge more than 10 per cent of the galaxy’s total stellar mass (NGC 1543 and NGC 4699, where it is 13.1 and 11.3 per cent, respectively), and the mean value is only 5.9 per cent. Hints Concerning the Frequency of Composite Bulges {#sec:frequency} -------------------------------------------------- How common are composite-bulge systems? The disadvantage of an existence-proof study such as this one is that it can do little, by itself, to answer this question. The galaxies discussed in this paper are unfortunately not drawn from any well-defined sample which has been consistently analysed with the same degree of spatial resolution in both imaging and spectroscopy, primarily because the necessary combined data (particularly high-resolution spectroscopy for stellar kinematics) are not available for most nearby galaxies. We *can* note that most of our galaxies are part of a sample of early-type barred galaxies originally studied by @erwin-sparke03 and expanded by @erwin05, which does allow us to put some very crude lower limits. Of the 25 barred S0 galaxies in the aforementioned sample with $i > 30\degr$, we can identify three which are composite bulge hosts (NGC 2859, NGC 3945 and NGC 4371). Of the 34 barred S0/a–Sb barred galaxies with the same inclination cutoff, there are three more galaxies from our composite-bulge set (NGC 1068, NGC 3368 and NGC 4699). This suggests that *at least* $\sim 10$ per cent of S0–Sb barred galaxies are probably composite-bulge systems. The true frequency could be much higher – though it clearly cannot be 100 per cent, as there are clearly some galaxies with classical bulges but no disky pseudobulges (e.g., NGC 1332 and NGC 7457, Sections \[sec:n7457\] and \[sec:n1332\]). If we take the presence of nuclear bars as indicators of disky pseudobulges, as is often done [e.g., @kormendy04; @fisher-drory08], then the recent review of @erwin11 suggests that disky pseudobulges (with or without classical bulges) can be found in at least 20 per cent of early-type (S0–Sab) disk galaxies. The presence of nuclear bars as indicators of disky pseudobulges also means that weaker, less luminous disky pseudobulges may be lurking within more dominant classical bulges. For example, at least four of the galaxies classified as having classical bulges in the recent studies of @fisher-drory08 and @fisher-drory10 have nuclear bars: NGC 3031, NGC 3992, NGC 4548 and NGC 6684 [@elmegreen95; @erwin04; @gutierrez11; @erwin14-db]. @fisher-drory10 argued that galaxies with composite bulges would not be clearly distinguishable from galaxies with only a disky pseudobulge when using 1-D surface-brightness profiles, unless the classical bulge were very luminous; they suggested that some of the galaxies with what they called “inactive pseudobulges”[^6] might harbor classical bulges as well, if the latter had relatively small Sérsic indices (e.g., $\la 3$). This is exactly what we find: all of our classical-bulge components have Sérsic indices $\la 2.2$. Of the four galaxies in common with their sample (NGC 1543, NGC 3368, NGC 3945 and NGC 4371), all are classified by Fisher & Drory as “inactive pseudobulges” based on the mid-IR colours. It is worth noting that their 1-D decompositions produced Sérsic indices for the *photometric bulges* of $n < 2$ for all but one of these galaxies (NGC 4371), so one cannot conclude from the Sérsic index of the photometric bulge alone that a galaxy is *lacking* a classical-bulge component. Embedded Classical Bulges Compared with Other “Spheroid” Systems {#sec:cb-plots} ---------------------------------------------------------------- Table \[tab:classical-parameters\] presents the structural characteristics of the classical bulges in our composite-bulge systems. The listed colours come from aperture photometry on SDSS images [or *HST* images in the case of NGC 3945; @erwin03-id]; we use the colour-to-$M/L$ calibrations of @bell03 to calculate the resulting stellar masses. The exception to this is NGC 1068, where the bright AGN point source makes accurate stellar colour measurements in the inner few arc seconds very difficult. Instead, we adopted a $K$-band $M/L$ ratio of 0.7, using the lower end of the age estimate (5–12 Gyr) from the near-IR spectroscopy of @storchi-bergmann12 and the average $M/L$ ratios of @longhetti09.[^7] In this paper, we refer to these compact, inner components as “classical bulges” because they fulfill at least some of the standard criteria for classical bulges. The fact that they have rounder isophotes than the outer disk, yet share similar or identical position angles, suggests they are approximately oblate spheroids (with little triaxiality) which are rounder (in an edge-on, vertical sense) than the disk. In addition, we have clear evidence that the classical bulges are kinematically hot, with [$V_{\rm dp} / \sigma$]{} always $< 1$, in the cases of NGC 1068, NGC 3368, NGC 3945, NGC 4371 and NGC 4699, where the available kinematic data comes from observations with high enough spatial resolution to resolve the classical bulge region. In other galaxies, such as NGC 2859. The one exception to the usual classical-bulge paradigm is in the luminosity profiles. Traditionally, classical bulges have been claimed to have $R^{1/4}$ profiles – i.e., a Sérsic index of $n = 4$. More recent formulations have suggested an approximate dividing line of $n = 2$, with classical bulges having higher values and pseudobulges (of whatever kind) having lower [e.g., @fisher-drory10]. Almost all of the objects we find have $n < 2$; only for NGC 3945 and NGC 4371 is $n \ga 2$. So one *could* argue, purely on the basis of the Sérsic indices, that most of our central structures are a peculiar species of round, kinematically hot “pseudobulge.” However, we prefer to consider the possibility that classical bulges can have a range of surface-brightness profiles, and that their primary characteristics remain their overall shape, their lack of disky substructure, and pressure-dominated stellar kinematics. There are other kinematically hot, round stellar systems which are not considered classical bulges, of course. As we have seen, the classical bulge parts of composite bulges span a range of sizes, with half-light radii from $\sim 400$ down to $\sim 25$ pc. The lower end of this range is sufficiently small that one might wonder whether some of these objects would really be better understood as “nuclear star clusters” (NSCs), which are extremely common in late-type spirals and at least moderately common in earlier-type disks [see, e.g., the review by @boker08]. In Figures \[fig:re-vs-mstar\] and \[fig:mue-vs-mstar\], we plot the classical bulge components of our composite bulges in the context of other “spheroids” – that is, stellar systems dominated by velocity dispersion – ranging from NSCs, dwarf spheroidals and ultracompact dwarfs to giant ellipticals. For the latter three classes of objects, we use the aggregated data compiled by @misgeld11. To those data we have added the (photometric) bulges of S0–Sa galaxies from Table 1 of @laurikainen10 (red stars). These are more modern measurements based on 2D decompositions of near-IR images, which helps ensure that light belonging to the bar is not counted as part of the bulge. (We note that these decompositions do not account for the possibility of composite bulges, and we exclude the seven galaxies in their sample which are studied in this paper.) We converted their $K$-band measurements to stellar masses using the $M/L$ ratios of @bell03 and colours from HyperLEDA. In particular, we used $(B - V)_{e}$ colours, which are a better approximation to the bulge colours than the total galaxy colour.[^8] We also include the bulges of unbarred S0–Sbc galaxies from @balcells07a, converting their $K$-band magnitudes to stellar masses in the same fashion (green stars). Finally, we plot NSCs using a compilation of masses from @erwin-gadotti12 and half-light radii primarily from @boker04 and @walcher05, with some additional values from @ho96, @graham09, @barth09, @kormendy10b and @seth10. What Figure \[fig:re-vs-mstar\] shows is that the structures we identify as classical bulges in the composite-bulge systems fall into the “bulge” section of the plot, not the “nuclear cluster” section. Elliptical galaxies[^9] and classical bulges form a clear, relatively tight sequence in the $R_{e}$–$M_{\star}$ plane, and our classical bulges either fall directly into this sequence or form a natural extension of it. Similarly, in Figure \[fig:mue-vs-mstar\], our classical bulges fall into the same (loose) sequence formed by giant ellipticals and larger classical bulges. Although the smallest classical bulges are undeniably small, and begin to approach the most massive NSCs in *size* (half-light radius), they are significantly more massive. Moreover, in at least two of the composite-bulge galaxies (NGC 1543 and NGC 1553) there is evidence for distinct nuclear star clusters with half-light radii $\sim 4$ pc located *inside* the classical bulge components. While we cannot rule out the possibility of an as-yet undetected population of central spheroids with masses $\sim 10^{8}$ [$\mathrm{M}_{\sun}$]{} and $R_{e} \sim 10$–20 pc, which would provide continuity between the classical bulges and NSCs, for the time being we consider them to be distinct phenomena. {width="6.5in"} {width="6.5in"} Disky Pseudobulges in the Mass-Size-Density Planes {#sec:pb-plots} -------------------------------------------------- As we have seen, the classical-bulge components of our composite-bulge systems appear to follow the same trends as (larger) classical bulge and ellipticals, at least in the $R_{e}$–$M_{\star}$ and $\langle \Sigma_{\star} \rangle_{e}$–$M_{\star}$ planes. What about the disky pseudobulges: do they resemble large-scale disks, classical bulges, or something else? Figure \[fig:re-vs-mstar-pseudo\] and \[fig:mue-vs-mstar-pseudo\] show where the disky pseudobulges (filled blue diamonds) fall in the same diagrams used for Figures \[fig:re-vs-mstar\] and \[fig:mue-vs-mstar\]; we have desaturated the previously plotted points to allow the pseudobulge data points to stand out more clearly. We also plot the (large-scale) disk components from the S0/spiral decompositions of @balcells07a and @laurikainen10 as filled grey diamonds. The stellar masses of our disky pseudobulges – and the main disks from the @laurikainen10 and @balcells07a samples – plotted in Figures \[fig:re-vs-mstar-pseudo\] and \[fig:mue-vs-mstar-pseudo\] come from the same general colour-to-$M/L$ estimates [based on @bell03] as used for the classical-bulge components in the preceding section. For most of the disky pseudobulges, we use the exponential component from the inner decomposition (e.g., panel b of Figure \[fig:n3945b\]) and assume that the disky pseudobulge is an exponential disk with an ellipticity equal to that of the outer disk. (For NGC 4371, we use the plotted broken-exponential fit from panel b of Figure \[fig:n4371b\].) In the cases of NGC 1068, NGC 1543 and NGC 2859, where the inner fits specifically exclude the contribution of bright inner bars, we include an additional flux term to account for the inner bar itself [e.g., @erwin11]. The half-light radii are computed using the best-fitting exponential scale lengths from the inner decompositions (e.g., panel b of Figure \[fig:n3945b\]) – i.e., excluding any contribution from the classical bulge component. (The half-light radii for the large-scale disks are also estimated this way.) For disky pseudobulges with significant inner bars, this may mis-estimate the true half-light radius, but probably by less than a factor of 2, which means a very small shift on the plots. What we can see from Figures \[fig:re-vs-mstar-pseudo\] and \[fig:mue-vs-mstar-pseudo\] is that in terms of size, stellar mass, and mean density, disky pseudobulges actually fall into the bulge/elliptical sequence, just like the classical bulges (albeit with masses and sizes that are on average larger than the classical bulges). Although some of the disky pseudobulges are as massive as large-scale stellar disks, they are roughly a factor of $\sim 5$ times more compact. This suggests that their formation mechanisms are probably not the same as those involved in large-scale disks. It also is an indication that otherwise unclassified objects which fall on the classical-bulge/elliptical sequence in the mass-size and mass-density planes may not always be kinematically hot spheroidal systems. (We note in passing that the precise location and distribution of large-scale disks on plots such as these may depend partly on Hubble type. For example, disks in very late type spirals may lie further to the left in the $R_{e}$–$M_{\star}$ plane; see, e.g., Fig. 9 of @laurikainen10 or Figs. 18 and 20 of @kormendy12.) {width="6.5in"} {width="6.5in"} What About Boxy/Peanut-Shaped Pseudobulges? {#sec:boxy} ------------------------------------------- @athanassoula05 argued that instead of lumping the central regions of disk galaxies into just two types – classical bulges versus pseudobulges – one could, from a theoretical point of view, distinguish at least three possible structures: classical bulges, “disc-like bulges” (i.e., what we have been calling disky pseudobulges), and boxy/peanut-shaped bulges, which are actually the vertically thickened inner parts of bars. She even suggested that some galaxies might contain all three at the same time. In this paper, we have identified a number of disk galaxies containing *both* classical bulges and disky pseudobulges; what about boxy/peanut-bulges? We have avoided dealing with the latter in order to concentrate on the contrasting disky and spheroidal nature of the pseudobulges and classical bulges we are interested in; moreover, the traditional approach to identifying boxy/peanut-shaped bulges relies on galaxies which are edge-on, not moderately inclined. None the less, all but one of the galaxies discussed in this paper are barred. The extensive survey of edge-on galaxies by @lutticke00a found that the frequency of boxy and peanut-shaped bulges was consistent with most if not all barred galaxies having a vertically thickened inner region (a box/peanut structure, or B/P structure). @bureau06 identified a number of edge-on galaxies which appeared to have both B/P structures *and* disky pseudobulges, so the coexistence of those two structures is certainly plausible. (Due to low spatial resolution and the presence of strong dust extinction in the central regions, small embedded classical bulges would have been difficult to find in their galaxies, if any such were present.) More recently, @erwin-debattista13 identified several morphological signatures of the B/P structure in barred galaxies which can be seen when the galaxy is only moderately inclined (i.e., $i \sim 40$–70), if the bar is favorably aligned with respect to the disk major axis. Since NGC 1068, NGC 1543, NGC 2859 and NGC 4262 are close to face-on (mostly $i \la 30\degr$), we would not expect to see signatures of the B/P structure. Additionally, the bars in NGC 3945 and NGC 4371 are oriented very close to the minor axes of their respective host galaxies. The latter orientation is one which minimizes the signature of the B/P structure; @erwin-debattista13 did not find this signature in those three galaxies. However, they *did* identify NGC 3368 as an example of a galaxy with a visible projected B/P structure; as we noted in Section \[sec:n3368\], this is probably responsible for the slightly boxy isophotes *outside* the disky pseudobulge – see Figure \[fig:n3368-boxy\]. Thus, NGC 3368 is a clear case of a galaxy matching the suggestion of @athanassoula05 that disk galaxies can simultaneously host classical bulges, disky pseudobulges, *and* box/peanut bulges. ![Disky pseudobulge inside a boxy/peanut-shaped bulge. Median-smoothed $K$-band isophotes for the interior of the outer bar of NGC 3368 (see Figure \[fig:n3368a\] for larger-scale context). The thicker green contour indicate the region of boxy isophotes, which are probably the projection of the inner, vertically thickened part of the bar – a boxy/peanut-shaped bulge. The region inside which is dominated by the disky pseudobulge is indicated by the thicker red contour. See also Fig. A1 of @erwin-debattista13.[]{data-label="fig:n3368-boxy"}](Erwin_fig12){width="3.2in"} Speculations on Formation ------------------------- ### Morphology: Disky Pseudobulges Are (Mostly) Found Inside Bars {#sec:pb-formation} Most discussions of pseudobulges [see, e.g., @kormendy04] portray them as secondary structures formed late in a galaxy’s history, usually via some form of bar-driven gas inflow and subsequent star formation in the central regions of the galaxy. The fact that almost all the galaxies discussed in this paper are barred is at least broadly consistent with the hypothesis of bar-driven formation. What is perhaps not yet clear is whether bar-driven inflow and star formation can readily create disky pseudobulges as large and as massive as some of those we identify. A possible additional issue is the fact that star-formation inside the bars of early-type disks in the local universe is usually observed taking place in nuclear *rings*. Several of our disky pseudobulges do feature nuclear rings (Table \[tab:disk-features\]), which argues for a connection; the question is whether the rest of the disky pseudobulge, both inside and outside the ring, was formed the same way. @wozniak09 studied a high-resolution disk galaxy simulation and noted the formation of an extended “nuclear disk” inside the bar, due to gas-driven bar inflow and star formation. This disk apparently gave rise to an independently rotating nuclear bar, with a gaseous nuclear ring of radius $\sim 400$ pc. The reported size of the disk – $\sim 500$ pc after 2 Gyr – is consistent with some of our disky pseudobulges, and the mass ($7 \times 10^9$ [$\mathrm{M}_{\sun}$]{}, 34 per cent of the galaxy’s total stellar mass) is as well. More recently, @cole14 have reported on a detailed disk-galaxy simulation in which a similar large nuclear disk formed inside a bar, and compared it to the disky pseudobulges in NGC 3368, NGC 3945, and NGC 4371. The nuclear disk in the simulation was more extended (relative to the size of the bar) than is the case for the three disky pseudobulges, but did have a similar mass fraction (29% of the simulated galaxy’s total stellar mass). So it appears that at least some detailed, high-resolution simulations *can* produce barred galaxies with disky pseudobulges similar to those seen in real galaxies. We note that three of the composite-bulge galaxies do pose some more general problems for the bar-driven formation scenario, because their bars are weak, missing, or simply too small. NGC 1068, though formally unbarred, does at least appear to have a large-scale bar surrounding its disky pseudobulge [@erwin04 and references therein]; however, this bar is quite weak, and is sometimes considered an “oval disk” rather than a true bar [@kormendy79b; @kormendy04]. It is not clear whether such a large disky pseudobulge (44 per cent of the NGC 1068’s stellar mass) would be formed by such a weak bar. NGC 1553 is classified as unbarred, though it might be possible that its “lens” is either the remnant of a once-strong bar [as argued by, e.g., @kormendy04], or is the projection of the B/P structure of a still-existing, albeit rather weak, bar (see Section \[sec:n1553\]). The most difficult case is NGC 4699, where the disky pseudobulge is *larger* than the sole bar in the system. This implies that not *all* disky pseudobulges can be formed by bar-driven gas inflow. The alternative would be to postulate a previous, large bar in NGC 4699 which has since vanished. Unfortunately, theories of bar destruction either require unnaturally high central mass concentrations (which should prevent the existence of the *current* small bar in this galaxy), or don’t yield testable predictions that would allow us to unambiguously identify whether a galaxy once had a bar, as opposed to never having had one. A different formation scenario, which could potentially account for cases like NGC 4699, might be that some disky pseudobulges form *early*, possibly before any large-scale bar, as part of a general inside-out process. For example, @guedes13 have presented a cosmologically motivated disk-galaxy simulation which formed with a “pseudobulge” inside a larger disk; their analysis showed that most of the stars in the pseudobulge formed *in situ* and *prior to* most of the outer disk. One way of distinguishing between these scenarios would be to look for evidence of significant differences in ages between the stars making up the disky pseudobulge and those making up the bar (if present) or disk outside. Younger ages in the disky pseudobulge would argue for the bar-driven formation mechanism, while a predominantly *older* population in the disky pseudobulge would suggest something more like the inside-out scenario of @guedes13. ### Possible Clues from Stellar Populations Unfortunately, we are rather lacking in detailed stellar population analyses for most of our composite-bulge galaxies. Published analyses tend to be patchy and limited, either spatially or in terms of models with multiple populations. De Lorenzo-C[á]{}ceres et al. (2012, 2013) presented an analysis of stellar populations near the centres of five double-barred galaxies, one of which is the composite-bulge system NGC 2859. They found minimal age differences between the stars of the inner and outer bars, with the former being either coeval with or slightly younger than the latter. This seems to suggest that the inner bars – and by implication the rest of the disky pseudobulges – in these galaxies could indeed be (somewhat) younger than the outer bars, and thus formed after the outer bar formed, in agreement with the bar-driven formation model. For the classical bulges, the existing data are (perhaps) contradictory. In the case of NGC 2859, @delorenzo-caceres13 found that they youngest part of the central region of NGC 2859 was what we identify as the classical bulge. While this could be interpreted as evidence that classical bulges form *after* the disky pseudobulges, one should note that de Lorenzo-Caceres et al. measured luminosity-weighted ages from comparisons with single stellar population (SSP) models, so there is the possibility that the measured ages are contaminated by more recent stellar populations. The relevance of this concern was shown by @sarzi05, who used *HST* STIS spectroscopy to estimate the stellar populations in the central $0.2 \times 0.25$ arcsec of a number of galaxies, including the composite-bulge system NGC 3368. Their best-fitting SSP models indicated a mean age of $\sim 1$ Gyr for the nuclear region (dominated by the classical bulge) of NGC 3368, but their best-fitting *multiple*-population models included both younger and older components, with a 10 Gyr population contributing 27 per cent of the nuclear light (and the $\la 1$ Gyr components accounting for only 1.4 per cent of the nuclear stellar mass). @storchi-bergmann12 modeled the near-IR spectra of the central $4.5 \times 5.0$ arcsec of NGC 1068 with multiple-age populations and found that the *oldest* component (5–13 Gyr in age) was concentrated in the inner $r < 1$ arcsec, corresponding to what we identify as the classical bulge (Section \[sec:n1068\]). So the limited stellar-population evidence is, overall, broadly consistent with the disky pseudobulges being younger than both the bars outside and the classical bulges inside, but there is clearly room for further work. ### Classical Bulge Formation? In the standard picture of galaxy formation, classical bulges are formed at high redshift from the violent merger of smaller galaxies or proto-galactic sub-clumps, resulting in a compact, centrally concentrated object, with stellar kinematics dominated by dispersion and an old, metal-rich stellar population. Although we know of no reason why this couldn’t be the case for our classical bulges, the latter *do* tend to be smaller and lower in mass (and certainly in $B/T$) than what is usually assumed for classical bulges. It is unclear whether this process would produce such *small* structures, or ones with Sérsic indices of $\sim 1$–2.5. Another possibility is the more recent scenario of clump-cluster mergers at high redshift [e.g., @noguchi00; @immeli04; @elmegreen08]. In this model, massive star clusters form in turbulent, gas-rich disks and then, due to dynamical friction, spiral in towards the center, where they merge in a fashion not entirely unlike the standard classical-bulge merger-formation. Although some simulations have suggested that the resulting bulge would be relatively massive, with high Sérsic indices [e.g., @elmegreen08], it seems possible that smaller, shorter-lived versions of this scenario might form lower-mass, lower-Sérsic-index classical bulges like those we find in the composite-bulge galaxies. There is also the possibility that the structures we identify as classical bulges in these galaxies are actually byproducts of secular evolution – i.e., that they are produced in some fashion *after* the main disk forms, or even after the disky pseudobulge forms. We know that nuclear star clusters can host multiple episodes of star formation [e.g., @walcher06], and the center of the galaxy is where gas that loses angular momentum will tend to accumulate, which might suggest that compact classical bulges form through a similar process. However, the classical bulges in our galaxies tend to be significanly larger than – and even coexist with – nuclear star clusters (Section \[sec:cb-plots\]), which argues against that idea. Our classical bulges might conceivably form directly out of disky pseudobulges via some kind of instability, though realistic models for such a process are lacking. We *can* rule out standard bar instabilities, since these produce elongated, boxy/peanut-shaped structures elongated along the parent bar [e.g., @erwin-debattista13 and references therein], in contrast to the round, approximately axisymmetric structures we see, and also because some of the strongest cases, such as NGC 4371, do not have nuclear bars. (The B/P structures of the *large-scale* bars would be far larger than the classical bulges we find, as is clearly the case for at least NGC 3368; see Section \[sec:boxy\]). Similarly, the presence of nuclear bars in many of these galaxies (e.g., NGC 1068, NGC 1543, NGC 1553, NGC 2859, NGC 3945, and possibly NGC 4699) rules out scenarios in which a nuclear bar is “destroyed” in order to form a classical bulge. We also note that the apparent high frequency of pseudobulges in late-type spirals [e.g., @fisher-drory11] suggests that pseudobulges by themselves probably do not *always* give rise to classical bulges. Summary {#sec:summary} ======= We have presented a morphological and kinematic analysis of nine disk galaxies (S0 or spiral) in which the photometric bulge region – that is, the excess stellar light above an inward extrapolation of the outer disk profile – is composed of at least two distinct components: 1. A *disky pseudobulge*: a structure with flattening similar to that of the outer disk, one or more morphological features characteristics of disks (nuclear rings, bars and/or spirals) and a stellar kinematics which is dominated by rotation. 2. A *classical bulge*: a component which is rounder (more spheroidal) than the disky pseudobulge and which has kinematics dominated by velocity dispersion. These components do, however, tend to have profiles best fitted with Sérsic indices of $n = 1$–2.2, so they are not classical in the sense of having $R^{1/4}$ profiles. In at least one galaxy there is also evidence for a boxy/peanut-shaped bulge (the vertically thick inner part of a bar), demonstrating that all three types of “bulge” can coexist, as suggested by, e.g., @athanassoula05. Using Schwarzschild orbit-superposition modeling for three of these galaxies (NGC 3368, NGC 4371 and NGC 4699), taken from @erwin14-smbh, we investigated the 3D stellar orbital structure of the best-fitting models for each galaxy. We found that the stellar velocity dispersion is approximately isotropic within the classical-bulge regions but is equatorially biased in the disky pseudobulges (as expected for a highly flattened system), and also that the ratio of azimuthal velocity to total velocity dispersion ([$V_{\varphi} / \langle\sigma\rangle$]{}) is typically $\la 0.5$ in the classical bulge and increases towards values $\ga 1$ in the disky pseudobulge. Although we are currently unable to put strong limits on the frequency of composite-bulge galaxies, they are probably present in *at least* $\sim 10$ per cent of barred S0 and early-type spiral galaxies. Plotting the classical-bulge components of the composite-bulge galaxies in the $R_{e}$–$M_{\star}$ and $\langle \Sigma_{\star} \rangle_{e}$–$M_{\star}$ planes shows that they fall into the same general sequence as elliptical galaxies and the (larger) bulges of disk galaxies. Even though some of the classical-bulge components are quite small, with half-light radii $\sim 30$ pc, they remain distinct from nuclear star clusters (and some in fact harbor nuclear star clusters in their centres). Curiously, the disky pseudobulge components *also* lie along the elliptical/classical-bulge sequences in these planes. While some disky pseudobulges are as massive as the main (outer) disks of S0 and spiral galaxies, they are considerably more compact. Since almost all of our composite-bulge galaxies are barred, the disky pseudobulge components could plausibly be the result of bar-driven gas inflow and star formation. We note that the “nuclear disks” formed in this fashion in the simulations of @wozniak09 and @cole14 are similar in size and relative stellar mass to the disky pseudobulges in our galaxies. Cases where disky pseudobulges are found in *unbarred* galaxies (or galaxies where the only bar is *smaller* than the disky pseudobulge) are more difficult to explain, unless disky pseudobulges can form early in a galaxy’s history as part of a general inside-out process [e.g., @guedes13]. Acknowledgements {#acknowledgements .unnumbered} ================ We enjoyed helpful and interesting conversations with a number of people, including Niv Drory, Witold Maciejewski and Victor Debattista. We are particularly grateful to Rick Davies, Joris Gerssen, Karl Gebhardt, Erin Hicks and Richard McDermid for supplying us with kinematic data, both long-slit and IFU, and to Jakob Walcher for help with nuclear star cluster data. We would also like to thank Bego[ñ]{}a Garc[í]{}a-Lorenzo for faithfully executing our WHT-ISIS service observations of NGC 4371, and the anonymous referee for comments which helped improve the manuscript. P.E. was supported in part by DFG Priority Programme 1177 (“Witnesses of Cosmic History: Formation and evolution of black holes, galaxies and their environment”). This research is (partially) based on data obtained with the William Herschel Telescope; the WHT and its service programme are operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrof[í]{}sica de Canarias. Based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 080.B-0336 and 082.B-0037. Funding for the creation and distribution of the SDSS Archive has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho and the Max Planck Society. The SDSS Web site is `http://www.sdss.org/`. The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, the Korean Scientist Group, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory and the University of Washington. This research made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. It also made use of the Lyon-Meudon Extragalactic Database (LEDA; part of HyperLeda at http://leda.univ-lyon1.fr/). [128]{} natexlab\#1[\#1]{} K. N. [et al.]{}, 2009, ApJS, 182, 543 Athanassoula E., 2005, MNRAS, 358, 1477 M., [Graham]{} A. W., [Peletier]{} R. F., 2007, ApJ, 665, 1084 A. J., [Strigari]{} L. E., [Bentz]{} M. C., [Greene]{} J. E., [Ho]{} L. C., 2009, ApJ, 690, 1031 E. F., [McIntosh]{} D. H., [Katz]{} N., [Weinberg]{} M. D., 2003, ApJS, 149, 289 R., 1990, A&A, 229, 441 R., [Burstein]{} D., [Faber]{} S. M., 1993, ApJ, 411, 153 R., [Saglia]{} R. P., [Gerhard]{} O. E., 1994, MNRAS, 269, 785 Binney J., 1978, MNRAS, 183, 501 Binney J., 2005, MNRAS, 363, 937 J. P. [et al.]{}, 2009, ApJ, 694, 556 T., 2008, Journal of Physics Conference Series, 131, 012043 T., [Sarzi]{} M., [McLaughlin]{} D. E., [van der Marel]{} R. P., [Rix]{} H.-W., [Ho]{} L. C., [Shields]{} J. C., 2004, AJ, 127, 105 N. A. [et al.]{}, 2010, ApJ, 716, 1 G. A., [Richstone]{} D. O., [Bothun]{} G. D., [Heckman]{} T. M., 1993, ApJ, 402, 76 Bureau M., Aronica G., Athanassoula E., Dettmar R.-J., Bosma A., Freeman K. C., 2006, MNRAS, 370, 753 Cappellari M. [et al.]{}, 2007, MNRAS, 379, 418 D. [et al.]{}, 2010, ApJ, 712, 692 D. R., [Debattista]{} V. P., [Erwin]{} P., [Earp]{} S. W. F., [Roškar]{} R., 2014, MNRAS S., [Knapen]{} J. H., [Beckman]{} J. E., [Laurikainen]{} E., [Salo]{} H., [Mart[í]{}nez-Valpuesta]{} I., [Buta]{} R. J., 2010, MNRAS, 402, 2462 D. M., [Kraemer]{} S. B., 2000, ApJ, 532, 247 R. I., [M[ü]{}ller S[á]{}nchez]{} F., [Genzel]{} R., [Tacconi]{} L. J., [Hicks]{} E. K. S., [Friedrich]{} S., [Sternberg]{} A., 2007, ApJ, 671, 1388 A., [Falc[ó]{}n-Barroso]{} J., [Vazdekis]{} A., 2013, MNRAS, 431, 2397 A., [Falc[ó]{}n-Barroso]{} J., [Vazdekis]{} A., [Mart[í]{}nez-Valpuesta]{} I., 2008, ApJL, 684, L83 A., [Vazdekis]{} A., [Aguerri]{} J. A. L., [Corsini]{} E. M., [Debattista]{} V. P., 2012, MNRAS, 420, 1092 de Vaucouleurs G., de Vaucouleurs A., Corwin H. G., Buta R. J., Paturel G., Fouqu[é]{} P., 1993, Third Reference Catalog of Bright Galaxies. New York: Springer-Verlag B. G., [Bournaud]{} F., [Elmegreen]{} D. M., 2008, ApJ, 688, 67 D. M., [Chromey]{} F. R., [Johnson]{} C. O., 1995, AJ, 110, 2102 E. [et al.]{}, 2011, MNRAS, 414, 888 E. [et al.]{}, 2004, MNRAS, 352, 721 E., [Fathi]{} K., [Wozniak]{} H., [Ferruit]{} P., [Mundell]{} C. G., [Schinnerer]{} E., 2006, MNRAS, 365, 367 Erwin P., 2004, A&A, 415, 941 P., 2011, Mem. S. A. It. Suppl., 18, 145 P., 2014, MNRAS, in prep P., 2014, MNRAS, in prep Erwin P., Beckman J. E., Pohlen M., 2005, ApJL, 626, L81 P., [Debattista]{} V. P., 2013, MNRAS, 431, 3060 P., [Gadotti]{} D. A., 2012, Advances in Astronomy, 2012 P., [Pohlen]{} M., [Beckman]{} J. E., 2008, AJ, 135, 20 Erwin P., Sparke L. S., 1999, ApJL, 521, L37 Erwin P., Sparke L. S., 2003, ApJS, 146, 299 P. [et al.]{}, 2014, MNRAS, in prep P., [Vega Beltr[á]{}n]{} J. C., [Beckman]{} J. E., 2001, in The Central Kiloparsec of Starbursts and AGN: The La Palma Connection, [J. H. Knapen, J. E. Beckman, I. Shlosman, & T. J. Mahoney]{}, ed., p. 171 Erwin P., Vega Beltr[á]{}n J. C., Graham A. W., Beckman J. E., 2003, ApJ, 597, 929 P. B. [et al.]{}, 2002, ApJS, 143, 73 M. H., [Saglia]{} R. P., [Fisher]{} D. B., [Drory]{} N., [Bender]{} R., [Hopp]{} U., 2012, ApJ, 754, 67 L. [et al.]{}, 2006, ApJS, 164, 334 D. B., [Drory]{} N., 2008, AJ, 136, 773 D. B., [Drory]{} N., 2010, ApJ, 716, 942 D. B., [Drory]{} N., 2011, ApJL, 733, L47 D. B., [Drory]{} N., [Fabricius]{} M. H., 2009, ApJ, 697, 630 Freedman W. L. [et al.]{}, 2001, ApJ, 553, 47 K. C., 1975, in IAU Symposium, Vol. 69, Dynamics of the Solar Systems, [Hayli]{} A., ed., p. 367 D. A., 2009, MNRAS, 393, 1531 K. [et al.]{}, 2003, ApJ, 583, 92 J., [Allington-Smith]{} J., [Miller]{} B. W., [Turner]{} J. E. H., [Walker]{} A., 2006, MNRAS, 365, 29 A. W., [Spitler]{} L. R., 2009, MNRAS, 397, 2148 J. E. [et al.]{}, 2010, ApJ, 721, 26 J., [Mayer]{} L., [Carollo]{} M., [Madau]{} P., 2013, ApJ, 772, 36 K. [et al.]{}, 2009, ApJ, 695, 1577 L., [Erwin]{} P., [Aladro]{} R., [Beckman]{} J. E., 2011, AJ, 142, 145 E. K. S., [Davies]{} R. I., [Maciejewski]{} W., [Emsellem]{} E., [Malkan]{} M. A., [Dumas]{} G., [M[ü]{}ller-S[á]{}nchez]{} F., [Rivers]{} A., 2013, ApJ, 768, 107 L. C., [Filippenko]{} A. V., 1996, ApJ, 472, 600 L. C., [Greene]{} J. E., [Filippenko]{} A. V., [Sargent]{} W. L. W., 2009, ApJS, 183, 1 P. F., [Kere[š]{}]{} D., [Murray]{} N., [Quataert]{} E., [Hernquist]{} L., 2012, MNRAS, 427, 968 J., 2008, MNRAS, 386, 2242 G., 1977, ApJL, 218, L43 A., [Samland]{} M., [Gerhard]{} O., [Westera]{} P., 2004, A&A, 413, 547 S., [Saitoh]{} T. R., 2012, MNRAS, 2819 Jarvis B. J., Dubath P., Martinet L., Bacon R., 1988, A&AS, 74, 513 Knapen J. H., de Jong R. S., Stedman S., Bramich D. M., 2003, MNRAS, 344, 527 Kormendy J., 1979, ApJ, 227, 714 J., 1982, ApJ, 257, 75 Kormendy J., 1984, ApJ, 286, 116 Kormendy J., 1993, in IAU Symp. 153: Galactic Bulges, Dejonghe H., Habing H. J., eds., Kluwer, Dordrecht, p. 209 J., [Bender]{} R., 2012, ApJS, 198, 2 J., [Bender]{} R., [Cornell]{} M. E., 2011, Nature, 469, 374 J., [Drory]{} N., [Bender]{} R., [Cornell]{} M. E., 2010, ApJ, 723, 54 J., [Kennicutt]{}, Jr. R. C., 2004, ARA&A, 42, 603 J., [Norman]{} C. A., 1979, ApJ, 233, 539 K., [Fisher]{} D., [Merrifield]{} M. R., 1996, MNRAS, 283, 543 E., [Salo]{} H., [Buta]{} R., [Knapen]{} J., [Speltincx]{} T., [Block]{} D., 2006, AJ, 132, 2634 E., [Salo]{} H., [Buta]{} R., [Knapen]{} J. H., 2009, ApJL, 692, L34 E., [Salo]{} H., [Buta]{} R., [Knapen]{} J. H., [Comer[ó]{}n]{} S., 2010, MNRAS, 405, 1089 Laurikainen E., Salo H., Buta R., Vasylyev S., 2004, MNRAS, 355, 1251 M., [Saracco]{} P., 2009, MNRAS, 394, 774 G., [Zaggia]{} S. R., [Busarello]{} G., [Richter]{} G., 1994, A&AS, 105, 433 L[ü]{}tticke R., Dettmar R.-J., Pohlen M., 2000, A&AS, 145, 405 R. M. [et al.]{}, 2006, MNRAS, 373, 906 S. [et al.]{}, 2005, ApJ, 625, 121 I., [Hilker]{} M., 2011, MNRAS, 414, 3699 I., [Hilker]{} M., [Mieske]{} S., 2009, A&A, 496, 683 I., [Mieske]{} S., [Hilker]{} M., 2008, A&A, 486, 697 M., 2000, MNRAS, 312, 194 N., [Saglia]{} R. P., [Thomas]{} J., [Bender]{} R., [Davies]{} R. I., [Gebhardt]{} K., 2008, MNRAS, 391, 1629 N., [Saglia]{} R. P., [Thomas]{} J., [Bender]{} R., [Pannella]{} M., [Gebhardt]{} K., [Davies]{} R. I., 2007, MNRAS, 379, 909 N., [Thomas]{} J., [Erwin]{} P., [Saglia]{} R. P., [Bender]{} R., [Davies]{} R. I., 2010, MNRAS, 403, 646 Ohta K., Hamabe M., Wakamatsu K.-I., 1990, ApJ, 357, 71 J. [et al.]{}, 2003, ApJ, 596, 903 M., [Trujillo]{} I., 2006, A&A, 454, 759 J. [et al.]{}, 2009, MNRAS, 971 M., [Gottesman]{} S. T., [Aguerri]{} J.-A. L., [Varela]{} A.-M., 1997, AJ, 114, 1413 A., 1999, in The Formation of Galactic Bulges, [Carollo]{} C. M., [Ferguson]{} H. C., [Wyse]{} R. F. G., eds., p. 9 S. P., [Thomas]{} J., [Erwin]{} P., [Saglia]{} R. P., [Nowak]{} N., [Bender]{} R., 2011, MNRAS, 410, 1223 S. P. [et al.]{}, 2013, AJ, 146, 45 Sandage A., Bedke J., 1994, The Carnegie atlas of galaxies. Washington, DC: Carnegie Institution of Washington with The Flintridge Foundation M., [Rix]{} H.-W., [Shields]{} J. C., [Ho]{} L. C., [Barth]{} A. J., [Rudnick]{} G., [Filippenko]{} A. V., [Sargent]{} W. L. W., 2005, ApJ, 628, 169 E., [Eckart]{} A., [Tacconi]{} L. J., [Genzel]{} R., [Downes]{} D., 2000, ApJ, 533, 850 N. Z., [Matthews]{} K., [Carico]{} D. P., [Sanders]{} D. B., 1988, ApJL, 327, L61 A. C. [et al.]{}, 2010, ApJ, 714, 713 K. L., [Gerssen]{} J., [van der Marel]{} R. P., 2003, AJ, 126, 2707 M., [Axon]{} D., [Probst]{} R., [Gatley]{} I., 1995, MNRAS, 274, 369 F., [Prugniel]{} P., 1997, A&AS, 126, 519 M., [Hau]{} G. K. T., [Forbes]{} D. A., [Couch]{} W. J., 2010, MNRAS, 408, 254 T., [Riffel]{} R. A., [Riffel]{} R., [Diniz]{} M. R., [Borges Vale]{} T., [McGregor]{} P. J., 2012, ApJ, 755, 87 J., [Saglia]{} R. P., [Bender]{} R., [Erwin]{} P., [Fabricius]{} M., 2014, ApJ, 782, 39 J., [Saglia]{} R. P., [Bender]{} R., [Thomas]{} D., [Gebhardt]{} K., [Magorrian]{} J., [Richstone]{} D., 2004, MNRAS, 353, 391 Tonry J. L., Dressler A., Blakeslee J. P., Ajhar E. A., Fletcher A. B., Luppino G. A., Metzger M. R., Moore C. B., 2001, ApJ, 546, 681 S. C., [Worthey]{} G., [Faber]{} S. M., [Burstein]{} D., [González]{} J. J., 1998, ApJS, 116, 1 J. C., [Pizzella]{} A., [Corsini]{} E. M., [Funes]{} J. G., [Zeilinger]{} W. W., [Beckman]{} J. E., [Bertola]{} F., 2001, A&A, 374, 394 C. J., [B[ö]{}ker]{} T., [Charlot]{} S., [Ho]{} L. C., [Rix]{} H.-W., [Rossa]{} J., [Shields]{} J. C., [van der Marel]{} R. P., 2006, ApJ, 649, 692 C. J. [et al.]{}, 2005, ApJ, 618, 237 G., et al., 2003, AJ, 126, 2268 T., [Jogee]{} S., [Khochfar]{} S., [Burkert]{} A., [Kormendy]{} J., 2009, ApJ, 696, 411 H., [Friedli]{} D., [Martinet]{} L., [Martin]{} P., [Bratschi]{} P., 1995, A&AS, 111, 115 H., [Michel-Dansac]{} L., 2009, A&A, 494, 11 R. F. G., [Gilmore]{} G., [Franx]{} M., 1997, ARAA, 35, 637 D. G. [et al.]{}, 2000, AJ, 120, 1579 Other Composite Bulges {#sec:others-full} ====================== NGC 1553 {#sec:n1553} -------- NGC 1553 is a nominally unbarred S0 galaxy, notable for its prominent lens [e.g., @freeman75; @kormendy84a]. @kormendy04 called attention to it as an example of a pseudobulge in an unbarred galaxy, based on its rather high [$V_{\rm max} / \sigma_{0}$]{} value. As Kormendy & Kennicutt noted, this galaxy’s surface-brightness profile looks very similar to that of many barred galaxies: an extended, shallow-surface-brightness “shelf” with a much steeper falloff at its outer edge (located in between the central bulge and the outer exponential disk). The lens *is* more elliptical than the outer disk, and the presence of weak boxyness in the isophotes (strongest at $a \sim 25$ arcsec) might suggest we are seeing the projected box/peanut structure of a (weak) bar, rather than a flattened structure. This would imply an unusually large ratio of B/P size to bar size, since we see little or no sign of the “spurs” which are the projection of the vertically thin outer part of the bar [see @erwin-debattista13]. Regardless of its true nature, the presence of the lens leads us to model it as a separate, broken-exponential component in our global decomposition; the result is a relatively good fit to the major-axis profile (panel b of Figure \[fig:n1553a\]). The photometric bulge region ($r < 16$ arcsec) shows clear evidence for a disky pseudobulge, as previously suggested by @kormendy04 and @laurikainen06: the ellipticity of the isophotes is similar to or greater than that of the main disk (panels c and e of Figure \[fig:n1553a\]), and the stellar kinematics are dominated by rotation, with [$V_{\rm dp} / \sigma$]{} reaching a maximum of $\sim 1.4$ at $r \sim 8$ arcsec (panel f). Analysis of the *HST* images reveals a previously unrecognized nuclear bar, with surrounding stellar spiral arms (panel d), further evidence that the photometric bulge is disklike. Inside the nuclear bar, the isophotes are clearly rounder than the outer disk. We also find a clear central excess in the surface-brightness profile, and are able to fit the inner $r < 30$ arcsec major-axis profile quite well using the sum of an inner exponential + Sérsic, added to the contribution from the lens component we used in the global decomposition (panel b of Figure \[fig:n1553b\]). As was the case for NGC 3945 and NGC 4371, the inner photometric bulge (where the inner Sérsic component dominates, at $r < 1.5$ arcsec) corresponds to the round central isophotes. Although the ground-based stellar kinematics of @kormendy84a and @longo94 which we use to determine [$V_{\rm dp} / \sigma$]{} may not fully resolve our proposed classical-bulge region,[^10] the fact that ${\ensuremath{V_{\rm dp} / \sigma}}< 0.5$ for $r \la 3$ arcsec suggests this region is indeed dominated by velocity dispersion, allowing us to classify the inner photometric bulge as a classical bulge. {width="6.0in"} {width="6.0in"} NGC 2859 {#sec:n2859} -------- NGC 2859 is a strongly double-barred galaxy, as originally noted by @kormendy79a; @kormendy82 observed a relatively high degree of stellar rotation in the photometric bulge region, with ${\ensuremath{(V_{\rm max} / \sigma_{0})^{\star}}}= 1.2$. This makes the galaxy potentially rather similar to NGC 3945. The similarity with NGC 3945 begins, in fact, with a luminous outer ring (panel a of Figure \[fig:n2859a\]) creating a strongly non-exponential outer-disk profile. We choose to fit the region *interior* to the outer ring ($r \la 70$ arcsec, panel b). This produces a reasonable photometric bulge-disk decomposition, with ${\ensuremath{R_{bd}}}\approx 12$ arcsec. If we had tried including the outer ring, the photometric bulge region would only have become larger; e.g., the decomposition in @fabricius12 gives ${\ensuremath{R_{bd}}}= 30$ arcsec. Interior to this we find a disky pseudobulge, as evidenced morphologically by the strong inner bar (Figure \[fig:n2859a\], panels c and d) and kinematically by the [$V_{\rm dp} / \sigma$]{} curve (panel f), which reaches a maximum of $\sim 1.3$–1.5 at $r \sim 7$ arcsec. The kinematic data combines our WHT-ISIS long-slit observations (Appendix \[sec:data-spec\]) and a pseudo-longslit constructed from the SAURON data of @delorenzo-caceres08; the latter profile was previously presented in @fabricius12. We note that @delorenzo-caceres13 have also argued for the existence of a distinct inner disk in this galaxy from their 2D analysis of the SAURON kinematics. {width="6.0in"} Unlike the case of NGC 3945, where the inner bar was a small perturbation within the disky pseudobulge and could thus be ignored in the fitting process (particularly as it was oriented almost perpendicular to the major axis), the inner bar in NGC 2859 is much larger and stronger relative to the disky pseudobulge; indeed, it could perhaps be argued that the disky pseudobulge is largely just the inner bar plus its surrounding lens. However, we can take advantage of this fact by using the inner bar itself to guide our decomposition. A more detailed discussion of this approach, along with an analysis of the inner bar’s structure, will be presented elsewhere [@erwin14-dissect]. As Figure \[fig:n2859b\] shows, we can use a cut along the inner bar’s major axis and decompose this into an underlying exponential, a broken-exponential profile for the bar itself and a central Sérsic component. The latter defines the inner photometric bulge, with ${\ensuremath{R_{bd,i}}}= 0.94$ arcsec. This region is associated with a position angle close to that of the outer disk and an ellipticity of $\sim 0.05$ (panel c of Figure \[fig:n2859b\]), so we have morphological evidence for a classical bulge. Finally, the stellar kinematics (panel d of the figure) show that ${\ensuremath{V_{\rm dp} / \sigma}}$ remains $< 1$ for $r \la 3$ arcsec, in both the SAURON and the (higher-resolution) WHT-ISIS data. {width="6.0in"} NGC 3368 {#sec:n3368} -------- The case of NGC 3368 was originally considered in @nowak10; our analysis here is very similar. The large-scale major-axis decomposition (panel b of Figure \[fig:n3368a\]) shows a fairly plausible Sérsic + exponential fit, with ${\ensuremath{R_{bd}}}= 52$ arcsec. We note that this is rather larger than the value of ${\ensuremath{R_{bd}}}= 23.3$ arcsec which @fabricius12 found from their 1-D decomposition, which may reflect the effects of masking intermediate parts of the profile in the latter study. A similar fit to a partially masked 1D profile of this galaxy by @fisher-drory08 resulted in ${\ensuremath{R_{bd}}}\sim 35$ arcsec. Adopting either of these smaller radii as the outer boundary of the photometric bulge region has no significant effect on our analysis, however. The isophotes and ellipse fits (panels c and e) show two regions of high ellipticity, corresponding to the “inner disk” and inner bar of @erwin04; unsharp masking (panel d) shows the inner bar more clearly (panel d). The stellar kinematics from @fabricius12 clearly shows a peak in the [$V_{\rm dp} / \sigma$]{} profile (panel f), which reaches a maximum of $\sim 1.3$ at $r \approx 7$ arcsec; the improved S/N of the HET spectrum shows this more clearly that the older kinematics from @vega-beltran01 used by @nowak10. Thus we appear to have reasonable evidence for a disky pseudobulge, as argued in @nowak10. As in the cases of NGC 1553 and NGC 2859, the ellipticity profile (panel e of Figure \[fig:n3368a\]) is more complicated than those of NGC 3945 and NGC 4371, both because of the strong contribution from the inner bar (which is indirect evidence for disk-like structures, but is not a disk itself) and because the intermediate ellipticity peak (at $a \sim 21$ arcsec) is probably associated with the structure of the outer bar – specifically, with the projection of the vertically thick inner part of the bar (see Section \[sec:boxy\]). {width="6.0in"} Figure \[fig:n3368b\] shows the evidence for a classical bulge inside the disky pseudobulge of this galaxy. The inner decomposition presented here is somewhat different from that of @nowak10, because we are now including an additional (exponential) component to represent the contribution of the (outer) bar and the lens; consequently, the parameters of the best-fitting inner exponential and Sérsic components are somewhat different. None the less, our main conclusions are unchanged: we find evidence that the light at $r \la 1$ arcsec is due to a separate photometric component with much rounder isophotes than either the disky pseudobulge or the outer disk. The [$V_{\rm dp} / \sigma$]{} plot incorporates major-axis values from the SINFONI datacubes of @nowak10 and @hicks13 and the lower-resolution major-axis long-slit spectroscopy of @fabricius12. The innermost SINFONI kinematics from @nowak10, obtained with an AO-corrected seeing of $\sim 0.17$ arcsec FWHM, clearly show that this inner component has dispersion-dominated stellar kinematics, with [$V_{\rm dp} / \sigma$]{} significantly $< 1$. {width="6.0in"} NGC 4262 -------- NGC 4262 is a low-inclination barred S0 galaxy in the Virgo Cluster, classified by @erwin04 as possessing an inner disk inside the bar, based in part on observations by @shaw95. Although the galaxy is quite close to face-on, we do find evidence for both a (possibly) disky pseudobulge and a very small classical bulge. The decomposition of the major-axis profile (Figure \[fig:n4262a\], panel b) is relatively simple and clean, with the resulting photometric bulge region defined by $r < {\ensuremath{R_{bd}}}= 7.2$ arcsec. The main morphological evidence for a disky pseudobulge is a distinct stellar nuclear ring with $a \sim 2.9$ arcsec (panel d), also noted by @comeron10; the isophotes in this region have an ellipticity only marginally less than that of the outer disk, and greater than that of the inner $a \la 1$ arcsec (see also panel c of Figure \[fig:n4262b\]). Kinematically, we find that [$V_{\rm dp} / \sigma$]{} reaches a peak value of almost 1.6 at approximately the same radius as the nuclear ring. The stellar kinematic data are from major-axis cuts through SAURON IFU data from the SAURON public archive,[^11] originally published in @emsellem04, and higher-resolution OASIS IFU data from @mcdermid06. {width="6.0in"} Closer inspection of the ACS-WFC isophotes shows that the isophotes become gradually rounder interior to the stellar nuclear ring, reaching values of $\sim 0.05$ before PSF effects dominate (Figure \[fig:n4262b\]). Decomposition of the inner major-axis profile (panel b of Figure \[fig:n4262b\]) shows a small central excess, reasonably well fit by a Sérsic component with $n \sim 0.9$. Stellar kinematics with the necessary resolution for this inner photometric-bulge region ($r < 0.32$ arcsec) are unavailable; however, the fact that [$V_{\rm dp} / \sigma$]{} is $< 1$ for radii $\la 2$ arcsec suggests that the inner stellar kinematics are probably dispersion-dominated, and that the central photometric excess corresponds to a (very compact) classical bulge. {width="6.0in"} NGC 4699 -------- This is a luminous, intermediate-type (Sb) spiral galaxy with a very extended, Type III outer disk [@erwin08]. In our analysis, we ignore the very outer part of this disk (visible in the azimuthally averaged profile of Erwin et al. at $r \ga 120$ arcsec), and so our “outer” decomposition (panel b of Figure \[fig:n4699a\]) is restricted to $r < 120$ arcsec. This fit yields a significant photometric bulge which dominates the light at $r \la 45$ arcsec; this clearly corresponds to what @carnegie call “the bulge” of this galaxy. Inspection of the photometric bulge region shows that it is as elliptical as the outer disk, or even more so (panels c and e of Figure \[fig:n4699a\]), and is dominated by tightly wrapped spiral arms and a relatively strong bar (panels c and d of the same figure). Although the long-slit stellar kinematics of @bower93 only extend to $r \sim 20$ arcsec, they clearly show that [$V_{\rm dp} / \sigma$]{} reaches values $> 1$ in the photometric-bulge region, so we are confident in identifying a disky pseudobulge in NGC 4699. Figure \[fig:n4699b\] shows the previously identified photometric bulge region. In the case of this galaxy, the major axis is close to the bar’s major axis, but this produces only a very weak bump in the major-axis profile, similar to that due to spiral structure further out, so we do not include a separate component for the bar. The resulting fit (including two exponential components for the outer disk) is rather good, with a clear inner excess (modeled as a Sérsic component with $n = 1.4$) creating an inner photometric-bulge zone with ${\ensuremath{R_{bd,i}}}= 2.8$ arcsec. The stellar kinematics from the ground-based data of @bower93 suggest a relatively low value of [$V_{\rm dp} / \sigma$]{} within [$R_{bd,i}$]{}, and the SINFONI AO data appear to confirm this, since the SINFONI [$V_{\rm dp} / \sigma$]{} values reach a plateau of only $\sim 0.5$. Strictly speaking, we cannot rule out the possibility that [$V_{\rm dp} / \sigma$]{} might reach values $\sim 1$ between $r \approx 1.5$ and 3 arcsec, since this is beyond the range of our SINFONI data but within a region where seeing effects might reduce [$V_{\rm dp} / \sigma$]{} from the ground-based data. None the less, we suggest that the inner photometric bulge region of NGC 4699, associated with very round isophotes (panel c of Figure \[fig:n4699b\]) and [$V_{\rm dp} / \sigma$]{} values $\la 0.5$, is most likely another example of a compact classical bulge. Our decomposition and identification of the classical bulge is similar to that of @weinzirl09, who performed a 2D bulge/bar/disk decomposition of an $H$-band image of NGC 4699 from the OSU Bright Spiral Galaxy Survey @eskridge02. (Inspection of the OSU-BSGS $H$-band image shows that the main disk of the galaxy is not visible in that image, so Weinzirl et al. treated the disky pseudobulge as the galaxy’s only disk.) They found a bulge with $n = 2.08$ and $r_e = 2.62$ arcsec – values within $\sim 35$ per cent of the corresponding values from our fit. {width="6.0in"} {width="6.0in"} Ambiguous Case: NGC 1068 {#sec:n1068} ------------------------ In most respects, the Sb galaxy NGC 1068 appears to be similar to the previous composite-bulge galaxies: it has a large, disky pseudobulge combined with a much smaller central component. The sole ambiguity concerns the morphology of the central component – is it rounder than the galaxy disk, or is it more of a nuclear disk? – and stems from uncertainty about the galaxy’s inclination. Both @schinnerer00, using [H[i]{}]{} data, and @davies07, using stellar kinemetry analysis of their SINFONI data, suggest an inclination of $\approx 40\degr$, which implies an observed ellipticity of $\approx 0.22$ for an early-type disk. On the other hand, @gutierrez11 used SDSS images and found an outer-disk ellipticity of $\sim 0.14$, implying an inclination of $\sim 31\degr$. Since, as we will see below, the inner component has an ellipticity of $\sim 0.15$, the question of whether the inner component is disklike or spheroidal depends on the adopted galaxy inclination – thus, we consider this galaxy a somewhat ambiguous case, though we include it with the other composite-bulge galaxies in our analysis. ### Disky Pseudobulge Our global analysis of NGC 1068’s surface-brightness profile uses the light at $r \la 100$ arcsec and is very similar to that carried out by @shapiro03 for this galaxy (their Fig. 2), As both @pt06 and @gutierrez11 showed, there is considerable stellar light further out in a shallower profile, much of which is due to the outer pseudoring. The global fit for this galaxy is thus similar to those for NGC 3945 and NGC 2859, where we deliberately exclude light due to extended outer rings. The decomposition (Figure \[fig:n1068a\], panel b) is relatively straightforward, and the resulting photometric bulge has ${\ensuremath{R_{bd}}}= 24$ arcsec. The morphology interior to this radius is strikingly disklike: the near-IR stellar light is dominated by the very strong (inner) bar first noted by @scoville88, with star-forming spiral arms forming a pseudoring just outside the bar (panels c and d); due to this bar, the isophotes have an ellipticity much higher than that of the outer disk (panel e). The long-slit stellar-kinematic data of @shapiro03 show ${\ensuremath{V_{\rm dp} / \sigma}}> 1$ at all radii $\ga 1$ arcsec, and in fact the ratio reaches a value of almost $\sim 3$ at $r \sim 15$ arcsec (panel f), so this an even more extreme case of a photometric bulge with disklike kinematics than NGC 3945. We are certainly not the first to suggest that the kinematics in this part of the galaxy are more disklike than bulgelike: in particular, @emsellem06 pointed this out using 2D stellar kinematics from the SAURON instrument, in conjunction with $N$-body/hydrodynamical modeling. {width="6.0in"} ### Nuclear Disk or Classical Bulge? {width="6.0in"} Optical and near-IR images of NGC 1068 are strongly contaminated by light from the bright AGN. To get around this problem, we make use of an $H$-band image from the 100mas SINFONI observations (1.5 arcsec field of view) of @davies07, where the AGN emission (assumed to come mostly from hot dust in the circumnuclear torus) has been spectroscopically modeled and removed, leaving behind a “pure stellar” continuum which is then collapsed along the wavelength axis to form an image. We matched the surface-brightness profile extracted from this image to a profile from a larger-scale *HST* NICMOS3 F200N image. For the stellar kinematics, we supplement the large-scale (but low-spatial-resolution) data of @shapiro03 with data from the Gemini GMOS IFU observations of @gerssen06, which had a seeing of FWHM $= 0.5$ arcsec, and with the VLT-SINFONI kinematics of @davies07, observed in AO mode with a corrected FWHM of 0.10 arcsec. We perform our inner decomposition by choosing a position angle which avoids most of the inner bar (PA $= 137\degr$, perpendicular to the bar). Since the galaxy is seen at a relatively low inclination, the projection effects are relatively small; however, this does mean that our decomposition does not exactly mirror the major-axis stellar kinematics. Comparison of the NICMOS3 image with the PA $= 137\degr$ profile shows that the increase in surface brightness starting at $r \sim 6$ arcsec is actually due to the profile crossing into the bright part of the bar, along its minor axis. Since we consider bars to be disk phenomena, this excess light should be treated as part of the disky pseudobulge. There is a second, much steeper increase in the profile which sets in for $r \la 1$ arcsec, associated with rounder isophotes; this is the best candidate for an embedded bulge. Our best fit to the surface-brightness profile thus uses *three* components: an exponential for the main part of the pseudobulge outside the bar itself (in analogy with large-scale bars, one could perhaps call this a “lens”); a Gaussian for the minor axis profile of the bar;[^12] and a Sérsic for the innermost component. The innermost (Sérsic) component corresponds to the “extra emission” at $r < 1$ arcsec noted by @davies07, although we identify their 1–5 arcsec “$R^{1/4}$ bulge” profile as being due to the inner bar, not to a larger-scale bulge. The final $R_{e}$ value recorded in Table \[tab:classical-parameters\] for this Sérsic component has been corrected to its major-axis value assuming an ellipticity of 0.15, based on the ellipse fits (Figure \[fig:n1068b\]). As in the other galaxies, the [$V_{\rm dp} / \sigma$]{} profile shows dispersion-dominated stellar kinematics in the inner $r < {\ensuremath{R_{bd,i}}}$ region: [$V_{\rm dp} / \sigma$]{} reaches a maximum of only $\sim 0.7$. Is the innermost component a compact classical bulge, or is it something more disklike? @davies07 argued for a disk, and suggested (based on indirect arguments about the estimated $M/L$ ratio of the inner component) that its stellar population was relatively young: $\sim 300$ Myr. If we adopt an inclination of 31 for the galaxy, then the inner ellipticity of $\sim 0.15$ is consistent with that of a disk having a flattening similar to the outer disk’s. However, @crenshaw00 and [@storchi-bergmann12] have both presented high-resolution spectroscopic evidence – including fits to the near-nuclear spectra – indicating that the $r < 1$ arcsec region is actually dominated by *old* stars, with an age of at least 2 Gyr according to Crenshaw & Kraemer’s optical STIS spectroscopy, and an age of 5–15 Gyr according to the near-IR spectroscopy of Storchi-Bergmann et al. If we combine this with the dispersion-dominated kinematics *and* assume an inclination of 40 for the galaxy, then the central stellar component is rounder than a disk, kinematically hot, *and* made up of old stars – a good case for a classical bulge, albeit one with a nearly-exponential profile. Possible Case: NGC 1543 ----------------------- NGC 1543 is something of a borderline case, primarily because we do not have a good handle on the stellar kinematics. The galaxy is so close to face-on that it is difficult to determine the major axis with any accuracy, and thus long-slit data cannot really be used. (We need to know the major axis accurately in order to properly deproject the observed velocities to their major-axis, edge-on values.) The only available stellar kinematics are two long-slit observations by @jarvis88. Since both their “major-axis” and “minor-axis” profiles (at position angles of 90 and 0, respectively) show stellar rotation, all we really can say is that the major axis is not close to 0 or 90. Thus we are unable to perform a proper kinematic analysis for NGC 1543. None the less, the *morphology* of the galaxy is very suggestive of a composite-bulge system (Figure \[fig:n1543\]). Full details of the disky pseudobulge decomposition, including analysis of the structure of the inner bar, will be presented elsewhere [@erwin14-dissect]; see @erwin11 for a preliminary discussion. Because the the major axis of the inner bar is relatively close to the minor axis of the outer bar, we use a profile along the position angle of the inner bar for our decomposition (as in the case of NGC 2859; see Section \[sec:n2859\]). Figure \[fig:n1543\] summarizes both the global, “naive” bulge/disk decomposition (panel b) and the inner, composite-bulge decomposition (panel d). We construct our surface-brightness profile using an *HST* WFPC2 F814W image at small radii and a Spitzer IRAC2 image at large radii (we use the IRAC2 image instead of the IRAC1 image because the galaxy position on the former is slightly better for purposes of measuring the sky background). Excluding the broad outer ring (similar to those of NGC 2859 and NGC 3945), a decomposition of the inner $\sim 100$ arcsec (panel a) yields a photometric bulge with $n = 1.6$, $r_e = 9.4$ arcsec and ${\ensuremath{R_{bd}}}\approx 26$ arcsec; this is similar to, but slightly larger than, the “pseudobulge” reported by @fisher-drory10 from their 1-D IRAC1 decomposition ($n = 1.51$, $r_e = 6.5$ arcsec). The photometric bulge region thus defined ($r \la 26$ arcsec) is dominated by a very strong inner bar, with a very weak stellar nuclear ring surrounding it [@erwin14-dissect]. We take this as strongly suggestive evidence for a disky pseudobulge, though as noted above we lack the necessary kinematic data for full confirmation. In the central regions, the isophotes become very round (panels c and f of Figure \[fig:n1543\]), and there is a central excess of light above the inner bar. Simultaneous decompositions along the major and minor axes of the inner bar support the presence of an inner Sérsic component with $n = 1.5$, $r_e = 2.7$ arcsec and ellipticity $\la 0.05$ (panel d of the figure shows the decomposition along the inner-bar major axis). So there is morphological and photometric evidence for a distinct, classical-bulge-like component in the centre of this galaxy as well. There is in addition evidence for a very compact, distinct feature inside the classical bulge, which is probably a nuclear star cluster (see @erwin11 and @erwin14-dissect). {width="6.0in"} The Uncertain Case of NGC 3489 {#sec:n3489} ------------------------------ @nowak10 considered two cases of composite-bulge galaxies: the double-barred spiral NGC 3368 and the barred S0 galaxy NGC 3489. Although we confirm the previous classification of NGC 3368 (see Section \[sec:n3368\], above), NGC 3489 has proved to be more ambiguous (Figure \[fig:n3489\]). Careful analysis of the isophotes of NGC 3489 suggests that the broad, slightly boxy region inside the bar may well be an example of a projected box/peanut (B/P) structure (outlined isophote in panel a of Figure \[fig:n3489\]), even though @erwin-debattista13 did not classify this galaxy as such. The presence of a B/P structure certainly does not prevent the simultaneous existence of a composite bulge in this galaxy – indeed, Section \[sec:boxy\] argues that NGC 3368 has a clear B/P structure in addition to its composite bulge – but it does mean that the major-axis decomposition in @nowak10 needs to be revised, since what they considered to be the inner-disk component, dominating the major-axis light from $\approx 4$–12 arcsec, is more likely to be a combination of the B/P structure and the lens region immediately outside the bar. For our revised inner decomposition (panel d of Figure \[fig:n3489\]), we include the contribution from the outer disk component (panel b) and treat the B/P structure as a separate component with a Sérsic profile; we also include an additional Sérsic component for the nuclear luminosity excess which @nowak10 modeled with a Gaussian. The resulting fit has an inner exponential with scale length $\sim 55$ pc, corresponding to the original “classical bulge” component in @nowak10, as well as an extremely compact Sérsic component with $n = 0.7$ and $R_{e} = 0.19$ arcsec. The [$R_{bd}$]{} radii for these two components are 2.7 and 0.2 arcsec, respectively. As shown by the ellipticity plot (panel e), the inner exponential component, which dominates the light at $r \la 2.7$ arcsec, is in fact highly elliptical – almost as elliptical as the outer disk. This is very similar to the disky pseudobulges identified in our other galaxies. On the other hand, the stellar kinematics in this region (panel f) are rather dispersion-dominated, with ${\ensuremath{V_{\rm dp} / \sigma}}\sim 0.7$; even at the outer edge of this region, [$V_{\rm dp} / \sigma$]{} is never larger than $\sim 0.9$. So this structure is both flattened like a disk and at the same time kinematically *hot*, making it unlike any of the disky pseudobulges in the other galaxies of our sample. The innermost (Sérsic) component is associated with rounder isophotes and a lower [$V_{\rm dp} / \sigma$]{} value, but it is so small[^13] that it could more plausibly be classified as a nuclear star cluster than as a genuine classical bulge. We are thus left with a curiosity: our revised analysis identifies a structure in NGC 3489 which morphologically resembles a disky pseudobulge, but is kinematically hot, along with a distinct nuclear component that might best be classified as a nuclear star cluster. NGC 3489 may thus be a kind of transition object, and an indication that the central regions of early-type disk galaxies can be even more diverse and complicated than our main “composite-bulge” argument suggests. {width="6.0in"} WHT-ISIS Spectroscopy {#sec:data-spec} ===================== Parameter NGC 2859 NGC 4371 ----------------------------------------- ---------------- ---------------- Date 2000 Dec 13 2001 Jun 4 Grating ($\rm lines\;mm^{-1}$) R600B R600B Central wavelength (Å) 5498 5498 Detector EEV12 EEV12 Pixel binning $1\times1$ $1\times1$ Scale (arcsec pix$^{-1}$) 0.36 0.36 Reciprocal dispersion (Å pix$^{-1}$) 0.45 0.45 Slit width (arcsec) 0.93 1.2 Slit length () 4.0 4.0 Slit position angle () 85 92 Instrumental $\sigma$ ([km s$^{-1}$]{}) 60 60 Seeing FWHM (arcsec) 1.0 1.0 Galaxy exposure 3$\times$1200s 3000s + 2$\times$2400s Sky exposure 900s 1200s : WHT-ISIS Instrumental Setup[]{data-label="tab:isis-setup"} Instrumental setup for the blue arm of the WHT-ISIS spectrograph, as used for observations of NGC 2859 and NGC 4371. We obtained long-slit spectroscopy of NGC 2959 with the ISIS spectrograph of the William Herschel Telescope on 2000 December 13, using the blue arm with the R600B grating; further details of the observing setup are listed in Table \[tab:isis-setup\]. Because the galaxy was large enough to fill most of the slit, we obtained a separate 900s exposure of the nearby blank sky, offset 6 to the east of the galaxy centre. We also observed two kinematic standard stars (HR941 and HR2660) with the same setup. NGC 4371 was observed with an almost identical instrumental setup as part of ING service-time observations on 2001 June 4 (see Table \[tab:isis-setup\]). A separate 1200s sky exposure was taken with a 3 offset to the north of the galaxy centre, and observations of four kinematic standard stars (HR5200, HR5966, HD5340, HR5340) were also obtained. Following standard <span style="font-variant:small-caps;">midas</span>[^14] reduction of the ISIS observations, including bias-subtraction, flat-fielding and wavelength calibration using CuAr and CuNe+CuAr lamp exposures, the extracted galaxy spectra were analysed using the Fourier Correlation Quotient method [@bender90; @bender94] in order to determine the stellar kinematics. The resulting kinematic values – radial velocity, velocity dispersion and the Gauss-Hermite coefficients $h_{3}$ and $h_{4}$ – are presented in Table \[tab:isis-kinematics\]. ---------- ---- ---------- ------------------- ------------------ -------------------- -------------------- Galaxy PA $R$ $V$ $\sigma$ $h_{3}$ $h_{4}$ arcsec [km s$^{-1}$]{} [km s$^{-1}$]{} NGC 2859 85 $-$14.70 $1554.6 \pm 39.5$ $203.1 \pm 35.2$ $-0.124 \pm 0.081$ $-0.072 \pm 0.149$ NGC 2859 85 $-$9.03 $1586.1 \pm 32.6$ $194.3 \pm 28.1$ $-0.125 \pm 0.067$ $-0.026 \pm 0.120$ NGC 2859 85 $-$6.03 $1552.8 \pm 13.4$ $160.4 \pm 10.3$ $-0.083 \pm 0.050$ $-0.019 \pm 0.065$ NGC 2859 85 $-$4.50 $1575.0 \pm 11.2$ $151.2 \pm 8.2$ $-0.076 \pm 0.044$ $0.085 \pm 0.057$ ---------- ---- ---------- ------------------- ------------------ -------------------- -------------------- Stellar kinematics along the major axes of NGC 2859 and NGC 4371, as determined from spectra obtained with WHT-ISIS; see Table \[tab:isis-setup\] for the instrumental setup. This table is published in its entirety in the online edition of this paper; a sample is provided here for guidance in terms of form and content. [^1]: Very compact central concentrations – e.g., nuclear star clusters – are considered separate objects and are excluded from this definition. [^2]: $V$ is usually taken to be the maximum stellar rotation velocity $V_{\rm max}$ and $\sigma$ is some “central” value; the ellipticity is sometimes the maximum observed value and sometimes a mean value. [^3]: Based on the SBF distance of @tonry01 and the @mei05 correction. [^4]: This is the ratio of the observed [$V_{\rm max} / \sigma_{0}$]{} to the value expected for an isotropic oblate rotator having the same ellipticity; the value for NGC 3945 was 1.82. [^5]: The main exception is NGC 4371, where the disky pseudobulge is better fit with a broken-exponential profile; see Section \[sec:n4371b\]. [^6]: Low Sérsic indices and/or disky morphologies combined with mid-IR colours indicative of low star-formation rates. [^7]: http://www.brera.inaf.it/utenti/marcella/ mtol.html [^8]: For two galaxies without $(B - V)_{e}$ colours (NGC 5636 and NGC 5953), we used $g - r$ colours measured within an aperture of 5 arcsec radius on SDSS images; we were unable to find or measure colours for NGC 3892, which is excluded from the plots. [^9]: Here we are excluding dE galaxies – roughly, those “ellipticals” with $M_{\star} \la 5 \times 10^{9} {\ensuremath{\mathrm{M}_{\sun}}}$ – which tend to form a separate sequence merging into the dwarf spheroidals; see, e.g., arguments in @kormendy12, where dE and dSph are together termed “spheroidals”. [^10]: Note, however, that Longo et al. reported seeing of $< 1$ arcsec for their observations of this galaxy [^11]: http://www.strw.leidenuniv.nl/sauron/ [^12]: Evidence that the minor-axis profiles of at least some bars are Gaussian can be found in @ohta90 and @prieto97. [^13]: The Sérsic $R_{e}$ is $\approx 9$ pc, but this does not account for PSF convolution. [^14]: <span style="font-variant:small-caps;">midas</span> is developed and maintained by the European Southern Observatory.

--- abstract: | We consider the dynamical Gross-Pitaevskii (GP) hierarchy on $\R^d$, $d\geq1$, for cubic, quintic, focusing and defocusing interactions. For both the focusing and defocusing case, and any $d\geq1$, we prove local existence and uniqueness of solutions in certain Sobolev type spaces $\cH_\xi^\alpha$ of sequences of marginal density matrices. The regularity is accounted for by $$\alpha \, \left\{ \begin{array}{rcl} > &\frac12& {\rm if} \; d=1 \\ >&\frac d2-\frac{1}{2(p-1)} & {\rm if} \; d\geq2 \; {\rm and} \; (d,p)\neq(3,2)\\ \geq & 1 & {\rm if} \; (d,p)=(3,2) \,, \end{array} \right.$$ where $p=2$ for the cubic, and $p=4$ for the quintic GP hierarchy; the parameter $\xi>0$ is arbitrary and determines the energy scale of the problem. This result includes the proof of an a priori spacetime bound conjectured by Klainerman and Machedon for the cubic GP hierarchy in $d=3$. In the defocusing case, we prove the existence and uniqueness of solutions globally in time for the cubic GP hierarchy for $1\leq d\leq3$, and of the quintic GP hierarchy for $1\leq d\leq 2$, in an appropriate space of Sobolev type, and under the assumption of an a priori energy bound. For the focusing GP hierarchies, we prove lower bounds on the blowup rate. Also pseudoconformal invariance is established in the cases corresponding to $L^2$ criticality, both in the focusing and defocusing context. All of these results hold without the assumption of factorized initial conditions. address: - 'T. Chen, Department of Mathematics, University of Texas at Austin.' - 'N. Pavlović, Department of Mathematics, University of Texas at Austin.' author: - Thomas Chen - Nataša Pavlović title: 'On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies' --- Introduction ============ The derivation of the nonlinear Schrödinger equation as the dynamical mean field limit of the manybody quantum dynamics of interacting Bose gases is a research area that is recently experiencing remarkable progress, see [@esy1; @esy2; @ey; @kiscst; @klma; @rosc] and the references therein, and also [@adgote; @eesy; @frgrsc; @frknpi; @frknsc; @he; @sp]. A main motivation to investigate this problem is to understand the dynamical behavior of Bose-Einstein condensates. For recent developments in the mathematical analysis of Bose gases and their condensation, we refer to the fundamental work of Lieb, Seiringer, Yngvason, et al.; see [@ailisesoyn; @lise; @lisesoyn; @liseyn] and the references therein. The procedure developed in the landmark works of Erdös, Schlein, and Yau, [@esy1; @esy2; @ey], to obtain the dynamical mean field limit of an interacting Bose gas, comprises the following main ingredients. One determines the BBGKY hierarchy of marginal density matrices for particle number $N$, and derives the Gross-Pitaevskii (GP) hierarchy in the limit $N\rightarrow\infty$, for a scaling where the particle interaction potential tends to a delta distribution; see also [@kiscst; @sc]. For factorized initial data, the solutions of the GP hierarchy are governed by a cubic NLS for systems with 2-body interactions, [@esy1; @esy2; @ey; @kiscst], and quintic NLS for systems with 3-body interactions, [@chpa]. The proof of the uniqueness of solutions of the GP hierarchy is the most difficult part of this analysis, and is obtained in [@esy1; @esy2; @ey] by use of highly sophisticated Feynman graph expansion methods inspired by quantum field theory. Recently, an alternative method to prove the uniqueness of solutions in the $d=3$ case has been developed by Klainerman and Machedon in [@klma], using spacetime bounds on the density matrices in the GP hierarchy; this result makes the assumption of a particular a priori spacetime bound on the density matrices which has so far remained conjectural. In the work [@kiscst] of Kirkpatrick, Schlein, and Staffilani, the corresponding problem in $d=2$ is solved, and the assumption made in [@klma] is replaced by a spatial a priori bound which is proven in [@kiscst]. Alternative methods to obtain dynamical mean field limits of interacting Bose gases using operator-theoretic methods are developed by Fröhlich et al in [@frgrsc; @frknpi; @frknsc]. All of the above mentioned works discuss Bose gases with [*repulsive*]{} interactions; it is currently not known how to obtain a GP hierarchy from the $N\rightarrow\infty$ limit of a BBGKY hierarchy with attractive interactions. In the work at hand, we have nothing to add to this issue. Instead, we start here directly from the level of the GP hierarchy, and are thus free to also consider [*attractive*]{} interactions within this context. Accordingly, we will refer to the corresponding GP hierarchies as [*cubic*]{}, [*quintic*]{}, [*focusing*]{}, or [*defocusing GP hierarchies*]{}, depending on the type of the NLS governing the solutions obtained from factorized initial conditions. In the present work, we investigate the Cauchy problem for the cubic and quintic GP hierarchy with focusing and defocusing interactions. Our results do not assume any factorization of the initial data. As a crucial ingredient of our arguments, we introduce Banach spaces $\cH_\xi^\alpha=\{ \, \Gamma\in\Gspace \, | \, \| \, \Gamma \, \|_{\cH_\xi^\alpha} <\infty \, \}$ where \[bigG\] = { = ( \^[(k)]{}(x\_1,…,x\_k;x\_1’,…,x\_k’) )\_[k]{} | \^[(k)]{} &lt; } is the space of sequences of $k$-particle density matrices, and \_[\_\^]{} := \_[k]{} \^k \^[(k)]{} \_[H\^(\^[dk]{}\^[dk]{})]{} . The parameter $\xi>0$ is determined by the initial condition, and it sets the energy scale of a given Cauchy problem. If $\Gamma\in\cH_\xi^\alpha$, then $\xi^{-1}$ is the typical $H^\alpha$-energy per particle. The parameter $\alpha$ determines the regularity of the solution, and our results hold for $\alpha\in\alphaset(d,p)$ where \[eq-alphaset-def-0\] (d,p) := { [cc]{} (12,) & [if]{} d=1\ (d2-, ) & [if]{} d2 (d,p)(3,2)\ \[1,) & [if]{} (d,p)=(3,2) , . in dimensions $d\geq1$, and where $p=2$ for the cubic, and $p=4$ for the quintic GP hierarchy. The parameter $\xi>0$ determines the energy scale of the problem. The main results proven in this paper are: 1. We prove local existence and uniqueness of solutions for the cubic and quintic GP hierarchy with focusing or defocusing interactions, in $\cH_\xi^\alpha$, for $\alpha\in \alphaset(d,p)$, which satisfy a spacetime bound $\|\opB\Gamma\|_{L^1_{t\in I}\cH^{\alpha}_{\xi}}<\infty$ for some $\xi>0$ (the operator $\opB$ is defined in Section \[sec-defandresults-1\] below). This spacetime bound has been conjectured by Klainerman and Machedon in [@klma]. It is of Strichartz-type, and is proven in Section \[sec-locwp-1\] using a Picard-type fixed point argument on the space $L^1_{t\in [0,T]}\cH_\xi^\alpha$; see inequality (\[eqn-BGamma-spacetime-bd-0\]) and Remark \[rem-Strichartz-1\] below. Accordingly, we conclude that a solution of the GP hierarchy in $\cH_\xi^\alpha$ is unique if and only if this spacetime bound holds.\ 2. We prove the global existence and uniqueness of solutions in $\cH_\xi^1$ satisfying the above noted spacetime bound, for the defocusing cubic GP hierarchy for $1\leq d\leq3$, and the defocusing quintic GP hierarchy for $1\leq d\leq 2$, provided that an a priori bound $\|\Gamma(t)\|_{\cH_\xi^1}<c$ holds for $\xi>0$ sufficiently small.\ 3. We indroduce generalized pseudoconformal transformations, and prove the invariance of the cubic GP hierarchy in $d=2$, and of the quintic GP hierarchy in $d=1$, under their application. Because the NLS obtained from factorized initial data in these cases are $L^2$-critical, we will, for brevity, refer to these GP hierarchies as being $L^2$-critical.\ 4. For the focusing cubic or quintic GP hierarchy, we prove lower bounds on the blowup rate in $\cH^\alpha_\xi$ and $\cL^r_\xi$, where both spaces are defined in Section \[sec-defandresults-1\] below. An important ingredient of our proof of the local existence and uniqueness of solutions is the use of certain spacetime bounds for the non-interacting GP hierarchy established in [@klma] for the cubic GP hierarchy in $d=3$ (which were generalized to cubic in $d=2$ in [@kiscst], and to the quintic GP hierarchy in [@chpa]), and the “boardgame estimates" developed in [@klma] (and generalized to the quintic case in [@chpa]), which were motivated by the Feynman graph expansion techniques of [@esy1; @esy2]. For our discussion of blowup solutions of the focusing (cubic or quintic) GP hierarchy, we make extensive use of a quantity that controls the average $H^\alpha$-energy per particle, and, in a different form, the average $L^r$-norm per particle. It is introduced in Definition \[def-AvHLp-1\] below, and turns out to be the key observable for our discussion of blowup solutions.\ Organization of the paper {#organization-of-the-paper .unnumbered} ------------------------- In Section \[sec-defandresults-1\], we introduce the cubic and quintic GP hierarchy, and state our main theorems. In Section \[sec-locwp-1\], we prove the local wellposedness of the Cauchy problem for the cubic and quintic GP hierarchy, for both focusing and defocusing interactions. In Section \[sec-gwp-GP-1\], the local wellposedness is enhanced to global wellposedness for the cubic and quintic defocusing GP hierarchies, using energy conservation. In Section \[sec-Proof-blowuprate-1\], we prove lower bounds on the blowup rate of blowup solutions in the spaces $\cH_\xi^\alpha$ and $\cL^r_\xi$ (see below for their definitions). In Section \[sec-confinv-1\], we prove the pseudoconformal invariance of the $L^2$-critical cubic (in $d=2$) and quintic (in $d=1$) GP hierarchies. In the Appendix, we reformulate the Klainerman-Machedon spacetime bounds in a form convenient for our work. Definition of the model and statement of the main results {#sec-defandresults-1} ========================================================= We introduce the space := \_[k=1]{}\^L\^2(\^[dk]{}\^[dk]{}) of sequences of density matrices := ( \^[(k)]{} )\_[k]{} where $\gamma^{(k)}\geq0$, $\tr\gamma^{(k)} =1$, and where every $\gamma^{(k)}(\ux_k,\ux_k')$ is symmetric in all components of $\ux_k$, and in all components of $\ux_k'$, respectively. We call $\Gamma=(\gamma^{(k)})_{k\in\N}$ admissible if\ && = dx\_[k+1]{} dx\_[k+p2]{} \^[(k+)]{}(\_[k]{},x\_[k+1]{},…,x\_[k+p2]{};\_k’,x\_[k+1]{},…,x\_[k+p2]{}) for all $k\in\N$. Let $0<\xi<1$ and $r>1$. We define \_\^r := { | \_[\_\^r]{} &lt; } where \_[\_\^r]{} := \_[k=1]{}\^\^[ k]{} \^[(k)]{} \_[L\^r(\^[dk]{}\^[dk]{})]{} . Furthermore, we define \_\^ := { | \_[\_\^]{} &lt; } where \_[\_\^]{} = \_[k=1]{}\^\^[ k]{} \^[(k)]{} \_[H\^(\^[dk]{}\^[dk]{})]{} , with \^[(k)]{} \_[H\^(\^[dk]{}\^[dk]{})]{} = S\^[(k,)]{} \^[(k)]{} \_[L\^2(\^[dk]{}\^[dk]{})]{} , and $S^{(k,\alpha)}:=\prod_{j=1}^k\langle\nabla_{x_j}\rangle^\alpha\langle\nabla_{x_j'}\rangle^\alpha$. Clearly, $\cL_\xi^r$, $\cH_\xi^\alpha$ are Banach spaces. We note that Banach spaces of integral kernels of a similar type as those introduced above are, for instance, used for operator-theoretic renormalization group methods in the spectral analysis of quantum electrodynamics, [@bcfs]. Let $p\in\{2,4\}$. We consider the $p$-GP (Gross-Pitaevskii) hierarchy given by \[eq-def-b0-2\] i\_t \^[(k)]{} = \_[j=1]{}\^k \[-\_[x\_j]{},\^[(k)]{}\] + B\_[k+p2]{} \^[(k+p2)]{} in $d$ dimensions, for $k\in\N$. Here,\ && := \_[j=1]{}\^k (B\_[j;k+1,…,k+p2]{}\^[(k+)]{})(t,x\_1,…,x\_k;x\_1’,…,x\_k’)\ && := \_[j=1]{}\^kdx\_[k+1]{}dx\_[k+p2]{} dx\_[k+1]{}’dx\_[k+p2]{}’\ &&\ && \^[(k+p2)]{}(t,x\_1,…,x\_[k+p2]{};x\_1’,…,x\_[k+p2]{}’) accounts for $\frac p2+1$-body interactions between the Bose particles. For a factorized initial condition \[eq-initcond-fact-1\] \^[(k)]{}(0) = | \_0 \_0 |\^[k]{} with $\phi_0\in H^\alpha$, one obtains that \^[(k)]{}(t) = | (t) (t) |\^[k]{} is a solution of if $\phi_t$ satisfies the NLS \[eq-NLS-p-1\] i\_t \_t + \_x \_t - |\_t|\^p \_t = 0 with initial condition $\phi(0)=\phi_0$, where $\mu\in\{1,-1\}$. For $p=2$, this is the cubic NLS, and for $p=4$, this is the quintic NLS. The NLS is defocusing for $\mu=1$, and focusing for $\mu=-1$. Accordingly, we refer to (\[eq-def-b0-2\]) as the [*cubic GP hierarchy*]{} if $p=2$, and as the [*quintic GP hierarchy*]{} if $p=4$. Moreover, for $\mu=1$ or $\mu=-1$ we refer to the GP hierarchies as being defocusing or focusing, respectively. We recall the definition of the set $\alphaset(d,p)$, for $p=2,4$ and $d\geq1$, \[eq-alphaset-def-1\] (d,p) = { [cc]{} (12,) & [if]{} d=1\ (d2-, ) & [if]{} d2 (d,p)(3,2)\ \[1,) & [if]{} (d,p)=(3,2) . Our main result in this paper is the following theorem. \[thm-main-0\] Let $0<\xi_2=\eta\xi_1\leq\xi_1<1$. Assume that $\alpha\in\alphaset(d,p)$ where $d\geq1$ and $p\in\{2,4\}$, and $0<\eta<1$ sufficiently small. Then, the following hold. - For every $\Gamma_0\in\cH_{{\xi_1}}^\alpha$, there exist constants $T>0$ and $0<\xi_2\leq\xi_1$ such that there exists a unique solution $\Gamma(t)\in\cH_{\xi_2}^\alpha$ for $t\in[0,T]$ with $\|\opB\Gamma\|_{L^1_{t\in[0,T]}\cH_{\xi_2}^\alpha}<\infty$.\ - Assume that given $\Gamma_0\in\cH_{{\xi_1}}^\alpha$, there are constants $T>0$ and $0<\xi_2\leq\xi_1$ such that for $t\in I=[0,T]$, there exists a solution $\Gamma(t)$ of the $p$-GP hierarchy (\[seclwp-pGP\]) in the space $L^\infty_{t\in I}\cH_{\xi_2}^\alpha$.\ \ Then, the solution $\Gamma(t)\in L^\infty_{t\in I}\cH_{\xi_2}^\alpha$ is [*unique*]{} if and only if $\| \, \opB \Gamma \, \|_{L^1_{t\in I}\cH_{\xi}^\alpha}<\infty$ holds for some $\xi>0$.\ \ If the latter is satisfied, then in fact, the Strichartz-type bound \[eqn-BGamma-spacetime-bd-0\] \_[L\^1\_[tI]{}\_[\_2]{}\^]{} &&C(d,p,\_1,\_2) \_0\_[\_[\_1]{}\^]{} holds.\ \[rem-BBGKYlim-1\] An immediate implication of part (ii) of Theorem \[thm-main-0\] is that every solution $\Gamma(t)$ extracted by a diagonal argument from the $N\rightarrow\infty$ limit of the $N$-particle BBGKY hierarchies (with repulsive interactions) studied by Erdös-Schlein-Yau in [@esy1; @esy2], Kirkpatrick-Schlein-Staffilani in [@kiscst], and Chen-Pavlović in [@chpa], satisfies $\| \, \opB \Gamma \, \|_{L^1_{t\in I}\cH_{\xi}^\alpha}<\infty$ for some $\xi>0$. This is true because uniqueness of those solutions has been established in these works with independent methods. The role of the parameters $\xi_1,\xi_2$ is as follows: Given initial data $\Gamma_0=(\gamma^{(k)})_{k\in\N}$ with $\|\gamma^{(k)}\|_{H^\alpha(\R^{dk}\times\R^{dk})}<\infty$ for all $k$, we determine $\xi_1>0$ sufficiently small such that $\Gamma_0\in\cH_{\xi_1}^\alpha$. This means that the energy per particle in $\Gamma_0$ is bounded by $\xi_1^{-1}$. In cases of physical interest, $\xi_1>0$; the notion of an energy per particle will be quantified below. Then, we find a suitable $\xi_2=\eta\xi_1\ll\xi_1$ such that the Cauchy problem for the the GP hierarchy can be solved in a sufficiently large space $\cH^\alpha_{\xi_2}$. The requirement $\xi_2 \ll \xi_1$ is used to ensure that a solution $\Gamma(t)$ does not drift out of $\cH^\alpha_{\xi_2}$ for $t\in I=[0,T]$ with $T=T(\xi_2)>0$; we thereby impose the assumption that the energy per particle does not exceed $\xi_2^{-1}$ while $t\in I$, but once this assumption is violated, we may choose $0<\xi_2'<\xi_2$ to continue the solution to $T(\xi_2')>T(\xi_2)$. In particular, there is no implication of the size of $\xi_2$ on the regularity accounted for by $\alpha$. For factorized initial data, the statement that the solution of the NLS remains in $H^\alpha$ for $t\in I$ is equivalent to the statement that the solution of the GP hierarchy remains in $\cH^\alpha_{\xi}$ for [*an arbitrary*]{} nonzero $\xi>0$. \[rem-Strichartz-1\] We note that the estimate (\[eqn-BGamma-spacetime-bd-0\]), for the cubic GP hierarchy with $d=3$ and $\alpha=1$, proves the a priori spacetime bound conjectured in [@klma]. For factorized initial data $\Gamma=(|\phi_0\rangle\langle\phi_0|^{\otimes k})_{k\in\N}$ in the cubic case, so that $\Gamma=(|\phi(t)\rangle\langle\phi(t)|^{\otimes k})_{k\in\N}$ where $i\partial_t\phi+\Delta\phi-\mu|\phi|^2\phi=0$, it corresponds to the inequality ||\^2 \_[L\^1\_[tI]{}H\^]{}\^[13]{} C(T) \_0 \_[H\^]{} which is of Strichartz type. The example of factorized solutions with $\phi(t)\in H^1$, $t\in I$, is discussed in detail in [@klma]. We say that a solution $\Gamma(t)$ of the GP hierarchy blows up in finite time with respect to $H^\alpha$ if there exists $T^*<\infty$ such that for every $\xi>0$ there exists $T_{\xi,\Gamma}^*<T^*$ such that $\|\Gamma(t)\|_{\cH_{\xi}^\alpha}\rightarrow\infty$ as $t\nearrow T^{*}_{\xi,\Gamma}$, and $T_{\xi,\Gamma}^*\nearrow T^*$ as $\xi\rightarrow0$. For the study of blowup solutions, it is convenient to introduce the following quantity. \[def-AvHLp-1\] We refer to \_[H\^]{}() := \^[-1]{} , \_[L\^r]{}() := \^[-1]{} , respectively, as the typical (or average) $H^\alpha$-energy and the typical $L^r$-norm per particle. We note that = ( | |\^[k]{} )\_[k]{} \_[H\^]{}() = \_[H\^]{}\^2 \_[L\^r]{}() = \_[L\^r]{}\^2 in the factorized case. The fact that $\Gamma\in\cH_\xi^\alpha$ means that the typical energy per particle is bounded by $\Av_{H^\alpha}(\Gamma )<\xi^{-1}$. Therefore, the parameter $\xi$ determines the $H^\alpha$-energy scale in the problem. While solutions with a bounded $H^\alpha$-energy remain in the same $\cH_\xi^\alpha$, blowup solutions undergo transitions $\cH_{\xi_1}^\alpha\rightarrow\cH_{\xi_2}^\alpha\rightarrow\cH_{\xi_3}^\alpha\rightarrow\cdots$ where the sequence $\xi_1>\xi_2>\cdots$ converges to zero as $t\rightarrow T^*$. It is easy to see that blowup in finite time of $\Gamma(t)$ with respect to $H^\alpha$ is equivalent to the statement that $\Av_{H^\alpha}(\Gamma(t))\rightarrow\infty$ as $t\nearrow T^*$. Clearly, $(\Av_{N}(\Gamma ))^{-1}$ is the convergence radius of $\|\Gamma\|_{\cN_\xi}$ as a power series in $\xi$, for the norms $N=H^\alpha,L^r$ and $\cN_\xi=\cH^\alpha_\xi ,\cL^r_\xi$, respectively. \[thm-blowuprate-L2crit-1\] Assume that $\Gamma(t)$ is a solution of the (cubic $p=2$ or $p=4$ quintic) $p$-GP hierarchy with initial condition $\Gamma(t_0)=\Gamma_{0}\in\cH_{\xi}^\alpha$, for some $\xi>0$, which blows up in finite time. Then, the following lower bounds on the blowup rate hold: 1. Assume that $\frac4d\leq p < \frac{4}{d-2\alpha}$. Then, ( \_[H\^]{}((t)) )\^[12]{} &gt; . Thus specifically, for the cubic GP hierarchy in $d=2$, and for the quintic GP hierarchy in $d=1$, ( \_[H\^1]{}((t )) )\^[12]{} , with respect to the Sobolev spaces $H^\alpha$, $\cH_\xi^\alpha$. 2. ( \_[L\^r]{}((t )) )\^[12]{} , &lt; r. We note that in the factorized case, the above lower bounds on the blow-up rate coincide with the known lower bounds on the blow-up rate for solutions to the NLS (see, for example, [@ca]). The cubic GP hierarchy in $d=2$, and the quintic GP hierarchy in $d=1$ are distinguished by being invariant under a class of generalized pseudoconformal transformations, as presented below. Let us first recall pseudoconformal invariance on the level of the NLS (\[eq-NLS-p-1\]). If the NLS (\[eq-NLS-p-1\]) is $L^2$-critical, that is, $p=\frac 4d$, it is invariant under the pseudoconformal transformations \[eq-NLSpc-def-1\] [P]{}\_t (x) := e\^[-i]{} \_( ) , for $b\in\R\setminus\{0\}$. That is, i\_t[P]{}\_t + \_t - |[P]{}\_t|\^p [P]{}\_t = 0 ; see for instance [@ca]. There are two cases of $L^2$-critical NLS with $p\in\N$: The cubic ($p=2$) NLS in $d=2$, and the quintic ($p=4$) NLS in $d=1$. For the GP hierarchy, one can likewise introduce pseudoconformal transformations, and as we prove in this paper, the GP hierarchy is pseudoconformally invariant when $p=2$ and $d=2$ (cubic), or $p=4$ and $d=1$ (quintic). This property is independent of whether the GP hierarchy is defocusing, $\mu=1$, or focusing, $\mu=-1$. \[thm-pseudoconf-gamma-1\] For $d=2$ and $p=2$ (cubic), or $d=1$ and $p=4$ (quintic), the focusing or defocusing ($\mu\in\{1,-1\}$) GP hierarchy is invariant under the pseudoconformal transformations \[eq-pseudoconf-gamma-1\]\ && := e\^[-i]{} \^[(k)]{}( , ; ) , for $b\in\R\setminus\{0\}$. That is, \[eq-infhierarch-pseudoconf-1\] i\_t \^[(k)]{} + \_\^[(k)]{}\^[(k)]{} - B\_[k+p2]{} \^[(k+2)]{} = 0 , for all $k\geq1$. The proof is given in Section \[sec-confinv-1\]. For a survey of related matters for the NLS, see for instance [@ca; @ra; @ta]. Of course, the following is immediately clear. Assume that $\alpha\in\alphaset(d,p)$ where $d\geq1$ and $p\in\{2,4\}$. Moreover, assume that $\Gamma(t)\in\cH_{\xi_2}^\alpha$ solves the (cubic or quintic) focusing ($\mu=-1$) GP hierarchy with factorized initial condition $\Gamma_0=(|\phi_0\rangle\langle\phi_0|^{\otimes k})_{k\in\N}\in\cH_{\xi }^\alpha$ for some $\xi>0$, where $\phi_0\in H^\alpha $.\ \ Then, if there exists $T^*<\infty$ such that $\|\phi(t)\|_{H^\alpha}\rightarrow\infty$ as $t\nearrow T^*$, it follows that also $\Av_{H^\alpha}(\Gamma(t))\rightarrow\infty$ as $t\nearrow T^*$. This follows from $\Av_{H^\alpha}(\Gamma(t))=\|\phi(t)\|_{H^\alpha}^2$ for product states. For various scenarios in which blowup occurs for solutions of the cubic or quintic NLS, we refer to the literature; see for instance [@ca; @ra] for surveys. Local existence and uniqueness of solutions for the focusing and defocusing GP hierarchy {#sec-locwp-1} ======================================================================================== In this section, we prove a local existence and uniqueness result for the cubic and quintic GP hierarchy for both focusing and defocusing interactions. We formulate all arguments for the cubic hierarchy ($p=2$). For the quintic hierarchy ($p=4$), the generalizations are straightforward, and will only be briefly described. We introduce the notation \_\^[(k)]{} = \_[\_k]{} - \_[’\_k]{} with \_[\_k]{} = \_[j=1]{}\^k\_[x\_j]{} and \_[, x\_j]{} = \_[x\_j]{} - \_[x\_j’]{} . Moreover, we write \_ := ( \^[(k)]{}\_\^[(k)]{} )\_[k]{} and := ( B\_[k+1]{} \^[(k+1)]{} )\_[k]{} . Then, the $p$-GP hierarchy can be written as \[seclwp-pGP\] i\_t + \_ = . In integral formulation, it is formally given by \[int-pGP\] (t) = e\^[it\_]{}\_0 - i \_0\^t ds e\^[i(t-s)\_]{} (s) . We will prove the local existence and uniqueness of such a solution in the following manner. We note that (\[int-pGP\]) can be formally written as a system of integral equations (t) & = & e\^[it\_]{}\_0 - i \_0\^t ds e\^[i(t-s)\_]{} (s) \[int-pGP-sys-1\]\ (t) & = & e\^[it\_]{}\_0 - i \_0\^t ds e\^[i(t-s)\_]{} (s) , \[int-pGP-sys-2\] where we note that the second line is formally a self-consistent fixed point equation for $\opB\Gamma$. Let $I:=[0,T]$. We introduce the product space \[def-Wspace-1\] := L\^\_[tI]{}\_\^ L\^1\_[tI]{}\_\^ , which we endow with the norm (, ) \_ := \_[L\^\_[tI]{}\_\^]{} + \_[L\^1\_[tI]{}\_\^]{} . Clearly, $( \Wspacexi , \| \, \cdot \, \|_{\Wspacexi} )$ is a Banach space. Then, we introduce the system (t) & = & e\^[it\_]{}\_0 - i \_0\^t ds e\^[i(t-s)\_]{} (s) \[int-pGP-sys-Wspace-1\]\ (t) & = & e\^[it\_]{}\_0 - i \_0\^t ds e\^[i(t-s)\_]{} (s) , \[int-pGP-sys-Wspace-2\] which is formally equivalent to the system (\[int-pGP-sys-1\]), (\[int-pGP-sys-2\]). Our main result in this section is the following existence and uniqueness theorem, which corresponds to part (i) in Theorem \[thm-main-0\]. \[thm-locwp-main-1\] Assume that $\alpha\in\alphaset(d,p)$ where $d\geq1$ and $p\in\{2,4\}$. Then, the following holds. For every $\Gamma_0\in\cH_{{\xi_1}}^\alpha$, there exist constants $T>0$ and $0<\xi_2\leq\xi_1$ such that for $t\in I=[0,T]$, there exists a unique solution $(\Gamma(t),\Xi(t))$ of the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]), in the space $\Wspace$. Moreover, the relation (t)=(t)L\^1\_[t I]{}\_[\_2]{}\^, holds for the solution, and the component $\Gamma(t)$ satisfies the $p$-GP hierarchy (\[seclwp-pGP\]), \[eq-GPhier-thm-1\] i\_t + \_ - = 0, with initial condition $\Gamma(0)=\Gamma_0\in\cH_{\xi_1}^\alpha$. Moreover, there exists a constant $C(T,d,p,\xi_1,\xi_2)$ such that \_[L\^1\_[tI]{}\_[\_2]{}\^]{} C(T,d,p,\_1,\_2) \_0 \_[\_[\_1]{}\^]{} holds. We note that our local existence and uniqueness result differs than the one proven in [@esy1; @esy2] in that performing the contraction principle on the space $\Wspace$ presumes finiteness of $\| \, \opB \Gamma \, \|_{L^1_{t\in I}\cH_{\xi_2}^\alpha}$. However, [^1] it is a priori conceivable that there exist solutions of the $p$-GP hierarchy $\Gamma(t)\in \cH_{\xi_2}^\alpha$ for which $\| \, \opB \Gamma \, \|_{L^1_{t\in I}\cH_{\xi_2}^\alpha}$ is not finite. We note that finiteness of $\| \, \opB \Gamma \, \|_{L^1_{t\in I}\cH_{\xi_2}^\alpha}$ is an essential element of this analysis, and has been assumed without proof in [@klma] as a key ingredient. It was previously unknown whether this condition is generally satisfied for solutions of the GP hierarchy. Using Theorem \[thm-locwp-main-1\], we can prove that $\| \, \opB \Gamma \, \|_{L^1_{t\in I}\cH_{\xi_2}^\alpha}<\infty$ is a sufficient and necessary condition for a solution $\Gamma(t)\in\cH_{\xi_2}^\alpha$ of (\[seclwp-pGP\]), with initial condition $\Gamma(0)\in\cH_{\xi_1}^\alpha$, to be unique. Accordingly, we arrive at the following result which corresponds to part (ii) in Theorem \[thm-main-0\]. \[thm-locwp-main-1-L1bd\] Assume that $\alpha\in\alphaset(d,p)$ where $d\geq1$ and $p\in\{2,4\}$. Assume moreover that for $\Gamma_0\in\cH_{{\xi_1}}^\alpha$ there exist constants $T>0$ and $0<\xi_2\leq\xi_1$ such that for $t\in I=[0,T]$, there exists a solution $\Gamma(t)$ of the $p$-GP hierarchy (\[seclwp-pGP\]) in the space $L^\infty_{t\in I}\cH_{\xi_2}^\alpha$. Then, the solution $\Gamma(t)\in L^\infty_{t\in I}\cH_{\xi_2}^\alpha$ is [*unique*]{} if and only if $\| \, \opB \Gamma \, \|_{L^1_{t\in I}\cH_{\xi}^\alpha}<\infty$ holds for some $\xi>0$. If the latter holds, then in fact, there exists a constant $C(T,d,p,\xi_1,\xi_2)$ such that \[eq-Strichartz-Wspace-1-1\] \_[L\^1\_[tI]{}\_[\_2]{}\^]{} C(T,d,p,\_1,\_2) \_0 \_[\_[\_1]{}\^]{} is satisfied. We first prove Theorem \[thm-locwp-main-1\]. We make the key observation that the fixed point equation (\[int-pGP-sys-Wspace-2\]) determining the component $\Xi(t)$ of the desired solution $(\Gamma(t),\Xi(t))$ is self-contained, and independent of the component $\Gamma(t)$. Therefore, we can invoke the Picard fixed point principle on the space $L^1_{t\in I}\cH_{\xi_2}^\alpha$ (the second factor of the product space $\Wspace$) to first find a unique $\Xi\in L^1_{t\in I}\cH_{\xi_2}^\alpha$ solving (\[int-pGP-sys-Wspace-2\]). We recall that the fixed point equation for $\Xi(t)$ is given by \[eq-opBGamma-fixedpt-1\] (t) = e\^[it\_]{} \_0 - i \_0\^t ds e\^[i(t-s)\_]{}(s), in the space $L^1\cH_{\xi_2}^\alpha$. We define, for an admissible sequence of density matrices $\widetilde\Gamma=(\widetilde\gamma^{(k)})_{k\in\N}$, \[eq-Duh-j-def-1\]\ & := & (-i)\^j\_0\^t dt\_1 \_0\^[t\_[j-1]{}]{}dt\_j e\^[i(t-t\_1)\_\^[(k+1)]{}]{}B\_[k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && B\_[k+3]{} B\_[k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}(t\_j) . Then, any solution of (\[eq-opBGamma-fixedpt-1\]) satisfies the fixed point equation (obtained from iterating the Duhamel formula $k$ times for the $k$-th component of $\opB\Gamma$) ()\^[(k)]{}(t) = \_[j=1]{}\^[k-1]{}B\_[k+1]{}\_j(\_0)\^[(k+1)]{}(t) + B\_[k+1]{}\_[k]{}()\^[(k+1)]{}(t) . To formulate a Picard-type fixed point argument, we define ()=( ()\^[(k)]{})\_[k]{} where the $k$-th component is given by \[eq-PhiGamma-fixed-pt\] ()\^[(k)]{}(t) = \_[j=1]{}\^[k-1]{}B\_[k+1]{}\_j(\_0)\^[(k+1)]{}(t) + B\_[k+1]{}\_[k]{}()\^[(k+1)]{}(t) . Similarly as in [@chpa], we use different approaches when $d\geq2$ and when $d=1$. In dimension $d=1$, and for both the cubic and quintic GP hierarchy, we use a spatial a priori bound as in [@chpa] where we refer for details. In dimensions $d\geq2$, we apply the Klainerman-Machedon spacetime bounds similarly to [@klma] and [@kiscst]. This is explained in detail in the Appendix. We first consider the case $d\geq2$. In the sequel, we will often abbreviate H\^\_k := H\^(\^[dk]{}\^[dk]{}) . We invoke Propositions \[prp-spacetime-bd-1\] and \[prp-spacetime-bd-2\] in the Appendix. They generalize the spacetime bounds and “board game" arguments developed in [@klma]. Proposition \[prp-spacetime-bd-2\] implies that for $\Gamma_0=(\gamma_0^{(k)})_{k\in\N}$,\ & &lt; & k C\^k \_[j=1]{}\^[k-1]{} (c T)\^[2]{} \^[(k+j+1)]{}\_0\_[H\^\_[k+j+1]{}]{}\ & &lt; & k (C \_1\^[-1]{})\^[k]{} \_[j=1]{}\^[k-1]{} (c T \_1\^[-2]{})\^[2]{} \_1\^[k+j+1]{} \^[(k+j+1)]{}\_0\_[H\^\_[k+j+1]{}]{}\ & &lt; & (c T \_1\^[-2]{}) k (C \_1\^[-1]{})\^[k]{} \_[j=1]{}\^[k-1]{} \_1\^[k+j+1]{} \^[(k+j+1)]{}\_0\_[H\^\_[k+j+1]{}]{} . \[eq-Duhexp-aux-1\] Therefore,\ & &lt; & (c T \_1\^[-2]{}) \_[k=1]{}\^k (C )\^k \_[j=1]{}\^[k-1]{} \_1\^[k+j+1]{} \^[(k+j+1)]{}\_0\_[H\^\_[k+j+1]{}]{}\ & &lt; & (c T \_1\^[-2]{}) \_[k=1]{}\^k (C )\^k \_[=1]{}\^[2k]{} \_1\^ \^[()]{}\_0\_[H\^\_]{}\ & &lt; & (c T \_1\^[-2]{}) \_[k=1]{}\^k (C )\^k \_0\_[\_[\_1]{}\^]{}\ & &lt; & (c T \_1\^[-2]{}) \_0\_[\_[\_1]{}\^]{} \[eq-Duhexp-aux-2\] for $\xi_2=\eta\xi_1$, with $0<\eta\leq1$ sufficiently small. This implies that, for $I=[0,T]$, and any $T>0$, ( \_[j=1]{}\^[k-1]{}B\_[k]{}\_j(\_0)\^[(k+1)]{}(t) )\_[k]{} L\^1\_[tI]{}\^\_[\_2]{} , provided that $\Gamma_0\in\cH_{\xi_1}^\alpha$, and $\xi_2=\eta\xi_1$ with $\eta>0$ sufficiently small. Our next step is to prove that $\Phi$ is a contraction on $L^1_{t\in I}\cH^\alpha_{\xi_2}$. To this end, we use the bound (\_1)\^[(k)]{} -(\_2)\^[(k)]{} \_[L\_[tI]{}\^1H\^\_k]{} k (C T)\^ \_1\^[(2k)]{} - \_2\^[(2k)]{} \_[L\^1\_[tI]{}H\^\_[2k]{}]{} which is proven in Proposition \[prp-spacetime-bd-2\] in the Appendix. We obtain \[eq-PhiGamma-contr-1\]\ & = & \_[k=1]{}\^\_2\^k (\_1)\^[(k)]{} -(\_2)\^[(k)]{} \_[L\_[tI]{}\^1H\^\_k]{}\ && \_[k=1]{}\^k (C T \_2\^[-2]{})\^ \_2\^[2k]{} \_1\^[(2k)]{} - \_2\^[(2k)]{} \_[L\^1\_[tI]{}H\^\_[2k]{}]{}\ && \_[k1]{}{ k (C T \_2\^[-2]{})\^ } \_[k=1]{}\^ \_2\^[2k]{} \_1\^[(2k)]{} - \_2\^[(2k)]{} \_[L\^1\_[tI]{}H\^\_[2k]{}]{}\ && ( C T \_2\^[-2]{})\^[12]{} \_1 - \_2 \_[L\^1\_[tI]{}\_[\_2]{}\^]{}, for $T>0$ sufficiently small. This implies that $\Phi$ is a contraction on $L^1_{t\in I}\cH^\alpha_{\xi_2}$ provided that $T$ is sufficiently small. Consequently, for a every initial condition $\Gamma_0\in\cH_{\xi_1}^\alpha$, there exist $T>0$ and $\xi_2=\eta\xi_1$ with $0<\eta\leq1$ such that there exists a unique solution $\Xi(t)$ of (\[eq-opBGamma-fixedpt-1\]) in $L^1_{t \in I}\cH^\alpha_{\xi_2}$, Repeating the above arguments, we find that this solution satisfies \[eq-Strichartz-cH-1\] \_[L\^1\_[tI]{}\_[\_2]{}\^]{} &&(c T \_1\^[-2]{}) \_0\_[\_[\_1]{}\^]{} + ( C T \_2\^[-2]{} )\^[12]{} \_[L\^1\_[tI]{}\_[\_2]{}\^]{} . Hence, in particular, \[eqn-BGamma-spacetime-bd-1\] \_[L\^1\_[tI]{}\_[\_2]{}]{} && \_0\_[\_[\_1]{}\^]{} for sufficiently small $T>0$. Given the solution $\Xi$ in $L^1_{t \in I}\cH^\alpha_{\xi_2}$ obtained from the contraction argument (\[eq-PhiGamma-contr-1\]), the right hand side of the integral equation (\[int-pGP-sys-Wspace-1\]) is completely determined by $\Gamma_0$ and $\Xi(s)$ where $s\in I$. Accordingly, $\Gamma(t)$ is given by \[int-pGP-sys-Wspace-1-2\] (t) = e\^[it\_]{}\_0 - i \_0\^t ds e\^[i(t-s)\_]{} (s) where the solution $\Xi(s)$ found in step [*(1)*]{} above has been substituted into the integral. Evidently, \[eq-Gamma-Duh-opBGamma-1\] (t) \_[ \_[\_2]{}\^]{} & & \_0 \_[\_[\_2]{}\^]{} + \_0\^t ds e\^[i(t-s)\_]{}(s)\_[\_[\_2]{}\^]{}\ & & \_0 \_[\_[\_2]{}\^]{} + \_0\^t ds (s)\_[\_[\_2]{}\^]{}\ & & \_0 \_[\_[\_2]{}\^]{} + (s) \_[L\^1\_[sI]{}\_[\_2]{}\^]{} , using the unitarity of $e^{i(t-s)\opDelta_\pm}$ with respect to $\cH_{\xi_2}^\alpha$. By (\[eqn-BGamma-spacetime-bd-1\]), we conclude that the last line is bounded, hence $\Gamma(t)$ determined by (\[int-pGP-sys-Wspace-1-2\]) indeed lies in $L^\infty_{t\in I}\cH_{\xi_2}^\alpha$. To prove uniqueness, we assume that $(\Gamma_1(t),\Xi_1(t))$ and $(\Gamma_2(t),\Xi_2(t))\in\Wspace$ correspond to solutions of the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]) to the same initial condition $\Gamma_1(0)=\Gamma_2(0)\in\cH_{\xi_1}^\alpha$, for $t\in[0,T]$. Then, linearity implies that $(\Gamma_1(t)-\Gamma_2(t),\Xi_1(t))-\Xi_2(t))\in\Wspace$ solves the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]) with initial condition $\Gamma_1(0)-\Gamma_2(0)=(0)_{k\in\N}$. Hence, (\[eq-Gamma-Duh-opBGamma-1\]) implies that \_1(t)-\_2(t) \_[ \_[\_2]{}\^]{} & & ( 1 + ) \_1(0)-\_2(0)\_[\_[\_1]{}\^]{}\ &=&0. This implies that $\Gamma_1(t)=\Gamma_2(t)$, for all $t<T$. This proves local existence and uniqueness of solutions for the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]). Next, we observe that applying $\opB$ to $\Gamma(t)$ produces the rhs of the fixed point equation (\[eq-opBGamma-fixedpt-1\]) for $\Xi(t)$. Therefore, we immediately find that \[eq-XiopBGammadiff-1\] = L\^1\_[tI]{}\_[\_2]{}\^. Finally, it is easy to verify for the unique solution $(\Gamma(t),\Xi(t))\in\Wspace$ of (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]) that i\_t (t) + \_(t) = (t) . Therefore, using (\[eq-XiopBGammadiff-1\]), i\_t (t) + \_(t) - (t) \_[L\^1\_[tI]{}\_[\_2]{}\^]{} = 0 , which implies (\[eq-GPhier-thm-1\]). For the quintic GP hierarchy, all steps of the above proof can be adopted with minor modifications. A key difference is the fact that (\[eq-Gamma-Duh-opBGamma-1\]) is replaced by \[eq-Gamma-Duh-opBGamma-quintic-1\] (t) \_[ \_[\_2]{}\^]{} & & \_0 \_[\_[\_2]{}\^]{} + \_[L\^1\_[tI]{}\_[\_2]{}\^]{}\ & & ( 1 + ) \_0\_[\_[\_1]{}\^]{} . This concluded the proof of Theorem \[thm-locwp-main-1\] for $d\geq2$. In the case $d=1$, we can straightforwardly adapt the proof given in [@chpa] of the uniqueness of solutions of the quintic GP hierarchy in $d=1$. The spacetime bounds of Proposition \[prp-spacetime-bd-1\] is not available in $d=1$ since it would produce divergent bounds. However, the spatial bounds in $d=1$ proven in [@chpa] apply for both the cubic and the quintic GP hierarchy, under the assumption that $\alpha>\frac12$. The result is that we get a factor $t$ instead of $t^{\frac12}$, in all of the bounds found above for the cubic GP hierarchy that produced a factor $t^{\frac12}$. Accordingly, we find \[eq-Gamma-Duh-opBGamma-1D-1\] (t) \_[ \_[\_2]{}\^]{} & & ( 1 + ) \_0\_[\_[\_1]{}\^]{} for the cubic GP hierarchy instead of (\[eq-Gamma-Duh-opBGamma-1\]), and \[eq-Gamma-Duh-opBGamma-quintic-1D-1\] (t) \_[ \_[\_2]{}\^]{} & & ( 1 + ) \_0\_[\_[\_1]{}\^]{} for the quintic GP hierarchy instead of (\[eq-Gamma-Duh-opBGamma-quintic-1\]), respectively. Hence, we obtain local wellposedness for sufficiently small $T>0$. In conclusion, we have proved the existence and uniqueness of solutions of the GP hierarchy (\[seclwp-pGP\]) in the space $\Wspace$ defined in (\[def-Wspace-1\]). It is now straightforward to prove Theorem \[thm-locwp-main-1-L1bd\]. $"\Rightarrow"$. We first prove that uniqueness of $\Gamma\in L^\infty_{t\in I}\cH_{\xi_2}^\alpha$ implies $\| \opB\Gamma \|_{L^1_{t \in I} \cH^\alpha_{\xi}}<\infty$ for some $\xi>0$. Let $t'\in I_1=[0,T_1]$. Assume $\Gamma(t+t')$ is the unique solution in $\cH_{\xi_2}^1$ of the $p$-GP hierarchy for the initial condition $\Gamma(t)\in \cH_{\xi_1}^\alpha$, for $t'\in I_1$, and $0<\xi_1<1$ chosen appropriately. We may also use $\Gamma(t)$ as an initial condition for the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]) in $\mathfrak{W}_{\xi_2}^\alpha(I_2)$. Thereby, we obtain a unique solution $( \Gamma'(t+t') , \Xi(t+t') $ in $\mathfrak{W}_{\xi_2}^\alpha(I_2)$, for $t' \in I_2 = [0,T_2]$, for $T_2>0$ and $0<\xi_2\leq\xi_1$ chosen small enough. Let $I:=[0,\min\{T_1,T_2\}]$. From Theorem \[thm-locwp-main-1\], we infer that $\Gamma'(t+t')$ is also an element of $\cH_{\xi_2}^\alpha$, for $t\in I$, and solves the GP hierarchy equation with initial condition $\Gamma'(t)=\Gamma(t)$ for $t'=0$. Since by assumption, $\Gamma(t+t')$ is the unique solution of the GP hierarchy in $\cH_{\xi_2}^\alpha$, solving the same GP hierarchy equation, and with the same initial condition $\Gamma(t)$ at $t'=0$. Thus, $\Gamma(t+t')=\Gamma'(t+t')$. But since the solution $( \Gamma'(t+t') , \Xi(t+t') )$ in $\Wspace$ for the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]) has the property that $\| \opB\Gamma'(t + . ) \|_{L^1_{t'\in I} \cH^\alpha_{\xi_2}} \leq C(T,d,p,\xi_1,\xi_2) \, \| \Gamma(t) \|_{\cH_{\xi_1}^\alpha}$ is bounded, we conclude that also, $\| \opB\Gamma(t + . ) \|_{L^1_{t'\in I} \cH^\alpha_{\xi_2}} \leq C(T,d,p,\xi_1,\xi_2) \, \| \Gamma(t) \|_{\cH_{\xi_1}^\alpha}$. $"\Leftarrow"$. Next, we prove that $\| \opB\Gamma \|_{L^1_{t \in I} \cH^\alpha_{\xi}}<\infty$ for some $\xi>0$ implies uniqueness of $\Gamma\in L^\infty_{t\in I}\cH_{\xi_2}^\alpha$, for some $\xi_2$. We may set $\xi=\xi_2$ by choosing $\min\{\xi,\xi_2\}$. We note that since $\Gamma\in L^\infty_{t\in I}\cH_{\xi_2}^\alpha$ and $\| \opB\Gamma \|_{L^1_{t \in I} \cH^\alpha_{\xi}}<\infty$, it is clear that $(\Gamma,\opB\Gamma)\in\Wspace$. Accordingly, Theorem \[thm-locwp-main-1\] implies that $\Gamma$ is the unique solution in $\Wspace$ for the initial condition $\Gamma(0)=\Gamma_0\in\cH_{\xi_1}^\alpha$ for which $\| \opB\Gamma \|_{L^1_{t \in I} \cH^\alpha_{\xi}}<\infty$ is satisfied. Finally, Theorem \[thm-locwp-main-1\] implies that whenever $\| \opB\Gamma \|_{L^1_{t \in I} \cH^\alpha_{\xi}}<\infty$ holds for some $\xi>0$, the stronger estimate (\[eq-Strichartz-Wspace-1-1\]) is satisfied. $\;$\ \ On the global existence and uniqueness of solutions for defocusing GP hierarchies {#sec-gwp-GP-1} ================================================================================= It is proved for the $d=2,3$ defocusing cubic case in [@esy1; @esy2; @kiscst], and for the $d=1,2$ defocusing quintic case in [@chpa], that whenever $\Gamma(t)=(\gamma^{(k)}(t))_{k\in\N}$ is obtained from the $N\rightarrow\infty$ limit of a bosonic $N$-particle BBGKY hierarchy with repulsive interactions, \[eq-GPsol-defoc-apriorien-1\] ( S\^[(k,1)]{} \^[(k)]{}\_0 ) &lt; C\^k for some constant $C$, then \[eq-GPsol-defoc-apriorien-2\] ( S\^[(k,1)]{} \^[(k)]{}(t) ) &lt; C\_0\^k with $C_0$ independent of $t\geq0$. This follows from energy conservation in the $N$-particle system of which the GP hierarchy is the $N\rightarrow\infty$ limit. We first claim that this implies that $\Gamma(t)\in\cH_{\xi_1}^1$ for some $\xi_1>0$, and all $t\in\R$. To see this, we consider a fixed $k$. Let $\gamma^{(k)}$ be non-negative, normalized trace class, $\tr(\gamma^{(k)})=1$, and hermitean. Then, we have that \^[(k)]{}(\_k;\_k’) = \_j \_j |\_j(\_k’)\_j(\_k)| for an orthogonal basis $\psi_j$ of $L^2(\R^{dk})$ with $\lambda_j\geq0$ and $\sum\lambda_j=1$. Then, \^[(k)]{} \_[H\^1]{} & = & \_[j,j’]{} \_j \_[j’]{} | \_[\_k]{}\_j | \_[\_k]{}\_[j’]{} |\^2\ && (\_j \_j \_[\_k]{}\_j \^2 )\^2\ &=& ( ( S\^[(k,1)]{} \^[(k)]{} ) )\^2 . Thus, for a solution $\gamma^{(k)}(t)$ of the (cubic or quintic) GP hierarchy with initial condition satisfying (\[eq-GPsol-defoc-apriorien-1\]), we have that \^[(k)]{}(t) \_[H\^1]{} &lt; C\_0\^k with $C_0$ independent of $t$. Thus, for $\xi_1$ sufficiently small, we obtain the a priori bound (t)\_[\_[\_1]{}\^1]{} ( \_[k=1]{}\^(C\_0)\^[ 2k]{} )\^[12]{} &lt; , for all $t\in\R$. This implies the existence of a solution $\Gamma(t)\in\cH_{\xi_1}^1$ globally in time. The key question is whether $\Gamma(t)\in \cH_{\xi_1}^1$ is the unique solution of the $p$-GP hierarchy with initial condition $\Gamma_0\in\cH_{\xi_1}^1$. To this end, we prove the following uniqueness result. \[thm-defocGP-globalwp-1\] Assume that $1\leq d\leq3$ for $p=2$, and in $1\leq d\leq 2$ for $p=4$ such that $1 \in\alphaset(d,p)$. Assume that $\Gamma(t)\in\cH_{\xi_1}^1$, $t\in\R$, is a solution of the defocusing $p$-GP hierarchy for the initial condition $\Gamma(0)=\Gamma_0$, and satisfies the a priori bound $\|\Gamma(t)\|_{\cH_{\xi_1}^1}<c_0<\infty$ for $t\in\R$. Moreover, assume that for every sufficiently short open interval $I\in\R$, there exists a constant $\xi=\xi(I)$ with $0<\xi\leq \xi_1$ such that $\|\opB\Gamma\|_{L^1_{t\in I}\cH_{\xi}^1}<\infty$. Then, there exists a constant $\xi_2=\xi_2(\xi_1)$ depending only on $\xi_1$, and a constant $T_0=T_0(d,p,\xi_1,\xi_2)$ such that the pair $(\Gamma,\opB\Gamma)$ is the unique solution of the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]) associated to the defocusing ($\mu=+1$) $p$-GP hierarchy with initial condition $\Gamma(0)=\Gamma_0\in\cH_{\xi_1}^1$, in the space := \_[I,|I|&lt;T\_0]{}ace , for $0<\xi_2\leq\xi_1$ sufficiently small. Hence in particular, \_[L\^1\_[tI]{}\_[\_2]{}\^1]{} C(T\_0,d,p,\_1,\_2) c\_0 for all $I\in\R$ with $|I|<T_0$. Assume that $I'\subset\R$ is such that $\|\opB\Gamma\|_{L^1_{t\in I'}\cH_{\xi}^1}<\infty$. Let $I:=[\tau,T]\subset I'$ Then, Theorem \[thm-locwp-main-1-L1bd\] implies that in fact, there exists a constant $\xi_2\leq\xi_1$ such that \_[L\^1\_[tI ]{}\_[\_2]{}\^1]{} &lt; C\_0 () \_[\_[\_1]{}\^1]{} &lt; C\_0 c\_0 , for a constant $C_0=C_0(|I|,d,p,\xi_1,\xi_2)$. This estimate holds independently of the value of $\tau\in\R$. Therefore, we can cover $\R$ with intervals $\R=\cup_j I_j$ where $I_j=[\tau_j,\tau_{j+1}]$, $j\in\Z$, and $|I_j|=|I|=: T_0$. Assume that the initial time $t=0\in I_0$. Then, Theorem \[thm-locwp-main-1-L1bd\] implies that there exists a unique (forward and backward in time) solution $(\Gamma,\opB\Gamma)$ of the system (\[int-pGP-sys-Wspace-1\]), (\[int-pGP-sys-Wspace-2\]) in $\mathfrak{W}_{\xi_2}^1(I_0)$ for the initial condition initial condition $\Gamma_0$. By iteration, we use $\Gamma(\tau_{1})$ as the initial condition for $t\in I_1$, and $\Gamma(\tau_0)$ as the terminal condition for $t\in I_{-1}$, and continue recursively, thereby covering $\R=\cup I_j$. In conclusion, we obtain that there exists a unique global in time solution in $\Wspaceglob$ with initial condition $\Gamma_0\in\cH_{\xi_1}^1$, under the assumptions of the theorem. As noted in Remark \[rem-BBGKYlim-1\], every solution $\Gamma(t)\in\cH_{\xi_1}^1$ extracted by a diagonal argument from the $N\rightarrow\infty$ limit of the $N$-particle BBGKY hierarchies (with repulsive interactions) studied by Erdös-Schlein-Yau in [@esy1; @esy2], Kirkpatrick-Schlein-Staffilani in [@kiscst], and Chen-Pavlović in [@chpa], satisfies $\| \, \opB \Gamma \, \|_{L^1_{t\in I}\cH_{\xi}^\alpha}<\infty$ for some $\xi>0$, and all sufficiently short intervals $I\subset\R$. This follows a posteriori from the uniqueness of those solutions, which has been established in these works with independent methods, combined with Theorem \[thm-locwp-main-1-L1bd\] in the work at hand. Accordingly, $(\Gamma,\opB\Gamma)$ is automatically an element of $\Wspaceglob$ whenever $\Gamma(t)$ is obtained from an $N\rightarrow\infty$ limit of the above noted type. $\;$\ \ Lower bound on the blowup rates {#sec-Proof-blowuprate-1} =============================== In this section, we establish Theorem \[thm-blowuprate-L2crit-1\]. We adapt a standard proof given for $L^2$-critical focusing NLS to the GP hierarchy; see for instance [@ra]. Let $p \in \{2,4\}$. Similarly as in (\[eq-pseudoconf-gamma-1\]), one finds that the $p$-GP hierarchy is invariant under the rescaling \[eq-pseudoconf-gamma-2\]\ && := \^[(k)]{}( t +((t))\^[-2]{} , ((t))\^[-1]{}\_k ; ((t))\^[-1]{}\_k’ ) If $\Gamma(t)=(\gamma^{(k)}(t))_{k\in\N}$ solves the $p$-GP hierarchy, then $\cS_{\lambda,t}\Gamma=(\cS_{\lambda,t}\gamma^{(k)} )_{k\in\N}$ is also a solution of the $p$-GP hierarchy. The proof can be straightforwardly adapted from the one given in Section \[sec-confinv-1\]. \ \ Blowup in finite time means that there exists $T^*<\infty$ such that $\Av_{H^\alpha}(\Gamma(t))\rightarrow\infty$ as $t\rightarrow T^*$. To prove a lower bound on the blowup rate, we may assume that $1<\Av_{H^\alpha}(\Gamma(t))<\infty$ at a fixed time $t$, and choose (t) = (\_[H\^]{}((t)))\^ &gt; 1 . We note that $\frac{4}{d}<p<\frac{4}{d-2\alpha}$ implies that $2\alpha-d+\frac4p>0$. Let S\^[(k,)]{}\_[(t)]{} := \_[j=1]{}\^k ((t))\^[-1]{} \_[x\_j]{}\^ ((t))\^[-1]{} \_[x\_j’]{} \^ where $\langle b\nabla_{x}\rangle=\sqrt{1-b^2\Delta_{x}}$ for any $b\in\R$. Clearly, \[eq-Sdot-quasiblowup-L2-aux-1\]\ & = & ((t))\^[k(d-)]{} ( S\^[(k,)]{}\_[(t)]{} \^[(k)]{} ) (t+((t))\^[-2]{}) \_[L\^2\_[\_k,\_k’]{}]{} , and \[eq-scaleSkalpha-bd-1\] ((t))\^[-2k]{} S\^[(k,)]{} S\^[(k,)]{}\_[(t)]{} S\^[(k,)]{} since we are assuming that $\lambda(t)>1$. We define \_&lt;(,t,) := ((t))\^[-4p+d-2]{} = (\_[H\^]{}((t)))\^[-1]{} and \[eq-xibigxi-id-1\] \_&gt;(,t,) := ((t))\^[-4p+d ]{} = (\_[H\^]{}((t)))\^[ ]{} . Clearly, (\[eq-Sdot-quasiblowup-L2-aux-1\]) and (\[eq-scaleSkalpha-bd-1\]) imply that \[eq-sandwich-bd-1\] (t+((t))\^[-2]{}) \_[\_[\_&lt;(,t,)]{}\^]{} & & \_[,t]{}() \_[ \^\_]{}\ && (t+((t))\^[-2]{}) \_[\_[\_&gt;(,t,)]{}\^]{} . As a consequence of the definition of $\Av_{H^\alpha}(\Gamma(t))$, it follows that for $\tau=0$, 0 &lt; (t ) \_[\_[\_&lt;(,t,)]{}\^]{} &lt; c for any $0<\xi<1$. To ensure that $\| \Gamma(t+(\lambda(t))^{-2}\tau) \|_{\cH_{\xi_>(\xi,t,\lambda)}^\alpha}<c$, we use the fact that according to (\[eq-xibigxi-id-1\]), $\xi_>(\xi,t,\lambda) $ can be made arbitrarily small by choosing $\xi$ small. We note that our assumption $\frac{4}{d}<p<\frac{4}{d-2\alpha}$ implies that $2\alpha-d+\frac4p>0$ and $d-\frac4p>0$, so that the exponent on the rhs of (\[eq-xibigxi-id-1\]) is positive. If blowup occurs, such that $\Av_{H^\alpha}(\Gamma(t))\rightarrow\infty$ as $t\nearrow T^*$, the above considerations necessitate the choice of values of $\xi$ (whose reciprocal determines the energy scale) tending to zero as $t\nearrow T^*$. Thus, for $\xi_1>0$ sufficiently small, \_[,t]{}(0) \_[ \^\_[\_1]{}]{} (t ) \_[\_[\_&gt;(\_1,t,)]{}\^]{} Due to Theorem \[thm-locwp-main-1\], we may pick $0<\xi_2=\eta\xi_1\ll\xi_1<1$, such that there exists a solution $\cS_{\lambda,t}\gamma^{(k)} (\tau)\in\cH_{\xi_2}^\alpha$ if $\tau\in[0,\tau_{\max}]$, for $\tau_{\max}>0$ sufficiently small. But this implies that (t+((t))\^[-2]{}) \_[\_[\_&lt;(\_2,t,)]{}\^]{} \_[,t]{}() \_[ \^\_[\_2]{}]{} &lt; for $\tau\in[0,\tau_{\max}]$ so that there is no blowup if $\tau$ lies in that interval. Therefore, the blowup time $T^*$ is bounded from below by T\^\* &gt; t+((t))\^[-2]{}\_ , and hence, ( \_[H\^]{}((t)) )\^[12]{} = (t)\^[(-+)]{} &gt; . This proves ($a$). \ \ It is easy to see that \[eq-Lp-rescale\] \_[,t]{}\^[(k)]{} () \_[L\^r\_[\_k,\_k’]{}]{} = ((t))\^[-2k(2p-)]{} \^[(k)]{} (t+((t))\^[-2]{}) \_[L\^r\_[\_k,\_k’]{}]{} , which, in turn, implies that \_[,t]{}\^[(k)]{} (0) \_[\^r\_]{} &=& \_[k1]{}\^k \_[,t]{}\^[(k)]{} (0) \_[L\^r\_[\_k,\_k’]{}]{}\ & = & \_[k1]{} \^k ((t))\^[-2k(2p-)]{} \^[(k)]{} (t) \_[L\^r\_[\_k,\_k’]{}]{}\ & = & \_[k1]{} ( )\^[k]{} \^[(k)]{} (t) \_[L\^r\_[\_k,\_k’]{}]{} . \[bigLp-lambda\] However is bounded for every $\xi<1$, if we choose (t) = ( \_[L\^r]{}((t)) )\^ . Now we argue as in the proof of the part ($a$) by using the local well-posedness Theorem \[thm-locwp-main-1\] to conclude that the $H^\alpha$ blowup time $T^*$ is bounded from below by T\^\* &gt; t+((t))\^[-2]{}\_ . Therefore ( \_[L\^r]{}((t)) )\^[12]{} = (t)\^[(-)]{} &gt; . Hence ($b$) is proved. Proof of pseudoconformal invariance {#sec-confinv-1} =================================== In this section, we prove Theorem \[thm-pseudoconf-gamma-1\]. We recall the pseudoconformal transformations \[eq-pseudoconf-gamma-1\]\ && := e\^[-i]{} \^[(k)]{}( , ; ) , for any $b\in\R\setminus\{0\}$. Similarly as in the case of NLS, one can verify that\ &=& e\^[-i]{}\ && ((i\_t+ \_[\_k]{}-\_[\_k’]{})\^[(k)]{}) ( , ; ) . Now we shall prove the pseudoconformal invariance of the quintic GP hieararchy when $d=1$. In particular, we find that\ &=& e\^[-i]{} dx\_[k+1]{} dx\_[k+2]{} dx\_[k+1]{}’ dx\_[k+2]{}’\ &&(x\_j-x\_[k+1]{})(x\_j-x\_[k+1]{}’) (x\_j-x\_[k+2]{})(x\_j-x\_[k+2]{}’)\ && \^[(k)]{}( , ; )\ &=& e\^[-i]{}\ && \^[(k)]{}( , ; ) ,\ &=& e\^[-i]{}\ && B\_[j;k+1,k+2]{}\^1\^[(k+2)]{} ( , ; ) with $B_{j;k+1,k+2}=B_{j;k+1,k+2}^1-B_{j;k+1,k+2}^2$; in $B_{j;k+1,k+2}^2$, the variable $x_j$ in $B_{j;k+1,k+2}^1$ is replaced by $x_j'$. Notably, we have used that e\^[-i]{}|\_[x\_[k+1]{}=x\_[k+2]{}=x\_[k+1]{}’=x\_[k+2]{}’=x\_j]{} = 1 . Thus, when $d=1$, we obtain\ &=& e\^[-i]{}\ && ( (i\_t+ \_[\_k]{}-\_[\_k’]{})\^[(k)]{} - \_[j=1]{}\^k B\_[j;k+1,k+2]{}\^[(k+2)]{} ) ( , ; )\ &=&0 . This proves pseudoconformal invariance of the quintic GP hierarchy in dimension $d=1$. For the cubic GP hierarchy, the operators $B_{j;k+1,k+2}$ are replaced by operators $B_{j;k+1}$ which contract $x_j$, $x_j'$ only with $x_{k+1}$ and $x_{k+1}'$, (i\_t+ \_[\_k]{}-\_[\_k’]{})\^[(k)]{} - \_[j=1]{}\^k B\_[j;k+1]{}\^[(k+1)]{} = 0 . The same considerations as above then produce\ &=& e\^[-i]{}\ && ( (i\_t+ \_[\_k]{}-\_[\_k’]{})\^[(k)]{} - \_[j=1]{}\^k B\_[j;k+1]{}\^[(k+1)]{} ) ( , ; )\ &=&0 if $d=2$. This proves Theorem \[thm-pseudoconf-gamma-1\]. The Klainerman-Machedon spacetime bounds {#sect-boundsfreeGP-1} ======================================== We present the Klainerman-Machedon spacetime bounds in dimensions $d\geq2$ in the form required for this paper, with $\alpha\in\alphaset(d,p)$; see (\[eq-alphaset-def-1\]). In the regime $\alpha>\frac d2-\frac{1}{2(p-1)}$, we present a simple argument to prove the result. In the endpoint case $(d,p)=(3,2)$ and $\alpha=1$, we invoke a result of [@klma]. \[prp-spacetime-bd-1\] Let $p=2,4$ account for the cubic and quintic GP hierarchy, respectively, and assume that $\alpha\in\alphaset(d,p)$. Let $\gamma^{(k+\frac p2)}$ be the solution of i\_t\^[(k+p2)]{}(t,\_[k+p2]{};\_[k+p2]{}’) + (\_[\_[k+p2]{}]{}-\_[\_[k+p2]{}’]{}) \^[(k+p2)]{}(t,\_[k+p2]{};\_[k+p2]{}’) = 0 with initial condition \^[(k+p2)]{}(0,) = \_0\^[(k+p2)]{}\^. Then, there exists a constant $C$ such that\ && C S\^[(k+p2,)]{} \_0\^[(k+p2)]{} \_[L\^2 (\^[d(k+p2)]{}\^[d(k+p2)]{})]{} holds. For notational convenience, we discuss the proof for the quintic GP hierarchy where $p=4$. Let $(\tau,\uu_k,\uu_k')$, $\uq:=(q_1,q_2)$, and $\uq':=(q_1',q_2')$ denote the Fourier conjugate variables corresponding to $(t,\ux_k,\ux_k')$, $(x_{k+1},x_{k+2})$, and $(x_{k+1}',x_{k+2}')$, respectively. Without any loss of generality, we may assume that $j=1$ in $B_{j;k+1,k+2}$. Then, abbreviating () := ( + (u\_1+q\_1+q\_2-q\_1’-q\_2’)\^2 + \_[j=2]{}\^k u\_j\^2 + ||\^2 - |\_k’|\^2 - |’|\^2 ) we find\ & = & \_dd\_k d\_k’ \_[j=1]{}\^ku\_j\^[2]{} u\_j’\^[2]{}\ && ( dd’ () \^[(k+2)]{}(,u\_1+q\_1+q\_2-q\_1’-q\_2’,u\_2,…,u\_k,;\_k’,’) )\^2 . Using the Schwarz estimate, this is bounded by &&\_dd\_k d\_k’ I\_(,\_k,\_k’) dd’ ()\ && u\_1+q\_1+q\_2-q\_1’-q\_2’ \^[2]{} q\_1\^[2]{} q\_2\^[2]{} q\_1’\^[2]{} q\_2’ \^[2]{} \_[j=2]{}\^ku\_j\^[2]{} \_[j’=1]{}\^ku\_[j’]{}’\^[2]{}\ && | \^[(k+2)]{}(,u\_1+q\_1+q\_2-q\_1’-q\_2’,u\_2,…,u\_k,;\_k’,’) |\^2 where \[eq-Ialpha-def-1\]\ && := d d’ . Similarly as in [@klma; @kiscst], we observe that $$\label{eq-u1-aux-bd-1} \bra u_1\ket^{2\alpha} \, \leq \, C\Big[ \, \bra u_1+q_1+q_2-q_1'-q_2'\ket^{2\alpha} + \bra q_1\ket^{2\alpha} + \bra q_2\ket^{2\alpha} + \bra q_1'\ket^{2\alpha} + \bra q_2'\ket^{2\alpha} \, \Big] \,, $$ so that I\_(,\_k,\_k’) \_[=1]{}\^5 J\_where $J_\ell$ is obtained from bounding the numerator of (\[eq-Ialpha-def-1\]) using (\[eq-u1-aux-bd-1\]), and from canceling the $\ell$-th term on the rhs of (\[eq-u1-aux-bd-1\]) with the corresponding term in the denominator of (\[eq-Ialpha-def-1\]). Thus, for instance, J\_1 &lt; d d’ , and each of the terms $J_\ell$ with $\ell=2,\dots,5$ can be brought into a similar form by appropriately translating one of the momenta $q_i$, $q_j'$. Further following [@klma; @kiscst], we observe that the argument of the delta distribution equals + (u\_1+q\_1+q\_2-q\_1’)\^2 + \_[j=2]{}\^k u\_j\^2 + ||\^2 - |\_k’|\^2 - (q\_1’)\^2 - 2(u\_1+q\_1+q\_2-q\_1’)q\_2’ , and we integrate out the delta distribution using the component of $q_2'$ parallel to $(u_1+q_1+q_2-q_1')$. This leads to the bound \[eq-J1-bound-aux-1\] J\_1 & &lt; & C\_C dd q\_1’ where C\_ := \_ . Clearly, $C_\alpha$ is finite for any $\alpha>\frac12$. Moreover, it is clear that $J_1$ is monotonically decreasing in $\alpha$. For the cubic GP hierarchy, the above arguments lead to the condition that instead of (\[eq-J1-bound-aux-1\]), the integral \[eq-J1-bound-aux-2\] d q\_1 must be bounded. \ \ We first consider the case $p=4$ corresponding to the quintic GP hierarchy, and argue as follows. To bound (\[eq-J1-bound-aux-1\]), we pick a spherically symmetric function $h\geq0$ with rapid decay away from the unit ball in $\R^d$, such that $h^\vee(x)\geq0$ decays rapidly outside of the unit ball in $\R^d$, and \[eq-convuppbd-1\] &lt; h\*( \_[B\_1]{} + )(q) . (for example, $h(u)= c_1 e^{-c_2 u^2}$, for suitable constants $c_1,c_2$), where $\chi_{B_1} +\chi_{B_1}^c=1$ is a smooth partition of unity with $\chi_{B_1}$ supported on the unit ball, with $\chi_{B_1}(u)=1$ for $|u|\leq\frac12$, and $\chi_{B_1}(u)=0$ for $|u|>\frac12$. Clearly, $h* \chi_{B_1}$ and $h*\frac{\chi_{B_1}^c}{| \, \cdot \, |^{2\alpha}}$ are both in $L^\infty$, for any $\alpha>0$. Then, assuming that $\alpha<\frac d2$, inserting this into (\[eq-J1-bound-aux-1\]), the most singular part is given by \[eq-J1-convFTprod-bd-1\]\ &=& C\_C dx ( )\^(x) ( (h\*)\^(x) )\^3\ &=& C\_C’ dx (h\^(x))\^3 ( ( (\_[B\_1]{}\^c)\^ ) (x) )\^3\ &&lt;& C\_C’ dx (h\^(x))\^3 ( )\^3 . For sufficiently large $C'$, this is an upper bound on all of the remaining terms that are obtained from substituting the bound (\[eq-convuppbd-1\]) into (\[eq-J1-bound-aux-1\]). We have here used that $(\chi_{B_1}^c)^\vee=1^\vee-\chi_{B_1}^\vee=\delta-\chi_{B_1}^\vee$, so that $|((\chi_{B_1}^c)^\vee*\frac{1}{| \, \cdot \, |^{d-2\alpha}})(x)| \leq C \, \frac{1}{|x|^{d-2\alpha}}$ holds for $\alpha<\frac d2$. We conclude that (\[eq-J1-convFTprod-bd-1\]) is finite provided that the singularity at $x=0$ is integrable, since $h^\vee(x)$ falls off rapidly as $|x|\rightarrow\infty$. In dimension $d$, this is the case if the exponents in the denominator satisfy d-1 + 3d - 6 &lt; d , such that \[eq-alpha-lowbd-1\] &gt; d2-16 . This proves the claim for the quintic GP hierarchy, i.e., for $p=4$. In order to prove the lower bound (\[eq-alpha-lowbd-1\]) on $\alpha$, we have assumed that $\alpha<\frac d2$, which is consistent with it. Now, since $J_1$ is monotonically decreasing in $\alpha$, we arrive at the asserted result. For the cubic GP hierarchy, the same considerations lead to the condition that $(\ref{eq-J1-bound-aux-1}) \, < \, \infty$ if $\alpha \, > \, \frac d2-\frac12$. For a general $p$-GP hierarchy, one obtains the condition $\alpha>\frac d2-\frac{1}{2(p-1)}$. \ \ In the situation $d=3$ and $p=2$ of the cubic GP hierarchy in 3 dimensions, we have the endpoint case $\frac d2-\frac1{2(p-1)}=1$. Klainerman and Machedon have proven in [@klma] that (\[eq-Ialpha-def-1\]) is bounded in this case. Next, we prove the iterated spacetime estimates for the cubic GP hierarchy used in Section \[sec-locwp-1\], which involve the “boardgame estimates” of [@klma], which are motivated by the Feynman graph techniques in [@esy1; @esy2; @ey]. The corresponding results for the quintic GP hierarchy are obtained in an analogous manner, and we refer to [@chpa] for details. \[prp-spacetime-bd-2\] Assume $\alpha$ as in Proposition \[prp-spacetime-bd-1\], for the cubic GP hierarchy ($p=2$). For brevity, let H\^\_k := H\^(\^[dk]{}\^[dk]{}) . Then, for $k\geq1$ and $j\leq k$, and $t\in I=[0,T]$, B\_[k+1]{}\_j(\_0)\^[(k+1)]{}(t) \_[L\^1\_[tI]{}H\^\_k]{} &lt; k C\^k (c T)\^[2]{} \^[(k+j+1)]{}\_0\_[H\^\_[k+j+1]{}]{} . Moreover, \[eq-Boardgame-rem-est-1\] B\_[k+1]{}\_[k]{}()\^[(k+1)]{} \_[L\^1\_[tI]{}H\^\_k]{} &lt; k C\^k (c T)\^[k2]{} ()\^[(2k)]{} \_[L\^1\_[t I]{}H\^\_[2k]{}]{} , where $\duh_j( \, \cdot \,)$ is defined in (\[eq-Duh-j-def-1\]). Let $I=[0,T]$. Using an argument presented as a “board game", it is proven in [@klma] that the following holds. Let ${\mathcal E}_{j,k+1}$ denote the space of sequences $\umu=(\mus(1),\dots,\mus(j))$ where $\mu(i)\in\{1,\dots,k+i\}$, where for every $i\in\{1,\dots,j\}$, one has $\mu(i)\geq\mu(i')$ for all $i'>i$. The elements of ${\mathcal E}_{j,k+1}$ parametrize $(k+j)\times j$ matrices in so-called “special upper echelon form" (see [@klma] for definitions). The cardinality of this set satisfies $|{\mathcal E}_{j,k+1}|\leq C^{j+k}$. For every $\umu\in{\mathcal E}_{j,k+1}$, one associates the term\ &:=&\_[D]{} dt\_1 dt\_[j]{} e\^[i(t-t\_1)\_\^[(k+1)]{}]{}B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{}e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 for a measurable subset $D\subset[0,t]^j$. Then, it is proven in [@klma] that \[eq-klma-boardgame-dec-1\]\ && \_[\_[j,k+1]{}]{} B\_[k+1]{} ( \_j(\_0)\^[(k+1)]{}(t) )\_ \_[L\^2\_[tI]{}H\^]{} . For the proof of (\[eq-klma-boardgame-dec-1\]) in the case of the cubic GP hierarchy, we refer to [@klma]. For the case of the quintic GP hierarchy, we refer to [@chpa]. We have, under the given assumptions on $\alpha$, that for $I=[0,T]$ and $D\subset I^j$,\ && \_[=1]{}\^k \_[D]{} dt\_1 dt\_[j]{} B\_[,k+1]{}e\^[i(t-t\_1) \_\^[(k+1)]{}]{} B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[L\^1\_[tI]{}H\^\_k]{}\ && k \_[I]{} dt \_D dt\_1 dt\_[j]{} B\_[,k+1]{}e\^[i(t - t\_1)\_\^[(k+1)]{}]{}B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[ H\^\_k]{}\ && k \_[I\^[j+1]{}]{} dt dt\_1 dt\_[j]{} B\_[,k+1]{}e\^[i(t - t\_1)\_\^[(k+1)]{}]{}B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{}\^[(k+j+1)]{}\_0 \_[ H\^\_k]{} . \[eq-BGamma-Duhj-aux-1\] By Cauchy-Schwarz with respect to the integral in $t$, this is bounded by && k T\^[12]{} \_[I\^[j]{}]{} dt\_1 dt\_[j]{} B\_[,k+1]{}e\^[i(t - t\_1)\_\^[(k+1)]{}]{}B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{} e\^[i t\_j\_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[ L\^2\_t(I)H\^\_k]{}\ && k T\^[12]{} \_[I\^[j]{}]{} dt\_1 dt\_[j]{} B\_[,k+1]{}e\^[i(t - t\_1)\_\^[(k+1)]{}]{}B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[ L\^2\_t()H\^\_k]{} . Using Proposition \[prp-spacetime-bd-1\] and unitarity of $e^{-i t_1\Delta_\pm^{(k+1)}}$, this is bounded by && k (cT)\^[12]{} \_[I\^[j]{}]{} dt\_1 dt\_[j]{} B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[ H\^\_[k+1]{}]{} \[ap-usingKM1\]\ &=& k (c T)\^[12]{} \_[I\^[j-1]{}]{} dt\_2 dt\_[j]{} B\_[(1),k+2]{} e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} B\_[(j),k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[L\^1\_[t\_1]{}(I)H\^\_[k+1]{}]{} . \[ap-Hol\] Iterating the same steps as above, we find, after $j$ steps, & &\ && k (c T)\^ \_[I]{} dt\_[j]{} B\_[(j-1),k+j]{} e\^[i(t\_[j-1]{}-t\_[j]{})\_\^[(k+j)]{}]{} \[ap-KMiterated\]\ && B\_[(j),k+j+1]{} e\^[i t\_j \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[L\^2\_[t\_[j-1]{}I]{}H\^\_[k+j-1]{}]{}\ && k (c T)\^ \_[I]{} dt\_[j]{} B\_[(j),k+j+1]{} e\^[i t\_[j]{} \_\^[(k+j+1)]{}]{} \^[(k+j+1)]{}\_0 \_[H\^\_[k+j]{}]{}\ & & k (c T)\^ \^[(k+j+1)]{}\_0 \_[H\^\_[k+j+1]{}]{} . \[A2final\] In the last step, we used Cauchy-Schwarz in $t_j$, and Proposition \[prp-spacetime-bd-1\]. Then, estimating by $C^{j+k}$ the number of terms in the sum over $\umu\in {\mathcal E}_{j,k+1}$, \[eq-BGamma-Duhj-combin-bd-1\] B\_[k+1]{}\_j(\_0)\^[(k+1)]{}(t) \_[L\^1\_[tI]{}H\^]{} k C\^k (c T)\^[2]{} \^[(k+j+1)]{}\_0\_[H\^]{} , as claimed. In the same manner, we prove (\[eq-Boardgame-rem-est-1\]). In this case, we have\ && \_[=1]{}\^k \_[D]{} dt\_1 dt\_[k]{} B\_[,k+1]{}e\^[i(t-t\_1) \_\^[(k+1)]{}]{} B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && e\^[i t\_[k]{} \_\^[(2k)]{}]{} B\_[ 2k+ 1]{} \^[(2k+ 1)]{}(t\_k) \_[L\^1\_[tI]{}H\^\_k]{}\ && k \_[I]{}dt \_[D]{} dt\_1 dt\_[k]{} B\_[,k+1]{}e\^[i(t-t\_1) \_\^[(k+1)]{}]{} B\_[(1),k+2]{}e\^[i(t\_1-t\_2)\_\^[(k+2)]{}]{}\ && e\^[i t\_[k]{} \_\^[(2k)]{}]{} B\_[ 2k+ 1]{} \^[(2k+ 1)]{}(t\_k) \_[ H\^\_k]{} Applying the same arguments as above between (\[eq-BGamma-Duhj-aux-1\]) and (\[ap-KMiterated\]), with $j=k$, one finds the upper bound & &\ && k (c T)\^ \_[I]{} dt\_[k]{} B\_[(k-1),2k ]{} e\^[i t\_[k]{} \_\^[(2k )]{}]{} \[ap-KMiterated-BGamma\]\ && B\_[ 2k+ 1]{} \^[(2k+ 1)]{}(t\_k) \_[L\^2\_[t\_[k-1]{}I]{}H\^\_[2k -1]{}]{}\ && k (c T)\^ \_[I]{} dt\_[k]{} B\_[ 2k+ 1]{} \^[(2k+ 1)]{}(t\_k) \_[H\^\_[2k]{}]{}\ & = & k (c T)\^ B\_[ 2k+ 1]{} \^[(2k+ 1)]{}(t\_k) \_[L\_[tI]{}\^1H\^\_[2k ]{}]{}\ & = & k (c T)\^ ()\^[(2k)]{}(t\_k) \_[L\_[tI]{}\^1H\^\_[2k ]{}]{} . \[A2final-BGamma\] Invoking the argument used to establish (\[eq-BGamma-Duhj-combin-bd-1\]), we arrive at (\[eq-Boardgame-rem-est-1\]). For more details, we refer to [@chpa]. Acknowledgements {#acknowledgements .unnumbered} ---------------- We thank S. Klainerman and N. Tzirakis for inspiring discussions. We are deeply indebted to B. Schlein for extremely useful comments that helped us to significantly improve a previous version. The work of T.C. was supported by NSF grant DMS-0704031. The work of N.P. was supported NSF grant number DMS 0758247 and an Alfred P. Sloan Research Fellowship. [99]{} R. Adami, G. Golse, A. Teta, [*Rigorous derivation of the cubic NLS in dimension one*]{}, J. Stat. Phys. [**127**]{}, no. 6, 1194–1220 (2007). M. Aizenman, E.H. Lieb, R. Seiringer, J.P. Solovej, J. Yngvason [*Bose-Einstein Quantum Phase Transition in an Optical Lattice Model*]{}, Phys. Rev. A [**70**]{}, 023612 (2004). V. Bach, T. Chen, J. Fröhlich and I. M. Sigal, [*Smooth Feshbach map and operator-theoretic renormalization group methods*]{}, J. Funct. Anal., [**203**]{} (1), 44-92 (2003). T. Cazenave, [*Semilinear Schrödinger equations*]{}, Courant lecture notes [**10**]{}, Amer. Math. Soc. (2003). T. Chen, N. Pavlović, [*The quintic NLS as the mean field limit of a Boson gas with three-body interactions*]{}, Preprint http://arxiv.org/abs/0812.2740. A. Elgart, L. Erdös, B. Schlein, H.-T. Yau, [*Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons*]{}, Arch. Rat. Mech. Anal. [**179**]{}, no. 2, 265–283 (2006). L. Erdös, B. Schlein, H.-T. Yau, [*Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate*]{}, Comm. Pure Appl. Math. [**59**]{} (12), 1659–1741 (2006). L. Erdös, B. Schlein, H.-T. Yau, [*Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems*]{}, Invent. Math. [**167**]{} (2007), 515–614. L. Erdös, H.-T. Yau, [*Derivation of the nonlinear Schrödinger equation from a many body Coulomb system*]{}, Adv. Theor. Math. Phys. [**5**]{}, no. 6, 1169–1205 (2001). J. Fröhlich, S. Graffi, S. Schwarz, [*Mean-field- and classical limit of many-body Schrödinger dynamics for bosons*]{}, Comm. Math. Phys. [**271**]{}, no. 3, 681–697 (2007). J. Fröhlich, A. Knowles, A. Pizzo, [*Atomism and quantization*]{}, J. Phys. A [**40**]{}, no. 12, 3033–3045 (2007). J. Fröhlich, A. Knowles, S. Schwarz [*On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction*]{}, Preprint arXiv:0805.4299. K. Hepp, [*The classical limit for quantum mechanical correlation functions*]{}, Comm. Math. Phys. [**35**]{}, 265–277 (1974). S. Klainerman, M. Machedon, [*On the uniqueness of solutions to the Gross-Pitaevskii hierarchy*]{}, Commun. Math. Phys. [**279**]{}, no. 1, 169–185 (2008). K. Kirkpatrick, B. Schlein, G. Staffilani, [*Derivation of the two dimensional nonlinear Schrödinger equation from many body quantum dynamics*]{}, Preprint arXiv:0808.0505. E.H. Lieb, R. Seiringer, [*Proof of Bose-Einstein condensation for dilute trapped gases*]{}, Phys. Rev. Lett. [**88**]{}, 170409 (2002). E.H. Lieb, R. Seiringer, J.P. Solovej, J. Yngvason, [*The mathematics of the Bose gas and its condensation*]{}, Birkhäuser (2005). E.H. Lieb, R. Seiringer, J. Yngvason, [*A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas*]{}, Commun. Math. Phys. [**224**]{} (2001). G. Perelman, [*On the formation of singularities in solutions of the critial nonlinear Schrödinger equation*]{}, Ann. H. Poinc., [**2**]{} (4), 605–673 (2001). P. Raphael, [*Stability and blow up for the non linear Schrödinger Equation*]{}, Lecture notes for CMI workshop, Zürich (2008). I. Rodnianski, B. Schlein, [*Quantum fluctuations and rate of convergence towards mean field dynamics*]{}, preprint arXiv:math-ph/0711.3087. B. Schlein, [*Derivation of Effective Evolution Equations from Microscopic Quantum Dynamics*]{}, Lecture notes for the minicourse held at the 2008 CMI Summer School in Zurich. H. Spohn, [*Kinetic Equations from Hamiltonian Dynamics*]{}, Rev. Mod. Phys. [**52**]{}, no. 3, 569–615 (1980). T. Tao, [*Nonlinear dispersive equations. Local and global analysis*]{}, CBMS 106, eds: AMS, 2006. [^1]: We thank B. Schlein for calling our attention to this fact.

--- abstract: 'Recently Marcus, Spielman and Srivastava gave a spectacular proof of a theorem which implies a positive solution to the Kadison–Singer problem. We extend (and slightly sharpen) this theorem to the realm of hyperbolic polynomials. A benefit of the extension is that the proof becomes coherent in its general form, and fits naturally in the theory of hyperbolic polynomials. We also study the sharpness of the bound in the theorem, and in the final section we describe how the hyperbolic Marcus–Spielman–Srivastava theorem may be interpreted in terms of strong Rayleigh measures. We use this to derive sufficient conditions for a weak half-plane property matroid to have $k$ disjoint bases.' address: 'Department of Mathematics, Royal Institute of Technology, SE-100 44 Stockholm, Sweden' author: - Petter Brändén title: 'Hyperbolic polynomials and the Marcus–Spielman–Srivastava theorem' --- This work is based on notes from a graduate course focused on hyperbolic polynomials and the recent papers [@MSS1; @MSS2] of Marcus, Spielman and Srivastava, given by the author at the Royal Institute of Technology (Stockholm) in the fall of 2013. Introduction ============ Recently Marcus, Spielman and Srivastava [@MSS2] gave a spectacular proof of Theorem \[MSSmain\] below, which implies a positive solution to the infamous Kadison–Singer problem [@KS]. One purpose of this work is to extend Theorem \[MSSmain\] to the realm of hyperbolic polynomials. Although our proof essentially follows the setup in [@MSS2], a benefit of the extension (Theorem \[t1\]) is that the proof becomes coherent in its general form, and fits naturally in the theory of hyperbolic polynomials. We study the sharpness of the bound in Theorem \[t1\]. We prove that a conjecture in [@MSS2] on the sharpness of the bound (Conjecture \[maxmax\] in this paper) is equivalent to the seemingly weaker Conjecture \[maxmax2\]. Using known results about the asymptotic behavior of the largest zero of Jacobi polynomials, we prove in Section \[sbound\] that the bound is close to being optimal in the hyperbolic setting, see Proposition \[lowprop\]. In the final section we describe how Theorem \[t1\] may be interpreted in terms of strong Rayleigh measures. We use this to derive sufficient conditions for a weak half-plane property matroid to have $k$ disjoint bases. These conditions are very different from Edmonds characterization in terms of the rank function of the matroid [@Edm]. The following theorem is a stronger version of Weaver’s $KS_k$ conjecture [@We] which is known to imply a positive solution to the Kadison–Singer problem [@KS]. See [@Cas] for a review of the many consequences of Theorem \[MSSmain\]. \[MSSmain\] Let $k \geq 2$ be an integer. Suppose ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_m \in {\mathbb{C}}^d$ satisfy $\sum_{i=1}^m {\mathbf{v}}_i{\mathbf{v}}_i^* = I$, where $I$ is the identity matrix. If $\|{\mathbf{v}}_i \|^2 \leq \epsilon$ for all $1\leq i \leq m$, then there is a partition of $S_1\cup S_2 \cup \cdots \cup S_k=[m]:=\{1,2,\ldots,m\}$ such that $$\label{sqbound1} \left\| \sum_{i \in S_j} {\mathbf{v}}_i{\mathbf{v}}_i^* \right\| \leq \frac {(1+\sqrt{k\epsilon})^2} k,$$ for each $j \in [k]$, where $\|\cdot \|$ denotes the operator matrix norm. Hyperbolic polynomials are multivariate generalizations of real–rooted polynomials, which have their origin in PDE theory where they were studied by Petrovsky, Gårding, Bott, Atiyah and Hörmander, see [@ABG; @Ga; @Horm]. During recent years hyperbolic polynomials have been studied in diverse areas such as control theory, optimization, real algebraic geometry, probability theory, computer science and combinatorics, see [@Pem; @Ren; @Vin; @Wag] and the references therein. A homogeneous polynomial $h({\mathbf{x}}) \in {\mathbb{R}}[x_1, \ldots, x_n]$ is *hyperbolic* with respect to a vector ${\mathbf{e}}\in {\mathbb{R}}^n$ if $h({\mathbf{e}}) \neq 0$, and if for all ${\mathbf{x}}\in {\mathbb{R}}^n$ the univariate polynomial $t \mapsto h(t{\mathbf{e}}-{\mathbf{x}})$ has only real zeros. Here are some examples of hyperbolic polynomials: 1. Let $h({\mathbf{x}})= x_1\cdots x_n$. Then $h({\mathbf{x}})$ is hyperbolic with respect to any vector ${\mathbf{e}}\in {\mathbb{R}}_{++}^n=(0,\infty)^n$: $$h(t{\mathbf{e}}-{\mathbf{x}}) = \prod_{j=1}^n (te_j-x_j).$$ 2. Let $X=(x_{ij})_{i,j=1}^n$ be a matrix of $n(n+1)/2$ variables where we impose $x_{ij}=x_{ji}$. Then $\det(X)$ is hyperbolic with respect to $I=\diag(1, \ldots, 1)$. Indeed $t \mapsto \det(tI-X)$ is the characteristic polynomial of the symmetric matrix $X$, so it has only real zeros. More generally we may consider complex hermitian $Z=(x_{jk}+iy_{jk})_{j,k=1}^n$ (where $i = \sqrt{-1}$) of $n^2$ real variables where we impose $x_{jk}=x_{kj}$ and $y_{jk}=-y_{kj}$, for all $1\leq j,k \leq n$. Then $\det(Z)$ is a real polynomial which is hyperbolic with respect to $I$. 3. Let $h({\mathbf{x}})=x_1^2-x_2^2-\cdots-x_n^2$. Then $h$ is hyperbolic with respect to $(1,0,\ldots,0)^T$. Suppose $h$ is hyperbolic with respect to ${\mathbf{e}}\in {\mathbb{R}}^n$. We may write $$\label{dalambdas} h(t{\mathbf{e}}-{\mathbf{x}}) = h({\mathbf{e}})\prod_{j=1}^d (t - \lambda_j({\mathbf{x}})),$$ where ${\lambda_{\rm max}}({\mathbf{x}})=\lambda_1({\mathbf{x}}) \geq \cdots \geq \lambda_d({\mathbf{x}})={\lambda_{\rm min}}({\mathbf{x}})$ are called the *eigenvalues* of ${\mathbf{x}}$ (with respect to ${\mathbf{e}}$), and $d$ is the degree of $h$. In particular $$\label{prolambda} h({\mathbf{x}}) = h({\mathbf{e}})\lambda_1({\mathbf{x}}) \cdots \lambda_d({\mathbf{x}}).$$ By homogeneity $$\label{dilambdas} \lambda_j(s{\mathbf{x}}+t{\mathbf{e}})= \begin{cases} s\lambda_j({\mathbf{x}})+t &\mbox{ if } s\geq 0 \mbox{ and } \\ s\lambda_{d-j}({\mathbf{x}})+t &\mbox{ if } s \leq 0 \end{cases},$$ for all $s,t \in {\mathbb{R}}$ and ${\mathbf{x}}\in {\mathbb{R}}^n$. The (open) *hyperbolicity cone* is the set $$\Lambda_{\tiny{++}}= \Lambda_{\tiny{++}}({\mathbf{e}})= \{ {\mathbf{x}}\in {\mathbb{R}}^n : {\lambda_{\rm min}}({\mathbf{x}}) >0\}.$$ We denote its closure by $\Lambda_{\tiny{+}}= \Lambda_{\tiny{+}}({\mathbf{e}})=\{ {\mathbf{x}}\in {\mathbb{R}}^n : {\lambda_{\rm min}}({\mathbf{x}}) \geq 0\}$. Since $h(t{\mathbf{e}}-{\mathbf{e}})=h({\mathbf{e}})(t-1)^d$ we see that ${\mathbf{e}}\in \Lambda_{\tiny{++}}$. The hyperbolicity cones for the examples above are: 1. $\Lambda_{\tiny{++}}({\mathbf{e}})= {\mathbb{R}}_{++}^n$. 2. $\Lambda_{\tiny{++}}(I)$ is the cone of positive definite matrices. 3. $\Lambda_{\tiny{++}}(1,0,\ldots,0)$ is the *Lorentz cone* $$\left\{{\mathbf{x}}\in {\mathbb{R}}^n : x_1 > \sqrt{x_2^2+\cdots+x_n^2}\right\}.$$ The following theorem collects a few fundamental facts about hyperbolic polynomials and their hyperbolicity cones. For proofs see [@Ga; @Ren]. \[hypfund\] Suppose $h$ is hyperbolic with respect to ${\mathbf{e}}\in {\mathbb{R}}^n$. 1. $\Lambda_+({\mathbf{e}})$ and $\Lambda_{++}({\mathbf{e}})$ are convex cones. 2. $\Lambda_{++}({\mathbf{e}})$ is the connected component of $$\{ {\mathbf{x}}\in {\mathbb{R}}^n : h({\mathbf{x}}) \neq 0\}$$ which contains ${\mathbf{e}}$. 3. ${\lambda_{\rm min}}: {\mathbb{R}}^n \rightarrow {\mathbb{R}}$ is a concave function, and ${\lambda_{\rm max}}: {\mathbb{R}}^n \rightarrow {\mathbb{R}}$ is a convex function. 4. If ${\mathbf{e}}' \in \Lambda_{++}({\mathbf{e}})$, then $h$ is hyperbolic with respect to ${\mathbf{e}}'$ and $\Lambda_{++}({\mathbf{e}}')=\Lambda_{++}({\mathbf{e}})$. Recall that the *lineality space*, $L(C)$, of a convex cone $C$ is $C \cap -C$, i.e., the largest linear space contained in $C$. It follows that $L(\Lambda_+)= \{{\mathbf{x}}: \lambda_i({\mathbf{x}})=0 \mbox{ for all } i\}$, see e.g. [@Ren]. The *trace*, *rank* and *spectral radius* (with respect to ${\mathbf{e}}$) of ${\mathbf{x}}\in {\mathbb{R}}^n$ are defined as for matrices: $$\tr({\mathbf{x}}) = \sum_{i=1}^d\lambda_i({\mathbf{x}}), \ \ \rk({\mathbf{x}})= \#\{ i : \lambda_i({\mathbf{x}})\neq 0\} \ \ \mbox{ and } \ \ \|{\mathbf{x}}\| = \max_{1\leq i\leq d} |\lambda_i({\mathbf{x}})|.$$ Note that $\| {\mathbf{x}}\| = \max\{ {\lambda_{\rm max}}({\mathbf{x}}), -{\lambda_{\rm min}}({\mathbf{x}})\}$ and hence $ \| \cdot \|$ is convex by Theorem \[hypfund\] (3). It follows that $\| \cdot \|$ is a seminorm and that $\| {\mathbf{x}}\|=0$ if and only if ${\mathbf{x}}\in L(\Lambda_+)$. Hence $\| \cdot \|$ is a norm if and only if $L(\Lambda_+)=\{0\}$. The following theorem is a generalization of Theorem \[MSSmain\] to hyperbolic polynomials. \[t1\] Let $k\geq 2$ be an integer and $\epsilon$ a positive real number. Suppose $h$ is hyperbolic with respect to ${\mathbf{e}}\in {\mathbb{R}}^n$, and let ${\mathbf{u}}_1, \ldots, {\mathbf{u}}_m \in \Lambda_{+}$ be such that - $\rk({\mathbf{u}}_i) \leq 1$ for all $1\leq i \leq m$, - $\tr({\mathbf{u}}_i) \leq \epsilon$ for all $1\leq i \leq m$, and - ${\mathbf{u}}_1+ {\mathbf{u}}_2+\cdots+ {\mathbf{u}}_m={\mathbf{e}}$. Then there is a partition of $S_1\cup S_2 \cup \cdots \cup S_k=[m]$ such that $$\label{sqbound} \left\| \sum_{i \in S_j} {\mathbf{u}}_i \right\| \leq \frac 1 k \delta(k\epsilon, m),$$ for each $j \in [k]$, where $$\delta(\alpha, m):=\left( 1-\frac 1 m +\sqrt{\alpha - \frac 1 m \left(1-\frac 1 m\right)}\right)^2.$$ We recover (a slightly improved) Theorem \[MSSmain\] when $h= \det$ in Theorem \[t1\]. Compatible families of polynomials ================================== Let $f$ and $g$ be two real–rooted polynomials of degree $n-1$ and $n$, respectively. We say that $f$ is an *interleaver* of $g$ if $$\beta_1 \leq \alpha_1\leq \beta_2 \leq \alpha_2 \leq \cdots \leq \alpha_{n-1} \leq \beta_n,$$ where $\alpha_1 \leq \cdots \leq \alpha_{n-1}$ and $\beta_1 \leq \cdots \leq \beta_{n}$ are the zeros of $f$ and $g$, respectively. A family of polynomials $\{f_1(x), \ldots, f_m(x)\}$ of real–rooted polynomials of the same degree and the same sign of leading coefficients is called *compatible* if it satisfies any of the equivalent conditions in the next theorem. Theorem \[CS\] has been discovered several times. We refer to [@CS Theorem 3.6] for a proof. \[CS\] Let $f_1(x), \ldots, f_m(x)$ be real–rooted polynomials of the same degree and with positive leading coefficients. The following are equivalent. 1. $f_1(x), \ldots, f_m(x)$ have a common interleaver. 2. for all $p_1, \ldots, p_m \geq 0$, $\sum_{i}p_i=1$, the polynomial $$p_1f_1(x)+ \cdots+ p_mf_m(x)$$ is real–rooted. \[largestz\] Let $f_1,\ldots, f_m$ be real–rooted polynomials that have the same degree and positive leading coefficients, and suppose $p_1, \ldots, p_m \geq 0$ sum to one. If $\{f_1,\ldots, f_m\}$ is compatible, then for some $1 \leq i \leq m$ with $p_i >0$ the largest zero of $f_i$ is smaller or equal to the largest zero of the polynomial $$f=p_1f_1 + p_2f_2 + \cdots + p_mf_m.$$ If $\alpha$ is the largest zero of the common interleaver, then $f_i(\alpha) \leq 0$ for all $i$, so that the largest zero, $\beta$, of $f(x)$ is located in the interval $[\alpha, \infty)$, as are the largest zeros of $f_i$ for each $1\leq i \leq m$. Since $f(\beta)=0$, there is an index $i$ with $p_i >0$ such that $f_i(\beta) \geq 0$. Hence the largest zero of $f_i$ is at most $\beta$. Suppose $S_1, \ldots, S_m$ are finite sets. A family of polynomials, $\{f({\mathbf{s}};t)\}_{{\mathbf{s}}\in S_1 \times \cdots \times S_m}$, for which all non-zero members are of the same degree and have the same signs of their leading coefficients is called *compatible* if for all choices of independent random variables ${\mathsf{X}}_1 \in S_1, \ldots, {\mathsf{X}}_m \in S_m$, the polynomial $ {\mathbb{E}}f({\mathsf{X}}_1,\ldots, {\mathsf{X}}_n;t) $ is real–rooted. The notion of compatible families of polynomials is less general than that of *interlacing families of polynomials* in [@MSS1; @MSS2]. However since all families appearing here (and in [@MSS1; @MSS2]) are compatible we find it more convenient to work with these. The following theorem is in essence from [@MSS1]. \[expfam\] Let $\{f({\mathbf{s}};t)\}_{{\mathbf{s}}\in S_1 \times \cdots \times S_m}$ be a compatible family, and let  ${\mathsf{X}}_1 \in S_1, \ldots, {\mathsf{X}}_m \in S_m$ be independent random variables such that ${\mathbb{E}}f({\mathsf{X}}_1,\ldots, {\mathsf{X}}_m;t) \not \equiv 0$. Then there is a tuple ${\mathbf{s}}=(s_1, \ldots, s_n) \in S_1 \times \cdots \times S_m$, with ${\mathbb{P}}[{\mathsf{X}}_i=s_i]>0$ for each $1\leq i \leq m$, such that the largest zero of $f(s_1,\ldots, s_m;t)$ is smaller or equal to the largest zero of ${\mathbb{E}}f({\mathsf{X}}_1,\ldots, {\mathsf{X}}_m;t)$. The proof is by induction over $m$. The case when $m=1$ is Lemma \[largestz\], so suppose $m>1$. If $S_m=\{c_1,\ldots, c_k\}$, then $${\mathbb{E}}f({\mathsf{X}}_1,\ldots, {\mathsf{X}}_m;t)= \sum_{i=1}^k q_i {\mathbb{E}}f({\mathsf{X}}_1,\ldots, {\mathsf{X}}_{m-1}, c_i;t),$$ for some $q_i \geq 0$. However $$\sum_{i=1}^k p_i {\mathbb{E}}f({\mathsf{X}}_1,\ldots, {\mathsf{X}}_{m-1}, c_i;t)$$ is real–rooted for all choices of $p_i \geq 0$ such that $\sum_ip_i=1$. By Lemma \[largestz\] and Theorem \[CS\] there is an index $j$ with $q_j>0$ such that ${\mathbb{E}}f({\mathsf{X}}_1,\ldots, {\mathsf{X}}_{m-1}, c_j;t) \not \equiv 0$ and such that the largest zero of ${\mathbb{E}}f({\mathsf{X}}_1,\ldots, {\mathsf{X}}_{m-1}, c_j;t)$ is no larger than the largest zero of ${\mathbb{E}}f({\mathsf{X}}_1,\ldots, {\mathsf{X}}_m;t)$. The theorem now follows by induction. Mixed hyperbolic polynomials ============================ Recall that the *directional derivative* of $h({\mathbf{x}}) \in {\mathbb{R}}[x_1,\ldots, x_n]$ with respect to ${\mathbf{v}}=(v_1,\ldots, v_n)^T \in {\mathbb{R}}^n$ is defined as $$D_{\mathbf{v}}h({\mathbf{x}}) := \sum_{k=0}^n v_k \frac{ \partial h }{\partial x_k}({\mathbf{x}}),$$ and note that $$\label{dvalt} (D_{\mathbf{v}}h)({\mathbf{x}}+t{\mathbf{v}}) = \frac d {dt} h({\mathbf{x}}+ t {\mathbf{v}}) .$$ If $h$ is hyperbolic with respect to ${\mathbf{e}}$, then $$\tr({\mathbf{v}})= \frac {D_{\mathbf{v}}h({\mathbf{e}})}{h({\mathbf{e}})},$$ by . Hence ${\mathbf{v}}\rightarrow \tr({\mathbf{v}})$ is linear. The following theorem is essentially known, see e.g. [@BGLS; @Ga; @Ren]. However we need slightly more general results, so we provide proofs below, when necessary. \[direct\] Let $h$ be a hyperbolic polynomial and let ${\mathbf{v}}\in \Lambda_+$ be such that $D_{\mathbf{v}}h \not \equiv 0$. Then 1. $D_{\mathbf{v}}h$ is hyperbolic with hyperbolicity cone containing $\Lambda_{++}$. 2. The polynomial $h({\mathbf{x}})-yD_{\mathbf{v}}h({\mathbf{x}}) \in {\mathbb{R}}[x_1,\ldots, x_n,y]$ is hyperbolic with hyperbolicity cone containing $\Lambda_{++} \times \{y: y \leq 0\}$. 3. The rational function $${\mathbf{x}}\mapsto \frac {h({\mathbf{x}})}{D_{\mathbf{v}}h({\mathbf{x}})}$$ is concave on $\Lambda_{++}$. (1). See [@BrOp Lemma 4]. (2). The polynomial $h({\mathbf{x}})y$ is hyperbolic with hyperbolicity cone containing $\Lambda_{++} \times \{y : y<0\}$. Hence so is $H({\mathbf{x}},y):= D_{({\mathbf{v}},-1)} h({\mathbf{x}})y= h({\mathbf{x}})- y D_{\mathbf{v}}h({\mathbf{x}})$ by (1). Since $H({\mathbf{e}}',0) = h({\mathbf{e}}') \neq 0$ for each ${\mathbf{e}}' \in \Lambda_{++}$, we see that also $\Lambda_{++}\times \{0\}$ is a subset of the hyperbolicity cone (by Theorem \[hypfund\] (2)) of $H$. (3). If ${\mathbf{x}}\in \Lambda_{++}$, then (by Theorem \[hypfund\] (2)) $({\mathbf{x}},y)$ is in the closure of the hyperbolicity cone of $H({\mathbf{x}},y)$ if and only if $$y \leq \frac {h({\mathbf{x}})}{D_{\mathbf{v}}h({\mathbf{x}})}.$$ Since hyperbolicity cones are convex $$y_1 \leq \frac {h({\mathbf{x}}_1)}{D_{\mathbf{v}}h({\mathbf{x}}_1)} \mbox{ and } y_2 \leq \frac {h({\mathbf{x}}_2)}{D_{\mathbf{v}}h({\mathbf{x}}_2)} \mbox{ imply } y_1+y_2 \leq \frac {h({\mathbf{x}}_1+{\mathbf{x}}_2)}{D_{\mathbf{v}}h({\mathbf{x}}_1+{\mathbf{x}}_2)},$$ for all ${\mathbf{x}}_1,{\mathbf{x}}_2 \in \Lambda_{++}$, from which (3) follows. \[rankalt\] Let $h$ be hyperbolic with hyperbolicity cone $\Lambda_{++}\subseteq {\mathbb{R}}^n$. The rank function does not depend on the choice of ${\mathbf{e}}\in \Lambda_{++}$, and $$\rk({\mathbf{v}})= \max\{ k : D_{\mathbf{v}}^kh \not \equiv 0\}, \quad \mbox{ for all } {\mathbf{v}}\in {\mathbb{R}}^n.$$ That the rank does not depend on the choice of ${\mathbf{e}}\in \Lambda_{++}$ is known, see [@Ren Prop. 22] or [@BrObs Lemma 4.4]. By $$\label{mag} h({\mathbf{x}}-y{\mathbf{v}}) = \left( \sum_{k=0}^{\infty} \frac {(-y)^k D_{\mathbf{v}}^k}{k!} \right) h({\mathbf{x}}).$$ Thus $$h({\mathbf{e}}-t{\mathbf{v}}) = h({\mathbf{e}})\prod_{j=1}^d(1-t\lambda_j({\mathbf{v}}))= \sum_{k=0}^d (-1)^k\frac {D^k_{\mathbf{v}}h({\mathbf{e}})} {k!} t^k,$$ and hence $\rk({\mathbf{v}})= \deg h({\mathbf{e}}-t{\mathbf{v}})= \max\{k : D^k_{\mathbf{v}}h({\mathbf{e}})\neq 0\}$. Since the rank does not depend on the choice of ${\mathbf{e}}\in \Lambda_{++}$, if $D^{k+1}_{\mathbf{v}}h({\mathbf{e}})=D^{k+2}_{\mathbf{v}}h({\mathbf{e}})=\cdots =0$ for some ${\mathbf{e}}\in \Lambda_{++}$, then $D^{k+1}_{\mathbf{v}}h({\mathbf{e}}')=D^{k+2}_{\mathbf{v}}h({\mathbf{e}}')=\cdots =0$ for all ${\mathbf{e}}' \in \Lambda_{++}$. Since $ \Lambda_{++}$ has non-empty interior this means $D^{k+1}_{\mathbf{v}}h \equiv 0$. If $h({\mathbf{x}}) \in {\mathbb{R}}[x_1,\ldots, x_n]$ and ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_m \in {\mathbb{R}}^n$ let $h[{\mathbf{v}}_1, \ldots, {\mathbf{v}}_m]$ be the polynomial in ${\mathbb{R}}[x_1,\ldots, x_n,y_1,\ldots, y_m]$ defined by $$h[{\mathbf{v}}_1, \ldots, {\mathbf{v}}_m] = \prod_{j=1}^m \left(1-y_jD_{{\mathbf{v}}_j}\right) h({\mathbf{x}}).$$ By iterating Theorem \[direct\] (2) we get: \[mixhyp\] If $h({\mathbf{x}})$ is hyperbolic with hyperbolicity cone $\Lambda_{++}$ and ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_m \in \Lambda_+$, then $h[{\mathbf{v}}_1, \ldots, {\mathbf{v}}_m]$ is hyperbolic with hyperbolicity cone containing $\Lambda_{++} \times (-{\mathbb{R}}_+^m)$, where ${\mathbb{R}}_+ := [0,\infty)$. \[rk1le\] Suppose $h$ is hyperbolic. If ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_m \in \Lambda_+$ have rank at most one, then $$h[{\mathbf{v}}_1, \ldots, {\mathbf{v}}_m] = h({\mathbf{x}}-y_1{\mathbf{v}}_1 - \cdots - y_m {\mathbf{v}}_m).$$ If ${\mathbf{v}}$ has rank at most one, then $D_{\mathbf{v}}^k h \equiv 0$ for all $k \geq 2$ by Lemma \[rankalt\]. Hence, by , $$h({\mathbf{x}}-y{\mathbf{v}}) = \left( \sum_{k=0}^{\infty} \frac {(-y)^k D_{\mathbf{v}}^k}{k!} \right) h({\mathbf{x}})= (1-yD_{{\mathbf{v}}})h({\mathbf{x}}),$$ from which the lemma follows. Note that $({\mathbf{v}}_1,\ldots,{\mathbf{v}}_m) \mapsto h[{\mathbf{v}}_1,\ldots,{\mathbf{v}}_m]$ is affine linear in each coordinate, i.e., for all $p \in {\mathbb{R}}$ and $1\leq i \leq m$: $$\begin{aligned} & h[{\mathbf{v}}_1,\ldots,(1-p){\mathbf{v}}_i+p{\mathbf{v}}_i',\ldots, {\mathbf{v}}_m] \\ = &(1-p)h[{\mathbf{v}}_1,\ldots,{\mathbf{v}}_i,\ldots, {\mathbf{v}}_m] +ph[{\mathbf{v}}_1,\ldots,{\mathbf{v}}_i',\ldots, {\mathbf{v}}_m].\end{aligned}$$ Hence if ${\mathsf{X}}_1, \ldots, {\mathsf{X}}_m$ are independent random variables in ${\mathbb{R}}^n$, then $$\label{mixedexp} {\mathbb{E}}h[{\mathsf{X}}_1,\ldots,{\mathsf{X}}_m] = h[{\mathbb{E}}{\mathsf{X}}_1,\ldots,{\mathbb{E}}{\mathsf{X}}_m].$$ \[mixedchar\] Let $h({\mathbf{x}})$ be hyperbolic with respect to ${\mathbf{e}}\in {\mathbb{R}}^n$, let $V_1, \ldots, V_m$ be finite sets of vectors in $\Lambda_+$, and let ${\mathbf{w}}\in {\mathbb{R}}^{n+m}$. For ${\mathbf{V}}=({\mathbf{v}}_1,\ldots, {\mathbf{v}}_m) \in V_1\times \cdots \times V_m$, let $$f({\mathbf{V}};t) := h[{\mathbf{v}}_1,\ldots, {\mathbf{v}}_m](t{\mathbf{e}}+{\mathbf{w}}).$$ Then $\{f({\mathbf{V}};t)\}_{{\mathbf{V}}\in V_1\times \cdots \times V_m}$ is a compatible family. In particular if in addition all vectors in $V_1 \cup \cdots \cup V_m$ have rank at most one, and $$g({\mathbf{V}};t) := h(t{\mathbf{e}}+{\mathbf{w}}- \alpha_1{\mathbf{v}}_1-\cdots-\alpha_m{\mathbf{v}}_m),$$ where ${\mathbf{w}}\in {\mathbb{R}}^n$ and $(\alpha_1,\ldots, \alpha_m)\in {\mathbb{R}}^m$, then $\{g({\mathbf{V}};t)\}_{{\mathbf{V}}\in V_1\times \cdots \times V_m}$ is a compatible family. Let ${\mathsf{X}}_1 \in V_1, \ldots, {\mathsf{X}}_m \in V_m$ be independent random variables. Then the polynomial ${\mathbb{E}}h[{\mathsf{X}}_1, \ldots, {\mathsf{X}}_m]= h[{\mathbb{E}}{\mathsf{X}}_1,\ldots,{\mathbb{E}}{\mathsf{X}}_m]$ is hyperbolic with respect to $({\mathbf{e}}, 0,\ldots,0)$ by Theorem \[mixhyp\] (since ${\mathbb{E}}{\mathbf{v}}_i \in \Lambda_+$ for all $i$ by convexity). In particular the polynomial ${\mathbb{E}}f({\mathsf{X}}_1,\ldots, {\mathsf{X}}_m;t)$ is real–rooted. The second assertion is an immediate consequence of the first combined with Lemma \[rk1le\]. Bounds on zeros of mixed characteristic polynomials =================================================== To prove Theorem \[hypprob\], we want to bound the zeros of the *mixed characteristic polynomial* $$\label{mip} t \mapsto h[{\mathbf{v}}_1, \ldots, {\mathbf{v}}_m](t{\mathbf{e}}+{\mathbf{1}}),$$ where $h$ is hyperbolic with respect to ${\mathbf{e}}\in {\mathbb{R}}^n$, ${\mathbf{1}}\in {\mathbb{R}}^m$ is the all ones vector, and ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_m \in \Lambda_+({\mathbf{e}})$ satisfy ${\mathbf{v}}_1+\cdots+{\mathbf{v}}_m ={\mathbf{e}}$ and $\tr({\mathbf{v}}_i) \leq \epsilon$ for all $1\leq i \leq m$. \[hypid\] Note that a real number $\rho$ is larger than the maximum zero of if and only if $\rho {\mathbf{e}}+{\mathbf{1}}$ is in the hyperbolicity cone $\Gamma_{++}$ of $h[{\mathbf{v}}_1, \ldots, {\mathbf{v}}_m]$. Hence the maximal zero of is equal to $$\inf \{ \rho >0 : \rho {\mathbf{e}}+{\mathbf{1}}\in \Gamma_{++}\}.$$ For the remainder of this section, let $h \in {\mathbb{R}}[x_1,\ldots, x_n]$ be hyperbolic with respect to ${\mathbf{e}}$, and let ${\mathbf{v}}_1,\ldots, {\mathbf{v}}_m \in \Lambda_{++}$. To enhance readability in the computations to come, let $\partial_j := D_{{\mathbf{v}}_j}$ and $$\xi_j[g] := \frac {g}{\partial_j g}.$$ Note that a continuously differentiable concave function $f : (0,\infty) \to {\mathbb{R}}$ satisfies $$f(t+\delta) \geq f(t)+ \delta f'(t+\delta), \quad \mbox{ for all } \delta \geq 0.$$ Hence by Theorem \[direct\] $$\label{concon} \xi_i[h]({\mathbf{x}}+\delta {\mathbf{v}}_j) \geq \xi_i[h]({\mathbf{x}}) + \delta \partial_j \xi_i[h]({\mathbf{x}}+\delta {\mathbf{v}}_j)$$ for all ${\mathbf{x}}\in \Lambda_{+}$ and $\delta \geq 0$. The following elementary identity is left for the reader to verify. \[tech\] $$\xi_i[h-\partial_jh]=\xi_i[h]-\frac {\partial_j \xi_i[h]\cdot \xi_j[\partial_ih]}{\xi_j[\partial_ih]-1}.$$ \[engine\] If ${\mathbf{x}}\in \Lambda_{++}$, $1\leq i,j \leq n$, $\delta > 1$ and $$\xi_j[h]({\mathbf{x}}) \geq \frac \delta {\delta-1},$$ then $$\xi_i[h-\partial_jh]({\mathbf{x}}+\delta {\mathbf{v}}_j) \geq \xi_i[h]({\mathbf{x}}).$$ Since $\xi_i[h]$ is concave on $\Lambda_{++}$ (Theorem \[direct\] (3)) and homogeneous of degree one: $$\frac {\xi_i[h]({\mathbf{z}}+\delta {\mathbf{v}}_j) - \xi_i[h]({\mathbf{z}})}{\delta} \geq \xi_i[h]({\mathbf{v}}_j), \ \ \ \mbox{ for all } {\mathbf{z}}\in \Lambda_{++}.$$ Hence $$\label{parat} \partial_j \xi_i[h]({\mathbf{z}}) \geq \xi_i[h]({\mathbf{v}}_j)\geq 0, \ \ \ \mbox{ for all } {\mathbf{z}}\in \Lambda_{++}.$$ If ${\mathbf{z}}\in \Lambda_{++}$, then (by Theorem \[hypfund\] (2)) $({\mathbf{z}}, t)$ is in the closure of the hyperbolicity cone of $h-y\partial_j h$ if and only if $t \leq \xi_j[h]({\mathbf{z}})$. By Theorem \[direct\] the polynomial $$D_{({\mathbf{v}}_i,0)}(h-y\partial_j h) = \partial_i h -y \partial_j \partial_i h$$ is hyperbolic with hyperbolicity cone containing the hyperbolicity cone of $h-y\partial_j h$. Hence if ${\mathbf{z}}\in \Lambda_{++}$ and $t \leq \xi_j[h]({\mathbf{z}})$, then $t \leq \xi_j[\partial_ih]({\mathbf{z}})$, and thus $$\label{parata} \xi_j[\partial_ih]({\mathbf{z}}) \geq \xi_j[h]({\mathbf{z}}), \ \ \ \mbox{ for all } {\mathbf{z}}\in \Lambda_{++}.$$ Let ${\mathbf{x}}$ be as in the statement of the lemma. By Lemma \[tech\] and $$\begin{aligned} \xi_i[h-\partial_jh]({\mathbf{x}}+\delta {\mathbf{v}}_j) - \xi_i[h]({\mathbf{x}}) &= \xi_i[h]({\mathbf{x}}+\delta {\mathbf{v}}_j)-\xi_i[h]({\mathbf{x}})- \frac {\partial_j \xi_i[h]\cdot \xi_j[\partial_ih]}{\xi_j[\partial_ih]-1}({\mathbf{x}}+\delta {\mathbf{v}}_j)\\ &\geq \partial_j \xi_i[h]({\mathbf{x}}+\delta {\mathbf{v}}_j) \left( \delta - \frac { \xi_j[\partial_ih]({\mathbf{x}}+\delta {\mathbf{v}}_j) }{\xi_j[\partial_ih]({\mathbf{x}}+\delta {\mathbf{v}}_j)-1}\right)\\ &\geq \xi_i[h]({\mathbf{v}}_j) \left( \delta - \frac { \delta/(\delta-1)}{ \delta/(\delta-1)-1}\right) =0, \end{aligned}$$ where the last inequality follows from , and the concavity of ${\mathbf{z}}\rightarrow \xi_j[h]({\mathbf{z}})$. Consider ${\mathbb{R}}^{n+m}={\mathbb{R}}^n\oplus {\mathbb{R}}^m$ and let ${\mathbf{e}}_1,\ldots, {\mathbf{e}}_m$ be the standard bases of ${\mathbb{R}}^m$ (inside ${\mathbb{R}}^n\oplus {\mathbb{R}}^m$). \[corbond\] Suppose $h$ is hyperbolic with respect to ${\mathbf{e}}\in {\mathbb{R}}^n$, and let $\Gamma_+$ be the (closed) hyperbolicity cone of $h[{\mathbf{v}}_1,\ldots, {\mathbf{v}}_m]$, where ${\mathbf{v}}_1,\ldots, {\mathbf{v}}_m \in \Lambda_{+}({\mathbf{e}})$. Suppose $t_i, t_j > 1$ and ${\mathbf{x}}\in \Lambda_{+}({\mathbf{e}})$ are such that $${\mathbf{x}}+t_k {\mathbf{e}}_k \in \Gamma_+, \quad \mbox{ for } k \in \{i,j\}.$$ Then $${\mathbf{x}}+\frac {t_j}{t_j-1}{\mathbf{v}}_j + {\mathbf{e}}_j + t_i {\mathbf{e}}_i \in \Gamma_+.$$ Moreover if ${\mathbf{x}}+t_k {\mathbf{e}}_k \in \Gamma_+$ for all $k \in [m]$, then $${\mathbf{x}}+ \left(1-\frac 1 m\right) \sum_{i=1}^m \frac {t_i}{t_i-1}{\mathbf{v}}_i +\left(1-\frac 1 m\right)\sum_{i=1}^m {\mathbf{e}}_i+ \frac 1 m\sum_{i=1}^m t_i{\mathbf{e}}_i \in \Gamma_+.$$ By continuity we may assume ${\mathbf{x}},{\mathbf{v}}_1,\ldots, {\mathbf{v}}_m \in \Lambda_{++}({\mathbf{e}})$. Let $\delta_k = t_k/(t_k-1)$. Then $${\mathbf{x}}+ t_k {\mathbf{e}}_k \in \Gamma_+ \mbox{ if and only if } \xi_k[h] \geq \frac {\delta_k}{\delta_k-1}.$$ Also $\xi_i[h-\partial_jh]({\mathbf{x}}+\delta_j {\mathbf{v}}_j) \geq \delta_i/(\delta_i-1)$ is equivalent to $${\mathbf{x}}+\delta_j{\mathbf{v}}_j + {\mathbf{e}}_j+\frac {\delta_i} {\delta_i-1} {\mathbf{e}}_i \in \Gamma_+.$$ Hence the first part follows from Lemma \[engine\]. Suppose ${\mathbf{x}}+t_k {\mathbf{e}}_k \in \Gamma_+$ for all $k \in [m]$. Since ${\mathbf{x}}+s{\mathbf{e}}_1, {\mathbf{v}}_1 \in \Gamma_+$ for all $s \leq t_1$, the vector $${\mathbf{x}}' := {\mathbf{x}}+\frac {t_1}{t_1-1}{\mathbf{v}}_1 + {\mathbf{e}}_1$$ is in the hyperbolicity cone of $(1-y_1D_{{\mathbf{v}}_1})h$. By the first part we have ${\mathbf{x}}'+t_2{\mathbf{e}}_2, {\mathbf{x}}'+t_3{\mathbf{e}}_3\in \Gamma_+$. Hence we may apply the first part of the theorem with $h$ replaced by $(1-y_1D_{{\mathbf{v}}_1})h$ to conclude $${\mathbf{x}}'+ \frac {t_2}{t_2-1}{\mathbf{v}}_2 + {\mathbf{e}}_2+ t_3{\mathbf{e}}_3={\mathbf{x}}+\frac {t_1}{t_1-1}{\mathbf{v}}_1 + \frac {t_2}{t_2-1}{\mathbf{v}}_2+{\mathbf{e}}_1 +{\mathbf{e}}_2 + t_3{\mathbf{e}}_3\in \Gamma_+.$$ By continuing this procedure with different orderings we may conclude that $${\mathbf{x}}+ \left(\sum_{i=1}^m \frac {t_i}{t_i-1}{\mathbf{v}}_i\right)-\frac {t_j}{t_j-1}{\mathbf{v}}_j +\left(\sum_{i=1}^m {\mathbf{e}}_i\right)-{\mathbf{e}}_j+t_j{\mathbf{e}}_j \in \Gamma_+,$$ for each $1\leq j \leq m$. The second part now follows from convexity of $\Gamma_+$ upon taking the convex sum of these vectors. \[mainbound\] Suppose $h$ is hyperbolic with respect to ${\mathbf{e}}\in {\mathbb{R}}^n$ and suppose ${\mathbf{v}}_1,\ldots, {\mathbf{v}}_m \in \Lambda_{+}({\mathbf{e}})$ are such that ${\mathbf{e}}= {\mathbf{v}}_1+\cdots+{\mathbf{v}}_m$, where $\tr({\mathbf{v}}_j) \leq \epsilon$ for each $1\leq j \leq m$. Then the largest zero of the polynomial $$t \mapsto h[{\mathbf{v}}_1, \ldots, {\mathbf{v}}_m](t{\mathbf{e}}+{\mathbf{1}})$$ is at most $$\delta(\epsilon, m):=\left( 1-\frac 1 m +\sqrt{\epsilon - \frac 1 m \left(1-\frac 1 m\right)}\right)^2.$$ Let $t >1$ and set ${\mathbf{x}}=\epsilon t{\mathbf{e}}$ and $t_i=t$ for $1\leq i \leq m$. Then ${\mathbf{x}}+t_i {\mathbf{e}}_i = t(\epsilon {\mathbf{e}}+ {\mathbf{e}}_i) \in \Lambda_+$ since $$h[{\mathbf{v}}_1,\ldots, {\mathbf{v}}_m](\epsilon {\mathbf{e}}+ {\mathbf{e}}_i)= \epsilon h({\mathbf{e}})- D_{{\mathbf{v}}_i}h({\mathbf{e}}) = h({\mathbf{e}})(\epsilon -\tr({\mathbf{v}}_i)) \geq 0.$$ Apply Corollary \[corbond\] to conclude that for each $t>1$: $$\left(\epsilon t+ \left(1-\frac 1 m\right)\frac t {t-1}\right) {\mathbf{e}}+ \left(1-\frac 1 m + \frac t m \right) {\mathbf{1}}\in \Gamma_+.$$ Hence by (the homogeneity of $\Gamma_+$ and) Remark \[hypid\], the maximal zero is at most $$\inf \left\{ \frac {\epsilon t+ \left(1-\frac 1 m\right)\frac t {t-1}} {1-\frac 1 m + \frac t m } : t >1\right\}.$$ It is a simple exercise to deduce that the infimum is exactly what is displayed in the statement of the theorem. Proof of Theorem \[t1\] ======================= To prove Theorem \[t1\] we use the following theorem which for $h=\det$ appears in [@MSS1; @MSS2]: \[hypprob\] Suppose $h$ is hyperbolic with respect to ${\mathbf{e}}$. Let ${\mathsf{X}}_1, \ldots, {\mathsf{X}}_m$ be independent random vectors in $\Lambda_+$ of rank at most one and with finite supports such that $$\label{hypeta2} {\mathbb{E}}\sum_{i=1}^m {\mathsf{X}}_i ={\mathbf{e}},$$ and $$\label{hyptr} \tr({\mathbb{E}}{\mathsf{X}}_i) \leq \epsilon \mbox{ for all } 1\leq i \leq m.$$ Then $$\label{hypbig} {\mathbb{P}}\left[ {\lambda_{\rm max}}\left(\sum_{i=1}^m {\mathsf{X}}_i \right) \leq \delta(\epsilon,m) \right] >0.$$ Let $V_i$ be the support of ${\mathsf{X}}_i$, for each $1 \leq i \leq m$. By Theorem \[mixedchar\], the family $$\{h(t{\mathbf{e}}- {\mathbf{v}}_1-\cdots-{\mathbf{v}}_m)\}_{{\mathbf{v}}_i \in V_i}$$ is compatible. By Theorem \[expfam\] there are vectors ${\mathbf{v}}_i \in V_i$, $1\leq i \leq m$, such that the largest zero of $h(t{\mathbf{e}}- {\mathbf{v}}_1-\ldots-{\mathbf{v}}_m)$ is smaller or equal to the largest zero of $${\mathbb{E}}h(t{\mathbf{e}}- {\mathsf{X}}_1-\cdots-{\mathsf{X}}_m)= {\mathbb{E}}h[{\mathsf{X}}_1,\ldots, {\mathsf{X}}_m](t{\mathbf{e}}+{\mathbf{1}})= h[{\mathbb{E}}{\mathsf{X}}_1,\ldots, {\mathbb{E}}{\mathsf{X}}_m](t{\mathbf{e}}+{\mathbf{1}}).$$ The theorem now follows from Theorem \[mainbound\]. For $1\leq i \leq k$, let ${\mathbf{x}}^i=(x_{i1},\ldots,x_{in})$ where ${\mathbf{y}}=\{x_{ij} : 1\leq i \leq k, 1\leq j \leq k\}$ are independent variables. Consider the polynomial $$g({\mathbf{y}}) = h({\mathbf{x}}^1)h({\mathbf{x}}^2) \cdots h({\mathbf{x}}^k) \in {\mathbb{R}}[{\mathbf{y}}],$$ which is hyperbolic with respect to ${\mathbf{e}}^1\oplus \cdots \oplus {\mathbf{e}}^k$, where ${\mathbf{e}}^i$ is a copy of ${\mathbf{e}}$ in the variables ${\mathbf{x}}^i$, for all $1 \leq i \leq k$. The hyperbolicity cone of $g$ is the direct sum $\Lambda_+:=\Lambda_+({\mathbf{e}}^1) \oplus \cdots \oplus \Lambda_+({\mathbf{e}}^k)$, where $\Lambda_+({\mathbf{e}}^i)$ is a copy of $\Lambda_+({\mathbf{e}})$ in the variables ${\mathbf{x}}^i$, for all $1 \leq i \leq k$. Let ${\mathsf{X}}_1, \ldots, {\mathsf{X}}_m$ be independent random vectors in $\Lambda_+$ such that for all $1\leq i \leq k$ and $1\leq j \leq m$: $${\mathbb{P}}\left[ {\mathsf{X}}_j = k{\mathbf{u}}_j^i\right] = \frac 1 k,$$ where ${\mathbf{u}}_1^i, \ldots, {\mathbf{u}}_m^i$ are copies in $\Lambda_+({\mathbf{e}}^i)$ of ${\mathbf{u}}_1, \ldots, {\mathbf{u}}_m$. Then $$\begin{aligned} {\mathbb{E}}{\mathsf{X}}_j &= {\mathbf{u}}_j^1 \oplus {\mathbf{u}}_j^2 \oplus \cdots \oplus {\mathbf{u}}_j^k, \\ \tr({\mathbb{E}}{\mathsf{X}}_j) &= k\tr({\mathbf{u}}_j) \leq k\epsilon, \mbox{ and } \\ {\mathbb{E}}\sum_{j=1}^m {\mathsf{X}}_j &= {\mathbf{e}}^1\oplus \cdots \oplus {\mathbf{e}}^k,\end{aligned}$$ for all $1\leq j \leq k$. By Theorem \[hypprob\] there is a partition $S_1\cup \cdots \cup S_k =[m]$ such that $${\lambda_{\rm max}}\left(\sum_{i \in S_1}k{\mathbf{u}}_i^1+\cdots + \sum_{i \in S_k}k{\mathbf{u}}_i^k \right)\leq \delta(k\epsilon,m).$$ However $${\lambda_{\rm max}}\! \left(\sum_{i \in S_1}k{\mathbf{u}}_i^1+\cdots + \sum_{i \in S_k}k{\mathbf{u}}_i^k \right) = k \! \max_{1\leq j \leq k} {\lambda_{\rm max}}\! \left(\sum_{i \in S_j}{\mathbf{u}}_i^j \right) = k \! \max_{1\leq j \leq k} {\lambda_{\rm max}}\! \left(\sum_{i \in S_j}{\mathbf{u}}_i \right),$$ and the theorem follows. On a conjecture on the optimal bound ==================================== We have seen that the core of the proof of Theorem \[t1\] is to bound the zeros of mixed characteristic polynomials. To achieve better bounds in Theorem \[t1\] we are therefore motivated to look closer at the following problem. \[central\] Let $h$ be a polynomial of degree $d$ which is hyperbolic with respect to ${\mathbf{e}}$, and let $\epsilon >0$ and $m \in {\mathbb{Z}}_+$ be given. Determine the largest possible maximal zero, $\rho=\rho(h,{\mathbf{e}},\epsilon,m)$, of mixed characteristic polynomials: $$\chi[{\mathbf{v}}_1,\ldots, {\mathbf{v}}_m](t):=h[{\mathbf{v}}_1, \ldots, {\mathbf{v}}_m](t{\mathbf{e}}+ {\mathbf{1}})$$ subject to the conditions 1. ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_m \in \Lambda_+$, 2. ${\mathbf{v}}_1 + \cdots + {\mathbf{v}}_m = {\mathbf{e}}$, and 3. $\tr({\mathbf{v}}_i) \leq \epsilon$ for all $1 \leq i \leq m$. The following conjecture was made by Marcus *et al.* [@MSS2] in the case when $h = \det$, but we take the liberty to extend the conjecture to any hyperbolic polynomial. \[maxmax\] The maximal zero in Problem \[central\] is achieved for $${\mathbf{v}}_1=\cdots ={\mathbf{v}}_k= \frac \epsilon d {\mathbf{e}}, {\mathbf{v}}_{k+1}= \left(1-\frac k d \epsilon\right){\mathbf{e}}, {\mathbf{v}}_{k+2}={\mathbf{v}}_{k+3}= \cdots= {\mathbf{v}}_{m}=0,$$ where $k= \lfloor d/\epsilon \rfloor$. We will prove here that Conjecture \[maxmax\] is equivalent to the following seemingly weaker conjecture. \[maxmax2\] The maximal zero in Problem \[central\] is achieved for some ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_m$ where ${\mathbf{v}}_i \in \Lambda_{++}\cup \{0\}$ for each $i \in [m]$. We start by proving that there is a solution to Problem \[central\] for which the ${\mathbf{v}}_i$’s have correct traces, i.e., as those in Conjecture \[maxmax\]. By a “solution" to Problem \[central\] we mean a list of vectors ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_m$, as in Problem \[central\], which realize the maximal zero. First a useful lemma. \[nicein\] Suppose ${\mathbf{u}}, {\mathbf{v}}, {\mathbf{w}}\in \Lambda_+$. Then $$(D_{\mathbf{u}}D_{\mathbf{v}}h({\mathbf{w}}))^2 \geq D_{\mathbf{u}}^2 h({\mathbf{w}}) \cdot D_{\mathbf{v}}^2 h({\mathbf{w}}),$$ and hence $$D_{\mathbf{u}}D_{\mathbf{v}}h({\mathbf{w}}) \geq \min\{ D_{\mathbf{u}}^2 h({\mathbf{w}}), D_{\mathbf{v}}^2 h({\mathbf{w}})\}.$$ By continuity we may assume ${\mathbf{u}}, {\mathbf{v}}, {\mathbf{w}}\in \Lambda_{++}$. Then the polynomial $$\begin{aligned} & g(x,y,z):=h(x{\mathbf{u}}+y{\mathbf{v}}+z{\mathbf{w}}) = h({\mathbf{w}})z^d+ \big(D_{\mathbf{u}}h({\mathbf{w}}) x + D_{\mathbf{v}}h({\mathbf{w}})y\big)z^{d-1}+ \\ &+ \left( D_{\mathbf{u}}D_{\mathbf{v}}h({\mathbf{w}}) xy + \frac 1 2 D_{\mathbf{u}}^2 h({\mathbf{w}}) x^2 + \frac 1 2 D_{\mathbf{v}}^2 h({\mathbf{w}})y^2\right)z^{d-2}+ \cdots\end{aligned}$$ is hyperbolic with hyperbolicity cone containing the positive orthant. By Theorem \[direct\] (1) so is $\partial^{d-2} g /\partial z^{d-2}$, and hence the polynomial $$2\frac {\partial^{d-2} g} {\partial z^{d-2}} \big( (1,0,0)+ t(0,0,1) \big) = D_{\mathbf{u}}^2 h({\mathbf{w}}) + 2D_{\mathbf{u}}D_{\mathbf{v}}h({\mathbf{w}})t + D_{\mathbf{v}}^2 h({\mathbf{w}})t^2$$ is real–rooted. Thus its discriminant is nonnegative, which yields the desired inequality. \[righttrace\] There is a solution to Problem \[central\] such that all but at most one of the ${\mathbf{v}}_i$’s have trace either zero or $\epsilon$. Moreover, if there is a solution to Problem \[central\] which satisfies the condition in Conjecture \[maxmax2\], then there is such a solution such that all but at most one of the ${\mathbf{v}}_i$’s have trace either zero or $\epsilon$. Let ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_m$ be a solution to Problem \[central\], and let $\rho$ be the maximal zero. Suppose $0<\tr({\mathbf{v}}_1), \tr({\mathbf{v}}_2) <\epsilon$. By Remark \[hypid\] $\rho {\mathbf{e}}+ {\mathbf{1}}$ is in the hyperbolicity cone $\Gamma_+$ of $h[{\mathbf{v}}_1, \ldots, {\mathbf{v}}_m]$. Since also $-{\mathbf{e}}_1, -{\mathbf{e}}_2 \in \Gamma_+$ we have ${\mathbf{w}}:= \rho{\mathbf{e}}+{\mathbf{1}}-{\mathbf{e}}_1-{\mathbf{e}}_2 \in \Gamma_+$, and hence ${\mathbf{w}}$ is in the (closed) hyperbolicity cone of $g= h[{\mathbf{v}}_3, \ldots, {\mathbf{v}}_m]$. By Lemma \[nicein\] we may assume $$D_{{\mathbf{v}}_1}D_{{\mathbf{v}}_2} g({\mathbf{w}}) \geq D_{{\mathbf{v}}_1}^2g({\mathbf{w}}) \geq 0,$$ since otherwise change the indices $1$ and $2$. For $$0 \leq s \leq \min\left\{1, \frac {\epsilon -\tr({\mathbf{v}}_2)} {\tr({\mathbf{v}}_1)}\right\},$$ we have (since $(1-D_{{\mathbf{v}}_1})(1-D_{{\mathbf{v}}_2})g({\mathbf{w}})=0$): $$\begin{aligned} & h[{\mathbf{v}}_1-s{\mathbf{v}}_1, {\mathbf{v}}_2+s{\mathbf{v}}_1, {\mathbf{v}}_3, \ldots, {\mathbf{v}}_m](\rho {\mathbf{e}}+{\mathbf{1}}) \\ =& -s(D_{{\mathbf{v}}_1}D_{{\mathbf{v}}_2} g({\mathbf{w}})-D_{{\mathbf{v}}_1}^2g({\mathbf{w}}))-s^2D_{{\mathbf{v}}_1}^2g({\mathbf{w}}) \leq 0. \end{aligned}$$ Hence the maximal zero of $\chi[{\mathbf{v}}_1-s{\mathbf{v}}_1, {\mathbf{v}}_2+s{\mathbf{v}}_1, {\mathbf{v}}_3, \ldots, {\mathbf{v}}_m](t)$ is at least $\rho$, and since $\rho$ is the largest possible maximal zero $$\chi[{\mathbf{v}}_1-s{\mathbf{v}}_1, {\mathbf{v}}_2+s{\mathbf{v}}_1, {\mathbf{v}}_3, \ldots, {\mathbf{v}}_m](\rho)=0.$$ We may therefore alter ${\mathbf{v}}_1, {\mathbf{v}}_2$ so that either ${\mathbf{v}}_1=0$ or $\tr({\mathbf{v}}_2) = \epsilon$, while retaining the maximal zero $\rho$. Continuing this process we arrive at a solution of the desired form. \[average\] Suppose ${\mathbf{v}}_1,\ldots, {\mathbf{v}}_m$ is a solution to Problem \[central\] such that $\tr({\mathbf{v}}_1)= \tr({\mathbf{v}}_2)$ and ${\mathbf{v}}_1, {\mathbf{v}}_2 \in \Lambda_{++}$. Then $({\mathbf{v}}_1+{\mathbf{v}}_2)/2, ({\mathbf{v}}_1+{\mathbf{v}}_2)/2, {\mathbf{v}}_3, \ldots, {\mathbf{v}}_m$ is also a solution to Problem \[central\]. Let ${\mathbf{v}}_1(s)= (1-s){\mathbf{v}}_1+s{\mathbf{v}}_2$ and ${\mathbf{v}}_2(s)= (1-s){\mathbf{v}}_2+s{\mathbf{v}}_1$. Then $\tr({\mathbf{v}}_1(s))=\tr({\mathbf{v}}_2(s)) = \tr({\mathbf{v}}_1)$, ${\mathbf{v}}_1(s)+{\mathbf{v}}_2(s)={\mathbf{v}}_1+{\mathbf{v}}_2$, and ${\mathbf{v}}_1(s), {\mathbf{v}}_2(s) \in \Lambda_{++}$ for all $s \in (-\delta, 1+\delta)$ for some $\delta>0$. Let $\rho$ be the maximal zero in Problem \[central\]. Then the function $$(-\delta, 1+\delta) \ni s \mapsto \chi[{\mathbf{v}}_1(s),{\mathbf{v}}_2(s), {\mathbf{v}}_3, \ldots, {\mathbf{v}}_m](\rho)$$ is a degree at most two polynomial which has local minima at $s=0$ and $s=1$. Hence this function is identically zero, and thus $$\chi[{\mathbf{v}}_1(1/2),{\mathbf{v}}_2(1/2), {\mathbf{v}}_3, \ldots, {\mathbf{v}}_m](\rho) =0$$ as desired. \[infav\] Suppose ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_m$ is a solution to Problem \[central\] such that ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_k \in \Lambda_{++}$ all have the same trace, and let $${\mathbf{v}}= \frac 1 k ({\mathbf{v}}_1 + \cdots + {\mathbf{v}}_k).$$ By applying Lemma \[average\] infinitely many times (and invoking Hurwitz’ theorem on the continuity of zeros) we see that also ${\mathbf{v}}, \ldots, {\mathbf{v}}, {\mathbf{v}}_{k+1}, \ldots, {\mathbf{v}}_m$ is a solution to Problem \[central\]. Clearly Conjecture \[maxmax\] implies Conjecture \[maxmax2\]. To prove the other implication assume Conjecture \[maxmax2\]. Then by Proposition \[righttrace\] and Remark \[infav\] we may assume that we have a solution of the form ${\mathbf{v}}_1, \ldots, {\mathbf{v}}_m$, where ${\mathbf{v}}_1=\cdots={\mathbf{v}}_k={\mathbf{v}}$, ${\mathbf{v}}_{k+1}={\mathbf{e}}-k{\mathbf{v}}$, ${\mathbf{v}}_{k+2}=\cdots={\mathbf{v}}_m=0$, and where ${\mathbf{v}}, {\mathbf{e}}-k{\mathbf{v}}\in \Lambda_{++}$ and $\tr({\mathbf{v}})= \epsilon$ and $0<d-k\epsilon = \tr({\mathbf{e}}-k{\mathbf{v}})<\epsilon$. Hence we want to maximize the largest zero of $$\label{gv} g_{\mathbf{v}}(t):=(1-D_{\mathbf{v}})^{k} (1-D_{\mathbf{e}}+kD_{\mathbf{v}}) h(t{\mathbf{e}})$$ where - ${\mathbf{v}}, {\mathbf{e}}-k{\mathbf{v}}\in \Lambda_{++}$ - $\tr({\mathbf{v}})=\epsilon$, where $0<d-k\epsilon <\epsilon$. Let $I \subseteq {\mathbb{R}}$ be an interval. We say that a univariate polynomial is $I$–*rooted* if all its zeros lie in $I$. \[tgv\] Let $g_{\mathbf{v}}(t)$ be given by . Then $g_{\mathbf{v}}(t)= T_{k,d}(h(t{\mathbf{e}}-{\mathbf{v}}))$ where $T_{k,d}: {\mathbb{R}}[t] \rightarrow {\mathbb{R}}[t]$ is the linear operator defined by $$T_{k,d}\left(\sum_{j\geq 0} a_jt^j\right) = -\sum_{j=0}^d \left( \frac {j+1} {k+1} a_{j+1} +(d-1-j)a_j\right) (d-j)! \binom {k+1}{d-j}t^j.$$ Moreover if $f$ is a $[0,1/k]$–rooted polynomial of degree $d$, then $T_{k,d}(f)$ is real–rooted. By $$\label{collect} h(t{\mathbf{e}}-{\mathbf{v}})= \sum_{j=0}^d (-1)^j \frac 1 {j!} D_{\mathbf{v}}^jh({\mathbf{e}})t^{d-j}=:\sum_{j \geq 0} a_j t^j.$$ Note that $D_{\mathbf{v}}^j h(t{\mathbf{e}})= D_{\mathbf{v}}^j h({\mathbf{e}})t^{d-j}$ and $D_{\mathbf{v}}^j D_{\mathbf{e}}h(t{\mathbf{e}})= D_{\mathbf{e}}D_{\mathbf{v}}^j h(t{\mathbf{e}})= (d-j)D_{\mathbf{v}}^j h({\mathbf{e}})t^{d-j-1}$. Expanding and comparing coefficients with one sees that $g_{\mathbf{v}}(t)=T_{k,d}(h(t{\mathbf{e}}-{\mathbf{v}})$. To prove the final statement of the lemma we may by Hurwitz’ theorem on the continuity of zeros assume that $f$ is a $(0,1/k)$–rooted polynomial of degree $d$. We may choose a hyperbolic degree $d$ polynomial $h$ and a vector ${\mathbf{v}}$ such that $f(t)= h(t{\mathbf{e}}-{\mathbf{v}})$, for example $h(x,y)= (-y)^df(-x/y)$, ${\mathbf{e}}=(1,0)$ and ${\mathbf{v}}=(0,1)$. Then ${\mathbf{v}}\in \Lambda_{++}$ and ${\mathbf{w}}={\mathbf{e}}-k{\mathbf{v}}\in \Lambda_{++}$ by e.g. . Hence $$T_{k,d}(f)(t) = \chi[{\mathbf{v}},{\mathbf{v}}, \ldots, {\mathbf{v}}, {\mathbf{w}}](t)$$ is real–rooted. The *trace*, $\tr(f)$, of a non-constant polynomial is the sum of the the zeros of $f$ (counted with multiplicity). Let ${\mathcal{M}}_d$ be the affine space of all monic real polynomials of degree $d$. \[affine\] Let $T : {\mathcal{M}}_d \rightarrow {\mathcal{M}}_m$ be an affine linear operator, and let $\epsilon>0$. Suppose $T$ sends $[a,b]$–rooted polynomials to real–rooted polynomials. Consider the problem of maximizing the largest zero of $T(f)$ over all $[a,b]$–rooted polynomials $f \in {\mathcal{M}}_d$ with $\tr(f) = \epsilon$. Then this (maximal) zero is achieved for some $T(f)$, where $f$ has at most one distinct zero in $(a,b)$. Moreover, if the maximal zero above is achieved for some $T(f)$, where $f \in {\mathcal{M}}_d$ is $(a,b)$–rooted, then the maximal zero is also achieved for $T ((t-\epsilon/d)^d)$. Let ${\mathcal{A}}={\mathcal{A}}(a,b,d,\epsilon)$ be the set of all $[a,b]$–rooted polynomials $f \in {\mathcal{M}}_d$ with $\tr(f)=\epsilon$. Note that continuity, compactness and Hurwitz’ theorem the maximum zero (say $\rho$) is achieved for some $T(f)$, where $f \in {\mathcal{A}}$. We argue that we may move zeros of $f$ to the boundary of $[a,b]$, while retaining $\tr(f)$ and the maximal zero of $T(f)$ as long as $f$ has at least two distinct zeros in $(a,b)$. Suppose $a<\alpha<\beta<b$ are two zeros of $f\in {\mathcal{A}}$ and that the maximal zero is realized for $T(f)$. For $0<|s| \leq \min(b-\beta, \alpha-a, \beta-\alpha)$, let $$f_s(x) := \frac {(x-\alpha-s)(x-\beta+s)}{(x-\alpha)(x-\beta)} f(x),$$ and note that $f_s \in {\mathcal{A}}$ and $$f= (1-\theta)f_{s}+ \theta f_{-s}, \quad \mbox{ where } \quad \theta = \frac 1 2 \left(1- \frac s {\beta-\alpha}\right)\in [0,1].$$ By assumption $T(f_s)(\rho) \geq 0$. Since $0=T(f)(\rho)= (1-\theta)T(f_{s})(\rho)+ \theta T(f_{-s})(\rho)$, we conclude that $T(f_s)(\rho) = T(f_{-s})(\rho)=0$. Hence the maximal zero $\rho$ is realized also for $T(f_s)$ where $s= -\min(b-\beta, \alpha-a, \beta-\alpha)$. By possible iterating this process a few times we will have moved at least one interior zero to the boundary. We can continue until there is at most one distinct zero in $(a,b)$. Suppose the maximal zero $\rho$ above is achieved for some $f \in {\mathcal{M}}_d$ which is $(a,b)$–rooted. Then $\rho$ is also attained for the same problem when we replace $[a,b]$ by $[r,s]$ where $a<r<s<b$ and $r-a$ and $b-s$ are sufficiently small. Hence, by what we have just proved, for each such $r,s$ there are nonnegative integers $i,j$ with $i+j \leq d$ such that $$\label{abs} T\left( (t-r)^i (t-s)^j \left(t - \frac {\epsilon -ir-js}{d-i-j}\right)^{d-i-j}\right) (\rho)=0.$$ The left–hand–side of is a polynomial, say $P_{i,j}(r,s) \in {\mathbb{R}}[r,s]$. Hence the polynomial $\prod_{i,j}P_{i,j}(r,s)$, where the product is over all $i,j$ which are realized for some such $r,s$, vanishes on a set with nonempty interior, so it is identically zero. Hence $P_{i,j}(r,s) \equiv 0$ for some $i,j$. But then $0=P_{i,j}(\epsilon/d, \epsilon/d)=T((t-\epsilon/d)^d)(\rho)$ as desired. We may now finish the proof of that Conjecture \[maxmax2\] implies Conjecture \[maxmax\]. It remains to prove that the largest possible zero of $g_{\mathbf{v}}(t)$, where ${\mathbf{v}}$ satisfies (a) and (b) above is achieved when $h(t{\mathbf{e}}-{\mathbf{v}})= (t-\epsilon/d)^d$, assuming (we assume Conjecture \[maxmax2\]) that the maximum is achieved for some ${\mathbf{v}}$ where $h(t{\mathbf{e}}-{\mathbf{v}})$ is $(0,1/k)$–rooted. By e.g. considering $h=\det$ on $d \times d$–matrices this is equivalent to proving that the maximal zero of $T_{k,d}(f)$ where $f$ ranges over all monic $[0,1/k]$–rooted polynomials of degree $d$ with trace $\epsilon$ is achieved when $f=(t-\epsilon/d)^d$, under the assumption that the maximal zero is achieved for some $(0,1/k)$–rooted $f$. This follows from the last part of Lemma \[affine\]. Sharpness of the bound in Theorem \[t1\] {#sbound} ======================================== We will in this section use results known about the asymptotic behavior of the largest zero of Jacobi polynomials to see that the bound in Theorem \[t1\] is close to being optimal. Consider the degree $d$ elementary symmetric polynomial in $mk$ variables: $$e_d(x_1,\ldots, x_{mk}) = \sum_{|S|=d} \prod_{i \in S}x_i,$$ which is hyperbolic with respect to the all ones vector ${\mathbf{1}}\in {\mathbb{R}}^{mk}$, see e.g. [@BrOp; @COSW]. Since the coefficients of $e_d({\mathbf{x}})$ are nonnegative, its hyperbolicity cone contains the positive orthant. If ${\mathbf{e}}_i$ denotes the $i$th standard basis vector, then $$\tr({\mathbf{e}}_i) = \frac {d} {mk}, \ \ \rk({\mathbf{e}}_i)=1 \ \ \mbox{ and } \ \ {\mathbf{e}}_1+\cdots+{\mathbf{e}}_{mk}={\mathbf{1}},$$ for all $1\leq i \leq mk$. By symmetry, the partition $$S_1=\{1,\ldots, m\}, S_2=\{m+1, \ldots, 2m\}, \ldots, S_k=\{(k-1)m+1,\ldots, km\}$$ minimizes the bound in . Now $$\begin{aligned} e_d\left(t{\mathbf{1}}-\sum_{i \in S_1}{\mathbf{e}}_i\right) &= \sum_{|A|=d} (t- 1)^{|A\cap S_1|}t^{d- |A\cap S_1|}\\ &= \sum_{j=0}^d \binom {m(k-1)} {j} \binom m {d-j}(t-1)^{d-j}t^{j}\\ &= P^{(mk-m-d,m-d)}_d(2t-1),\end{aligned}$$ where $P^{(\alpha,\beta)}_k(t)$ is a Jacobi polynomial. The asymptotic behavior of the largest zero of Jacobi polynomials is well studied, see e.g. [@Is; @Kra]. For example, if $\alpha_d, \beta_d >-1$ satisfy $$\frac {\alpha_d}{\alpha_d +\beta_d +2d} \to a \mbox{ and } \frac {\beta_d}{\alpha_d +\beta_d +2d} \to b \mbox{ as } d \to \infty,$$ then the largest zero of $P_d^{(\alpha_d,\beta_d)}(t)$ converges to $$\label{abby} b^2-a^2+\sqrt{(a^2+b^2-1)^2-4a^2b^2},$$ as $d \to \infty$, see [@Is Theorem 8]. Fix $\epsilon$ and $k$, and let $m:=m(d)=\lceil d/(\epsilon k) \rceil $ and $\alpha_d=mk-m-d$, $\beta_d=m-d$. Then $a=1-1/k-\epsilon$ and $b=1/k-\epsilon$, and so by the largest zero of $P^{(\alpha_d, \beta_d)}_d(2t-1)$ converges to $$\frac 1 k + \epsilon \frac {k-2} k + 2 \frac{\sqrt{k-1}}{k} \sqrt{\epsilon-\epsilon^2},$$ which should be compared to the bound achieved by Theorem \[t1\] (as $m\to \infty$): $$\frac 1 k + \epsilon + 2\frac {\sqrt{k}} k \sqrt{\epsilon}.$$ We conclude: \[lowprop\] There is no version of Theorem \[t1\] with an ($m,d$-independent) bound in the right–hand–side of which is smaller than $$\label{bbound} \frac 1 k + \epsilon \frac {k-2} k + 2 \frac{\sqrt{k-1}}{k} \sqrt{\epsilon-\epsilon^2},$$ for $\epsilon \leq 1-1/k$. It is known that if $1<d<n-1$, then $e_d(x_1, \ldots, x_n)$ is *not* a determinantal polynomial, i.e., there is no tuple of positive semidefinite matrices $A_1, \ldots, A_n$ such that $$e_d(x_1, \ldots, x_n) = \det(x_1A_1+ \cdots+x_nA_n).$$ Thus we cannot directly derive an analog of Proposition \[lowprop\] for Theorem \[MSSmain\]. Consequences for strong Rayleigh measures and weak half-plane property matroids =============================================================================== A discrete probability measure, $\mu$, on $2^{[n]}$ is called *strong Rayleigh* if its *multivariate partition function* $$P_\mu({\mathbf{x}}) := \sum_{S \subseteq [n]} \mu(\{S\}) \prod_{j \in S}x_j,$$ is *stable*, i.e., if $P_\mu({\mathbf{x}}) \neq 0$ whenever ${{\rm Im}}(x_j)>0$ for all $1 \leq j \leq n$. Strong Rayleigh measures were investigated in [@BBL], see also [@Pem; @Wag]. We shall now reformulate Theorem \[t1\] in terms of strong Rayleigh measures. The measure $\mu$ is of *constant sum* $d$ if $|S|=d$ whenever $\mu(\{S\}) \neq 0$, i.e., if $P_\mu({\mathbf{x}})$ is homogeneous of degree $d$. It is not hard to see that a constant sum measure $\mu$ is strong Rayleigh if and only if $P_\mu({\mathbf{x}})$ is hyperbolic with respect to the all ones vector ${\mathbf{1}}$ and ${\mathbb{R}}_+^n \subseteq \Lambda_+({\mathbf{1}})$, see [@BBL]. Note that if ${\mathbf{e}}_i$ is the $i$th standard basis vector then $$\tr({\mathbf{e}}_i) = \sum_{S \ni i} \mu(\{S\})= {\mathbb{P}}[S : i \in S],$$ where the trace is defined as in the introduction for the hyperbolic polynomial $P_\mu$, with ${\mathbf{e}}={\mathbf{1}}$. If $S \subseteq [n]$ we write ${\mathbf{e}}_S:=\sum_{i \in S}{\mathbf{e}}_i$. The following theorem is now an immediate consequence of Theorem \[t1\]. \[t11\] Let $k\geq 2$ be an integer and $\epsilon$ a positive real number. Suppose $\mu$ is a constant sum strong Rayleigh measure on $2^{[n]}$ such that ${\mathbb{P}}[S : i \in S] \leq \epsilon$ for all $1\leq i \leq n$. Then there is a partition $S_1 \cup \cdots \cup S_k=[n]$ such that $$\| e_{S_i} \| = {\lambda_{\rm max}}({\mathbf{e}}_{S_i}) \leq \frac 1 k \delta(k\epsilon,n)$$ for each $1 \leq i \leq n$. Let us also see that Theorem \[t1\] easily follows from Theorem \[t11\]. Assume the hypothesis in Theorem \[t1\], and form the polynomial $$P({\mathbf{x}})= h(x_1{\mathbf{u}}_1+\cdots+x_m{\mathbf{u}}_m)/h({\mathbf{e}}).$$ It follows that $P({\mathbf{x}})$ is hyperbolic with hyperbolicity cone containing the positive orthant. Since $\rk({\mathbf{u}}_i) \leq 1$ for all $1\leq i \leq m$ we may expand $P({\mathbf{x}})$ as $$P({\mathbf{x}})= \sum_{S \subseteq [m]} \mu(\{S\}) \prod_{j \in S}x_j,$$ where $\mu(\{S\}) \geq 0$ for all $S \subseteq [m]$. Since $\tr_h({\mathbf{u}}_i)= \tr_P({\mathbf{e}}_i)$ for all $1\leq i \leq m$, the conclusion in Theorem \[t1\] now follows from Theorem \[t11\]. The *support* of $\mu$ is $\{S: \mu(\{S\})>0\}$. Choe *et al.* [@COSW] proved that the support of a constant sum strong Rayleigh measure is the set of bases of matroid. Such matroids are called *weak half-plane property matroids*. The rank function, $r$, of such a matroid is given by $$r(S) = \rk\left(\sum_{i \in S}{\mathbf{e}}_i\right),$$ where $\rk$ is the rank function associated to the hyperbolic polynomial $P_\mu$ as defined in the introduction, see [@BrObs; @Gu]. Edmonds Base Packing Theorem [@Edm] characterizes, in terms of the rank function, when a matroid contains $k$ disjoint bases. Namely if and only if $$r(S) \geq d -\frac {n-|S|} k, \quad \mbox{ for all } S \subseteq [n],$$ where $r$ is the rank function of a rank $d$ matroid on $n$ elements. Using Theorem \[t11\] we may deduce a sufficient condition (of a totally different form) for a matroid with the weak half-plane property to have $k$ disjoint bases: \[packing\] Let $k\geq 2$ be an integer. Suppose $\mu$ is a constant sum strong Rayleigh measure such that $${\mathbb{P}}[S : i \in S] \leq \left(\frac 1 {\sqrt{k-1}} - \frac 1 {\sqrt{k}}\right)^2$$ for all $1\leq i \leq n$. Then the support of $\mu$ contains $k$ disjoint bases. Suppose $\tr({\mathbf{e}}_i) \leq \epsilon$ for all $i$. Let $S_1 \cup \cdots \cup S_k=[n]$ be a partition afforded by Theorem \[t11\], and let ${\mathbf{v}}_j= \sum_{i \in S_j}{\mathbf{e}}_i$ for each $j \in [n]$. If we can prove that ${\lambda_{\rm min}}({\mathbf{v}}_j)>0$, then $\rk({\mathbf{v}}_j)=\rk({\mathbf{1}})$ and so $S_j$ contains a basis. Now, by , Theorem \[t11\], and the convexity of ${\lambda_{\rm max}}$: $$\begin{aligned} {\lambda_{\rm min}}({\mathbf{v}}_j) &= 1-{\lambda_{\rm max}}({\mathbf{1}}-{\mathbf{v}}_j) =1-{\lambda_{\rm max}}\left(\sum_{i \neq j}{\mathbf{v}}_i\right) \\ &\geq 1- \sum_{i \neq j}{\lambda_{\rm max}}({\mathbf{v}}_i) \geq 1-\frac {k-1} {k} \delta(k\epsilon,n) \\ &> 1- \frac {k-1} {k} \left(1+\sqrt{\epsilon k}\right)^2.\end{aligned}$$ Hence we want the quantity on the left hand side to be nonnegative, which is equivalent to $$\epsilon \leq \left(\frac 1 {\sqrt{k-1}} - \frac 1 {\sqrt{k}}\right)^2.$$ We have not investigated the sharpness of Theorem \[packing\], nor if it is possible to prove analogous versions for arbitrary matroids. For an arbitrary matroid on $[n]$ one could take the uniform measure on the set of bases of the matroid and define $\tr(i) = {\mathbb{P}}[S: i\in S]$. What trace bounds guarantees the existence of $k$ disjoint bases? It would be interesting to see if other theorems on matroids have analogs for weak half-plane property matroids using Theorem \[t11\]. Also, can we find continuous versions of theorems in matroid theory using the analogy that Theorem \[t1\] can be seen as a continuous version of Edmonds Base Packing Theorem? [99]{} M. F. Atiyah, R. Bott, L Gårding, Lacunas for hyperbolic differential operators with constant coefficients. I, Acta Math. [**124**]{} (1970), 109–189. H. H. Bauschke, O. Güler, A. S. Lewis, H. S. Sendov, Hyperbolic polynomials and convex analysis, Canad. J. Math. [**53**]{} (2001), 470–488. J. Borcea, P. Brändén, T. M. Liggett, [Negative dependence and the geometry of polynomials]{}, J. Amer. Math. Soc. [**22**]{} (2009), 521–567, <http://arxiv.org/abs/0707.2340>. P. Brändén, [ Obstructions to determinantal representability,]{} Adv. Math., [**226**]{} (2011), 1202–1212, <http://arxiv.org/pdf/1004.1382.pdf>. P. Brändén, Hyperbolicity cones of elementary symmetric polynomials are spectrahedral, Optim. Lett. [**8**]{} (2014), 1773–1782, <http://arxiv.org/abs/1204.2997>. Y. Choe, J. Oxley, A. Sokal, D. G. Wagner, [ Homogeneous multivariate polynomials with the half-plane property]{}. Adv. Appl. Math. [**32**]{} (2004), 88–187, <http://arxiv.org/pdf/math/0202034.pdf>. P. G. Casazza, Consequences of the Marcus/Spielman/Srivastava solution to the Kadison–Singer Problem, <http://arxiv.org/abs/1407.4768>. M. Chudnovsky, P. Seymour, The roots of the independence polynomial of a clawfree graph, J. Combin. Theory Ser. B [**97**]{} (2007), 350–357. J. Edmonds, Lehman’s switching game and a theorem of Tutte and Nash-Williams, J. Res. Nat. Bur. Standards Sect. B [**69 B**]{} (1965), 73–77. L. Gårding, [An inequality for hyperbolic polynomials]{}, J. Math. Mech. [**8**]{} (1959), 957–965. L. Gurvits, Combinatorial and algorithmic aspects of hyperbolic polynomials, <http://arxiv.org/abs/math/0404474>. L. Hörmander, The analysis of linear partial differential operators. II. Differential operators with constant coefficients, Springer-Verlag, Berlin, 1983. M. E. H. Ismael, X. Li, Bound on the extreme zeros of orthogonal polynomials, Proc. Amer. Math. Soc. [**115**]{} (1992), 131–140. R. V. Kadison, I. M. Singer, Extensions of pure states, Amer. J. Math. [**81**]{} (1959), 383–400. I. Krasikov, On extreme zeros of classical orthogonal polynomials, J. Comput. Appl. Math. [**193**]{} (2006), 168–182. A. W. Marcus, D. A. Spielman, N. Srivastava, Interlacing families I: Bipartite Ramanujan graphs of all degrees, Ann. of Math. (to appear), [http://arxiv.org/abs/1304.4132 ](http://arxiv.org/abs/1304.4132 ). A. W. Marcus, D. A. Spielman, N. Srivastava, Interlacing families II: Mixed characteristic polynomials and the Kadison-Singer problem, Ann. of Math. (to appear), <http://arxiv.org/abs/1306.3969>. R. Pemantle, Hyperbolicity and stable polynomials in combinatorics and probability, Current developments in mathematics, 2011, 57–123, Int. Press, Somerville, MA, 2012, <http://arxiv.org/abs/1210.3231>. J. Renegar, Hyperbolic programs, and their derivative relaxations, Found. Comput. Math., [**6**]{} (2006), 59–79. V. Vinnikov, LMI representations of convex semialgebraic sets and determinantal representations of algebraic hypersurfaces: past, present, and future, Mathematical methods in systems, optimization, and control, 325–349, Oper. Theory Adv. Appl., [**222**]{}, BirkhŠuser/Springer Basel AG, Basel, 2012, <http://arxiv.org/abs/1205.2286>. D. G. Wagner, Multivariate stable polynomials: theory and applications, Bull. Amer. Math. Soc. [**48**]{} (2011), 53–84, <http://arxiv.org/abs/0911.3569>. N. Weaver, The Kadison-Singer problem in discrepancy theory, Discrete Math. [**278**]{} (2004), 227–239.

--- abstract: 'We perform full 3D numerical simulations of compact objects, such as black holes or neutron stars, boosted through an ambient force-free plasma that posses a uniform magnetization. We study jet formation and energy extraction from the resulting stationary late time solutions. The implementation of appropriate boundary conditions has allowed us to explore a wide range of boost velocities, finding the jet power scales as $\gamma v^2$ (being $\gamma$ the Lorentz factor). We also explore other parameters of the problem like the orientation of the motion respect to the asymptotic magnetic field or the inclusion of black hole spin. Additionally, by comparing a black hole with a perfectly conducting sphere in flat spacetime, we manage to disentangle curvature effects from those produced by the perfect conducting surface. It is shown that when the stellar compactness is increased these two effects act in combination, further enhancing the luminosity produced by the neutron star.' author: - Ramiro Cayuso - Federico Carrasco - Barbara Sbarato - Oscar Reula bibliography: - 'FFE.bib' title: Astrophysical jets from boosted compact objects --- Introduction ============ Enormous amounts of energy, in the form of Poynting winds or highly collimated relativistic jets, are often observed in various astrophysical scenarios. Such energetic phenomena are believed to be powered by compact objects like black holes (BH) and neutron stars (NS), from the interactions with strong and large-scale magnetic fields on their surrounding magnetospheres. In the seminal works of Goldreich & Julian [@goldreich] and Blandford & Znajek [@Blandford] (describing pulsars and active galactic nuclei, respectively), it was first demonstrated that the vicinity of these spinning compact objects would be filled with a tenuous plasma. In such rarefied environments, the electromagnetic force dominates over particle inertia and leads to a great simplification in the problem, allowing to capture the basic mechanism that taps rotational energy by means of the electromagnetic field. While pulsars admits a classical interpretation as Faraday disks [@faraday1832], in the black hole scenario the energy is instead extracted in a form of generalized Penrose process (see e.g. [@lasota2014]) known as the Blandford-Znajek mechanism. This low-inertia limit of relativistic magnetohydrodynamics, referred as force-free electrodynamics (FFE), has been –since then– widely used to study global properties of neutron stars and black holes magnetospheres, like for instance Refs. [@contopoulos1999; @Komissarov2004b; @mckinney2006relativistic; @timokhin2006force; @spitkovsky2006; @Palenzuela2010Mag; @Gralla2014]. In the force-free approximation, when there is a perturbation of an otherwise constant magnetic field, the dynamics makes these perturbation travel preponderantly along the magnetic field lines, thus carrying energy with them along this direction. In this work, we simulate a couple of astrophysically relevant situations where this happens, which consist on a black hole or a neutron star moving through a plasma-filled region of constant magnetic field. Galactic mergers could provide a likely scenario for the black hole case [@begelman1980; @milosavljevic2005], since the resulting circumbinary disk of the merged galaxy will anchor magnetic field lines, some of which traverse the central region where the binary –and eventually the final supermassive black hole– moves. Another example could be a BH-NS binary, in which the black hole would move through the magnetic field of a neutron star [@mcwilliams2011; @paschalidis2013]. In such cases, we expect the black hole to loose some kinetic energy, transforming it by enlarging its mass but also into electromagnetic energy propagated away by the jets. There has been a number of previous numerical studies on this scenario, [@palenzuela2010dual; @Palenzuela2010Mag; @Luis2011], which we use as starting point for the present work. All of them analyze the problem from the point of view of the stationary magnetic field, namely, in their numerical grid the black hole moves and creates the jets which carry the energy. The advantage is that they can readily compute an approximate -since their time direction is a not Killing direction for the background geometry- energy flux. It is precisely this absence of a timelike Killing vector field and the correspondingly lack of a conserved positive-definite energy, which permits to have energy transport via jets in this approximation where the background is fixed. The disadvantage is that they can not model high speed black holes for they move too quickly outside the grid. In our case we choose to describe the problem from the black hole static geometry. The black hole sees a boosted magnetic field and the corresponding electric field, the interaction of its geometry with that electromagnetic configuration generates a stationary solution which takes energy away through jets. In our case we do have a background Killing vector field, and so conservation of the energy it defines, but this is not the energy an observer for which the (uniform) magnetic field is at rest would see. Thus, we also have to define approximate energy fluxes corresponding to these –for our description– moving observers. The energies so defined are transported away, as expected. The other situation we model is that of a neutron star, also moving on a region of uniformly magnetized plasma. This could happen if a neutron star orbits near an active supermassive black hole, where both strong magnetic fields and force-free plasma are expected around the central region. It could also be relevant in the context of electromagnetic precursor signals from neutron stars mergers [@palenzuela2013electromagnetic; @palenzuela2013linking; @ponce2014], the likely progenitors of gamma-ray burst. We consider here an idealized setting where the neutron star is represented by a perfectly conducting spherical surface and there is no field generated at the stellar interior. This might be regarded as the limiting case in which the exterior magnetic field is much stronger than the one associated to the star, so that the later can be neglected. We defer the inclusion of the star’s own magnetic field and rotation to a more detailed analysis on a future work. A similar behavior to the boosted (nonspinning) black hole case is found, although the details of the operating mechanisms are not the same. Here, kinetic energy from the motion is transformed into Alfvén waves, sourced by the boundary conditions at the conductor. One nice aspect of this problem is that it allows to take the flat spacetime limit, in which the boosted time direction is also a Killing direction. This means there is no ambiguity on defining the energy fluxes used for the description; and thus, it might help gaining some insight into the previous –more delicate– black hole scenario. In section II, we describe the setup for both cases: the numerical scheme, geometry and evolution equations; the initial data, the boundary conditions and, finally, the energy fluxes definitions. With all of these information one should be able to reproduce our results unambiguously. Except for the boundary conditions and energy fluxes, the setting is very similar to the one in [@palenzuela2010dual; @Luis2011]. In section III we present the results of our simulations, where different aspects of the problem were explored. Conclusions and perspectives are drawn on Section IV. Throughout, we adopt geometrized units in which $c=G=1$ and Lorentz-Heaviside units for the electromagnetic field. Setup ===== We are interested on modeling the magnetosphere of a compact object (BH or NS) that travels across a uniform magnetic field by solving the equations of force-free electrodynamics. The code used here was first described in [@FFE2] for black holes and later extended in [@NS], by developing appropriate boundary conditions for the perfectly conducting surface of a neutron star. Since we adopt the reference frame of the central object, its motion relative to the uniform magnetic field will be accomplished through suitable boundary conditions at the external surface of the domain. We shall look for stationary solutions obtained by evolving the fields until they do not change appreciably. The resulting state is determined only by boundary conditions, the background geometry, and to some extent on the handling of the electric field growth on the current sheets that the dynamics generates. Although a detailed description of our numerical implementation can be found on previous works [@FFE2; @NS], we shall start this section by briefly summarizing its basic features along with information about the metric and the set of evolution equations employed. Then, we shall describe initial data and boundary conditions for the two scenarios we want to study. And finally, we shall discuss the energy fluxes definitions used to analyze the results. Numerical Implementation ------------------------ We evolve a particular version of force-free electrodynamics derived at [@FFE], which has some improved properties in terms of well posedness and involves the full force-free current density [^1]. More concretely, we shall consider the evolution system given by Eqs. (8)-(9)-(10) in [@NS]. The numerical scheme to solve these equations is based on the *multi-block approach* [@Leco_1; @Carpenter1994; @Carpenter1999; @Carpenter2001], in which the numerical domain is built from several non-overlapping grids where only grid-points at their boundaries are sheared. The equations are discretized at each individual subdomain by using difference operators constructed to satisfy summation by parts (SBP). In particular, we employ difference operators which are sixth-order accurate on the interior and third-order at the boundaries. Numerical dissipation is incorporated through the use of adapted Kreiss-Oliger operators. These compatible difference and dissipation operators were both taken from Ref. [@Tiglio2007]. *Penalty terms* [@Carpenter1994; @Carpenter1999; @Carpenter2001] are added to the evolution equations at boundary points. These terms penalize possible mismatches between the different values the characteristic fields take at the interfaces, providing a consistent way of communicate information between the different blocks: essentially, the outgoing characteristic modes of one grid are matched onto the ingoing modes of its neighboring grids. At each subdomain, it is possible to find a semi-discrete energy defined by both a symmetrizer of the system at the continuum and a discrete scalar product (with respect to which SBP holds). The summation by parts property of the operators allows one to obtain an energy estimate, up to outer boundary and interface terms left after SBP. The penalties are constructed so that they make a contribution to the energy estimate which cancels inconvenient interface terms, thus providing an energy estimate which covers the whole integration region across grids. Such semi-discrete energy estimates –provided an appropriate time integrator is chosen– guarantee the stability of the numerical scheme [@Kreiss; @Carpenter1994; @Carpenter1999; @Carpenter2001; @Leco_2]. A classical fourth order Runge-Kutta method is used for time integration in our code. We use a particular multiple patch infrastructure that has been equipped with the Kerr metric, as in Ref. [@Leco_1]. This provides a numerical domain that is perfectly adapted to the geometry of the problems, having two global inner/outer boundaries with spherical topology [^2]. The Kerr metric is parametrized by the mass $M$ and spin $a$, and can be written in the Kerr-Schild form as $g_{ab} = \eta_{a b} + H \, \ell_{a } \ell_{b } $, where $\eta_{ab}$ is the flat metric and $\ell_{a}$ is a null co-vector with respect to both $\eta_{ab}$ and $g_{ab}$. For visual representation, throughout this article, we will present our results in the Cartesian coordinates $\{t,x,y,z\}$ associated with the flat part of the metric[^3]. In these coordinates, the metric function $H$ takes the form $$\begin{aligned} H &=& \frac{2 M r}{r^2 + a^2 z^2 /r^2}\\ r^2 &=& \frac{1}{2}(\rho ^2-a^2) + \sqrt{\frac{1}{4}(\rho ^2-a^2)^2 + a^2 z^2} \\ \rho^2 &=& x^2+y^2+z^2 $$ and the co-vector $\ell_{a}$ reads $$\ell_{a} = \left\lbrace 1, \frac{rx + ay}{r^2+a^2}, \frac{ry-ax}{r^2+a^2}, \frac{z}{r} \right\rbrace .$$ Typically we solve in a region between an interior sphere whose radio is either inside the black hole or represents a perfectly conducting boundary, and an exterior spherical surface at $r=162M$. This region is covered by a total of $9\times 6$ grids, being $9$ the number of layers. The typical resolution used on each of these grids is of $41 \times 41$ grid-points in the angular directions and $N_r =101$ points for the radial one. The grids layers do not cover regions of identical radial extension, having more resolution near the inner boundary than in the asymptotic region: from layer to layer we decrease the effective radial resolution by a factor $1.3$. In some cases, we have increased the resolution of the individual grids to $61 \times 61$ and $N_r =151$. In order to handle current sheets, we use a rather standard approach in which electric field is effectively dissipated to maintain the condition that the plasma is magnetically dominated (i.e. $B^2 -E^2 >0$), discussed in [@FFE2]. At the current sheets the magnetic field presents a jump discontinuity in the vertical direction in the $y$ component while the electric field has a spike in the $x$ component. When high order finite difference operators are used in such discontinuous regions, the fields behave in an unsatisfactory manner. Indeed, high order operators tend to give a noisier results. To overcome this issue, the precision of the finite difference operators is reduced from 6th to 2nd order for those grids covering the region where current sheets form. Thus, providing a substantial improvement in the quality of the numerical approximation. Initial Data ------------ ### Boosted Black Hole We consider a black hole moving with velocity $v_{o}$ respect to a reference frame in which the magnetic field is asymptotically uniform and along the $z$-axis, while the electric field vanishes. The boost direction is not necessarily orthogonal to this magnetic field, so we shall study the evolution of the system for different alignments between the black hole velocity and the asymptotic magnetic field. Since we adopt the reference frame in which the black hole is at rest, the electric and magnetic fields should arise from a Lorentz transformation of the electromagnetic field from the frame in which the magnetic field is uniform and the electric field vanishes. That is, $$\label{B_prime} \vec{B}=\gamma \, \vec{B}^{\prime} - \frac{\gamma^{2}}{\gamma + 1}(\vec{v} \cdot \vec{B}^{\prime}) \, \vec{v}$$ $$\label{E_prime} \vec{E}= \gamma(\vec{v}\times \vec{B}^{\prime}).$$ where $\gamma=\frac{1}{\sqrt{1-v^{2}}}$ is the Lorentz factor. In Kerr-Schild Cartesian coordinates, the uniform magnetic field would read $$B^{\prime x}=B^{\prime y}=0 \quad , \quad B^{\prime z}= \frac{B_{0}}{\sqrt{h}}$$ while the velocity can be generically written as $$\label{velocity} v^{x}= 0 , \quad v^{y}=v_{0}\cos(\chi), \quad v^{z}=v_{0}\sin(\chi)$$ being $\chi$ the angle among the velocity and the $y$-axis. We emphasize the fact that the steady state solutions would only depend on the boundary conditions, namely on the asymptotic boosted fields, and not on the particular way we have chosen to set the initial configuration in the interior.\ ### Boosted perfectly conducting sphere We shall consider a perfectly conducting surface as an idealized model of a neutron star. In the present work, we have assumed that the interior magnetic field of the star is several orders of magnitude weaker than the exterior uniform magnetic field. Thus, we propose an initial data that comes from the configuration of an asymptotically uniform magnetic field around a superconducting sphere, which ensures a vanishing normal component of the magnetic field at the stellar surface. Concretely, the initial condition is given by $$\label{ini_cond_conf} \vec{B} = B_{0} \hat{z} - \frac{B_{0}R^{3}}{2r^{3}}\left( \frac{3z}{r}\hat{r} - \hat{z}\right)$$\ where $R$ is the radius of the star. Hence the field is tangent to the stellar surface and asymptotically matches a uniform magnetic field along the $z$-axis, as we wanted. Notice, however, that it does not satisfies the magnetically dominated plasma condition at the poles where the magnetic field vanishes. Since the force-free equations would break down at these points, we simply chose a grid that does not contain them, which has proven to be enough for achieving well-behaved solutions. Boundary Conditions ------------------- As mentioned before, our numerical domain is bounded by two inner/outer spherical surfaces, where boundary conditions needs to be specified. The treatment given in the present article to these boundaries was previously described and employed in Refs. [@FFE2; @NS]. However, we find important to briefly summarize here the main aspects[^4] and explain how these boundary conditions are applied in this new astrophysical context. Generally speaking, physical conditions are imposed by fixing appropriately all the incoming characteristic (physical) modes via the penalty method. Whereas for the constraints, in this case the divergence-free condition $\nabla \cdot \vec{B}=0$, we adopt a method presented in [@FFE2] (see also [@Mari]) which restricts possible incoming violations at both boundary surfaces. Our implementation of the outer boundary conditions consist on setting the incoming physical modes according to a fixed source $U_{ext} = (\vec{E}_{ext}, \vec{B}_{ext})$ that we control. This idea appear motivated on the interfaces treatment and was already employed in Ref. [@FFE2], where $U_{ext}$ represented a uniform magnetic field sourced by a distant accretion disk. We shall use this strategy again here, with $U_{ext}$ now being the boosted electromagnetic configurations that threatens the compact object magnetosphere. Thus, for the black hole case, $U_{ext}$ is given by the boosted uniform magnetic field (see eqns. -). While for the neutron star, the source is given by the boost of its initial configuration evaluated at the boundary. In this later case, we shall introduce the boost smoothly to its final boost velocity $v_0$ (at time $t_f$) by using a time-dependent function $$\frac{v_0}{2} \left[ 1 - \cos (\pi t / t_f ) \right] \quad \text{if} \quad 0 \leq t \leq t_f$$ The treatment of the inner boundary is, on the other hand, very different between the black hole and neutron star scenarios. In the first case, the inner edge of the domain is simply placed inside the black hole horizon where all characteristic modes points inward (i.e. they are all outgoing from our numerical domain perspective), and hence, there are no incoming modes to be prescribed. In the neutron star case, the inner edge of our domain is placed at the stellar surface which is assumed to behave as an idealized perfect conductor. In the present work, we have further assumed that the interior magnetic field of the star is negligibly small with respect to the one of the external magnetized plasma and also that the star is not rotating. Thus, the boundary conditions reduces to $$B^r = 0 , \quad \alpha E_{\theta} = \sqrt{h} \beta^r B^{\phi}, \quad \alpha E_{\phi} = -\sqrt{h} \beta^r B^{\theta}$$ with $\alpha$, $\beta^i$ and $h_{ij}$ being the *lapse function*, *shift vector* and the *intrinsic metric* on the spatial slices, respectively. The normal magnetic field is keep fixed to zero by enforcing it at each Runge-Kutta substep, as done in Ref. [@PHAEDRA]. While the electric field components are imposed through the *penalty method*, by fixing the incoming physical modes to a rather involved combination of outgoing modes. We refer the interest reader to [@NS] (in particular Sec. II-C and the Appendix) for further details on this. Fluxes ------ In this section we shall discuss the relevance of different quantities needed to describe the jets and the energy extraction process. Before computing any quantity it is important to recall that the electromagnetic field does not by itself define any four-momentum, a four-vector and the energy-momentum tensor are necessary to build this quantity. The different choices of that four-vector define different four-momenta that can be thought to be related to different observers. Thus, the result obtained by computing the electromagnetic flux (as an spatial component of the four-momentum) in the BH’s frame will certainly be different from the one obtained in the plasma frame of reference. The electromagnetic flux expression for the BH’s frame comes naturally, it is associated to the time-like Killing vector field of the background spacetime geometry that allows to construct a conserved four-momentum current. To obtain fluxes this vector is contracted to the normal to some space-like hypersurface, usually the boundary of a sphere at some radius $r=R$ in Kerr-Schild coordinates. We will refer to this quantity as Poynting flux, defined as follows, $$\label{flux_pf} \Phi_{ \mathcal{E}}:= \sqrt{-g} \, p^r$$ where $p^{a}$ is the four-momentum defined by, $$\label{cuadri-momento} p^{a} := - T^{ab}k_{b}$$ where $T^{ab}$ is the Energy-momentum tensor and $k^{a}$ is the Killing vector field related to stationarity (see for instance Appendix B of [@FFE2]). The factor $\sqrt{-g}$ appears from the normalization at the surface $r=R$. Now the task is to find a quantity that is representative of the electromagnetic energy flux in the reference frame in which the BH is not static, namely the frame where the asymptotic magnetic field is constant and at rest. To represent an observer that moves relative to the BH with velocity $v^{\prime a}$, we can take its four-velocity $n^{\prime a}$ to be, $$\label{tprima} n^{\prime a} = \gamma ( n^{a} + v^{\prime a})$$ One can thus define the four-momentum $p^{\prime a}$ for this observer as, $$\label{cuadri-momento_prima} p^{\prime a} := - T^{ab}n^{\prime}_{b} \quad$$ where $n^{a}$ is the normal vector to the equal-time hypersurface $\Sigma_{t} $ of the space-time foliation. Since the background geometry describes a curved spacetime, this boosted quantity is rather arbitrary: it does not have the same normalization as the Killing vector field nor gives a conserved current. Nevertheless it acquires meaning as an asymptotic quantity, far away from the BH we can use the fact that the geometry is approximately flat there and so the boosted time direction would approach a boosted Killing vector of the underlying asymptotic geometry. Thus, for large distances we can define the four-momentum of an observer that is in the plasma rest frame (i.e. with velocity $v^{\prime a} = - v^{a}$) as in . Now using this four-momentum we can define a quantity that is representative of the electromagnetic flux in this frame, we can do so by contracting $p^{\prime a}$ with a vector $N^{\prime a}$ that is both of norm unity and orthogonal to $n^{\prime a}$. By proposing an *Ansatz* of the form $N^{\prime a} = ( N^{a} + \zeta v^{\prime a} + \xi k^{a} )$ (where $k^{a}$ is the Killing vector field associated with stationarity and $N^{a}$ is the radial unit vector) and using the conditions: $$\label{conditions} N^{\prime a}N^{\prime}_{a} = 1 \quad ; \quad n^{\prime a}N^{\prime}_{a} = 0$$ we can obtain the value of the constants $\xi$ and $\zeta$ as, $$\label{a_m2} \xi = -\frac{N_{\beta}}{\alpha} + v^{\prime r} + \zeta v^{\prime 2}{\alpha - v^{\prime}_{\beta}}$$ $$\label{b_m2} \begin{split} \zeta = & \frac{\sqrt{\alpha ^{2}(\alpha - v^{\prime}_{\beta})^{2}((v^{\prime r}\alpha^{2} - \beta ^{r}v^{\prime 2}_{\beta} )^{2} + v^{\prime 2}q(v^{\prime r 2}\alpha ^{2} - \beta ^{r 2} ) )}}{\alpha ^{2} v^{\prime 2}(\alpha ^{2} + v^{\prime 2}q - v^{\prime 2}_{\beta})}\\ & + \frac{ (\alpha - v^{\prime}_{\beta})(\beta ^{r}\alpha v^{\prime}_{\beta} - v^{\prime r}\alpha^{3}) + v^{\prime 2}q \alpha ( \beta ^{r} - \alpha v^{\prime r}) }{\alpha ^{2} v^{\prime 2}(\alpha ^{2} + v^{\prime 2}q - v^{\prime 2}_{\beta})} \end{split}$$ where $q = (-\alpha ^{2} + \beta ^{2}) $, $ v^{\prime}_{\beta} = v^{\prime i} \beta _{i}$ and $\alpha$ $\&$ $\beta$ are respectively the Lapse and Shift of the spacetime foliation. It can be easily checked that for large values of r (i.e. where $\alpha \rightarrow 1$ and $\beta \rightarrow 0$) the vector $N^{\prime a}$ is simply the Lorentz transformation of the $N^{a}$ vector, and so the surface determined by the vectors $N^{\prime a}$ is asymptotically a boosted sphere. Now we can finally define the electromagnetic energy flux for the boosted frame as $$\label{PF_boost} \Phi^{\prime}_{ \mathcal{E}}:= \sqrt{h }p^{\prime a}N^{\prime}_{a}$$ where $h$ is the determinant of induced metric to the surface in which this flux is computed. It is very important to stress that, even though the expression can be evaluated in the whole numerical domain, it’s value will only be representative of the electromagnetic flux far away from the BH, where $n^{\prime a}$ approaches an asymptotic Killing vector field and consequently the $p^{\prime a}$ is an asymptotically conserved four-momentum. This downside of not being able to properly define the electromagnetic flux (for the plasma’s reference frame) in the whole numerical domain is not a consequence of our choice of the BH rest reference frame. In previous works, [@Luis2011; @alic2012; @palenzuela2010dual] a measurement of the electromagnetic flux is also given accurately only far away from the Black Hole(s) system since the Killing field needed to construct the conserved four momentum is only an asymptotic concept. Another aspect that has to be taken into account is the fact that actually this system is not isolated, since uniform magnetic and electric fields are present as a background in the whole numerical domain and they would remain so as a consequence of the boundary conditions imposed at the outer boundary. Special care has to be taken in order to distinguish the electromagnetic flow generated by the BH’s interaction with the fields, from the ubiquitous flow generated by the background. In order to subtract adequately this background radiation we take the same approach as in [@Luis2011; @alic2012], that is, we subtract the value of the field’s initial condition to the stationary values of the electromagnetic field (final configuration) before computing the value of the electromagnetic energy flux. Another interesting approach to this problem is to compute asymptotic fluxes using the plane wave structure of force-free electrodynamics, that is, its fast and Alfvén propagation modes. Mensurable quantities can be constructed for each plane wave mode and study them separately. For instance, we shall look at the radial fluxes $\Phi^{\pm}_A$ (Alfvén modes) and $\Phi^{\pm}_T$ (fast magnetosonic) of the final stationary solutions. Following appendix A of Ref. [@NS] and assuming our solutions reasonably satisfy the constraint, i.e. $\vec{E}\cdot \vec{B} \approx 0$, we get: $$\begin{aligned} \Phi^{\pm}_A &:=& \lambda^{\pm}_A (\Theta^{\pm}_A (U))^2 (u^{\pm}_A)^2 \nonumber \\ &=& (\beta_m + \alpha \sigma^{\pm}_A ) \frac{E_{m}^2}{\left[ 1-(\sigma^{\pm}_A)^2 \right] } \label{A-flux}\\ \Phi^{\pm}_T &:=& \lambda^{\pm}_T (\Theta^{\pm}_T (U))^2 (u^{\pm}_T)^2 \nonumber\\ &=& (\beta_m \pm \alpha ) \frac{\left[ B_{p}^2 - E^2 \right] ^2}{\left[ B_{p}^2 + E_{p}^2 \mp 2 S_m \right] } \label{F-flux}\end{aligned}$$ where $\sigma_{A}^{\pm} := \frac{1}{B^2} \left( S_m \pm \sqrt{B^2 - E^2} B_m \right) $, $A_m$ represents contractions of vectors on the radial unitary direction $m^i$ and, $A_{p}^i := A^i - A_m m^i$, its perpendicular projections. Numerical Results ================= Boosted black hole ------------------ In this section we shall explore different parameters of the problem such as boost velocity $v$, black hole spin $a$ and inclination angle $\chi$ among the boost direction and the $y$-axis (i.e. $\chi$ represents the departure of the direction of the motion from the case in which it is orthogonal to the asymptotic magnetic field). Lets start first by the case where the asymptotic magnetic field is orthogonal to the boost velocity $v^{a}$ (e.i. $\chi=0$). We shall study some aspects of the solution for different magnitudes of this velocity, such as the topology of the electric and magnetic field configurations, the power of the electromagnetic flux and the development of a current sheet. Later, we shall study the dependence of the luminosity on the misalignment between the boost direction and the asymptotic magnetic field. And finally, we will analyze the effects that including rotation has on the electromagnetic flux. ### Orthogonal velocity We present a late time configuration for the case of a Schwarzschild black hole, boosted with velocity $ v=0.5$ orthogonal to the asymptotic magnetic field. The general structure of the solution is depicted on Fig. \[fig:fieldlines\], where it can be seen that magnetic field lines are disturbed by the passage of the black hole, leaving a trail behind it. Similarly, electric field lines, shown at the $z=0$ plane, are also dragged by the black hole as it moves along the $y$-axis, but from the opposite side. The way in which the magnetic field is pulled towards the black hole, result in a discontinuity of the $y$-component at the equatorial plane, thus producing a current sheet with strong electric fields. ![ Late time numerical solution ($t = 400M)$ for a black hole moving at speed $v=0.5$ along the $y$-direction. Streamlines of the magnetic field at the $x=0$ plane (top) and of the electric field at the $z=0$ plane (bottom) are illustrated. []{data-label="fig:fieldlines"}](v05_B_field_lines0000.png "fig:") ![ Late time numerical solution ($t = 400M)$ for a black hole moving at speed $v=0.5$ along the $y$-direction. Streamlines of the magnetic field at the $x=0$ plane (top) and of the electric field at the $z=0$ plane (bottom) are illustrated. []{data-label="fig:fieldlines"}](v05_E_field_lines00000.png "fig:") In Fig. \[Current\_sheet\] we have plotted the quantity $\frac{B^{2}-E^{2}}{B^{2}}$, which when close to zero signals the location this current sheet. In such regions, the numerical mechanism that effectively dissipates electric field is actively operating to avoid violations of the magnetic domination condition. We see that for the present case, the current sheet extends behind the black hole’s motion, up to approximately $6M$. ![Surface where $\frac{B^{2} - E^{2}}{B^{2}}=0.1 $, for a late time solution of a black hole moving at speed $v=0.5$ along the $y$-direction. It signals the presence of a strong current sheet, where electric fields is effectively dissipated.[]{data-label="Current_sheet"}](CS_3d0000.png) Figure \[pfn\_v05\] displays the radial electromagnetic energy flux density on the $x=0$ plane, as measured in the two relevant reference frames of the problem, namely: the one where the observer is at rest with respect to the asymptotic magnetic field (plasma frame, top image) and the one in which the black hole is at rest (BH frame, bottom image). Both the flux in the plasma frame, i.e. $p^{\prime a}N^{\prime}_{a}$, and the flux measured in the BH frame, $p^{r}$, exhibit a pair of highly collimated jets emerging from the black hole. These jets form an angle with the $z$-axis given by, $\theta_{jet} = \tan^{-1} (\gamma v)$, in the co-moving frame; and equivalently, $\theta_{jet}^{\prime} = \tan^{-1} (v)$, in the plasma frame. Such misalignment between the collimated energy flux and the original magnetic field orientation is expected and has been reported previously in [@Luis2011]. At a first glance, we see the main difference between the two fluxes is at their non-collimated components, especially in front of the black hole where it gives negatives values in the BH frame. ![Electromagnetic flux density $p^{r}$ in the $x=0$ plane for the orthogonal boost velocity $v=0.5$, as measured in the black hole frame when the background initial field is not subtracted from the final stationary field configuration.[]{data-label="PFN_back"}](PF_v05_cold_n_hot0000.png) Figure \[PFN\_back\] presents again the radial electromagnetic energy flux density as measured in the BH’s frame, but now the initial background fields has not been subtracted from the final stationary configuration. The same pair of jets as in Fig. \[pfn\_v05\] can be seen, except that they are somewhat hidden now by a mainly dipolar flux density distribution arising from the background electromagnetic field being boosted against the black hole. It is worth emphasizing that integrating this flux around the BH horizon gives a rather small but negative value, thus showing that there is no energy extraction from the black hole. This is consistent with the fact that there is no ergoregion here and, hence, the Blandford-Znajek mechanism is not possible. The net positive flow of electromagnetic energy in the plasma frame, on the other hand, must arise from the available energy due to the relative motion between the magnetized plasma and the black hole. We turn next to a more quantitative analysis and consider, first, how does the emitted jet power changes with the distance from the black hole horizon. Thus, we measure the integrated flux in the plasma frame, $\Phi^{\prime}$, as a function of radius. The integration is performed on a surface determined by the normal vector $N^{\prime a}$ (that in this frame represents spheres), within a cylindrical region of $60M$ diameter enclosing the collimated jet. Following the notation of Refs. [@palenzuela2010dual; @Luis2011], we shall compute this quantity in physical units, respect to a representative system in which a black hole of mass $M=10^8 M_{\odot}$ is immersed on a magnetic field of strength $B_o = 10^{4} G$. That is, the results will be expressed proportional to $(M_{8} B_{4})^2 \equiv (\frac{M}{10^8 M_{\odot}})^2 (\frac{B}{10^4 G})^2$, allowing for an easy translation to any pair of physical values $M$ and $B$. Figure \[R\_dependence\] presents the behavior of $\Phi^{\prime}$ in the range $r=70-150 M$, for a black hole moving at speed $v=0.5$. It can be seen that the emitted power drops approximately $20\%$ between $r=70M$ and $r=150M$. A function of the form $\Phi^{\prime} \sim \Phi_{\infty}(1 + \sigma r^{-1})$, also shown in the plot, fits the numerical data very well. The asymptotic value for the collimated flux is $\Phi_{\infty} = 3.44 \times 10^{44} \textit{erg} / s$, and $\sigma = 28.1 M$. The expression $\Phi^{\prime}$ we propose to compute this flux is an approximation that relies on an asymptotic Killing vector field and is only a local integration. Hence, it is prone to errors and should be considered only as a guidance. Too close to the BH the uncertainties on the approximation to the asymptotic Killing vector field are important and, far away, there are dispersion effects. Thus, we fixed an intermediate radius $r=90M$ as the integration surface to measure the collimated energy flux $\Phi^{\prime}$. ![Dependence of the net collimated flux $\Phi^{\prime}$ with integration radius $r$, for the black hole moving at speed $v=0.5$.[]{data-label="R_dependence"}](comp_r.png) The results obtained for several boost velocities for both reference frames are summarized on Fig.\[pf\_v\]. For the plasma frame (top figure), as expected from previous numerical experiments in the regime $v \leq 0.2$ [@Luis2011], further supported on theoretical arguments [@morozova2014; @penna2015] later, the emitted power for non-relativistic speeds goes as $\propto v^{2}$ (see red dashed curve). However, we find that for larger boost velocities the correct dependence is instead given by $\propto \gamma v^{2}$ (solid black line), which fits the numerical values very well for the whole range of velocities explored. To the best of our knowledge, no one has pointed out this behavior before, which may have important observational consequences on astrophysical scenarios were such relativistic speeds are plausible. Meanwhile, from the BH’s frame, the behavior with the boost velocity is instead given by $\propto \gamma ^{4} v^{2}$. ### Misaligned case Now, we consider situations in which the exterior magnetic field and the black hole velocity are not orthogonal. In order to do that, we take the boost velocity to lay on the $y-z$ plane and parametrize different orientations by the angle $\chi$ it forms with the $y$-axis (see expression ). For this particular scenario, we have focused in the case of a black hole moving at speed $v=0.5$. A representative late time configuration of the magnetic field is shown in Fig. \[tilted\_streamlines\], corresponding to a black hole traveling with inclination angle $\chi=-\pi / 4$. Notice that the field topology is similar to the one of the orthogonal case (shown in Fig. \[fig:fieldlines\]), but now the asymptotic field is rotated an angle $ \Theta = \tan^{-1} (\frac{B^{\prime y}}{B^{\prime z}}) = \tan^{-1} (\frac{v^2 \cos \chi \sin\chi}{\sqrt{1-v^2}+1-v^2 \sin^2 \chi})$ within the $y-z$ plane, as seen from the black hole’s frame. Such rotation is induced by the Lorentz transformation and can be straightforwardly computed from equation . ![Representative streamlines of the magnetic field for the $\chi=-\pi / 4$ case for the late time numerical solution (t=400M) for a black hole with $v=0.5$.[]{data-label="tilted_streamlines"}](visit_tilt0000.png) Figure \[pfn\_over4\] shows the EM energy flux density $p^{\prime a}N^{\prime}_{a}$ in the $x=0$ plane, corresponding to a late time solution of a black hole moving with a direction determined by $\chi=\pi/4$. In contrast to Fig. \[pfn\_v05\], we see that –as expected– the solution has lost the reflection symmetry respect to the $z=0$ plane, exhibiting now a pair of asymmetric jets. To study the dependence of the jet power on inclination, we vary the angle $\chi$ from $0$ to $\pi/2$. It can be seen, in Fig. \[angles\_pfn\], that the power for each individual upper/lower jet highly depends on the inclination angle: they can be up to $\approx 17 \%$ higher than in the orthogonal case ($\chi=0$) and vanishing for $\chi = \pi/2$. Figure \[angles\_pfn\] also shows the net collimated power, i.e. the sum of the contributions from both jets, along with a fitting $\propto \cos^{2}(\chi)$, which fits very well with the numerical data. The same $\cos^{2}(\chi)$ behaviour was observed in [@FFE2] for the stationary Kerr BH case, with the exception that $\chi$ represents the inclination angle between the asymptotic magnetic field and the BH rotation axis. This behaviour is expected, since the electric field in the BH frame is $\propto \cos (\chi)$. By performing simulations for negative inclination angles we observed that, as expected, there is a symmetry between the lower and upper jets, in which the upper jet power for a given angle $\chi_o$ equals the power of the lower jet at the opposite angle, i.e. $-\chi_o$. The simulations performed for different angles $\chi$ also show that the direction of each jet is shifted respect to the $\chi=0$ case, this displacement is shown in Fig.\[displacement\_angle\], which also presents a fitting of the numerical data with functions $\propto \cos(\chi - \delta)$ (for the upper jet) and $\propto \cos(\chi + \delta)$ (for the lower one). For the boost velocity employed here (i.e. $v=0.5$), we find that $\delta = 0.17 \pm 0.01$. ![Electromagnetic flux density $p^{\prime a}N^{\prime}_{a}$ for the $x=0$ plane for late time numerical solution ($t = 400M)$ for a black hole with velocity $v=0.5$ and $\chi = \pi / 4$.[]{data-label="pfn_over4"}](piover4_fixed0000.png) ![Dependence of the EM flux $\Phi^{\prime}$, for a black hole with $v=0.5$, on the angle $\chi$. The squares (dots) correspond to the net flux integrated near the lower (upper) jet, while the green diamonds correspond to the sum of these quantities. The blue doted curve corresponds to a fitting $\propto \cos^{2}(\chi)$.[]{data-label="angles_pfn"}](angles.png) ![The angle $\Theta$ between the Upper (Lower) jet and the $z$ ($-z$) axis, is given as a function of the inclination angle $\chi$ for a black hole moving at speed $v=0.5$.[]{data-label="displacement_angle"}](disp_angles.png) ### Spinning black hole In this section, we present the results of the simulations performed on a Kerr background, focusing on the effects the black hole rotation has in the emitted power. In Ref. [@morozova2014], analytic vacuum Maxwell solutions were found for the field configurations in the vicinity of black hole which is both moving and spinning. The estimated luminosities from these solutions have shown that the effect of rotation would be subdominant respect to the one associated with the translation motion. This scenario has also been studied numerically in Ref. [@Luis2011], within the force-free approximation. Considering boost velocities up to $v=0.2$, the authors of [@Luis2011] have proposed a decomposition of the total luminosity as, $$\label{PF_boost_sep} \Phi = \Phi_{spin} + \Phi_{boost}v^2$$ where $\Phi_{spin}$ and $\Phi_{boost}$ represent the contributions of spin and linear motion, respectively. Thus, suggesting that the two mechanisms acts separately, with $\Phi_{spin}$ being independent of the velocity $v$ (only depending on $a$) and $\Phi_{boost}$ being a constant which does not depend on spin. ![Collimated EM flux $\Phi^{\prime}$ for black holes moving at low velocities, one with $a=0.0$ (black dots) and other with $a=0.6$ (red squares). The curves are fits of the form, $ \Phi_{spin} + \Phi_{boost}\gamma v^2$.[]{data-label="a06"}](a06.png) In Fig. \[a06\] we show the luminosities at two different spin values: $a=0$ (i.e. non spinning case) and $a=0.6$, for velocities up to $v=0.2$. The plot reproduces very well the results of Ref. [@Luis2011] (specifically, their figure 2), which illustrates –through the fitting curves– the above mentioned behavior of the two separate contributions. This serves two purposes: it confirm their results by an independent approach to the problem, and on the other hand, it further validates our numerical implementation. Now, we shall explore what happens if one goes to larger speeds. To that end, we present in Fig. \[a00\_vs\_a09\] the resulting luminosities at two spin values, $a=0$ and $a=0.9$, for velocities that ranges from $v=0.1$ to $v=0.7$. Surprisingly, we find that the curves that represent the non-spinning and the highly-spinning black holes tend to overlap for highly-relativistic boost velocities, $v \gtrsim 0.5$. ![Dependence on the BH’s velocity of the total $\Phi^{\prime}$ EM collimated flux for black holes moving with a boost velocity $v$, one with $a=0.0$(red squares) and other with $a=0.9$ (black dots). The two curves approach as the velocity increases.[]{data-label="a00_vs_a09"}](a09.png) It means the decomposition made above no longer holds for such speeds, where $\Phi_{spin} $ is seen to decrease (see Fig. \[a09\_minus\_a09\]). This can be explained, at least in part, by noticing that the power on the BZ mechanism diminishes with the inclination angle among the rotation axis and the asymptotic magnetic field. Thus, the angle $\theta_{jet}(v)$ produced by the motion would tend to reduce the spin contribution from the total luminosity. ![Difference of the two curve from Fig.\[a00\_vs\_a09\] e.i. $\Phi^{\prime}_{a=0.9}-\Phi^{\prime}_{a=0.0}$ and its dependence with the boost velocity $v$.[]{data-label="a09_minus_a09"}](dif_a09_a00.png) Indeed, we have confirmed numerically that by aligning the spin axis to the upper jet (since it can not be aligned simultaneously to both jets) one gets a larger spin contribution $\Phi_{spin} $ than compared to the case in which the rotation axis is perpendicular to the motion and, moreover, that this difference increments with boost velocity. Boosted perfectly conducting sphere ----------------------------------- We turn now to the idealized setup of a neutron star that is moving across a uniformly magnetized plasma in the orthogonal direction. The star has been modeled by a perfectly conducting spherical surface on a Schwarzschild spacetime, and it was further assumed to have no magnetic field on its own[^5]. The star is smoothly brought to relative motion by gradually boosting the initial electromagnetic configuration at the outer boundary. After an initial dynamical transient, the numerical solutions reach a steady state showing collimated electromagnetic jets. We will analyze these solutions and how their jet power vary with the boost velocity $v$ and stellar compactness $\mathcal{C} \equiv M/R$. Of particular interest is the flat spacetime limit, $M=0$, for that in such case the existence of the Killing vectors makes the notion of the boost and fluxes well defined everywhere, and thus allows for an interesting comparison with the previous black hole scenario. In a force-free environment, conductors are shown to act as sources by imposing boundary conditions on the surrounding fields [@gralla2016]. A similar setting to our star embedded in flat spacetime has been studied in the context of satellites (see e.g. [@1965drag]), where the problem is fairly well understood. The motion ($\hat{y}$) across a uniform magnetic field ($\hat{z}$) induces charge separation along the transverse ($\hat{x}$) direction, which is conducted away through the plasma in the form of Alfvén waves. An stationary electric circuit, as the one depicted on Fig. \[fig:circuit\], is then established. Such configurations gives rise to a significant damping on the motion of the object, as mechanical energy is converted to Alfvén radiation [@1965drag]. ![Late time numerical solution for a perfectly conducting sphere moving at speed $v=0.5$ along the $y$-axis. Magnetic fieldlines and flux density $p^{\prime a}N^{\prime}_{a}$ (in color scale) at the $x=0$ plane are represented in the top and bottom panels for stellar compactness of $\mathcal{C}=0$ and $\mathcal{C}=0.2$, respectively. The stellar surface is depicted by the gray disk in the center. []{data-label="fig:neutron_star"}](conducting_sphere_v05m00_todo0000.png "fig:") ![Late time numerical solution for a perfectly conducting sphere moving at speed $v=0.5$ along the $y$-axis. Magnetic fieldlines and flux density $p^{\prime a}N^{\prime}_{a}$ (in color scale) at the $x=0$ plane are represented in the top and bottom panels for stellar compactness of $\mathcal{C}=0$ and $\mathcal{C}=0.2$, respectively. The stellar surface is depicted by the gray disk in the center. []{data-label="fig:neutron_star"}](conducting_sphere_v05_todo0000.png "fig:")    In Fig.\[fig:neutron\_star\] we show the Poynting fluxes produced by the neutron star moving at speed $v=0.5$, for the case discussed so far in flat spacetime (top panel), and also when including curvature effects setting a stellar compactness $\mathcal{C}=0.2$ (bottom panel). The overall qualitative picture is similar among these two cases, and also when comparing with the black hole scenario: there are positive electromagnetic fluxes along the (same) jet directions, where plasma currents are sustained by two counter-oriented twisted bundles of magnetic field lines (see right panel of Fig. \[fig:circuit\]). Even though the distortions on the magnetic field by the strong curvature of the BH are similar to the ones produced by the NS in flat spacetime, the underlying mechanism operating is quite different. As one might expect, the currents at the black hole horizon does not look like those at the conducting surface of the star. Moreover, there is no current sheet forming behind the star, as the one shown in Fig. \[Current\_sheet\] for the BH, when there is no curvature (i.e. $M=0$). But when the mass of the neutron star is tuned-on (i.e. $M\neq0$), an analogous current sheet emerges and the emitted power gets enhanced. Thus, indicating that a composition of the two effects is acting; namely, the one associated with the perfect conductor condition and the one due to spacetime curvature. Quantitatively, the dependence of the collimated jet power on the boost speed is again $\propto \gamma v^2$ as in the black hole scenario (see Fig. \[conducting\_sphere\]). ![Dependence of the total collimated flux $\Phi^{\prime}$ on the speed $v$, for the late time solution of a NS of compactness $\mathcal{C}=0.2$. The red curve represents a fit of the form $\Phi$ $\propto$ $ \gamma v^{2}$.[]{data-label="conducting_sphere"}](NS_v.png) Even though there is no unambiguous way to compare luminosities among a black hole and a perfectly conducting sphere in flat spacetime, we choose to relate the stellar radius $R$ with the BH mass $M$ by setting $R=2M$ in geometric units. This way, we will be comparing the plasma frame luminosity produced by the black hole with the one of a NS whose surface is placed at the Schwarzschild radius of the BH, exploring different stellar compactness at this fixed radius (Fig. \[compacticity\]). We find that the emitted power is now larger by a factor between $1.4$ and $3$, depending on the compactness. A similar enhanced luminosity for the NS scenario was found for rotating compact objects [@gralla2016electromagnetic], with the explanation there being traced to distinct effective resistances of their electronic circuits. This argument, reminiscent from the membrane paradigm, indicates that the perfect conductor (placed at the BH horizon) in flat spacetime would produce larger values ($\sim 40$% in this case) simply because the black hole behaves effectively as a poorer conductor. We observe that increasing the stellar compactness then leads to an interesting interplay between the effects related with the conducting surface and those of gravity. The luminosity quickly rises (up to $\sim 3 L_{BH}$ by $\mathcal{C}=0.1$) and then smoothly begins to drop, presumably approaching the value $L_{BH}$ at compactness $\mathcal{C}\approx 0.5$. ![Luminosity produced by the NS, normalized by the one of the BH (i.e. $L_{NS}/L_{BH}$), as a function of the stellar compactness $\mathcal{C} \equiv M_{*}/R$ when moving at speed $v=0.3$ . []{data-label="compacticity"}](compactness.png) Decomposing the solutions in physical modes ------------------------------------------- Here we want to analyze the radial fluxes associated with the physical propagation modes, as defined by equations and . To that end, the fluxes are plotted in figure \[fig:modes\], for the three main cases under consideration. Namely, the black hole and the perfectly conducting sphere in flat/Schwazschild spacetime, moving at speed $v=0.5$ along the $y$-axis. Fast magnetosonic modes move at the speed of light without reference to the background magnetic field, thus here they seems to carry the contribution from the relative motion between the magnetized plasma and the object. On the other hand, Alfvén modes show funnels of positive radiation along the jets. But they also show similarly collimated regions of negative (i.e. incoming) flux at the opposite side of the $y=0$ plane. This is a bit puzzling, considering there is no electric charge density nor currents at those regions, as opposed to what happens with the positive components (outgoing) along the jets. It might be related, however, with the fact that we are computing these fluxes on the frame of the moving objects and not in the plasma frame; in that frame the net sum of the different contributions to the energy should be approximately zero when integrating on (e.g. spherical) surfaces enclosing the object.   \    Discussion ========== In this paper, we have studied jets arising from black holes and neutron stars moving across a magnetized force-free plasma environment. Even though, as one may expect, there is no conserved (Killing) energy being extracted from the moving objects –like e.g. emerging from the BH horizon–, we find however a positive (outgoing) net flux of approximate energy at the plasma frame. Such energy represents the one measured by an observer at rest with respect to the (asymptotic) uniform magnetic field. We see that part of the available energy from the relative motion between the magnetized plasma and the object gets transferred to the electromagnetic field, producing two counter-oriented twisted bundles of magnetic fieldlines which induce stationary currents and support the highly collimated energy fluxes found; namely, the jets. By working on the co-moving reference frame, we were able to explore a wide range of boost velocities, finding a dependence of the luminosity of the form, $L \propto \gamma \, v^2$, in both scenarios. Would there be an astrophysical situation where relative high velocities appear between compact objects and magnetic fields, the gamma factor would become preponderant. We have also explored in some detail the other relevant parameters of the problem, like the orientation of the motion respect to the asymptotic magnetic field or the inclusion of black hole spin. We looked not only to the total energy flux, but also the contributions from each of the different force-free modes, namely Alven and fast modes. This way we were able to better understand the character of the process. Comparing a black hole with a perfectly conducting sphere on flat spacetime, we have concluded that the overall effects are quite similar, although there are subtle but important differences among the two mechanisms. Clearly, the horizon does not behave as a perfect conductor[^6] and, moreover, there is a strong current sheet emerging and playing an important role in the black hole case. We find that the perfect conductor generates about $40$% larger luminosity than a black hole –when placed at the BH horizon–, in agreement with the familiar arguments from the membrane paradigm which states a BH would posses an effective finite conductivity. Furthermore, we saw that when the mass of the neutron star is turned-on, a nontrivial superposition of these two mechanisms operates; interestingly, producing larger luminosities at intermediate values of the stellar compactness. As mentioned before, we left the inclusion of the stellar magnetic field and rotation to a future work, where the interaction among this field and the external one can be explored in detail. This configurations could be used to mimic the presence of a NS companion orbiting in the context of a binary merger and to systematically study precursor electromagnetic signals along the lines of Refs. [@palenzuela2013electromagnetic; @ponce2014]. Acknowledgments =============== We acknowledge financial support from CONICET, SeCyT-UNC and MinCyT-Argentina. F.C. acknowledge support from the Spanish Ministry of Economy and Competitiveness grants AYA2016-80289-P and AYA2017-82089-ERC (AEI/FEDER, UE). This work used computational resources from Mendieta Cluster (Centro de Computación de Alto Desempeño, Universidad Nacional de Córdoba), Pirayu Cluster (supported by the Agencia Santafesina de Ciencia, Tecnología e Innovación, Gobierno de la Provincia de Santa Fe, Proyecto AC-00010-18) and Centro de Cómputos de Alto Rendimiento (CeCAR). Which are all part of the Sistema Nacional de Computación de Alto Desempeño – MinCyT-Argentina. [^1]: Similar hyperbolic formulations were presented in Refs. [@pfeiffer; @Pfeiffer2015]. [^2]: See figure 2 of [@FFE2] for an illustration of the numerical domain. [^3]: Sometimes referred as the Kerr-Schild Cartesian coordinates, or the Kerr-Schild frame (see e.g. [@Visser2007]). [^4]: We refer to [@FFE2; @NS] for a complete discussion and technical details. [^5]: Physically, it would correspond to the limit in which the exterior magnetic field totally overwhelms the internal field of the star. [^6]: Notice that the spacetime might act as an effective conductor with finite resistivity, as suggested by the membrane paradigm approach (see e.g. Ref. [@thorne], and also [@penna2015] in the present context).

--- abstract: 'A general method is developed for deriving Quantum First and Second Fundamental Theorems of Coinvariant Theory from classical analogs in Invariant Theory, in the case that the quantization parameter $q$ is transcendental over a base field. Several examples are given illustrating the utility of the method; these recover earlier results of various researchers including Domokos, Fioresi, Hacon, Rigal, Strickland, and the present authors.' author: - 'K R Goodearl and T H Lenagan[^1]' title: Quantized coinvariants at transcendental $q$ --- 2000 Mathematics Subject Classification: 16W35, 16W30, 20G42, 17B37, 81R50. Keywords: coinvariants, First Fundamental Theorem, Second Fundamental Theorem, quantum group, quantized coordinate ring. Introduction and background {#introduction-and-background .unnumbered} =========================== In the classic terminology of Hermann Weyl [@Weyl], a full solution to any invariant theory problem should incorporate a *First Fundamental Theorem*, giving a set of generators (finite, where possible) for the ring of invariants, and a *Second Fundamental Theorem*, giving generators for the ideal of relations among the generators of the ring of invariants. Many of the classical settings of Invariant Theory have quantized analogs, and one seeks corresponding analogs of the classical First and Second Fundamental Theorems. However, the setting must be dualized before potential quantized analogs can be framed, since there are no quantum analogs of the original objects, only quantum analogs of their coordinate rings. Hence, one rephrases the classical results in terms of rings of invariants (see below), and then seeks quantized versions of these. A first stumbling block is that in general these coactions are not algebra homomorphisms, and so at the outset it is not even obvious that the coinvariants form a subalgebra. However, this can often be established; cf [@GLRrims], Proposition 1.1; [@DLjpaa], Proposition 1.3. Typically, a classical invariant theoretical setting is quantized uniformly with respect to a parameter $q$, that is, there is a family of rings of coinvariants to be determined, parametrized by nonzero scalars in a base field, such that the case $q=1$ is the classical one. (Many authors, however, restrict attention to special values of $q$, such as those transcendental over the rational number field, and do not address the general case.) As in the classical setting, one usually has identified natural candidates to generate the ring of coinvariants. Effectively, then, one has a parametrized inclusion of algebras (candidate subalgebras inside the algebras of coinvariants), which is an equality at $q=1$, and one seeks equality for other values of $q$. In the best of all worlds, the equality at $q=1$ could be “lifted” by some general process to equality at arbitrary $q$. Lifting to transcendental $q$ has been done succesfully in some cases, by ad hoc methods – see, for example, [@FiHa], [@Str]. Also, an early version of [@GLRrims] for the transcendental case was obtained in this way. Quantum Second Fundamental Theorems can be approached in a similar manner. We develop here a general method for lifting equalities of inclusions from $q=1$ to transcendental $q$, which applies to many analyses of quantized coinvariants. In order to apply classical results as indicated above, we must be able to transform invariants to coinvariants. For morphic actions of algebraic groups, the setting of interest to us, invariants and coinvariants are related as follows. Suppose that $\gamma: G\times V\rightarrow V$ is a morphic action of an algebraic group $G$ on a variety $V$. This action induces an action of $G$ on $\O(V)$, where $(x.f)(v)= f(x^{-1}.v)$ for $x\in G$, $f\in \O(V)$, and $v\in V$. The invariants for this action are, of course, those functions in $\O(V)$ which are constant on $G$-orbits. The comorphism of $\gamma$ is an algebra homomorphism $\gamma^*: \O(V) \rightarrow \O(G)\otimes \O(V)$, with respect to which $\O(V)$ becomes a left $\O(G)$-comodule. Now a function $f\in \O(V)$ is a coinvariant in this comodule when $\gamma^*(f)=1\otimes f$. Since $1\otimes f$ corresponds to the function $(x,v) \mapsto f(v)$ on $G\times V$, we see that $\gamma^*(f)=1\otimes f$ if and only if $f(x.v)= f(v)$ for all $x\in G$ and $v\in V$, that is, if and only if $f$ is an invariant function. To summarize: $$\O(V)^{\co\O(G)}= \O(V)^G.$$ Quantized coordinate rings have been constructed for all complex semisimple algebraic groups $G$. These quantized coordinate rings are Hopf algebras, which we denote $\Oq(G)$, since we are concentrating on the single parameter versions. In those cases where a morphic action of $G$ on a variety $V$ has been quantized, we have a quantized coordinate ring $\Oq(V)$ which supports an $\Oq(G)$-coaction. The coaction is often not an algebra homomorphism, but nonetheless – as mentioned above – the set of $\Oq(G)$-coinvariants in $\Oq(V)$ is typically a subalgebra. The goal of Quantum First and Second Fundamental Theorems for this setting is to give generators and relations for the algebra $\Oq(V)^{\co\Oq(G)}$ of $\Oq(G)$-coinvariants in $\Oq(V)$. We discuss several standard settings in later sections of the paper, and outline how our general method applies. We recover known Quantum First and Second Fundamental Theorems at transcendental $q$ in these settings, with some simplifications to the original proofs, and in some cases extending the range of the theorems. Throughout, $k$ will denote a field, which may be of arbitrary characteristic and need not be algebraically closed. Reduction modulo $q-1$ ====================== Throughout this section, we work with a field extension $\kc \subset k$ and a scalar $q\in k\setminus \kc$ which is transcendental over $\kc$. Thus, the $\kc$-subalgebra $R= \kc[q,q^{-1}] \subset k$ is a Laurent polynomial ring. Let us denote reduction modulo $q-1$ by overbars, that is, given any $R$-module homomorphism $\phi:A\rightarrow B$, we write $\phibar: \Abar\rightarrow \Bbar$ for the induced map $A/(q-1)A \rightarrow B/(q-1)B$. Let $A \stackrel{\phi}\goesto B \stackrel{\psi}\goesto C$ be a complex of $R$-modules, such that $C$ is torsionfree. Suppose that there are $R$-module decompositions $$A= \bigoplus_{j\in J} A_j, \qquad\qquad B= \bigoplus_{j\in J} B_j, \qquad\qquad C= \bigoplus_{j\in J} C_j$$ such that $B_j$ is finitely generated, $\phi(A_j)\subseteq B_j$, and $\psi(B_j)\subseteq C_j$ for all $j\in J$. If the reduced complex $\Abar \stackrel{\phibar}\goesto \Bbar \stackrel{\psibar}\goesto \Cbar$ is exact, then so is $$\xymatrixcolsep{3pc} \xymatrix{ k\otimes_R A \ar[r]^{\id\otimes\phi} &k\otimes_R B \ar[r]^{\,\id\otimes\psi\,} &k\otimes_R C. }$$ The hypotheses and the conclusions all reduce to the direct sum components of the given decompositions, so it is enough to work in one component. Hence, there is no loss of generality in assuming that $B$ is finitely generated. Let $S$ denote the localization of $R$ at the maximal ideal $(q-1)R$. Set $$\Atil= S\otimes_R A, \qquad\qquad \Btil= S\otimes_R B, \qquad\qquad \Ctil= S\otimes_R C,$$ and let $\Atil \stackrel{\phitil}\goesto \Btil \stackrel{\psitil}\goesto \Ctil$ denote the induced complex of $S$-modules. Since $\Atil/(q-1)\Atil$ is naturally isomorphic to $A/(q-1)A= \Abar$, and similarly for $B$ and $C$, there is a commutative diagram $$\xymatrixcolsep{4pc} \xymatrix{ \fiddle{\Atil} \ar[r]^{\phitil} \ar[d]_{\alpha} &\fiddle{\Btil} \ar[r]^{\psitil} \ar[d]_{\beta} &\fiddle{\Ctil} \ar[d]^{\gamma}\\ \fiddle{\Abar} \ar[r] ^{\phibar} &\fiddle{\Bbar} \ar[r]^{\psibar} &\fiddle{\Cbar} }$$ where $\alpha$, $\beta$, $\gamma$ are epimorphisms with kernels $(q-1)\Atil$ etc. The bottom row of the diagram is exact by hypothesis, and we claim that the top row is exact. Consider an element $x\in \ker \psitil$. Chasing $x$ around the diagram, we see that $x-\phitil(y)= (q-1)z$ for some $y\in \Atil$ and $z\in \Btil$. Note that $(q-1)z \in \ker\psitil$. Since $\Ctil$ is torsionfree, it follows that $z\in \ker\psitil$. Thus $\ker\psitil \subseteq \phitil(\Atil) +(q-1)\ker\psitil$, whence $$(\ker\psitil)/\phitil(\Atil) = (q-1) \bigl[ (\ker\psitil)/\phitil(\Atil) \bigr].$$ Since $\ker\psitil$ is a finitely generated $S$-module, it follows from Nakayama’s Lemma that $(\ker\psitil)/\phitil(\Atil) =0$, establishing the claim. Since $k$ is a flat $S$-module, the sequence $\xymatrixcolsep{3pc} \xymatrix{ \fiddle{k\otimes_S \Atil} \ar[r]^{\id\otimes\phi} &\fiddle{k\otimes_S \Btil} \ar[r]^{\,\id\otimes\psi\,} &\fiddle{k\otimes_S \Ctil} }$ is exact. This is isomorphic to the sequence $$\xymatrixcolsep{3pc} \xymatrix{ k\otimes_R A \ar[r]^{\id\otimes\phi} &k\otimes_R B \ar[r]^{\,\id\otimes\psi\,} &k\otimes_R C }$$ and therefore the latter is exact. Proposition 1.1 is useful in obtaining quantized versions of both First and Second Fundamental Theorems. For ease of application, we write out general versions of both situations with appropriate notation, as follows. Let $H$ be an $R$-bialgebra and $B$ a left $H$-comodule with structure map $\lambda: B\rightarrow H\otimes_R B$, and let $\phi:A\rightarrow B$ be an $R$-module homomorphism whose image is contained in $B^{\co H}$. Assume that $B$ and $H$ are torsionfree $R$-modules, and suppose that there exist $R$-module decompositions $A= \bigoplus_{j\in J} A_j$ and $B= \bigoplus_{j\in J} B_j$ such that $B_j$ is finitely generated, $\phi(A_j)\subseteq B_j$, and $\lambda(B_j)\subseteq H\otimes_R B_j$ for all $j\in J$. Now $\Bbar$ is a left comodule over the $\kc$-bialgebra $\Hbar$, and $k\otimes_R B$ is a left comodule over the $k$-bialgebra $k\otimes_R H$. If $\Bbar^{\co \Hbar}$ equals the image of the induced map $\phibar : \Abar \rightarrow \Bbar$, then $[k\otimes_R B]^{\co k\otimes_R H}$ equals the image of the induced map $\id\otimes\phi: k\otimes_R A \rightarrow k\otimes_R B$. Since $R$ is a principal ideal domain, $H\otimes_R B$ is a torsionfree $R$-module. We may identify $\overline{H\otimes_R B}$ with $\Hbar\otimes_{\kc} \Bbar$ and $k\otimes_R (H\otimes_R B)$ with $(k\otimes_R H) \otimes_k (k\otimes_R B)$. Now apply Proposition 1.1 to the complex $A \stackrel{\,\phi\,}\goesto B \stackrel{\,\lambda -\gamma\,}\goesto H\otimes_R B$, where $\gamma:B \rightarrow H\otimes_R B$ is the map $b\mapsto 1\otimes b$. Let $\phi:A\rightarrow B$ be a homomorphism of $R$-algebras, and $I$ an ideal of $A$ contained in $\ker\phi$. Assume that $B$ is a torsionfree $R$-module, and suppose that there exist $R$-module decompositions $$I= \bigoplus_{j\in J} I_j, \qquad\qquad A= \bigoplus_{j\in J} A_j, \qquad\qquad B= \bigoplus_{j\in J} B_j$$ such that $A_j$ is finitely generated, $I_j\subseteq A_j$, and $\phi(A_j) \subseteq B_j$ for all $j\in J$. If $\ker \phibar$ equals the image of $\overline{I}$, then the kernel of the $k$-algebra homomorphism $\id\otimes\phi: k\otimes_R A \rightarrow k\otimes_R B$ equals the image of $k\otimes_R I$. Apply Proposition 1.1 to the complex $I\stackrel{\,\eta\,}\goesto A\stackrel{\,\phi\,}\goesto B$, where $\eta$ is the inclusion map. Proposition 1.1 can easily be adapted to yield other exactness conclusions. In particular, suppose that, in addition to the given hypotheses, the cokernels of $\phi$ and $\psi$ are torsionfree $R$-modules. Then one can show that for all nonzero scalars $\lambda\in \kc$, the induced complex $$A/(q-\lambda)A \longrightarrow B/(q-\lambda)B \longrightarrow C/(q-\lambda)C$$ is exact. Unfortunately, in the applications to quantized coinvariants, it appears to be a difficult task to verify the above additional hypotheses. The interior action of $GL_t$ on $M_{m,t}\times M_{t,n}$ {#interior} =============================================== Let $m,n,t$ be positive integers with $t\le \min\{m,n\}$. The group $GL_t(k)$ acts on the variety $V:= M_{m,t}(k)\times M_{n,t}(k)$ via the rule $$g\cdot(A,B)= (Ag^{-1},gB),$$ and consequently on the coordinate ring $\O(V) \cong \O(M_{m,t}(k)) \otimes \O(M_{t,n}(k))$. The invariant theory of this action is closely related to the matrix multiplication map $$\mu: V \longrightarrow M_{m,n}(k)$$ and its comorphism $\mu^*$. The classical Fundamental Theorems for this situation ([@DeCP], Theorems 3.1, 3.4) are 1. The ring of invariants $\O(V)^{GL_t(k)}$ equals the image of $\mu^*$. 2. The kernel of $\mu^*$ is the ideal of $\O(M_{m,n}(k))$ generated by all $(t+1)\times (t+1)$ minors. As discussed in the introduction, the comorphism of the action map $\gamma: GL_t(k) \times V \rightarrow V$ is an algebra homomorphism $\gamma^*: \O(V) \rightarrow \O(GL_t(k)) \otimes \O(V)$ which makes $\O(V)$ into a left comodule over $\O(GL_t(k))$, and the coinvariants for this coaction equal the invariants for the action above: $\O(V)^{GL_t(k)}= \O(V)^{\co \O(GL_t(k))}$. We may consider quantum versions of this situation, relative to a parameter $q\in k^\times$, using the standard quantized coordinate rings $\OqMmt$, $\OqMtn$, and $\OqGLt$. We write $X_{ij}$ for the standard generators in each of these algebras; see, e.g., [@GLduke], (1.1), for the standard relations. To quantize the classical coaction above, one combines quantized versions of the actions of $GL_t(k)$ on $M_{m,t}(k)$ and $M_{n,t}(k)$ by right and left multiplication, respectively. The latter quantizations are algebra morphisms $$\begin{aligned} \rho_q^* : \OqMmt &\longrightarrow \OqMmt \otimes \OqGLt \\ \lambda_q^*: \OqMtn &\longrightarrow \OqGLt\otimes \OqMtn \end{aligned}$$ which are both determined by $X_{ij} \mapsto \sum_{l=1}^t X_{il}\otimes X_{lj}$ for all $i,j$. The right comodule structure on $\OqMmt$ determines a left comodule structure in the standard way, via the map $\tau\circ (\id\otimes S)\circ \rho_q^*$, where $\tau$ is the flip and $S$ is the antipode in $\OqGLt$. We set $\Oq(V)= \OqMmt\otimes \OqMtn$, which is a left comodule over $\OqGLt$ in the standard way. The resulting structure map $\gamma_q^*: \Oq(V)\rightarrow \OqGLt \otimes\Oq(V)$ is determined by the rule $$\gamma_q^*(a\otimes b)= \sum_{(a),(b)} S(a_1)b_{-1} \otimes a_0\otimes b_0$$ for $a\in \OqMmt$ and $b\in \OqMtn$, where $\rho_q^*(a)= \sum_{(a)} a_0\otimes a_1$ and $\lambda_q^*(b)= \sum_{(b)} b_{-1}\otimes b_0$. It is, unfortunately, more involved to work with $\gamma_q^*$ than with its classical predecessor $\gamma^*$ because $\gamma_q^*$ is not an algebra homomorphism. The analog of $\mu^*$ is the algebra homomorphism $\mu_q^*: \OqMmn \rightarrow \Oq(V)$ such that $X_{ij} \mapsto \sum_{l=1}^t X_{il}\otimes X_{lj}$ for all $i,j$. We can now state the Fundamental Theorems for the quantized coinvariants in the current situation: 1. The set of coinvariants $\Oq(V)^{\co\OqGLt}$ equals the image of $\mu_q^*$ (and so is a subalgebra of $\Oq(V)$). 2. The kernel of $\mu_q^*$ is the ideal of $\OqMmn$ generated by all $(t+1)\times (t+1)$ quantum minors. These theorems were proved for arbitrary $q$ in [@GLRrims], Theorem 4.5, and [@GLduke], Proposition 2.4, respectively. To illustrate the use of Theorems 1.2 and 1.3, we specialize to the case that $q$ is transcendental over a subfield $\kc\subset k$. Set $R= \kc[q,q^{-1}]$ as in the previous section. Since the construction of quantum matrix algebras requires only a commutative base ring and an invertible element in that ring, we can form $\OqMmtR$ and $\OqMtnR$. These $R$-algebras are iterated skew polynomial extensions of $R$, and thus are free $R$-modules, as is the algebra $\Oq(V_R) := \OqMmtR \otimes_R \OqMtnR$. Similarly, $\OqMtR$ is a free $R$-module and an integral domain, and so the localization $\OqGLtR$, obtained by inverting the quantum determinant, is a torsionfree $R$-module, as well as a Hopf $R$-algebra. Since the restrictions of $\rho_q^*$ and $\lambda_q^*$ make $\OqMmtR$ and $\OqMtnR$ into right and left comodules over $\OqGLtR$, respectively, the restriction $\gamma_R^*$ of $\gamma_q^*$ makes $\Oq(V_R)$ into a left $\OqGLtR$-comodule. Finally, $\mu_q^*$ restricts to an $R$-algebra homomorphism $\mu_R^*: \OqMmnR \rightarrow \Oq(V_R)$. It is easily checked that the image of $\mu_R^*$ is contained in $\Oq(V_R)^{\co \OqGLtR}$, and that the kernel of $\mu_R^*$ contains the ideal $I$ generated by all $(t+1)\times(t+1)$ quantum minors (cf [@GLRrims], Proposition 2.3 and [@GLduke], (2.1)). All of the quantum matrix algebras $\Oq(M_{\bullet,\bullet}(R))$ are positively graded $R$-algebras, with the generators $X_{ij}$ having degree $1$, and $\Oq(V_R)$ inherits a positive grading from its two factors. In each of these algebras, the homogeneous components are finitely generated free $R$-modules. It is easily checked that $$\mu_R^* \bigl( \OqMmnR_j \bigr) \subseteq \Oq(V_R)_{2j} \qquad\text{and}\qquad \gamma_R^* \bigl( \Oq(V_R)_j \bigr) \subseteq \OqGLtR \otimes_R \Oq(V_R)_j$$ for all $j\ge 0$. Moreover, the ideal $I$ of $\OqMmnR$ is homogeneous with respect to this grading. Note that when we come to apply Theorem 1.3, we should replace the grading on $\Oq(V_R)$ by the vector space decomposition $\bigoplus_{j=0}^\infty \bigl( \Oq(V_R)_{2j} \oplus \Oq(V_R)_{2j+1} \bigr)$, for instance. The classical First and Second Fundamental Theorems (1) and (2) say that $\overline{\mu_R^*}$ maps $\overline{\OqMmnR}$ onto the coinvariants of $\overline{\Oq(V_R)}$, and that the kernel of $\overline{\mu_R^*}$ equals the image of $\overline{I}$. Therefore Theorems 1.2 and 1.3 yield the quantized Fundamental Theorems $(1_q)$ and $(2_q)$ with no further work in the transcendental case. The right action of $SL_r$ on $M_{n,r}$ ======================================= Fix positive integers $r<n$. In this section, we consider the right action of $SL_r(k)$ on $M_{n,r}(k)$ by multiplication: $A.g= Ag$ for $A\in M_{n,r}(k)$ and $g\in SL_r(k)$. The First Fundamental Theorem for this case (cf [@Ful], Proposition 2, p 138) says that 1. The ring of invariants $\O(M_{n,r}(k))^{SL_r(k)}$ equals the subalgebra of $\O(M_{n,r}(k))$ generated by all $r\times r$ minors. To state the Second Fundamental Theorem for this case, let $k[X]$ be the polynomial ring in a set of variables $X_I$, where $I$ runs over all $r$-element subsets of $\{1,\dots,n\}$. In view of the theorem above, there is a natural homomorphism of $\phi: k[X]\rightarrow \O(M_{n,r}(k))^{SL_r(k)}$. 2. The kernel of $\phi$ is generated by the Plücker relations ([@Ful], Proposition 2, p 138). To quantize this situation, we make $\OqMnr$ into a right comodule over $\OqSLr$ via the algebra homomorphism $\rho: \OqMnr \rightarrow \OqMnr\otimes \OqSLr$ such that $\rho(X_{ij})= \sum_{l=1}^r X_{il}\otimes X_{lr}$ for all $i,j$. The First Fundamental Theorem for the quantized coinvariants is the statement 1. The set of coinvariants $\OqMnr^{\co\OqSLr}$ equals the subalgebra of $\OqMnr$ generated by all $r\times r$ quantum minors. This follows from work of Fioresi and Hacon under the assumptions that $k$ is algebraically closed of characteristic $0$ and $q$ is transcendental over some subfield of $k$ ([@FiHa], Theorem 3.12) – in fact, they prove (in our notation) that $(1_q)$ holds with $k$ replaced by a Laurent polynomial ring $\kc[q,q^{-1}]$. The result is obtained for any nonzero $q$ in an arbitrary field $k$ in [@KLR], Theorem 5.2. The Fioresi-Hacon version of the Second Fundamental Theorem yields a presentation of $\OqMnr^{\co\OqSLr}$ in terms of generators $\lambda_{i_1\cdots i_r}$ where $i_1\cdots i_r$ runs over all unordered sequences of $r$ distinct integers from $\{1,\dots,n\}$. Each $\lambda_{i_1\cdots i_r}$ is mapped to $(-q)^{\ell(\pi)}D_{\{i_1,\dots,i_r\}}$ where $\ell(\pi)$ denotes the length of the permutation $\pi$ sending $l\mapsto i_l$ for $l=1,\dots,r$, and $D_{\{i_1,\dots,i_r\}}$ denotes the $r\times r$ quantum minor with row index set $\{i_1,\dots,i_r\}$. 1. The algebra $\OqMnr^{\co\OqSLr}$ is isomorphic to the quotient of the free algebra $k\langle \lambda_{i_1\cdots i_r}\rangle$ modulo the ideal generated by the relations (c) and (y) given in [@FiHa], Theorem 3.14. The hypotheses on $k$ and $q$ are as above. Now let $\kc$, $k$, $q$, $R$ be as in Section \[interior\], and define everything over $R$. In particular, $\OqMnrR$ is a right comodule algebra over $\OqSLrR$. As already mentioned, Fioresi and Hacon have proved the $R$-algebra versions of $(1_q)$ and $(2_q)$, assuming that $k$ is algebraically closed of characteristic zero. The easier parts of their work – proving inclusions rather than equalities in these $R$-algebra forms – lead directly to the $k$-algebra forms of these theorems, as follows. Let $R\langle \lambda_\bullet\rangle$ denote the free $R$-algebra on generators $\lambda_{i_1\cdots i_r}$, and let $\phi_q: R\langle \lambda_\bullet \rangle \rightarrow \OqMnrR$ be the $R$-algebra homomorphism sending each $\lambda_{i_1\cdots i_r}$ to $(-q)^{\ell(\pi)}D_{\{i_1,\dots,i_r\}}$. The image of $\phi_q$ is the $R$-subalgebra generated by the $D_{\{i_1,\dots,i_r\}}$, which is contained in $\OqMnrR^{\co \OqSLrR}$ by [@FiHa], Lemma 3.5. Let $I_q$ be the ideal of $R\langle \lambda_\bullet \rangle$ generated by the relations (c) and (y) of [@FiHa], Theorem 3.14; that $I_q\subseteq \ker \phi_q$ was checked by Fioresi ([@Fio], Proposition 2.21 and Theorem 3.6). Now $\OqMnrR$ and $R\langle \lambda_\bullet \rangle$ are positively graded $R$-algebras, with the $X_{ij}$ having degree $1$ and the $\lambda_\bullet$ having degree $r$. The homogeneous components with respect to these gradings are finitely generated free $R$-modules, $I_q$ is a homogeneous ideal of $R\langle \lambda_\bullet \rangle$, and the map $\phi_q$ is homogeneous of degree $0$. The classical Fundamental Theorems (1) and (2) say that $\overline{\phi_q}$ maps $\overline{R\langle \lambda_\bullet \rangle}$ onto the coinvariants of $\overline{\OqMnrR}$, and the kernel of $\overline{\phi_q}$ equals $\overline{I_q}$. (The latter statement requires a change of relations using the results of [@Ful], Chapter 8, as observed in [@FiHa], p 435.) Therefore Theorems 1.2 and 1.3 yield the quantized Fundamental Theorems $(1_q)$ and $(2_q)$. Note that the hypotheses of algebraic closure and characteristic zero on $k$ are not needed, although $q$ is still assumed to be transcendental over a subfield of $k$. The right action of $Sp_{2n}$ on $M_{m,2n}$ =========================================== Fix positive integers $m$ and $n$, and consider the right action of $Sp_{2n}(k)$ on $M_{m,2n}(k)$ by multiplication. Here we take $Sp_{2n}(k)$ to preserve the standard alternating bilinear form on $k^{2n}$, which we denote $\langle-,-\rangle$, and we view $M_{m,2n}(k)$ as the variety of $m$ vectors of length $2n$. Thus, each row $x_i = (X_{i1},\dots,X_{i,2n})$ of generators in $\O(M_{m,2n}(k))$ corresponds to the $i$-th of $m$ generic $2n$-vectors. We shall need the functions describing the values of $\langle-,-\rangle$ on two of these generic vectors: $$z_{ij} := \langle x_i,x_j\rangle= \sum_{l=1}^n (X_{i,2l-1}X_{j,2l}- X_{i,2l}X_{j,2l-1})$$ for $i,j=1,\dots,m$. Next, consider the variety $\Alt_m(k)$ of alternating $m\times m$ matrices over $k$; its coordinate ring is the algebra $$\O(\Alt_m(k))= \O(M_m(k))/ \langle X_{ii},\, X_{ij}+ X_{ji} \mid i,j= 1,\dots,m \rangle.$$ Let us write $Y_{ij}$ for the coset of $X_{ij}$ in $\O(\Alt_m(k))$. There is a morphism $\nu: M_{m,2n}(k) \rightarrow \Alt_m(k)$ given by the rule $\nu(A)= ABA^{\tr}$, where $B$ is the matrix of the symplectic form $\langle-,-\rangle$, and the comorphism $\nu^*: \O(\Alt_m(k)) \rightarrow \O(M_m(k))$ sends $Y_{ij} \mapsto z_{ij}$ for all $i,j$. For any even number $h$ of distinct indices $i_1,\dots,i_h$ from $\{1,\dots,m\}$, let $[i_1,\dots,i_h] \in \O(\Alt_m(k))$ be the Pfaffian of the submatrix of $(Y_{ij})$ obtained by taking the rows and columns with indices $i_1,\dots,i_h$. The First and Second Fundamental Theorems for the present situation ([@DeCP], Theorems 6.6, 6.7) state that 1. The ring of invariants $\O(M_{m,2n}(k))^{Sp_{2n}(k)}$ equals the subalgebra of $\O(M_{m,2n}(k))$ generated by the $z_{ij}$ for $i<j$, that is, the image of $\nu^*$. 2. If $m\le 2n+1$, the kernel of $\nu^*$ is zero, while if $m\ge 2n+2$, the kernel of $\nu^*$ is the ideal of $\O(\Alt_m(k))$ generated by all the Pfaffians $[i_1,\dots,i_{2n+2}]$. A quantized version of this situation was studied by Strickland [@Str], who quantized the $U(\spn(k))$-module action on $\O(M_{2n,m}(k))$ rather than its dual, the $\O(Sp_{2n}(k))$ coaction. We transpose the matrices and indices from her paper in order to match the notation for the classical situation used above. Since the approaches to quantized enveloping algebras for $\spn(k)$ and quantized coordinate rings for $Sp_n(k)$ become more complicated at roots of unity, let us restrict our discussion to a quantum parameter $q\in k^\times$ which is not a root of unity. Recall that the quantized coordinate rings of the different classical groups coact on different quantized coordinate rings of affine spaces. In particular, $\OqSpn$ coacts on an algebra that we denote $\Oqspn$, the [*quantized coordinate ring of symplectic $2n$-space*]{} ([@RTF], Definition 14; see [@Mus], §1.1, for a simpler set of relations). Thus, $\O(M_{m,2n}(k))$ needs to be quantized by a suitable algebra with $m\cdot2n$ generators, each row of which generates a copy of $\Oqspn$. Let us call this algebra $\Oqspnm$ and take it to be the $k$-algebra with generators $X_{ij}= x_{j,i}$ (for $i=1,\dots,m$ and $j=1,\dots,2n$) satisfying the relations given by Strickland in [@Str], Equations (2.1), (2.2), (2.3). (The $x_{j,i}$ are Strickland’s generators for the algebra she denotes $B$; we transpose the indices for the reasons indicated above.) Similarly, let us write $\OqAltm$ for the $k$-algebra with generators $Y_{ij}= a_{j,i}$ for $1\le j<i\le m$ satisfying the relations given in [@Str], Equations (1.1). As in [@Str], Theorem 2.5(2), there is a $k$-algebra homomorphism $\phi: \OqAltm \rightarrow \Oqspnm$ such that $$\phi(Y_{ts})= \sum_{l=1}^n q^{l-1-n}X_{sl}X_{t,2n-l+1}- \sum_{l=1}^n q^{n-l+1} X_{s,2n-l+1}X_{tl}$$ for $s<t$. Set $B_q= \Oqspnm$ for the time being. Strickland defines an action of $\Uqspn$ on $B_q$ ([@Str], p 87). It is clear from the definition of this action that $B_q$ is a locally finite dimensional left $\Uqspn$-module. Hence, $B_q$ becomes a right comodule over the Hopf dual $\Uqspn^\circ$ in the standard way ([@Mon], Lemma 1.6.4(2)), and the $\Uqspn$-invariants coincide with the $\Uqspn^\circ$-coinvariants ([@Mon], Lemma 1.7.2(2)). The standard quantized coordinate ring of $Sp_{2n}(k)$, which we denote $\OqSpn$, is a sub-Hopf-algebra of $\Uqspn^\circ$, and one can check that the coaction $B_q\rightarrow B_q\otimes \Uqspn^\circ$ actually maps $B_q$ into $B_q\otimes \OqSpn$. Consequently, $B_q$ is a right comodule over $\OqSpn$, and its $\OqSpn$-coinvariants coincide with its $\Uqspn^\circ$-coinvariants. Statements of First and Second Fundamental Theorems for quantized coinvariants in the symplectic situation above can be given as follows, where the [*$q$-Pfaffians*]{} $[i_1,\dots,i_h]$ are defined as in [@Str], p 82: 1. The set of coinvariants $\Oqspnm^{\co\OqSpn}$ equals the image of $\phi$. 2. If $m\le 2n+1$, the kernel of $\phi$ is zero, while if $m\ge 2n+2$, the kernel of $\phi$ is the ideal of $\OqAltm$ generated by all the $q$-Pfaffians $[i_1,\dots,i_{2n+2}]$. Both statements have been proved by Strickland (modulo the changes of notation discussed above) under the assumptions that ${\rm char}(k)=0$ and $q$ is transcendental over a subfield of $k$ ([@Str], Theorem 2.5). Via Theorems 1.2 and 1.3, we obtain the transcendental cases of $(1_q)$ and $(2_q)$ from the classical results (1) and (2) in arbitrary characteristic. (We note that part of Strickland’s development also involves reduction modulo $q-1$. See the proof of [@Str], Theorem 1.5.) As far as we are aware, it is an open question whether $(1_q)$ and $(2_q)$ hold when $q$ is algebraic over the prime subfield of $k$. The conjugation action of $GL_n$ on $M_n$ ========================================= Fix a positive integer $n$, and assume that $k$ has characteristic zero. In this section, we consider the quantum analogue of the classical conjugation action of $GL_n(k)$ on $M_n(k)$. We shall need the trace functions $\tr_i$ for $i= 1, \dots, n$, where $\tr_i$ is the sum of the $i\times i$ principal minors. Note that $\tr_1$ is the usual trace function, and that $\tr_n$ is the determinant function. The First and Second Fundamental Theorems for this situation ([@Kra], Satz 3.1) can be stated as follows: 1. The ring of invariants $\O(M_n(k))^{GL_n(k)}$ equals the subalgebra of $\O(M_n(k))$ generated by $\tr_1,\dots,\tr_n$. 2. There are no relations among the $\tr_i$, that is, $\tr_1,\dots,\tr_n$ are algebraically independent over $k$. The coinvariants of a quantum analogue of the conjugation action have been studied in [@DL]. The [*right conjugation coaction*]{} of $\OqGLn$ on $\OqMn$ is the right coaction $\beta:\OqMn\goesto \OqMn \otimes \OqGLn$ given by $\beta(u):= \sum_{(u)} u_2\otimes S(u_1)u_3$, where we are using the Sweedler notation. In particular, $\beta(X_{ij}) = \sum_{l,m} X_{lm}\otimes S(X_{il})X_{mj}$ for all $i,j$. However, as with the interior coaction studied in Section \[interior\], the map $\beta$ is not an algebra homomorphism. Recall that if $I$ and $J$ are subsets of $\{1, \dots, n\}$ of the same size, then the quantum determinant of the quantum matrix subalgebra generated by the $X_{ij}$ with $i\in I$ and $j\in J$ is denoted by $[I|J]$ and called a [*quantum minor*]{} of the relevant quantum matrix algebra. For each $i = 1, \dots, n$, define the weighted sums of principal quantum minors by $\tau_i:= \sum_{I} q^{-2w(I)}[I|I]$, where $I$ runs through all $i$ element subsets of $\{1, \dots, n\}$ and $w(I)$ is the sum of the entries in the index set $I$. These weighted sums of principal quantum minors provide quantized coinvariants for $\beta$ by [@DL], Proposition 7.2. One can state First and Second Fundamental Theorems for quantized coinvariants in this situation as follows: 1. The set of coinvariants $\OqMn^{\co\OqGLn}$ is equal to the subalgebra of $\OqMn$ generated by $\tau_1, \dots, \tau_n$. 2. The subalgebra $k[\tau_1, \dots, \tau_n]$ is a commutative polynomial algebra of degree $n$. These statements were proved for $k= \mc$ and $q$ not a root of unity in [@DL], Theorem 7.3, by using the corepresentation theory of the cosemisimple Hopf algebra $\OqGLnC$. We give a partial extension below. Now let $\kc = \mq$. Suppose that $q\in k^\times$ is transcendental over $\mq$, set $R= \mq[q,q^{-1}]$ and define everything over $R$. Then $\OqMnR$ is a right comodule over $\OqGLnR$ by using $\beta$ as above. A straightforward calculation shows that the $R$-subalgebra generated by $\tau_1, \dots, \tau_n$ is contained in $\OqMnR^{\co \OqGLnR}$; see, for example, [@DL], Proposition 7.2. Also, note that when $q=1$ the coinvariants $\tau_1, \dots, \tau_n$ coincide with the classical traces $\tr_1, \dots, \tr_n$. Let $F(R)$ denote the free $R$-algebra on generators $\gamma_1, \dots, \gamma_n$, and consider $F(R)$ to be graded by setting $\deg(\gamma_i) = i$. Let $\phi_q$ be the $R$-algebra homomorphism sending $\gamma_i$ to $\tau_i$. Then $\phi_q$ is homogeneous of degree $0$ and the image of $\phi_q$ is contained in $\OqMnR^{\co \OqGLnR}$. The homogeneous components of $\OqMnR$ and $F(R)$ are finitely generated free $R$-modules with $\phi_q(F(R)_j) \subseteq \OqMnR_j$ and $\beta(\OqMnR_j) \subseteq \OqMnR_j\otimes\OqGLnR$. The classical First Fundamental Theorem (1) says that $\overline{\phi_q}$ maps $\overline{F(R)}$ onto the coinvariants of $\overline{\OqMnR}$. Theorem 1.2 now yields the quantized First Fundamental Theorem $(1_q)$ for the case that $\rm{char}(k)=0$ and $q$ is transcendental over $\mq$. In [@CW], Corollary 2.3, Cohen and Westreich show that the $\tau_i$ commute, by exploiting the coquasitriangular structure of $\OqGLn$ (see, e.g., [@Hay], Theorem 3.1 and Proposition 4.1; a more detailed proof for the case $k=\mc$ is given in [@KlSc], Theorem 10.9). Thus, if we set $I$ to be the ideal of $F(R)$ generated by the commutators $\gamma_i\gamma_j - \gamma_j\gamma_i$, we see that $I\subseteq \ker(\phi_q)$. It is obvious that $I$ is a homogeneous ideal. The classical theory shows that $\ker(\overline{\phi_q}) = \overline{I}$. Thus, Theorem 1.3 yields the Second Fundamental Theorem $(2_q)$ in the case under discussion. [15]{} M Cohen and S Westreich, *Some interrelations between Hopf algebras and their duals*, preprint Ben Gurion University (2002). C De Concini and C Procesi, *A characteristic free approach to invariant theory*, Advances in Math. [**21**]{} (1976) 330-354. M Domokos and T H Lenagan, *Conjugation coinvariants of quantum matrices*, Bull. London Math. Soc. [**35**]{} (2003) 117-127. M Domokos and T H Lenagan, *Weakly multiplicative coactions of quantized function algebras*, to appear in J Pure and Applied Algebra. R Fioresi, *Quantum deformation of the Grassmannian manifold*, J. Algebra [**214**]{} (1999) 418-447. R Fioresi and C Hacon, *Quantum coinvariant theory for the quantum special linear group and quantum Schubert varieties*, J. Algebra [**242**]{} (2001) 433-446. W Fulton, *Young Tableaux*, London Math. Soc. Student Texts 35, Cambridge (1997) Cambridge Univ. Press. K R Goodearl and T H Lenagan, *Quantum determinantal ideals*, Duke Math J 103 (2000) 165-190. K R Goodearl, T H Lenagan and L Rigal, *The first fundamental theorem of coinvariant theory for the quantum general linear group*, Publ. RIMS (Kyoto) [**36**]{} (2000) 269-296. T Hayashi, *Quantum groups and quantum determinants*, J. Algebra [**152**]{} (1992) 146-165. A Kelly, L Rigal and T H Lenagan, *Ring theoretic properties of quantum grassmannians*, posted at http://arXiv.org/math.QA/0208152. A Klimyk and K Schmüdgen, *Quantum Groups and Their Representations*, Berlin (1997) Springer-Verlag. H Kraft, *Klassische Invariantentheorie: Eine Einführung*, in Algebraische Transformationsgruppen und Invariantentheorie (H Kraft, P Slodowy, and T A Springer, Eds), Basel (1989) Birkhäuser, pp 41-62. S Montgomery, *Hopf Algebras and Their Actions on Rings*, CBMS Regional Conf. Series in Math. 82, Providence (1993) Amer. Math. Soc. I M Musson, *Ring-theoretic properties of the coordinate rings of quantum symplectic and Euclidean space*, in Ring Theory, Proc. Biennial Ohio State–Denison Conf., 1992 (S K Jain and S T Rizvi, Eds), River Edge, NJ (1993) World Scientific, pp 248-258. N Yu Reshetikhin, L A Takhtadzhyan, and L D Faddeev, *Quantization of Lie groups and Lie algebras*, Leningrad Math. J. [**1**]{} (1990) 193-225. E Strickland, *Classical invariant theory for the quantum symplectic group*, Advances in Math. [**123**]{} (1996) 78-90. H Weyl, *The Classical Groups*, Princeton (1939) Princeton Univ. Press. K R Goodearl,\ Department of Mathematics,\ University of California, Santa Barbara,\ CA 93106,\ USA\ email: goodearl@math.ucsb.edu\ \ T H Lenagan,\ School of Mathematics,\ James Clerk Maxwell Building,\ Kings Buildings,\ Mayfield Road,\ Edinburgh EH9 3JZ,\ Scotland\ email: tom@maths.ed.ac.uk [^1]: This research was partially supported by NSF research grant DMS-9622876 and by NATO Collaborative Research Grant CRG.960250.

--- abstract: 'We study the mechanism of the ‘pearling’ instability seen recently in experiments on lipid tubules under a local applied laser intensity. We argue that the correct boundary conditions are fixed chemical potentials, or surface tensions $\Sigma$, at the laser spot and the reservoir in contact with the tubule. We support this with a microscopic picture which includes the intensity profile of the laser beam, and show how this leads to a steady-state flow of lipid along the surface and gradients in the local lipid concentration and surface tension (or chemical potential). This leads to a natural explanation for front propagation and makes several predictions based on the tubule length. While most of the qualitative conclusions of previous studies remain the same, the ‘ramped’ control parameter (surface tension) implies several new qualitative results. We also explore some of the consequences of front propagation into a noisy (due to pre-existing thermal fluctuations) unstable medium.' author: - | Peter D. Olmsted[^1] and F. C. MacKintosh\ [Department of Physics, University of Michigan, Ann Arbor MI 48109-2210]{} title: 'Instability and front propagation in laser-tweezed lipid bilayer tubules' --- =10000 5truemm PACS: 47.20.Dr, 47.20.-k, 47.20.Hw, 82.65.Dp 5truemm Short Title:   [**Dynamic instability in bilayer tubules.**]{} [2]{} Introduction ============ A recent series of exciting experiments [@barziv94] demonstrated a dynamic instability induced on tubules of single lipid bilayers by application of laser ‘tweezers’, whereby the cylindrical tubule of radius $R_0$ modulates with a wavenumber given by $q^{\ast}R_0\simeq 0.8$. This instability has been attributed to an excess surface tension due to the gain in electrostatic energy when surfactant molecules, of higher dielectric constant than water, displace water in the electric field of the laser. The starting point for understanding this phenomenon is the Rayleigh instability [@rayleigh1; @rayleigh2; @tomotika] of a thin cylindrical thread of liquid with positive surface tension, whereby the thread can reduce its surface area at fixed volume by modulating and evolving towards a string of beads. Rayleigh calculated the preferred wavelength of a cylinder of fluid in air in the inviscid [@rayleigh1] and non-inertial (viscous) [@rayleigh2] limits, finding in the former case a characteristic non-zero wavenumber and in the latter case a preferred wavenumber of zero (or infinite wavelength). Later, Tomotika [@tomotika] calculated the instability for a viscous fluid surrounded by another viscous fluid, again in the non-inertial regime, finding that the change in boundary conditions restores a finite characteristic wavelength. See Olami and Granek [@granek95] for a discussion of this point. The present problem, however, requires a much different detailed dynamical analysis which relates the flow of lipid molecules in the interface to the bulk flow in the surrounding fluid. An important physical ingredient is a new conserved quantity, the lipid on the surface. At present there are (at least) two theoretical treatments of the experiments of Ref.[@barziv94]. Bar-Ziv and Moses [@barziv94] and Nelson and co-workers [@nelsontube95a] have proposed the picture that the surface tension rapidly equilibrates everywhere to an induced value $\Sigma_0$, and the instability proceeds from this state. In contrast, Granek and Olami [@granek95] have postulated that the correct treatment of the problem is to impose a constant rate at which lipid molecules are drawn into the trap from the tubule. This loss of lipid is accommodated by stretching out small wavelength surface fluctuations and the result is again a uniform surface tension $\Sigma_0$. Goldstein, [*et al.*]{} (GNPS) [@nelsontube95b] demonstrated quantitatively how the equilibration of the tension in the tube stays ‘ahead’ of a shape change, so that a treatment with a constant (in time) surface tension is reasonable; and argued that the primary loss of area is in the shape instability itself, rather than through the removal of small-scale wrinkles. We propose a slightly different picture of the steady state before the onset of the instability, which follows from consideration of the experimental configuration. The tubules, as formed, are several hundred microns long and are attached at either end to ‘massive lipid globules’ [@barziv94] of order $10\mu\hbox{m}$ in diameter. Hence, the tubules must be in contact with a reservoir which fixes the lipid chemical potential (or, equivalently, the surface tension). If we assume the system is equilibrated, it follows that the chemical potential for exchange between the tubule, reservoir, and solvent/lipid bath vanishes [@schulman61], and we may assume a reference chemical potential of zero or, equivalently, zero surface tension. This coincides with the experimental observation of visible thermal fluctuations on the tubules [@barziv94]. Now imagine applying a laser to the tubule. In the electric field of the chemical potential of a lipid molecule is lowered by an amount $\delta\varepsilon{\cal E\/} D a$, where $D$ is the molecular length, $\delta\varepsilon$ is the dielectric constant relative to water, $a$ the area of the lipid, and ${\cal E\/}$ the energy density deposited in the trap. Nelson [*et al.*]{} [@nelsontube95b] calculated that this yields an energy gain per area of bilayer of $\Sigma_0\sim 2 \cdot 10^{-3} \,\hbox{erg cm}^{-2}$, for a laser power of $50\,\hbox{mW}$. Hence there is a large reduction in the local chemical potential as the lipid suddenly finds it advantageous to move into the laser spot. The surface tension in the adjacent portion of the tubule increases as lipids start to move out of the surface. Since the other end of the tubule is in contact with a reservoir at zero chemical potential, the final state (prohibiting, for the moment, surface undulations) must be a non-equilibrium steady state in which: 1. Lipid is transported at constant velocity from the reservoir at zero chemical potential to the laser trap at a negative chemical potential. 2. The chemical potential drops linearly along the tubule, with a gradient that balances the frictional drag of the bulk fluid in steady state. 3. The local lipid concentration also varies linearly, since the two-dimensional lipid fluid membrane is compressible. This differs significantly from the treatments of Nelson [*et al.*]{} and Granek and Olami in that [*lipid must flow out of the anchoring globules*]{} and the chemical potential (or surface tension) never attains a non-zero constant over the duration of the experiment. In fact, prohibiting the shape instability, the boundary conditions specified by both Olami and Granek [*and*]{} Nelson [*et al.*]{} yield a tense final state as (a small amount of) area is drawn out of surface fluctuations, while the treatment of the anchoring globules as reservoirs yields the steady-state described below.[^2] Several consequences follow from this observation. First, a chemical potential gradient suggests a mechanism for front propagation [@nelsontube95b; @sarloos88]. The front starts at the laser spot where the surface tension is largest, and ‘propagates’ outward toward the anchoring globule simply because the amplitude of the instability grows at different rates along the tube. Our results predict a speed of front propagation which is inversely proportional to the length of the tube, and is largest near the laser spot, decreasing to zero somewhere near the anchoring reservoirs; and a characteristic wavenumber which also decreases (much slower, see Fig. \[fig:dispersion2\] below) away from the laser spot. The outline of this paper is as follows. In Section 2 we derive the linear concentration gradient in the absence of surface undulations. We predict a ‘ramped’, or spatially-varying control parameter, the effective surface tension, which is in fact the two-dimensional pressure whose gradient drives the flow of lipid against the viscous drag of the bulk fluid. In Section 3 we present a detailed microscopic picture of the uptake of surfactant by the trap, and argue that a competition between bending and compression energies modifies the effective surface tension of the trap. This leads to a prediction of a critical laser power for the onset of an instability. While this section may safely be omitted in reading this paper, it illuminates the nature of the instability by treating a realistic scenario for how the trap buckles to initiate flow. In Section 4 we discuss the implications of a slowly varying surface tension on the detailed calculation of Goldstein, [*et al.*]{} [@nelsontube95a; @nelsontube95b]. We also discuss front propagation within the picture of a surface tension gradient, which relates the problem to a large body of work on front propagation with ‘ramped’ parameters [@kramer82]. The issue of front propagation in this system is delicate [@nelsontube95b], and our results suggest at least two possibilities, which we briefly raise in this work and pose for further investigation. Depending on whether noise ([*i.e.*]{} existing thermal fluctuations in the tubule) is present, we expect front propagation which is either (a) characteristic of that predicted by the so-called Marginal Stability Criteria (MSC) [@nelsontube95b; @kramer82], or (b) dominated by amplification of existing ‘noise’, which can lead to behavior reminiscent of front propagation for a steep enough ramp. We conclude in Section 5 by recalling the relevant timescales and frequencies, and summarizing our predictions and the differences from previous treatments. Steady state ============ In this section we calculate the steady-state configuration of a tubule under the action of an applied laser intensity, assuming the laser supplies a chemical potential $- \Sigma_0$ at the laser spot. Note that this implies a reservoir in which to pack lipid molecules. In Section 3 we support this with a microscopic picture which leads to virtually the same results that we obtain in this section, with a prefactor of order one which depends on the laser shape. Note that there are several possible microscopic scenarios for initiating flow into the trap, and Section 3 addresses only one of these. Equations of motion ------------------- Changes in chemical potential $\delta\!\mu$ are related to changes in surface tension $\delta\Sigma$ by $\delta\mu = - \phi^{-1}\delta\Sigma $, where $\phi$ is the lipid concentration (and hence $\phi^{-1}=a$ is the area per lipid). Also, $$\Sigma=-p,$$ where $p$ is the 2-dimensional pressure of the fluid of lipid molecules. The geometry of the system is taken as shown in Figure \[fig:geom\], with the cylinder aligned parallel to the $z$-axis and $r$ the radial coordinate. The boundary conditions are $$\begin{aligned} p(z=0) = 0 && \qquad(\hbox{reservoir}) \\ p(z=L) = - \Sigma_0 && \qquad(\hbox{laser spot}),\end{aligned}$$ where $\Sigma_0$ is the surface tension induced by the laser. 20.5pc =3truein The Navier-Stokes and continuity equations for the 2D fluid of lipid molecules are $$\begin{aligned} \partial_t \phi&=& -\nabla\!\cdot\!(\phi{\rm\bf v}) \label{eq:continuity}\\ \rho_s (\partial_t + {\rm\bf v}\!\cdot\!\nabla){\rm\bf v} &=& \eta_s \nabla^2 {\rm\bf v} + (\case{1}{3}\eta_s + \gamma_s) \nabla (\nabla\cdot{\rm\bf v}) \label{eq:NS}\\ && - \nabla p + \Delta{\bf T}^b\!\!\cdot\!{\bf\hat{r}}. \nonumber\end{aligned}$$ Here $\eta_s$ and $\gamma_s$ are 2D shear and bulk viscosities, $\rho_s$ is 2D lipid mass density, and $\Delta{\bf T}^b\cdot{\bf\hat{r}}$ is the viscous drag acting on the surface from the dissipative stress tensor $T^b_{\alpha\beta} = \case12\eta(\nabla_{\alpha}u_{\beta} + \nabla_{\beta}u_{\alpha})$ in the surrounding fluid. This flow is established in a vorticity diffusion time $\tau_{v}$ which is much smaller than other times in the problem. We ignore drag from outside the cylinder for the moment, since this flow essentially moves with the surfactant molecules and contributes relatively little to the boundary stress[^3]. With the above approximation, the boundary stress is given by the shear stress in the tube. For a [*uniform*]{} flow of lipid ${\rm\bf v} = v {\bf\hat z}$, the interior flow is Poiseuille [@landaufluids], $${\rm\bf u}(r) = v {2r^2 - R_0^2\over R_0^2}{\bf\hat z},$$ where $R_0$ is the tube radius and we use the no-slip boundary condition $u(r=R_0)=v$. Hence the stress acting on the surface is $$\Delta{\bf T}^b\!\cdot\!{\bf\hat{r}} = -{2\eta v \over R_0}. \label{eq:stress}$$ Gradients of ${\rm\bf v}$ in the $z$-direction change the flow profile from simple Poiseuille, but this has only a very small effect on the dynamics of establishing the steady state, primarily in the region of the laser spot, which we ignore for now. The final ingredient we need is the compressibility of the film, through the constitutive relation $$p = p_0 - \chi^{-1} \delta a \label{eq:pressure}$$ where $p_0$ is the equilibrium pressure. Now we specialize to the problem at hand. We linearize the dynamic equations in ${\rm\bf v}$ and $\delta\phi = \phi - \phi_0$, assume a velocity of the form ${\rm\bf v} = v(z){\bf\hat z}$, and ignore the inertial term in the Navier-Stokes equation. Employing Eq. (\[eq:pressure\]), we obtain $$\begin{aligned} \partial_t \delta\phi&=& -\phi_0 \nabla_z v \label{eq:phidot}\\ 0 &=& \hat{\eta} \nabla_z^2 v - B \phi_0^{-1} \nabla_z \delta \phi + {2\eta\over R_0} v,\label{eq:vdot}\end{aligned}$$ where $\hat{\eta}=\case43 \eta_s + \gamma_s$. The boundary conditions are $$\begin{aligned} \delta \phi(z=0)&=& 0 \\ \delta \phi(z=L)&=& \phi_0\Sigma_0/B.\end{aligned}$$ where $B=\chi^{-1}\phi_0^{-1}$ is the two dimensional bulk modulus. Dynamics of equilibration ------------------------- Assigning $\delta\phi(z,t) = \delta\hat{\phi}(q,\omega) e^{i (q z - \omega t)}$ and similarly for $v(z,t)$, we obtain the following dispersion relation: $$\omega = {i B q^2 \over q^2 \hat{\eta} + 2\eta/R_0}. \label{eq:dispersion}$$ This yields $\omega\simeq i R_0 B q^2 / \eta$ for $q\ll q^{\ast}$ and $\omega \simeq B / \hat{\eta}$ for $q\gg q^{\ast}$, where $q^{\ast} = \sqrt{2 \eta / (R_0 \hat{\eta})}$. Hence, at long wavelengths we have diffusive behavior governed by the friction against the bulk fluid, while at short wavelengths the dynamics is dominated by the 2D viscosity. The crossover length is given by $1/q^{\ast} \sim 0.1\mu\hbox{m}$, where we have taken $\eta \sim 10^{-2} \,\hbox{g cm}^{-1}\hbox{s}^{-1}, \hat{\eta} \sim 10^{-6} \,\hbox{g s}^{-1}$, and $R_0\sim 0.5 \mu\hbox{m}$. Hence in most cases of interest we are in the regime dominated by bulk fluid dissipation and may ignore $\hat{\eta}$. We can now estimate (within linear response) the time to attain steady state after imposing the localized tension by the laser as, roughly, the relaxation time of the slowest mode given by the dispersion relation Eq. (\[eq:dispersion\]). Taking $q = 2\pi/L$, we have $$\tau_{\mit ss} \sim {2 L^2\eta\over (2\pi)^2 R_0 B}\sim 10^{-5}\,\hbox{s},$$ for $B\sim 150 \,\hbox{erg cm}^{-2}$ [@evans87] and $L\sim 100\mu\hbox{m}$. This estimate of $B$ is a zero-temperature estimate and ignores small-scale thermal fluctuations which soften this modulus considerably [@helfrich84; @nelsontube95b]. As we discuss in the conclusion and as shown in Reference [@nelsontube95b], this effect can reduce $B$ by up to three orders of magnitude, increasing $\tau_{\mit ss}$ accordingly, to of order $10^{-2}\,\hbox{s}$. We can compare this to the vorticity diffusion time, $$\tau_{\it v} = {\rho R_0^2 \over \eta} \sim 10^{-7}\hbox{s},$$ where we take $\rho=1\,\hbox{g cm}^{-3}$. Since $\tau_{\mit ss}>\tau_v$ our assumption above of a uniform shear stress is reasonable. Steady State ------------ To find the steady state we equate the left hand side of Eq. (\[eq:phidot\]) to zero, which yields a constant velocity $\bar{v}$. From Eq (\[eq:vdot\]) $\nabla_z \delta \phi$ is also a constant and, applying the boundary conditions and the pressure constitutive equation, Eq. (\[eq:pressure\]), we find the following steady-state profile $$\begin{aligned} \bar{v} &=& { R_0 \Sigma_0\over 2 \eta L} \label{eq:vbar}\\ \delta \phi &=& -\phi_0 {\Sigma_0 \over B} {z\over L} \\ \delta\Sigma &=& \Sigma_0 {z\over L} \label{eq:sigma}.\end{aligned}$$ Thus we find that the steady state, excluding modulations of the cylinder, is a non-equilibrium steady state, where the lipid molecules run down a chemical potential gradient and the molecular spacing increases to reflect this changing local potential. An estimate above yields $\bar{v} \sim 1 \mu\hbox{m s}^{-1}$, where we use $\Sigma_0 \sim 10^{-3} \hbox{erg cm}^{-2}$. The effective surface tension (or two dimensional pressure) is induced by the applied laser, and is non-zero [*only in the presence of flow*]{}. Microscopic Picture =================== We have shown how lipid flow, which is a necessary condition for an effective non-zero surface tension far from the trap, follows from a boundary condition of fixed chemical potential at the trap. In this section we argue that this boundary condition requires an instability in the trap, and we present detailed calculations for a possible scenario for the trap to initiate flow. We stress that there are several possible mechanisms, including buckling, ejection of micelles or bilayer structures, and growth of ‘cancerous’ membranes. This is surely not an exhaustive list. Basic Considerations -------------------- We first note that a laser spot centered at $z=L$ typically has a Gaussian intensity profile [@webb81], which leads to an energy gain per area of lipid ${\cal U\/}(z) = - \Sigma_0\,\zeta(z)$, with $$\zeta(z) = e^{-(z-L)^2/2\Delta^2}.$$ The spot radius was estimated to be $\Delta\simeq 0.15 \mu\hbox{m}$ in the experiments of Bar-Ziv and Moses [@barziv94]. We can envision two scenarios after applying the laser: - Lipid can be sucked into the trap until the electrostatic energy gain balances the cost of compressing the molecules in the bilayer. At this point the trap is full, flow stops, and the chemical potential (and surface tension) of the entire tube reverts back to zero. - For a critical tension $\Sigma^{\ast}$ (Eq. \[eq:deltabar0\]) we expect the compressed section of the tubule to become unstable with respect to buckling. For higher intensities the trap continues to fold to accommodate more lipid, initiating a flow along the tubule. This flow must be accompanied by a chemical potential (or surface tension) gradient, which drives the instability seen in the experiments. Our discussion suggests that the trap boundary condition should contain the physics that, at a certain distance from the center of the trap, lipid is incorporated into folds to relieve the in-layer compression. A reasonable choice is $$a(z=L-\bar{\Delta})=a_0, \label{eq:BCtrap}$$ where $\bar{\Delta}$ is a distance to be determined. This asserts that the area per head assumes its preferred equilibrium value at the point where the folding begins. In the next two subsections we derive the steady-state flow into the trap (prohibiting for the moment the ‘pearling’ shape change). We first obtain a general relation for the local area per lipid $a(z)$, which depends on the trap boundary condition. We then deduce a crude criterion for the position $z=L-\bar{\Delta}$ at which the trap buckles. Applying the [*assumption*]{} of Eq. (\[eq:BCtrap\]) at this position then yields the desired profile and steady-state flow. Detailed Steady State --------------------- Let us examine the steady state. The continuity equation, Eq. (\[eq:continuity\]), yields the condition $$v(z) = C a(z),$$ where $C$ is a constant to be determined. Hence the steady-state Navier-Stokes equation, ignoring the 2D viscosity, becomes $$0 = {2\eta C a(z) \over R} + \chi^{-1} \nabla_z a(z) - \nabla {\cal U\/}(z).$$ 20.5pc =3truein The solution of this equation with the boundary condition $a(0)=a_0$, which follows from contact with a reservoir at ambient pressure, is $$\begin{aligned} {\delta a \over a_0} &=& e^{\lambda z} - 1 - \gamma\int_0^z\!ds\, e^{-\lambda(s - z)} {\partial \over\partial s} \zeta(s) \label{eq:solution}\\ \lambda &=& {2\eta\chi C\over R_0}, \\ \gamma &=& {\Sigma_0\over B},\end{aligned}$$ where $\lambda$ is determined by the boundary condition at the trap. Note that $\gamma \sim 10^{-5}$, since $\Sigma_0\sim 10^{-3} \,\hbox{erg cm}^{-2}$ and $B\sim 150 \,\hbox{erg cm}^{-2}$. Using Eq. (\[eq:BCtrap\]), we find $$\lambda= {- \gamma \,\zeta(L-\bar{\Delta})\over L - \bar{\Delta} - \int_0^{L-\bar{\Delta}}\!ds\, \zeta(s)},$$ where we have expanded for small $\lambda$ and made use of $\zeta(0)=0$ far from the trap. For all practical purposes $\lambda= - \gamma \,\zeta(L-\bar{\Delta})/L$. Since $\lambda L \ll 1$ for most cases of experimental interest, the calculation of Section 2 applies in the region outside the trap (see Fig. \[fig:profiles\]) with the surface tension replaced by $$\Sigma_0 \rightarrow \bar{\Sigma} = \Sigma_0\,\zeta(L-\bar{\Delta}). \label{eq:barsig}$$ For a Gaussian shape $\zeta(L-\bar{\Delta}) \alt 1$ \[$\zeta(L-\bar{\Delta})\simeq 0.31$ in Fig. \[fig:buckle\]\], so the modification to the naive boundary condition is rather minimal. Trap Boundary Condition ----------------------- To complete our discussion we estimate the stability against folding inside the trap. This determines the position $\bar{\Delta}$ at which the boundary condition (\[eq:BCtrap\]) applies, as well as a critical effective surface tension parameter (or laser intensity) $\Sigma^{\ast}$ at which the system initiates flow. We imagine that the system has attained a steady state in the absence of buckling and flow, given by Eq. (\[eq:solution\]) with $\lambda=0$: $$a(z) = a_0\left[1 - \gamma\,\zeta\left(z\right)\right].$$ Here $\gamma\,\zeta\left(z\right)$ is a measure of the compression. Rather than calculating the stability of a patch with a non-uniform area per head $a(z)$, we calculate the stability against buckling of a patch with uniform $a=\psi a_0$, with $\psi= 1 - \epsilon $, and use the resulting critical strain $\epsilon^{\ast}$ to determine $\bar{\Delta}$ through $$\gamma\,\zeta(L-\bar{\Delta}) \equiv \epsilon^{\ast}. \label{eq:deltabar}$$ We consider perturbations $R(z) = R_0(1 + u(z))$ which preserve the volume of the fluid. This constraint yields the condition $\int [u(z)^2 + 2 R_0 u(z)] =0$ [@safran]. The free energy, which includes in-plane compression and bending, is [^4] $$2 F = \int d^2\!r \left[ B \left({\psi a - a_0 \over a_0}\right)^2 + \kappa H^2 \right],$$ where $H$ is the mean curvature, $a_0 = dz R_0$, and $a = \psi d^2\!r$. For the perturbation above [@granek95], $$\begin{aligned} d^2\!r &=& dz (R_0 + u(z)) \sqrt{1 + u'(z)^2} \\ H &=& {(1 + u'(z)^2)^{-1/2}\over R_0 + u(z)} - {u''(z)\over(1 + u'(z)^2)^{3/2}},\end{aligned}$$ where $u'(z) = du/dz$. To quadratic order in $u(z) = \sum_q \hat{u}(q) e^{iqz}$ the energy per unit area $A$ becomes [@granek95] $$\begin{aligned} {2 F\over A} &=& \sum_q \hat{u}(q)^2 \left[ {3\kappa\over R_0} - 5 B R_0 \epsilon\right. \nonumber\\ && - \hat{q}^2 \left.\left(3B R_0 \epsilon + {\kappa\over R_0}\right) + 2 \hat{q}^4 {\kappa\over R_0}\right],\end{aligned}$$ where $\hat{q}=q R_0$. The vanishing of the term in square brackets defines $\epsilon^{\ast}$, the minimum strain above which this energy is unstable to undulations. Combining this condition with our estimate for how $\epsilon(z)$ varies away from the trap, Eq. (\[eq:deltabar\]), we find the following relation which determines $\bar{\Delta}$: $$\begin{aligned} \sigma^{\ast}\equiv{\Sigma^{\ast} R_{0}^2 \over \kappa} &=& {1\over \zeta(L-\bar{\Delta})} {3 + 2 \bar{q}^4 - \bar{q}^2 \over 5 + 3 \bar{q}^2} \label{eq:deltabar0}\\ &\equiv& g(\bar{q})\end{aligned}$$ where $\bar{q} \simeq \pi R_0/\bar{\Delta}$. For $\sigma>\sigma^{\ast}$ the tubule should buckle inside the trap. Fig. \[fig:buckle\] shows $g(\bar{q})$ for the Gaussian laser spot. 20.5pc =3truein Our criterion depends on the trap shape, but not the compression modulus $B$. This happens because, while the strain induced in packing lipid in the trap varies as $1/B$, the critical strain at which buckling occurs is also inversely proportional to $B$, and the $B$ dependence cancels out. In fact, the same order of magnitude estimate emerges from a comparison of bending and effective surface tension. What we have gained, however, is a picture of the forces at play that induce the buckling. While our estimate apparently fails to predict buckling for typical values, relaxing a few approximations we have made should change this picture. First, we have assumed uniform coverage at the lowest value of the actual nonuniform coverage in the quiescent trap. Second, we have assumed an axisymmetric deformation. This is obviously not the case if the laser spot size is smaller that the tubule diameter. In addition, the volume constraint must be handled differently. Removing these approximations should, in both cases, result in a smaller $\Sigma^{\ast}$. For example, in the limit of small trap sizes we can ignore curvature and ask about the stability of a flat interface against buckling, for which the criterion above becomes $$\Sigma^{\ast} = {\kappa \bar{q}^2\over\zeta(L-\bar{\Delta})}.$$ demanding that the characteristic buckling wavevector $\bar{q}$ be roughly the inverse of the trap size $\bar{\Delta}$ determines the critical trap size and intensity. This is essentially the same estimate given by Bar-Ziv, Frisch, and Moses in a somewhat different context [@barziv95]. Dispersion Relation and Front Propagation ========================================= Dispersion Relation ------------------- We have shown thus far that, under steady-state conditions before any macroscopic shape instability occurs, the proper boundary conditions imply a surface tension gradient along the tubule which supports lipid flow. We now turn to the effects of this gradient. Rather than repeating the the analysis of GNPS [@nelsontube95b] with a non-uniform surface tension, we note that the characteristic wavenumber at which the instability occurs is typically $q^{\ast}R_0\simeq 0.8$. Since $R_0\ll L$, we suspect that the assumption of a locally constant surface tension along the tubule is a good first step. This allows us to transcribe the results of Ref. [@nelsontube95b]. The primary result of interest is the growth rate $\omega(q)$ of an undulation $u(q,t)$, where $q$ denotes a Fourier mode along the tubule. This frequency is defined through $$({\partial\over \partial t} + i q \bar{v}) u(q,t) = \omega(q) u(q,t),$$ where the convective term arises because the lipids have an average velocity. In the original instability calculation presented by Rayleigh [@rayleigh1], and as has been emphasized in Refs.[@barziv94; @nelsontube95a; @granek95; @nelsontube95b], the structure of $\omega(q)$ is as follows: $$\omega(q) = \Phi(q) T(qR_0).$$ The function $T(qR_0)$ is determined by the energetics of the problem and, in our case, is non-zero for $\Sigma R_0^2/\kappa$ greater than a critical value of order unity [@nelsontube95b; @granek95]. In the Rayleigh case the instability occurs for $\Sigma>0$. The function $\Phi(q)$ is determined by the dynamics of the problem, and it is here that much of the interesting and surprising physics lies. Energetics tells us that the most unstable modes are at low $q$, where undulations are the least ‘violent’, while dynamic considerations severely penalize the growth of modes in the limit $q\rightarrow 0$. Goldstein [*et al.*]{} [@nelsontube95b] calculated $\omega(q)$ for a uniform surface tension $\bar{\Sigma}$, including the effects of bending as well as friction between the two bilayer leaves. Changing the boundary conditions of their work to allow for flux from the reservoir adds a convective term to the dynamics, and, aside from our local approximation above, changes nothing else. A plot of $\omega(q)$ is shown in Figure \[fig:dispersion\] for several values of the surface tension $\sigma =\bar{\Sigma} R_0^2 / \kappa$, with values for the bilayer friction and bulk modulus taken as in GNPS. 20.5pc =3truein Since $\sigma$ is $z$-dependent (Eq. \[eq:sigma\]), the growth rate $\omega^{\ast}$ and wavenumber $q^{\ast}$ of the fastest growing mode are $z$-dependent, and are greatest near the laser spot, as in Fig. \[fig:dispersion2\]. A single Fourier modulation has, locally, the following form: [@nelsontube95b] $$u(z,t) = u_0 e^{i q (z - \bar{v} t) + \omega(q,z)t},$$ where $\omega(q,z)$ is the function plotted as $\omega(q)$ in Fig. 5 of Reference [@nelsontube95b] and is reproduced here in Fig. \[fig:dispersion\]. The $z$-dependence comes through the $z$-dependence of $\sigma$. Note that we rely [*strongly*]{} on the condition $q L \gg 1$ \[Note that in the experiments [@barziv94] with, say, $L\simeq 200\,\mu\hbox{m}$ and a diameter of $1\,\mu\hbox{m}$, this condition is easily satisfied, $qL\simeq 160$.\] Given a dispersion relation which depends on position, there are several immediate naive predictions: The local wavenumber and apparent growth rate of the pattern should decrease as the anchoring globules are approached, with the instability vanishing at a point close to the reservoir where the induced surface tension is not strong enough to overcome the barrier due to bending, $\Sigma^{\ast} \simeq \kappa/R_0^2$. Hence, in the experiments with $\sigma\simeq 20$ [@nelsontube95a], this occurs at $1/20th$ of the distance from the anchoring globule to the laser trap. At points closer to the globule any undulation is a decaying remnant of the pattern developed closer to the trap. 20.5pc =3truein Front Propagation ----------------- We now confront the issues of front propagation and wavelength selection. GNPS argued that the Marginal Stability Criterion hypothesis provides a reasonable estimate for both the propagation speed $v_f$ and the selected wavenumber $k^{\ast}$ [@nelsontube95b]. A naive extension of this calculation, again assuming a local dispersion relation, predicts a spatially varying front speed and selected wavenumber. However, this prediction and the qualitative picture relies on two assumptions: (1) the existence of a propagating front, and (2) the absence of noise—i.e., that the propagation occurs into a uniform, unstable medium with no thermal fluctuations. The latter assumption is obviously not correct in detail, as thermal fluctuations, including modes of wavelengths comparable to the most unstable modes, are apparent in the experiments prior to the onset of the instability. Here we discuss these issues in the context of a spatially varying control parameter. Because a full treatment of the problem does not yet exist in the literature and is beyond the scope of this paper, we limit ourselves in this work to some numerical experiments and suggestions which, we hope, will stimulate further research on both this specific problem and the general aspects of front propagation into spatially-varying media in the presence of noise. For the rest of this paper we refer to ‘noise’ as a set of random initial conditions which obey a Boltzmann distribution, and do not consider temporally fluctuating noise. Kramer [*et al.*]{} [@kramer82], followed by others [@buell86; @kramer85; @riecke87], showed that, in the presence of a ‘ramped’ control parameter that becomes subcritical at some point, as happens near the reservoir, the uniquely-selected wavenumber need not correspond to that determined by the MSC. This effect is expected to take precedence over the MSC-determined wavenumber at times after which non-linearities become important and the ‘phase’ of the pattern has time to diffuse of order the system size. However, we are concerned with the more fundamental issue of the [*existence*]{} of a propagating front. In the presence of ‘noise’ which, in the present experiment, corresponds to existing thermal fluctuations around the reference smooth cylindrical state, a propagating front can be expected to exist for times less than the characteristic growth times of existing fluctuations in the vicinity of the most unstable mode. Hence, given a quench into an unstable ‘ramped’ state with an initial perturbation near the laser spot, propagation away from the perturbation occurs for an initial period of time, followed by rapid growth all along the cylinder as the initial conditions (‘noise’ of unstable wavelengths) are amplified to visible length scales. An initial perturbation near the laser spot is natural because, in practice, the laser spot diameter is smaller than the tubule diameter and a ‘pinching’ effect results whereby surfactant flows around the circumference of the tubule (as well as along the cylinder diameter) to fill the trap. The effect of a ‘ramp’ in the control parameter should be most dramatic after the noise overwhelms the front propagation: for a flat control parameter (no ramp) the noise grows randomly everywhere, and the ‘front’ should break down when the noise has grown to visible amplitudes. However, for a steep enough ramp the non-uniform amplification of the noise could resemble front propagation. To check these conjectures we have employed a simple caricature of the tubule dynamics, specified by $$({\partial\over \partial t} + \bar{v} \partial_x) u(x,t) = \left[a(x) 2 k_0^2 \partial_x^2 + \partial_x^4 \right] u(x,t) - g u(x,t)^3, \label{eq:caricature}$$ where $a(x)$ is a spatially varying control parameter chosen to mimic the dispersion relation and position dependencies in Figs. \[fig:dispersion\] and \[fig:dispersion2\]. A choice which gives reasonable qualitative agreement is $$a(x) = {|x-x_0|^{\alpha+1}\over (x-x_0)}, \label{eq:trial}$$ where $x_0$ is the point at which the system is absolutely unstable. We emphasize that this is a toy model whose details do not correspond to the Bar-Ziv [*et al.*]{} experiments, but which we believe contains the essential physics of front propagation into an unstable inhomogeneous medium, as occurs in these experiments. For Fig. \[fig:dispersion2\], $x_0\sim 0.05 L$ and $\alpha=1/8$ are reasonable. Fig. \[fig:mimic\] shows the local dispersion relation $\omega^{\ast}(x)$ for various $\alpha$. 20.5pc =3truein We have chosen the simplest possible non-linearity to stabilize the system. One may choose a more physical non-linearity such as the driving force arising from terms of higher than quadratic order in the mean curvature [@nelsontube95b], but our purpose here is primarily to illustrate some qualitative behavior of a front propagating into a noisy, non-uniform media. Figs. \[fig:snapshots1\] and \[fig:snapshots2\] show snapshots in the evolution of the system, given by Eq. (\[eq:caricature\]), for an initial perturbation at the trap of $1\%$ of the final amplitude (as determined by the non-linear term $g$) and an initial condition (or ‘noise’) which is taken to be a superposition of $300$ harmonics weighted with a Boltzmann weight corresponding to a non-zero surface tension ([*i.e.*]{} with an energy proportional to $q^2$). We have chosen a system size of $150$ wavelengths, and arbitrarily chosen the vertical scale to fill the figures. The general features are as described above: a front ‘propagates’ for an initial time from the initial perturbation, after which the ‘noise’ takes over and a very irregular growth quickly overtakes the system. 20.5pc =3truein The steeper ramp ($\alpha=1/2$, Fig. \[fig:snapshots1\]) has a better defined growth in the ‘noise’ regime, and could almost be called a ‘front’. In contrast, the growth into the shallow ramp ($\alpha=1/8$, Fig. \[fig:snapshots2\]) is more ragged and it would be charitable to call this a front. The shallower ramp has a very slightly faster propagation speed in the initial regime, and an obviously faster ‘propagation’ speed in the noise-dominated regime. Both of these behaviors may be traced to the faster overall growth rate for a shallower ramp (where a larger fraction of the tubule is more unstable). 20.5pc =3truein Fig. \[fig:noise\] shows the the results of fixing the ramp and varying the noise amplitude. For a noisier system the effective propagation speed in the noise-dominated regime is faster, and the breakdown of the simple propagating regime occurs earlier. The propagation velocity before the noise takes over is independent of the noise amplitude. The delay before noise-dominance increases logarithmically with increasing noise amplitude, consistent with the simple argument that the propagating solution exists until the noise has grown to a given amplitude, since this initial growth is exponential. To summarize, we have performed exploratory numerical calculations to investigate some of the consequences of a ramped control parameter, with an initial localized perturbation and initial global ‘noise’ for an initial condition, finding: 1. At early times a front propagates away from the localized perturbation. We find a dimensionless front velocity of $v\,\omega(k_0)/k_0 = 3.3$, while the Marginal Stability Criteria [@sarloos88] predict $4.6$. A similar agreement was found in the simulations of GNPS [@nelsontube95b]. 20.5pc =3truein 2. In this first regime the propagation velocity is independent of noise amplitude. 3. After a time, which may be taken to be the time required for the ‘noise’ to grow to an observable amplitude, the unstable pattern rapidly develops everywhere on the tubule. 4. The speed and qualitative character of this growth depend on the noise and ramp characteristics. The growth is faster for a shallower ramp and/or stronger noise, and looks reminiscent of a front for a steeper ramp; and in all cases is much faster than the simple front propagation from the initial localized perturbation at the laser trap. We have also performed calculations with no initial localized perturbation but, as this is probably not physically relevant, we do not report the results here. This initial study raises several questions which we feel are worth pursuing. In cases where the noise is weak enough and an apparent front exists, can this be understood quantitatively in terms of the gradient in the control parameter, and how does this relate to previous investigations of ‘ramped’ control parameters [@kramer82; @buell86; @kramer85; @riecke87]? Can a ramp stabilize an advancing front? Conclusion ========== Physical Picture ---------------- We have given the following picture of the action of lipid tubules upon the application of laser tweezers. In the absence of buckling, the laser induces a local compression of lipid molecules in the laser spot. This takes place in a time of order $\tau_{\mit ss}\sim 10^{-5} \hbox{s}$. A sufficiently large laser intensity induces a local buckling of the membrane in the trap, which initiates flow down the tubule from the reservoir. We do not have an estimate of the delay time for the instability in the trap. In the absence of undulations [*outside*]{} the trap, this flow would build up to a steady-state value $\bar{v}\sim 1\,\mu\hbox{m s}^{-1}$ in a time order $\tau_{\mit ss}$. The physics of this flow is a balance between drag against the bulk fluid and a force due to the gradient imposed by the chemical potential drop between the reservoir and the trap. Given the steady-state tension profile within the membrane and the reasonable assumption that the gradient occurs over a length (L) much larger than the critical wavelength ($\sim R_0$), the analysis of GNPS [@nelsontube95b] leads to a Rayleigh-like instability to undulations. This instability initiates near the laser spot where the chemical potential (or surface tension) is lowest, and propagates away from the spot to a point along the tubule at which the local surface tension falls below the critical tension $\Sigma_{cr}=\kappa/R_0^2$ which characterizes the instability. Typical growth frequencies are $\omega\sim 25\,\hbox{s}^{-1}$ [@nelsontube95b], which corresponds to times $\tau_{\omega}\sim 10^{-1}\hbox{s}$. We note that the experiments find a significant time delay of order seconds before the instability, [@mosesprivate], a still-unexplained observation. The estimate above for $\tau_{\mit ss}$ assumes a bare $2D$ compression modulus $B$, while GNPS (see [@helfrich84]) pointed out that $B$ undergoes significant softening at the lengthscale of the tubule, due to the thermal fluctuations at low surface tension, and estimated a decrease of up to three orders of magnitude. This, correspondingly, would increase the value of $\tau_{\mit ss}$ to $10^{-2}\,\hbox{s}$, so that an accurate quantitative calculation must include the dynamics of the increase in surface tension. This leads to, effectively, a smaller applied tension $\sigma$ and hence a slower propagation speed. This spirit was followed in the approach of Granek and Olami [@granek95]. Our interpretation assumes that the trap accommodates material by folding, or some other means. Our analysis suggests that the proper boundary condition should be a fixed surface tension $\bar{\Sigma}<\Sigma_0$ at the laser spot, where the laser shape determines $\bar{\Sigma}$ through Eq. (\[eq:barsig\]) and $\bar{\Delta}$ through Eqs.(\[eq:deltabar\],\[eq:deltabar0\]). This reservoir picture suggests that, upon turning off the laser, the system can revert to the original tubule by unfolding or, if severe topological changes have occurred (by, for example, budding in the laser spot or the creation of metastable ‘pearls’ as seen in the experiments), attain some other long-lived metastable state. Discussion ---------- In this work we have made several assumptions. The assumption that we can treat the anchoring globules as reservoirs presupposes that any damping processes retarding the transfer of lipid to and from the globules is negligible relative to other dynamical processes. We expect this to arise from the same source as the two-dimensional surface viscosity, which we have argued in Section 2.2 to be negligible. We have given a simplified picture of the scenario of trap buckling, where we take a single characteristic buckling wavevector, and treat the trap as uniform. This ‘single-mode’ approximation may be naive, and preliminary calculations suggest that the system is in fact less stable than this simple analysis would suggest [@pdofcmunpub]. There are also several other possible modes of instability which we have only mentioned but which could certainly play a role. We have also specified a boundary condition at the trap whereby the the lipid relaxes to its preferred area per head group, which seems reasonable but is not otherwise justified. Finally, we have made a local approximation for the variation of the surface tension so that we may use the results of GNPS. This applies for sufficiently long tubules, $L/R_0\gg 1$. Front propagation and the detailed effects of propagating into a spatially-varying medium have only been touched upon in our numerical treatment. This study still leaves much to be resolved; one important question is how to accurately treat the non-linear regime. This has been treated in different ways by Olami and Granek [@granek95] who considered the non-linear effect of removing surfactant from the membrane in the [*absence*]{} of a gradient, and by Goldstein [*et al.*]{} [@nelsontube95b], who added the correct non-linear terms in the bending energy to examine the propagation of the pearls. The primary new ingredients in our theory are (1) our treatment of the anchoring lipid globules as reservoirs and (2) our exploratory treatment of the role of pre-existing thermal fluctuations (noise) in determining the ‘front-like’ characteristics of the instability. Both Nelson and co-workers [@nelsontube95a; @nelsontube95b] and Olami and Granek [@granek95] ‘turn off’ the reservoir. In the latter case material is drawn out of the existing thermal fluctuations, while Nelson and co-workers attribute the area change primarily to the shape instability itself. Olami and Granek impose a constant flux boundary condition at the trap, while Nelson and co-workers impose a fixed chemical potential $-\bar{\Sigma}$ at the trap which, fairly rapidly, reduces the chemical potential everywhere to $-\bar{\Sigma}$. Our picture essentially gives the same boundary condition at the trap, but the treatment of the globule as a reservoir changes the qualitative picture dramatically. Our theory differs from previous theories in several respects, and there are many consequences which may be checked experimentally. Obviously, we expect flow when an instability develops. This could be visualized by, for example, fluorescence spectroscopy with a very dilute fraction of labelled lipids. The inhomogeneous surface tension implies that the local dispersion relation is also spatially-dependent, as in Fig \[fig:dispersion2\], which implies that the velocity of front propagation $v_f$ (which is proportional to $\omega^{\ast}$ [@nelsontube95b]) and characteristic wavenumbers should decrease farther away from the laser spot. Note that the characteristic wavelength changes very gradually compared to the speed of propagation, and as such would be more difficult to detect. It would also be interesting to see, experimentally, whether fluctuations are actually strong enough to destroy the front-like character, or whether two characteristic regimes exist in the experiments, as indicated in Fig. \[fig:noise\]. Finally, we mention that the opportunity of using laser pulses to control flow within lipid and other systems presents amusing possibilities and applications. It is a pleasure to thank E. Moses, R. Granek, P. Nelson, T. Powers, C.-M. Chen, S. Milner, and W. van Saarloos for helpful conversations and correspondence. This work was supported in part by NSF Grant No. DMR 92-57544 and by The Donors of the Petroleum Research Fund, administered by the American Chemical Society. [10]{} R. Bar-Ziv and E. Moses, [*Instability and “pearling”states produced in tubular membranes by competition of curvature and tension*]{}, Phys. Rev. Lett. [**73**]{} (1994) 1392. , [*On the instability of jets*]{}, Proc. Lond. Math. Soc. [ **10**]{} (1879) 4. , [*On the instability of a cylinder of viscous liquid under capillary force*]{}, Phil. Mag. [**34**]{} (1892) 145. S. Tomotika, [*On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid*]{}, Proc. Roy. Soc. Lond. [**A150**]{} (1932) 322. R. Granek and Z. Olami, [*Dynamics of Rayleigh-like instability induced by laser tweezers in tubular vesicles of self-assembled membranes*]{}, J. Phys. II (France) [**5**]{} (1995) 1349. P. Nelson, T. Powers, and U. Seifert, [*Dynamical theory of the pearling instability in cylindrical vesicles*]{}, Phys. Rev. Lett. [**74**]{} (1995) 3384. R. E. Goldstein, P. Nelson, T. Powers, and U. Seifert, [*Front propagation in the pearling instability of tubular vesicles*]{}, J. Phys. II (France) [ **6**]{} (1996) 767. J. H. Schulman and J. B. Montagne, [*Formation of microemulsions by amino alkyl alcohols*]{}, Ann. N.Y. Acad. Sci. [**92**]{} (1961) 366. W. van Sarloos, [*Front propagation into unstable states*]{}, Phys. Rev. [ **A37**]{} (1988) 211. L. Kramer, E. Ben-Jacob, H. Brand, and M. C. Cross, [*Wavelength selection in systems far from equilibrium*]{}, Phys. Rev. Lett. [**49**]{} (1982) 1891. L. D. Landau and E. M. Lifschitz, [*Fluid Mechanics*]{} (Pergamon, Oxford, 1959). E. Evans and D. Needham, [*Physical properties of surfactant bilayer membranes: thermal transitions; elasticity; rigidity; cohesion; and colloidal interactions*]{}, J. Phys. Chem. [**91**]{} (1987) 4219. W. Helfrich and R.-M. Servuss, [*Undulations steric interaction and cohesion of fluid membranes*]{}, Nuovo Cim. [**3D**]{} (1984) 137. M. B. Schneider and W. W. Webb, [*Measurement of submicron laser beam radii*]{}, Appl. Opt. [**20**]{} (1981) 1382. S. A. Safran, [*Statistical Thermodynamics of Surfaces, Interfaces, and Membranes*]{} (Addison-Wesley, Reading, MA, 1994). R. Bar-Ziv, T. Frisch, and E. Moses, [*Entropic expulsion in vesicles*]{}, Phys. Rev. Lett. [**75**]{} (1995) 3481. J. B. Buell and I. Catton, [*Wavenumber selection in ramped Rayleigh-[B]{}énard convection*]{}, J. Fluid Mech. [**171**]{} (1986) 477. L. Kramer and H. Riecke, [*Wavelength selection in Rayleigh-[B]{}énard convection*]{}, Z. Phys. B - Cond. Matt. [**59**]{} (1985) 245. H. Riecke and H.-G. Paap, [*Perfect wave-number selection and drifting patterns in ramped Taylor vortex flow*]{}, Phys. Rev. Lett. [**59**]{} (1987) 2570. E. Moses, private communucation. P. D. Olmsted and F. C. MacKintosh, unpublished (1996). [^1]: e-mail: [*phy6pdo@irc.leeds.ac.uk*]{}; permanent address, Dept. of Physics, University of Leeds, Leeds LS2 9JT, UK [^2]: If we wait long enough the trap will ‘fill up’ with surfactant and the chemical potential return to zero everywhere. However, in the present case of strong laser power the surface instability will have occurred by this time. See Section 3.3. [^3]: We may include this as, for example, the Stokes drag on a cylinder, which increases the right hand side of Equation \[eq:stress\] by of order 10% [@landaufluids]. [^4]: We do not include a term involving interaction with the laser, since we are interested in an instability at fixed particle number on the membrane.

--- abstract: 'The maximum genus $\gamma_M(G)$ of a graph $G$ is the largest genus of an orientable surface into which $G$ has a cellular embedding. Combinatorially, it coincides with the maximum number of disjoint pairs of adjacent edges of $G$ whose removal results in a connected spanning subgraph of $G$. In this paper we prove that removing pairs of adjacent edges from $G$ arbitrarily while retaining connectedness leads to at least $\gamma_M(G)/2$ pairs of edges removed. This allows us to describe a greedy algorithm for the maximum genus of a graph; our algorithm returns an integer $k$ such that $\gamma_M(G)/2\le k \le \gamma_M(G)$, providing a simple method to efficiently approximate maximum genus. As a consequence of our approach we obtain a $2$-approximate counterpart of Xuong’s combinatorial characterisation of maximum genus.' author: - 'Michal Kotrbčík[^1]' - 'Martin Škoviera[^2]' bibliography: - 'bibl.bib' title: 'Simple greedy $2$-approximation algorithm for the maximum genus of a graph' --- [**Keywords:**]{} maximum genus, embedding, graph, greedy algorithm. [**AMS subject classification.**]{} Primary: 05C10. Secondary: 05C85, 05C40. Introduction ============ The *maximum genus* $\gamma_M(G)$ of a graph $G$ is the maximum integer $g$ such that $G$ has a cellular embedding in the orientable surface of genus $g$. A result of Duke [@duke] implies that a graph $G$ has a cellular embedding in the orientable surface of genus $g$ if and only if $\gamma(G) \le g \le \gamma_M(G)$ where $\gamma(G)$ denotes the (minimum) genus of $G$. The problem of determining the set of genera of orientable surfaces upon which $G$ can be embedded thus reduces to calculation of $\gamma(G)$ and $\gamma_M(G)$. Computing the minimum genus of a graph is a notoriously difficult problem, which is known to be NP-complete even for cubic graphs (see [@Th; @T2]). Nevertheless, the minimum genus can be calculated in linear time for graphs with bounded genus or bounded treewidth by [@KMR:2008]. Moreover, for graphs with fixed treewidth and bounded maximum degree [@gross:2014] provides a polynomial-time algorithm obtaining the complete genus distribution $\{g_i\}$ of the graph $G$, where $g_i$ denotes the number of cellular embeddings of $G$ into the orientable surface of genus $i$. For graphs of bounded maximum degree [@CS:2013] has recently proposed a polynomial-time algorithm constructing an embedding with genus at most $O(\gamma(G)^{c_1}\log^{c_2} n)$ where $c_1$ and $c_2$ are constants. On the other hand, for every $\epsilon>0$ and every function $f(n) = O(n^{1-\epsilon})$ there is no polynomial-time algorithm that constructs an embedding of any graph $G$ with $n$ vertices into the surface of genus at most $\gamma(G) + f(n)$ unless P $=$ NP (see [@CKK; @CKK2]). For maximum genus, the situation is quite different, as maximum genus admits a good (min-max) characterisation by Xuong’s and Nebeský’s theorems, see [@KOK; @X] and [@KG; @N], respectively. From among these results the best known is Xuong’s theorem stating that $\gamma_M(G) = (\beta(G) - \min_T \odd(G-E(T)))/2$, where $\beta(G)$ is the cycle rank of $G$, $\odd(G-E(T))$ is the number of components of $G-E(T)$ with an odd number of edges, and the minimum is taken over all spanning trees $T$ of $G$. Building on these results, Furst et al. [@FGM] and Glukhov [@G] independently devised polynomial-time algorithms for determining the maximum genus of an arbitrary graph. The algorithm of [@FGM] uses Xuong’s characterisation of maximum genus and exploits a reduction to the linear matroid parity on an auxiliary graph; its running time is bounded by $O(mn\Delta\log^6m)$, where $n,m$, and $\Delta$ are the number of vertices, edges, and the maximum degree of the graph, respectively. A matroidal structure is also in the backgroung of the algorithm derived in [@G], albeit in a different way. Starting with any spanning tree $T$ of $G$, the algorithm greedily finds a sequence of graphs $F_i$ such that $T = F_0 \subseteq \cdots \subseteq F_n \subseteq G$, $|E(F_{i+1}) - E(F_i)| = 2$, and $\gamma_M(F_i) = i$ for all $i$, and $\gamma_M(F_n) = \gamma_M(G)$. The running time of this algorithm is bounded by $O(m^6)$. Although two polynomial-time algorithms for the maximum genus problem are known, both are relatively complicated. It is therefore desirable to have a simpler way to determine the maximum genus, at least approximately. A greedy approximation algorithm for the maximum genus of a graph was proposed by Chen [@chen]. The algorithm has two main phases. First, it modifies a given graph $G$ into a $3$-regular graph $H$ by vertex splitting, chooses an arbitrary spanning tree $T$ of $H$, and finds a set $P$ of disjoint pairs of adjacent edges in $H-E(T)$ with the maximum possible size. Second, it constructs a single-face embedding of $T\cup P$ and then inserts the remaining edges into the embedding while trying to raise the genus as much as possible. A high-genus embedding of $G$ in the same surface is then constructed by contracting the edges created by vertex splitting. The algorithm constructs an embedding of $G$ with genus at least $\gamma_M(G)/4$ and its running time $O(m\log n)$ is dominated by the second phase, that is, by operations on an embedded spanning subgraph of $H$. In this paper we show that there is a much simpler way to approximate maximum genus. Our algorithm repeatedly removes arbitrary pairs of adjacent edges from $G$ while keeping the graph connected. We prove that this simple idea leads to at least $\gamma_M(G)/2$ pairs removed, providing an algorithm that returns an integer $k$ such that $\gamma_M(G)/2 \le k \le \gamma_M(G)$. This process can be implemented with running time $O(m^2\log^2n/(n\log\log n))$. The algorithms developed in [@chen] can then be used to efficiently construct an embedding with genus $k$. Our result provides the first method to approximate maximum genus that can be easily implemented and improves the previous more complicated algorithm of Chen [@chen], which can guarantee embedding with genus only $\gamma_M(G)/4$. Structurally, our approach yields a natural $2$-approximate counterpart of Xuong’s theorem. Background ========== In this section we present definitions and results that provide the background for our algorithm. Our terminology is standard and consistent with [@MT]. By a graph we mean a finite undirected graph with loops and parallel edges permitted. Throughout, all embeddings into surfaces are cellular, forcing our graphs to be connected, and the surfaces are orientable. For more details and the necessary background we refer the reader to [@GT] or [@MT]; a recent survey of maximum genus can be found in [@topics Chapter 2]. One of the earliest results on embeddings of graphs is the following observation, which is sometimes called Ringeisen’s edge-addition lemma. Although it is implicit in [@NSW], Ringeisen [@ringeisen:1972] was perhaps the first to draw an explicit attention to it. \[lemma:edge-addition-technique\] Let $\Pi$ be an embedding of a connected graph $G$ and let $e$ be an edge not contained in $G$, but incident with vertices in $G$ - If both ends of $e$ are inserted into the same face of $\Pi$, then this face splits into two faces of the extended embedding of $G+e$ and the genus does not change. - If the ends of $e$ are inserted into two distinct faces of $\Pi$, then in the extended embedding of $G+e$ these faces are merged into one and the genus raises by one. The next lemma, independently obtained in [@KOK], [@jungerman:1978], and [@X], constitutes the cornerstone of proofs of Xuong’s theorem. It follows easily from Lemma \[lemma:edge-addition-technique\]. \[lemma:adding-pairs-single-face\] Let $G$ be a connected graph and $\{e,f\}$ a pair of adjacent edges not contained in $G$, but incident with vertices in $G$. If $G$ has an embedding with a single face, then so does $G\cup\{e,f\}$. Recall that by Xuong’s theorem $\gamma_M(G) = (\beta(G) - \min_T \odd(G-E(T)))/2$, where $\odd(G-E(T))$ is the number of components of the cotree $G-E(T)$ with an odd number of edges. It is not difficult to see that every cotree component with an even number of edges can be partitioned into pairs of adjacent edges, and that every cotree component with an odd number of edges can be partitioned into pairs of adjacent edges and one unpaired edge. Therefore, any spanning tree $S$ minimising $\odd(G-E(T))$ maximises the number of pairs in the above partition of the cotree. The proof strategy of Xuong’s theorem can now be summarised as follows. First, embed $S$ in the $2$-sphere arbitrarily. Then repeatedly apply Lemma \[lemma:adding-pairs-single-face\] to pairs obtained from the partition of the components of $G-E(S)$, each time rising the genus by one. Finally, add the remaining edges. Lemma \[lemma:edge-addition-technique\] guarantees that the addition cannot lower the genus. The result of this process is an embedding of $G$ with genus at least $(\beta(G) - \min_T \odd(G-E(T)))/2$. The fact that a spanning tree minimising $\odd(G-E(T))$ maximises the number of pairs of adjacent edges in the cotree suggests a slightly different combinatorial characterisation of maximum genus. It is due to Khomenko et al. [@KOK] and in fact is older than Xuong’s theorem itself. \[thm:KOK\] The maximum genus of a connected graph equals the maximum number of disjoint pairs of adjacent edges whose removal leaves a connected graph. The following useful lemma, found for example in [@CKK], is an extension of Lemma \[lemma:adding-pairs-single-face\] to embeddings with more than one face. It can either be proved directly by using Ringeisen’s edge-adding technique or can be derived from Xuongs’s theorem. We may note in passing that this lemma was used in [@CKK] to devise an algorithm that constructs an embedding of genus $\gamma_M(G)-1$ whenever such an embedding exists. \[lemma:adding-pairs\] Let $G$ be a connected graph and $\{e,f\}$ a pair of adjacent edges not contained in $G$, but incident with vertices in $G$. Then $\gamma_M(G\cup\{e,f\})\ge \gamma_M(G) + 1$. Our main observation is that Lemma \[lemma:adding-pairs\] can be applied to sets of pairs of adjacent edges which do not necessarily have the maximum possible size. Indeed, if we find any $k$ pairs of adjacent edges $(e_i,f_i)_{i=1}^{k}$ in a graph $G$ such that $G-\bigcup_{i=1}^{k} \{e_i,f_i\}$ is connected, then by Lemma \[lemma:adding-pairs\] we can assert that the maximum genus of $G$ is at least $k$. This suggests that identifying a large number of pairs of adjacent edges whose removal leaves a connected subgraph can be utilised to obtain a simple approximation algorithm for the maximum genus. Indeed, in the following section we show that choosing the pairs of adjacent edges arbitrarily yields an effective approximation of maximum genus. Algorithm ========= In this section we present a greedy algorithm for finding at least $\gamma_M(G)/2$ pairs of adjacent edges while the rest of the graph remains connected. The idea is simple: if the removal of a pair of adjacent edges does not disconnect the graph, then we remove it. To prove that the set output by Greedy-Max-Genus Algorithm always contains at least $\gamma_M(G)/2$ pairs of adjacent edges we employ the following lemma, which can be easily proved either using Xuong’s theorem or directly from Lemma \[lemma:edge-addition-technique\]. \[lemma:edge-removal\] Let $G$ be a connected graph and let $e$ be an arbitrary edge of $G$ such that $G-e$ is connected. Then $$\gamma_M(G)-1\le\gamma_M(G-e)\le \gamma_M(G).$$ The final ingredient for our Greedy-Max-Genus Algorithm is the following characterisation of graphs with maximum genus $0$. \[thm:mgZero\] The following statements are equivalent for every connected graph $G$. - $\gamma_M(G)=0$ - No two cycles of $G$ have a vertex in common. - $G$ contains no pair of adjacent edges whose removal leaves a connected graph. The equivalence (i) $\Leftrightarrow$ (ii) in Theorem \[thm:mgZero\] was first proved by Nordhaus et al. in [@NRSW]. The equivalence (ii) $\Leftrightarrow$ (iii) is easy to see, nevertheless it is its appropriate combination with Lemma \[lemma:edge-removal\] which yields the desired performance guarantee for Greedy-Max-Genus algorithm, as shown in the following theorem. \[thm:alg\] For every connected graph $G$, the set of pairs output by Greedy-Max-Genus Algorithm run on $G$ contains at least $\gamma_M(G)/2$ pairs of adjacent edges. Assume that the algorithm stops after the removal of $k$ disjoint pairs of adjacent edges from $G$. For $i\in\{0,1,\dots,k\}$ let $H_i$ denote the graph obtained from $G$ by the removal of the the first $i$ pairs of edges. By Lemma \[lemma:edge-removal\], the removal of a single edge from a graph can lower its maximum genus by at most one. Therefore, the removal of two edges can lower the maximum genus by at most two. It follows that $\gamma_M(H_i) \ge \gamma_M(G) -2i$ for each $i$; in particular, $\gamma_M(H_k) \ge \gamma_M(G) -2k$. From Theorem \[thm:mgZero\] we get that $\gamma_M(H_k) = 0$. By combining these expressions we get $2k \ge \gamma_M(G)$, which yields $k\ge \gamma_M(G)/2$, as desired. Let $n$ and $m$ denote the number of vertices and edges of $G$, respectively, and let $k$ be the number of pairs of adjacent edges produced by Greedy-Max-Genus Algorithm run on $G$. An embedding of $G$ with genus at least $k$ can be constructed from the set of pairs of adjacent edges in time $O(n+k\log n)$, see the proof of Theorem 4.5 in [@chen] for details. Observe that any maximal set of pairs of adjacent edges of $G$ whose removal from $G$ yields a connected graph can be output by Greedy-Max-Genus Algorithm run on $G$. Hence, as a corollary of Theorem \[thm:alg\] we obtain the following 2-approximate counterpart of Xuong’s theorem. \[thm:approx\] Let $G$ be a connected graph and let $P$ be any inclusion-wise maximal set of disjoint pairs of adjacent edges of $G$ whose removal leaves a connected subgraph. Then $|P|\ge\gamma_M(G)/2$. The following example shows that the bound of Theorem \[thm:approx\] is tight and Greedy-Max-Genus Algorithm can output the value $\gamma_M(G)/2$ for infinitely many graphs $G$. \[ex:1\] Take the star $K_{1,2n}$ where $n$ is an arbitrary positive integer, replace every edge with a pair of parallel edges, and add a loop to every vertex of degree $2$. Denote the resulting graph by $G_n$. Using Theorem \[thm:KOK\] it is easy to see that $\gamma_M(G_n)= 2n$. Indeed, take a set $P$ of $2n$ disjoint pairs of adjacent edges, each consisting of a loop and one of its adjacent edges. Since the edges of $G_n$ not in $P$ form a spanning tree, $P$ has maximum size with respect to the property that $G-P$ is connected. Thus $\gamma_M(G_n) = 2n$ by Theorem \[thm:KOK\]. On the other hand, consider a set $P'$ of $n$ disjoint pairs of adjacent edges that include only edges incident with the central vertex. The removal of these pairs from $G_n$ leaves a spanning tree of $G_n$ with a loop attached to every pendant vertex, so $P'$ is a maximal set of pairs for which $G_n-P'$ is connected. Since $|P'|=n=\gamma_M(G_n)/2$, our example confirms that the bound in Theorems \[thm:alg\] and \[thm:approx\] is best possible. Example \[ex:1\] also implies that processing vertices in the decreasing order with respect to their degrees does not necessarily lead to a better performance of the algorithm. Implementation ============== To implement the algorithm it is clearly sufficient to consider all pairs of edges $\{e,f\}$ with a common end-vertex and test whether removing the pair does not disconnect the graph. The running time is thus $O((\tau+\rho)\sum_{i=1}^n d_i^2)$, where $\tau$ is the time required to test the connectivity, $\rho$ is the time required to update the underlying data structure, and $d_i$ is the degree of the $i$-th vertex. If the input graph is simple, then $\sum_{i=1}^n d_i^2 = O(m^2/n)$ by [@caen:1998]. If the input graph is not simple, we can preprocess it as follows. Let $P=\emptyset$. From every set $F$ of pairwise parallel edges we repeatedly remove pairs of edges and add them into $P$ until $F$ contains at most two parallel edges. Simirarly, from every set $F$ of loops incident with a single vertex we remove pairs of loops and add them into $P$ until $F$ contains at most one loop. Let $G'$ denote the resulting graph. Finally, the set $P$ of pairs of adjacent edges is added to the set $P'$ produced by Greedy-Max-Genus Algorithm on $G'$. It can be easily seen that in $G'$ the sum of squares of the degrees is again in $O(m^2/n)$ and that the preprocessing phase can be done in $O(m)$, where $m$ and $n$ is the number of edges and vertices of the input graph. Regarding the testing of connectivity, we have $\tau=O(m)$ for instance by using DFS, in which case there is no need for additional updates of data structure and thus $\rho=O(m)$. Therefore, we obtain an implementation which reduces essentially to a series of connectivity tests and has running time $O(m^3/n)$. Using the dynamic graph algorithm for connectivity from [@WN:2013] it is possible to support updates in $\rho = O(\log^2 n/\log\log n)$ amortized time and queries in $\tau = O(\log n/\log\log n)$ worst-case time. This yields the total running time $O(m^2\log^2n/(n\log\log n))$. Discussion ========== We have presented an approximation algorithm for the maximum genus problem that for any connected graph $G$ outputs an integer $k$ such that $\gamma_M(G)/2 \le k \le \gamma_M(G)$. This result shows that the classical ideas dating back to Norhaus et al. [@NRSW] and Ringeisen [@ringeisen:1972], and to characterisations of maximum genus by Xuong [@X] and Khomenko et al. [@KOK], can be used also for efficient approximation of maximum genus. Our algorithm is much simpler than both the $4$-approximation algorithm of Chen [@chen] and the precise polynomial-time algorithms for the maximum genus problem of Furst et al. [@FGM] and Glukhov [@G], outperforms the existing $4$-approximation algorithm [@chen], and provides the first approximation of maximum genus that can be easily implemented. On the structural side, we have obtained a natural $2$-approimation counterpart of Xuong’s theorem. Acknowledgement {#acknowledgement .unnumbered} =============== The first author was partially supported by Ministry of Education, Youth, and Sport of Czech Republic, Project No. CZ.1.07/2.3.00/30.0009. The second author was partially supported by VEGA grant 1/0474/15. The authors would like to thank Rastislav Královič and Jana Višňovská for reading preliminary versions of this paper and making useful suggestions. [^1]: Department of Computer Science, Faculty of Informatics, Masaryk University, Botanická 68a, 602 00 Brno, Czech Republic (kotrbcik@fi.muni.cz) [^2]: Department of Computer Science Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina, 842 48 Bratislava, Slovakia (skoviera@dcs.fmph.uniba.sk)

--- abstract: 'We investigate interfacial properties between two highly incompatible polymers of different stiffness. The extensive Monte Carlo simulations of the binary polymer melt yield detailed interfacial profiles and the interfacial tension via an analysis of capillary fluctuations. We extract an effective Flory-Huggins parameter from the simulations, which is used in self-consistent field calculations. These take due account of the chain architecture via a partial enumeration of the single chain partition function, using chain conformations obtained by Monte Carlo simulations of the pure phases. The agreement between the simulations and self-consistent field calculations is almost quantitative, however we find deviations from the predictions of the Gaussian chain model for high incompatibilities or large stiffness. The interfacial width at very high incompatibilities is smaller than the prediction of the Gaussian chain model, and decreases upon increasing the statistical segment length of the semi-flexible component.' author: - | M. Müller${}^{1,2}$ and A. Werner${}^{1}$\ [${}^1$ Institut f[ü]{}r Physik, Johannes Gutenberg Universit[ä]{}t]{}\ [D-55099 Mainz, Germany]{}\ [${}^2$Department of Physics, Box 351560, University of Washington,]{}\ [Seattle, Washington 98195-1560]{} title: | Interfaces between highly incompatible polymers of different stiffness:\ Monte Carlo simulations and self-consistent field calculations --- epsf Introduction ============== Melt blending of polymers has proven useful in designing new composite materials with improved application properties. In many practical situations the constituents of the blend are characterized by some degree of structural asymmetry. For example, a flexible component might contribute to a higher resistance to fracture, while blending it with a stiffer polymer can increase the tensile strength of the material. Since the entropy of mixing in polymeric systems decreases with increasing degree of polymerization, a small unfavorable mismatch in enthalpic interactions, entropic packing effects or the combination of both, generally leads to materials which are not homogeneous on mesoscopic scales, but rather fine dispersions of one component in another. Therefore properties of interfaces between unmixed phases are crucial in controlling the application properties of composites[@GENERAL] and have found abiding experimental interest[@FRIEND1; @FRIEND2; @FRIEND3; @FRIEND4]. Recently, the bulk phase behavior and surface properties[@WU] of polyolefins[@BATES; @OLEFINS] with varying microstructure has attracted considerable experimental and theoretical interest. These mixtures are often modeled[@BATES; @LIU; @SCHWEIZER] as blends of polymers with different bending rigidities, the less branched polymer corresponding to the more flexible component. For pure hard core interactions, field theoretical calculations by Fredrickson, Liu and Bates[@LIU], polymer reference interaction site model (P-RISM) computations by Singh and Schweizer[@SCHWEIZER], lattice cluster theories by Freed and Dudowicz[@FREED] and Monte Carlo simulations[@M1] find a small positive contribution to the Flory-Huggins parameter $\chi$. Monte Carlo simulations which include a repulsion between unlike species reveal an additional increase of the effective Flory-Huggins parameter with chain stiffness, because a back folding of chains becomes less probable with increasing stiffness and the number of intermolecular contacts increases[@M1] respectively. Qualitatively similar effects were found analytically in P-RISM[@SCHWEIZER] and lattice cluster[@FREED] theories. In spite of their ubiquitous occurrence, interfacial properties in asymmetric blends have attracted comparably little interest. When entropic packing contributions to the Flory-Huggins parameter $\chi$ are small and composition fluctuations are negligible, the self-consistent field theory is expected to yield an adequate quantitative description. Helfand and Sapse[@HS] extended the self-consistent field theory to Gaussian chains with different statistical segment lengths. In the limit of infinite long Gaussian chains and strong segregation, they obtained analytical expressions for the interfacial width $w$ and the interfacial tension $\sigma$. Both increase upon increasing the statistical segment length of one component, leaving $\chi$ and the architecture of the other component unaltered. However, there are other models, that incorporate structural disparities on the monomer level. Freed and coworkers model monomers as clusters of various shape on a lattice[@FREED2] and have explored corrections to the energy of mixing and entropic contributions to the Flory-Huggins parameter. Stiffness disparities have also been investigated using the worm-like chain model[@WORM], which captures the crossover between rod-like behavior on small length scales and Gaussian statistics on length scales much larger than the persistence length. Morse and Fredrickson[@MORSE] extended the self-consistent field calculation to a symmetric blend of worm-like chains. For vanishing bending rigidity $\kappa$ they reproduced the Gaussian chain result. In the limit of high bending rigidities and strong segregation ($\kappa\chi \gg 1$), however, they found that the width $w$ of the monomer density profile can be considerably smaller than for a Gaussian chain with the same long distance behavior. At large $\kappa \chi$, increasing the statistical segment length even leads to a decrease of the interfacial width in qualitative contrast to the Gaussian chain result. They also observed that the width of the bond orientation profile is of the order of the persistence length, which is much larger than $w$ in that limit. Thus the interfacial width $w$ and the persistence length constitute two independent length scales of the interfacial profiles. A reduction of the interfacial width in the case of small bending rigidities was obtained numerically by Schmid and Müller[@SCHMID1]. They noted that the local structure might become important if its length scale is comparable to the interfacial width; a situation which occurs at rather large incompatibilities. In the present study we extend our Monte Carlo studies[@M1] of structural asymmetric blends to the investigation of interfacial properties between well segregated phases of flexible and semi-flexible polymers. We consider rather small bending rigidities of the semi-flexible component, so that the long distance behavior of both species is Gaussian. However, we chose the incompatibility $\chi$ high enough, such that the interfacial width and the persistence length are comparable for the higher bending rigidities. The Monte Carlo simulations highlight the architectural influences and give a detailed picture of interfaces between structural asymmetric polymers. They yield density and orientation profiles for bonds and chains as a whole. Extracting an effective Flory-Huggins parameter $\chi$ from the simulation data, we compare our Monte Carlo results to self-consistent field calculations which take due account of the chain architecture via a partial enumeration procedure[@SZLEIFER; @M2; @M2A], and to Gaussian chain results. Therefore we can assess the importance of the level of coarse graining on the interfacial properties. Our paper is organized as follows: In the next section we describe our polymer model, especially the dependence of single chain properties on the stiffness. We comment on some computational aspects of the Monte Carlo simulations and describe the measurement of the interfacial tension. We also introduce the salient features of our self-consistent field calculations for arbitrary molecular architecture. In the following, we present our simulational results and compare them to the self-consistent field calculations. We close with a brief discussion of our findings and an outlook on future work. Model and technical details ============================= Bond fluctuation model and single chain properties -------------------------------------------------- In the framework of our coarse grained lattice model, a small number of chemical repeat units, say 3-5, is mapped onto a lattice monomer, such that the relevant features - chain connectivity and excluded volume interaction between monomeric units - are retained. We use the three dimensional bond fluctuation model (BFM)[@BFM], which has found widely spread application in computer simulations, because it combines the computational efficiency of lattice models with a rather faithful approximation of continuous space properties. Each effective monomer blocks a cube of 8 neighboring sites from further occupancy on a simple cubic lattice. Due to the extended monomer size, the model captures some nontrivial packing effects. We consider a blend of $n_A$ flexible polymers of length $N_A$ and $n_B$ semi-flexible B-polymers comprising $N_B$ monomers in a volume $V$. At a total monomer density $\Phi_0 = (N_An_A+N_Bn_B)/V= 0.5/8$, the model reproduces many properties of a dense polymeric melt. We use chain lengths $N=N_A=N_B=32$ and $64$, which correspond to a degree of polymerization of the order 120 and 240 in more chemically realistic polymer models. Monomers are connected via one of 108 bond vectors with lengths $2,\sqrt{5},\sqrt{6},3$ or $\sqrt{10}$, where here, and henceforth, all lengths are measured in units of the lattice spacing. The large number of bond vectors permits 87 different bond angles. The persistence length of the semi-flexible B-polymers is tuned by imposing an intermolecular potential, which favors straight bond angles. We use a particular simple choice[@M1]: $E(\theta) = f k_BT \cos(\theta)$ where $\theta$ denotes the complementary angle to two successive bonds. Previous Monte Carlo simulations[@M1] of the bulk thermodynamics for $N=32$ and $f=1.0$ revealed a purely entropic Flory-Huggins parameter $\Delta \chi = 0.0018(2)$ for the athermal blend. This small value is in good quantitative agreement with theories[@LIU; @SCHWEIZER]. These packing effects result in a slight increase of the osmotic pressure with the bending energy, which gives rise to a monomer density difference of about $1\%$ between the coexisting phases. Since $\Delta \chi \ll 2/N = 0.0625$ for this combination of chain length and stiffness disparity, we introduce an additional enthalpic repulsion to induce phase separation. For simplicity, these thermal interactions are modeled as a square well potential comprising all 54 neighbor sites up to a distance $\sqrt{6}$. The contact of monomers of the same species lowers the energy by $\epsilon k_BT$, whereas the contact of different monomers increases the energy by the same amount. A finite size scaling analysis yields accurate estimates for the critical point of the binary blends ($N=32$): $\epsilon_c=0.01442(6), \phi_{Ac}=0.5$ and $\epsilon_c=0.0127(1), \phi_{Ac}=0.516(10)$ for $f=0$ and $1$[@M1], respectively. In the present study we chose $\epsilon=0.05$ which corresponds roughly to $\chi \approx 0.27$. This value is much higher than typical values for polyolefin blends[@BATES]. Our results correspond to rather strongly immiscible blends (e.g. interfaces between polystyrene (PS) and polyvinylpropylene PVP [@FRIEND4]). The conformational data for $N=32$ and $\epsilon=0.05$ as a function of the bending energy $f$ are presented in Fig. \[fig:conf\] and Table \[tab:konf\]. The inset shows the growth of the chain extension upon increasing $f$. The ratio between the square end-to-end distance $R^2$ and the square radius of gyration $R_g^2$ remains very close to the Gaussian value $6$ (within $5\%$ even for $f=2$). Also the small wave vector regime of the single chain structure function $$S(q)=\frac{1}{N} \left \langle \left| \sum_{i=1}^{N} \exp(i\vec{q}\vec{r}_i)\right|^2 \right \rangle$$ is well describable by a Debye function $S_D(q)/N = 2[\exp(-q^2R_g^2)-1+q^2R_g^2]/(q^2R_g^2)^2$[@FLORY] for $q<0.3$. Thus the [*long*]{} range behavior of our chains is characterized by Gaussian statistics for all values of the bending energy $f$ studied and we define the statistical segment length $b$ according $b^2 = R^2/(N-1)$. Note that the statistical segment length grows from 3.06 for $f=0$ to $4.63$ for $f=2$. This asymmetry in the statistical segment length is of similar magnitude as in polyolefin blends[@BATES]. However, for length scales of the order of the statistical segment length, we find deviations from the Gaussian behavior. The plateau $q^2S(q)=12/b^2$ for large $q$ in the Kratky Porod plot is only observed for flexible chains ($f=0$) and yields a slightly higher estimate for the statistical segment length $b=3.4$. For the semi-flexible chains the slope of $q^2S(q)$ in the range $0.3<q<1$ increases upon increasing the bending energy. Defining an effective bending rigidity of an equivalent worm-like chain $\kappa = R^2/2\langle b^2\rangle (N-1)$ ($\langle b^2\rangle$: mean squared bond length), $\kappa$ grows from 0.68 to 1.57 ($\kappa\chi = 0.18 \cdots 0.47$) upon increasing the bending energy $f$. For wave vectors $q R_g\approx 2\pi R_g/2 w$ (denoted by the arrows in the Fig. \[fig:conf\]), where $w=3.4$ corresponds roughly to the width of the monomer density profile in the self-consistent field (SCF) calculations, we find deviations from the Gaussian behavior for higher bending energies and anticipate corrections to the predictions of the Gaussian model. Local fluid structure and effective Flory-Huggins parameter $\chi$ ------------------------------------------------------------------ In order to compare our simulational results to self-consistent field (SCF) calculations, which cannot account for the local fluid structure of our model, we have to identify an effective Flory-Huggins parameter $\chi$. For the bulk behavior in the one phase region this has been discussed in ref. [@M1; @M0]: We define a dimensionless monomer density $\phi_{A(B)}$ as the ratio between the local number density of A(B)-monomers and the total monomer density $\Phi_0$. Then, the density of intermolecular contacts $n_{AB}$ takes the form: $$\frac{2n_{AA}}{\Phi_0\phi_A^2} = \Phi_0 \int_{r \leq \sqrt{6}}d^3r\; g_{AA}(r) \equiv z_{AA} \qquad \mbox{and} \qquad \frac{ n_{AB}}{\Phi_0\phi_A\phi_B} = \Phi_0 \int_{r \leq \sqrt{6}}d^3r\; g_{AB}(r) \equiv z_{AB}$$ where $g_{IJ}$ denotes the $IJ$ interchain correlation function, which is normalized such that $g_{IJ}(r \to \infty)=1$. The integration is extended over the spatial extension of the square well potential and $z_{IJ}$ corresponds to the effective coordination number of the Flory-Huggins lattice. If the coupling between chain conformations and effective monomer repulsion is negligible, only the [*inter*]{}molecular energy drives the phase separation. In this case (as we shall see in the next subsection), the $\chi$ parameter takes the form: $\chi=\epsilon(z_{AA}+2z_{AB}+z_{BB})/2k_BT$, where $z_{IJ}$ denote the coordination numbers obtained from the [*inter*]{}molecular pair-correlation functions. At the critical temperatures the coordination numbers have been measured in the simulations at composition $\phi_A=\phi_B=1/2$[@M1]. From its very definition the Flory-Huggins parameter $\chi$ accurately describes the intermolecular interaction energy, and it agrees nicely with values obtained from the semi-grandcanonical equation of state and the estimate from the long wavelength behavior of the collective structure factor. It also yields estimates of the critical temperature, which agree with the Monte Carlo results up to $1/\sqrt{N}$ corrections due to composition fluctuations[@M0]. In the pure system, the intermolecular coordination number of the flexible component is lower than the corresponding value for the semi-flexible chains[@M1]. The number of intramolecular contacts[@C2] is higher for the flexible chains. Therefore, the $\chi$-parameter grows upon increasing $f$[@M1]. Due to the larger chain extension for the semi-flexible component, the correlation hole has a larger spatial extent, but is more shallow. The intermolecular pair correlation function is presented in the inset of Fig. \[fig:ginter\]. Due to the extended monomer size $g(r)$ vanishes for distances $r<2$. At short distances, the presence of single site vacancies introduces local packing effects, which gives rise to several neighbor shells in the fluid. The extended structure of the polymer manifests itself in a reduction of contacts with [*other*]{} chains on the length scale of the end-to-end distance. On short distances, the intermolecular pair-correlation function for the stiffer chains is larger than for the flexible ones. For flexible chains it is possible to separate the monomeric packing effect from the polymeric correlation hole by dividing $g(r)$ by its monomeric equivalent[@M0], which exhibits only packing effects. The ratio $g(r)/g_{N=1}(r)$ presents the conditional probability of finding a monomer of a different chain at a distance $r$, if there would be one in the monomer system. This ratio, presented in Fig. \[fig:ginter\], is a rather smooth function, indicating, that the chain connectivity hardly affects the monomeric packing. If the correlation hole would be characterized by a single length scale, i.e. the end-to-end distance $R$ in the Gaussian chain model, one expects a scaling behavior of the form: $$1-\frac{g(r)}{g_{N=1}(r)} = \frac{N}{R^3} f\left(\frac{r}{R}\right)$$ Such a scaling plot is shown in Fig. \[fig:ginter\]. The data collapse well for the different bending rigidities at large distances, whereas there are deviations for small distances. This is a further indication, that the chain structure is characterized by two independent length scales, the end-to-end distance and the persistence length. In the well segregated regime (far below the critical temperature), it is very difficult to measure the $AB$ intermolecular correlation function in the bulk. Therefore, unlike ref. [@M1], we make an [*additional*]{} ad-hoc assumption: $z_{AB}=(z_{AA}+z_{BB})/2$. For symmetric blends near the critical point P-RISM calculations[@YETH] predict that deviations from this behavior die out with growing chain length like $1/\sqrt{N}$. However, the validity of this random-packing like assumption for highly incompatible structural asymmetric blends is not obvious. We explore the interfacial structure by simulating a system in a $L\times L\times 2L$ geometry with $L=64$ and periodic boundary conditions in the canonical ensemble. The system contains two interfaces parallel to the $xy$ plane. The chain conformations are generated via local monomer displacements and slithering snake moves, which are applied at a ratio 1:3 (except for $f=0$, where only local monomer displacements were employed). The systems were equilibrated over 125,000 attempted local moves per monomer (AMM) and 375,000 slithering snake tries per chain (SS). Every 12,500 AMM and 37,500 SS movements a configuration was stored for detailed analysis, at least 898 configuration were generated. We use a trivial parallelization strategy on a CRAY T3E, running typically 8 or 32 configurations in parallel. Profiles across the interface are measured according to the following procedures: “Apparent” profiles are obtained by locating the instantaneous position of the interface across the whole lateral system extension in each snapshot and averaging over profiles with respect to the instantaneous, but laterally averaged midpoint. These profiles exhibit a system size dependent broadening due to capillary fluctuations, which is not accounted for in the SCF calculations. To avoid this broadening, we define “reduced” profiles by laterally dividing the system into subsystems of size $B \times B$. We choose $B=16$. One could reduce the effect of capillary fluctuations further by chosing a smaller block size $B$, however, one should take care not to cut off “intrinsic” fluctuations[@SEM_CAP]. Since on the scale $B$ fluctuations are still reasonably described by a Helfrich Hamiltonian[@HELFRICH](see below), our block size $B$ is larger than the length scale of “intrinsic” fluctuations. This is consistent with Semenov’s[@SEM_CAP] estimate for the corresponding length scale $L_{\rm cutoff}=\pi w \approx 10<B$. Thus this averaging procedure reduces the influence of capillary fluctuations, but does not eliminate it completely[@AW]. The presence of an interface gives rise to a spatial dependence of the local monomer densities and chain conformations, which in turn is reflected in the intermolecular pair correlation functions. In Fig. \[fig:zcoord\] we present the reduced profiles of the intermolecular and intramolecular coordination numbers as a function of the distance from the center of the interface for the bending energies $f=0$ and $2$. The individual coordination numbers exhibit a considerable spatial dependence; this is however partially due to the spatial range of interactions and the remaining capillary fluctuations[@C1]. To illustrate the effect we plot the apparent and reduced profile of the AB intermolecular coordination number. The value at the center of the interface increases upon reducing $B$; the intrinsic value can not be estimated from these data with high precision. However, the average value of the “reduced” profile is close to $z_{AB}=(z_{AA}+z_{BB})/2$, the value used in the SCF calculations. The total number of intermolecular contacts $z^{inter}=(n_{AA}^{inter}+n_{BB}^{inter}+n_{AB}^{inter})/ \Phi_0(\phi_A+\phi_B)^2$ is much less sensitive to the intrinsic (local) profiles and shows a gradual transition between the corresponding bulk values, with a reduction at the center of the interface[@M4] of about $8\%$. This spatial dependence of the effective Flory Huggins parameter is neglected. Interestingly, the sum of all contacts $z^{all}$ (both intermolecular and intramolecular) is largely independent of the stiffness or the distance from the interface, i.e. the bending energy or the unfavorable interactions at the interface causes the chains to rearrange (e.g. exchange unfavorable interchain contacts by energetic favorable intrachain contacts) but hardly affect the structure of the underlying monomer fluid. Therefore, the local fluid structure is dominated by the packing constraints and the excluded volume interactions. The chain connectivity, bending energies, and the thermal interactions are of minor importance for the monomer fluid. The chain conformations are strongly influenced by the bending energies but depend only slightly on the thermal interactions. The Flory-Huggins parameter is determined by the thermal interactions and also depends on the bending energies via the correlation hole effect. The disparity in the packing behavior of the flexible and the stiff polymers is of minor importance for $\chi$ for the chain lengths studied. Measuring the surface tension via the capillary fluctuation spectrum -------------------------------------------------------------------- Due to the stiffness disparity between the species, straightforward application of semi-grandcanonical identity changes between different polymer types are rather inefficient (note that the efficiency drops by about 3 orders of magnitude[@M1] upon increasing $f$ from 0 to 1 for $N=32$) and therefore limited to small chain length and stiffness. Measurement of the interfacial tension via the reweighting of the composition distribution, which has been successfully applied to structural symmetric blends ($f=0$), is therefore difficult. In principle, the interfacial tension can be determined via the anisotropy of the pressure tensor. This method has been successfully applied in off-lattice simulations[@PRESSURE], but the generalization to lattice models is difficult[@CIFRA]. However, the spectrum of capillary fluctuations offers an alternative[@M3] for measuring the interfacial tension; a method which does not rely on identity switches. Let $u(x,y)$ be the local interfacial position. Then the free energy cost for deviations from a flat planar interface is given by the Helfrich expression[@HELFRICH]: $${\cal H} = \int dxdy\; \frac{\sigma}{2} (\nabla u)^2+ \cdots$$ where higher order gradient terms are neglected. In our simulation, we define local $x$- and $y$-averaged interface positions by minimizing the quantity $$\left| \sum_{z=u(y)-6}^{u(y)+6}\sum_{x=0}^{x=L-1} {\Large (}\phi_A(x,y,z)-\phi_B(x,y,z) {\Large )}\right|$$ for the $x$-averaged position $u(y)$ and a similar expression for the $y$-averaged one. This averaged interfacial position is Fourier decomposed according to: $ u(y)=\frac{a_0}{2} + \sum_{l=0}^{L/2} a(q_l)\cos(q_ly) + b(q_l)\sin(q_ly) $ with $q_l=2\pi l/L$. The Helfrich Hamiltonian predicts that the Fourier components $a(q_l)$ and $b(q_l)$ are Gaussian distributed with a width $$\frac{2}{L^2\left\langle a^2(q)\right\rangle} = \frac{2}{L^2\left\langle b^2(q)\right\rangle} = \frac{\sigma}{k_BT} q^2$$ In Fig. \[fig:sigma\_all\] we present the distribution of the Fourier components for two different bending energies $f=0,2$ and the 4 smallest wave vectors $q$. This long wavelength part of the fluctuation spectrum is well described by the quadratic Helfrich expression. The straight line marks the expected Gaussian distribution for the Fourier amplitudes, to which the simulation data comply. The inverse width of the distribution determines the interfacial tension. The extracted value for the symmetric blend agrees with the independent measurement obtained via the reweighting scheme[@M4]. (The latter scheme measures the interfacial free energy via the ratio of the probability for finding the system in a homogeneous bulk state or a configuration comprising two interfaces. ) To estimate the errors of measuring the interfacial tension via the capillary fluctuation spectrum, it would be desirable to increase the lateral system size. However, the error in extrapolating the simulation data to $q \to 0$ is smaller than $7\%$. Thus the analysis of the capillary fluctuation spectrum is an efficient alternative for measuring interfacial tensions in structurally asymmetric systems; the results are compared to the predictions of the SCF calculations in Sec. IIIa. Self-consistent field calculations ---------------------------------- The mean field approach is similar to Helfand[@HT; @HS], Noolandi[@NOOLANDI], and Shull[@SHULL], except for the treatment of the chain architecture[@M2]. The partition function of a binary polymer blend has the general form[@H75]: $${\cal Z} \sim \frac{1}{n_A!n_B!} \int \Pi_{\alpha=1}^{n_A} {\cal D}[r_\alpha] {\cal P}_A[r_\alpha] \Pi_{\beta=1}^{n_B} {\cal D}[r_\beta] {\cal P}_B[r_\beta] \exp \left( -\frac{\Phi_0}{k_BT} \int d^3r\; {\cal E}(\hat{\phi}_A,\hat{\phi}_B)\right)$$ where the functional integrals ${\cal D}[r]$ sum over all polymer conformations and ${\cal P}[r]$ denotes the probability distribution characterizing the noninteracting, single chain conformations. ${\cal E}$ represents a segmental interaction free energy, and the dimensionless monomer density takes the form[@H75]: $$\hat{\phi}_A(r) = \frac{1}{\Phi_0} \sum_{\alpha=1}^{n_A}\sum_{i_A=1}^{N_A} \delta(r-r_{\alpha,i_A})$$ where the sum runs over all monomers in the A-polymer $\alpha$. A similar expression holds for $\hat{\phi}_B(r)$. The segment free energy ${\cal E}$ comprises two contributions: a free volume part arising from hard core interactions and an energetic term from the thermal interactions. Since the melt is nearly incompressible, we approximate the free volume part by a simple quadratic expression introduced by Helfand[@HT], which reproduces the relative reduction of the total monomer density by about $4\%$[@SCHMID1]. However the difference of the bulk densities of the coexisting phases has a different sign in the simulations and than in the SCF calculations. In the simulations the higher osmotic pressure of the semi-flexible component results in a slightly lower bulk density of the semi-flexible component in the simulations, an effect neglected in the SCF calculations. Moreover, in the SCF calculations the more negative intermolecular energy density (see below) of the B component results in a slightly higher density of semi-flexible polymers. The total density differences between the coexisting phases is however only about $1\%$. The pairwise intermolecular interactions $V_{IJ}(r)$ ($I,J$=A,B) are treated as point interactions of strength $\epsilon z_{IJ} \delta(r)/\Phi_0$. $z_{IJ}$ parameterizes the local fluid structure of the underlying microscopic model, as discussed above. The coupling between individual chain conformations and the coordination numbers, which results in the spatial dependence of the $\chi$-parameter observed in the simulations, is neglected. Furthermore, we ignore purely entropic contributions (which have been determined to be small by Monte Carlo simulations) and do not include orientation dependent segmental interactions, which will eventually lead to a nematic phase at much higher bending energies $f$. Thus we take the interactions to be $$\frac{{\cal E}(\phi_A,\phi_B)}{k_BT} = \frac{\zeta}{2} \left( \phi_A + \phi_B - 1\right)^2 - \frac{\epsilon z_{AA}}{2} \phi_A^2 - \frac{\epsilon z_{BB}}{2} \phi_B^2 + \epsilon z_{AB} \phi_A \phi_B$$ The inverse compressibility $\zeta$ has been measured in simulations of the athermal model; $\zeta=4.1$[@WM1]. A Hubbard-Stratonovich transformation rewrites the many chain problem in terms of independent chains in external, fluctuating fields $W_A$ and $W_B$. $${\cal Z} \sim \int {\cal D}[W_A,W_B,\Phi_A,\Phi_B] \exp \left(-{\cal F}[W_A,W_B,\Phi_A,\Phi_B]/k_BT\right)$$ where the free energy functional is defined by $$\begin{aligned} \frac{{\cal F}[W_A,W_B,\Phi_A,\Phi_B]}{\Phi_0 k_BT V} &=& \frac{\bar{\phi}_A}{N_A} \ln \bar{\phi}_A + \frac{\bar{\phi}_B}{N_B} \ln \bar{\phi}_B + \frac{1}{V} \int d^3r \; {\cal E}(\Phi_A,\Phi_B) \nonumber \\ && - \frac{1}{V} \int d^3r \left\{ W_A\Phi_A + W_B\Phi_B \right\} - \frac{\bar{\phi}_A}{N_A} \ln q_A[W_A] - \frac{\bar{\phi}_B}{N_B} \ln q_B[W_B]\end{aligned}$$ $\bar{\phi}_A = \frac{n_A N_A}{\Phi_0 V} = 1 -\bar{\phi}_B$ denotes the average A-monomer density and $q_A[W_A]$ the single chain partition function in the external field $W_A$ $$q_A[W_A] = \frac{1}{V} \int {\cal D}_1[r] {\cal P}_A[r] \exp \left(- \Phi_0 \int d^3r\; \hat{\phi}_A W_A \right)$$ respectively. The leading contributions to the partition function stem from those values $\phi_A,\phi_B,w_A,w_B$ of the collective variables which extremize the free energy functional, and the mean field approximation amounts to retaining only these contributions. The values are determined by: $$\begin{aligned} \frac{\delta {\cal F}}{\delta \phi_A} = 0 &\Rightarrow & w_a = \frac{\delta}{\delta \phi_A} \int d^3r\; {\cal E}(\phi_A,\phi_B) = \zeta (\phi_A + \phi_B -1) - \epsilon z_{AA} \phi_A + \epsilon z_{AB} \phi_B \\ \frac{\delta {\cal F}}{\delta w_A} = 0 &\Rightarrow & \phi_A = \frac{\bar{\phi}_A V}{N_A q_A} \frac{\delta q_A}{\delta w_A} \label{eq:d}\end{aligned}$$ and similar expressions for $w_B$ and $\phi_B$. The saddle point integration approximates the original problem of mutually interacting chains by one of a single chain in an external field, which is determined, in turn, by the monomer density. Composition fluctuations are ignored, but the coupling between chain conformations (e.g. orientations) and the monomer density is retained. The free energy of a homogeneous system takes the Flory-Huggins form: $$\frac{{\cal F}}{\Phi_0k_BTV} = \frac{\bar{\phi}_A}{N_A}\ln\left(\bar{\phi}_A\right) +\frac{1-\bar{\phi}_A}{N_B}\ln \left(1-\bar{\phi}_A\right) -\frac{1}{2}\epsilon\left\{ \left( z_{AA}+2z_{AB}+z_{BB}\right)\bar{\phi}_A^2 -2\left(z_{AB}+z_{BB}\right)\bar{\phi}_A +z_{BB} \right\}$$ where we identify the Flory-Huggins parameter $\chi=( z_{AA}+2z_{AB}+z_{BB} )\epsilon/2$. In the strongly segregated regime, the free energy of a system containing one interface is given by: ${\cal F}=-\epsilon(\bar{\phi}_Az_{AA}+(1-\bar{\phi}_A)z_{BB})\Phi_0k_BTV/2+\sigma k_BTL^2$. The definition of the interfacial tension as the difference of the free energy of a system containing an interface and the homogeneous bulk system corresponds literally to the measurement of the interfacial tension via the reweighting scheme[@M4] in the Monte Carlo simulations. As shown in Sec. IIc for the symmetric blend, these values agree with the measurement of the interfacial tension via the capillary fluctuation spectrum, so that we can compare the results of the SCF calculations and the values extracted from the capillary wave spectrum quantitatively. For the special cases of Gaussian chains[@HT; @EDWARDS] and worm-like polymers[@WORM; @MORSE] one can treat the single chain problem in an arbitrary external field in limiting cases (e.g $N \to \infty$) analytically. For general parameters, however, one has to resort to numerical procedures even for these simple models. The BFM chains used in the simulations are characterized by structure on different length scales. The conformations are rod-like for length smaller than the persistence length, which depends on the bending energy $f$. On intermediate length scales, they obey self-avoiding walk statistics, while on the largest scale, the excluded volume interactions are screened in the melt, and the chains exhibit Gaussian statistics. Since we want to explore dependence on the explicit chain structure, we evaluate the single chain partition function via a partial enumeration scheme, introduced by Szleifer and coworkers[@SZLEIFER]. The method is conceptually straightforward and applicable to [*arbitrary*]{} architecture[@M2; @M2A]. It can use experimental or simulational data as input. Note that no adjustable parameters are involved in the chain structure (such as the statistical segment length in the Gaussian model or the bond length and the bending rigidity in the worm-like polymer model) and the chain structure is correctly represented on [*all*]{} length scales. Using Monte Carlo simulations of the pure melt, we generated 40,960 independent polymer conformations for each bending energy. Rotating and translating those original conformations, we obtain a sample of 7,864,320 polymer conformations for chain length $N=32$. (Note only the $z$ coordinates of the chains are employed for a flat interface parallel to the $xy$ plane.) For $N=64$ we use twice as many conformations. Within this framework, the A-monomer density (c.f. eq. \[eq:d\]) is simply the statistical average of independent A-polymers in the external field $w_A$: $$\phi_A = \bar{\phi}_A \frac{ \sum_{\alpha=1}^{C} \frac{1}{N_A} \sum_{i=1}^{N_A} V \delta(r-r_{\alpha,i}) \exp \left( -\sum_{i=1}^{N_A} w_A(r_{\alpha,i}) \right) } { \sum_{\alpha=1}^{C} \exp \left( -\sum_{i=1}^{N_A} w_A(r_{\alpha,i}) \right) }$$ Other single chain quantities are given by corresponding averages over independent chains in the fields $w_A$ and $w_B$. The set of nonlinear equations is expanded in a Fourier series[@MATSEN] and solved by a Newton-Raphson like method. Convergence is usually reached within 3-6 steps. The evaluation of the partition function [@M2A] in the external fields poses rather high memory demands (several Gbytes). Therefore we employ a CRAY T3E, assigning a subset of conformations to each processing element. Typically we use 64 or 128 processors in parallel, and the program scales very well with the number of processors employed[@M2A]. One needs about 1200 seconds for each set of parameters, which is roughly 2 orders of magnitude less than for the detailed Monte Carlo simulations. Comparison between Monte Carlo simulations and self-consistent field (SCF) calculations ========================================================================================= In the following we compare our Monte Carlo simulations to the results of the SCF calculations. Both, large length scale thermodynamic properties (e.g. interfacial tension) as well as the local interfacial structure (e.g. orientation of individual bonds) are investigated. The temperature dependence of most quantities for symmetric blends ($f=0$) has been studied previously[@M4] and compared to predictions of the Gaussian and worm-like chain model[@SCHMID1]. The results for vanishing bending energy compare well to our calculations. Interfacial tension ------------------- The interfacial tension $\sigma$ between the coexisting phases has an important impact on the morphology of the compound material[@MORPHOLOGY]. The control of domain size and shape is a key to tailoring the application properties of the blend. The size of minority droplets often is the smaller, the smaller the interfacial tension between the coexisting phases[@MORPHOLOGY; @MILNER]. In the strong segregation limit, Helfand and Sapse[@HS] obtained for infinite long Gaussian chains in an incompressible blend the analytic expression: $$\sigma = \Phi_0 \sqrt{ \chi /6 } \left( \frac{2}{3} \frac{b_A^2+b_A b_B+b_B^2}{b_A+b_B} \right)$$ The interfacial tension $\sigma$ grows upon increasing the bending energy (i.e. the statistical segment length) of the semi-flexible component. This behavior is presented in Fig. \[fig:Ssigma\], as well as our simulation results and the SCF calculations, which take account of the detailed chain architecture. All data exhibit an increase of the interfacial tension of about $30\%$. The simulation data and the SCF calculations agree nicely on the growth of the interfacial tension upon increasing the bending energy. The almost quantitative agreement indicates that our identification of the $\chi$-parameter yields reasonable results for structural asymmetric mixtures. However, the Gaussian chain result is about a factor $1.3$ higher than the simulation data. Recently, Ermoshkin and Semenov[@ER] calculated corrections to the interfacial tension due to effects of finite chain length $N$. For symmetric blends, they found that chain end effects reduce the interfacial tension by a factor $(1-4\ln2/\chi N) \approx 0.67$, which accounts well for the discrepancies between the Helfand-Sapse result and the Monte Carlo data. Similar reductions are found in numerical SCF calculations[@SCHMID1; @SHULL]. Note that purely entropic, packing contributions to the Flory-Huggins parameter $\Delta\chi$ are less than $1\%$ of the total $\chi$ value for $f=1$. That is somewhat smaller than the uncertainties in identifying the enthalpic contributions of $\chi$ and the accuracy of our interfacial tension measurement in the simulations. Therefore, purely entropic effects derived from packing are irrelevant to the interfacial behavior for the chain lengths, stiffness asymmetries, and temperatures investigated in the present study. Monomer density profiles ------------------------ Another important characterization of the interface are the density profiles of the individual components. Experiments[@ENTANGLE] indicate, that entanglements in the interfacial zone are of major importance for the mechanical properties of the blend. Of course, our chain lengths are too small to observe entanglements, however static properties can be extracted from our simulation data. The density profiles obtained from the SCF calculation are presented in Fig. \[fig:profile\], as well as the “apparent” profiles in the Monte Carlo simulation. The width of the apparent profile in the Monte Carlo simulations is about a factor 1.5 larger than the SCF result; this is not unexpected, because capillary fluctuations increase the squared width by a term proportional to $\ln(L)/\sigma$. However, the profiles are qualitative similar: both data show a reduction of the total monomer density at the center of the interface (the relative reduction is roughly $\chi/2\zeta$[@M4]) and almost no dependence on the bending energy $f$. The dependence of the interfacial width on the bending energy is shown in Fig. \[fig:width\][@C3]. The width $w_r$ of the reduced profile is smaller than the apparent width $w_a$, and agrees better with the SCF results. Due to remaining capillary wave effects it is an upper bound on the intrinsic width. All profiles presented below are obtained by the reduced averaging procedure. The excess energy density of the interface can also be used to estimate the intrinsic width. Since the relative increase in interfacial area due to capillary fluctuations is of the order $\sigma\ln L/L^2$, this quantity (as well as the interfacial tension) is not strongly affected by fluctuations of the local interfacial position. A tanh-shaped profile $\phi_A = \frac{1}{2}(1+\tanh\frac{z}{w})$ yields in the SCF framework: $$\frac{e_s}{k_BT} = \Phi_0 \int dz\; \left\{ -\frac{\epsilon z_{AA}}{2}\phi_A^2 -\frac{\epsilon z_{BB}}{2}\phi_B^2 + \epsilon z_{AB}\phi_A\phi_B \right\} -\frac{\Phi_0 L (z_{AA}+z_{BB})}{4} \approx \frac{1}{2} w_e\Phi_0\chi$$ where we have assumed incompressibility and neglected the finite range of interactions and any contribution of the intramolecular interactions to the excess energy density. Of course, this measure relies crucially on the identification of the Flory-Huggins parameter. However, the method is computational very convenient and can be combined[@M3] with the reweighting methods of measuring interfacial tensions. It results in values which are between the reduced width and the SCF results, which shows again the consistent parameterization of the local fluid structure. The Gaussian chain model for $N \to \infty$ predicts for symmetric blends ($f=0$) a width which is about $20\%$ smaller than the SCF result. SCF calculations[@SCHMID1] of Gaussian chains with the same long distance behavior and which include chain end effects and the finite compressibility agree within $2\%$ with our results for symmetric, flexible mixtures. An increase of the chain length from $N=32$ to $64$ reduces the effective $\chi$-parameter by $4\%$ and reduces the broadening due to finite chain length effects. The latter effect is stronger, such that the width decreases slightly. Most notably, the apparent width of the Monte Carlo data, the energetic width $w_e$ and the results of the SCF calculations show almost no dependence on the bending energy $f$, whereas the analytic expression obtained by Helfand and Sapse $$w = \sqrt{ \frac{b_A^2+b_B^2}{12 \chi} }$$ predicts an increase of about $28\%$ due to the variation of $b_B$. Taking account of the stiffness dependence of the effective Flory-Huggins parameter $\chi$, the formula above predicts an increase of $21\%$. Qualitatively, calculations for worm-like chains[@MORSE; @SCHMID1] indicate that increasing the bending rigidity results in a reduction of the interfacial width compared to the Gaussian chain result. For the present combination of parameters, both effects seem to cancel, resulting in an interfacial width, which is nearly independent of the bending rigidity. For lower incompatibilities, the interfacial width is larger, the Gaussian description on the length scale of the interfacial width becomes more appropriate. Therefore the difference between the width in the flexible/semi-flexible blend and the width in the symmetric flexible mixture increases in accord with the Helfand Sapse description. This is confirmed by SCF calculations (c.f. Fig. \[fig:dw\]), where we have assumed that the effective coordination numbers are temperature independent. However, upon increasing the incompatibility further ($\epsilon>0.082$), one finds that an increase of the statistical segment length results in a [*smaller*]{} interfacial width of the asymmetric blend in qualitative contrast to the predictions of the Gaussian model. Distribution of chain ends and orientations ------------------------------------------- The enrichment of chain ends at the center of the interface[@M4] and at hard walls[@SKKUMAR; @BITSANIS; @SCHMID; @JOERG] has attracted abiding interest. Chain ends are important for the interdiffusion and healing properties at interfaces between long polymers[@WU]. They also play an important role for reactions at interfaces. In many experimental systems, chain ends have slightly different interactions than inner chain segments,which might result in a modification of the interfacial properties. On the theoretical side, the behavior of chain ends is related to corrections to the ground state approximation. Therefore it is a sensitive test for a quantitative theoretical description. Chain end effects give rise to large corrections to the interfacial width and tension, and they also play an important role for long range interactions between interfaces[@ER]. The distribution of chain ends for symmetric blends has been investigated by Monte Carlo simulations[@M4], and in the framework of SCFT for Gaussian chains[@WU; @SCHMID1]. In Fig. \[fig:end\] the simulational results and the SCF calculations are presented; both agree almost quantitatively. As in symmetric blends, chain ends are enriched at the center of the interface, and this effect goes along with a depletion away from the interface. The fact that the depletion zone in the wings shifts outwards with increasing chain length, indicates that the length scale of the rearrangement of chain ends is the radius of gyration. A-polymers stick their ends into the B-rich phase and vice versa. The effect on the semi-flexible chains becomes more pronounced with growing stiffness, while the A-polymers are hardly influenced by the stiffness of the B-polymers. The instantaneous shape of a polymer coil is a prolate ellipsoid[@M4]. Polymers orient themselves by putting their ends preferentially at the center of the interface. This is quantified by the orientational parameter[@M4] for the end-to-end vector (cf. Fig. \[fig:qe\]): $$q_e(z) = \frac{3\langle R^2_z\rangle_z-\langle \vec{R}^2\rangle_z}{2\langle \vec{R}^2\rangle_z}$$ where the outer index $z$ at the brackets denotes the z coordinate of the midpoint of the end-to-end vector $\vec{R}$, and the inner indices its Cartesian components. The chains align their two long axis parallel to the interface in their majority phases, similar to the behavior at a hard wall. The chain orientation of semi-flexible polymers increases for growing stiffness, while the flexible A-polymers are not affected. The agreement between Monte Carlo simulation and SCF calculations is again almost quantitative. In the SCF framework, the orientations of the chains in the minority phase is accessible. The polymers align perpendicular to the interface, as to reach with one end their corresponding bulk phase. The length scale of the ordering increases with the bending energy $f$ and with chain length $N$. The orientation of individual bond vectors $q_b$ shows a similar behavior. Bonds align parallel to the interface; the effect for the semi-flexible component grows with increasing bending energy and its range is largely independent of the chain length. The agreement between simulations and SCF calculations is very good. The Gaussian chain model cannot predict any nonzero orientation of the bonds. The orientation of bonds in our model is, in fact, much smaller than for the end-to-end distance[@M4]. In contrast to the width of the density profile, the spatial range over which the orientation of bonds extends grows upon increasing the bending energy. Therefore, the orientational width and the width of the composition profile are two independent microscopic length scales. Conclusions and outlook ======================== In summary, we have presented extensive simulations of highly incompatible polymers with different stiffness. The local structure of the interface has been characterized by density profiles of different monomer species and chain ends and orientational profiles of whole chains and individual bonds. The interfacial tension has been measured via analyzing the spectrum of capillary fluctuations. Using the pair correlation functions of the pure components and a random-packing like assumption for the intermolecular contacts between different species, we have extracted an effective Flory-Huggins parameter, which takes account of the stiffness dependence of the structure of the polymeric fluid. The effective Flory-Huggins parameter grows upon increasing the stiffness, because back folding is less probable and the number of intermolecular contacts increases respectively. This effective Flory Huggins parameter was then employed in SCF calculations, as well as the chain conformations in the pure melt. These calculation incorporate the chain structure on [*all*]{} length scales via a partial enumeration scheme; there is no free parameter in describing the chain architecture. Using the detailed local structure of the bulk (as obtained by simulations) in the SCF calculations, we predict the interfacial properties. Monte Carlo results and SCF calculations for the interfacial tension, the excess interfacial energy, the redistribution of chain ends and orientations of whole chains and individual bonds agree very well provided that the analysis accounts for capillary fluctuations. However, comparing our results to the analytical predictions of the Gaussian chain model for infinite chain length, we find qualitative deviations, especially for the dependence of the interfacial width on the chain stiffness. This finding might be important for extracting the Flory-Huggins parameter from interfacial profiles in highly incompatible polymer blends. Therefore, our results emphasize that the local structure, both of the underlying monomer fluid and of the chain architecture, is important for a quantitative description. The radius of gyration determines the range of orientation of whole chains and the distribution of chain ends. Furthermore, we identify two independent microscopic length scales of the interfacial profile; one controls the width of the monomer density profile, the other corresponds to the range of orientations. This behavior resembles the findings in symmetric blends of worm-like chains in the limit $\kappa\chi \gg 1$[@MORSE] and the behavior of a homopolymer melt at a hard wall which is the limiting case for infinite incompatibility. However, in the present study this behavior is found for a different model which can be described neither by Gaussian nor by worm-like statistics on small length scales. Deviations from the Gaussian model occur under rather mild conditions which correspond roughly to $\kappa\chi=0.18 \cdots 0.47$ in the equivalent worm-like chain model. Furthermore our self consistent field approach as well as the simulation techniques are applicable to arbitrary chain architecture[@M2]. Assuming that the chain conformations and the local fluid structure are approximately independent of temperature, we have extended the self consistent field calculations to other incompatibilities. The results indicate that chain architecture becomes important when its length scale is comparable with the interfacial width. At very high incompatibility, increasing the stiffness of the semi-flexible component results in a decrease of the interfacial width. However, the Gaussian chain results and our calculations, which take account of the explicit chain architecture on all length scales, agree better for lower incompatibilities, where the interfacial width is much larger than the persistence length. Acknowledgment {#acknowledgment .unnumbered} -------------- It is a great pleasure to thank K. Binder, G.S. Grest and F. Schmid for helpful and stimulating discussion, and M. Schick for critical reading of the manuscript. Generous access to the CRAY T3E at the San Diego Supercomputer Center (through a grant to M. Schick) is also gratefully acknowledged. M.M. thanks the Bundesministerium für Forschung, Technologie, Bildung und Wissenschaft(BMBF) for support under grant No. 03N8008C. A.W. thanks the Deutsche Forschungsgemeinschaft for support under grant number Bi 314/3. [99]{} D.S. Walsh, J.S. Higgins, A. Maconnachie (eds.) [*Polymer Blends and Mixtures*]{}, Martinus Nijhoff, Dordrecht (1985). L.A. Kleintjens and P.J. Lemstra (eds.) [*Integration of Fundamental Polymer Science and Technology*]{}, Elsevier, Amsterdam (1986). M. Stamm and D.W. Schubert, Ann. Rev. Mat. Sci [**25**]{}, 326 (1995). D.W. Schubert, V. Abetz, M. Stamm , T. Hack, W. Siol, Macromolecules [**28**]{}, 2519 (1995). B. Löwenhaupt and G.P. Hellmann, Colloid & Polymer Sci. [**268**]{}, 885 (1990). H.E. Hermes, J.S. Higgins and D.G. Bucknall, Polymer [**38**]{}, 985 (1997). K.H. Dai and E.J. Kramer, Polymer [**35**]{}, 157 (1994). [*ibid*]{} Macromolecules [**25**]{}, 220 (1992). D.T. Wu, G.H. Fredrickson, and J.P. Carton, J.Chem.Phys [**104**]{}, 6387 (1996). D.T. Wu, G.H. Fredrickson, J.P. Carton, A. Ajdari, and L. Leibler, J.Poly.Sci [**33B**]{}, 2373 (1996). F.S. Bates and G.H. Fredrickson, Macromolecules [**27**]{}, 1065 (1994). F.S. Bates, M.F. Schulz, and J.H. Rosedale, Macromolecules [**25**]{}, 5547 (1992). W.W. Graessley, R. Krishnamoorti, G.C. Reichart, N.P.Balsara, L.J. Fetters, and D.J. Lohse, Macromolecules [**28**]{}, 1260 (1995) R. Krishnamoorti, W.W. Graessley, L.J. Fetters, R.G. Garner, and D.J. Lohse, Macromolecules [**28**]{}, 1252 (1995). G.H. Fredrickson, A.J. Liu, and F.S. Bates, Macromolecules [**27**]{}, 2503 (1994). G.H. Fredrickson and A.J. Liu, J.Polym.Sci. [**B33**]{}, 1203 (1995). C. Singh and K.S. Schweizer, J.Chem.Phys. [**103**]{}, 5814 (1995). C. Singh and K.S. Schweizer, Macromolecules [**30**]{}, 1490 (1997). J.D. Weinhold, S.K. Kumar, C. Singh and K.S. Schweizer, J.Chem.Phys. [**103**]{}, 9460 (1995). K.F. Freed and J. Dudowicz, Macromolecules [**29**]{}, 625 (1996). M. Müller, Macromolecules [**28**]{}, 6556 (1995). E. Helfand and A.M. Sapse, J.Chem.Phys. [**62**]{}, 1329 (1975). A.I. Pesci and K.F. Freed, J.Chem.Phys. [**90**]{}, 2017 (1989). N. Saito, K. Takahashi, and Y. Yunoki, J.Phys.Soc.Jpn. [**22**]{}, 219 (1967). D.C. Morse and G.H. Fredrickson, Phys.Rev.Lett. [**73**]{}, 3235 (1994). F. Schmid and M. Müller, Macromolecules [**28**]{}, 8639 (1995). I. Szleifer, A. Ben-Shaul, and W.M. Gelbhart, J.Chem.Phys. [**85**]{}, 5345 (1986), [*ibid.*]{} [**86**]{}, 7094 (1987). M. Müller and M. Schick, Macromolecules [**29**]{}, 8900 (1996). M. Müller and M. Schick, in preparation. I. Carmesin, K. Kremer, Macromolecules [**21**]{}, 2819 (1988). H.P. Deutsch and K. Binder, J.Chem.Phys. [**94**]{}, 2294 (1991). P.J. Flory, Statistical Mechanics of Chain Molecules, Interscience Publishers, N.Y. (1969). M. Müller and K. Binder, Macromolecules [**28**]{}, 1825 (1995). The correlation hole effect also results in a reduction of the effective coordination number with increasing chain length[@M0]. A. Yethiraj and K.S. Schweizer, J.Chem.Phys. [**97**]{}, 5927 (1992). K.S. Schweizer and A. Yethiraj, J.Chem.Phys. [**98**]{}, 9053 and 9080 (1993). A.N. Semenov, Macromolecules [**26**]{}, 6617 (1993). [*ibid.*]{} [**27**]{}, 2732 (1994). W. Helfrich, Z.Naturforsch. [**28c**]{}, 693 (1973). P.B. Canham, J.Teor.Biol. [**26**]{}, 61 (1970). E. Evans, Biophys.J. [**14**]{}, 923 (1974). A. Werner, F. Schmid, M. Müller, and K. Binder, in preparation. The reduction of the measured AB coordination number upon increasing the lateral system size can be rationalized as follows. Assume, that the [*intrinsic*]{} coordination number $z_{\rm intr}$ has no spatial dependence, the measured value has the form: $$\langle z \rangle = z_{\rm intr} \frac{\langle \phi_A \phi_B\rangle} {\langle \phi_A \rangle\langle \phi_B \rangle}$$ The average in the simulation corresponds to a folding with the distribution of the deviation of local interfacial position from its lateral mean[@SEM_CAP]. The effect of this folding becomes the smaller the smaller the lateral system size is. Right at the center of the interface $\phi_A$ and $\phi_B$ have almost no curvature, whereas the product $\phi_A\phi_B$ has a large negative curvature (a maximum). Therefore $\langle \phi_A \phi_B\rangle < \langle \phi_A \rangle\langle \phi_B \rangle$, i.e. the measured value is smaller than the intrinsic one. M. Müller, K. Binder, and W. Oed, Faraday Trans. [**91**]{}, 2369 (1995). G.S. Grest, M.D. Lacasse, K. Kremer, and A.M. Gupta, J.Chem.Phys. [**105**]{}, 10583 (1996). G.S. Grest, M.D. Lacasse, and M. Murat, MRS Bulletin [**22**]{}, 27 (1997). R.B. Pandey, A. Milchev, and K. Binder, Macromolecules [**29**]{},1194 (1997). G.J.A. Ypma, P. Cifra, E. Nies, and A.R.D. van Bergen, Macromolecules [**29**]{}, 1252 (1996). P. Cifra, E. Nies, and F.E. Karasz, Macromolecules [**27**]{}, 1166 (1994). M. Müller and M. Schick, J.Chem.Phys. [**105**]{}, 8885 (1996), [*ibid.*]{} [**105**]{}, 8282 (1996). E. Helfand and Y. Tagami, J.Polym.Sci.,Polym.Lett. [**9**]{}, 741 (1971). E. Helfand and Y. Tagami, J.Chem.Phys. [**56**]{}, 3592 (1972). J. Noolandi and K.M. Hong, Macromolecules [**14**]{}, 727 (1981), [*ibid*]{} [**15**]{}, 483 (1982). K.R. Shull, Macromolecules [**26**]{}, 2346 (1993). E. Helfand, J.Chem.Phys [**62**]{}, 999 (1975). The compressibility of the athermal system has been measured in M. Müller and W. Paul, J.Chem.Phys. [**100**]{}, 719 (1994). $\zeta = - \Phi \frac{\partial^2s}{\partial \Phi^2}$, where $s$ denotes the entropy density. S.F. Edwards, Proc.Phys.Soc. [**85**]{}, 613 (1965). M.W. Matsen and M. Schick, Phys.Rev.Lett. [**74**]{}, 4225 (1995). L. Leibler, Macromol.Chem.Macromol.Symp. [**16**]{}, 1 (1988). S. Wu, Polym.Eng.Sci. [**27**]{}, 335 (1987). S.T. Milner, MRS Bulletin [**22**]{}, 38 (1997). A.V. Ermoshkin and A.N Semenov, Macromolecules [**29**]{}, 6294 (1996). A.N. Semenov, J.Phys.II [**6**]{}, 1759 (1997). C. Creton, E.J. Kramer, and G. Hadziioannou, Macromolecules [**24**]{}, 1846 (1991). J. Washiyama, C. Creton, and E.J. Kramer, Macromolecules [**25**]{}, 4751 (1992). Upon increasing the stiffness disparity, the profiles of our compressible blend become slightly asymmetric. Fitting a tanh-profile to the A- and B-monomer density, one finds that the B-profile is about $1\%$ wider ($f=2$) than the A-profile. This effect is observed in simulations and SCF calculations. C. Jalbert, J. Koberstein, A. Hariharan and S.K. Kumar, Macromolecules [**30**]{}, 4481 (1997). I. Bitsanis, G. Hadziioannou, J.Chem.Phys. [**92**]{}, 3827 (1990). F. Schmid, J.Chem.Phys. [**104**]{}, 9191 (1996). J. Baschnagel and K. Binder, Macromolecules [**28**]{}, 6808 (1995). $N$ $f$ $\langle b^2\rangle$ $R^2$ $R_g^2$ $z_{BB}$ $\langle e \rangle/k_BT$ $e_s/k_BT$ $w_a$ $w_r$ $w_e$ $w_{\rm scf}$ ----- ----- ---------------------- ------- --------- ---------- -------------------------- ------------ ------- ------- ------- --------------- 32 0.0 6.92 290.4 48.8 2.65 -0.00732 0.0290 4.77 3.88 3.51 3.11 0.5 6.88 350.1 58.1 2.84 -0.00731 0.0293 4.63 3.83 3.40 3.13 1.0 6.86 431.8 70.8 3.02 -0.00730 0.0300 4.57 3.80 3.38 3.16 1.5 6.84 536.9 86.5 3.17 -0.00730 0.0304 4.49 3.77 3.34 3.19 2.0 6.84 665.2 105.1 3.29 -0.00730 0.0310 4.53 3.75 3.34 3.23 64 0.0 6.92 609.3 101.7 2.53 -0.00732 1.0 6.86 987.3 148.3 2.92 -0.00730 0.0265 4.13 3.43 3.11 2.99 : Single chain conformations and interfacial properties as a function of the bending energy $f$. Interfacial data refer to blends of flexible ($f=0$) and semi-flexible ($f$ as indicated) chains. $\langle b^2\rangle$: mean squared bond length, $R^2$: mean squared end-to-end distance, $R_g^2$ mean squared radius of gyration, $z_{BB}$ effective intermolecular coordination number as measured in simulations of the bulk system, $\langle e \rangle/k_BT$: bulk energy density, $e_s/k_BT$: interfacial energy excess per unit area, $w_a$ apparent interfacial with for $L=64$, $w_r$ reduced interfacial width for block size $B=16$, $w_e$ estimated interfacial with from the excess energy, $w_{\rm scf}$ interfacial width in the SCF calculations.[]{data-label="tab:konf"}

--- abstract: 'We introduce new motivic invariants of arbitrary varieties over a perfect field. These cohomological invariants take values in the category of one-motives (considered up to isogeny in positive characteristic). The algebraic definition of these invariants presented here proves a conjecture of Deligne. Other applications include some cases of conjectures of Serre, Katz, and Jannsen on the independence of $\ell$ of parts of the étale cohomology of arbitrary varieties over number fields and finite fields.' address: - 'Department of Mathematics, University of Maryland, College Park MD 20742, USA' - | Max-Planck-Institut für Mathematik\ Vivatsgasse 7\ D-53111 Bonn, Germany author: - Niranjan Ramachandran title: 'One-motives and a conjecture of Deligne' --- \[section\] \[thm\][Lemma]{} \[thm\][Corollary]{} \[thm\][Proposition]{} \[thm\][Conjecture]{} \[thm\][Question]{} \[thm\][Remark]{} \[thm\][Definition]{} \[thm\][Example]{} \[thm\][Observation]{} [^1] $\frak{Wenn~die~K\ddot{o}nige~bau'n,}$ $\frak{haben~die~K\ddot{a}rrner~zu~tun.}$ $\frak{F.~Schiller}$ [**Introduction.**]{} P. Deligne [@h 10.4.1] has attached one-motives to complex algebraic varieties using the theory of mixed Hodge structures. He has conjectured that these one-motives admit a *purely algebraic* definition. The aim of this article is to prove his conjecture (Theorem \[peddha\]). Recall the well known result of Riemann [@h2 4.4.3], presented here in modern guise: the “Hodge realization" $T_{{{\mathbb{Z}}}}$ — this is $A \mapsto H_1(A, {{\mathbb{Z}}})$ — defines an equivalence from the category of complex abelian varieties to the category of torsion-free polarizable Hodge structures of type $\{(0,-1), (-1,0)\}$. In particular, any such Hodge structure arises as the $H_1$ of an essentially unique complex abelian variety. Deligne [@h §10.1] has introduced the algebraic notion of a one-motive over a field $k$, generalizing that of an abelian variety — §\[rev\] contains the precise definitions; he has also generalized Riemann’s result by showing that the “Hodge realization” $T_{{{\mathbb{Z}}}}$ defines an equivalence from the category of one-motives over ${{\mathbb{C}}}$ to the category of torsion-free mixed Hodge structures $H$ of type $$(*) \qquad {} \qquad {} \qquad {} \qquad \{(-1, -1), (-1, 0), (0, -1), (0,0)\} \qquad {}$$ with $Gr^W_{-1}H$ polarizable. Thus, any such mixed Hodge structure $H$ arises from an essentially unique one-motive $I(H)$ over ${{\mathbb{C}}}$. The functor $I$ is a quasi-inverse to $T_{{{\mathbb{Z}}}}$. For any complex variety $V$ and any integer $n \ge 0$, consider the largest mixed Hodge substructure $t^n(V)$ of type $(*)$ of $H^n(V, {{\mathbb{Z}}}(1))/{\rm torsion}$; there exists a well-defined one-motive $I^n(V)$ over ${{\mathbb{C}}}$ whose Hodge realization is $t^n(V)$; so $I^n(V):= I(t^n(V))$. Deligne [@h 10.4.1] has conjectured that $I^n(V)$ admits a purely algebraic definition. His proof (ibid. 10.3. — Interprétation algébrique du $H^1$ mixte: cas des courbes) of his conjecture for arbitrary curves suggests a precise formulation of the conjecture. Namely, we have the following (this formulation is due to the referee): \[dc1\] [(Deligne)]{} For an arbitrary variety $V$ over an arbitrary field $k$ and integer $n$, define a one-motive $L^n(V/k)$ and homomorphisms[^2] $$\begin{aligned} T_{\ell}(L^n(V/k)) & \to & H^n(V \times \bar{k}, {{\mathbb{Z}}}_{\ell}(1))/{\rm torsion}, \\ T_{DR}(L^n(V/k)) & \to & H^n_{DR}(V/k)\end{aligned}$$ from the $\ell$-adic and de Rham realizations of $L^n(V/k)$. The definitions of $L^n(V/k)$ and the homomorphisms should be algebraic, canonical, and functorial in $V$ and $k$. Furthermore, $L^n(V/{{{\mathbb{C}}}})$ should be canonically isomorphic to $I^n(V)$. (Clearly, $V$ can be replaced by a simplicial scheme.) The prototype is A. Weil’s construction [@aweil] of the Jacobian; his construction proves the conjecture for smooth projective curves and $n=1$. The conjecture is true for smooth projective varieties (\[dcon\]): it amounts to an algebraic construction of the Picard variety and the Néron-Severi group. The case $n=1$ of (\[dc1\]) is known (up to $p$-isogeny in characteristic $p >0$) for arbitrary varieties over perfect fields [@bs3; @h; @ra; @jp2]; the case $n=2$ is known for complex proper surfaces [@ca; @ca2]. No general results were known for higher cohomology (i.e., for $n >2$). A natural approach to Conjecture \[dc1\] is to use proper hypercoverings [@h 6.2] by smooth simplicial schemes; namely, to mimic Deligne’s approach [@h] to the construction of the mixed Hodge structure on $H^*(V,{{\mathbb{Z}}})$ of a complex algebraic variety $V$. This approach, which we follow here, gives a two-step strategy to prove (\[dc1\]): [**Step 1.**]{} Construct one-motives $L^n$ ($n \ge 0$) for smooth simplicial schemes arising from simplicial pairs (\[simpxy\]) and show that they have the properties given in (\[dc1\]). [**Step 2.**]{} Prove cohomological descent for these one-motives; more precisely, show that the one-motives $L^n$, given by (i), of a proper hypercovering of a variety $V$ are “independent” of the proper hypercovering; and, thus, $L^n$ depend only on $V$. Sections \[constr\], \[Tmt\], \[oakland\] are devoted to the first step, but only for fields of characteristic zero; the case of positive characteristic is relegated to Section \[pos+\]. Our construction of the requisite one-motives $L^n$, inspired by [@ca], relies on the theory of the Picard scheme [@blr Chapter 8]; the techniques are those of [@ra] but here applied to truncated simplicial schemes. The realizations of $L^n$ are treated in Sections \[Tmt\] (Hodge, de Rham), \[oakland\] (étale); here a crucial use is made of the validity of the Hodge conjecture for divisors (\[h11\]). Section \[varieties\] is devoted to the second step. It turns out that, because an important spectral sequence [@h 8.1.19.1] degenerates only with rational coefficients, the method of proper hypercoverings only provides a theory of isogeny one-motives $L^*(-)\otimes{{\mathbb{Q}}}$. More precisely, given two proper hypercoverings $U_{{\bullet}}$ and $'U_{{\bullet}}$ of $V$, we can only show that the associated one-motives $L^n$ and $'L^n$ are isogenous; the isogeny one-motive $L^n\otimes{{\mathbb{Q}}}$ depends only on $V$. Thus, a new ingredient is necessary to complete the second step, i.e., to endow these isogeny one-motives with integral structures. This is done, as in [@milram], via the integral structure on étale cohomology. Thus, we provide a complete proof (\[peddha\]) of Conjecture \[dc1\] for an arbitrary field of characteristic zero. We now turn to the case of Conjecture \[dc1\] for a field $k$ of characteristic $p >0$; let us begin by indicating why the conjecture must be weakened slightly. First, in [@jamo Appendix], A. Grothendieck notes that, for a curve $C$ over $k$, the construction of Deligne [@h 10.3] provides a one-motive $H^1_m(C) = L^1(C/k)$ defined over the perfection $k^{perf}$ of $k$; thus, [@h 10.3] proves the case $n=1$ of (\[dc1\]) only for curves over a perfect field. Second, he (loc. cit) expresses doubts about the existence of a ${{\mathbb{Z}}}$-linear (i.e., integral) category of mixed motives over an imperfect field $k$; he anticipates only a ${{\mathbb{Z}}}[1/p]$-linear category, i.e., a category of mixed motives up to $p$-isogeny. Third, the existence of proper hypercoverings (by smooth simplicial schemes) for an arbitrary variety $V$ over $k$ is known only when $k$ is perfect [@dj]. If one expects that the method of proper hypercoverings provides, as in characteristic zero, one-motives up to isogeny associated with $V$, then the methods of [@milram] allow a refinement to one-motives defined up to $p$-isogeny: controlling $p$-isogeny requires an integral $p$-adic cohomology theory for arbitrary varieties over $k$. These considerations[^3] lead us to a weak version of (\[dc1\]) by only requiring one-motives $L^*(V/k)\otimes{{\mathbb{Z}}}[1/p]$ (up to $p$-isogeny) over $k^{perf}$. While the first step can be carried out in positive characteristic in the same way as in characteristic zero, the second step cannot be unless, as it seems, one assumes the Tate conjecture (\[tatec\]) for divisors. Note that our proof of (\[dc1\]) in characteristic zero depends on the validity of the Hodge conjecture (\[h11\]) for divisors; thus, the appearance of the Tate conjecture is rather natural. For a perfect field $k$, (\[dc1\]) — up to $p$-isogeny — holds (\[groth\]) under the assumption of the Tate conjecture (\[tatec\]) for surfaces. The analogous result is also valid for an imperfect field $k$ under the additional assumption of “resolution of singularities” over $k$. Using a suggestion of M. Marcolli, we provide an *unconditional* construction (\[mem\]) of $J^n(-)\otimes{{\mathbb{Z}}}[1/p]$ ($n \ge 0$) and $L^n(-)\otimes{{\mathbb{Z}}}[1/p]$ ($ 2\ge n \ge 0$) of one-motives (up to $p$-isogeny) for arbitrary varieties over a perfect field $k$. The $J^n(V)\otimes{{\mathbb{Z}}}[1/p]$ are good substitutes for the (as yet conditional) $L^n(V)\otimes{{\mathbb{Z}}}[1/p]$; for instance, $W_{-1}J^n(V) = W_{-1}L^n(V)$. In particular, we generalize Carlson’s results [@ca] on $L^2$ of a complex projective surface to any variety over a perfect field (and up to $p$-isogeny in characteristic $p>0$). In Section \[applix\], we use these new invariants $L^n$ and $J^n$ to provide affirmative answers to special cases of questions [@ja; @ka; @se2] in the motivic folklore. These concern “independence of $\ell$” of $\ell$-adic étale cohomology of *arbitrary* varieties over number fields and finite fields. Since the circulation of this manuscript (circa 1998), other authors [@brs] have independently obtained some of the results presented here; [@banff] is a leisurely introduction to our results. [**Notation.**]{} We work over $S:= $ Spec $k$; here $k$ is a field of characteristic $p$ (except in §\[rev\], \[pos+\], $k$ is perfect unless indicated otherwise); in sections §\[constr\], \[Tmt\], \[oakland\], and \[varieties\] we assume $p$ to be zero. In §\[pos+\], we assume $p>0$. We fix an algebraic closure $\bar{k}$ of $k$; $\bar{S}:=$ Spec $\bar{k}$ and ${{\mathbb{G}}}:= {\rm{Gal}}(\bar{k}/k)$. For any scheme $X$ over $S$, we set $\bar{X}:= X \times_S \bar{S}$. All schemes will be supposed to be separated and locally noetherian. A variety is a geometrically integral scheme of finite type over $S$. For any set $B$, ${{\mathbb{Z}}}(B)$ is the free abelian group generated by the elements of $B$. $\pi_0(X):= $ the set (with a ${{\mathbb{G}}}$-action) of connected components of $\bar{X}$. $D_X:= $ the étale group scheme corresponding to ${{\mathbb{Z}}}(\pi_0(\bar{X}))$. $T_X:= Hom (D_X,{{\mathbb{G}}_m})$, the algebraic torus associated with $D_X$. $w(X):= $ the set of irreducible components of $\bar{X}$. $W_X:= $ the étale group scheme corresponding to the ${{\mathbb{G}}}$-module ${{\mathbb{Z}}}({w(X)})$. For any group scheme $\mathcal G$, $\pi_0(\mathcal G)$ is a group with an action of ${{\mathbb{G}}}$; we shall also use $\pi_0(\mathcal G)$ for the corresponding étale group scheme. For a variety $V$ over ${{\mathbb{C}}}$, $V({{\mathbb{C}}})$ (resp. $V^{an}$) denotes the associated topological space with the classical topology (resp. analytic variety). Given $X$ over $S$ and an imbedding $\iota: k \hookrightarrow {{\mathbb{C}}}$, we denote by $X_{\iota}$ the scheme over ${{\mathbb{C}}}$ obtained by base change. $MHS:=$ the abelian category of (${{\mathbb{Z}}}$)-mixed Hodge structures. $\mathcal M_F:=$ the additive category of one-motives over a field $F$. $S_{fppf}$ (resp. $S_{fpqc}$) is the big site over $S$ with the $fppf$ (resp. $fpqc$) topology [@blr pp. 200-201]. $\hat{{{\mathbb{Z}}}}$ is the profinite completion ${\varprojlim}_r {{\mathbb{Z}}}/{r{{\mathbb{Z}}}}$ of ${{\mathbb{Z}}}$. $\mathbb A = \hat{{{\mathbb{Z}}}}\otimes {{\mathbb{Q}}}$ is the ring of finite adèles of ${{\mathbb{Q}}}$. ${{\mathbb{Z}}}^p: = {\varprojlim}_r {{\mathbb{Z}}}/{r{{\mathbb{Z}}}}$ with $r$ coprime to $p$ and ${{\mathbb{A}}}^p:= {{\mathbb{Z}}}^p\otimes{{\mathbb{Q}}}$. We refer to section x.y by §x.y and to specific results by Theorem x.y or Remark x.y or simply (x.y). We use $\square$ to denote the end of a remark or a proof. Preliminaries {#rev} ============= Let us begin by reviewing well-known results, which will be of use in the paper. [**One-motives.**]{} [@h §10] A one-motive $M:= [B \xrightarrow{u} G]$ over $S$ (or over $k$) is a two-term complex consisting of a semi-abelian variety $G$ over $k$ (i.e., $G$ is an extension of an abelian variety $A$ by a torus $T$), a finitely generated torsion-free abelian group $B$ with a structure of a discrete ${{\mathbb{G}}}$-module, and a homomorphism $u:B \rightarrow G(\bar{k})$ of ${{\mathbb{G}}}$-modules. In particular, if $k$ is algebraically closed, then $u$ is a homomorphism of abelian groups. It is convenient to regard $B$ as an étale group scheme (locally constant) on $S$. A morphism of one-motives is a morphism of complexes. From the category $M_k$ of one-motives over $k$, there are “realization” functors (Hodge) $T_{{{\mathbb{Z}}}}$ — for each $\iota: k {\hookrightarrow}{{\mathbb{C}}}$ — to (the category of) torsion-free ${{\mathbb{Z}}}$-mixed Hodge structures of type $(*)$, (étale) $T_{\ell}$ — for each $\ell \neq p$ — to ${{\mathbb{Z}}}_{\ell}$-modules with an action of ${{\mathbb{G}}}$, and (de Rham) $T_{DR}$ — if $p =0$ — to $k$-vector spaces. A morphism $\phi:M_1 \to M_2$ is called an isogeny if $\phi_B: B_1 \to B_2$ is injective with finite cokernel and $\phi_G: G_1 \to G_2$ is surjective with finite kernel. The (additive) category $\mathcal M_k$ of one-motives over $k$ enjoys Cartier duality [@h 10.2.11]. The dual of an isogeny is also an isogeny. A $p$-isogeny is an isogeny $\phi$ such that the orders of ${\operatorname{Coker}}(\phi_B)$ and ${\operatorname{Ker}}(\phi_G)$ are powers of $p$. The ${{\mathbb{Q}}}$-linear abelian category $\mathcal M_k\otimes {{\mathbb{Q}}}$ of isogeny one-motives over $k$ is obtained from $\mathcal M_k$ by inverting isogenies; $\mathcal M_k\otimes {{\mathbb{Q}}}$ inherits realization functors (Hodge) $T_{{{\mathbb{Z}}}}$ to ${{\mathbb{Q}}}$-mixed Hodge structures, (étale) $T_{\ell}$ to ${{\mathbb{Q}}}_{\ell}$-vector spaces, (de Rham) $T_{DR}$ to $k$-vector spaces, weight filtration $W$, and Cartier duality from $\mathcal M_k$. Every one-motive $M$ defines an isogeny one-motive $M \otimes{{\mathbb{Q}}}$. The weight filtration $W$ on $[B \xrightarrow{u} G]\otimes{{\mathbb{Q}}}$ is $W_{-3} =0$, $W_{-2} = [0 \to T]\otimes{{\mathbb{Q}}}$, $W_{-1} = [0 \to G]\otimes{{\mathbb{Q}}}$, and $W_0 = [B \xrightarrow{u} G]\otimes{{\mathbb{Q}}}$. Finally, if $p =0$, the functors $T_{\ell}$ can be combined to a functor $M \mapsto TM = M\otimes\hat{{{\mathbb{Z}}}}:= {\prod_{\ell}}~T_{\ell}M$; here $TM$ is a $\hat{{{\mathbb{Z}}}}$-module with a ${{\mathbb{G}}}$-action. The adèlic realization functor $M \mapsto TM\otimes{{\mathbb{Q}}}$ from $\mathcal M_k$ to ${{\mathbb{A}}}$-modules with a ${{\mathbb{G}}}$-action factorizes via $\mathcal M_k\otimes{{\mathbb{Q}}}$; this gives the adèlic realization functor of an isogeny one-motive: $M\otimes{{\mathbb{Q}}}\mapsto M\otimes{{\mathbb{A}}}$. If $p > 0$, then the ${{\mathbb{Z}}}[1/p]$-linear category $\mathcal M_k\otimes {{\mathbb{Z}}}[1/p]$ of one-motives up to $p$-isogeny over $k$ is obtained from $\mathcal M_k$ by inverting $p$-isogenies. The realization functor $M \mapsto T^pM = {\prod_{\ell \neq p}}~T_{\ell}M$ from $\mathcal M_k$ to the category of ${{\mathbb{Z}}}^p$-modules with a ${{\mathbb{G}}}$-action extends to the category $\mathcal M_k\otimes {{\mathbb{Z}}}[1/p]$. [**Relative representability.**]{} As indicated in [@blr pp.200-201], representability issues are best treated in the $fppf$-topology. The following simple lemma will be used often. \[repco\] [[(i)]{}]{} Let $F$ be a representable contravariant functor from the category of schemes over $S$ to sets. Then $F$ is a sheaf with respect to the ${fpqc}$-topology and, hence, with respect to the $fppf$, étale, and Zariski topologies. [[(ii)]{}]{} Let $0 \rightarrow F \rightarrow G \rightarrow H \rightarrow 0$ be an exact sequence of sheaves of abelian groups on $S_{fppf}$. Suppose $F,H$ are representable, and that $F \rightarrow S$ is an affine morphism [(]{}i.e. the scheme representing $F$ is affine over $S$[)]{}. Then $G$ is representable, necessarily by a commutative group scheme. \(i) [@blr Prop. 1, p.200]. \(ii) The proof of [@oort Prop. 17.4] for $S_{fpqc}$ also works for $S_{fppf}$. [**Picard functor.**]{} Let $f:X \rightarrow S$ be a smooth proper scheme. \[odb\] The sheaves $f_*\mathcal O$, $f_*\mathcal O^*$, and $R^1 f_*\mathcal O^*$ on $S_{fppf}$ are representable. The scheme $T_X:= Hom(D_X, {{\mathbb{G}}_m})$ represents $f_*\mathcal O^*$. The representability of $f_*\mathcal O$ and $f_*\mathcal O^*$ is rather elementary [@blr Cor. 8, Lem. 10, pp.207-208]. Each character of $D_X$ provides a non-zero function, constant (since $X$ is proper) on each connected component of $X$, i.e., on each irreducible component of $X$ (since $X$ is smooth). Thus $T_X$ represents $f_*\mathcal O^*$. The representability of $R^1 f_*\mathcal O^*$ is due to Murre-Oort [@blr Thm. 3, p.211]. The scheme representing $R^1 f_*\mathcal O^*$ is the Picard scheme $Pic_X$ of $X$. It is reduced in characteristic zero but it may not be so in positive characteristic. Its reduced neutral (= identity) component $Pic^{0, red}_X$ is the classical Picard variety $Pic(X)$. The Néron-Severi group scheme $NS_X$ is the étale group scheme corresponding to the ${{\mathbb{G}}}$-module $\pi_0(Pic^{red}_X)$; we often write $NS(X)$ for $NS_X(S)$. \[memu1\] [[($p=0$)]{}]{} The $S_{fppf}$-sheaves $R^if_*\Omega ^{j}$ and $\mathcal H^i_{DR}(X) = R^if_*\Omega$ given by Hodge and de Rham cohomology [(]{}$\Omega$ is the de Rham complex on $X$[)]{} as well as the sheaf $R^1f_*\Omega^*$ corresponding to the multiplicative de Rham complex [@mame 3.1.7, p.31] on $X$ $$\Omega^*:= [\mathcal O^* \xrightarrow{d~{\rm log}} \Omega^1 \rightarrow \Omega^2 \cdots ]$$ are all representable. The first two are representable by vector group schemes. The sheaves $R^if_*\Omega^j$, $\mathcal H^i_{DR}(X)$ are coherent, free, and commute with arbitrary base change [@katzdiff 1.4.1.8], [@grocry p.309-310]. For any $t: S' \to S$, we have $R^if_*\Omega^j (S') = t^* H^i(X,\Omega^j)$; similarly for $\mathcal H^i_{DR}(X)$. Any locally free $\mathcal O_T$-module $L$ on a scheme $T$ gives rise to a sheaf on $T_{fppf}$ which is representable by a vector group scheme [@mame p.1]. Thus, $R^if_*\Omega^j$, $\mathcal H^i_{DR}(X)$ are representable by vector group schemes. Similarly, the sheaves $R^if_*C$ — here $C$ is $[\Omega^1 \to \Omega^2 \cdots]$ — are also representable by vector group schemes. In the exact sequence[^4] (cf. (\[gumma\]))$$0 \to f_*C \to R^1f_*\Omega^* \to R^1f_* \mathcal O^* \to R^1f_*C,$$representability is already known for all the sheaves other than $R^1f_*\Omega^*$; and $f_*C$ is representable by an affine scheme. The representability of $R^1f_*\Omega^*$ follows from (\[repco\]). The scheme $Pic^{\natural}_X$ representing $R^1 f_*\Omega^*$ classifies [@mes 2.5] [@katzdiff 7.2.1] isomorphism classes of line bundles (=invertible sheaves) on $X$ endowed with an integrable connection; cf. (\[haccha1\]). \[memu2\] Assume that $k$ is of characteristic zero. [[(i)]{}]{} The neutral component $E^{\natural}$ of $Pic^{\natural}_X$ is the universal additive [(]{}= vectorial[)]{} extension of $Pic^0_X = Pic_X^{0, red}$. [[(ii)]{}]{} The additive group scheme $Lie~E^{\natural}$ represents the sheaf $\mathcal H^1_{DR}(X)$; [Lie]{} $E^{\natural} = {\rm Lie}~Pic^{\natural}_X \xrightarrow{\sim} H^1_{DR}(X):= {{\mathbb{H}}}^1(X,\Omega) \xleftarrow{\sim} {{\mathbb{H}}}^1(X,[\mathcal O \rightarrow \Omega^1])$. A proof of (i) for $X$ an abelian variety is in [@mes 2.1, 2.7, 2.8]; it can also be obtained by combining propositions 2.6.7, 3.2.3, and 4.2.1 of [@mame Chapter I]. For a general smooth proper $X$ over $S$, let $g: E^* \to E^{\natural}$ be the map induced from the universal additive extension $E^*$ of $Pic^0_X$. We need to show that $g$ is an isomorphism. Since both $E^*$ and $E^{\natural}$ are compatible with base change, we may assume $X(S) \neq \varnothing$; this provides an Albanese map $u: X \to Alb(X)$. We now argue as in [@mes 3.0] using the standard isomorphisms $u^*: H^0(Alb(X), \Omega^1) \xrightarrow{\sim} H^0(X, \Omega^1)$ and $u^*: Pic^0_{Alb(X)} \xrightarrow{\sim} Pic^0_X$. Part (ii) follows from (i) by [@mes Lem. 2.6.9]. [**Divisors on a smooth proper variety.**]{} We recall the classical properties of $Pic(X)$ and $NS(X) = NS_X({S})$ for $f:X \to S$ smooth proper. \[picmd\] One has \(i) ($\ell \neq p$) an isomorphism $H^1_{et}(\bar{X},{{\mathbb{Z}}}_{\ell}(1)) \xrightarrow{\sim} T_{\ell}Pic(X)$ [@mi p.125] of ${{\mathbb{G}}}$-modules provided by the Kummer sequence (\[kummer\]). \(ii) ($p =0$) Lie $Pic(X) \xrightarrow{\sim} H^1(X,\mathcal O)$ [@blr Thm. 1, p.231]. \(iii) ($p=0$) an exact sequence (\[memu2\]) $$0 \to H^0(X,\Omega^1) \to {\rm Lie}~E^{\natural} \to H^1(X,\mathcal O) \to 0.$$ \(iv) ($k = {{\mathbb{C}}}$) an isomorphism of pure Hodge structures [@cime pp.156-158]: $$\label{hdgpic} H^1(X({{\mathbb{C}}}),{{\mathbb{Z}}}(1)) \xrightarrow{\sim} H_1(Pic(X),{{\mathbb{Z}}})$$ provided by the exponential sequence $$\label{exp} 0 \to {{\mathbb{Z}}}(1) \to \mathcal O \xrightarrow{exp} \mathcal O^* \to 1;$$a commutative diagram [@dmos Thm. 1.4, p. 17] — vertical maps are [@gaga]: $$\begin{CD} 0 @>>> H^0(X,\Omega^1) @>>> H^1_{DR}(X) @>>> H^1(X,\mathcal O) @>>> 0\\ @. @V{\wr}VV @V{\wr}VV @V{\wr}VV @.\\ 0 @>>> H^0(X^{an},\Omega^1) @>>> H^1(X({{\mathbb{C}}}),{{\mathbb{C}}}) @>>> H^1(X^{an},\mathcal O) @>>>0.\\ \end{CD}$$ \(v) ($\ell \neq p$) the “cycle class map” [@mi VI §9] furnishes a ${{\mathbb{G}}}$-equivariant inclusion [@mi 3.2.9 (d), p.216]: $$\label{algs} NS(\bar{X})\otimes_{{{\mathbb{Z}}}} {{\mathbb{Z}}}_{\ell} \hookrightarrow H^2_{et}(\bar{X}, {{\mathbb{Z}}}_{\ell}(1));$$ numerical and homological equivalence coincide for divisors (with ${{\mathbb{Q}}}_{\ell}$-coefficients). \(vi) ($k ={{\mathbb{C}}}$) Lefschetz’s $(1,1)$-theorem = the integral Hodge conjecture for divisors [@dmot p.143] [@cime p. 156]: $$\label{h11} NS(X) \xrightarrow{\sim} {\mbox{Hom}}_{MHS}({{\mathbb{Z}}}, H^2(X({{\mathbb{C}}}),{{\mathbb{Z}}}(1))).$$ \(vii) ($k$ finitely generated) the Tate conjecture [@tate p.72] [@jamm 5.1] for divisors asks if $$\label{tatec} NS(X)\otimes_{{{\mathbb{Z}}}}{{\mathbb{Q}}}_{\ell} \xrightarrow{\sim} H^2_{et}({X}\times k^{sep},{{\mathbb{Q}}}_{\ell}(1))^{{{\mathbb{G}}}_{sep}} ?$$ Here $k^{sep}$ is a separable algebraic closure of $k$, ${{\mathbb{G}}}_{sep}$ the associated Galois group, and $M^{{{\mathbb{G}}}_{sep}}$ denotes the invariants of a ${{\mathbb{G}}}_{sep}$-module $M$. See [@tate §5] for the known cases of (\[tatec\]). \(viii) ($p=0$) the Chern class map [@h2 2.2.4] gives an injection $$\label{beagle} c_X: NS(X)\otimes k \hookrightarrow H^1(X, \Omega^1).$$ (The map $d~{\rm log}: \mathcal O^* \to \Omega^1$ [@h2 2.2.4] induces a map $R^1f_*\mathcal O^* \to R^1f_*\Omega^1$ of representable $S_{fppf}$-sheaves; since the second is affine and the identity component of the first is an abelian scheme, one gets the map $c_X$.) \[dcon\] Let us consider (\[dc1\]) for $X$. By the purity of $H^n(X, {{\mathbb{Z}}}(1))$ ($k = {{\mathbb{C}}}$), both $t^n(X)$ and $I^n(X)$ are zero for $n>2$. Parts (i)-(iv) of (\[picmd\]) identify $[0 \to Pic(X)]$ as the one-motive $L^1(X/k)$ whereas parts (v), (vi), and (viii) identify $[NS_X/{\rm torsion} \to 0]$ as the one-motive $L^2(X/k)$. Conjecture \[dc1\] is known for smooth proper varieties. From (\[h11\]), one obtains the \[1-1i\] Let $X$ be a smooth proper scheme over $S = {\rm Spec}~k$. The dimension of the ${{\mathbb{Q}}}$-vector space $$H^{1,1}_{{{\mathbb{Q}}}}(X_{\iota}):= {\rm Hom}_{MHS}({{\mathbb{Q}}}(-1), H^2(X_{\iota},{{\mathbb{Q}}}))$$ is independent of the map $\iota: k \hookrightarrow {{\mathbb{C}}}$. [**Smooth varieties.**]{} Let $U$ be the open complement of a strict divisor $Y$ (\[simpxy\]) with normal crossings in a smooth projective complex variety $X$. The normalization $\tilde{Y}$ of $Y$ is a smooth projective scheme. Consider the map $W_Y \to Pic_X$ which sends a divisor $E$ to the class of the invertible sheaf $\mathcal O(E)$ on $X$. Let $N$ be the cokernel of the induced map $\lambda: W_Y \to NS_X$. \[podade\] The $(0,0)$-part of $H^2(U, {{\mathbb{Q}}}(1))$ is $N\otimes{{\mathbb{Q}}}$. So $t^2(U)\otimes{{\mathbb{Q}}}\xrightarrow{\sim} N\otimes{{\mathbb{Q}}}$. Let $j: U \hookrightarrow X$ denote the inclusion. The cycle class map [@h2 2.2.4-5] and the Gysin sequence [@hp 3.3] provide the following commutative diagram $$\begin{CD} W_{Y} @>{\lambda}>> NS({X}) @>>> N @>>> 0\\ @| @VVV @VVV @.\\ H^0(\tilde{Y},{{\mathbb{Z}}}) @>>> H^2(X,{{\mathbb{Z}}}(1)) @>{j^*}>> H^2(U,{{\mathbb{Z}}}(1)) @. {}.\\ \end{CD}$$ By [@h2 3.2.17], we know that $W_0 H^2(U, {{\mathbb{Q}}}(1)) = j^*(H^2(X,{{\mathbb{Q}}}(1)))$. This implies that the $(0,0)$-part of $H^2(U,{{\mathbb{Q}}}(1))$ is the image under $j^*$ of the $(0,0)$-part of $H^2(X,{{\mathbb{Q}}}(1))$. From (\[h11\]), $NS(X)$ is the $(0,0)$-part of $H^2(X,{{\mathbb{Z}}}(1))$. This proves the first claim. The second follows immediately because $t^2(U)\otimes{{\mathbb{Q}}}$ is the $(0,0)$-part of $H^2(U, {{\mathbb{Q}}}(1))$: by [@h2 3.2.15 (ii)], $W_{-1}H^2(U, {{\mathbb{Q}}}(1)) =0$. \[toto\] Since $W_{m-1}H^m(U, {{\mathbb{Q}}}) =0$ [@h2 3.2.15] for any $m \ge 0$, only $t^1(U)$ and $t^2(U)$ can be nontrivial. Conjecture \[dc1\] for the case of $t^1(U)$ is classically known (\[curie\]); $L^1(U/k)$ can be identified as the Picard one-motive $[{\operatorname{Ker}}(\lambda) \to Pic(X)]$ of [@ra], the Cartier dual of the generalized Albanese variety of $U$ [@jp2]. The result (\[podade\]), an analog of (\[h11\]) for smooth varieties, identifies $[N \to 0]\otimes{{\mathbb{Q}}}$ as $L^2(U/k)\otimes{{\mathbb{Q}}}$. Thus, (\[dc1\]) is known (up to isogeny) for the smooth variety $U$. [**Simplicial objects.**]{} [@curt] A simplicial object $B_{{\bullet}}$ in a category $\mathcal C$ is a sequence of objects $$B_{{\bullet}} = \{B_0, B_1, B_2, ..., B_n, ...\},$$together with morphisms (face) $ d_i: B_n \rightarrow B_{n-1}$, (degeneracies) $ s_i: B_n \rightarrow B_{n+1}$ ($0 \le i \le n$) satisfying the simplicial identities of which we need only (the degeneracy maps will not play a role in our discussions) $$\label{att} d_i d_j = d_{j-1}d_i \qquad i <j.$$ A truncated simplicial object $B_{\ge m}$ is the subsequence of objects $\{B_m, B_{m+1}, B_{m+2}, ...\}$, together with all the maps $d_i$ and $s_j$ between them. Given a simplicial (resp. cosimplicial) commutative group scheme $A_{{\bullet}}$, then the pair $(A_{{\bullet}}, \delta)$ — here $\delta_n = \Sigma_{i=0}^{i=n} (-1)^i d_i$ [@curt 3.4] — becomes a chain complex of commutative group schemes: it follows from (\[att\]) that $\delta_n \delta_{n+1} = 0$ (resp. $\delta_{n+1} \delta_n = 0$). [**Simplicial pairs.**]{} Let $a:X \rightarrow S$ be a smooth projective morphism. Let $Y$ be a [*strict*]{} divisor with normal crossings on $X$ as in [@dj 2.4]; in particular, this means that $Y$ is reduced and its irreducible components $Y^i (i \in I)$ are regular schemes, and of codimension one in $X$. Let $U$ be the open subscheme of $X$ corresponding to the complement of $Y$. The normalization $\tilde{Y}$ of $Y$ is the disjoint sum of the $Y^i$’s. \[simpxy\] A simplicial pair $(X_{{\bullet}}, Y_{{\bullet}})$ consists of the data of \(i) a simplicial scheme $X_{{\bullet}}$ smooth and projective over $S$. \(ii) a strict divisor with normal crossings $Y_{{\bullet}}$ of $X_{{\bullet}}$; in particular, each $Y_m$ is a strict divisor (as defined above) of $X_m$. (These conditions are ($\alpha$), ($\beta$) of [@dj p. 51]; it follows from [@dj] that any variety over $S$ ($k$ perfect) admits a proper hypercovering (§\[varieties\]) corresponding to a simplicial pair.) In particular, $(X_{{\bullet}}, Y_{{\bullet}})$ is a simplicial object in the category of pairs. The schemes $U_{m}:= X_{m} - Y_{m}$ give a smooth simplicial subscheme $U_{{\bullet}}$ of $X_{{\bullet}}$ [@h 6.2.6]; let $j: U_{{\bullet}} \to X_{{\bullet}}$ be the natural map. A simplicial pair $(X_{{\bullet}}, Y_{{\bullet}})$ gives, for each $l\ge 0$, a truncated simplicial pair $(X_{\ge l},Y_{\ge l}) = (X_{{\bullet}},Y_{{\bullet}})_{\ge l}$ which consists of the schemes $(X_m,Y_m)$ (for ${m \ge l}$) and the maps $d, s$. One has an evident notion [@h 6.2.8] [@sga4 p. 75] of a morphism $\theta: (X_{{\bullet}},Y_{{\bullet}}) \to (Z_{{\bullet}},J_{{\bullet}})$ of simplicial pairs; $\theta$ satisfies [@jannbson 1.10] $\theta (Y_m) \subset J_m$ and $\theta (X_m - Y_m) \subset (Z_m - J_m)$. \[eggses\] Fix a simplicial pair $(X_{{\bullet}}, Y_{{\bullet}})$. The simplicial abelian group scheme $D_{{\bullet}}$ — here $D_i = D_{X_i}$ — gives upon normalization a chain complex still denoted $D_{{\bullet}}$. Similarly, one obtains the chain complexes of group schemes: $T_{{\bullet}}$ $Pic_{{\bullet}}$, $NS_{{\bullet}}$, $Pic^0_{{\bullet}}$, $W_{{\bullet}}$. The map $\lambda_i: W_i = W_{Y_i} \to NS_{X_i} = NS_i$ which sends a divisor $E$ of $X_i$ supported on $Y_i$ to the class $[\mathcal O(E)]$ of the invertible sheaf $\mathcal O(E)$ gives a map $\lambda: W_{{\bullet}} \to Pic_{{\bullet}} \to NS_{{\bullet}}$. [**Exact and spectral sequences.**]{} We summarize some results from [@h 5.1-5.3] about cohomology of sheaves on a simplicial scheme $Z_{{\bullet}}$. For any sheaf (or complex of sheaves) $F$ on $Z_{{\bullet}}$, there is a spectral sequence [@h 5.2.3.2, 5.1.12.2] [^5] $$\label{zsfbu} E_1^{p,q} = H^q(Z_p,F) \Rightarrow H^{p+q}(Z_{{\bullet}},F)$$ with associated low-degree exact sequence $$0 \rightarrow E^{1,0}_2(F) \rightarrow H^1(Z_{{\bullet}},F) \rightarrow E^{0,1}_2(F) \rightarrow E^{2,0}_2(F).$$ The “filtration bête” $\sigma$ of Deligne [@h2 1.4.7] is: $$\label{sigcon} \sigma_{\ge m}H^*(Z_{{\bullet}}, F):= {\operatorname{Im}}(H^*(Z_{{\bullet}}, \sigma_{\ge m} F ) \to H^*(Z_{{\bullet}}, F)).$$ The methods used to deduce (\[zsfbu\]) in [@h 5.2.3, 5.2.7] also work for truncated simplicial schemes. Given a complex of sheaves $C^{{\bullet}}$ on $Z_{\ge m}$, there is a spectral sequence $$\label{ssfbu} E_1^{p+m,q}(C^{{\bullet}}) = H^q(Z_{p+m},C^{{\bullet}}) \Rightarrow H^{p+q}(Z_{\ge m},C^{{\bullet}})$$ with associated low-degree exact sequence $$\label{ssylow} 0 \rightarrow E^{1+m,0}_2(C^{{\bullet}}) \rightarrow {{\mathbb{H}}}^1(Z_{\ge m},C^{{\bullet}}) \rightarrow E^{m,1}_2(C^{{\bullet}}) \rightarrow E^{2+m,0}_2(C^{{\bullet}}).$$Let $g_r: Z_r \to S$ and $f:Z_{\ge m} \rightarrow S$ be the structure maps. Analogs of (\[ssfbu\]) and (\[ssylow\]) for the associated $S_{fppf}$-sheaves hold as well: $$\label{fppfbu} E^{p+m,q}_1 (C^{{\bullet}})= R^qg_{p+m ~*}(C^{{\bullet}}) \Rightarrow R^{p+q}f_*(C^{{\bullet}})$$ $$\label{fppflow} 0 \rightarrow E^{1+m,0}_2(C^{{\bullet}}) \rightarrow R^1f_*(C^{{\bullet}}) \rightarrow E^{m,1}_2(C^{{\bullet}}) \rightarrow E^{2+m,0}_2(C^{{\bullet}}).$$ Construction of $L^n$ for a simplicial pair {#constr} =========================================== Throughout this section, $k$ will denote a field of characteristic zero. For each simplicial pair $(X_{{\bullet}},Y_{{\bullet}})$ and each non-negative integer $n$, we shall construct one-motives $$L^n =L^n(X_{{\bullet}},Y_{{\bullet}}) = [\mathcal B_n \xrightarrow{\phi_n} \tilde{\mathcal P}_n]$$ and $J^n(X_{{\bullet}},Y_{{\bullet}})$; these are contravariant functorial. Their realizations will be analyzed in subsequent sections. Our construction was obtained by a careful analysis of [@h 8.1.19.1]. Fix a simplicial pair $(X_{{\bullet}},Y_{{\bullet}})$; we have a diagram $U_{{\bullet}} \overset{j}{\hookrightarrow} X_{{\bullet}} \overset{i}{\hookleftarrow} Y_{{\bullet}}$. Let $a: X_{{\bullet}} \rightarrow S$, $a_m:X_m \rightarrow S$, and $f: X_{\ge n-1} \rightarrow S$ be the structure morphisms.[^6] We often write $\mathcal O^*_m$ for $\mathcal O^*_{X_m}$, $Pic_{m}$ for $Pic_{X_m}$, etc. [**Construction of $\mathcal P_n$.**]{} We begin with the construction of a semi-abelian variety $\mathcal P_n$ which is isogenous to the required $\tilde{\mathcal P}_n$; our method is similar to [@ra §3.1]. \[reps\] [[(i)]{}]{} The sheaf $R^1f_* \mathcal O^* = R^na_*\sigma_{\ge n-1} \mathcal O^*_{{\bullet}}$ is representable by a locally algebraic group scheme $\mathcal G'_n$ with neutral component $\mathcal G_n$. [[(ii)]{}]{} ${\rm Lie}~ \mathcal G_n {\xrightarrow{\sim}}H^1(X_{\ge n-1}, \mathcal O)$. [[(iii)]{}]{} [[(]{}]{}$k = {{\mathbb{C}}}$[[)]{}]{} $H^i(X_{\ge n-1},\mathcal F) \xrightarrow{\sim} H^i(X^{an}_{\ge n-1}, \mathcal F)$, $\mathcal F = \mathcal O, \mathcal O^*$ and $i =0,1$. \(i) By (\[fppflow\]), the sheaf $R^1f_*\mathcal O^*$ sits in an exact sequence $$0 \rightarrow E^{n,0}_2 \rightarrow R^1f_*\mathcal O^* \xrightarrow{\pi} E^{n-1,1}_2 \xrightarrow{\psi} E^{n+1,0}_2.$$ By (\[odb\]), the sheaf $a_{m~*} \mathcal O^*_m$ is representable by the torus $T_{X_m}$. The group (of multiplicative type) $Hom(H^m(D_{{\bullet}}), {{\mathbb{G}}_m})$ dual to the homology $H^m(D_{{\bullet}})$ (\[eggses\]) of $$D_{X_{m+1}} \xrightarrow{\delta_{m}} D_{X_{m}} \xrightarrow{\delta_{m-1}} D_{X_{m-1}}$$ represents $E^{m,0}_2$. Since $Pic_{X_m}$ represents $R^1a_{m~*}\mathcal O^*_m$, the scheme $ \mathcal R':= {\operatorname{Ker}}({\delta^*_{n-1}}:Pic_{X_{n-1}} \rightarrow Pic_{X_n})$ represents $E^{n-1,1}_2$. Since ${\operatorname{Ker}}(\psi)$ is representable and the scheme representing $E^{n,0}_2$ is affine, we can apply (\[repco\]). \(ii) It follows from (i) by [@mes Lem. 2.6.9]. \(iii) Use the GAGA isomorphisms $H^i(X_m, \mathcal F) \xrightarrow{\sim} H^i(X_m^{an}, \mathcal F) \quad (i = 0,1) $ [@gaga Prop. 17, 18] for $\mathcal F = \mathcal O, \mathcal O^*$ and (\[ssfbu\]), (\[ssylow\]) for $X_{\ge n-1}$ and $X_{\ge n-1}^{an}$. We call $\mathcal T'$ (resp. $\mathcal Q'$) the group scheme representing $E^{n,0}_2$ (resp. $E^{n+1,0}_2$); we denote its neutral component by $\mathcal T$ (resp. $\mathcal Q$). The neutral component $\mathcal R$ of $\mathcal R'$ is an abelian scheme (it is a subscheme of $Pic^0_{X_{n-1}}$) and $\mathcal Q$ is affine; so $\psi(\mathcal R) = 0$. Hence $\mathcal R$ is the neutral component of ${\operatorname{Ker}}(\mathcal R' \xrightarrow{\psi} \mathcal Q')$. Thus we have an exact sequence $$\label{lcw} 0 \rightarrow \mathcal T' \rightarrow \mathcal G_n \xrightarrow{\pi} \mathcal R \rightarrow 0.$$ \[avva\] (i) By definition [@gi 1.1], an invertible sheaf $L$ on a simplicial scheme $Z_{{\bullet}}$ is an invertible sheaf $L_m$ on each $Z_m$ such that: for each morphism $\tau: Z_m \to Z_r$ which is a composition of $s_i$’s and $d_j$’s, the map $\tau^* L_r \to L_m$ is an isomorphism. In (loc. cit), it is shown that $L$ is determined entirely by the data of $L_0$ and the isomorphism $\alpha: d_0^* L_0 {\xrightarrow{\sim}}d_1^*L_0$ satisfying $d_2^*(\alpha) \circ d_0^*(\alpha) = d_1^*(\alpha)$; (\[reps\]), for $n=1$, proves the representability of the Picard functor of $Z_{{\bullet}}$ which is smooth and proper over $S$. The scheme $\mathcal G'_n$ classifies isomorphism classes of pairs $(\mathcal L, \alpha)$ where \(a) $\mathcal L$ is an invertible sheaf on $X_{n-1}$ such that $\delta^*_{n-1} \mathcal L$ is isomorphic to $\mathcal O_{X_n}$ (by (\[att\]), $\delta^*_{n}\delta^*_{n-1} \mathcal L = \mathcal O_{X_{n+1}}$); \(b) $\alpha$ is a trivialization $\delta^*_{n-1} \mathcal L \xrightarrow{\sim} \mathcal O_{X_n}$ on $X_{n}$ satisfying a cocycle condition: $\delta^*_{n} \alpha = 1$; namely, $\delta_n^* \alpha$ is the identity isomorphism of $\mathcal O_{X_{n+1}}$. \(ii) For $n =1$, we can think of $\alpha$ as an isomorphism $d_0^* \mathcal L \xrightarrow{\sim} d_1^*\mathcal L$. The cocycle condition $d_2^*(\alpha)\circ d_0^*(\alpha) = d_1^*(\alpha)$ ($\Longleftrightarrow \delta_1^*\alpha =0$) becomes the commutativity of $$\begin{CD} d_0^* d_0^* \mathcal L @>{d_0^* (\alpha)}>> d_0^* d_1^* \mathcal L @ = d_2^* d_0^* \mathcal L \\ @| @. @V{d_2^* (\alpha)}VV \\ d_1^* d_0^* \mathcal L @>{d_1^* (\alpha)}>> d_1^* d_1^* \mathcal L @= d_2^* d_1^* \mathcal L\\ \end{CD} {}$$ (using the identities $d_0d_1=d_0d_0, d_0d_2 = d_1d_0, d_1d_2 = d_1d_1$). \[kaku\] Suppose given a simplicial scheme $Z_{{\bullet}}$, an invertible sheaf $\mathcal F$ on $Z_m$ and a nowhere vanishing section $s$ of $\mathcal F$. The identity (\[att\]) implies that (i) the sheaf $\delta^*_{m+1}\delta^*_m\mathcal F$ is naturally isomorphic to $\mathcal O_{Z_{m+2}}$; (ii) the section $\delta^*_{m+1}\delta^*_ms$ of $\delta^*_{m+1}\delta^*_m\mathcal F = \mathcal O_{Z_{m+2}}$ corresponds to the identity section of $\mathcal O_{Z_{m+2}}$. [**Definition of $\mathcal B_n$ and $\phi_n$.**]{} We now turn to the construction of the map $\phi_n$. This will take several steps. Let $\lambda'_m: W_m \to Pic_{X_m}$ be the map $E \mapsto \mathcal O(E)$. In general, there is no lifting of $\lambda'_{n-1}: W_{Y_{n-1}} \to Pic_{X_{n-1}}$ to a map $W_{Y_{n-1}} \to \mathcal G'_n \xrightarrow{\pi} R' \rightarrow Pic_{X_{n-1}}$. In other words, the invertible sheaf $\mathcal O(E)$ on $X_{n-1}$ corresponding to $E \in W_{n-1}$ may not always satisfy conditions (a), (b) of (\[avva\](i)). But a natural lifting does exist on the subgroup $K:= {\operatorname{Ker}}(\delta^*_{n-1}: W_{Y_{n-1}} \to W_{Y_n})$ of $W_{Y_{n-1}}$. \[liftss\] There exists a canonical and functorial map $\vartheta': K \rightarrow \mathcal G'_n$ which fits into a commutative diagram: $$\begin{CD} K @>{\vartheta'}>> \mathcal G'_n\\ @VVV @VVV\\ W_{Y_{n-1}} @>{{\lambda}'_{n-1}}>> Pic_{X_{n-1}}.\\ \end{CD}$$ The scheme $\mathcal G'_n$ classifies pairs $(\mathcal L, \alpha)$ where $\mathcal L$ is an invertible sheaf on $X_{n-1}$ and $\alpha$ is a trivialization $\delta^*_{n-1}\mathcal L \xrightarrow{\sim}\mathcal O_{X_n}$ satisfying the cocycle condition $\delta^*_{n}\alpha =1$. For any $E \in K$, pick a rational section $s_E$ of $\mathcal O(E)$ such that the divisor $div(s_E)= E$; the set of such sections forms a torsor over $H^0(X_{n-1},\mathcal O^*)$. Since $Y_{{\bullet}}$ is a simplicial divisor, the pull-back $\delta^*_{n-1}s_E$ is a rational section of $\delta^*\mathcal O(E)$ with divisor $\delta^*_{n-1}E$. As $E$ lies in $K$, we have $\delta^*_{n-1}E =0$. So $\delta^*_{n-1}s_E$ provides a trivialization $\alpha_E: \delta^*_{n-1}\mathcal O(E) \xrightarrow{\sim} \mathcal O$ on $X_n$. The trivialization $\delta^*_n \alpha_E$ of $\delta^*_n \delta^*_{n-1}\mathcal O(E)$ on $X_{n+1}$ is provided by the nowhere vanishing regular section $t_E:= \delta^*_n \delta^*_{n-1} s_E$. Now the rational section $s_E$ is a nowhere vanishing section of $\mathcal O(E)$ on the open subscheme $U_{n-1}$. Applying (\[kaku\]) to $U_{{\bullet}}$. we see that $t_E$ is the identity section of the sheaf $\mathcal O_{U_{n+1}}$. Since $t_E$ is regular on $X_{n+1}$, we deduce that $t_E$ is the identity section of $\mathcal O_{X_{n+1}}$. The map $\vartheta'$ is defined by $E \mapsto (\mathcal O(E), \alpha_E)$. It is clear that modifying $s_E$ by an element of $H^0(X_{n-1},\mathcal O^*)$ does not affect the isomorphism class of the pair $(\mathcal O(E), \alpha_E)$. The maps $d_i:X_{\ge n-1} {\rightarrow}X_{n-2}$ collectively provide the morphism $$\label{lift} \delta^*: Pic_{X_{n-2}} \to \mathcal G'_n.$$ More explicitly, given an invertible sheaf $\mathcal L$ on $X_{n-2}$, the pull-back $\mathcal L':= \delta^*_{n-2} \mathcal L$ is an invertible sheaf on $X_{n-1}$. As in (\[kaku\]), (\[att\]) implies that there is a canonical trivialization, call it $\beta_{\mathcal L'}$, of $\delta^*_{n-1} \mathcal L'$ on $X_{n}$ satisfying $\delta^*_n \beta_{\mathcal L'} = 1$. The map $\delta^*$ sends $\mathcal L$ to the (isomorphism class of the) pair $(\mathcal L', \beta_{\mathcal L'})$. \[tildel\] $\mathcal P'_n$ is the quotient of $\mathcal G'_n$ by ${\delta}^*(Pic^0_{X_{n-2}})$; its neutral component is $\mathcal P_n$. Note $\pi_0(\mathcal G'_n) \xrightarrow{\sim} \pi_0(\mathcal P_n')$. \[koil\] [[(]{}]{}$k = {{\mathbb{C}}}$[[)]{}]{} ${\rm Lie}~\mathcal P_n \xrightarrow{\sim} \frac{H^1(X_{\ge n-1}, \mathcal O)}{\delta^*H^1(X_{n-2}, \mathcal O)} \xrightarrow{\sim} \frac{H^1(X_{\ge n-1}^{an}, \mathcal O)}{\delta^*H^1(X^{an}_{n-2}, \mathcal O)}.$ Combine (\[reps\]) (iii) with (\[picmd\] (ii)). The map (\[lift\]), in turn, induces a map $NS_{n-2} \xrightarrow{\mu} \mathcal P'_n$. Combining this with $\vartheta: K \xrightarrow{\vartheta'} \mathcal G'_n \rightarrow \mathcal P'_n$, we get the map $$\rho: K \oplus NS_{X_{n-2}}\rightarrow \mathcal P'_n \qquad \rho(a,b) = \vartheta(a) + \mu(b).$$ We now turn to the introduction of several group schemes relevant for the definition of $\mathcal B_n$. The mapping cone complex of $\lambda: W_{{\bullet}} \to NS_{{\bullet}}$ (\[eggses\]) is: $$\begin{aligned} \label{iwas} W_{Y_{n-2}}\oplus NS_{X_{n-3}} \xrightarrow{\gamma_{n-3}} W_{Y_{n-1}}\oplus NS_{X_{n-2}} \xrightarrow{\gamma_{n-2}} W_{Y_{n}}\oplus NS_{X_{n-1}}\\ \gamma_{m}:(E,\beta) \mapsto (\delta^*_{m+1}E, \delta^*_{m}\beta - \lambda_{m+1}(E)). \end{aligned}$$ Since $W_{{\bullet}}$ is a complex, the image of $\gamma_{n-3}$ is contained in $K\oplus NS_{X_{n-2}}$. One checks that the composite map $\rho \circ \gamma_{n-3}$ is zero. \[auro\] (i) $\mathcal C'_n$ denotes $\frac{K \oplus NS_{X_{n-2}}}{\gamma_{n-3}(W_{n-2})}$; and $\mathcal C_n$ is the kernel of the composite map $ \rho': \mathcal C'_n \xrightarrow{\rho} \mathcal P'_n \rightarrow \pi_0(\mathcal P'_n)$. \(ii) $\mathcal A_n$ is the kernel of the composite map $\mathcal C'_n \xrightarrow{\rho'}\pi_0(\mathcal P'_n) \xrightarrow{\sim} \pi_0(\mathcal G'_n) \to NS_{X_{n-1}}$; this map — the middle isomorphism is the inverse of $\pi_0(\mathcal P'_n) \xleftarrow{\sim} \pi_0(\mathcal G'_n)$ — is the map $\mathcal C'_n \to NS_{X_{n-1}}$ induced by $\gamma_{n-2}$. Note $\mathcal C_n$ is a subgroup of $\mathcal A_n$. \(iii) $\mathcal B'_n$ is the kernel of the map $$\label{greece} \frac{K\oplus NS_{X_{n-2}}}{\gamma_{n-3}(W_{n-2}\oplus NS_{X_{n-3}})} \xrightarrow{\rho} \mathcal P'_n \rightarrow \pi_0(\mathcal P'_n),$$ which is analogous to the cycle class map; $\pi_0(\mathcal P'_n)$ plays the role of a relative Néron-Severi group. We have$$\label{bareesh} \frac{\mathcal C_n}{NS_{X_{n-3}}} {\xrightarrow{\sim}}\mathcal B'_n.$$ Restricting $\rho$ to $\mathcal B'_n$, we obtain a morphism of group schemes $\mathcal B'_n \xrightarrow{\rho} \mathcal P_n.$ Let $\tau_n$ be the torsion subgroup of $\mathcal B'_n$. Replacing $\mathcal B'_n$ by $\mathcal B_n:= \mathcal B'_n/{\tau_n}$ and $\mathcal P_n$ by $\tilde{\mathcal P}_n:= \mathcal P_n/{\rho(\tau_n)}$, the map $\rho$ induces a map $\phi_n: \mathcal B_n {\rightarrow}\tilde{\mathcal P}_n$. $L^n(X_{{\bullet}},Y_{{\bullet}})$ is the one-motive $$\label{phin} L^n= L^n(X_{{\bullet}},Y_{{\bullet}}):= [\mathcal B_n \xrightarrow{\phi_n} \tilde{\mathcal P}_n].$$ \[zeusp\] (i) The one-motive $L^n$ depends only on the schemes $X_m$, $Y_m$ for $m=n-3, \cdots, n+1$ (and the maps between them). \(ii) Let $k'$ be an extension of $k$ and write $S'=$ Spec $k'$. It is clear from the definitions that $L^n (X_{{\bullet}} \times_S S', Y'_{{\bullet}} \times_S S') = L^n \times_S S'$. \(iii) By (i) and (ii), there is no loss of generality in assuming that $k$ is finitely generated over ${{\mathbb{Q}}}$. Such a field always admits an embedding into ${{\mathbb{C}}}$. \(iv) $L^n$ is a contravariant functor for morphisms of simplicial pairs. [**The one-motive $J^n$.**]{} \[auro2\] $\mathcal V_m$ (resp. $N_m$) is the kernel (resp. cokernel) of $W_m \xrightarrow{\lambda_m} NS_m$; and $K^0$ (resp. $M$) is the kernel (resp. cokernel) of the composite map $K \xrightarrow{\vartheta} \mathcal P'_n \rightarrow \pi_0(\mathcal P'_n)$. The one-motive $J^n$ is $$\label{hosadu} J^n = J^n(X_{{\bullet}}, Y_{{\bullet}}):= [(\frac{{K^0}/{\mathcal V_{n-2}}}{\rm torsion} \xrightarrow{\phi_n} \tilde{\mathcal P}_n].$$ The natural morphism $J^n \to L^n$ is an isomorphism for $n \le 1$; $J^n$ is contravariant functorial for morphisms of simplicial pairs. The remainder of this section is devoted to results which will be used in §\[Tmt\]. \[codered\] One has an exact sequence$$0 \to \frac{\mathcal V_{n-1} \cap K}{\mathcal V_{n-2}} \to \mathcal A_n \to {\operatorname{Ker}}(\delta^*_{n-2}: N_{n-2} \to N_{n-1})\to 0.$$ Taking the quotient of $0 \to K \to K\oplus NS_{n-2} \to NS_{n-2} \to 0$ by the exact sequence $0 \to \mathcal V_{n-2} \to W_{Y_{n-2}} \to {\operatorname{Im}}(\lambda_{n-2}) \to 0$, we get the exact sequence in the first row of the diagram $$\begin{CD} 0 @>>> \frac{K}{\mathcal V_{n-2}} @>>> \frac{K\oplus NS_{n-2}}{\gamma_{n-3}(W_{n-2})} @>>> N_{n-2} @>>> 0\\ @. @VVV @VVV @VVV @.\\ 0 @>>> {\operatorname{Im}}(\lambda_{n-1}) @>>> NS_{n-1} @>>> N_{n-1} @>>> 0;\\ \end{CD}$$the required sequence is an easy consequence. [**The de Rham cohomology sheaves.**]{} We shall use the following complexes on a (simplicial) smooth projective scheme; the scheme could be either $X_{{\bullet}}$, $X_m$ or $X_{\ge m}$. We often omit the subscript from the complex if it is clear from the context which scheme it is on. $$\begin{aligned} \Omega & := &[ \mathcal O \xrightarrow{d} \Omega^1 \xrightarrow{d} \Omega^2 \dots]\\ \Omega(log~Y) & := &[\mathcal O \xrightarrow{d} \Omega^1(log~Y) \xrightarrow{d}...]\\ \Gamma_0 := [\mathcal O \xrightarrow{d} \Omega^1], & & \Gamma:= [\mathcal O \xrightarrow{d} \Omega^1(log~Y)]\\ \Gamma^*_0 := [\mathcal O^* \xrightarrow{d~{\rm log}} \Omega^1], & & \Gamma^*:= [\mathcal O^* \xrightarrow{d~{\rm log}} \Omega^1(log~Y)]\end{aligned}$$ Let $\Omega(log~Y)$ be the logarithmic de Rham complex [@h 3.1.3, 6.2.7] of $U_{{\bullet}}$ on $X_{{\bullet}}$. \[dlog\] The map $dl:\mathcal G'_n = R^1f_* \mathcal O^* \to R^1f_*\Omega^1$ induced by $d~{\rm log}: \mathcal O^* \to \Omega^1$ [@h2 2.2.4] yields a map $dlog: \pi_0( \mathcal G'_n) \to H^1(X_{\ge n-1}, \Omega^1)$. Since $\mathcal G_n$ is a semi-abelian variety and $R^1f_*\Omega^1$ is a vector scheme, the map $dl$ restricted to $\mathcal G_n$ is zero. \[gumma\] [[(i)]{}]{} The sheaf $R^1f_*\Gamma^*$ is representable by a group scheme $\mathcal G^{\diamondsuit}_n$ with ${{\mathbb{H}}}^1(X_{\ge n-1},\Gamma^*)$ as its group of $k$-rational points. [[(ii)]{}]{} Lie $\mathcal G^{\diamondsuit}_n {\xrightarrow{\sim}}{{\mathbb{H}}}^1(X_{\ge n-1}, \Gamma){\xrightarrow{\sim}}H^1_{DR}(U_{\ge n-1})$. [[(iii)]{}]{} The group $\mathcal G^{\diamondsuit}_n$ is an extension of a subgroup [(]{}containing $\mathcal G_n$[)]{} of $\mathcal G'_n$ by ${}f_*\Omega^1(log ~Y)$. \(i) In the exact sequence, given by the map $\Gamma^* \to \mathcal O^*$, $$f_* \Gamma^* \xrightarrow{\beta} f_* \mathcal O^* \to f_* \Omega^1(log~Y) \xrightarrow{h} R^1f_*\Gamma^* \to R^1f_*\mathcal O^* \to R^1f_* \Omega^1(log~Y),$$ $\beta$ is clearly an isomorphism: ${{\mathbb{H}}}^0(X_{\ge n-1}, \Gamma^*) = H^0(X_{\ge n-1}, \mathcal O^*)$. So $h$ is injective. The sheaves $R^if_* \Omega^j(log~Y)$ are locally free and commute with base change [@katzdiff 1.4.1.8]; so they are representable by vector group schemes (cf. the proof of (\[memu1\])). Lemma \[repco\] now provides the representability of $R^1f_*\Gamma^*$. \(ii) the first isomorphism follows from (i) [@mes Lemma 2.6.9] and the second isomorphism from [@katzdiff 1.0.3.7], [@h2 3.1.8]; cf. also [@jamm 3.4]. \(iii) It suffices to show that the map Lie$~\mathcal G^{\diamondsuit}_n {\rightarrow}$  Lie$~\mathcal G_n$ is onto. Under the identifications, Lie $\mathcal G^{\diamondsuit}_n {\xrightarrow{\sim}}{{\mathbb{H}}}^1(X_{\ge n-1}, \Gamma)$ and Lie$~\mathcal G_n {\xrightarrow{\sim}}H^1(X_{\ge n-1}, \mathcal O)$, we need the surjectivity of the forgetful map $ {{\mathbb{H}}}^1(X_{\ge n-1}, \Gamma) {\rightarrow}H^1(X_{\ge n-1}, \mathcal O)$. We can use (\[ssylow\]) for $\Gamma$ and $\mathcal O$ on $X_{\ge n-1}$; since ${{\mathbb{H}}}^0(X_m,\Gamma) = H^0(X_m,\mathcal O_{X_m})$, we have $E^{n,0}_2(\Gamma) \xrightarrow{\sim} E^{n,0}_2(\mathcal O)$. Thus we need the surjectivity of $E^{n-1,1}_2(\Gamma) {\rightarrow}E^{n-1,1}_2(\mathcal O)$. Let us show that even the composite map $\pi_0: E^{n-1,1}_2(\Gamma_0) {\rightarrow}E^{n-1,1}_2(\Gamma) {\rightarrow}E^{n-1,1}_2(\mathcal O)$ is onto. Write $E_m^{\natural}$ for the universal additive extension of $Pic^0_{X_m}$. We have $$\begin{aligned} E^{n-1,1}_2(\Gamma_0) \xleftarrow{\sim} {\operatorname{Ker}}(\delta^*_{n-1}: {\rm Lie}~E^{\natural}_{n-1} {\rightarrow}{\rm Lie}~E^{\natural}_{n})\\ E^{n-1,1}_2(\mathcal O) \xleftarrow{\sim} {\operatorname{Ker}}(\delta^*_{n-1}: {\rm Lie}~Pic^0_{X_{n-1}} {\rightarrow}{\rm Lie}~Pic^0_{X_{n}}) = {\rm Lie}~\mathcal R.\end{aligned}$$ For any abelian scheme $A$, the Lie algebra Lie $A^{\natural}$ of the universal additive (= vectorial) extension is canonically isomorphic to $H^1_{DR}(A^*)$ of the dual abelian scheme $A^*$ [@mame 4.1.7, p. 48] (\[memu2\]). So the functor ${\natural}: A \mapsto {\rm Lie}~A^{\natural}$ is exact. Thus $E^{n-1,1}_2(\Gamma_0) \xleftarrow{\sim} {\rm Lie}~\mathcal R^{\natural}$. Hence $\pi_0$ is the natural (surjective) projection Lie $\mathcal R^{\natural} {\rightarrow}{\rm Lie}~\mathcal R$. \[diksha\] (i) For any semi-abelian variety $G$ (with associated abelian variety $A$ and torus $T$), the universal additive (= vectorial) extension $G^{\natural}$ of $G$ is the pullback $A^{\natural} \times_A G$ [@h 10.1.7c] of the universal additive extension $A^{\natural}$ of $A$. The proof of (\[gumma\]) shows that (a) $R^1f_*\Gamma^*_0$ is a vectorial extension of a subgroup (containing $\mathcal G_n$) of $\mathcal G'_n$. (b) this vectorial extension restricted to $\mathcal G_n$ is the universal extension $\mathcal G^{\natural}_n = \mathcal R^{\natural} \times_{\mathcal R} \mathcal G_n$ of $\mathcal G_n$. The universal additive extension $\mathcal P^{\natural}_n$ of $\mathcal P_n$ is the quotient of $\mathcal G^{\natural}_n$ by $\delta^* E^{\natural}_{n-2}$, defined as in the proof of (\[gumma\] (iii)). \(ii) The analog of (\[gumma\]) is also true for a smooth proper variety $Z$ and a strict divisor $V$ of $Z$. For the choice of the strict divisor $Y_{m}$ of $X_{m}$, we obtain the existence of a group scheme $Pic^{\diamondsuit}_{X_{m}}$ with ${{\mathbb{H}}}^1(X_{m},\Gamma^*)$ as its group of $k$-rational points; one has Lie $Pic^{\diamondsuit}_{X_{m}} {\xrightarrow{\sim}}{{\mathbb{H}}}^1(X_{m},\Gamma) {\xrightarrow{\sim}}H^1_{DR}(U_m)$, the last isomorphism due to [@h2 3.1.8], [@katzdiff 1.0.3.7]. The cokernel $\mathcal P^{\diamondsuit}_n$ of ${\delta}^*: Pic^{\diamondsuit}_{X_{n-2}} {\rightarrow}\mathcal G^{\diamondsuit}_n$ is a vectorial extension of a subgroup (containing $\mathcal P_n$) of $\mathcal P'_n$. \[ohoh!\] [[(a)]{}]{} [(]{}$k ={{\mathbb{C}}}$[)]{} ${\rm Lie}~\mathcal P^{\diamondsuit}_n \xrightarrow{\sim} \sigma_{\ge n-1} H^n(U_{{\bullet}}, {{\mathbb{C}}})$. [[(b)]{}]{} [(]{}$k ={{\mathbb{C}}}$[)]{} ${\rm Lie}~\mathcal P^{\natural}_n \xrightarrow{\sim} \sigma_{\ge n-1} H^n(X_{{\bullet}}, {{\mathbb{C}}}) \xrightarrow{\sim} W_{-1}H^n(U_{{\bullet}}, {{\mathbb{C}}}) $. [[(c)]{}]{} ${\rm Lie}~\mathcal P^{\diamondsuit}_n \xrightarrow{\sim} \sigma_{\ge n-1} H^n_{DR}(U_{{\bullet}})$. Combine $H^1(U_*, {{\mathbb{C}}}) \xleftarrow{\sim} H^1(X^{an}_*, j_{*} {{\mathbb{C}}}_{U}) \xrightarrow{\sim} {{\mathbb{H}}}^1(X^{an}_*, \Omega(log~Y))$ (logarithmic Poincaré lemma [@h2 3.2.2]), ${{\mathbb{H}}}^1(X^{an}_*,\Gamma) \xrightarrow{\sim} {{\mathbb{H}}}^1(X_*^{an},\Omega(log~Y))$ ($X_* = X_m$ or $X_{\ge n-1}$) with (\[gumma\]), (\[diksha\]). Part (a) follows from the next isomorphism (the injectivity is a consequence of the degeneration of [@h 8.1.19.1] at $E_2$) $$\label{patagon} \frac{H^1(U_{\ge n-1}, {{\mathbb{C}}})}{{\delta}^*_{n-2}H^1(U_{n-2}, {{\mathbb{C}}})}\xrightarrow{\sim} \sigma_{\ge n-1} H^n(U_{{\bullet}},{{\mathbb{C}}});$$ the surjectivity is by definition of $\sigma$. The first (resp. second) isomorphism in (b) is a special case of (a), i.e., for $Y = \varnothing$ (resp. from (\[malli\])). Part (c) is proved similarly using (\[gumma\]), (\[diksha\]). Hodge and de Rham realizations of $L^n$ {#Tmt} ======================================= We retain the notations of the previous section but we now take $k ={{\mathbb{C}}}$. We have a diagram $U_{{\bullet}} \xrightarrow{j} X_{{\bullet}} \xleftarrow{i} Y_{{\bullet}}$ corresponding to our simplicial pair $(X_{{\bullet}}, Y_{{\bullet}})$. As before, $f: X_{\ge n-1} \to S$ is the structure map. The main results (\[main\]) (\[derham\]) of this section prove the Hodge and de Rham part of the conjecture (\[dc1\]) for $U_{{\bullet}}$; the étale realization will be treated in § \[oakland\]. [**Statement of the theorem.**]{} The mixed Hodge structures $H^*(U_{{\bullet}},{{\mathbb{Z}}})$ are polarizable. Since $Gr^W_rH^m(U_{{\bullet}},{{\mathbb{Q}}})$ is a direct sum of subquotients of the cohomology of smooth projective varieties [@h 8.1.19.1] and the cohomology $H^*(V,{{\mathbb{Q}}})$ of a smooth projective complex variety $V$ is polarizable [@h2 2.2.6], this is clear. Denote the largest sub-mixed Hodge structure of type $(*)$ of $H^n(U_{{\bullet}},{{\mathbb{Z}}}(1))/{\rm torsion}$ by $t^n(U_{{\bullet}})$; by the previous lemma, $Gr^W_{-1}t^n(U_{{\bullet}})\otimes{{\mathbb{Q}}}= {{\mathbb{Q}}}(1)\otimes Gr^W_1H^n(U_{{\bullet}}, {{\mathbb{Q}}})$ is polarizable. By [@h 10.1.3], the mixed Hodge structure $t^n(U_{{\bullet}})$ corresponds to a one-motive $I^n(U_{{\bullet}})$ over ${{\mathbb{C}}}$. \[main\] There is a canonical and functorial isogeny of one-motives $$I^n(U_{{\bullet}}) \rightarrow L^n (X_{{\bullet}}, Y_{{\bullet}})$$ over ${{\mathbb{C}}}$, i.e., there is a canonical functorial isomorphism of ${{\mathbb{Q}}}$-mixed Hodge structures: $$t^n(U_{{\bullet}})\otimes{{\mathbb{Q}}}\xrightarrow{\sim} T_{{{\mathbb{Z}}}}(L^n)\otimes{{\mathbb{Q}}}.$$ $L^n(X_{{\bullet}}, Y_{{\bullet}})\otimes{{\mathbb{Q}}}$ depends only upon $U_{{\bullet}}$. Clear. We follow, for the most part, Deligne’s arguments in [@h 10.3]. [**Proof of the $W_{-1}$-part of Theorem \[main\].**]{} We begin with the isogeny one-motive $W_{-1}L^n\otimes{{\mathbb{Q}}}= [ 0 \rightarrow \tilde{\mathcal P}_n]\otimes{{\mathbb{Q}}}= [ 0 \rightarrow {\mathcal P}_n]\otimes{{\mathbb{Q}}}$. \[kona\] $W_{-1}t^n(U_{{\bullet}})\otimes{{\mathbb{Q}}}\xrightarrow{\sim} T_{{{\mathbb{Z}}}} (W_{-1}L^n)\otimes{{\mathbb{Q}}}$. Since $$\mathcal P_n = \frac{\mathcal G_n}{Pic(X_{n-2})}, \quad H^1(X_{n-2}, {{\mathbb{Z}}}(1)) \underset{(\ref{hdgpic})}{{\xrightarrow{\sim}}} H_1(Pic(X_{n-2}), {{\mathbb{Z}}})$$and $W_{-1} t^n(U_{{\bullet}})\otimes{{\mathbb{Q}}}= W_{-1}H^n(U_{{\bullet}}, {{\mathbb{Q}}}(1))$, this follows from the next \[malli\] [[(i)]{}]{} $H':= H^1(X_{\ge n-1},{{\mathbb{Z}}}(1)) \xrightarrow{\sim} T_{{{\mathbb{Z}}}}([0 \rightarrow \mathcal G_n]) = H_1(\mathcal G_n, {{\mathbb{Z}}})$. Note $t^1(X_{\ge n-1}) = H^1(X_{\ge n-1},{{\mathbb{Z}}}(1))$. ${\rm{(ii)}}~\frac{H^1(X_{\ge n-1},{{\mathbb{Q}}}(1))}{H^1(X_{n-2}, {{\mathbb{Q}}}(1))} {\xrightarrow{\sim}}\sigma_{\ge n-1} H^n(X_{{\bullet}},{{\mathbb{Q}}}(1)) = W_{-1}H^n(X_{{\bullet}},{{\mathbb{Q}}}(1)) \xrightarrow{\sim} W_{-1}H^n(U_{{\bullet}},{{\mathbb{Q}}}(1)).$ (The last isomorphism is an analog of a theorem of Grothendieck-Deligne [@h2 3.2.16-17], [@gr 9.1-9.4].) \(ii) It follows from the definition of the weight filtration [@ez p.55] that the image of $H^n(X_{{\bullet}}, \sigma_{\ge n-1}, {{\mathbb{Q}}}(1))$ in $H^n(X_{{\bullet}}, {{\mathbb{Q}}}(1))$ is $W_{-1}H^n(X_{{\bullet}},{{\mathbb{Q}}}(1))$. This proves the equality. The rest of (ii) follows from an inspection of the spectral sequence [@h 8.1.19.1] and the fact (ibid. 8.1.20 (ii)) that it degenerates at $E_2$. The relevant $E_1$-terms of [@h 8.1.19.1] are those with $b= 1$ and $-a = n-1$ (and $r =0$, $p=1$, $q=n-1$) since $W_{-1}H^n(U_{{\bullet}}, {{\mathbb{Q}}}(1))$ is the Tate twist of $W_1H^n(U_{{\bullet}}, {{\mathbb{Q}}})$. \(i) From (\[exp\]), we get the exact sequence $$H^0(X^{an}_{\ge n-1}, \mathcal O) \xrightarrow{exp} H^0(X^{an}_{\ge n-1}, \mathcal O^*) {\rightarrow}H' \xrightarrow{\xi} H^1(X^{an}_{\ge n-1}, \mathcal O) \xrightarrow{exp} H^1(X^{an}_{\ge n-1},\mathcal O^*).$$ Since the first map is surjective ($exp: {{\mathbb{C}}}\rightarrow {{\mathbb{C}}}^*$ is surjective), $\xi$ is injective. Proposition \[reps\] (iii) now gives the required isomorphism $$\boxplus: H'= H^1(X_{\ge n-1}, {{\mathbb{Z}}}(1)) \xrightarrow{\sim} T_{{{\mathbb{Z}}}}([0 \rightarrow \mathcal G_n]):= H_1(\mathcal G_n, {{\mathbb{Z}}}).$$ Let us show that $\boxplus$ is compatible with the weight and Hodge filtrations. In the sequences (\[ssfbu\]), (\[ssylow\]) on $X_{\ge n-1}$, we have ${\bullet}$ isomorphisms of Hodge structures $$\eta_1: E^{n,0}_2({{\mathbb{Q}}}(1)) \xrightarrow{\sim} H_1(\mathcal T',{{\mathbb{Q}}}), \quad \eta_2: E^{n-1,1}_2({{\mathbb{Q}}}(1)) \xrightarrow{\sim} H_1(\mathcal R, {{\mathbb{Q}}}).$$ For $\eta_1$, use $H^0(X_m^{an}, {{\mathbb{Z}}}(1)) \xrightarrow{\sim} H_1(T_{X_m}, {{\mathbb{Z}}})$, a consequence of (\[exp\]). And $\eta_2$ follows from (\[picmd\] (ii),(iv)). ${\bullet}$ surjectivity of the map $\pi$ in the commutative diagram $$\begin{CD} 0 @>>> E^{n,0}_2({{\mathbb{Q}}}(1)) @>>{\tau}> H'\otimes {{\mathbb{Q}}}@>>{\pi}> E^{n-1,1}_2({{\mathbb{Q}}}(1)) @>>> 0\\ @. @V{\eta_1}VV @VV{\boxplus}V @V{\eta_2}VV @.\\ 0 @>>> H_1(\mathcal T,{{\mathbb{Q}}}) @>>> H_1(\mathcal G_n,{{\mathbb{Q}}}) @>>> H_1(\mathcal R,{{\mathbb{Q}}}) @>>> 0. \end{CD}$$ This follows from the degeneration [@h 8.1.20] at $E_2$ of (\[ssfbu\]) for ${{\mathbb{Q}}}(1)$. The bottom row of the diagram is the the Hodge realization of the exact sequence (\[lcw\]) of isogeny one-motives. The image of $\tau$ is $W_{-2}H'\otimes{{\mathbb{Q}}}$ [@ez p.55]. Since $W_{-2}[ 0 \to \mathcal G_n]\otimes{{\mathbb{Q}}}= [0 \rightarrow \mathcal T]\otimes{{\mathbb{Q}}}$ [@h 10.1.4], we find that $\boxplus$ is compatible with the weight filtration. Since $H'$ is of type $(*)$, there is only one nontrivial step in the Hodge filtration, viz., $F^0$; and, $F^0(H'\otimes{{\mathbb{C}}}) \xrightarrow{\sim} F^0((H'/{W_{-2}H'})\otimes {{\mathbb{C}}})$ since $F^0\cap W_{-2} H'\otimes {{\mathbb{C}}}=0$. Thus, to show that $\boxplus$ is compatible with $F$, it suffices to show that $\eta_2$ is a map of Hodge structures. This we have done. This finishes the proof of the $W_{-1}$-part of Theorem \[main\]. [**Interpretation of $H^1$ and its applications.**]{} \[haccha1\] (Interpretation of $H^1$) Let $d:F {\rightarrow}G$ be a morphism of abelian sheaves on a space $Z$. In [@h 10.3.10], Deligne notes that ${{\mathbb{H}}}^1(Z, [F \xrightarrow{d} G])$ can be identified with the set of isomorphism classes of pairs $(L, \alpha)$ where $L$ is a $F$-torsor and $\alpha$ is a trivialization of the $G$-torsor $dL$. This identification is based upon the sequence $$H^0(Z, F) \to H^0(Z, G) {\rightarrow}{{\mathbb{H}}}^1(Z,[F\xrightarrow{d} G]) {\rightarrow}H^1(Z,F) \xrightarrow{d} H^1(Z,G)\qed$$ Let $j^m_* \mathcal O^*_{U}$ [@h 10.3.9(i)] be the subsheaf of meromorphic sections of $j_*\mathcal O^*_{U}$ on $X^{an}_{{\bullet}}$. \[curie\] One can prove (\[malli\](i)) using (\[haccha1\]); namely, the group $H^1(X_{\ge n-1}, {{\mathbb{Z}}}(1)) {\xrightarrow{\sim}}{{\mathbb{H}}}^1(X_{\ge n-1}, [\mathcal O \xrightarrow{exp} \mathcal O^*])$ is the set of pairs $(L, \alpha)$ where $L$ is an $\mathcal O_{X_{\ge n-1}}$-torsor and $\alpha$ a trivialization of $exp(L)$. Since ${\rm Aut}(L) \to {\rm Aut}(exp(L))$ is onto, $H^1(X_{\ge n-1}, {{\mathbb{Z}}}(1))$ is the set of isomorphism classes $L$ of $\mathcal O_{X_{\ge n-1}}$-torsors (= elements of Lie $\mathcal G_n$) with $exp(L) = 0$ in $\mathcal G_n$; thus, $H^1(X_{\ge n-1}, {{\mathbb{Z}}}(1)) {\xrightarrow{\sim}}H_1(\mathcal G_n, {{\mathbb{Z}}})$. Similar results, based on (\[haccha1\]), are as follows. \(a) the mixed Hodge structure $H^1(U_{m},{{\mathbb{Z}}}(1))$ is isomorphic to $T_{{{\mathbb{Z}}}}[\mathcal V_{m} \to Pic(X_{m})]$ (\[toto\]). \(b) the mixed Hodge structure $H^1(U_{\ge n-1},{{\mathbb{Z}}}(1))$ is isomorphic to $T_{{{\mathbb{Z}}}}[K^0 \to \mathcal G_n]$; note that $[K^0 \to \mathcal G_1]$ ($n=1$) is the Picard one-motive [@ra] of $U_{{\bullet}}$. We refer to [@ra] for the details; a sketch of the proof of (a) is as follows: $H^1(U_{m},{{\mathbb{Z}}}(1)) = {{\mathbb{H}}}^1(X_{m}, [\mathcal O_{X} \xrightarrow{exp} j^m_*\mathcal O^*_{U}])$ can be identified — as in [@h 10.3.10c] — with the set of isomorphism classes of pairs $(L, \alpha)$ where $L$ is an $\mathcal O_{X_{m}}$-torsor and $\alpha$ is an isomorphism of the invertible sheaf $exp(L)$ with $\mathcal O_{X_{m}}(E)$ ($E$ is a divisor supported on $Y_{m}$). Since ${\rm Aut}(L) {\rightarrow}{\rm Aut}(exp(L))$ is surjective, $H^1(U_{m},{{\mathbb{Z}}}(1))$ is the set of pairs $(p,d)$ where $p$ is an isomorphism class of an $\mathcal O_{X_{m}}$-torsor, i.e., an element of Lie $Pic(X_{m})$, and $d\in \mathcal V_{m}$ with $exp(p)$ as image in $Pic(X_{m})$. This gives an isomorphism (as abelian groups) $$H^1(U_{m},{{\mathbb{Z}}}(1)) \xrightarrow{\sim} T_{{{\mathbb{Z}}}}[\mathcal V_{m} \to Pic(X_{m})].$$(Combining (a) and (b) yields — see (\[hosadu\]) for the definition of $J^n$ — $$\label{k3}T_{{{\mathbb{Z}}}}J^n\otimes{{\mathbb{C}}}{\xrightarrow{\sim}}\frac{H^1(U_{\ge n-1}, {{\mathbb{C}}})}{H^1(U_{n-2}, {{\mathbb{C}}})}\underset{(\ref{patagon})}{{\xrightarrow{\sim}}} \sigma_{\ge n-1}H^n(U_{{\bullet}}, {{\mathbb{C}}})\underset{(\ref{ohoh!})}{\xleftarrow{\sim}} {\rm Lie} \mathcal P^{\diamondsuit}_n.)$$ \(c) $ \mathcal G^{\diamondsuit}_n({{\mathbb{C}}}) {\xrightarrow{\sim}}{{\mathbb{H}}}^1(X_{\ge n-1}, \Gamma^*)$ (\[gumma\]) is the group of isomorphism classes of pairs $(\mathcal L, \omega)$ with $\mathcal L$ an invertible sheaf on $X_{\ge n-1}$ and $\omega \in H^0(X_{\ge n-1}, \Omega^1(log~Y))$. To relate to [@h 10.3.10a], one uses the “connections $\theta$ on invertible sheaves = one-forms $\omega$” dictionary [@mes 2.5] [@ev 1.5, p.47]. Thus, ${{\mathbb{H}}}^1(X_{\ge n-1}, \Gamma^*)$ (cf. [@katzdiff 7.2.1]) is the set of isomorphism classes of pairs $(\mathcal L, \theta)$ where $\mathcal L$ is as before and $\theta$ a connection on $\mathcal L$, holomorphic on $U_{\ge n-1}$ and allowed to have simple poles along $Y_{\ge n-1}$. \(d) ${\rm Lie}~ \mathcal G^{\diamondsuit}_n({{\mathbb{C}}}) {\xrightarrow{\sim}}{{\mathbb{H}}}^1(X_{\ge n-1}, \Gamma) {\xrightarrow{\sim}}H^1_{DR}(U_{\ge n-1})$ (\[gumma\]) is the set of isomorphism classes of pairs $(L, \theta)$ where $L$ is an $\mathcal O_{X_{\ge n-1}}$-torsor and $\theta$ a connection on $L$ as in (c). \(e) Similar results hold for ${\rm Lie}~Pic^{\diamondsuit}_{X_{m}}({{\mathbb{C}}}) {\xrightarrow{\sim}}{{\mathbb{H}}}^1(X_{m},\Gamma) {\xrightarrow{\sim}}H^1_{DR}(U_{m})$ (\[diksha\]) and $Pic^{\diamondsuit}_{X_{m}}({{\mathbb{C}}}) {\xrightarrow{\sim}}{{\mathbb{H}}}^1(X_{m}, \Gamma^*)$ (\[diksha\]). \(f) Parts (d) and (e) yield an interpretation of ${\rm Lie}~\mathcal P^{\diamondsuit}_n {\xrightarrow{\sim}} \sigma_{\ge n-1} {{\mathbb{H}}}^n(U_{{\bullet}},\Gamma)$ (\[ohoh!\]). While $W_{-1}t^n(U_{{\bullet}})\otimes{{\mathbb{Q}}}$ is a quotient of $H^1(X_{\ge n-1}, {{\mathbb{Q}}}(1)))$, $t^n(U_{{\bullet}})\otimes{{\mathbb{Q}}}$ is a subquotient of $H^2(U_{\ge n-2}, {{\mathbb{Q}}}(1))$. Thus, more work is necessary to complete the proof of (\[main\]). [**Proof of Theorem \[main\].**]{} This will be accomplished in the following three steps. ${\bullet}$ [**Step 1.**]{} Construction of a certain mixed Hodge structure $h^2_X$. ${\bullet}$ [**Step 2.**]{} Relating $h^2_X$ and $t^n(U_{{\bullet}})$. ${\bullet}$ [**Step 3.**]{} Interpretation of $h^2_X$ using (\[haccha1\]). [**Step 1.**]{} *Construction of $h^2_X$.* This uses the truncated complex $\tau_{\le 1} Rj_*{{\mathbb{Z}}}(1)_U$ [@h2 1.4.6]. As before, let $q:\tilde{Y}_m \to Y_m \to X_m$ denote the natural map from the normalization $\tilde{Y}_m$ of $Y_m$. Since, on each $X_m$, $R^1j_{m,*}{{\mathbb{Z}}}(1) = q_*{{\mathbb{Z}}}_{\tilde{Y}_m}$ [@h2 3.1.9], we get the triangle in the derived category of sheaves on $X^{an}_{m}$ $${{\mathbb{Z}}}(1)_X \to \tau_{\le 1} Rj_{m,*}{{\mathbb{Z}}}(1) \to q_* {{\mathbb{Z}}}[-1] \to {{\mathbb{Z}}}_X(1)[1] \to \cdots;$$the truncated complexes $\tau_{\le 1} Rj_{m,*}{{\mathbb{Z}}}(1)$ for each $m$ combine to give a complex on $X_{\ge n-2}$ which we denote by $\tau_{\le 1} Rj_*{{\mathbb{Z}}}(1)$. Thus, in the exact sequence on $X_{\ge n-2}$ $$0 \to \sigma_{\ge n-1}{{\mathbb{Z}}}(1)_X \to \tau_{\le 1} Rj_*{{\mathbb{Z}}}(1) \to \mathcal F \to 0,$$ $\mathcal F$ on $X_m$ is (quasi-isomorphic to) the complex $q_* {{\mathbb{Z}}}[-1]$ for $m > n-2$ and $q_*{{\mathbb{Z}}}[-1] \to {{\mathbb{Z}}}_{X_{n-2}}(1)[1]$ for $m =n-2$. The associated cohomology sequence (here we use that $H^i(X_{\ge n-2}, \sigma_{\ge n-1}{{\mathbb{Z}}}(1)) = H^{i-1}(X_{\ge n-1}, {{\mathbb{Z}}}(1))$) $$\label{c2} \frac{H^1(X_{\ge n-1}, {{\mathbb{Z}}}(1))}{H^1(X_{n-2}, {{\mathbb{Z}}}(1))} {\hookrightarrow}{{\mathbb{H}}}^2(X_{\ge n-2}, \tau_{\le 1} Rj_*{{\mathbb{Z}}}(1)) \to {{\mathbb{H}}}^2(X_{\ge n-2}, \mathcal F) \xrightarrow{\delta^*} H^2(X_{\ge n-1}, {{\mathbb{Z}}}(1))$$ fits into the commutative diagram (defining $h^2_X$ by pullback via $\nu$) $$\begin{CD} \frac{t^1(X_{\ge n-1})}{t^1(X_{n-2})} @>>> h^2_X @>>> \frac{K \oplus NS_{n-2}}{W_{n-2}}\\ @V{\wr}V{(\ref{malli})}V @VVV @V{\nu}VV\\ \frac{H^1(X_{\ge n-1}, {{\mathbb{Z}}}(1))}{H^1(X_{n-2}, {{\mathbb{Z}}}(1))} @>>> {{\mathbb{H}}}^2(X_{\ge n-2}, \tau_{\le 1} Rj_*{{\mathbb{Z}}}(1)) @>>> \frac{H^0(\tilde{Y}_{\ge n-1}, {{\mathbb{Z}}}) \oplus H^2(X_{n-2}, {{\mathbb{Z}}}(1))}{H^0(\tilde{Y}_{n-2}, {{\mathbb{Z}}})}\\ @VVV @VVV @VVV \\ \frac{{{\mathbb{H}}}^2(X_{\ge n-2}, [\sigma_{\ge n-1} \mathcal O \xrightarrow{exp} j_*^m\mathcal O^*_U])}{H^1(X_{n-2}, \mathcal O)} @>>> {{\mathbb{H}}}^2(X_{\ge n-2}, [\mathcal O \xrightarrow{exp} j_*^m\mathcal O^*_U]) @>>> H^2(X_{n-2}, \mathcal O)\\ @VVV @V{a}VV @|\\ \frac{{{\mathbb{H}}}^2(X_{\ge n-2}, \mathcal K)}{H^1(X_{n-2}, \mathcal O)} @>>> {{\mathbb{H}}}^2(X_{\ge n-2}, \Gamma) @>>> H^2(X_{n-2}, \mathcal O)\\ @V{b}VV @. @.\\ \frac{H^1(X_{\ge n-1}, \mathcal O)}{H^1(X_{n-2}, \mathcal O)} @<{\sim}<{(\ref{koil})}< {\rm Lie}~\mathcal P_n @. (**)\\ \end{CD}$$ \[penang\] (i) The map $\nu$, induced by (\[h11\]) and the isomorphism $H^0(\tilde{Y}_i, {{\mathbb{Z}}}) \cong W_{Y_i}$, is *injective*. \(ii) $h^2_X$ is a mixed Hodge structure of type $(*)$; note $W_{-1}h^2_X = \frac{H^1(X_{\ge n-1}, {{\mathbb{Z}}}(1))}{H^1(X_{n-2}, {{\mathbb{Z}}}(1))}$. (An argument similar to that in the proof of (\[kapadu\](i)) shows that the natural map $h^2_X\otimes{{\mathbb{Q}}}\to H^2(U_{\ge n-2}, {{\mathbb{Q}}}(1))$ is injective with image $t^2(U_{\ge n-2})\otimes{{\mathbb{Q}}}$.) \(iii) The map $a$ is induced by the (first) morphism of complexes $$\begin{CD} \mathcal O_{X} @>{exp}>> j^m_*\mathcal O^*_{U} @>>> 0\\ @| @VV{d~{\rm log}}V @.\\ \mathcal O_{X} @>{d}>> \Omega^1(log~ Y) @>>> 0\\ @| @| @.\\ \mathcal O_{X} @>{d}>> \Omega^1(log~ Y) @>{d}>> \Omega^2(log~Y) \rightarrow \cdots; \\ \end{CD}$$it is compatible with the natural map$${{\mathbb{H}}}^2(X_{\ge n-2}, Rj_*{{\mathbb{Z}}}(1)) {\xrightarrow{\sim}}H^2(U_{\ge n-2}, {{\mathbb{Z}}}(1)) {\hookrightarrow}{{\mathbb{H}}}^2(X_{\ge n-2}, \Omega(log~Y)) \xrightarrow{\sim} H^2(U_{\ge n-2},{{\mathbb{C}}}).$$The last isomorphism is the logarithmic Poincaré lemma [@h2 3.2.2]. \(iv) On $X_{\ge n-1}$, $\mathcal K = \Gamma$ and, on $X_{n-2}$, $\mathcal K = [ 0 \to \Omega^1(log~Y)]$. \(v) Since $\mathcal C'_n = \frac{K \oplus NS_{n-2}}{W_{n-2}}$ is of type $(0,0)$, the map $h^2_X \to H^2(X_{n-2}, \mathcal O)$ in $(**)$ is zero. We deduce an injection $h^2_X {\hookrightarrow}\frac{{{\mathbb{H}}}^2(X_{\ge n-2}, [\sigma_{\ge n-1} \mathcal O \xrightarrow{exp} j_*^m\mathcal O^*_U])}{H^1(X_{n-2}, \mathcal O)}$ (with finite cokernel — see (\[devaki\]) below) and a commutative diagram $$\begin{CD} h^2_X @>>> h^2_X\otimes{{\mathbb{C}}}@<<< W_{-1}h^2_X\otimes{{\mathbb{C}}}\\ @VVV @VVV @V{\wr}V{(ii)}V\\ \frac{{{\mathbb{H}}}^2(X_{\ge n-2}, [\sigma_{\ge n-1} \mathcal O \xrightarrow{exp} j_*^m\mathcal O^*_U])}{H^1(X_{n-2}, \mathcal O)} @>{a}>> \frac{{{\mathbb{H}}}^2(X_{\ge n-2}, \mathcal K)}{H^1(X_{n-2},\mathcal O)} @<<< \frac{{{\mathbb{H}}}^1(X_{\ge n-1}, \Gamma_0)}{{{\mathbb{H}}}^1(X_{n-2}, \Gamma_0)}\\ @VVV @VV{b}V @V{(\ref{ohoh!})}V{\wr}V\\ \frac{H^1(X_{\ge n-1}, \mathcal O)}{H^1(X_{n-2}, \mathcal O)} @<{\sim}<{(\ref{koil})}< {\rm Lie}~\mathcal P_n @<<< {\rm Lie}~\mathcal P^{\natural}_n.\\ \end{CD}$$ (vi) The Hodge filtration [@h2 3.2.2], restricted to $\mathcal K$, is $F^0 \mathcal K = \mathcal K$ and $F^1\mathcal K = [0 \to \Omega^1(log~Y)]$ on $X_{\ge n-2}$. It induces the map $b$ — see $(**)$ — thereby defining the Hodge filtration on $h^2_X\otimes{{\mathbb{C}}}$. Because $h^2_X$ is defined using cohomology with ${{\mathbb{Z}}}(1)$-coefficients, $F^1$ on $\mathcal K$ corresponds to $F^0$ on $h^2_X\otimes{{\mathbb{C}}}$. \[samos\] One has a commutative diagram $$\begin{CD} \mathcal C'_n @>{\rho}>> {\mathcal P}'_n @>>> \pi_0(\mathcal P'_n)\\ @V{\nu}VV @. @A{\wr}AA\\ \frac{H^0(\tilde{Y}_{\ge n-1}, {{\mathbb{Z}}}) \oplus H^2(X_{n-2}, {{\mathbb{Z}}}(1))}{H^0(\tilde{Y}_{n-2}, {{\mathbb{Z}}})} @>{\delta^*}>{(\ref{c2})}> H^2(X_{\ge n-1}, {{\mathbb{Z}}}(1)) @<<< \pi_0(\mathcal G'_n).\\ \end{CD}$$ Straightforward. Combining (\[samos\]), (\[kona\]) and (\[malli\]), we obtain the exact sequence $$\label{kvk} 0 \to H_1(\mathcal P_n, {{\mathbb{Q}}}) \to h^2_X\otimes{{\mathbb{Q}}}\to \mathcal C_n\otimes{{\mathbb{Q}}}\to 0.$$ [**Step 2.**]{} *Relating $h^2_X$ and $t^n(U_{{\bullet}})$.* Using the next lemma, we shall show that $t^n(U_{{\bullet}})$ is a quotient of $h^2_X$ by $NS_{n-3}$. \[kapadu\] [[(i)]{}]{} One has an isomorphism $\nu:\mathcal B_n\otimes{{\mathbb{Q}}}{\xrightarrow{\sim}}Gr^W_0 t^n(U_{{\bullet}})\otimes {{\mathbb{Q}}}.$ [[(ii)]{}]{} $\mathcal C_n$ has finite index in $\mathcal A_n$; cf. [(\[auro\])]{}. [[(iii)]{}]{} $K^0:= {\operatorname{Ker}}(K \xrightarrow{\vartheta} \pi_0(\mathcal G'_n))$ has finite index in $K \cap \mathcal V_{n-1}:= {\operatorname{Ker}}(K \xrightarrow{\lambda_{n-1}} NS_{n-1})$. \(i) By [@h 8.1.19.1], $Gr^W_{b}H^n(U_{{\bullet}},{{\mathbb{Q}}})$ is a subquotient of $E_1^{n-b, b}$. Since $Gr^W_0 t^n(U_{{\bullet}})\otimes{{\mathbb{Q}}}$ is a Tate twist of the ($1,1$)-part of $Gr^W_2 H^n(U_{{\bullet}},{{\mathbb{Q}}})$, the relevant $E_1$-terms correspond to $b=2$ and $-a= n-2$ (with $p,q,r$ satisfying $p+2r = 2$ and $q-r = n-2$). If written explicitly (as is done immediately after 8.1.19.1 in [@h])[^7], we obtain — using the degeneration [@h 8.1.20 (ii)] at $E_2$ of [@h 8.1.19.1] — that $Gr^W_2 H^n(U_{{\bullet}},{{\mathbb{Q}}})$ is (the coefficients in (\[soya\]) are ${{\mathbb{Q}}}$) $$\label{soya} \frac{{\operatorname{Ker}}(H^0(\tilde{Y}_{n-1})(-1) \oplus H^2(X_{n-2}) \xrightarrow{t_{n-1}} H^0(\tilde{Y}_{n})(-1) \oplus H^2(X_{n-1}))}{{\operatorname{Im}}(H^0(\tilde{Y}_{n-2})(-1) \oplus H^2(X_{n-3}) \xrightarrow{t_{n-2}} H^0(\tilde{Y}_{n-1})(-1) \oplus H^2(X_{n-2}))};$$ here $t_{m}(a,b) = (\delta^*_{m} a, \delta^*_{m-1}b -\lambda_m(a))$. This is easily compared with (\[iwas\]). By [@h 8.1.20 (ii)], the group in (\[soya\]) is unchanged if one replaces $H^2(X_{n-1})$ by $H^2(X_{\ge n-1})$ and uses the map $\delta^*$ of (\[c2\]). The $(1,1)$-part of $Gr^W_2H^n(U_{{\bullet}},{{\mathbb{Q}}})$ can be identified using (\[h11\]). The lemma now follows from $\pi_0(\mathcal P'_n) \cong \pi_0(\mathcal G'_n)$, $\pi_0(\mathcal G'_n) {\hookrightarrow}H^2(X_{\ge n-1}, {{\mathbb{Z}}}(1))$ — a consequence of (\[exp\]), and (\[samos\]). \(ii) and (iii) also follow from the degeneration [@h 8.1.20 (ii)] at $E_2$ of [@h 8.1.19.1]. Using (\[kapadu\](i)) and (\[kona\]), we can rewrite $$\label{palani} 0 \to W_{-1}t^n(U_{{\bullet}}) \to t^n(U_{{\bullet}}) \to Gr^W_0 t^n(U_{{\bullet}}) \to 0$$ as the sequence (exact modulo finite groups) of mixed Hodge structures $$\label{horanadu} 0 \to H_1(\mathcal P_n, {{\mathbb{Z}}}) \to t^n(U_{{\bullet}}) \to \mathcal B_n \to 0.$$ We have an isomorphism of mixed Hodge structures $$\label{nyaya} \frac{h^2_X\otimes{{\mathbb{Q}}}}{\delta^*NS(X_{n-3})\otimes{{\mathbb{Q}}}} {\xrightarrow{\sim}}t^n(U_{{\bullet}})\otimes{{\mathbb{Q}}}.$$ Since $h^2_X$ is of type $(*)$, the natural map $$H^2(X_{\ge n-2}, \tau_{\le 1} Rj_*{{\mathbb{Z}}}(1)) \to H^2(U_{\ge n-2},{{\mathbb{Z}}}(1)) \to H^n(U_{{\bullet}}, {{\mathbb{Z}}}(1))$$gives a map $h^2_X \to t^n(U_{{\bullet}})$. The proposition follows from (\[horanadu\]), (\[bareesh\]), (\[kapadu\]), and (\[kvk\]). [**Step 3.**]{} *Interpretation of $h^2_X$ using* (\[haccha1\]). One has a natural isomorphism $$\frac{{{\mathbb{H}}}^2(X_{\ge n-2}, [\sigma_{\ge n-1} \mathcal O \xrightarrow{exp} j_*^m\mathcal O^*_U])}{H^1(X_{n-2}, \mathcal O)}{\xrightarrow{\sim}}T_{{{\mathbb{Z}}}}[\mathcal C_n \xrightarrow{\rho} \mathcal P_n].$$ \[merext\] Every $j_*^m\mathcal O^*_{U_{r}}$-torsor $P$ gives an invertible sheaf on $U_{r}$ and this extends to an invertible sheaf $P'$ on $X_{r}$; given any two such extensions $P'$ and $P''$, one has an isomorphism of invertible sheaves $P'' {\xrightarrow{\sim}}P' \otimes \mathcal O(v)$ ($v\in W_{Y_{r}}$) on $X_{r}$. The group ${{\mathbb{H}}}^2(X_{\ge n-2}, [\sigma_{\ge n-1} \mathcal O \xrightarrow{exp} j_*^m\mathcal O^*_U])$ is actually a ${{\mathbb{H}}}^1$ in disguise (this shift occurs for reasons of degree for $j_*^m\mathcal O^*_U$ — this is in degree one — and of truncation for $\mathcal O$): it sits in an exact sequence $$\begin{gathered} \to {{\mathbb{H}}}^2(X_{\ge n-2}, [\sigma_{\ge n-1} \mathcal O \xrightarrow{exp} j_*^m\mathcal O^*_U]) \to H^1(X_{\ge n-1}, \mathcal O_X)\oplus H^1(X_{n-2}, j^m_*\mathcal O^*_U) \to\\ \to H^1(X_{\ge n-1}, j^m_*\mathcal O^*_U) \to \end{gathered}$$ Via (\[haccha1\]), we can interpret ${{\mathbb{H}}}^2(X_{\ge n-2}, [\sigma_{\ge n-1} \mathcal O \xrightarrow{exp} j_*^m\mathcal O^*_U])$ as the set of isomorphism classes of triples $(L, P, \alpha)$ where $L$ is an $\mathcal O_{X_{\ge n-1}}$-torsor and $P$ is a $j_*^m\mathcal O^*_{U_{n-2}}$-torsor and $\alpha$ is an isomorphism[^8] $exp_m(L) {\xrightarrow{\sim}}\delta^*P$ of $j_*^m\mathcal O^*_{U_{\ge n-1}}$-torsors. Via (\[merext\]), $\alpha$ can also be thought of as an isomorphism (as in [@h 10.3.10c]) $exp(L) {\xrightarrow{\sim}}\delta^*P'\otimes \mathcal O(D')$ of invertible sheaves on $X_{\ge n-1}$ with $D' \in K$. If $P''$ is another invertible sheaf on $X_{n-2}$ corresponding to $P$, then we have $\alpha'':exp(L) {\xrightarrow{\sim}}\delta^* P''\otimes \mathcal O(D'')$ and $P'' {\xrightarrow{\sim}}P' \otimes \mathcal O(v)$ ($v\in W_{n-2}$); so one can rewrite $\alpha$ as $exp(L) {\xrightarrow{\sim}}\delta^*(\mathcal O(-v)\otimes P'') \otimes \mathcal O(D''+ \delta^*v)$. As $(P',D')- (P'', D'') = \gamma_{n-3}(v) \in NS_{n-2}\oplus K$, the element $w= (P', D') \in \mathcal C'_n$ depends only on the isomorphism class of the triple $(L, P, \alpha)$. Thus, we may associate the pair $(w, [L])\in \mathcal C'_n\oplus {\rm Lie}~\mathcal G_n$ with the isomorphism class of $(L, P, \alpha)$. The map $\delta^*:H^1(X_{n-2}, \mathcal O) \to {{\mathbb{H}}}^2(X_{\ge n-2}, [\sigma_{\ge n-1} \mathcal O \xrightarrow{exp} j_*^m\mathcal O^*_U])$ can be described as follows. The class $[I]\in H^1(X_{n-2}, \mathcal O)$ of an $\mathcal O_{X_{n-2}}$-torsor $I$ is mapped to the class of the triple $(\delta^* I, P= exp_m(I), \alpha_I)$ with $\alpha_I: exp_m(\delta^*I) {\xrightarrow{\sim}}\delta^*(exp_m(I))$ the tautological isomorphism; associated with this triple is the invertible sheaf $P' =exp(I)$ (\[merext\]), the element $D =0$ of $K$ and the isomorphism $\alpha_I: exp(\delta^*I) {\xrightarrow{\sim}}\delta^*(exp (I))$. If $P''$ is another invertible sheaf corresponding to the $j_*^m\mathcal O^*_U$-torsor $exp_m(I)$, then as before $P'' {\xrightarrow{\sim}}P'\otimes \mathcal O(v)$ ($v\in W_{n-2}$); thus, to the triple $(\delta^* I, exp_m(I), \alpha_I)$, we may attach the pair $(0,[\delta^*I]) \in \mathcal C'_n\oplus {\rm Lie}~\mathcal G_n$. Taking into account the surjectivity of $H^0(X_{\ge n-1}, \mathcal O) \xrightarrow{exp} H^0(X_{\ge n-1}, \mathcal O^*)$, we find that elements of $$\frac{{{\mathbb{H}}}^2(X_{\ge n-2}, [\sigma_{\ge n-1} \mathcal O \xrightarrow{exp} j_*^m\mathcal O^*_U])}{H^1(X_{n-2}, \mathcal O)}$$ can be identified with pairs $(w, L)$ where $L$ is an element of Lie $\mathcal P_n$ and (as before) $w \in \mathcal C'_n$ with $exp(L) = \rho(w)$ in $\mathcal P_n$. This last equality forces, by (\[samos\]), $w$ to be an element of $\mathcal C_n$. Recalling the definition of $T_{{{\mathbb{Z}}}}$ [@h 10.1.3.1], we deduce the required isomorphism. Consider the composite inclusion $$\label{devaki} h^2_X {\hookrightarrow}\frac{{{\mathbb{H}}}^2(X_{\ge n-2}, [\sigma_{\ge n-1} \mathcal O \xrightarrow{exp} j_*^m\mathcal O^*_U])}{H^1(X_{n-2}, \mathcal O)}~ {{\xrightarrow{\sim}}}~ T_{{{\mathbb{Z}}}}[\mathcal C_n \xrightarrow{\rho} \mathcal P_n];$$$h^2_X\otimes{{\mathbb{Q}}}$ (resp. $T_{{{\mathbb{Z}}}}[\mathcal C_n \xrightarrow{\rho} \mathcal P_n]\otimes{{\mathbb{Q}}}$) is an extension (\[kvk\]) (resp. [@h 10.1.3.1]) of $\mathcal C_n\otimes{{\mathbb{Q}}}$ by $H_1(\mathcal P_n, {{\mathbb{Q}}})$. Thus, we obtain an isomorphism of abelian groups $$h^2_X \xrightarrow{\sim}~ T_{{{\mathbb{Z}}}}[\mathcal C_n \xrightarrow{\rho} \mathcal P_n] \quad \textrm{(modulo finite groups)}.$$From (\[bareesh\]) and (\[nyaya\]), we deduce an isomorphism of abelian groups $$\Lambda: t^n(U_{{\bullet}}) \xrightarrow{\sim}~ T_{{{\mathbb{Z}}}}L^n \quad \textrm{(modulo finite groups)}.$$ [**Compatibilities of $\Lambda$.**]{} ${\bullet}$ *Weight filtration*: Since $NS(X_{n-3})\otimes{{\mathbb{Q}}}= t^2(X_{n-3})\otimes{{\mathbb{Q}}}$ is of type $(0,0)$, from (\[nyaya\]) we have $W_{-1}h^2_X\otimes{{\mathbb{Q}}}\xrightarrow{\sim} W_{-1}t^n(U_{{\bullet}})\otimes{{\mathbb{Q}}}$. The elements of $W_{-1}h^2_X$ are characterized, by (\[palani\]), (\[horanadu\]) and (\[curie\]), as those corresponding to pairs $(L, \beta)$ with $exp(L) = 0 =\beta$. This proves the compatibility of $\Lambda$ with the weight filtration. ${\bullet}$ *Hodge filtration*: From [@h 10.1.3.1], the map $\alpha: T_{{{\mathbb{Z}}}}L^n {\rightarrow}{\rm Lie}~\mathcal P_n$ (used to construct $T_{{{\mathbb{Z}}}}L^n$) gives the Hodge filtration. Namely, $$F^0(T_{{{\mathbb{Z}}}}L^n \otimes {{\mathbb{C}}}) = {\operatorname{Ker}}(\alpha_{{{\mathbb{C}}}}: (T_{{{\mathbb{Z}}}}L^n)\otimes{{\mathbb{C}}}{\rightarrow}{\rm Lie}~\mathcal P_n).$$ Therefore, we obtain $$\alpha_{{{\mathbb{C}}}}: \frac{(T_{{{\mathbb{Z}}}}L^n)\otimes{{\mathbb{C}}}}{F^0} \xrightarrow{\sim} {\rm Lie}~\mathcal P_n.$$ Since $t^n(U_{{\bullet}})$ is of type $(*)$, we get [@h 10.1.3.3] the isomorphism $$\frac{t^n(U_{{\bullet}})\otimes{{\mathbb{C}}}}{F^0} \xleftarrow{\sim} \frac{W_{-1}t^n(U_{{\bullet}})\otimes{{\mathbb{C}}}}{F^0 \cap W_{-1}}.$$ Now (\[ohoh!\]) and (\[kona\]) together imply that $$W_{-1} t^n(U_{{\bullet}})\otimes{{\mathbb{C}}}\xrightarrow{\sim} \frac{{{\mathbb{H}}}^1(X_{\ge n-1}, \Gamma_0)}{{{\mathbb{H}}}^1(X_{n-2}, \Gamma_0)} \xleftarrow{\sim} {\rm Lie}~\mathcal P^{\natural}_n.$$ Noting that the Hodge filtration on $H^*(U_{{\bullet}}, {{\mathbb{C}}})$ is induced by the filtration ([@h2 3.2.2], [@h 8.1.8, 8.1.12]) $$F^i \Omega(log~Y):= 0 \to 0 \to \cdots \to \Omega^i(log~Y) \xrightarrow{d} \Omega^{i+1}(log~Y) \to \qquad ,$$we obtain — cf. (\[penang\] (v), (vi)) — the commutativity of the following diagram $$\begin{CD} {} @. W_{-1}t^n(U_{{\bullet}})\otimes{{\mathbb{C}}}@<{\sim}<< {\rm Lie}~\mathcal P^{\natural}_n @>{\sim}>{(\ref{ohoh!})}> \frac{{{\mathbb{H}}}^1(X_{\ge n-1}, \Gamma_0)}{{{\mathbb{H}}}^1(X_{n-2}, \Gamma_0)}\\ @. @VVV @VVV @VVV\\ \frac{t^n(U_{{\bullet}})\otimes{{\mathbb{C}}}}{F^0} @<{\sim}<< \frac{W_{-1}t^n(U_{{\bullet}})\otimes{{\mathbb{C}}}}{F^0 \cap W_{-1}} @<{\sim}<< {\rm Lie}~\mathcal P_n @>{\sim}>{(\ref{koil})}> \frac{H^1(X_{\ge n-1}, \mathcal O)}{H^1(X_{n-2}, \mathcal O)},\\ \end{CD}$$ which fits into a larger commutative diagram $$\begin{CD} H^n(U_{{\bullet}},{{\mathbb{Z}}}(1))/{\rm torsion} @<<< t^n(U_{{\bullet}}) @>{\Lambda}>> T_{{{\mathbb{Z}}}}L^n @>{\alpha}>> {\rm Lie}~\mathcal P_n\\ @VVV @VVV @. @| \\ H^n(U_{{\bullet}},{{\mathbb{C}}}) @<<< t^n(U_{{\bullet}})\otimes{{\mathbb{C}}}@>>> \frac{t^n(U_{{\bullet}})\otimes{{\mathbb{C}}}}{F^0} @>>> {\rm Lie}~\mathcal P_n.\\ \end{CD}$$ This diagram shows that $\Lambda$ is compatible with the Hodge filtration thereby finishing the proof of Theorem \[main\]. Theorem \[main\] partly proves Conjecture \[dc1\] (up to isogeny) for the simplicial scheme $U_{{\bullet}}$; it remains to prove the statements concerning the de Rham and étale realizations. We first treat the de Rham realization. The étale realization is dealt with in §\[oakland\]. [**The de Rham realization of $L^n$.**]{} Using $T_{DR}L^n {\xrightarrow{\sim}}T_{{{\mathbb{C}}}}L^n$ [@h 10.1.8], $T_{{{\mathbb{C}}}}L^n {\xrightarrow{\sim}}t^n(U_{{\bullet}})\otimes{{\mathbb{C}}}$ (\[main\]), and $t^n(U_{{\bullet}})\otimes{{\mathbb{C}}}{\hookrightarrow}H^n(U_{{\bullet}}, {{\mathbb{C}}}) {\xrightarrow{\sim}}H^n_{DR}(U_{{\bullet}})$ [@h2 3.1.8, 3.2.2] gives us a map $$T_{DR}L^n \to H^n_{DR}(U_{{\bullet}});$$ our next task is to show that this map can be constructed purely algebraically (\[derham\]) and this, over any field of characteristic zero. In the remainder of this section, $k$ denotes a field of characteristic zero. Our next main result (\[derham\]) of this section requires us to construct a group scheme $\tilde{\mathcal U}$, a map $\tilde{\psi}: \mathcal B_n \to \tilde{\mathcal U}$ such that, when $k ={{\mathbb{C}}}$, $\tilde{\psi}$ lifts to a compatible homomorphism $\psi: T_{{{\mathbb{Z}}}}L^n \to {\rm Lie}~\tilde{\mathcal U}$. Once these are acquired, the criterion [@h 10.1.9] may be applied to deduce that ${\rm Lie}~\tilde{\mathcal U}$ is the de Rham realization $T_{DR}L^n$ of $L^n$. [**Step 1.**]{} *Construction of a map* $\psi_1: K \to \mathcal G_n^{\diamondsuit}$. Recall the map $q:\tilde{Y}_m \to Y_m \to X_m$ from the normalization $\tilde{Y}_m$ of $Y_m$. The Poincaré residue sequence [@h2 3.1.5.2] on $X_m$ and $X_{\ge n-1}$ $$0 {\rightarrow}\Omega^1 \rightarrow \Omega^1(log~Y) \xrightarrow{{\rm Res}} q_*\mathcal O_{\tilde{Y}} \rightarrow 0$$ gives the exact sequences $$\label{eq:5} H^0(\tilde{Y}_{m}, \mathcal O) \to H^1(X_{m}, \Omega^1) \to H^1(X_{m}, \Omega^1(log ~Y)),$$ $$\label{eq:2} H^0(\tilde{Y}_{\ge n-1}, \mathcal O) \to H^1(X_{\ge n-1}, \Omega^1) \to H^1(X_{\ge n-1}, \Omega^1(log ~Y)).$$ Since the first map of (\[eq:5\]) is the composition of $$W_{Y_m}(S)\otimes k {\xrightarrow{\sim}}H^0(\tilde{Y}_{m}, \mathcal O) \xrightarrow{\lambda_m} NS_{X_m}(S)\otimes k \underset{(\ref{beagle})}{\xrightarrow{c_{X_m}}} H^1(X_m, \Omega^1),$$ there is a natural injection — recall $N_m:= {\operatorname{Coker}}(\lambda_m)$ (\[auro2\]) — $$\label{eq:7} \kappa_m: {N_m}(S) \otimes k {\hookrightarrow}H^1(X_{\ge n-1}, \Omega^1(log ~Y)).$$ The boundary map of (\[eq:2\]) is induced by the composite map $$K(S) \xrightarrow{\vartheta'} H^1(X_{\ge n-1}, \mathcal O^*) = \mathcal G'_n (S) \to \pi_0(\mathcal G'_n) \underset{(\ref{dlog})}{\xrightarrow{dlog}} H^1(X_{\ge n-1}, \Omega^1)$$via $K(S)\otimes k {\xrightarrow{\sim}}H^0(\tilde{Y}_{\ge n-1}, \mathcal O)$. Thus, in the exact sequence $${{\mathbb{H}}}^1(X_{\ge n-1}, [\mathcal O^* \xrightarrow{d~{\rm log}} \Omega^1(log ~Y)]) \to H^1(X_{\ge n-1}, \mathcal O^*) \to H^1(X_{\ge n-1}, \Omega^1(log ~Y)),$$ one finds, by the exactness of (\[eq:2\]), that the map $\vartheta'$ admits a lifting to ${{\mathbb{H}}}^1(X_{\ge n-1}, \Gamma^*)$. In fact, one can easily construct a natural lifting $\psi_1: K(S) \to {{\mathbb{H}}}^1(X_{\ge n-1}, \Gamma^*) = \mathcal G_n^{\diamondsuit}(S)$ of $\vartheta'$ using Čech cohomology; let $C^*(G)$ denote Čech cochains of $X_{\ge n-1}$ with coefficients in $G$ (relative to a suitable open cover $\{U_i\}$) and $\partial$ the Čech differential. By (\[curie\]c), for each element $E \in K$, we have to construct an invertible sheaf $L$ on $X_{\ge n-1}$ and a connection $\nabla$ with at most logarithmic poles along $Y_{\ge n-1}$. In terms of cocycles [@mes 2.5] [@katzdiff 7.2], using $C^1(\Gamma^*) = C^1(\mathcal O^*) \oplus C^0(\Omega^1(log~Y))$, if $\{s_{ij}\}\in C^1(\mathcal O^*)$ represents $L$ and $\{\omega_i\} \in C^0(\Omega^1(log~Y))$ represents $\nabla$, then these satisfy $ds_{ij}/{s_{ij}} = \omega_j - \omega_i$; $\nabla$ is integrable if and only if $\omega_i$ is closed. Let $E\in K$ be an effective divisor. If $\{f_i\}$ are local equations for $E$, consider the cochain $(s_{ij}, df_i/{f_i})\in C^1(\mathcal O^*)\oplus C^0(\Omega^1(log~Y))$ where $f_j = f_i s_{ij}$; clearly, $df_j/{f_j} - df_i/{f_i} = ds_{ij}/{s_{ij}}$. Thus, the pair $(s_{ij}, df_i/{f_i})$ is a cocycle; and, it represents an element of ${{\mathbb{H}}}^1(X_{\ge n-1}, \Gamma^*)$. If $\{g_i\}$ are different local equations for $E$, then one gets an element $(t_{ij}, dg_i/{g_i})\in C^1(\mathcal O^*)\oplus C^0(\Omega^1(log~Y))$ with $g_j = g_i t_{ij}$. There exist $u_i \in C^0(\mathcal O^*)$ with $f_i = u_i g_i$. Since $s_{ij}/t_{ij} = u_j/u_i$ and $ds_{ij}/{s_{ij}} - dt_{ij}/{t_{ij}} = du_j/{u_j} - du_i/{u_i}$, i.e., the cocycle $(t_{ij}, dg_i/{g_i}) -(s_{ij}, df_i/{f_i})$ is a coboundary ($\partial u$), the element $\psi_1(E) = (s_{ij}, df_i/{f_i})$ of ${{\mathbb{H}}}^1(X_{\ge n-1}, \Gamma^*)$ depends only on $E$ but not on the choice of the local defining equations. The association $E \mapsto \psi_1(E)$ defined for effective $E$ easily extends to a homomorphism $\psi_1: K \to \mathcal G_n^{\diamondsuit}$. [**Step 2.**]{} *Construction of the scheme* $\tilde{\mathcal U}$. Let $g$ be the structure map $X_{\ge n-2} \to S$; recall $a_m$ is the structure map $X_m \to S$. \[sec:de-rham-realization-1\] The $S_{fppf}$-sheaf $R^2g_*\mathcal K^*$ associated with the complex $\mathcal K^*:= [\sigma_{\ge n-1} \mathcal O^* \xrightarrow{d~{\rm log}} \Omega^1(log~Y)]$ on $X_{\ge n-2}$ is representable. One has $\xi_*: \pi_0(R^2g_*\mathcal K^*) {\hookrightarrow}\pi_0(\mathcal G'_n)$; here $R^2g_*\mathcal K^*$ denotes the associated representing group scheme. One has an exact sequence $$\begin{gathered} H^0(X_{\ge n-1}, \mathcal O^*) \to H^1(X_{\ge n-2}, \Omega^1(log~Y)) \to {{\mathbb{H}}}^2(X_{\ge n-2}, \mathcal K^*) \xrightarrow{\xi} \\ \to H^1(X_{\ge n-1}, \mathcal O^*) \to H^2(X_{\ge n-2}, \Omega^1(log~Y));\end{gathered}$$ the sheaf $R^1f_*\mathcal O^*$ is representable by $\mathcal G'_n$, $f_*\mathcal O^*$ by a torus, and $R^ig_* \Omega^1(log~Y)$ are representable by vector group schemes. Since $\Hom(f_*\mathcal O^*, R^1g_* \Omega^1(log~Y)) = 0$, one can now invoke (\[reps\]) to obtain the representability of $R^2g_*\mathcal K^*$. Since ${\operatorname{Ker}}(\xi)$ is connected, the map $\xi_*$ is injective. The inclusion $\sigma_{\ge n-1}\Gamma^* {\hookrightarrow}\mathcal K^*$ induces an exact sequence $$\label{eq:6} 0 \to \frac{\mathcal G_n^{\diamondsuit}}{a_{n-2~*} \Omega^1(log~Y)} \to R^2g_*\mathcal K^* \xrightarrow{\upsilon} R^1a_{n-2~*} \Omega^1(log~Y).$$ Consider the commutative diagram $$\begin{CD} H^1(X_{n-3}, \Omega^1(log~Y)) @>{\delta^*}>> H^1(X_{\ge n-2}, \Omega^1(log~Y))\\ @VV{\delta^*_{n-3}}V @VVV\\ H^1(X_{n-2}, \Omega^1(log~Y)) @<{\upsilon}<< {{\mathbb{H}}}^2(X_{\ge n-2}, \mathcal K^*),\\ \end{CD}$$and the map $$Pic^0_{X_{n-2}}(S) {\hookrightarrow}H^1(X_{n-2}, \mathcal O^*) \to {{\mathbb{H}}}^2(X_{\ge n-2}, \mathcal K^*)$$induced by the inclusion $\mathcal K^* {\hookrightarrow}\Gamma^*$ on $X_{\ge n-2}$. Taking the quotient of ${{\mathbb{H}}}^2(X_{\ge n-2}, \mathcal K^*)$ by the images of $H^1(X_{n-3}, \Omega^1(log~Y))$ and $Pic^0_{X_{n-2}}(S)$ under these maps, and pulling back via (cf. (\[eq:6\]) (\[eq:7\])) $$\label{eq:8}\frac{{\operatorname{Ker}}(\delta^*_{n-2}:N_{n-2}\to N_{n-1})(S)\otimes k}{NS_{n-3}(S)\otimes k} \to \frac{N_{n-2}(S)\otimes k}{NS_{n-3}(S)\otimes k} \to \frac{H^1(X_{n-2}, \Omega^1(log~Y))}{H^1(X_{n-3}, \Omega^1(log~Y))}$$ (actually we do this at the level of the group schemes which represent the associated sheaves) gives us a group scheme $\mathcal U'$. Its identity component is denoted $\mathcal U$. \[sec:de-rham-realization\] The map $\xi_*:\pi_0(\mathcal U') {\hookrightarrow}\pi_0(\mathcal P'_n)$ is injective. It is clear that the previous operations of quotient and pullback do not affect $\pi_0$ and thus $\pi_0(R^2g_*\mathcal K^*) \cong \pi_0(\mathcal U')$. The natural map $\xi: R^2 g_*\mathcal K^* \to \mathcal G'_n$ provides a map $\xi: \mathcal U' \to \mathcal P'_n$; by (\[sec:de-rham-realization-1\]), one has an injection $\xi_*: \pi_0(\mathcal U') {\hookrightarrow}\pi_0(\mathcal P'_n)$. One has an exact sequence $$\label{eq:9} 0 \to {\rm Lie}~\mathcal P^{\diamondsuit}_n \to {\rm Lie}~\mathcal U \xrightarrow{\upsilon} \frac{{\operatorname{Ker}}(\delta^*_{n-2}:N_{n-2}\to N_{n-1})(S)\otimes k}{NS_{n-3}(S)\otimes k}.$$ By (\[diksha\]), ${\rm Lie}~\mathcal P^{\diamondsuit}_n$ is a quotient of ${\rm Lie}~\mathcal G^{\diamondsuit}_n$ by ${\rm Lie}~Pic^{\diamondsuit}_{X_{n-2}}$. Since ${\rm Lie}~Pic^{\diamondsuit}_{X_{n-2}}$ is an extension of ${\rm Lie}~Pic^0_{X_{n-2}}$ by $a_{n-2~ *}\Omega^1(log ~Y)$, we obtain (\[eq:9\]) noting (\[eq:6\]). One just has to observe that the image of $Pic^0_{X_{n-2}}$ in $R^2g_*\mathcal K^*$ is equal to that of the neutral component of $Pic^{\diamondsuit}_{X_{n-2}}$ under the natural map $ Pic^{\diamondsuit}_{X_{n-2}} \xrightarrow{\delta^*} \mathcal G^{\diamondsuit}_n \to R^2g_*\mathcal K^*$. Since the composite map $$H^1(X_{n-2}, \mathcal O^*) \to {{\mathbb{H}}}^2(X_{\ge n-2}, \mathcal K^*) \xrightarrow{\upsilon}H^1(X_{n-2}, \Omega^1(log~Y))$$is the map induced by $d~{\rm log}: \mathcal O^* \to \Omega^1 \to \Omega^1(log~Y)$, the natural map $NS_{n-2} \to \frac{R^2g_*\mathcal K^*}{Pic^0_{n-2}}$induces a map $$\psi_2:NS_{n-2} \to \mathcal U'.$$By definition (\[eq:8\]) of $\mathcal U'$, ${\operatorname{Im}}(\delta^*_{n-3}: NS_{n-3} \to NS_{n-2})$ is contained in ${\operatorname{Ker}}({\psi_2})$. The map $\psi': K \oplus NS_{n-2} \to \mathcal U'$ defined by $(u,v) \mapsto \psi_1(u) + \psi_2(v)$ provides a map $\psi':\mathcal B'_n \to \mathcal U$. Since $\psi' = \psi_2$ on the subgroup $\gamma_{n-3}(NS_{n-3}) = \delta^*_{n-3}(NS_{n-3})$, $NS_{n-3}$ is contained in ${\operatorname{Ker}}({\psi'})$. It is straightforward to check that $\gamma_{n-3}(W_{Y_{n-2}})$ is in ${\operatorname{Ker}}({\psi'})$. This gives a map $\psi': \mathcal B'_n \to \mathcal U'$. Note that the composite map $$K \oplus NS_{n-2} \xrightarrow{\psi'} \mathcal U' \xrightarrow{\xi} \mathcal P'_n$$ is $\rho$. By (\[sec:de-rham-realization\]), the image under $\psi'$ of $\mathcal B'_n$ is contained in $\mathcal U$. Put $\tilde{\mathcal U} = \mathcal U/{\psi'(\tau_n)}$; here, $\tau_n$ is the torsion subgroup of $\mathcal B'_n$. The map $\psi'$ induces a map $\tilde{\psi}: \mathcal B_n \to \tilde{\mathcal U}$. \[derham\] [(i) ($k={{\mathbb{C}}}$)]{} There is a natural commutative diagram $$\begin{CD} \mathcal B_n @>{\tilde{\psi}}>> \tilde{\mathcal U} @>{\xi}>> \tilde{\mathcal P}_n\\ @AAA @AAA @AAA\\ T_{{{\mathbb{Z}}}}L^n @>{\psi}>> {\rm Lie}~\tilde{\mathcal U} @>{\xi}>> {\rm Lie}~\tilde{\mathcal P}_n\\ \end{CD}$$ whose exterior square is [@h 10.1.3.1] for $L^n$; further, ${\psi}$ induces an isomorphism $\psi_{{{\mathbb{C}}}}:T_{{{\mathbb{C}}}}L^n {\xrightarrow{\sim}}{\rm Lie}~\tilde{\mathcal U}$. [(ii)]{} $T_{DR} L^n {\xrightarrow{\sim}}{\rm Lie}~\tilde{\mathcal U}$. [(iii)]{} There is a natural map ${\rm Lie}~\tilde{\mathcal U} \to H^n_{DR}(U_{{\bullet}})$. \(i) ($k = {{\mathbb{C}}}$) From (\[codered\]), (\[kapadu\]) and (\[hosadu\]), we obtain an exact sequence of isogeny one-motives over ${{\mathbb{C}}}$ $$0 \to \frac{[K^0 \to \mathcal G_n]}{[\mathcal V_{n-2} \to Pic^0_{n-2}]}\otimes{{\mathbb{Q}}}\to [\mathcal C_n \xrightarrow{\rho} \mathcal P_n]\otimes{{\mathbb{Q}}}\to [{\operatorname{Ker}}(\delta^*_{n-2}: N_{n-2} \to N_{n-1}) \to 0]\otimes{{\mathbb{Q}}}\to 0.$$ The proof (in Step 3.) of Theorem \[main\] also shows that (\[devaki\]) is an isomorphism (modulo finite groups) of mixed Hodge structures. Combining this with (\[k3\]), we obtain that $h^2_X\otimes{{\mathbb{C}}}{\xrightarrow{\sim}}T_{{{\mathbb{C}}}}[\mathcal C_n \xrightarrow{\rho} \mathcal P_n]$ sits in an exact sequence $$0 \to {\rm Lie}~\mathcal P^{\diamondsuit}_n \to h^2_X\otimes{{\mathbb{C}}}\to {\operatorname{Ker}}(\delta^*_{n-2}: N_{n-2} \to N_{n-1})\otimes{{\mathbb{C}}}\to 0.$$ Recall that there is a map $h^2_X\otimes{{\mathbb{C}}}\to {{\mathbb{H}}}^2(X_{\ge n-2}, \mathcal K) = {\rm Lie}~R^2g_*\mathcal K^*$ and that the latter sits in an exact sequence $$0 \to \frac{{{\mathbb{H}}}^1(X_{\ge n-1}, \Gamma)}{H^0(X_{n-2}, \Omega^1(log~Y))} \to {{\mathbb{H}}}^2(X_{\ge n-2}, \mathcal K) \to H^1(X_{n-2}, \Omega^1(log~Y)) \to {{\mathbb{H}}}^2(X_{\ge n-1}, \Gamma).$$ Comparing with (\[eq:9\]), the map $h^2_X\otimes{{\mathbb{C}}}\to {\rm Lie}~R^2g_*\mathcal K^*$ gives a map $$\psi_{{{\mathbb{C}}}}:T_{{{\mathbb{C}}}}L^n\underset{(\ref{nyaya}), (\ref{main})}{\xleftarrow{\sim}} \frac{h^2_X\otimes{{\mathbb{C}}}}{NS(X_{n-3})\otimes{{\mathbb{C}}}} \to {\rm Lie}~\tilde{\mathcal U},$$whose composition with $T_{{{\mathbb{Z}}}}L^n {\hookrightarrow}T_{{{\mathbb{C}}}}L^n$ gives us the map $\psi$. Since the composite map $ K \oplus NS_{n-2} \xrightarrow{\psi'} \mathcal U' \xrightarrow{\xi} \mathcal P'_n$ is $\rho$ — see (\[greece\]), it follows easily that the composite map $\xi\circ \tilde{\psi}: \mathcal B_n \to \tilde{\mathcal P}_n$ is $\phi_n$. Thus, the exterior square is the one that intervenes in the definition [@h 10.1.3.1] of $T_{{{\mathbb{Z}}}}L^n$. We leave it to the reader to check that $\psi_{{{\mathbb{C}}}}$ is an isomorphism and that it is compatible with the weight and Hodge filtrations. \(ii) This follows from (i) by [@h 10.1.9]. \(iii) Compose the natural maps $${\rm Lie}~\tilde{\mathcal U} \to {{\mathbb{H}}}^2(X_{\ge n-2}, \mathcal K) \to {{\mathbb{H}}}^n(X_{{\bullet}}, \Omega(log~Y)) {\xrightarrow{\sim}}H^n_{DR}(U_{{\bullet}}),$$the last being a consequence of [@h2 3.1.8], [@katzdiff 1.0.3.7], [@jamm 6.11.4]. Etale realization of $L^n$ {#oakland} ========================== Fix a simplicial pair $(X_{{\bullet}}, Y_{{\bullet}})$ over a field $k$ of characteristic zero. When $k = {{\mathbb{C}}}$, we have a map (see below for notations) $$TL^n\otimes{{\mathbb{Q}}}{\xrightarrow{\sim}}(T_{{{\mathbb{Z}}}}L^n)\otimes_{{{\mathbb{Z}}}}{{\mathbb{A}}}\underset{(\ref{main})}{{\xrightarrow{\sim}}}T_{{{\mathbb{Z}}}}I^n(U_{{\bullet}})\otimes_{{{\mathbb{Z}}}}{{\mathbb{A}}}{\hookrightarrow}H^n(U_{{\bullet}}, {{\mathbb{Z}}}(1))\otimes_{{{\mathbb{Z}}}} {{\mathbb{A}}}\underset{(\ref{artgro})}{{\xrightarrow{\sim}}}H^n_{et}(U_{{\bullet}}, {{\mathbb{A}}}(1));$$ the first isomorphism is from [@h 10.1.6]. We shall show that this map can be constructed purely algebraically for all $k$; combined with (\[main\]), (\[derham\]), this will prove (\[dc1\]) up to isogeny for $U_{{\bullet}}$. By (\[zeusp\]), we may and do assume that $k$ is a finitely generated extension of ${{\mathbb{Q}}}$. We adopt the following notations: $r$ is a positive integer, $\ell$ is a positive prime integer, $H^i_{et}(\bar{V},\hat{{{\mathbb{Z}}}}(1))$ is ${\varprojlim}_r H^i_{et}(\bar{V},\1_r)$ and $H^i_{et}{}(\bar{V},\mathbb A(1))$ is $H^i_{et}{}(\bar{V},\hat{{{\mathbb{Z}}}}(1))\otimes{{\mathbb{Q}}}$. For any one-motive $M$, recall the finite ${{\mathbb{Z}}}/{r{{\mathbb{Z}}}}$-module $T_{{{\mathbb{Z}}}/{r{{\mathbb{Z}}}}}M:= H^0(M\stackrel{L}{\otimes}{{{\mathbb{Z}}}/{r{{\mathbb{Z}}}}})$ [@h 10.1.5]; note $T_{\ell}M:= {\varprojlim}_d T_{{{\mathbb{Z}}}/{\ell^d{{\mathbb{Z}}}}}M$, $M\otimes\hat{{{\mathbb{Z}}}} = TM = {\varprojlim}_r T_{{{\mathbb{Z}}}/{r{{\mathbb{Z}}}}}M$, and $M\otimes{{\mathbb{A}}}:= TM\otimes{{\mathbb{Q}}}$. For any commutative group (scheme) $\mathcal A$, $_r\mathcal A$ is the ${\operatorname{Ker}}(\mathcal A \xrightarrow{r}\mathcal A)(\bar{S})$ and $\mathcal A_{tor}$ the torsion subgroup (scheme). Let ${{\mathbb{Z}}}/{r{{\mathbb{Z}}}}$ denote the constant group. All maps in this section are ${{\mathbb{G}}}$-equivariant. Recall the Kummer sequence of étale sheaves on a (simplicial) scheme $V$ $$\label{kummer} 0 \rightarrow \1_{r} \rightarrow {{\mathbb{G}}_m}\xrightarrow{r} {{\mathbb{G}}_m}\rightarrow 0.$$ For proper $V$, this gives, by the divisibility of $H^0_{et}{}(\bar{V},{{\mathbb{G}}_m})$, $$\label{kap} H^1_{et}{}(\bar{V},\1_r) {\xrightarrow{\sim}}~_r H^1_{et}{}(\bar{V},{{\mathbb{G}}_m});$$ for $V$ smooth and proper, we also obtain $$\label{kumns} \kappa_V: NS(\bar{V})\otimes{{{\mathbb{Z}}}/{r{{\mathbb{Z}}}}} {\hookrightarrow}H^2_{et}{}(\bar{V},\1_r).$$ Similarly, using the divisibility of $\mathcal G_n(\bar{S})$, one obtains an injection $$\label{gandhi} \pi_0(\mathcal G'_n)(\bar{S})\otimes{{{\mathbb{Z}}}/{r{{\mathbb{Z}}}}} {\hookrightarrow}H^2_{et}{}(\bar{X}_{\ge n-1},\1_r).$$ \[specet\] The spectral sequence [@h 8.1.19.1] has an étale analogue $E_1^{et}$ [@dj Introduction] [@h1 §6] [@hp §14]; it calculates the étale cohomology of $\bar{U}_{{\bullet}}$ with ${{\mathbb{Q}}}_{\ell}$-coefficients. The degeneration of the étale analogue at $E_2$ (i.e., that $E_2^{et} = E_{\infty}^{et}$) and the definition of the weight filtration $W$ on $H^*_{et}(\bar{U}, {{\mathbb{Q}}}_{\ell}(1))$ is a consequence of [@weil]; see [@h1 §6,7], [@hp §14] for these and the fact that the weight filtrations in mixed Hodge theory are compatible with those in étale cohomology, compared via (\[artgro\]). Recall the Artin-Grothendieck comparison theorem [@art §4] [@ka p.23]: for any variety $V$ over ${{\mathbb{C}}}$, one has canonical isomorphisms $$\label{artgro} H^i_{et}(V, {{\mathbb{Z}}}/{r{{\mathbb{Z}}}}) {\xrightarrow{\sim}}H^i(V({{\mathbb{C}}}), {{\mathbb{Z}}}/{r{{\mathbb{Z}}}})$$ between the étale and classical (singular) cohomology of $V$. Any imbedding $\iota: k {\hookrightarrow}{{\mathbb{C}}}$ gives an isomorphism of $\bar{k}$ with the algebraic closure $\overline{\iota(k)}$ of $\iota(k)$ in ${{\mathbb{C}}}$; this, in turn, provides an isomorphism $H^*_{et}(U_{m}\times_k \bar{k}, {{\mathbb{Z}}}/{r{{\mathbb{Z}}}}) {\xrightarrow{\sim}}H^*_{et}(U_{m}\times_k \overline{\iota(k)}, {{\mathbb{Z}}}/{r{{\mathbb{Z}}}}) {\xrightarrow{\sim}}H^*_{et}(U_{m}\times_{\iota} {{\mathbb{C}}},{{\mathbb{Z}}}/{r{{\mathbb{Z}}}})$; the last isomorphism is from [@mi VI 2.6]. Now (\[artgro\]) and the degeneration [@h 8.1.20 (ii)] of [@h 8.1.19.1] at $E_2$ together provide another proof that $E_2^{et} = E_{\infty}^{et}$. \[bombay\] One has [[(i)]{}]{} $$\frac{H^1_{et}{}(\bar{X}_{\ge n-1},{{\mathbb{A}}}(1))} {H^1_{et}{}(\bar{X}_{n-2}, {{\mathbb{A}}}(1))} {\xrightarrow{\sim}}\sigma_{\ge n-1} H^n_{et}{}(\bar{X}_{{\bullet}},{{\mathbb{A}}}(1)) = W_{-1}H^n_{et}{}(\bar{X}_{{\bullet}},{{\mathbb{A}}}(1)) {\xrightarrow{\sim}}W_{-1} H^n_{et}{}(\bar{U}_{{\bullet}},{{\mathbb{A}}}(1)).$$ [[(ii)]{}]{} $H^1_{et}{}(\bar{X}_{\ge n-1},\hat{{{\mathbb{Z}}}}(1))\xrightarrow{\sim} T[0 \to {\mathcal G}_n]$ and $T[0 \to {\mathcal P}_n]\otimes{{\mathbb{Q}}}\xleftarrow{\sim} W_{-1} H^n_{et}{}(\bar{U}_{{\bullet}},{{\mathbb{A}}}(1))$. \(i) As in (\[malli\]), this follows from the degeneration at $E_2$ of the étale analogue of [@h 8.1.19.1] for ${{\mathbb{Q}}}_{\ell}(1)$ — see (\[specet\]). \(ii) The first isomorphism follows from (\[kap\]) for $V = X_{\ge n-1}$ by taking the inverse limit over $r$ (note $\pi_0(\mathcal G')_{tor}$ is finite). Similarly, one has (\[picmd\] (i)) $H^1_{et}{}(\bar{X}_{n-2},{{\mathbb{Z}}}_{\ell}(1)) \xrightarrow{\sim}T_{\ell}Pic(X_{n-2})$. Combining this with (i) gives the second isomorphism. \[finel\] [[(i)]{}]{} $H^0([W_{Y_m} \xrightarrow{\lambda_{m}} Pic_{X_{m}}]\stackrel{L}{\otimes} {{\mathbb{Z}}}/{r{{\mathbb{Z}}}})) \xleftarrow{\sim} H^1_{et}{}(\bar{U}_{m}, \1_{r})$. [[(ii)]{}]{} $H^0([K \xrightarrow{\vartheta'} \mathcal G'_n]\stackrel{L}{\otimes } {{\mathbb{Z}}}/{r{{\mathbb{Z}}}}) \xleftarrow{\sim} H^1_{et}{}(\bar{U}_{\ge n-1},\1_{r})$. It is exactly identical to [@h 10.3.6]; we repeat the proof of (i) here for the convenience of the reader. By (\[haccha1\]), $H^1_{et}{}(\bar{U}_{m}, \1_{r}) {\xrightarrow{\sim}}{{\mathbb{H}}}^1_{et}(\bar{U}_{m}, [{{\mathbb{G}}_m}\xrightarrow{r} {{\mathbb{G}}_m}])$ is the set of isomorphism classes of pairs $(\mathcal L, \alpha)$ where $\mathcal L$ is an invertible sheaf on $\bar{U}_m$ together with an isomorphism $\alpha: \mathcal L^{\otimes r} {\xrightarrow{\sim}}\mathcal O$. Let $(\mathcal L, \alpha)$ be such a pair. The invertible sheaf $\mathcal L$ extends to an invertible sheaf $\tilde{\mathcal L}$ on $\bar{X}_m$, and there exists a divisor $E$ of $\bar{X}_m$ with support in $\bar{Y}_m$ such that $\alpha$ extends to an isomorphism $\alpha': \tilde{\mathcal L}^{\otimes r} {\xrightarrow{\sim}}\mathcal O(E)$. If there exists an isomorphism $\beta$ of $\tilde{\mathcal L}^{\otimes r}$ with $\mathcal O(E)$, this isomorphism is uniquely determined up to multiplication by an element of the divisible group $H^0_{et}(\bar{X}_m, {{\mathbb{G}}_m})$. One deduces that the pair $(\tilde{\mathcal L}, E)$ determines $(\mathcal L, \alpha)$ up to isomorphism. For a pair $(\tilde{\mathcal L}, E)$ to come from a suitable $(\mathcal L, \alpha)$, it is necessary and sufficient that $r[\tilde{\mathcal L}] = [\mathcal O(E)]$ in $Pic_{\bar{X}_m}$. It comes from $(\mathcal O_{\bar{X}_m}, 0)$ if and only if it is of the form $(\mathcal O_{\bar{X}_m}(E), rE)$. This identifies $H^1_{et}{}(\bar{U}_{m}, \1_{r})$ with $H^0$ of the complex which is the tensor product of $[W_{Y_m}(\bar{S}) \xrightarrow{\lambda_{m}} Pic(\bar{X}_{m})]$ (degrees $0$ and $1$) and $ [{{\mathbb{Z}}}\xrightarrow{r} {{\mathbb{Z}}}]$ (degrees $-1$ and $0$): $$\begin{CD} W_{Y_m}(\bar{S}) @>{\lambda_{m}}>> Pic(\bar{X}_{m})\\ @AA{r}A @AA{-r}A\\ W_{Y_m}(\bar{S}) @>{\lambda_{m}}>> Pic(\bar{X}_{m})\\ \end{CD}$$ This proves the first isomorphism. \[adivasi\] [[(i)]{}]{} The maps $[\mathcal B_n \xrightarrow{\phi_{n}}\tilde{{\mathcal P}}_n] \xleftarrow{b} [\mathcal B'_n \xrightarrow{\rho} {\mathcal P}_n] \xrightarrow{a} [\frac{\mathcal C'_n}{NS_{n-3}} \xrightarrow{\rho} \mathcal P'_n]$ of complexes [(]{}concentrated in degrees $0$ and $1$[)]{} induce isomorphisms [(]{}modulo finite groups[)]{} $$\label{maharaja} TL^n \xleftarrow{\sim}~{\varprojlim}_rH^0([\mathcal B'_n \xrightarrow{\rho} {\mathcal P}_n]\stackrel{L} {\otimes} {{{\mathbb{Z}}}/{r{{\mathbb{Z}}}}}) {\xrightarrow{\sim}}~{\varprojlim}_rH^0([\frac{\mathcal C'_n}{NS_{n-3}} \xrightarrow{\rho} \mathcal P'_n]\stackrel{L} {\otimes} {{{\mathbb{Z}}}/{r{{\mathbb{Z}}}}}).$$ [[(ii)]{}]{} The map $[\mathcal C_n \xrightarrow{\rho} \mathcal P_n]$ to $[\mathcal C'_n \xrightarrow{\rho} \mathcal P'_n]$ induces an isomorphism [(]{}modulo finite groups[)]{} $$\label{maharani} {\varprojlim}_rH^0([\mathcal C_n \xrightarrow{\rho} {\mathcal P}_n]\stackrel{L} {\otimes} {{{\mathbb{Z}}}/{r{{\mathbb{Z}}}}}) {\xrightarrow{\sim}}~{\varprojlim}_rH^0([\mathcal C'_n \xrightarrow{\rho} \mathcal P'_n]\stackrel{L} {\otimes} {{{\mathbb{Z}}}/{r{{\mathbb{Z}}}}}).$$ (cf. [@h 10.3.5 ii]) (i) The first isomorphism is clear because $b$ is an isogeny; the cokernel $[{\operatorname{Im}}(\rho') \to \pi_0(\mathcal P'_n)]$ of the map $a$ is quasi-isomorphic to $[ 0 \to {\operatorname{Coker}}(\rho')]$ and ${\operatorname{Coker}}(\rho')_{tor}$ is finite. This yields the second isomorphism. \(ii) The argument is similar to that of (i). The map $[\sigma_{\ge n-1}{{\mathbb{G}}_m}\xrightarrow{r} {{\mathbb{G}}_m}] \to [{{\mathbb{G}}_m}\xrightarrow{r} {{\mathbb{G}}_m}]$ of étale complexes on $U_{\ge n-2}$ gives the exact sequence $$\label{hippo} \frac{{{\mathbb{H}}}^2_{et}{}(\bar{U}_{\ge n-2}, [\sigma_{\ge n-1} {{\mathbb{G}}_m}\xrightarrow{r} {{\mathbb{G}}_m}])}{H^1_{et}(\bar{U}_{n-2}, {{\mathbb{G}}_m})} \to {{\mathbb{H}}}^2_{et}{}(\bar{U}_{\ge n-2}, [{{\mathbb{G}}_m}\xrightarrow{r} {{\mathbb{G}}_m}]) \to H^2_{et}(\bar{U}_{n-2}, {{\mathbb{G}}_m})$$ \[biget\][[(i)]{}]{} One has a canonical isomorphism $$\Pi_r: \frac{{{\mathbb{H}}}^2_{et}{}(\bar{U}_{\ge n-2}, [\sigma_{\ge n-1} {{\mathbb{G}}_m}\xrightarrow{r} {{\mathbb{G}}_m}])}{H^1_{et}(\bar{U}_{n-2}, {{\mathbb{G}}_m})} {\xrightarrow{\sim}}H^0([\mathcal C'_n \xrightarrow{\rho} \mathcal P'_n]\stackrel{L} {\otimes} {{{\mathbb{Z}}}}/{r{{\mathbb{Z}}}}).$$This gives an inclusion $\Lambda': T[\mathcal C_n\xrightarrow{\rho} \mathcal P_n]{\hookrightarrow}H^2_{et}(\bar{U}_{\ge n-2}, \hat{{{\mathbb{Z}}}}(1))$. [[(ii)]{}]{} The map $\Lambda'$ of [[(i)]{}]{} induces an inclusion $\Lambda_{et}: TL^n\otimes{{\mathbb{Q}}}{\hookrightarrow}H^n_{et}{}(\bar{U}_{{\bullet}},{{\mathbb{A}}}(1)).$ [[(iii)]{}]{} One has $\Lambda_{et}(T(W_{j}L^n \otimes{{\mathbb{Q}}})) \subseteq W_j H^n_{et}{}(\bar{U}_{{\bullet}},{{\mathbb{A}}}(1))$ with equality for $j = -2, -1$. \(i) The proof follows that of (\[finel\]). Via (\[haccha1\]), ${{\mathbb{H}}}^2_{et}{}(\bar{U}_{\ge n-2}, [\sigma_{\ge n-1} {{\mathbb{G}}_m}\xrightarrow{r} {{\mathbb{G}}_m}])$ is the group of isomorphism classes of triples $(I, \mathcal L, \alpha)$ where $I$ is a ${{\mathbb{G}}_m}$-torsor on $\bar{U}_{n-2}$ and $\mathcal L$ is a ${{\mathbb{G}}_m}$-torsor on $\bar{U}_{\ge n-1}$ and $\alpha: \delta^*I {\xrightarrow{\sim}}\mathcal L^{\otimes r}$. The class $[L] \in H^1_{et}(\bar{U}_{n-2}, {{\mathbb{G}}_m})$ of a ${{\mathbb{G}}_m}$-torsor $L$ gets mapped to the class $\delta^*[L]$ of the triple $(L^{\otimes r}, \delta^*L, \alpha_L)$ where $\alpha_L$ is the canonical isomorphism $\delta^*(L^{\otimes r}) {\xrightarrow{\sim}}(\delta^*L)^{\otimes r}$. Fix the class $f$ of a triple $(I, \mathcal L, \alpha)$. The torsors ${I}$ and $\mathcal L$ extend to torsors $\tilde{I}$ and $\tilde{\mathcal L}$ on $\bar{X}_{n-2}$ and $\bar{X}_{\ge n-1}$ respectively and there exists a divisor $E$ of $\bar{X}_{\ge n-1}$ supported on $\bar{Y}_{\ge n-1}$ such that $\alpha$ extends to an isomorphism $\tilde{\alpha}: \delta^*\tilde{I}\otimes \mathcal O(E) {\xrightarrow{\sim}}\tilde{\mathcal L}^{\otimes r}$. This says that the elements $w= (E, [\tilde{I}]) \in K \oplus NS_{n-2}$ and $[\tilde{\mathcal L}] \in \mathcal P'_n$ satisfy $\rho(w) = r [\tilde{\mathcal L}]$; in other words, the pair $(w, [\tilde{\mathcal L}])$ defines an element $\Pi_r(f)$ of $H^0([\mathcal C'_n \xrightarrow{\rho} \mathcal P'_n]\stackrel{L} {\otimes} {{{\mathbb{Z}}}}/{r{{\mathbb{Z}}}}).$ This element $\Pi_r(f)$ is clearly zero if our triple is isomorphic to $\delta^*[L]$: if $L$ extends to a ${{\mathbb{G}}_m}$-torsor $\tilde{L}$ on $\bar{X}_{n-2}$, then $\Pi_r(f)$ is given by $w = (0, r[\tilde{L}])$ and the class of $\rho([\tilde{L}]) \in \mathcal P'_n$. If $\tilde{I}'$ is another extension of $I$, then there is an isomorphism $\tilde{I} {\xrightarrow{\sim}}\tilde{I'} \otimes\mathcal O(d)$ ($d \in W_{n-2}$). Using $\tilde{I}'$ instead of $\tilde{I}$ modifies $w$ by $\gamma_{n-3}(d)$. If $\tilde{\mathcal L'}$ is another extension of $\mathcal L$, then there is an isomorphism $\tilde{\mathcal L} {\xrightarrow{\sim}}\tilde{\mathcal L'}\otimes \mathcal O(F)$ ($F \in K$). Using $ \tilde{\mathcal L'}$ instead of $\tilde{\mathcal L}$ modifies $w$ by $\pm rF$. This shows that the map $\Pi_r$ is well-defined; on the image of the map $\frac{H^1_{et}(\bar{U}_{\ge n-1}, \1_r)}{H^1_{et}(\bar{U}_{n-2}, \1_r)} \to {{\mathbb{H}}}^2_{et}{}(\bar{U}_{\ge n-2}, [\sigma_{\ge n-1} {{\mathbb{G}}_m}\xrightarrow{r} {{\mathbb{G}}_m}]),$ the map $\Pi_r$ reduces to the maps (isomorphisms) of (\[finel\]). Given an element $g$ of $H^0([\mathcal C'_n \xrightarrow{\rho} \mathcal P'_n]\stackrel{L} {\otimes} {{{\mathbb{Z}}}}/{r{{\mathbb{Z}}}})$, pick representatives $u\in K$, $\mathcal J_1 \in Pic_{n-2}$, and $\mathcal J \in \mathcal G'_n$ for $g$. By definition, $\mathcal O(u)\otimes\delta^* \mathcal J_1 {\xrightarrow{\sim}}\mathcal J^{\otimes r}\otimes \mathcal J_2$ where $\mathcal J_2$ is an element of $Pic^0_{n-2}$; by the divisibility of $Pic^0_{n-2}(\bar{S})$, the relation can be rewritten as $\mathcal O(u)\otimes\delta^* \mathcal J_1 {\xrightarrow{\sim}}\mathcal J^{\otimes r}$ for an appropriate element of $\mathcal G'_n$. Restricting to $\bar{U}_{\ge n-1}$, we obtain an isomorphism $\beta: \delta^* \mathcal J_1 {\xrightarrow{\sim}}\mathcal J^{\otimes r}$; the triple $(\mathcal J_1, \mathcal J, \beta)$ defines an element of ${{\mathbb{H}}}^2_{et}{}(\bar{U}_{\ge n-2}, [\sigma_{\ge n-1} {{\mathbb{G}}_m}\xrightarrow{r} {{\mathbb{G}}_m}])$. One easily verifies that this is an inverse to the map $\Pi_r$ and that $\Pi_r$ is an isomorphism. The map $\Lambda'$ is obtained by composing the inverse of ${\varprojlim}_r\Pi_r$ with the maps in (\[hippo\]) and (\[maharani\]). \(ii) Since $\delta^*H^2_{et}(\bar{U}_{n-3}, {{\mathbb{A}}}(1))$ is contained in ${\operatorname{Ker}}(H^2_{et}(\bar{U}_{\ge n-2}, {{\mathbb{A}}}(1)) \to H^n_{et}(\bar{U}_{{\bullet}}, {{\mathbb{A}}}(1)))$, the following diagram gives us $\Lambda_{et}$: $$\begin{CD} H^2_{et}(\bar{U}_{n-3}, {{\mathbb{A}}}(1)) @>{\delta^*}>> H^2_{et}(\bar{U}_{\ge n-2}, {{\mathbb{A}}}(1))\! \! \! \! \! \! @>>> H^n_{et}(\bar{U}_{{\bullet}}, {{\mathbb{A}}}(1))\\ @A{(\ref{kumns})}AA @AA{{\rm(i)}}A @.\\ NS(\bar{X}_{n-3})\otimes {{\mathbb{A}}}@>{\delta^*}>> T[\mathcal C_n \xrightarrow{\rho} {\mathcal P}_n]\otimes{{\mathbb{Q}}};\\ \end{CD}$$note $NS(\bar{X}_{n-3}) \otimes{{\mathbb{A}}}{\xrightarrow{\sim}}T[ NS(\bar{X}_{n-3}) \to 0]\otimes{{\mathbb{Q}}}$. Since $W_{-1}H^2_{et}(\bar{U}_{n-3}, {{\mathbb{A}}}(1)) =0$ [@hp 7.2] [@h2 3.2.15], the intersection $$\delta^*H^2_{et}(\bar{U}_{n-3}, {{\mathbb{A}}}(1)) ~\cap~ W_{-1} H^2_{et}(\bar{U}_{\ge n-2}, {{\mathbb{A}}}(1))$$is zero. Now (\[bombay\] (ii)) shows that $\Lambda_{et}$ is injective on $W_{-1}TL^n\otimes{{\mathbb{Q}}}$. It remains to show the injectivity of either $$\Lambda_{et}: \mathcal B_n(S)\otimes{{\mathbb{A}}}\to Gr^W_{0}H^n_{et}(\bar{U}_{{\bullet}}, {{\mathbb{A}}}(1))$$ or its ${{\mathbb{Q}}}_{\ell}$-analogue. The base change of this map via any $\iota: k {\hookrightarrow}{{\mathbb{C}}}$ can be identified, via (\[artgro\]) and the compatibility of the étale and classical cycle class maps [@jamm 5.3], with the *injective* map $\nu$ of (\[kapadu\](i)):$$\mathcal B_n({{\mathbb{C}}})\otimes{{\mathbb{Q}}}_{\ell} \underset{\nu \otimes{id}}{\xrightarrow{\sim}} Gr^W_0t^n(U_{{\bullet}, \iota})\otimes{{\mathbb{Q}}}_{\ell} {\hookrightarrow}Gr^W_0H^n(U_{{\bullet}, \iota}, {{\mathbb{Q}}}(1))\otimes{{\mathbb{Q}}}_{\ell} \underset{(\ref{artgro})}{\xleftarrow{\sim}} Gr^W_0H^n_{et}(U_{{\bullet}, \iota}, {{\mathbb{Q}}}_{\ell}(1)).$$ \(iii) Follows from the definition of the filtration $W$ ([@h1], [@weil], [@hp §14]) and (\[bombay\]). $t^n_{\ell}(U_{{\bullet}}):= \Lambda_{et}(T_{\ell}L^n \otimes{{\mathbb{Q}}})$ is a ${{\mathbb{G}}}$-invariant ${{\mathbb{Q}}}_{\ell}$-subspace of $H^n_{et}{}(\bar{U}_{{\bullet}}; {{\mathbb{Q}}}_{\ell}(1))$. Similarly, $t^n_{{{\mathbb{A}}}}(U_{{\bullet}}):= \Lambda_{et}(TL^n\otimes{{\mathbb{Q}}})$ is a ${{\mathbb{G}}}$-invariant ${{\mathbb{A}}}$-submodule of $H^n_{et}{}(\bar{U}_{{\bullet}}; {{\mathbb{A}}}(1))$. \[smalet\] Over $k = {{\mathbb{C}}}$, one has a commutative diagram $$\begin{CD} TL^n\otimes{{\mathbb{Q}}}@>{\Lambda_{et}}>> H^n_{et}({U}_{{\bullet}},{{\mathbb{A}}}(1))\\ @V{\wr}VV @V{\wr}V(\ref{artgro})V\\ T_{{{\mathbb{Z}}}}L^n\otimes{{\mathbb{A}}}@>{\Lambda}>{(\ref{main})}> H^n(U_{{\bullet}}({{\mathbb{C}}}), {{\mathbb{A}}}(1));\\ \end{CD}$$ the left vertical map is from [@h 10.1.6]. This diagram, together with (\[main\]), also provides the injectivity of $\Lambda_{et}$ in (\[biget\](ii)). It also provides canonical isomorphisms $t^n_{\ell}(U_{{\bullet}}) {\xrightarrow{\sim}}t^n(U_{{\bullet}})\otimes{{\mathbb{Q}}}_{\ell}$ and $t^n_{{{\mathbb{A}}}}(U_{{\bullet}}) {\xrightarrow{\sim}}t^n(U_{{\bullet}})\otimes{{\mathbb{A}}}$. The one-motives $L^n(V)$ {#varieties} ======================== In this section, $k$ is any field of characteristic zero. Throughout the rest of the paper, $V$ will denote a variety (over $S$). We shall now translate our results for simplicial schemes into ones for varieties; in particular, we show how to construct one-motives $L^n(V)$ associated with any algebraic variety $V$ over $S$. We will also show that $L^n(-)$ is a contravariant functor. The main result of this section is the proof of Conjecture \[dc1\] for fields of characteristic zero. [**Proper hypercoverings.**]{} Let $V$ be a variety over $S$. By [@h 6.2.8], [@sga4 5.3], there exists a simplicial pair ($X_{{\bullet}},Y_{{\bullet}}$) with a proper map $\alpha: U_{{\bullet}}\rightarrow V$ which makes $U_{{\bullet}}$ into a proper hypercovering of $V$. Namely, we have \(i) for any $\iota: k \hookrightarrow {{\mathbb{C}}}$, an isomorphism of mixed Hodge structures [@h 8.2.2] $$\label{sonatree} \alpha^*: H^*(V_{\iota},{{\mathbb{Z}}}) {\xrightarrow{\sim}}H^*(U_{{\bullet}, \iota},{{\mathbb{Z}}});$$(ii) an isomorphism of ${{\mathbb{G}}}$-modules $$\label{sonaty} \alpha^*: H^*_{et}(\bar{V},\hat{{{\mathbb{Z}}}}(1)) {\xrightarrow{\sim}}H^*_{et}(\bar{U}_{{\bullet}},\hat{{{\mathbb{Z}}}}(1));$$(iii) and an isomorphism of $k$-vector spaces $$\label{sonati} \alpha^*: H^*_{DR}(V) \xrightarrow{\sim} H^*_{DR}(U_{{\bullet}}).$$ For (ii), one uses the result [@sga4 4.3.2] that a proper surjective morphism is a map of universal cohomological descent for étale torsion sheaves and proceeds as in [@h 6.2.5]; cf. [@be p.306]. In (iii), the definition of $H^*_{DR}(V)$ (as indicated in [@pgal p. 89], [@jamm 6.11.4]) is that of [@hart] or, by [@illucie 1.2], as the *crystalline* cohomology of $V$; see also [@guillen Expose III], especially 1.3, 1.13, 1.14. for a cubical variant. We denote by $\beta_U$ the inverse of any of these isomorphisms $\alpha^*$. \[bertie\] Given two such proper hypercoverings of $V$, one can find another such proper hypercovering which dominates both. More generally, given any morphism $h:V \rightarrow W$ between two schemes, one can find such proper hypercoverings of $V$ and $W$ and a morphism between them which lifts the morphism $h$; we refer to [@sga4 5.1.4], [@h 6.2.8] for details. Suppose given two proper hypercoverings of $V$ fitting into a diagram $$\begin{CD} V @<<< U_{{\bullet}} @>>> X_{{\bullet}} @<<< Y_{{\bullet}}\\ @| @V{\theta}VV @V{\theta}VV @.\\ V @<<< 'U_{{\bullet}} @>>> 'X_{{\bullet}} @<<< 'Y_{{\bullet}}\\ \end{CD}$$ which yields a morphism of one-motives $$\theta^*_L: L^n('X_{{\bullet}}, 'Y_{{\bullet}}) \rightarrow L^n(X_{{\bullet}}, Y_{{\bullet}}).$$ \[pehla\] The morphism $\theta^*_L$ is an isogeny. The construction of the one-motives $L^n(U_{{\bullet}})$ and $L^n('U_{{\bullet}})$ relies only on a finite number of schemes and a finite number of associated morphisms between them. Therefore, we may assume that these one-motives are defined over a finitely generated subfield $k'$ of $k$. Such a field $k'$ always admits an embedding into ${{\mathbb{C}}}$. For any such embedding $\iota: k' \hookrightarrow {{\mathbb{C}}}$, one has a commutative diagram $$\begin{CD} T_{{{\mathbb{Z}}}}L^n('X_{{\bullet}}, 'Y_{{\bullet}})\otimes{{\mathbb{Q}}}@>>> H^n('U_{{\bullet}}\times{{\mathbb{C}}}, {{\mathbb{Q}}}(1))\\ @V{T\theta^*_L}VV @V{\theta^*}V{\wr}V\\ T_{{{\mathbb{Z}}}}L^n(X_{{\bullet}}, Y_{{\bullet}})\otimes{{\mathbb{Q}}}@>>> H^n(U_{{\bullet}}\times{{\mathbb{C}}}, {{\mathbb{Q}}}(1));\\ \end{CD}$$the isomorphism $\theta^*$ [@h 8.2.2] of mixed Hodge structures provides an isomorphism $t^n('U_{{\bullet}}) {\xrightarrow{\sim}}t^n(U_{{\bullet}})$. By Theorem \[main\] applied to $'U_{{\bullet}}\times{{\mathbb{C}}}$ and $U_{{\bullet}}\times{{\mathbb{C}}}$, the map $T\theta^*_L$ is an isomorphism of ${{\mathbb{Q}}}$-mixed Hodge structures. By [@h 10.1.3], $\theta^*_L$ is an isogeny. The isogeny one-motive $L^n(V)\otimes{{\mathbb{Q}}}$ is the isogeny class of the one-motive $L^n(X_{{\bullet}},Y_{{\bullet}})$ of any simplicial pair $(X_{{\bullet}}, Y_{{\bullet}})$ corresponding to a proper hypercovering $U_{{\bullet}}$ of $V$. \[banff\] Let $t^n_{\ell}(V)$ be the image of $t^{n}_{\ell}(U_{{\bullet}})$ under the map $\beta_U$, inverse to the isomorphism $\alpha^*: H^n_{et}(\bar{V},{{\mathbb{Q}}}_{\ell}(1)) \xrightarrow{\sim} H^n_{et}(\bar{U}_{{\bullet}},{{\mathbb{Q}}}_{\ell}(1))$ for any proper hypercovering $\alpha: U_{{\bullet}} \rightarrow V$ corresponding to a simplicial pair $(X_{{\bullet}}, Y_{{\bullet}})$. Also define $t^n_{{{\mathbb{A}}}}(V) = \beta_U(t^n_{{{\mathbb{A}}}}(U_{{\bullet}}))$, a ${{\mathbb{A}}}$-submodule of $H^n_{et}(\bar{V}, {{\mathbb{A}}}(1))$. Note that $t^n_{{{\mathbb{A}}}}(V)$ and $t^n_{\ell}(V)$ are independent of the choice of the proper hypercovering $U_{{\bullet}}$ of $V$. By (\[bertie\]), any two proper hypercoverings of $V$ are dominated by a third one; thus, as in [@h §8.2], the isogeny one-motive $L^n(V)\otimes{{\mathbb{Q}}}$ is well-defined by [@sga4 5.1.4], [@h 6.2.8]. \[funter\] $L^n(-)\otimes{{\mathbb{Q}}}$, $t^n_{\ell}(-)$ and $t^n_{{{\mathbb{A}}}}(-)$ are contravariant functorial. Given a morphism $f:V \rightarrow W$, there exists a proper hypercovering $(X_{{\bullet}},Y_{{\bullet}})$ (resp. $(Z_{{\bullet}},I_{{\bullet}})$) of $V$ (resp. $W$) and a morphism $\theta$ between them which lifts $f$ [@h 6.2.8]. This induces a morphism $\theta^*:L^n(W)\otimes{{\mathbb{Q}}}\rightarrow L^n(V)\otimes{{\mathbb{Q}}}$. One has to check that this induced morphism does not depend upon the choices of the auxiliary hypercoverings and for this we refer to (loc. cit). The next lemma, inspired by [@milram], will be used to deduce an integral version (i.e., an actual one-motive $L^n(V)$) of the isogeny one-motive $L^n(V)\otimes{{\mathbb{Q}}}$. Let $\mathcal C$ be the category whose objects are pairs $(R,L)$ where $R$ is an isogeny one-motive over $k$ and $L$ is a ${{\mathbb{G}}}$-stable $\hat{{{\mathbb{Z}}}}$-lattice of the ${{\mathbb{A}}}$-module $TR$, and morphisms from $(R_1, L_1)$ to $(R_2, L_2)$ are those morphisms $\phi: R_1 \to R_2$ of isogeny one-motives such that the induced map $\phi_*: TR_1 \to TR_2$ satisfies $\phi_*(L_1) \subset L_2$. \[intero\] The natural functor $M \mapsto (M\otimes{{\mathbb{Q}}}, M\otimes\hat{{{\mathbb{Z}}}})$ from the category $\mathcal M_k$ of one-motives over $k$ to $\mathcal C$ is an equivalence of categories. This follows from [@milram 2.2]; the main point is as follows: as $\hat{{{\mathbb{Z}}}}$-lattices of the ${{\mathbb{A}}}$-module $TR$ are compact and open in $TR$, any two ${{\mathbb{G}}}$-stable $\hat{{{\mathbb{Z}}}}$-lattices in $TR$ are commensurable. Let $V$ be any variety over $k$. Consider the ${{\mathbb{G}}}$-stable $\hat{{{\mathbb{Z}}}}$-lattice $$t^n_{{{\mathbb{A}}}}(V) \cap H^n_{et}(\bar{V}, \hat{{{\mathbb{Z}}}}(1))/{\rm torsion} \subset H^n_{et}(\bar{V}, {{\mathbb{A}}}(1))$$ in $t^n_{{{\mathbb{A}}}}(V)$. Since the map $\Lambda_{et}: TL^n(V)\otimes{{\mathbb{Q}}}\to H^n_{et}(\bar{V}, {{\mathbb{A}}}(1))$ is injective with image $ t^n_{{{\mathbb{A}}}}(V)$, this defines a ${{\mathbb{G}}}$-stable $\hat{{{\mathbb{Z}}}}$-lattice of $TL^n(V)\otimes{{\mathbb{Q}}}$ which, by (\[intero\]), determines a one-motive $L^n(V) =:L^n(V/k)$ over $k$. \[peddha\] Conjecture [\[dc1\]]{} is true for fields of characteristic zero. The one-motives $L^n(V/k)$ defined for an arbitrary variety $V$ over a field $k$ of characteristic zero are functorial in $V$ and $k$. Furthermore, one has [(i)]{} an injection of ${{\mathbb{G}}}$-modules $\Lambda_{et}: TL^n(V/k) {\hookrightarrow}H^n_{et}(\bar{V}, \hat{{{\mathbb{Z}}}}(1))/{\rm torsion}$; [(ii)]{} an injection of $k$-vector spaces $T_{DR}L^n(V/k) {\hookrightarrow}H^n_{DR}(V)$; [(iii)]{} a canonical and functorial isomorphism $L^n(V/{{{\mathbb{C}}}}) {\xrightarrow{\sim}}I^n(V)$. If $k$ is finitely generated over ${{\mathbb{Q}}}$, then $\Lambda_{et}$ is compatible with the weight filtration: $\Lambda_{et}(W_iL^n(V/k)\otimes{{\mathbb{A}}}) \subset W_iH^n_{et}(\bar{V}, {{\mathbb{A}}}(1))$ for $i=-2,-1,0$ [(]{}equality for $i=-2,-1$[)]{}. The contravariant functoriality of $L^n(V)$ is proved as in (\[funter\]) and compatibility with base change is clear (\[zeusp\]). \(i) follows from the definition of $L^n(V/k)$, (\[pehla\]), (\[biget\]), (\[sonaty\]); and (ii) follows from (\[derham\]), (\[sonati\]). (iii) follows from the definition of $L^n(V/k)$, (\[main\]), (\[sonatree\]), and (\[artgro\]). The last statement follows from (\[biget\]). Let $N$ be the dimension of $V$. The one-motives $L^n(V)$ vanish for $n > N + 1$; this follows from weight considerations [@hp 7.3]. If $V$ is smooth, then $L^n(V)$ is zero for $n > 2$ (\[toto\]). Positive characteristic {#pos+} ======================= In this section, $k$ is a field of characteristic $p >0$, $k^{perf}$ its perfection, $k^{sep}$ a separable closure (${{\mathbb{G}}}_{sep}$ is the associated Galois group), $\bar{k}$ (an algebraic closure of $k$) the compositum of $k^{perf}$ and $k^{sep}$; $\ell$ is a prime distinct from $p$. Even though we do not assume $k$ to be perfect, all our results involve passage to $k^{perf}$. For any variety $V$ over a perfect field $k$ and any integer $n$, we use “neat hypercoverings” to construct $L^n(V)\otimes{{\mathbb{Z}}}[1/p]$ ($0 \le n \le 2$), $J^n(V)\otimes{{\mathbb{Z}}}[1/p]$ ($ n \ge 0$) — one-motives up to $p$-isogeny, i.e., objects of $\mathcal M_k \otimes{{\mathbb{Z}}}[1/p]$ — which are contravariant functorial; these provide a partial proof of (\[dc1\]) for $k$. Finally, we reduce Conjecture \[dc1\] (up to $p$-isogeny), if $k$ is perfect, to the validity of (\[tatec\]) for surfaces; an analogous result (\[imper\]) holds, even if $k$ is not perfect, under the additional assumption of “resolution of singularities”. [**Generalized one-motives.**]{} Let $(X_{{\bullet}}, Y_{{\bullet}})$ be a simplicial pair over $S$ and $f$ be the structure map of $X_{\ge n-1}$. The sheaf $R^1f_* \mathcal O^*$ on $S_{fppf}$ is representable (\[reps\]). The (reduced) neutral component of the corresponding group scheme is not guaranteed to be a semi-abelian variety unless $k$ is perfect. [[@dg Corollary 2.3, p.288]]{} Let $G$ be a locally algebraic group scheme over a field $F$. If $F$ is perfect, then the reduced scheme $G^{red}$ is a smooth group scheme. This motivates the following definition. \[rtp\] A generalized one-motive over $k$ is a two-term complex $M =[B \xrightarrow{u} G]$ of group schemes over $S$ such that, after base change to [Spec]{} $k^{perf}$, $[B \xrightarrow{u} G^{red}]$ is a one-motive over [Spec]{} $k^{perf}$. The category $\tilde{\mathcal M}_k$ of generalized one-motives over $k$ is an additive category. The functors $M \mapsto T_{\ell}M$ ($\ell \neq p$), $M\mapsto T^pM$ on $\mathcal M_k$ extend to the category $\tilde{\mathcal M}_k$. Our constructions in Section \[constr\] provide generalized one-motives $L^n$, $J^n$ over $k$ (and one-motives $L^n$, $J^n$ over $k^{perf}$) associated with the simplicial pair $(X_{{\bullet}}, Y_{{\bullet}})$ over $S$; these are contravariant functorial and they are compatible with base change. [**Relating $L^n$ to étale cohomology.**]{} For any (simplicial) variety $V$, set $H^*_{et}(\bar{V}, {{\mathbb{Z}}}^p(1)): = {\varprojlim}_r H^*_{et}(\bar{V}, \1_r)$ with $(r,p) = 1$, $H^*_{et}(\bar{V}, {{\mathbb{A}}}^p(1)):= H^*_{et}(\bar{V}, {{\mathbb{Z}}}^p(1))\otimes{{\mathbb{Q}}}$. Using the generalized one-motives $L^n$, one checks that the proofs of (\[finel\]), (\[adivasi\]), (\[biget\](i)) are valid with ${{\mathbb{A}}}^p$ instead of ${{\mathbb{A}}}$. Also, (\[bombay\]) is valid: the étale analogue $E_{*}^{et}$ of [@h 8.1.19.1] calculating the étale cohomology of $\bar{U}_{{\bullet}}$ with ${{\mathbb{Q}}}_{\ell}$-coefficients degenerates at $E_2$ (\[specet\]). This is a consequence of the weight filtration $W$ [@weil2 5.3.6] on $H^i(\bar{V}, {{\mathbb{Q}}}_{\ell})$ for any scheme $V$ over $S$. Compatibility with $W$ forces the vanishing of the differentials $d_r$ of $E_{*}^{et}$ for $r >1$; cf. [@weil2 5.3.7] for another application. The point is that given any finite set of (smooth) schemes $X_i$, $\tilde{Y}_j$, there exist an integral scheme $\mathcal V$ of finite type over Spec ${{\mathbb{F}}}_p$, a dominant morphism $\eta: S \to \mathcal V$, smooth schemes $\mathcal X_i$, $\tilde{\mathcal Y}_j$ over Spec ${{\mathbb{F}}}_p$ with smooth proper morphisms $\mathcal X_i \to \mathcal V$, $\tilde{\mathcal Y}_j \to \mathcal V$ such that $\mathcal X_i \times_{\eta} S = X_i$, $ \tilde{\mathcal Y}_j \times_{\eta}S = \tilde{Y}_j$. The terms of the spectral sequence $E_{*}^{et}$ are pull-backs via $\eta$ of pure lisse sheaves on $\mathcal V$ (purity follows from [@weil2 6.2.6]); cf. [@jamm p.89, pp.115-117]. [**Relations with the Tate conjecture.**]{} The proof of the injectivity of the map $\Lambda_{et}: T_{\ell}L^n\otimes{{\mathbb{Q}}}\to H^n_{et}(\bar{U}_{{\bullet}}, {{\mathbb{Q}}}_{\ell}(1))$ of (\[biget\](ii)) uses (\[kapadu\](i)) which, in turn, is based on (\[h11\]); the injectivity is necessary for a proof of (\[pehla\]) in this context. Thus we need a proof (valid in positive characteristic) of the injectivity of $$\label{limbs} \nu_{et}: \mathcal B_n({S})\otimes{{\mathbb{Q}}}_{\ell} \to Gr^W_{0}H^n_{et}(\bar{U}_{{\bullet}}, {{\mathbb{Q}}}_{\ell}(1))$$where, by the degeneration (\[specet\]) of $E_{*}^{et}$ at $E_2$, we have — (\[kapadu\]) — $$Gr^W_{0}H^n_{et}(\bar{U}_{{\bullet}},{{\mathbb{Q}}}_{\ell}(1)) = \frac{{\operatorname{Ker}}(H^0(\tilde{Y}_{n-1}) \oplus H^2(\bar{X}_{n-2}) \xrightarrow{t_{n-1}} H^0(\tilde{Y}_{n}) \oplus H^2(\bar{X}_{n-1}))}{{\operatorname{Im}}(H^0(\tilde{Y}_{n-2}) \oplus H^2(\bar{X}_{n-3}) \xrightarrow{t_{n-2}} H^0(\tilde{Y}_{n-1}) \oplus H^2(\bar{X}_{n-2}))};$$ here $H^0(\tilde{Y}_m) = H^0_{et}(\tilde{Y}_m\times \bar{S}, {{\mathbb{Q}}}_{\ell})$ and $H^2(\bar{X}_m) = H^2_{et}(\bar{X}_m, {{\mathbb{Q}}}_{\ell}(1))$ and the maps $t_m$ are as in (\[soya\]). By [@mi 2.6, p. 224], $H^2_{et}({X}_{m}\times k^{sep}, {{\mathbb{Q}}}_{\ell}(1))) {\xrightarrow{\sim}}H^2_{et}(\bar{X}_{m}, {{\mathbb{Q}}}_{\ell}(1)))$. \[maam\] Let $k$ be a finitely generated extension of the prime field ${{\mathbb{F}}}_p$. The map $\nu_{et}$ of [(\[limbs\])]{} is injective if either [(i)]{} $\delta^*_{n-3}H^2_{et}({X}_{n-3}\times k^{sep},{{\mathbb{Q}}}_{\ell}(1))^{{{\mathbb{G}}}_{sep}}$ is equal to the image of the map $$\delta^*_{n-3}NS_{n-3}(S)\otimes{{\mathbb{Q}}}_{\ell} {\hookrightarrow}NS_{n-2}(S)\otimes{{\mathbb{Q}}}_{\ell} {\hookrightarrow}H^2_{et}({X}_{n-2}\times k^{sep}, {{\mathbb{Q}}}_{\ell}(1)).$$or [(ii)]{} the Tate conjecture [(\[tatec\])]{} is true for either $X_{n-3}$ or $X_{n-2}$. In particular, $\nu_{et}$ is injective when $n \le 2$. It is straightforward to check that (i) implies the desired injectivity. Let us show that (ii) implies (i). The exact sequences of ${{\mathbb{G}}}_{sep}$-modules$$0 \to {\operatorname{Ker}}(\delta^*_{n-3}) \to H^2_{et}({X}_{n-3}\times k^{sep}, {{\mathbb{Q}}}_{\ell}(1)) \to \delta^*_{n-3} H^2_{et}({X}_{n-3}\times k^{sep}, {{\mathbb{Q}}}_{\ell}(1)) \to 0,$$ $$0 \to \delta^*_{n-3}H^2_{et}({X}_{n-3}\times k^{sep}, {{\mathbb{Q}}}_{\ell}(1)) \to H^2_{et}({X}_{n-2}\times k^{sep}, {{\mathbb{Q}}}_{\ell}(1)) \to {\operatorname{Coker}}(\delta^*_{n-3}) \to 0$$are split: this follows from [@tate 2.10] — see the proof of [@tate 5.2(b)]. Thus, as in (loc. cit), these exact sequences remain exact after taking ${{\mathbb{G}}}_{sep}$-invariants. This, by (\[picmd\](v)), suffices for the implication (ii) $\Rightarrow$ (i). Since the map $\frac{K^0(S)}{W_{n-2}(S)}\otimes{{\mathbb{Q}}}_{\ell} \to \mathcal B_n({S})\otimes{{\mathbb{Q}}}_{\ell} \xrightarrow{\nu_{et}} Gr^W_{0}H^n_{et}(\bar{U}_{{\bullet}}, {{\mathbb{Q}}}_{\ell}(1))$ is clearly injective, we obtain the injectivity of $$\label{lambs} \Lambda_{et}: T_{\ell}J^n(X_{{\bullet}}, Y_{{\bullet}})\otimes{{\mathbb{Q}}}{\hookrightarrow}H^n_{et}(\bar{U}_{{\bullet}}, {{\mathbb{Q}}}_{\ell}(1));$$ its image is denoted $s^n_{\ell}(X_{{\bullet}}, Y_{{\bullet}})$. A similar definition gives $s^n_{{{\mathbb{A}}}^p}(X_{{\bullet}}, Y_{{\bullet}})$. \(i) Without a condition such as the injectivity of $\nu_{et}$, it is hard to show that the Galois representations $\Lambda_{et}(T_{\ell}L^n\otimes{{\mathbb{Q}}})$ are “independent of $\ell$”; this is unclear already for their dimensions [@ka pp.27-29]. \(ii) If $n \le 2$, then $\nu_{et}$ is injective; set $t^n_{\ell}(X_{{\bullet}}, Y_{{\bullet}}):= {\operatorname{Im}}(\Lambda_{et})$, a ${{\mathbb{G}}}$-invariant subspace of $H^n_{et}(\bar{U}_{{\bullet}}, {{\mathbb{Q}}}_{\ell}(1))$. A similar definition gives $t^n_{{{\mathbb{A}}}^p}(X_{{\bullet}}, Y_{{\bullet}})$. [**Varieties over perfect fields.**]{} From now on, we assume that $k$ is perfect. Let $V$ be an arbitrary variety over $S$. The results of [@dj] show [@dj §1] [@be 6.3] the existence of a simplicial pair $(X_{{\bullet}}, Y_{{\bullet}})$ over $S$ with a morphism $\alpha: U_{{\bullet}} {\rightarrow}V$ which makes $U_{{\bullet}}$ a proper hypercovering of $V$. One has (\[sonaty\]) $\alpha^*: H^*_{et}(\bar{V},{{{\mathbb{Z}}}}^p(1)){{\xrightarrow{\sim}}} H^*_{et}(\bar{U}_{{\bullet}},{{{\mathbb{Z}}}}^p(1))$; as before, $\beta_U$ denotes the inverse of $\alpha^*$. The methods of [@h 6.2] imply, using [@dj], the abundance of smooth hypercoverings so that (\[bertie\]) is valid as well. Let[^9] $s^n_{\ell}(V)$ be the direct limit of $\beta_U(s^n_{\ell}(X_{{\bullet}}, Y_{{\bullet}}))$ over all proper hypercoverings $\alpha: U_{{\bullet}} \rightarrow V$. It is a ${{\mathbb{Q}}}_{\ell}$-subspace of $H^n_{et}(\bar{V},{{\mathbb{Q}}}_{\ell}(1))$, with an action of ${{\mathbb{G}}}$. A similar definition holds for $t^n_{\ell}(V)$ ($n \le 2$). Since $H^n_{et}(\bar{V},{{\mathbb{Q}}}_{\ell}(1))$ [@ka p. 24] is a finite dimensional vector space over ${{\mathbb{Q}}}_{\ell}$, there is actually a (by no means unique) proper hypercovering $U_{{\bullet}}$ of $V$ such that $\beta_U(s^n_{\ell}(X_{{\bullet}}, Y_{{\bullet}})) = s^n_{\ell}(V)$; let us call such hypercoverings “neat”. The notion of “neat” does not depend on the auxiliary prime $\ell \neq p$. Clear: the map $\Lambda_{et}$ of (\[lambs\]) is injective which means that the dimension of $s^n_{\ell}(X_{{\bullet}}, Y_{{\bullet}})$ is independent of $\ell$; similarly, the dimension of $s^n_{\ell}(V)$ is independent of $\ell$. \[special\] [(i)]{} Any proper hypercovering of $V$ which dominates a “neat” proper hypercovering of $V$ is “neat”. [(ii)]{} Any two “neat” proper hypercoverings are dominated by a “neat” proper hypercovering. [(iii)]{} More generally, given a pair of proper hypercoverings one of which is “neat”, there is a proper hypercovering (automatically “neat”) dominating them. [(iv)]{} Any proper hypercovering is dominated by a “neat” proper hypercovering. The advantage of “neat” hypercoverings is the For any morphism $\theta$ between two “neat” proper hypercoverings $U_{{\bullet}}$ and $'U_{{\bullet}}$ of $V$, the induced map $\theta^*: J^n('X_{{\bullet}}, 'Y_{{\bullet}})\otimes{{\mathbb{Q}}}\to J^n(X_{{\bullet}}, Y_{{\bullet}})\otimes{{\mathbb{Q}}}$ is an isomorphism. Here $J^n(-)\otimes{{\mathbb{Q}}}$ denotes the isogeny one-motive obtained from the generalized one-motive — see [(\[rtp\])]{}. It is enough to show that the map $ \theta^*: T_{\ell}J^n('X_{{\bullet}}, 'Y_{{\bullet}})\otimes {{\mathbb{Q}}}{\rightarrow}T_{\ell}J^n(X_{{\bullet}}, Y_{{\bullet}})\otimes {{\mathbb{Q}}}$ is an isomorphism. This map is always injective (for arbitrary proper hypercoverings). But if the hypercoverings are “neat”, then both terms are actually equal to $s^n_{\ell}(V)$. Our definition of “neat” depends upon the integer $n$. But since $H^i_{et}(\bar{V}, {{\mathbb{A}}}^p(1)) = 0$ for $i > 2$ dim $V$ [@ka pp. 23-24], there exist, by (\[special\]), “neat” hypercoverings such that $\beta_U(s^n_{\ell}(X_{{\bullet}}, Y_{{\bullet}})) = s^n_{\ell}(V)$ holds for all $n$. From now on, let us consider only such “neat” hypercoverings. We define $J^n(V)\otimes{{\mathbb{Q}}}$ to be the isogeny one-motive $J^n(X_{{\bullet}}, Y_{{\bullet}})\otimes{{\mathbb{Q}}}$ of any “neat” proper hypercovering $U_{{\bullet}}$ of $V$. The following theorem is a trivial consequence of the previous definition; the weight filtration on étale cohomology is given by [@weil2 5.3.6]. One has ${{\mathbb{G}}}$-equivariant injections $$\Lambda_{et}: T_{\ell} W_iJ^n(V)\otimes{{\mathbb{Q}}}{\hookrightarrow}W_iH^n_{et}(\bar{V}, {{\mathbb{Q}}}_{\ell}(1));$$ these are isomorphisms for $i = -2, -1$. Similar statement holds for $\Lambda_{et}:T^pJ^n(V)\otimes{{\mathbb{Q}}}{\hookrightarrow}H^n_{et}(\bar{V},{{\mathbb{A}}}^p(1))$. Note $s^n_{\ell}(V)$ is a subspace (possibly proper) of $H^n_{et}(\bar{V},{{\mathbb{Q}}}_{\ell}(1))$; the allowed weights on $s^n_{\ell}(V)$ are $-2$, $-1$ and $0$; the allowed weights on $H^n_{et}(\bar{V},{{\mathbb{Q}}}_{\ell}(1))$ lie between $-2$ and $2n-2$. A similar statement is true for $s^n_{{{\mathbb{A}}}^p}(V):= \Lambda_{et}(T^pJ^n(V)\otimes{{\mathbb{Q}}}) \subset H^n_{et}(\bar{V},{{\mathbb{A}}}^p(1))$. (Functoriality) Given any morphism $f: Z {\rightarrow}V$, there is an induced morphism $f^*: J^n(V)\otimes{{\mathbb{Q}}}{\rightarrow}J^n(Z)\otimes{{\mathbb{Q}}}$: given $f$, pick a “neat” proper hypercovering $U_{{\bullet}}$ of $V$. One can find a proper hypercovering $E_{{\bullet}}$ of $Z$ and a morphism $F: E_{{\bullet}} {\rightarrow}U_{{\bullet}}$ lifting $f$. By (\[special\]), “neat” proper hypercoverings of $Z$ are cofinal among proper hypercoverings of $Z$. So we may choose $E_{{\bullet}}$ to be “neat”. This yields the functoriality of $J^n(-)\otimes{{\mathbb{Q}}}$, $s^n_{\ell}(-)$, and $s^n_{{{\mathbb{A}}}^p}(-)$. \[mem\] (i) A variant of (\[intero\]) shows that the ${{\mathbb{G}}}$-invariant ${{\mathbb{Z}}}^p$-lattice $$(s^n_{{{\mathbb{A}}}^p}(V) \cap H^n_{et}(\bar{V}, {{\mathbb{Z}}}^p(1))/{\rm torsion}) \subset H^n_{et}(\bar{V},{{\mathbb{A}}}^p(1))$$ provides a ${{\mathbb{Z}}}[1/p]$-integral structure on $J^n(V)\otimes{{\mathbb{Q}}}$, i.e., determines $J^n(V)\otimes {{\mathbb{Z}}}[1/p]$, a one-motive up to $p$-isogeny, defined over $k$. Controlling $p$ would require an integral $p$-adic cohomology for arbitrary varieties. \(ii) It is probable that the $p$-adic/crystalline realization of $J^n(V)\otimes{{\mathbb{Q}}}$ is related to the rigid cohomology of $V$ [@tsu]. [**The one-motives up to $p$-isogeny $L^n(V)\otimes{{\mathbb{Z}}}[1/p]$.**]{} Our methods for the construction of $J^n(V)\otimes{{\mathbb{Z}}}[1/p]$ show the following \[groth\] If [(\[tatec\])]{} is true for any surface over any finitely generated [(]{}over ${{\mathbb{F}}}_p$[)]{} subfield of a perfect field $k$, then [(\[dc1\]) (up to $p$-isogeny)]{} is true for $k$. Using the injectivity of $\nu_{et}$ of (\[limbs\]) assured by (\[maam\]), we can apply the previous methods to define $L^n(V)\otimes{{\mathbb{Q}}}$ (using a variant of “neat” hypercoverings) and refine it, as in (\[mem\](i)), to $L^n(V)\otimes{{\mathbb{Z}}}[1/p]$. The fact that the Tate conjecture (\[tatec\]) for divisors reduces to the case of surfaces is well-known; this is proved along the ideas of the proof of (\[maam\]) using the weak Lefschetz theorem [@weil2 4.1.6]. \[groth2\] (i) By (\[maam\]), $\nu_{et}$ of (\[limbs\]) is injective for $n \le 2$. In this case, our methods provide $L^n(V)\otimes {{\mathbb{Z}}}[1/p]$ (for $n \le 2$), one-motives up to $p$-isogeny, associated with $V$ which are contravariant functorial. It follows from the definitions (\[phin\]), (\[hosadu\]) that $J^n(V)\otimes {{\mathbb{Z}}}[1/p] = L^n(V)\otimes {{\mathbb{Z}}}[1/p]$ for $n \le 1$. \(ii) If the field $k$ in (\[groth\]) is not taken to be perfect, then one obtains that $L^n(V)\otimes{{\mathbb{Z}}}[1/p]$ (attached to a variety $V$ over $k$) is defined over $k^{perf}$. \[imper\] Let $F$ be a field of characteristic $p >0$; write $F^{perf}$ for its perfection. Assume that “resolution of singularities” holds over $F$ and that [(\[tatec\])]{} holds for any surface over any finitely generated [(]{}over ${{\mathbb{F}}}_p$[)]{} subfield of $F$. Then, [(\[dc1\]) (up to $p$-isogeny)]{} is true for $F$. Straightforward variant of (\[groth\]). Applications and related results {#applix} ================================ [**Related work.**]{} For any curve $C$, Deligne [@h 10.3] constructed a one-motive $H^1_m(C)(1)$ (isomorphic to our $L^1(C)$) and used it to prove Theorem \[main\] for the $H^1$ of a curve over ${{\mathbb{C}}}$. The one-motive $L^2(V)$ of a projective complex surface $V$ was already obtained by J. Carlson [@ca]. Carlson mentions in [@ca2] that he has constructed other one-motives for a special class of varieties (over ${{\mathbb{C}}}$) but these results remain unpublished (email, 3 Dec 2001). The one-motive $L^1(X_{{\bullet}}, Y_{{\bullet}})$ is the Picard one-motive $Pic^{+}$ of [@bs3] and $M^1$ of [@ra]. Finally, [@brs] contains independent proofs of some of our results. [**Motivic principles: illustrations.**]{} [@ja 1.7, 2.5] Let $V$ be a variety over a field $k$ of characteristic zero; by (\[zeusp\]), we may assume $k$ to be finitely generated over ${{\mathbb{Q}}}$ without loss of generality. \[swaraj\] The rank of $$H^{1,1}_{{{\mathbb{Q}}}}(V_{\iota})^n:= \Hom_{MHS}({{\mathbb{Q}}}(0), Gr^W_0 H^n(V_{\iota},{{\mathbb{Q}}}(1))),$$ [(the so-called $(1,1)$-part)]{} is independent of the imbedding $\iota: k \hookrightarrow {{\mathbb{C}}}$. This follows from Theorem \[main\] since the dimension of the left hand side is the rank of $\mathcal B_n({{\mathbb{C}}})$ of the one-motive $L^n(V)$. Put $k ={{\mathbb{C}}}$; (\[swaraj\]) yields an algebraic characterization of the $(1,1)$-part of $H^n(V,{{\mathbb{Q}}})$ (\[1-1i\]). The analogous statement for the $(m,m)$-part of $H^n(V,{{\mathbb{Q}}})$ is not known (for $m >1$) in general; it is part of the (homological) Hodge conjecture [@jamm 7.2]. Consider the Galois representations $M_{\ell}:=H^n_{et}(\bar{V}, {{\mathbb{Q}}}_{\ell}(1))$ (these have a weight filtration $W$ by Galois submodules [@hp §13, §14]. For each prime $\ell$, $h_{\ell}(V):= W_{-1}M_{\ell}$ defines an element $\zeta_{\ell} \in Ext^1_{{{\mathbb{G}}}}(Gr^W_{-1}M_{\ell},Gr^W_{-2}M_{\ell})$. For each $\iota: k {\hookrightarrow}{{\mathbb{C}}}$, the substructure $h_{\iota}(V):= W_{-1}M_{\iota}(V)$ of the mixed Hodge structure $M_{\iota}:= H^n(V_{\iota},{{\mathbb{Q}}}(1))$ defines an element $\zeta_{\iota} \in Ext^1_{MHS}(Gr^W_{-1}M_{\iota},Gr^W_{-2}M_{\iota})$. The isogeny one-motive $W_{-1}L^n\otimes{{\mathbb{Q}}}$ can be viewed as extension $\zeta$ of two isogeny one-motives given by the exact sequence (\[lcw\]): $$\zeta: 0 \rightarrow [0 \rightarrow \mathcal T]\otimes{{\mathbb{Q}}}\rightarrow [0 \rightarrow \tilde{\mathcal P}_n]\otimes{{\mathbb{Q}}}\rightarrow [ 0 \rightarrow \mathcal R]\otimes{{\mathbb{Q}}}\rightarrow 0$$ The element $\zeta \in Ext^1(\mathcal R, \mathcal T)\otimes{{\mathbb{Q}}}$ is zero if and only if $\tilde{\mathcal P}_n$ is isogenous to the direct product $\mathcal R \times \mathcal Q$. \[jap\] One has [(a)]{} $\zeta$ is zero $\Leftrightarrow \zeta_{\iota}$ is zero for all $\iota \Leftrightarrow \zeta_{\iota}$ is zero for one $\iota:k\hookrightarrow {{\mathbb{C}}}$. [(b)]{} $\zeta$ is zero $\Leftrightarrow \zeta_{\ell}$ is zero for all primes $\ell \Leftrightarrow \zeta_{\ell}$ is zero for one $\ell$. The relation between $\zeta$, $\zeta_{\ell}$, and $\zeta_{\iota}$ is given by (\[biget\]) and (\[kona\]). Statements (a) and (b) follow easily from [@ja Thm. 4.3] and [@h 10.1.3] that the $\ell$-adic (or Hodge) realizations of an isogeny one-motive is a direct sum of pure objects if and only if the isogeny one-motive is isogenous to a direct product. [**Independence of the prime $\ell$ in étale cohomology.**]{} We now take $k$ to be a finite field; and let $\Phi \in {{\mathbb{G}}}$ be the Frobenius automorphism. The weight filtration $W$ on $H^n_{et}(\bar{V},{{\mathbb{Q}}}_{\ell})$ is defined via the endomorphism $\Phi^*_{\ell}$ of $H^n_{et}(\bar{V},{{\mathbb{Q}}}_{\ell})$ induced by $\Phi$. Let $b_{i,\ell,n}(V)$ be the dimension of the ${{\mathbb{Q}}}_{\ell}$-vector space $Gr^W_{i}H^n_{et}(\bar{V},{{\mathbb{Q}}}_{\ell})$; we recall the well-known results: \[wayout\] [(Deligne)]{} [@weil] [@weil2 3.3.9] If $V$ is smooth and proper, then [(1)]{} $b_{i,\ell,n}(V) =0$ if $i \neq n$. [(2)]{} $b_{n,\ell,n}(V)$ is independent of $\ell$; thus, we may set $b_n(V) = b_{n,\ell,n}(V)$. [(3)]{} the characteristic polynomial of $\Phi^*_{\ell}$ on $H^n_{et}(\bar{V},{{\mathbb{Q}}}_{\ell})$ has coefficients in ${{\mathbb{Q}}}$ independent of the prime $\ell$. [(4)]{} [@weil2 4.1.5] [(]{}$V$ projective[)]{} $b_{2n+1}(V)$ is even.[^10] [(N. Katz)]{} [@ka pp. 27-29] Which parts of [(\[wayout\])]{} are valid for $b_{i,\ell,n}(V)$, $Gr^W_{i}H^n_{et}(\bar{V},{{\mathbb{Q}}}_{\ell})$ for general $V$, i.e., for $V$ possibly singular and non-proper? This is not answered by a formal application of [@weil] and resolution of singularities; cf. [@dj §1]. Genuinely new ingredients are needed for an answer in general. Katz [@ka3] has proved it for certain smooth varieties; no general results are known for singular varieties. A consequence of (\[zund\](v)) is: Fix an arbitrary variety $V$ and an integer $n \ge 0$. The following systems of Galois representations satisfy [(2)]{} and [(3)]{} of [(\[wayout\])]{}: [(i)]{} $W_{-2}H^n_{et}(\bar{V},{{\mathbb{Q}}}_{\ell}(1))$. [(ii)]{} $Gr^W_{-1}H^n_{et}(\bar{V},{{\mathbb{Q}}}_{\ell}(1))$. [(iii)]{} $W_{-1}H^n_{et}(\bar{V},{{\mathbb{Q}}}_{\ell}(1))$. Let $c_n(V)$ denote the rank (over ${{\mathbb{Z}}}$) of the Galois module $I_n(\bar{S})$; here $I_n$ is the “lattice” in the one-motive $J^n(V) = [I_n {\rightarrow}\mathcal P_n]$. \[zund\] [(i)]{} The integers $b_{-2,\ell ,n}(V)$ and $b_{-1,\ell ,n}(V)$ are independent of $\ell\neq p$. [(ii)]{} $b_{-1, \ell ,n}(V)$ is an [even]{} integer; cf. [@hp 16.1]. [(iii)]{} $b_{0, \ell , n(V)} \ge c_n(V)$. [(iv)]{} Let $f:V {\rightarrow}V$ be any morphism. The characteristic polynomial of the induced endomorphism $f^*_{\ell}: s^n_{\ell}(V) {\rightarrow}s^n_{\ell}(V)$ has ${{\mathbb{Q}}}$-coefficients which are independent of $\ell$. [(v)]{} The characteristic polynomial of the automorphism $\Phi_{\ell}$ on the ${{\mathbb{G}}}$-submodule $s^n_{\ell}(V)$ of $H^n_{et}(\bar{V},{{\mathbb{Q}}}_{\ell}(1))$ has ${{\mathbb{Q}}}$-coefficients which are independent of $\ell$. [(vi)]{} For any morphism $g: V {\rightarrow}W$, we have a commutative diagram $$\begin{CD} s^n_{\ell}(W) @>{g^*_{\ell}}>> s^n_{\ell}(V) \\ @VVV @VVV\\ H^n_{et}(\bar{W},{{\mathbb{Q}}}_{\ell}(1)) @>>> H^n_{et}(\bar{V},{{\mathbb{Q}}}_{\ell}(1));\\ \end{CD}$$ the characteristic polynomials of $\Phi_{\ell}$ on the ${{\mathbb{G}}}$-modules ${\operatorname{Ker}}(g^*_{\ell})$ and ${\operatorname{Coker}}(g^*_{\ell})$ have ${{\mathbb{Q}}}$-coefficients which are independent of $\ell$. Write $\mathcal P$ as an extension of an abelian variety $\mathcal A$ by a torus $\mathcal T$; it follows from (\[biget\]) that $b_{-2,\ell ,n}(V)$ (resp. $b_{-1,\ell ,n}(V)$) are the dimensions of $\mathcal T$ (resp. $\mathcal A$). This proves (i) and (ii). (iii) also follows from (\[biget\]) since we know (\[lambs\]) that $I(\bar{S})\otimes_{{{\mathbb{Z}}}}{{\mathbb{Q}}}_{\ell}$ injects into $Gr^W_{0}H^n_{et}(\bar{V}, {{\mathbb{Q}}}_{\ell}(1))$. The endomorphism algebra ${\operatorname{End}}(J^n(V)\otimes{{\mathbb{Q}}})$ is a finite dimensional ${{\mathbb{Q}}}$-algebra. By functoriality of $J^n(-)\otimes{{\mathbb{Q}}}$, the morphism $f$ induces an element $f^* \in {\operatorname{End}}(J^n(V)\otimes{{\mathbb{Q}}})$. The characteristic polynomial of $f^*$ is a polynomial with ${{\mathbb{Q}}}$-coefficients. Since $s^n_{\ell}(V)$ is the $\ell$-adic realization of $J^n(V)\otimes{{\mathbb{Q}}}$, by functoriality, the map $f^*_{\ell}$ on $s^n_{\ell}(V)$ has the same characteristic polynomial. This proves (iv). The same argument proves (v): the Frobenius morphism $F_V$ of $V$ and the geometric Frobenius $\Phi^{-1}$ induce the same endomorphism on $H^*(\bar{V},{{\mathbb{Q}}}_{\ell}(1))$ [@weil 1.15]. The map $g: V \to W$ induces a map $g^*: J^n(W)\otimes{{\mathbb{Q}}}\to J^n(V)\otimes{{\mathbb{Q}}}$. Since the category of isogeny one-motives is abelian, we have the isogeny one-motives ${\operatorname{Ker}}(g^*)$ and ${\operatorname{Coker}}(g^*)$. Their $\ell$-adic realizations are the Galois modules ${\operatorname{Ker}}(g^*_{\ell})$ and ${\operatorname{Coker}}(g^*_{\ell})$. Apply the argument in the previous paragraph to ${\operatorname{End}}({\operatorname{Ker}}(g^*))$ and ${\operatorname{End}}({\operatorname{Coker}}(g^*))$. This proves (vi). [**Rationality of systems of certain $\ell$-adic Galois representations.**]{} [(J.-P. Serre)]{} [@jp I-10] [@se2 12.5?] For a fixed variety $V$ over a number field $k$ and integer $n$, is the system of Galois representations $H^n_{et}(\bar{V},{{\mathbb{Q}}}_{\ell})$ “rational”? The same question for the systems $W_iH^n_{et}(\bar{V},{{\mathbb{Q}}}_{\ell}(1))$ clearly refines the above one. For smooth proper $V$, Deligne’s theorem (Weil conjectures) [@weil2] provides an affirmative answer (see [@se2] Exemple after 12.5? on page 393). For any imbedding $\iota: k \hookrightarrow {{\mathbb{C}}}$, the weight filtrations on $H^n(V_{\iota}, {{\mathbb{Q}}}(1))$ and $H^n_{et}(\bar{V},{{\mathbb{Q}}}_{\ell}(1))$ are compatible [@hp §14]. But this does not imply the “rationality” of the system $W_iH^n_{et}(\bar{V},{{\mathbb{Q}}}_{\ell}(1))$ in general. However, the combination of (loc. cit) and the proof of (\[zund\] (v)) yields The system of Galois representations $W_j H^n_{et}(\bar{V},{{\mathbb{Q}}}_{\ell}(1))$ is “rational” for $j =-2,-1$ as is the system $t^n_{\ell}(V)$. *Acknowledgements.* I heartily thank S. Lichtenbaum for his constant encouragement, guidance, and support. I would like to express my gratitude to S. Bloch, P. Deligne, H. Gillet, J. Gordon, M. Marcolli, J. Milne, M. Nori, and B. Totaro. The debt to the work of Deligne and Carlson should be evident. I would like to thank the referees for many constructive remarks and suggestions. “iyaM visRSTiryata AbabhUva yadi vA dadhe yadi vA na yo asyAdhyakSaH parame vyoman so aNga veda yadi vA naveda” Nasadiya Sukta (Rigveda X 129). [10]{} M. Artin, *Théorème de changement de base par un morphisme lisse, et applications*, Théorie des topos et cohomologie étale des schémas. [T]{}ome 3, Springer-Verlag, Berlin, 1973, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat, Lecture Notes in Mathematics, Vol. 305, pp. 206–249. [MR ]{}[50 \#7132]{} L. Barbieri-Viale, A. Rosenschon, and M. Saito, *Deligne’s conjecture on $1$-motives*, (2001), http://xxx.lanl.gov/abs/math.AG/0102150 (to appear in Annals of Mathematics). L. Barbieri-Viale and V. Srinivas, *Albanese and [P]{}icard 1-motives*, Mém. Soc. Math. Fr. (N.S.) (2001), no. 87, vi+104. [MR ]{}[1 891 270]{} P. Berthelot, *Altérations de variétés algébriques (d’après [A]{}. [J]{}. de [J]{}ong)*, Astérisque (1997), no. 241, Exp. No. 815, 5, 273–311, Séminaire Bourbaki, Vol. 1995/96. [MR ]{}[98m:14021]{} S. Bosch, W. L[ü]{}tkebohmert, and M. Raynaud, *Néron models*, Springer-Verlag, Berlin, 1990. [MR ]{}[91i:14034]{} J. A. Carlson, *The one-motif of an algebraic surface*, Compositio Math. **56** (1985), no. 3, 271–314. [MR ]{}[87g:14005]{} [to3em]{}, *The geometry of the extension class of a mixed [H]{}odge structure*, Algebraic geometry, (Proc. Sympos. Pure Math., Vol. 46, Part 2, Bowdoin, 1985), Amer. Math. Soc., Providence, RI, 1987, pp. 199–222. [MR ]{}[89b:32036]{} E. B. Curtis, *Simplicial homotopy theory*, Advances in Math. **6** (1971), 107–209 (1971). [MR ]{}[43 \#5529]{} A. J. de Jong, *Smoothness, semi-stability and alterations*, Inst. Hautes Études Sci. Publ. Math. (1996), no. 83, 51–93. [MR ]{}[98e:14011]{} P. Deligne, Letter to the author. 15 April 1998. [to3em]{}, *Théorie de [H]{}odge. [I]{}*, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, Gauthier-Villars, Paris, 1971, pp. 425–430. [MR ]{}[56 \#356]{} [to3em]{}, *Théorie de [H]{}odge. [I]{}[I]{}*, Inst. Hautes Études Sci. Publ. Math. (1971), no. 40, 5–57. [MR ]{}[58 \#16653a]{} [to3em]{}, *La conjecture de [W]{}eil. [I]{}*, Inst. Hautes Études Sci. Publ. Math. (1974), no. 43, 273–307. [MR ]{}[49 \#5013]{} [to3em]{}, *Théorie de [H]{}odge. [I]{}[I]{}[I]{}*, Inst. Hautes Études Sci. Publ. Math. (1974), no. 44, 5–77. [MR ]{}[58 \#16653b]{} [to3em]{}, *Poids dans la cohomologie des variétés algébriques*, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, Canad. Math. Congress, Montreal, Que., 1975, pp. 79–85. [MR ]{}[55 \#5634]{} [to3em]{}, *La conjecture de [W]{}eil. [I]{}[I]{}*, Inst. Hautes Études Sci. Publ. Math. (1980), no. 52, 137–252. [MR ]{}[83c:14017]{} [to3em]{}, *Le groupe fondamental de la droite projective moins trois points*, Galois groups over ${\bf Q}$ (Berkeley, CA, 1987), Springer, New York, 1989, pp. 79–297. [MR ]{}[90m:14016]{} [to3em]{}, *À quoi servent les motifs?*, Motives (Proc. Sympos. Pure Math., Vol. 55, Part 1, Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1994, pp. 143–161. [MR ]{}[95c:14013]{} P. Deligne, J. S. Milne, A. Ogus, and K-y. Shih, *Hodge cycles, motives, and [S]{}himura varieties*, Springer-Verlag, Berlin, 1982, Lecture Notes in Mathematics, Vol. 900. [MR ]{}[84m:14046]{} M. Demazure and P. Gabriel, *Introduction to algebraic geometry and algebraic groups*, North-Holland Publishing Co., Amsterdam, 1980. [MR ]{}[82e:14001]{} F. El Zein, *Introduction à la théorie de [H]{}odge mixte*, Hermann, Paris, 1991. [MR ]{}[93c:14013]{} H. Esnault and E. Viehweg, *Deligne-[B]{}eĭlinson cohomology*, Beĭ linson’s conjectures on special values of $L$-functions, Academic Press, Boston, MA, 1988, pp. 43–91. [MR ]{}[89k:14008]{} H. Gillet, *Universal cycle classes*, Compositio Math. **49** (1983), no. 1, 3–49. [MR ]{}[84j:14016]{} M. Green, J. Murre, and C. Voisin, *Algebraic cycles and [H]{}odge theory*, Springer-Verlag, Berlin, 1994, Lecture Notes in Mathematics, Vol. 1594. [MR ]{}[96c:14001]{} A. Grothendieck, *Crystals and the de [R]{}ham cohomology of schemes*, Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam, 1968, pp. 306–358. [MR ]{}[42 \#4558]{} [to3em]{}, *Le groupe de [B]{}rauer. [I]{}[I]{}[I]{}. [E]{}xemples et compléments*, Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam, 1968, pp. 88–188. [MR ]{}[39 \#5586c]{} F. Guill[é]{}n, V. Navarro Aznar, P. Pascual Gainza, and F. Puerta, *Hyperrésolutions cubiques et descente cohomologique*, Springer-Verlag, Berlin, 1988, Lecture Notes in Mathematics, Vol. 1335. [MR ]{}[90a:14024]{} R. Hartshorne, *On the [d]{}e [R]{}ham cohomology of algebraic varieties*, Inst. Hautes Études Sci. Publ. Math. (1975), no. 45, 5–99. [MR ]{}[55 \#5633]{} L. Illusie, *Report on crystalline cohomology*, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974), Amer. Math. Soc., Providence, R.I., 1975, pp. 459–478. [MR ]{}[52 \#13845]{} U. Jannsen, *Deligne homology, [H]{}odge-[$\mathcal D$]{}-conjecture, and motives*, Beĭlinson’s conjectures on special values of $L$-functions, Academic Press, Boston, MA, 1988, pp. 305–372. [MR ]{}[89h:14016]{} [to3em]{}, *Mixed motives and algebraic ${K}$-theory*, Springer-Verlag, Berlin, 1990, Lecture Notes in Mathematics, Vol. 1400. [MR ]{}[91g:14008]{} [to3em]{}, *Motivic sheaves and filtrations on [C]{}how groups*, Motives (Proc. Sympos. Pure Math., Vol. 55, Part 1, Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1994, pp. 245–302. [MR ]{}[95c:14006]{} [to3em]{}, *Mixed motives, motivic cohomology, and [E]{}xt-groups*, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) (Basel), Birkhäuser, 1995, pp. 667–679. [MR ]{}[97j:14022]{} N. M. Katz, *Algebraic solutions of differential equations ($p$-curvature and the [H]{}odge filtration)*, Invent. Math. **18** (1972), 1–118. [MR ]{}[49 \#2728]{} [to3em]{}, *Wild ramification and some problems of “independence of $l$”*, Amer. J. Math. **105** (1983), no. 1, 201–227. [MR ]{}[85f:14021]{} [to3em]{}, *Review of $l$-adic cohomology*, Motives (Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1994, pp. 21–30. [MR ]{}[95j:14021]{} B. Mazur and W. Messing, *Universal extensions and one dimensional crystalline cohomology*, Springer-Verlag, Berlin, 1974, Lecture Notes in Mathematics, Vol. 370. [MR ]{}[51 \#10350]{} W. Messing, *The universal extension of an abelian variety by a vector group*, Symposia Mathematica, Vol. XI (Convegno di Geometria, INDAM, Rome, 1972), Academic Press, London, 1973, pp. 359–372. [MR ]{}[49 \#2760]{} J. S. Milne, *Étale cohomology*, Princeton University Press, Princeton, N.J., 1980. [MR ]{}[81j:14002]{} J. S. Milne and N. Ramachandran, *Integral motives and special values of zeta functions*, (2002), Preprint. Available at http://xxx.lanl.gov/abs/math.NT/0204065. F. Oort, *Commutative group schemes*, Springer-Verlag, Berlin, 1966. [MR ]{}[35 \#4229]{} N. Ramachandran, *From [J]{}acobians to one-motives: exposition of a conjecture of [D]{}eligne*, The arithmetic and geometry of algebraic cycles (Banff, AB, 1998), Amer. Math. Soc., Providence, RI, 2000, pp. 215–234. [MR ]{}[2001k:14022]{} [to3em]{}, *Duality of [A]{}lbanese and [P]{}icard 1-motives*, $K$-Theory **22** (2001), no. 3, 271–301. [MR ]{}[2002e:14013]{} B. Saint-Donat, *Techniques de descente cohomologique*, Théorie des topos et cohomologie étale des schémas. [T]{}ome 2, Springer-Verlag, Berlin, 1972, (SGA 4), Lecture Notes in Mathematics, Vol. 270, pp. 83–162. [MR ]{}[50 \#7131]{} J.-P. Serre, *Géométrie algébrique et géométrie analytique*, Ann. Inst. Fourier, Grenoble **6** (1955–1956), 1–42. [MR ]{}[18,511a]{} [to3em]{}, *Abelian $l$-adic representations and elliptic curves*, second ed., Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. [MR ]{}[91b:11071]{} [to3em]{}, *Propriétés conjecturales des groupes de [G]{}alois motiviques et des représentations $l$-adiques*, Motives (Proc. Sympos. Pure Math., Vol. 55, Part 1, Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1994, pp. 377–400. [MR ]{}[95m:11059]{} [to3em]{}, *Morphismes universels et différentiels de troisième éspèce*, Exposés de Séminaires (1950-1999), Documents Mathématiques 1, Soc. Math. France, Paris, 2001. J. Tate, *Conjectures on algebraic cycles in $l$-adic cohomology*, Motives (Proc. Sympos. Pure Math., Vol. 55, Part 1, Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1994, pp. 71–83. [MR ]{}[95a:14010]{} N. Tsuzuki, *Cohomological descent of rigid cohomology for proper coverings*, (2001), Preprint. V. Voevodsky, *Triangulated categories of motives over a field*, Cycles, transfers, and motivic homology theories, Princeton Univ. Press, Princeton, NJ, 2000, pp. 188–238. [MR ]{}[1 764 202]{} A. Weil, *Variétés abéliennes et courbes algébriques*, Hermann & Cie., Paris, 1948. [MR ]{}[10,621d]{} [^1]: Funded in part by grants from Hewlett-Packard, Graduate Research Board (U. Maryland) and MPIM (Bonn) [^2]: Here $\bar{k}$ is an algebraic closure of $k$. [^3]: V. Voevodsky [@vv] works over perfect fields and neglects $p$-torsion in characteristic $p >0$. [^4]: The natural map $f_* \mathcal O^* \to f_*C$ is zero: the former is represented by a torus and the latter by a vector group scheme. [^5]: Note that the map $d_i$ of [@h 5.2.3.2, 5.3.3.2] is denoted here by $\delta_i$ and vice-versa. [^6]: We adopt the notation: $X_{\ge -1} = X_{{\bullet}}$. [^7]: We note a harmless typo in (loc. cit): Gysin maps go from $H^{b-2r}$ to $H^{b-2(r-1)}$ and not $H^{b-2(r-2)}$. [^8]: We write $exp_m(L)$ for the $j_*^m\mathcal O^*_{U_{\ge n-1}}$-torsor and $exp(L)$ for the $\mathcal O^*_X$-torsor associated with $L$. [^9]: This follows a suggestion of M. Marcolli. [^10]: Deligne (loc. cit) remarks that this is not yet known for $V$ proper smooth.

--- abstract: 'The interband $\pi$ and $\pi+\sigma$ plasmons in pristine graphene and the Dirac plasmon in doped graphene are not applicable, since they are broad or weak, and weakly couple to an external longitudinal or electromagnetic probe. Therefore, the [*ab initio*]{} Density Function Theory is used to demonstrate that the chemical doping of the graphene by the alkali or alkaline earth atoms dramatically changes the poor graphene excitation spectrum in the ultra-violet frequency range ($4 - 10$ eV). Four prominent modes are detected. Two of them are the intra-layer plasmons with the square-root dispersion, characteristic for the two-dimensional modes. The remaining two are the inter-layer plasmons, very strong in the long-wavelength limit but damped for larger wave-vectors. The optical absorption calculations show that the inter-layer plasmons are both optically active, which makes these materials suitable for small organic molecule sensing. This is particularly intriguing because the optically active two-dimensional plasmons have not been detected in other materials.' author: - 'V. Despoja$^{1,2}$' - 'L. Maruši'' c$^{3}$' title: UV active plasmons in alkali and alkaline earth intercalated graphene --- Extensive research of electronic excitations in graphene showed the existence of several two-dimensional (2D) plasmon modes: the intraband (Dirac) plasmon existing only in the doped graphene [@DasSarma; @PlPhT; @grafen; @Measurenanoribb; @Tip; @IR2], and the interband plasmons, which exist in pristine and doped graphene and originate from the interband electron-hole transition between the $\pi$ and $\pi^*$ bands and between the $\pi$ and $\sigma^*$ bands [@grafen; @politano1; @politano2; @Dino; @Eberlein]. These investigations also showed that the interband $\pi$ and $\pi+\sigma$ plasmons are broad and weak resonances, so their interaction with the external longitudinal or electromagnetic probes is weak as well, which makes them inadequate for most practical applications. The ’tunable’ Dirac plasmon in the doped graphene is also weak (for experimentally feasible doping), and in addition to that, it does not couple to an incident electromagnetic field directly. In the systems proposed so far, light could be coupled to the Dirac plasmon only indirectly, e.g. by using metallic tips, gratings or prisms, or by arranging graphene into nanoribbons[@Measurenanoribb; @Tip; @PRLGNR], which is all hard to fabricate. Also, such indirect coupling additionally reduces the intensity of the plasmon, thus reducing the efficiency of its application. The alkali or alkaline earth intercalated graphene is much easier to fabricate and offers a broader variety of plasmons, both intraband and (especially) interband. Such systems have recently been extensively studied, both theoretically and experimentally [@exp1; @exp2; @exp3; @exp1SuperC; @exp2SuperC; @theory; @lazic], but the attention has not been on the electronic excitations. Intercalating any alkali or alkaline earth metal to a single graphene layer causes the natural doping of the graphene and results in the formation of two quasi two-dimensional (q2D) plasmas. This supports the existence of two 2D intraband plasmons, acoustic and Dirac, with frequencies up to 4 eV [@2Dplasmons], as well as several interband and even inter-layer modes occurring at higher frequencies. Some of these modes are optically active and some of them can be manipulated by doping, which opens possibilities for their application in various fields, such as plasmonics, photonics, transformation optics, optoelectronics, light emitters, detectors and photovoltaic devices [@IR3; @APP1; @chinos; @appl1; @appl3; @appl4; @appl5; @appl6; @appl7; @appl8]. Moreover, ’tunable’ 2D plasmons could be very useful in the area of chemical or biological sensing [@APP2; @photopto; @appl2; @appl9], which is one of our main suggestions for the potential application of the results of this research. We performed calculations for several alkali and alkaline earth metals, with different coverages, and found that the effects which are the focus of this letter are valid for all of them. In all these cases, in addition to the graphene $\pi$ and $\sigma$ bands, there are also the $\pi$ and $\sigma$ bands of the intercalated metal. This opens possibilities for various electron-hole (e-h) transitions which may be the origins of the interband plasmons. We limit our investigation to the frequencies between 4 and 10 eV(the UV region), where the dominant interband plasmons occur, and identify four significant modes within this range. Two of them are not very well defined in the long-wavelength limit but they exist at larger wave-vectors as well, and show the square-root dispersion characteristic for the surface and 2D modes. These modes are the intra-layer modes, one located in the graphene layer and the other located in the metallic layer. The other two are very prominent in the long-wavelength limit, but at higher wave-vectors their intensities rapidly decrease, which makes them potentially interesting for optical applications [@IR3; @APP1; @APP2; @stauber; @PRLGNR]. Their dispersions are different from those typical for the 2D modes, indicating that they are different from the usual 2D plasmons. Detailed inspection (including retardation, i.e. finite speed of light, and tensorical response) shows that they are dipolar inter-layer modes (the electric field they produce oscillates perpendicular to the crystal plane), i.e. optically active q2D plasmons, contrary to the widely studied q2D plasmons which produce electric field parallel to the crystal plane, and are not optically active. The extensively studied graphene $\pi$ and $\pi+\sigma$ modes are optically active, but in the long wavelength limit ($Q \rightarrow 0$) they are not plasmons but electron-hole excitations [@Dino]. The theoretical formulation of the electronic response in various q2D systems has already been presented[@grafen; @Duncan2; @wake; @Rukelj], so here we only point out some details of the calculation important for the understanding of the result we want to present. We define the Electron Energy Loss Spectroscopy (EELS) local spectral function as the imaginary part of the excitation propagator $$S_{z_0}({\bf Q},\omega)=-Im D_{z_0}({\bf Q},\omega), \label{spectrum}$$ where $$D_{z_0}({\bf Q},\omega) = W^{ind}_{\textbf{G}_{\parallel}=0}(\textbf{Q},\omega,z_0,z_0). \label{propagator}$$ The $S_{z_0}({\bf Q},\omega)$ is also proportional to the probability density for the parallel momentum transfer ${\bf Q}=(Q_x,Q_y)$ and the energy loss $\omega$ of the reflected electron in the Reflection Electron Energy Loss Spectroscopy (REELS)[@REELS]. The induced dynamically screened Coulomb interaction is $W^{ind}=v^{2\textrm{D}}\otimes\chi\otimes v^{2\textrm{D}}$, where $v^{2\textrm{D}} = \frac{2\pi}{Q}e^{-Q\left|z-z'\right|}$ is the 2D Fourier transform of the bare Coulomb interaction and $\otimes=\int^{L/2}_{-L/2}dz$[@Leo]. The response function is obtained as the solution of the matrix Dyson equation $\hat{\chi}=\hat{\chi}^0 +\hat{\chi}^0\hat{v}^{2\textrm{D}}\hat{\chi}$ in the reciprocal space plane-wave basis ${\bf G}=({\bf G}_{\parallel},G_z)$. The non-interacting electrons response matrix is $\hat{\chi}^{0}=\frac{2}{\Omega}\sum_{i,j}(f_i-f_j)/(\omega+i\eta+E_i-E_j)\rho_{{\bf{G}},ij}\rho^*_{{\bf G}',ij}$, where $f_i$ is the Fermi-Dirac distribution, $\rho_{{\bf{G}},ij}$ are charge vertices  [@grafen], $\Omega$ is the normalization volume, and $i=(n,\bf{K})$ and $j=(m,{\bf K+Q})$ are Kohn-Sham-Bloch states. The Coulomb interaction with the surrounding supercells in the superlattice arrangement is excluded, as described in detail in Ref.[@Rukelj]. ![(color online) The intensity of the electronic excitations in (a) CsC$_8$, (b) CaC$_6$, (c) LiC$_6$ and (d) LiC$_2$. The white and green dotted lines in (d) show the boundaries of the e-h excitation gaps for the graphene $\pi$ bands around the Dirac point and the Li $\sigma$ bands around the $\Gamma$ point, respectively.[]{data-label="Fig1"}](EELS.pdf){width="\textwidth"} To calculate the Kohn-Sham (KS) wave functions $\phi_{n{\bf K}}$ and energy levels $E_{n{\bf K}}$, i.e. the band structure, of the LiC$_2$, LiC$_6$, CaC$_6$ and CsC$_8$ slabs, we use the plane-wave self-consistent field DFT code (PWSCF) within the QUANTUM ESPRESSO (QE) package [@QE]. The core-electron interaction is approximated by the norm-conserving pseudopotentials [@pseudopotentials], and the exchange correlation (XC) potential by the Perdew-Zunger local density approximation (LDA) [@LDA]. For the slab unit cell constant we use the graphene value of $a_{uc} = 4.651$ a.u. [@lattice], and we separate the slabs by $L=5a_{uc}=23.255$a.u. The equilibrium separations between the metallic and carbon layers within a slab for these four systems is $d = $ 4.1a.u.(2.17Å), 3.28a.u.(1.74Å), 4.46a.u. (2.36Å) and 5.8a.u. (3.08Å), respectively, as proposed in Ref.[@lazic; @Nanolett-Dino]. Our reference frame is chosen so that the graphene layer is positioned at $z=0$, and the metallic layer is at $z=d$. The ground state electronic densities of the slabs are calculated by using the $12\times12\times1$ Monkhorst-Pack K-point mesh[@MPmesh] of the first Brillouin zone (BZ). For the plane-wave cut-off energy we choose $50$Ry ($680$eV). The Fermi levels of these systems (measured from the Dirac point, i.e. from the pristine graphene Fermi level) are: $E_F$ = $1.78$, $1.55$, $1.375$ and $1.24$eV, respectively. For the response matrix $\hat{\chi}^{0}$ calculation in the long-wavelength ($Q<0.01$a.u.) limit we use $601\times601\times1$ $K$-point mesh and the damping parameter $\eta=10meV$, while for the larger $Q$s we use $201\times201\times1$ $K$-point mesh sampling and $\eta=30meV$. In all cases the band summation is performed over $30$ bands and the perpendicular crystal local field energy cut off is $10$Ry ($136$eV), which corresponds with $23$ $G_z$ wave vectors. Figs. \[Fig1\] shows the excitation spectra in (a) CsC$_8$, (b) CaC$_6$, (c) LiC$_6$ and (d) LiC$_2$ slabs, calculated from Eg.\[spectrum\]. The spectral intensities are shown as functions of $\omega$ and Q, using the color scheme, which enables us to see the dispersions of the modes. We can see that, in addition to the well known modes present in the doped graphene (Dirac (DP) and $\pi$ (C($\pi$)) plasmon[@grafen]), there are a few other modes, strong in the long-wavelength limit (indicating their optical activity) and more pronounced in the systems with higher electronic doping, especially for LiC$_6$ and LiC$_2$. Therefore, we put emphasis on the system with the highest doping, i.e. to the LiC$_2$. Fig.\[Fig1\](d) shows the intensities of the electronic excitations in the $\Gamma$M and $\Gamma$K direction for the LiC$_2$. The spectra in these two directions are very similar, so here we focus only on the $\Gamma$M direction. At lower frequencies (up to 4 eV) we can see the intra-band q2D plasmons, which have already been discussed in detail for the LiC$_2$[@2Dplasmons]. At frequencies between 4 and 10 eV we can see four significant inter-band modes (denoted as C($\pi$), Li($\pi + \sigma$), ILP$_1$ and ILP$_2$), two with the square-root dispersion, characteristic for the q2D systems, which exist for the larger wavevectors as well, and the other two which are strongly damped for larger wavevectors. These modes, which exist in all these systems (at similar frequencies), are the focus of this letter. To understand them we shall explore the band structure and the spectra of electronic excitations in the LiC$_2$ in more detail. However, our conclusions about the origins and characteristic of the modes obtained for the LiC$_2$, are valid for the other three systems as well. Fig.\[Fig2\] shows spectra $S(\omega)$ for the LiC$_2$ for various wave-vectors Q (denoted in graphs) in the $\Gamma$M direction. The lower panel contains the spectra of the n-doped graphene (dashed red lines), with the same doping as in the LiC$_2$ ($E_F=1.78$eV), for comparison. The doped graphene spectra show only two modes, the very prominent intraband Dirac plasmon, roughly matching the LiC$_2$ Dirac plasmon, and the interband $\pi$ plasmon around 5eV, which is very weak due to heavy doping. In the LiC$_2$ spectra, in addition to the already described q2D intraband acoustic and Dirac plasmon (AP and DP)[@2Dplasmons], we can notice a barely visible broad peak between 4.5 and 5eV, which corresponds with the graphene $\pi$ plasmon, plus three other modes which cannot be related to any of the graphene modes. This means that these modes are either the lithium q2D intra-layer modes, or the inter-layer modes, which represent charge oscillations perpendicular to the crystal plane. ![(color online) The spectra of the electronic excitations in the LiC$_2$ for the $\Gamma$M direction (full black lines). The lower panel contains the comparison with the doped graphene spectra (dashed red lines) with the matching Fermi levels. Wave-vector ranges for each figure are indicated on the figure.[]{data-label="Fig2"}](LiC2-gr.pdf){width="45.00000%"} Fig.\[Fig3\](a) shows the band structure of the LiC$_2$ slab, with the color scheme indicating the predominant origins of particular bands. Blue and turquoise indicate predominant lithium $\pi$ and $\sigma$ orbitals, respectively, while red and pink indicate predominant graphene $\pi$ and $\sigma$ orbitals, respectively. The arrows indicate the e-h transitions which are the potential origins of the four modes. However, since the transition energies are very similar, it is impossible to reach definite conclusions about the origins of the particular modes from the band structure itself. Fig.\[Fig3\](b) shows the imaginary (thick solid black line) and real (thick dashed black line) part of the excitation propagator (\[propagator\]) in the LiC$_2$ for several characteristic wave-vectors $Q$. The thin red line is the unscreened (single particle) spectrum obtained by replacing $\chi$ with $\chi^0$ in $W^{ind}$ used in (\[spectrum\]). By comparing these three lines we can distinguish the collective modes from the single particle excitations. Furthermore, comparing the frequencies of these excitations with the transitions in the band structure in the Fig.\[Fig3\](a) can help us identify the origins of some of the modes. ![(color online) (a) The band structure of the LiC$_2$, with the color scheme indicating the predominant origins of particular bands (blue - Li($\pi$), turquoise - Li($\sigma$), pink - C($\sigma$), red C($\pi$)).(b) Development of the interband plasmons with the increase of the wavevector. Thick full black lines - $Im D$, thick dashed black lines - $Re D$, thin red lines - spectra of the single particle excitations only. (c) Angle resolved optical absorption spectra in LiC$_2$, in ultraviolet frequency range.[]{data-label="Fig3"}](Bands.pdf "fig:"){width="29.00000%"} ![(color online) (a) The band structure of the LiC$_2$, with the color scheme indicating the predominant origins of particular bands (blue - Li($\pi$), turquoise - Li($\sigma$), pink - C($\sigma$), red C($\pi$)).(b) Development of the interband plasmons with the increase of the wavevector. Thick full black lines - $Im D$, thick dashed black lines - $Re D$, thin red lines - spectra of the single particle excitations only. (c) Angle resolved optical absorption spectra in LiC$_2$, in ultraviolet frequency range.[]{data-label="Fig3"}](IP.pdf "fig:"){width="40.00000%"} ![(color online) (a) The band structure of the LiC$_2$, with the color scheme indicating the predominant origins of particular bands (blue - Li($\pi$), turquoise - Li($\sigma$), pink - C($\sigma$), red C($\pi$)).(b) Development of the interband plasmons with the increase of the wavevector. Thick full black lines - $Im D$, thick dashed black lines - $Re D$, thin red lines - spectra of the single particle excitations only. (c) Angle resolved optical absorption spectra in LiC$_2$, in ultraviolet frequency range.[]{data-label="Fig3"}](LiC2_absorption.pdf "fig:"){width="22.00000%"} In the unscreened spectra (thin red lines) we can see two prominent peaks, one around 4ev which exist for all wavevectors, and the other around 6eV which disappears for larger Q. This indicates that the first one is the origin of the modes denoted as C($\pi$) and Li($\pi+\sigma$), while the second one is the origin of the remaining two modes (IL$_1$ and IL$_2$). Further analysis consists of applying the p and n doping to our system (i.e. changing the position of the Fermi level and causing some occupied bands to become unoccupied, and vice versa), and omitting particular bands from the calculation of the response function ${\chi}^{0}$, to determine the exact role of each band. By doing that, we found out that the first peak in the single particle spectra is actually an overlap of two peaks. One is coming from the transition between the $\pi$ and $\pi^*$ graphene bands around the M point (red arrow in Fig.\[Fig3\](a)), and that one is the origin of the mode denoted as C($\pi$), i.e. the graphene $\pi$ plasmon. The other is coming from the transition between the $\sigma$ and $\pi$ lithium bands around the $\Gamma$ point (blue arrow in Fig.\[Fig3\](a)), and that one is the origin of the mode denoted as Li($\pi+\sigma$), i.e. the lithium $\pi+\sigma$ plasmon. The second peak in the single particle spectra comes from the transitions between the graphene $\pi$ bands and the lithium $\sigma$ bands (black arrows in Fig.\[Fig3\](a)), and it is the origin of the remaining two modes, denoted as IL$_1$ and IL$_2$. Their dispersion is not square root like, which is the consequence of their inter-layer nature. All the presented spectra are calculated for the probe positioned at $z_0=L/2$, but we can change the position of the probe and monitor the changes in the spectra to determine the symmetry of the particular modes. By doing that we confirmed that one of the peaks (IL$_1$) is even, while the other one (IL$_2$) is odd. Contrary to the intra-layer plasmons C($\pi$) and Li($\pi+\sigma$) the inter-layer plasmons IL$_1$ and IL$_2$ are sharp, well defined resonances which could be especially suitable for the sensing of small organic molecules with excitonic spectra in UV frequency range. However, the crucial question is: can the IL$_1$ and IL$_2$ plasmons be excited by an external electromagnetic field, i.e, are they optically active? If that is the case, then it seems that the intercalated graphene may become the technologically simplest platform for biosensing. In the systems proposed so far, light could be coupled to plasmon resonances only indirectly, e.g. by using the metallic nanoparticles, gratings or prisms, or by arranging graphene into nanoribbons, which is much more difficult to fabricate. Another exciting aspect of this issue is that the optically active q2D plasmons have not been discovered in other systems. In order to answer this crucial questions we performed a sophisticated angle resolved optical absorption calculation which includes the retardation and the tensoric character of the LiC$_2$ dynamical response [@DSV]. The mathematical formulation of this theoretical tool and its application to the molybdenum disulfide (MoS$_2$) monolayer is presented in Ref.[@MoS2]. Fig.\[Fig3\](c) shows the absorption of $p$ polarized light in the LiC$_2$, as a function of the incident light with the frequency $\omega$ and angle $\theta$ (as sketched). For the normal incidence ($\theta=0^o$) the electric field is parallel with the crystal plane and there are no peaks corresponding to IL$_1$ and IL$_2$. However, as the incident angle increases the IL$_1$ and IL$_2$ peaks appear, and finally for the almost grazing incidence ($\theta=80^o$), i.e. for the the electrical field almost perpendicular to the crystal plane, they become very intensive. This undoubtedly confirms not only that these modes are optically active, but also their inter-layer character. The grazing spectra show some additional peaks (ex$_1$, ex$_2$ and ex$_3$) which do not appear in the EELS spectra, i.e. which can not be excited by an external longitudinal probe, which means that they are probably not plasmons but UV active excitons. In the remaining three systems presented in this letter, the electronic doping is weaker than in the LiC$_2$ which has two important consequences. First, the two inter-layer plasmons are not nearly as strong and sharp as they are in the LiC$_2$, but they still exist, and are still optically active in the UV region. Second, due to the weaker doping the Fermi level is lower (with respect to the LiC$_2$ Fermi level) by 0.23eV for LiC$_6$, 0.405eV for CaC$_6$ and 0.54eV for CsC$_8$. Considering that the plateau in the graphene $\pi^*$ band is only 0.1 - 0.2eV below the LiC$_2$ Fermi level, this means that in the three other systems that plateau is unoccupied, which makes the graphene $\pi$ plasmon much stronger. Therefore, by changing the doping (by changing the dopant or the coverage, or by applying the gate voltage) we can tune these modes, i.e. increase or decrease their intensities. In conclusion, we showed that doping the graphene by the alkali or alkaline earth atoms dramatically modifies the graphene plasmonics, especially in UV parts of the spectra, where we obtain four interband plasmons. This effect is the strongest in the full coverage lithium doped graphene (LiC$_2$), due to the highest doping. Two of the modes, not very strong in the long wavelength limit, exist for larger wavevectors, with the square-root dispersion characteristic for the 2D plasmons. They turned out to be the intra-layer modes, one within the graphene layer (the well known graphene $\pi$ plasmon), and the other within the intercalated metal layer. The other two plasmons ILP$_1$ and ILP$_2$ are strong and sharp in the long wavelength limit, but damped for the larger wave-vectors. They turned out to be the inter-layer optically active plasmons, i.e. they couple directly to the electromagnetic field. Such unusual and poorly explored optically active 2D plasmons can be used as the efficient sensor in chemical sensing and biosensing. This work was supported by the QuantiXLie Centre of Excellence, a project co-financed by the Croatian Government and European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004). Computational resources were provided by the Donostia International Physic Center (DIPC) computing center. [99]{} E. H. Hwang and S. Das Sarma, Phys. Rev. B [**75**]{}, 205418 (2007), Hwang E. H. and Das Sarma, Phys. Rev. B [**80**]{}, 205405 (2009) E. H. Hwang, R. Sensarma, and S. Das Sarma, Phys. Rev. B [**82**]{}, 195406 (2010) V. Despoja, D. Novko, K. Dekanić, M. Šunjić and L. Marušić, Phys. Rev. B [**87**]{}, 075447 (2013) H. Yan, T. Low, W. Zhu, Y. Wu, M. Freitag, X. Li, F. Guinea, P. Avouris and F. Xia, Nat Photonics [**7**]{}, 394 (2013) Z. Fei, G. O. Andreev, W. Bao, L. M. Zhang, A. S. McLeod, C. Wang, M. K. Stewart, Z. Zhao, G. Dominguez, M. Thiemens, M. M. Fogler, M. J. Tauber, A. H. Castro-Neto, C. N. Lau, F. Keilmann and D. N. Basov, Nano Lett. [**11**]{}, 4701 (2011) M. Jablan, H. Buljan and M. Soljači' c, Phys. Rev. B [**80**]{}, 245435 (2009) A. Politano, I. Radović, D. Borka, Z.L. Mišković, G. Chiarello, Carbon [**96**]{}, 91 (2016) A. Politano, I. Radović, D. Borka, Z.L. Mišković, H.K. Yu, D. Farías, G. Chiarello, Carbon [**114**]{}, 70 (2017) D. Novko, V. Despoja, and M. Šunjić, Phys. Rev. B [**91**]{}, 195407 (2015) T. Eberlein, U. Bangert, R. R. Nair, R. Jones, M. Gass, A. L. Bleloch, K. S. Novoselov, A. Geim, and P. R. Briddon, Phys. Rev. B [**77**]{}, 233406 (2008) C. Vacacela Gomez, M. Pisarra, M. Gravina, J.M. Pitarke, and A. Sindona, Phys. Rev. Lett. [**117**]{}, 116801 (2016) A. Kumar, A. L. M. Reddy, A. Mukherjee, M. Dubey, X. Zhan, N. Singh, L. Ci, W. E. Billups, J. Nagurny, G. Mital and P. M. Ajayan, ACS Nano [**5**]{}, 4345 (2011) S. L. Yang, J. A. Sobota, C. A. Howard, C. J. Pickard, M. Hashimoto, D. H. Lu, S. K. Mo, P. S. Kirchmann and Z. X. Shen, Nature Communications [**5**]{}, 3493 (2014) N. M. Caffrey, L. I. Johansson, C. Xia, R. Armiento, I. A. Abrikosov and C. Jacobi, Phys. Rev. B [**93**]{} 195421 (2016) S. Ichinokura, K. Sugawara, A. Takayama, T. Takahashi, and S. Hasegawa, ACS Nano [**10**]{}, 2761 (2016) K. Li, X. Feng, W. Zhang, Y. Ou, L. Chen, Ke. He, Li-Li Wang, L. Guo, G. Liu, Qi-Kun Xue and X. Ma, Appl. Phys. Lett. [**103**]{}, 062601 (2013) M. Khantha, N. A. Cordero, L. M. Molina, J. A. Alonso, and L. A. Girifalco, Phys. Rev. B [**70**]{}, 125422 (2004) P. Pervan, P. Lazić, M. Petrović, I. Šrut Rakić, I. Pletikosić, M. Kralj, M. Milun and T. Valla, Phys. Rev. B [**92**]{}, 245415 (2015) D. Novko, Nano Lett. [**17**]{}, 6991 (2017) L. Marušić and V. Despoja, Phys.Rev B [**95**]{}, 201408(R) (2017) F. Bonaccorso, Z. Sun, T. Hasan and A. C. Ferrari, Nature Photonics [**4**]{}, 611 (2010) A. Vakil and N. Engheta, Science [**332**]{}, 1291 (2011) Wu Hua-Qiang, Linghu Chang-Yang, L ̈Hong-Ming, and Qian He, Chin. Phys. B [**22**]{}, 098106 (2013) A. Pospischil, M. M. Furchi, and T. Mueller, Nat. Nanotechnol. [**9**]{}, 257 (2014) J. S. Ross, P. Klement, A. M. Jones, N. J. Ghimire, J. Yan, D. G. Mandrus, T. Taniguchi, K. Watanabe, K. Kitamura, W. Yao, D. H. Cobden, and X. Xu, Nat. Nanotechnol. [**9**]{}, 268 (2014) F. H. L. Koppens, T. Mueller, P. Avouris, A. C. Ferrari, M. S. Vitiello, and M. Polini, Nat. Nanotechnol. [**9**]{}, 780 (2014) S. Jo, N. Ubrig, H. Berger, A. B. Kuzmenko, and A. F. Morpurgo, Nano Lett. [**14**]{}, 2019 (2014) O. Lopez-Sanchez, E. Alarcon Llado, V. Koman, A. Fontcuberta i Morral, A. Radenovic, and A. Kis, ACS Nano [**8**]{}, 3042 (2014) C.-H. Lee, G.-H. Lee, A. M. van der Zande, W. Chen, Y. Li, M. Han, X. Cui, G. Arefe, C. Nuckolls, T. F. Heinz, J. Guo, J. Hone, and P. Kim, Nat. Nanotechnol. [**9**]{}, 676 (2014) B. W. H. Baugher, H. O. H. Churchill, Y. Yang, and P. Jarillo-Herrero, Nat. Nanotechnol. [**9**]{}, 262 (2014) T. Low and P. Avouris, ACS Nano [**8**]{}, 1086 (2014) F. Bonaccorso, Z. Sun, T. Hasan, A. C. Ferrari, Nature Photonics [**4**]{}, 611 (2010) L. Britnell, R. M. Ribeiro, A. Eckmann, R. Jalil, B. D. Belle, A. Mishchenko, Y. J. Kim, R. V. Gorbachev, T. Georgiou, S. V. Morozov, A. N. Grigorenko, A. K. Geim, C. Casiraghi, A. H. C. Neto, and K. S. Novoselov, Science [**340**]{}, 1311 (2013) C. Zhu, D. Du, Y. Lin, 2D Mater. [**2**]{}, 032004 (2015), A. Y. Zhu, E. Cubukcu, 2D Mater. [**2**]{}, 032005 (2015) T. Stauber and H. Kohler, Nano Lett. [**16**]{}, 6844 (2016) V. Despoja, Z. Rukelj and L. Marušić, Phys.Rev B [**94**]{}, 165446 (2016) V.Despoja, D. J. Mowbray, D. Vlahovi' c and L. Maruši' c, Phys. Rev. B [**86**]{}, 195429 (2012) V. Despoja, K. Dekani' c, M. Šunji' c, and L. Maruši' c, Phys. Rev. B [**86**]{}, 165419 (2012) Z.L.Wang, J.M.Cowley, Surface Science, [**193**]{}, 501 (1988) L. Marušić and M. Šunjić, Phys. Scr. [**63**]{}, 336 (2001) P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, [*et.al.*]{}, J. Phys.: Conden. Matter [**21**]{}, 395502 (2009) N. Troullier and J. L. Martins, Phys. Rev. B [**43**]{}, 1993 (1991) J.P. Perdew and A. Zunger, Phys. Rev. B [**23**]{}, 5048 (1981) R. Saito, G. Dresselhaus, and M. S. Dresselhaus, *Physical Properties of Carbon Nanotubes* (Imperial College Press, London, (1998) H.J. Monkhorst and J.D. Pack, Phys. Rev. B [**13**]{}, 5188 (1976) D. Novko, M. Šunji' c and V. Despoja, Phys. Rev B [**93**]{}, 125413 (2016) Z. Rukelj, A. Štrkalj, V. Despoja, Phys. Rev. B [**94**]{}, 115428 (2016)

--- abstract: 'We present an extension of the tunneling theory for scanning tunneling microcopy (STM) to include different types of vibrational-electronic couplings responsible for inelastic contributions to the tunnel current in the strong-coupling limit. It allows for a better understanding of more complex scanning tunneling spectra of molecules on a metallic substrate in separating elastic and inelastic contributions. The starting point is the exact solution of the spectral functions for the electronic active local orbitals in the absence of the STM tip. This includes electron-phonon coupling in the coupled system comprising the molecule and the substrate to arbitrary order including the anti-adiabatic strong coupling regime as well as the Kondo effect on a free electron spin of the molecule. The tunneling current is derived in second order of the tunneling matrix element which is expanded in powers of the relevant vibrational displacements. We use the results of an ab-initio calculation for the single-particle electronic properties as an adapted material-specific input for a numerical renormalization group approach for accurately determining the electronic properties of a NTCDA molecule on Ag(111) as a challenging sample system for our theory. Our analysis shows that the mismatch between the ab-initio many-body calculation of the tunnel current in the absence of any electron-phonon coupling to the experiment scanning tunneling spectra can be resolved by including two mechanisms: (i) a strong unconventional Holstein term on the local substrate orbital leads to reduction of the Kondo temperature and (ii) a different electron-vibrational coupling to the tunneling matrix element is responsible for inelastic steps in the $dI/dV$ curve at finite frequencies.' author: - Fabian Eickhoff - Elena Kolodzeiski - Taner Esat - Norman Fournier - Christian Wagner - Thorsten Deilmann - Ruslan Temirov - Michael Rohlfing - 'F. Stefan Tautz' - 'Frithjof B. Anders' title: 'Inelastic electron tunneling spectroscopy for probing strongly correlated many-body systems by scanning tunneling microscopy' --- Introduction ============ The investigation of phonons and molecular vibrations by inelastic electron tunneling spectroscopy dates back more than 50 years [@JaklevicLambe1966; @MolecularVibrationTunnel1968]. For example, point contact spectroscopy [@PCSreview1989] has been successfully used to measure the electron-phonon coupling function that enters the Migdal-Eliashberg theory [@McMillanTc1968; @AllenMitrovic] of superconductivity. Recently, the increasing relevance of quantum nanoscience [@Khajetoorians2011; @Baumann2015; @Donati2016; @Natterer2017; @Esat2018; @Cocker2016; @Doppagne2018; @Kimura2019; @Wagner2019] revitalizes the interest in vibrational inelastic electron tunneling spectroscopy (IETS) of molecules adsorbed on solid surfaces [@Stipe1998; @Guo2016; @Wegner2013; @Burema2013] or contacted in transport junctions [@Kim2011a; @Vitali2010; @Meierott2017; @Bruot2012; @Sukegawa2014]. While the fundamental mechanisms of the electron-phonon and electron-vibron interactions are well-understood (for simplicity, we will refer to both as electron-phonon interaction from now on), a quantitative theory with predicting power beyond a simplified picture comprising independent electronic degrees of freedoms and bosonic excitations is lacking. Even modern reviews [@REED2008] on this subject present the inelastic tunnel process only on the original level of understanding [@JaklevicLambe1966; @MolecularVibrationTunnel1968], i.e. the emission or absorption of a single phonon when a single electron is tunneling, as depicted in Fig. 1 of Ref. [@MolecularVibrationTunnel1968] or Fig. 1(a) of Ref. [@REED2008]. This commonly accepted picture is very adequate in the weak coupling limit [@MolecularVibrationTunnel1968] of the adiabatic regime [@EntelGrewe1979; @galperinNitzanRatner2006; @EidelsteinSchiller2013; @JovchevAnders2013], whence the electron-phonon coupling is small on the energy scale of the hybridization between the relevant molecular orbital(s) and the surface (or electrode in a transport experiment), and provides a basic understanding of the relevant physical processes. However, it becomes problematic in systems dominated by polaron formation, or for systems in the crossover region between the adiabatic and the anti-adiabatic regimes [@EntelGrewe1979; @galperinNitzanRatner2006; @EidelsteinSchiller2013]. This calls for a more general treatment of the inelastic tunneling process. In this paper we provide such a theory, focussing in particular on the case of scanning tunneling spectroscopy (STS). We generalize the original picture [@JaklevicLambe1966; @MolecularVibrationTunnel1968] to strongly correlated electron systems but maintain the notion that inelastic contributions to the tunneling current require absorption or emission of a phonon while the electron is crossing the tunnel barrier. We treat the STM tip and the system of interest as initially decoupled and fully characterized by their exact Green’s functions. After specifying the tunneling Hamiltonian $\hat H_T$, the tunnel current operator is derived from the charge conservation. Then the coupling between the system and the STM tip, $\hat H_T$, is switched on, and the evolving steady-state current is evaluated in second order of the tunneling matrix elements. All material-dependent spectral properties are encoded in the equilibrium spectral functions of the system. Combining an accurate determination of the molecular spectral function using Wilson’s numerical renormalization group (NRG) approach [@Wilson75; @BullaCostiPruschke2008] with a density functional approach [@RevModPhys.74.601] provides a theoretical approach to strongly coupled system with predicting power. STS is an established technique and its theoretical background is well-understood [@TersoffHamann1983; @TersoffHamann1985]. Setting aside more challenging situations, commonly a featureless density of states in the STM tip is assumed, and the STM is operated in the tunneling regime such that the measured $dI/dV$ curve may be interpreted as being proportional to the local energy-dependent density of states (LDOS) of the sample at the given bias voltage. Using spin-polarized tips [@SplittingRKKY2012] allows for the detection of the spin-dependent LDOS. Since electrons usually can tunnel from the STM tip to different orbitals in the target system, the quantum mechanical interference of different paths [@SchillerHershfield2000a] may lead to Fano line shapes [@FanoResonance1961] in the tunneling spectra. The interpretation of electron tunneling becomes more complicated if the spectrum is dominated by the Kondo effect. The Kondo effect, originally discovered as resistance anomaly in metals containing magnetic impurities [@Kondo62; @kondo_effect], has been studied experimentally in quantum dots [@Kondo_QD0; @Kondo_QD], atoms and molecules on surfaces [@kondo_atom; @LiSchneider1998; @Manoharan2000; @AgamSchiller2001; @Kondo_molecule; @kondo_molecule2], and molecular junctions [@kondo_SM]. A comprehensive understanding has been developed [@Wilson75; @kondo_anderson]: briefly, the at low temperatures logarithmically diverging antiferromagnetic exchange coupling between the unpaired spin and the itinerant electron states in the substrate (or leads) produces a singlet ground state with a low-energy single-particle excitation spectrum that is characterized by a resonance at zero energy. In such systems with their intrinsically highly non-linear LDOS in the vicinity of the chemical potential, it becomes very challenging to distinguish between elastic tunneling processes governed by the energy-dependent transfer matrix and additional inelastic contributions generated by the presence of an additional electron-phonon coupling. For example, in such systems so-called Kondo replica at vibrational frequencies have been observed [@Kondo_vib_SM; @kondo_vib_bjunc; @kondo_vib_bjunc2; @kondo_vib_bjunc3; @vib_kondo_stm; @vib_kondo_stm2; @vib_kondo_stm3], whose precise nature is, however, not yet understood. The interplay between Kondo physics and electron-vibron coupling has also been studied theoretically [@PaakeFlensberg2005; @vib_kondo_theo3; @vib_kondo_theo]. Since only the total tunneling current is accessible in experiments, its decomposition into individual processes requires guidance by a theory. In this paper, we present an approach providing this guidance. Specifically, we derive an extension to the comprehensive theory of the tunneling current in STM that was originally formulated by Schiller and Hershfield [@SchillerHershfield2000a] in the context of a magnetic adatom and generalized Fano’s analysis [@FanoResonance1961] to inelastic contributions in the tunneling Hamiltonian which includes the calculation of the current operator from the local continuity equation. Notably, our theory accounts for two different types of electron-phonon interactions: (i) the intrinsic electron-phonon coupling in the system in the absence the STM tip and (ii) vibrationally induced fluctuations of the distance between tip and molecule or substrate. The former is included in the system’s Green’s functions and only contribute to the elastic current. The latter enter the tunneling $H_T$ and, therefore, are the origin of the inelastic current contributions. Having developed said theory, we demonstrate its capabilities by applying the approach to explain the experimental data. To this end, we have chosen the experimental system of naphthalene-tetracarboxylic-acid-dianhydride (NTCDA) molecules adsorbed on the Ag(111) surface. Similar systems, PTCDA/Ag(111) [@ptcda_kondo_cleavage; @gated_molecular_wire; @PTCDAAgMove; @gated_wire_spectral] and PTCDA-Au complexes on Au(111) [@AU-PTCDA-monomer; @AU-PTCDA-dimer], have been investigated before but without the necessity of including phononic contributions. There, we applied a combination of density functional theory and many-body perturbation theory (DFT-MBPT) and used the ensuing quasiparticle spectrum as input to a NRG calculation [@BullaCostiPruschke2008] to comprehensively understand the STS spectra. However, despite the similarity between NTCDA and PTCDA, STM experiments on NTCDA/Ag(111) cannot be explained using the same methodology. Specifically, the theory predicts a zero-bias resonance whose width is significantly too large compared to the experiment. The origin of this deviation is not clear, as DFT-MBPT are expected to provide reliable input parameters for accurate NRG-calculated spectra [@PTCDAAgMove; @gated_wire_spectral; @AU-PTCDA-monomer; @AU-PTCDA-dimer]. Moreover, the calculated spectra lack additional features that are present in the experiment and hint towards inelastic electron-phonon contributions. The NTCDA/Ag(111) system, therefore, seems a good candidate as a first application case of our theory. Indeed, we argue below that the NTCDA/Ag(111) experiments can be interpreted in a conclusive way by incorporating the very different effects of two vibrational modes into the description. One mode couples strongly to the local substrate electrons, thus dynamically modifying the hybridization function between the substrate and the molecule; this results in a substantial reduction of the Kondo temperature of the NTCDA molecule/substrate system. In contrast, the second mode couples only weakly to the electronic system. Both modes, however, cause modulations of the tunneling distance. While the second mode induces rather accentuated inelastic side peaks close to the vibrational frequency as consequence of a second-order phonon absorption/emission process, the polaronic entanglement of the first mode with the electronic system gives rise to two inelastic current contribution: A first order term involving only a single phonon process is responsible for an asymmetric term while the second order contribution generates only very weak inelastic features located in the spectrum at an renormalized phonon frequency. The strength of the theory presented here is the inclusion of both rather different mechanisms on an equal footing. It demonstrates how different vibrational modes with similar frequencies can nevertheless lead to distinctly different spectroscopic signatures. While, based on the commonly accepted level of understanding of electron-phonon effects in electron tunneling, the step-like structures are easily identified as vibration-related, the vibrational sharpening of the Kondo resonance in the presence of only marginal side peaks would be impossible to pinpoint without theoretical guidance. Before going *in medias res*, we briefly review the relevant literature regarding inelastic tunneling in STM and STS. Many papers in the literature focus on the theory of inelastic contributions to the tunneling current and, therefore, modifications to STS spectra. In particular, the influence of vibrational modes has been addressed [@MolecularVibrationTunnel1968; @LorentePersson2000; @REED2008; @Leijnse2010]. Most of these theories [@Zawadowski1967; @TersoffHamann1983; @TersoffHamann1985] are based on the seminal many-body approach to tunneling by Bardeen [@Bardeen1961] that allowed to derive the Josephson current as a tunneling current between two superconductors [@Ambegaokar1963]. Higher-order electron-phonon processes in tunneling theories were investigated by Zawadwoski [@Zawadowski1967], while Caroli and collaborators [@Caroli71; @Caroli72] employed the Keldysh approach to calculate the inelastic electron-phonon terms. Paaske and Flensberg investigated the influence of vibrational effects onto the dynamics of a Kondo impurity [@PaakeFlensberg2005]. They combined a Schrieffer-Wolff transformation [@SchriefferWol66] with a third-order perturbation theory that is valid in the high-temperature regime well above the Kondo temperature and is limited to the anti-adiabatic regime. In their approximation, the atomic solution of a Holstein model [@LangFirsov1962] derived by Lang and Firsov – see also Mahan’s text book [@Mahan81] – modifies the Kondo coupling $J$ in the weak tunneling, large $U$ limit. This Kondo coupling $J$ matrix becomes energy dependent due to polaron formation, inducing steps in the transmission matrix at multiples of the phonon frequency. Lorente and Persson [@LorentePersson2000] combined the Keldysh approach of Caroli et al. [@Caroli72] with density functional theory, both relying on the free-electron picture and decoupled vibrational modes. Such an approach is only applicable in the adiabatic regime [@galperinNitzanRatner2006; @EidelsteinSchiller2013], but cannot address the anti-adiabatic regime that was considered by Paaske at al.  [@PaakeFlensberg2005]. In two recent letters [@Wehling2008; @PhysRevLett.110.026802], electron-phonon effects included in the single-particle self-energy have been attributed to an inelastic electron tunneling contribution. If these self-energy corrections are only evaluated in the adiabatic regime, the effect on the current is so small that it becomes visible only in the second derivative of the tunneling current $d^2I/dV^2$ [@WehlingTunnel2017]. This paper is organized as follows. In section \[sec:tunnel-current-general\] we present our theory of the tunneling current. The theory is independent of the system Hamiltonian and therefore of general nature. In particular, we discuss the inelastic and elastic contributions to the current, suggesting a partitioning that is based strictly on the question whether energy is transferred *during* the tunneling process. As a necessary step towards the application of the tunneling theory to an actual physical system, we specify a system Hamiltonian in section \[sec:Modeling the system\]. The choice of this Hamiltonian, consisting of a single impurity Anderson model (SIAM) and two distinct types of Holstein couplings, is motivated by the experimental system of NTCDA/Ag(111) which we introduce in section \[sec:experiment-NTCDA\]. One of the Holstein couplings is unconventional in the sense that it couples vibrations of the adsorbed molecule to electronic states in the substrate. In section \[sec:application-to-NTCDA\], we apply the tunneling theory to the NTCDA/Ag(111) system. To this end, we present NRG calculations of differential conductance spectra and compare them in detail to experimental scanning tunneling spectra (STS). As a result, we are able to present a model of the NTCDA/Ag(111) system that provides a comprehensive understanding of all generic features in the STS spectra. In section \[sec:STS-anti-adiabatic-regime\], we consider the STS spectra that are to be expected for a Kondo impurity in the anti-adiabatic regime. In particular, we show that Kondo replica that are naively expected do not show up, at least in the parameter regime which we consider. The paper closes with a conclusion (section \[sec:conclusion\]). Theory of the tunnel current {#sec:tunnel-current-general} ============================ In this section of the paper, we derive a generalized tunneling theory for STS spectroscopy that incorporates previous approaches [@SchillerHershfield2000a; @LorentePersson2000; @PaakeFlensberg2005] as limiting cases in certain parameter regimes. We differentiate between, first, vibrational contributions that modify the electronic single-particle Green’s function (GF) of the system even in the absence of the STM tip from, second, true inelastic contributions that are introduced during the electron tunneling process from the STM tip into the system, as illustrated in Ref. [@MolecularVibrationTunnel1968]. While the former enter the self-energy of the Green’s function for arbitrary order in the electron-phonon coupling, the latter are included in the perturbative treatment of the tunneling Hamiltonian. In essence, our approach is a generalization of the theory by Caroli et al. [@Caroli72] to arbitrary correlations in the system of interest, but with the limitation that it is strictly only correct up to second order in the tunnel matrix element. Higher-order corrections, as addressed by Zawadovski [@Zawadowski1967] for oxide interfaces, require a proper Keldysh theory that incorporates the feedback process from the system to the STM tip and vica versa. In such an approach, non-equilibrium distribution functions replace the Fermi functions that we use in our theory. In this so-called quantum point contact regime [@PCSreview1989] the STM tip is not any more a weak probe, and the STS spectra would not only contain information about the system of interest, but also about its coupling to the tip. Therefore, we exclude these considerations here and restrict ourselves to the tunneling limit for the system-STM coupling. Tunneling Hamiltonian --------------------- ![Cartoon of the setup: a system comprising a molecule and a substrate is coupled to an STM tip. The arrows indicate the two transmission paths for electrons tunneling from the tip to the system S. The interference of such multi-orbital tunneling paths is responsible for Fano lineshapes in the tunneling spectra [@SchillerHershfield2000a]. []{data-label="fig:1-system-tip"}](fig1){width="40.00000%"} We start from the most general situation for deriving the theory by dividing the total Hamiltonian of the coupled problem as depicted in Fig. \[fig:1-system-tip\] into three parts, $$\begin{aligned} \hat H &=& \hat H_{S} + \hat H_{\rm tip} + \hat H_{T},\end{aligned}$$ where $\hat H_{S}$ is the system Hamiltonian of the sample, comprising the adsorbed molecule and the substrate surface, $\hat H_{\rm tip}$ denotes the Hamiltonian of the STM tip and $\hat H_{T}$ accounts for all tunneling processes between the tip and the sample system $\mathrm{S}$. We will keep the system Hamiltonian $\hat H_{S}$ unspecified without any restrictions. In particular, we do not make any assumptions about its electronic, vibronic or even magnetic excitations. Therefore, the strong coupling limit, polaron formation or any other many-body effect, such as the Kondo effect or any kind of magnetism, may be included in the system $\mathrm{S}$. The STM tip, however, is modeled by a simple free electron gas $$\begin{aligned} \hat H_{\rm tip} &=& \sum_{\k \sigma} \e_{\k\sigma} c^\dagger_{\k\sigma,\rm tip} c_{\k\sigma,\rm tip}, \end{aligned}$$ where $c^\dagger_{\k\sigma,\rm tip}$ creates a tip electron with spin $\sigma $ and energy $\e_{\k\sigma}$. If relevant, the Hamiltonian can be extended to a multi-band description, thus interpreting the index $\sigma$ as a combined spin and band label. Assuming some appropriately chosen orbital basis in the sample system S and a single-electron tunneling process, the most general bilinear tunneling Hamiltonian is given by [@Bardeen1961; @TersoffHamann1985] $$\begin{aligned} \label{eq:general-tunneling-HT} \hat H_{T} &=& \sum_{\mu \k \sigma \sigma'} \left(T^{\sigma\sigma'}_{\mu,\k} d^\dagger_{\mu\sigma} c_{\k\sigma',\rm tip} +h.c. \right) \, \, \, ,\end{aligned}$$ where $d^\dagger_{\mu\sigma}$ creates an electron in the as yet unspecified localized orbital $\mu$ of the system $\mathrm{S}$. $\mu=0,\cdots, M-1$ labels different orbitals in the system. While only two tunneling paths are included in the cartoon Fig. \[fig:1-system-tip\], in general, electrons thus tunnel from the STM tip to $M$ different orthogonal orbitals of the system $\mathrm{S}$. These orbitals can be the orbitals of a molecule adsorbed on the substrate surface, or substrate Wannier orbitals in the vicinity of the STM tip. The shape of the STM tip has an influence which orbitals $\mu$ have to be included in Eq. . If $M\ge 2$, the quantum mechanical interference of the different tunneling paths includes the possibility of a Fano resonance [@SchillerHershfield2000a]. An additional capacitive coupling between the STM tip and the system [@Temirov2018] is ignored, since we target the low bias regime of STM junctions. Such charging terms become relevant at large bias voltages which are not considered in this paper. A difficulty arises because even if $T^{\sigma\sigma'}_{\mu,\k}$ is spin-diagonal, it still depends on the details of the tip shape, which is unknown in experiment. Therefore, one usually makes several approximations that effectively absorb the details of the tip shape into an unknown ratio of tunneling matrix elements, but nevertheless turn out to be helpful for the understanding of spectroscopy data. For example, Tersoff and Hamann [@TersoffHamann1985] assume that the STM tip electrons are described by plane waves, leading essentially to a factorization of the matrix elements $$\begin{aligned} \label{eq:Tersoff-Haman} T^{\sigma\sigma'}_{\mu,\k} = a_{\k} t_{\mu}^{\sigma\sigma'}. \end{aligned}$$ Then a fictitious STM tip orbital can be introduced as $$\begin{aligned} \label{eq:stm-tip-c0} c_{0\sigma,\rm tip} &=& \sum_{\k} a_{\k} c_{\k\sigma,\rm tip} \end{aligned}$$ that we label with $i=0$. The local annihilation operator $c_{0\sigma,\rm tip} $ of an electron with spin $\sigma$ in this orbital is expanded in the free electron operators $c_{\k\sigma,\rm tip}$ with some expansion coefficients $a_{\k}$ whose details are not of interest and do not enter the theory, unless the STM tip is characterized by a strongly non-linear DOS in the relevant energy range. In the approximation Eq.  and , the STM tip shape has disappeared in some overall tunneling matrix elements $t_{\mu}^{\sigma\sigma'}$. However, we have to be aware that the STM tip breaks the local point group symmetry of the molecule. Therefore, one has to be careful when excluding tunneling channels purely on the basis of the symmetries of $\hat H_{S}$. For frozen nuclear positions $\{\vec{R}_i\}$, the above bilinear tunneling Hamiltonian is given by $$\begin{aligned} \label{eq:H_T-bilinear} \hat H_{T} &=& \sum_{\mu\sigma\sigma'} t_{\mu}^{\sigma\sigma'}(\{\vec{R}_i\}) d^\dagger_{\mu\sigma} c_{0\sigma',\rm tip} + h.c.\end{aligned}$$ Here $t_{\mu}^{\sigma\sigma'}(\{\vec{R}_i\})$ denotes a tunnel matrix element that depends on some parameter set that is related to the atomic positions $\{\vec{R}_i\}$ in the system S and the tip, as well as on the spin. Since we are interested in the influence of molecular vibrations on the tunneling current, we must account for the change of the tunneling matrix element between the system S and the STM tip due to the vibrationally induced changes of the tip distance. To be more specific, let us assume that $d^\dagger_{\mu\sigma}$ creates an electron in some extended molecular orbital spread over the entire surface-adsorbed molecule, or in a local Wannier state of the substrate in the vicinity of the STM tip. The molecule will have some vibrational eigenmodes, labelled by $\nu$, that deform the orbital. Imagining a perfectly rigid STM tip without any intrinsic vibrations, the tip-orbital distance will change as a function of this displacement. Since the tunnel matrix elements are exponentially dependent on the distance, we model the tunneling matrix element by $$\begin{aligned} \label{eq:6} t^{\sigma \sigma'}_{\mu}(\{\vec{R}_i\}) &\to& t^{\sigma \sigma'}_{\mu}(\{\vec{R}^0_i\},\vec{R}_{\rm tip}) e^{f_\mu(\{ \hat X_\nu\}) }\end{aligned}$$ where $t^{\sigma \sigma'}_{\mu}(\{\vec{R}^0_i\},\vec{R}_{\rm tip})$ denotes the tunneling matrix element between the STM tip and the orbital $\mu$ if all atoms of the molecule are in their equilibrium positions $\{\vec{R}^0_i\}$ and the STM tip is located at position $\vec{R}_{\rm tip}$. The unknown function $f_\mu$ depends on the superposition of all individual dimensionless displacement operators $\hat X _\nu = b_\nu +b^\dagger_\nu $ of each molecular eigenmode. Since we also allow for an electron-phonon coupling in the system (the system Hamiltonian $\hat H_{S}$ is as yet unspecified), the equilibrium position of atoms within the molecule might be shifted with respect to the equilibrium positions in the absence of this coupling [@galperinNitzanRatner2006; @EidelsteinSchiller2013]. Therefore, it is useful to subtract the equilibrium position $x_{\nu 0}=\langle\hat X_{\nu}\rangle$ from $\hat X_{\nu}$ and define $\hat X_{\nu}'\equiv \hat X_{\nu} - x_{\nu0}$ in Eq. . We assume that vibration-induced changes in the tunneling matrix element are small and expand $\exp(f_\mu(\{ \hat X'_\nu\}))$ up to first order in the displacement. This leads to the simplification $$\begin{aligned} \label{eq:7} t^{\sigma \sigma'}_{\mu}(\{\vec{R}_i\}) & \approx & t^{\sigma \sigma'}_{\mu}(\{\vec{R}^0_i\})(1 +\sum_\nu \lambda^{\rm tip}_{\mu\nu} \hat X'_\nu) \label{eq:e-ph-tunnel}\end{aligned}$$ where $\lambda^{\rm tip}_{\mu\nu}$ parameterizes the change of the tunnel coupling of the STM tip to the orbital $\mu$, induced by the excitation of the molecular vibration $\nu$. Similar terms have been considered in the context of Heavy Fermion superconductivity [@Grewe84] where the ionic breathing mode couples to the lattice phonons. Although this parameter $\lambda^{\rm tip}_{\mu\nu}$ can be spin-dependent in the case of a magnetically ordered surface, we have dropped this spin dependency. From now on we also drop the argument $\{\vec{R}^0_i\}$ on the right hand side of Eq.  and use $t^{\sigma \sigma'}_{\mu}$ to refer to $t^{\sigma \sigma'}_{\mu}(\{\vec{R}^0_i\})$. The matrix form of $t^{\sigma\sigma'}_{\mu}$ in spin space can be expressed in general as $$\begin{aligned} \mat{t}_{\mu}&=& t^0_{\mu} \mat{I} + \vec{t}_{\mu} \vec{\mat{\sigma}} \label{spinmatrix}\end{aligned}$$ by specifying four parameters $t^0_{\mu}$, $\vec{t}_{\mu}$ as spin-dependent tunnel matrix elements. The spinor matrix $\vec{\mat{\sigma}}$ could for example represent a free localized spin in the system S, or on the STM tip, that can couple to magnetic excitations such as magnons in a magnetic system S, which may cause additional magnetic inelastic contributions. In this paper, however, we assume that $\mat{t}_{\mu}$ is diagonal with diagonal elements $$\begin{aligned} t_{\mu\sigma}= t^0_{\mu} +\sigma t^z_{\mu}. \label{diagonalspinmatrixelements}\end{aligned}$$ This allows spin-dependent tunneling matrix elements $t_{\mu\sigma}$, as they occur, e.g., for spin-polarized tips. If both the tip and the system S are paramagnetic, then $ \vec{t}_\mu=0$ in Eq. \[spinmatrix\] and $t_{\mu,+1/2}=t_{\mu,-1/2}= t^0_{\mu}$. Note that a small $\lambda^{\rm tip}_{\mu\nu}$ in Eq. \[eq:e-ph-tunnel\] does not imply that the electron-phonon coupling in the system S comprising substrate and molecule must be weak, because it is included in $\hat H_{S}$ and not connected to the parameters $\lambda^{\rm tip}_{\mu\nu}$ in the STM tunneling theory. In fact, the electron-phonon coupling in S can be arbitrarily strong [@PaakeFlensberg2005] since the theory that we will present below only requires that $t^{\sigma \sigma'}_{\mu}$ is a very small energy scale and, therefore, the STM must be operated in the tunneling limit. In the following we drop the prime in $\hat X'$ and demand $\langle{\hat X}\rangle=0$. We discuss the generic case of $\langle{\hat X}\rangle=x_0\not = 0$ below in Sec. \[sec:e-phonon-displacement\], where we show that in leading order the total tunneling current is independent of $x_0$, as expected, although the partitioning between elastic and inelastic contributions is not unique. This is not surprising, since the notion of an inelastic process requires the definition of the underlying phonon basis sets. Tunnel current operator ----------------------- As the next step, we explicitly derive the analytic form of the tunnel current operator from charge conservation. This approach has the advantage that it allows the construction of the total current operator of the problem systematically and without adding terms by hand. We will show below that the derived total current operator contains all elastic and inelastic contributions. Since the tunnel current changes the number of electrons on the STM tip, the current operator $\hat j_{\rm STM}$ is related to the change of the charge $\hat Q_{\rm tip}=e \hat N_{\rm tip}$ on the tip, i.e. $$\begin{aligned} \label{eqn:current-operator-derivation} \hat {j}_{\rm STM} &=& \frac{d\hat Q_{\rm tip}}{dt} = i \frac{e}{\hbar} [\hat H,\hat N_{\rm tip}] = i\frac{e}{\hbar} [\hat H_{T}, \hat N_{\rm tip}] \\ \nonumber &=& i\frac{e}{\hbar} \sum_{\mu\sigma} t_{\mu \sigma}(\{\vec{R}_i\}) \left( d^\dagger_{\mu\sigma} c_{0\sigma,\rm tip} - c^\dagger _{0\sigma,\rm tip} d_{\mu\sigma} \right) , %\end{aligned}$$ in order to enforce charge conservation in the total system, consisting of the tip and the sample system S. Because the total particle number operator of the STM tip, $\hat N_{\rm tip}$, commutes with $\hat H_0=\hat H_{S}+ \hat H_{\rm tip}$, the current operator is generated by the tunneling Hamiltonian $\hat H_{T}$ only. Here we also have assumed that the tunneling matrix elements are real, which can always be achieved by a local gauge transformation. Note that in Eq.  we use $t_{\mu \sigma}(\{\vec{R}_i\})$ for an arbitrary but fixed set of atomic positions $\{\vec{R}_i\}$. The inelastic contributions to the tunnel current will become transparent once we substitute the linear expansion of $t_{\mu \sigma}(\{\vec{R}_i\}) $ in the displacements, Eq.  , into Eq. . Eq.  demonstrates that the current operator depends only on the coupling between the two subsystems, $\hat H_{T}$, which is intuitively clear. A different $\hat H_{T}$, for example in the case of a magnetic interface, will modify the current operator derived in Eq. . Depending on the physics included in $\hat H_{T}$, this could include inelastic magnetic spin-flip contributions. STM tunnel current {#sec:tunnel-current} ------------------ Since in the tunneling limit $\hat H_{T}$ defines the smallest energy scale of the system, we proceed in the interaction picture. We assume that the tunnel Hamiltonian $\hat H_{T}$ is switched on at time $t_0$. Then, the current evaluated at time $t>t_0$ is given by $$\begin{aligned} I(t) &=& \langle \hat j_{\rm STM}(t) \rangle = {\rm Tr}[\hat \rho_0 e^{i\hat H (t-t_0)} \hat{j}_{\rm STM} e^{-i\hat H (t-t_0)}] \non &=& \langle \hat U^\dagger(t,t_0) \hat j_{I}(t) \hat U(t,t_0) \rangle_0 \label{eq:current-1}\end{aligned}$$ where $\hat j_{I}(t)= \exp[i \hat H_0 (t-t_0) ] \hat{j}_{\rm STM} \exp[-i \hat H_0 (t-t_0) ]$ is the STM current operator $ \hat j_{\rm STM}$ transformed into the interaction picture. Note that we absorb $\hbar$ in the time $t$, i.e. measure the time in units of inverse energy. The time evolution operator $\hat U(t,t_0)$ obeys the standard equation of motion $$\begin{aligned} \partial_t \hat U(t,t_0) &=& - i\hat V(t) \hat U(t,t_0)\end{aligned}$$ which is formally integrated to the time ordered operator $$\begin{aligned} \hat U(t,t_0) &=&T e^{ -i \int_{t_0}^t d t' \hat V(t') }.\end{aligned}$$ $\hat V(t)$ denotes $\hat H_{T}$ in the interaction picture, $$\begin{aligned} \hat V(t) &=& e^{i \hat H_0 (t-t_0)} \hat H_{T} e^{-i \hat H_0 (t-t_0)} .\end{aligned}$$ All expectation values have to be calculated with respect to the two decoupled system, $\langle \hat A \rangle_0 = {\rm Tr}\left[\hat \rho_0 \hat A \right]$, assuming thermodynamic equilibrium in each of the two uncoupled subsystems (S and the STM tip). Then, the density operator $\hat \rho_0$ factorizes into two independent contributions, $$\begin{aligned} \label{eq:rho-0} \hat \rho_0 &=& \hat \rho_{S}\hat \rho_{\rm tip}\end{aligned}$$ with $$\begin{aligned} \hat \rho_{S} &=& \frac{1}{Z_{S}}e^{-\beta(\hat H_{S} -\mu_{S} \hat N_{S})} \non \hat \rho_{\rm tip} &=& \frac{1}{Z_{\rm tip}}e^{-\beta(\hat H_{\rm tip} -\mu_{\rm tip} \hat N_{\rm tip})}.\end{aligned}$$ Here, we have introduced different chemical potentials for each subsystem: $\mu_{S}$ for the sample system S and $\mu_{\rm tip}$ for the STM tip. The bias voltage $V$ enters through the difference $\mu_{\rm tip}-\mu_S=eV$. For convenience, we define the chemical potential $\mu_{\rm S}$ of the system S as a reference energy and absorb it into the definition of the single particle energy. Evaluating the current up quadratic order in the tunneling matrix elements yields $$\begin{aligned} \label{eq:13} I(t) &=& i \int_{t_0}^t d t' \left[ \langle \hat V(t') \hat j_{I}(t) \rangle_0 - \langle \hat j_{I}(t) \hat V(t') \rangle_0 \right] \non && + O(t^3_s), \label{eqn:8}\end{aligned}$$ where $t_s$ is a measure of the order of magnitude of the largest tunneling matrix element $t_{\mu\sigma}$. Note that $\langle \hat V(t') \hat j_{I}(t) \rangle_0=\langle \hat j_{I}(t) \hat V(t') \rangle_0^*$, ensuring that the current is real. Substituting the linear expansion of tunneling matrix elements in the displacements of the vibrational modes $\nu$, Eq. , into $\langle \hat V(t') \hat j_{I}(t) \rangle_0$ yields $$\begin{aligned} \label{eq:Vj} \langle \hat V(t') \hat j_{I}(t) \rangle_0&=& \sum_{\mu\sigma\mu'\sigma'} t_{\mu \sigma}t_{\mu' \sigma'} \Big \langle (1 +\sum_\nu \lambda^{\rm tip}_{\mu\nu}\hat X_\nu (t'))(1 + \sum_{\nu'} \lambda^{\rm tip}_{\mu'\nu'}\hat X_{\nu'} (t) ) \left( d^\dagger_{\mu\sigma}(t') c_{0\sigma,\rm tip}(t') + c^\dagger_{0\sigma,\rm tip}(t') d_{\mu\sigma} (t') \right) \non && \times \left( d^\dagger_{\mu'\sigma'}(t) c_{0\sigma',\rm tip}(t) - c^\dagger_{0\sigma',\rm tip}(t) d_{\mu'\sigma'} (t) \right) \Big \rangle_0\end{aligned}$$ Since $\hat H_{S}$ and $\hat H_{\rm tip}$ as well as the corresponding density operators commute, the expectation values factorize into products of the sample system S and the STM tip, and we arrive at $$\begin{aligned} \label{eq:16} \langle \hat V(t') \hat j_I(t) \rangle_0&=& \sum_{\mu \mu'\sigma} t_{\mu \sigma}t_{\mu' \sigma} \Big \langle (1 + \sum_\nu \lambda^{\rm tip}_{\mu\nu}\hat X_\nu (t')) (1 +\sum_{\nu'} \lambda^{\rm tip}_{\mu\nu'}\hat X_{\nu'} (t) ) d_{\mu\sigma} (t') d^\dagger_{\mu'\sigma}(t) %% \Big \rangle_0 \Big \langle c^\dagger_{0\sigma,\rm tip}(t') c_{0\sigma,\rm tip}(t) \Big \rangle_0 \non && - \sum_{\mu\mu'\sigma} t_{\mu \sigma}t_{\mu' \sigma} \Big \langle (1 + \sum_\nu \lambda^{\rm tip}_{\mu\nu}\hat X_\nu (t')) (1 +\sum_{\nu'} \lambda^{\rm tip}_{\mu\nu'}\hat X_{\nu'} (t) ) d^\dagger_{\mu\sigma}(t') d_{\mu'\sigma}(t) %% \Big\rangle_0 \Big\langle c_{0\sigma,\rm tip}(t') c^\dagger_{0\sigma,\rm tip}(t) \Big\rangle_0, \nonumber \\\end{aligned}$$ under the assumption that the system S and the STM tip are in a normal conducting state. This factorization does not require a Wick’s theorem, and, therefore, the Hamiltonian $\hat H_0$ remains fully general. Note that the electronic correlation function of the STM tip is spin-diagonal, and hence the double sum over $\sigma \sigma'$ in Eq.  collapses to a single sum over $\sigma$ in Eq. . It is clear that the displacements terms $\hat X_\nu$ and the electronic orbital operators $d_{\mu \sigma}$ do not factorize, because we explicitly allow for a strong electron-phonon coupling and thus polaron formation in the system S. Finally, the terms of type $\langle d^\dagger_{\mu\sigma} d^\dagger_{\mu'\sigma }\rangle_0$ that we have neglected in Eq.  must be included if either the STM tip or the sample system S are superconducting. In this case, our approach reproduces the well-known derivation of the Josephson current by Ambegaokar and Baratoff [@Ambegaokar1963]. Eq.  can be divided into elastic and inelastic contributions. The former are obtained by setting all $ \lambda^{\rm tip}_{\mu\nu}=0$, while the inelastic terms are given by the difference between Eq.  for non-vanishing $\lambda^{\rm tip}_{\mu\nu}$ and for $\lambda^{\rm tip}_{\mu\nu}=0$. Similarly, the total current decomposes into the sum $$\begin{aligned} \label{eq:totalcurrent} I_{\rm tot} &=& I_{\rm el} +I_{\rm inel}, \end{aligned}$$ comprising an elastic and an inelastic current. This naturally defines the terminology used throughout the rest of the paper. In summary, we have related the total current to products involving a greater Green’s function $G^>\sim \langle C(\lambda^{\rm tip}) d^\dagger_{\mu\sigma}(t') d_{\mu'\sigma}(t) \rangle_0$ of the system S and a lesser Green’s function $G^< \sim \langle c_{0\sigma,\rm tip}(t') c^\dagger_{0\sigma,\rm tip}(t)\rangle_0$ of the STM tip and vice versa [@Keldysh65]. Most importantly, the Keldysh Green’s functions of a fully interacting system entangling, in general, vibrational and electronic operators, are employed for the system S. Therefore, the expressions derived and analyzed in the following sections go well beyond the standard literature. ### Elastic tunnel current {#sec:elastic tunnel current} Since we are interested in the asymptotic steady-state current, we perform the limit $t_0\to -\infty$ and calculate the current at the time $t=0$. Evaluating the greater and lesser Green functions with the equilibrium density operator in Eq.  and calculating the steady-state current for $\lambda^{\rm tip}_{\mu\nu}=0$ in Eq. , we obtain the well-known expression for the elastic tunnel current $$\begin{aligned} \label{eqn:el-current} I_{\rm el}(t=0) &=& \frac{2\pi e}{\hbar} \sum_{\sigma} \int_{-\infty}^{\infty} d\w \rho_{\sigma,\rm tip}(\w) \tau^{(0)}_{\sigma}(\w) \nonumber \\ &&\times \left[ f_{\rm tip}(\w) - f_{S}(\w) \right], \label{eqn:elasticCurrent}\end{aligned}$$ where $f_{\rm tip}(\w)=f(\w-eV)$ and $f_{S}(\w)=f(\w)$, with $f(\w)=[\exp(\beta \w)+1]^{-1}$ being the Fermi function. $$\begin{aligned} \tau^{(0)}_{\sigma}(\w) &=& \sum_{\mu\mu'}^M t_{\mu \sigma}t_{\mu' \sigma} \lim_{\delta\to 0^+} \frac{1}{\pi} \Im G_{d_{\mu\sigma}, d^\dagger_{\mu'\sigma}} (\w-i\delta), \label{eqn:tauzero}\end{aligned}$$ is the spin-dependent transmission function from the STM tip to the system S, and $\rho_{\sigma,\rm STM}(\w)$ denotes the spectral density of the STM tip, $$\begin{aligned} \rho_{\sigma,\rm tip}(\w) = \lim_{\delta\to 0^+} \frac{1}{\pi} \Im G_{c_{0\sigma,\rm tip}, c^\dagger_{0\sigma,\rm tip}} (\w-i\delta).\end{aligned}$$ As usual, $G_{A,B}(z)$ refers to the equilibrium Green’s function [@Rickayzen1980] defined in the complex frequency plane $z$ except on the real axis. Throughout the paper we moreover use the notation $\rho_{A,B}(\w)\equiv \Im G_{A,B}(\w-i0^+)/\pi$ to connect a Green’s function of the two operators $A,B$ with its spectral function $\rho_{A,B}(\w)$. Note, however, that we have written Eq.  in terms of a transmission function $\tau^{(0)}_{\sigma}(\w)$ which includes the tunnel matrix elements $t_{\mu \sigma}$ as well as the spectral properties $\rho_{\mu \sigma, \mu' \sigma}(\w)$ of S, the advantage being that then all contributions to the current in Eq. , i.e. Eqs. , and have the same overall structure. In fact, the transmission function $\tau^{(0)}_{\sigma}(\w) $ can also be interpreted as the fermionic Green’s function of the operator $A_\sigma= \sum_{\mu}^{M-1} t_{\mu \sigma}d_{\mu\sigma}$. The usual assumption in STM experiments is that the density of states $\rho_{\sigma,\rm tip}(\w)$ is featureless in the energy (voltage) interval of interest. Then, it can be replaced by a constant $\rho_{\sigma,\rm STM}$ that only enters the prefactor in Eq. . This confirms that a detailed knowledge of the expansion coefficients in Eq.  is not required, since these coefficients can be absorbed into this prefactor. We note that the result in Eq.  also allows for interferences among more than one elastic transport channels. For example, it is straightforward to show that Eq.  reproduces the result of Eq. (6) of Ref. [@SchillerHershfield2000a], if we set $M=2$ and replace $d_{1\sigma}$ by the local surface conduction electron operator $\psi(\vec{R}_s)$. The Fano resonance [@FanoResonance1961] is generated by the quantum interference between the two or more elastic tunneling channels. ### Inelastic tunnel current {#sec:I-inelastic} The contributions to the inelastic tunnel current are classified by the power of the electron-phonon coupling $ \lambda^{\rm tip}_{\mu\nu}$ in the tunneling Hamiltonian. We note again that the additional inelastic contributions arise from phonon absorption and emission *during* the tunneling process, while all electron scattering processes within the system $S$ are included in $I_{\rm el}$. In first order in $ \lambda^{\rm tip}_{\mu\nu}$, we obtain an inelastic tunnel current $$\begin{aligned} I^{(1)}_{\rm inel} &=& \frac{2\pi e}{\hbar} \sum_{\sigma} \int_{-\infty}^{\infty} d\w \rho_{\sigma,\rm tip}(\w) \tau^{(1)}_{\sigma}(\w) \non && \times \left[ f_{\rm tip}(\w) - f_{S}(\w) \right] \label{eqn:inelasticCurrent}\end{aligned}$$ from Eq. , where following the general notation in this paper (see above) the transmission function $\tau^{(1)}_{\sigma}(\w)$ is defined as the spectral function of a composite Green’s function $G^{(1)}_{d\sigma}$ which in turn involves tunneling matrix elements $t_{\mu \sigma}$ and correlation functions $G_{X_\nu d_{\mu\sigma}, d^\dagger_{\mu'\sigma}}$ and $G_{d_{\mu\sigma}, X_\nu d^\dagger_{\mu'\sigma}}$ as $$\begin{aligned} \label{eq:rho1explicit} \tau^{(1)}_{\sigma}(\w)&=& \frac{1}{\pi}\lim_{\delta\to 0^+} \Im G^{(1)}_{d\sigma} (\w-i\delta) \nonumber \\ &=&\sum_{\mu\mu'} t_{\mu \sigma}t_{\mu' \sigma} \\ \times &\bigg(&\sum_{\nu}^{N_\nu}\lambda^{\rm tip}_{\mu\nu}\lim_{\delta\to 0^+} \frac{1}{\pi} \Im G_{\hat X_\nu d_{\mu\sigma}, d^\dagger_{\mu'\sigma}} (\w-i\delta) \nonumber \\ &+& \sum_{\nu}^{N_\nu}\lambda^{\rm tip}_{\mu'\nu}\lim_{\delta\to 0^+} \frac{1}{\pi} \Im G_{d_{\mu\sigma}, \hat X_\nu d^\dagger_{\mu'\sigma}} (\w-i\delta)\bigg). \nonumber\end{aligned}$$ Since the expectation value of the anticommutator of a Green’s function $G_{A,B}(z)$ equals the frequency integral of the corresponding spectrum $\rho_{A,B}(\w)$, we can conclude that the spectra of $G_{d_{\mu \sigma}, \hat X_\nu d^\dagger_{\mu' \sigma}}$ and $G_{\hat X_\nu d_{\mu \sigma}, d^\dagger_{\mu' \sigma}}$ both individually integrate to $\langle \hat X_\nu\rangle \delta_{\mu,\mu'}$. Hence, for a vanishing displacement $\langle\hat X_\nu \rangle=0$, either the Green’s function $G^{(1)}_{d\sigma}$ is identically zero, or its spectrum (i.e., the transmission function $\tau^{(1)}_{\sigma}(\w)$) has equal positive and negative spectral contributions. The first statement is true in the limit of vanishing electron-phonon coupling. For non-vanishing electron-phonon coupling, however, the correlation function $G^{(1)}_{d\sigma}$ does not vanish, since quantum fluctuations and hence non-zero correlators $G_{d_{\mu \sigma}, \hat X_\nu d^\dagger_{\mu' \sigma}}$ and $G_{\hat X_\nu d_{\mu \sigma}, d^\dagger_{\mu' \sigma}}(z)$ are allowed even if $\langle\hat X_\nu \rangle=0$. In second-order in $\lambda^{\rm tip}_{\mu \nu}$, the inelastic tunnel current $$\begin{aligned} I^{(2)}_{\rm inel} &=& \frac{2\pi e}{\hbar} \sum_{\sigma} \int_{-\infty}^{\infty} d\w \rho_{\sigma,\rm tip}(\w) \tau^{(2)}_{\sigma}(\w) \non && \times \left[ f_{\rm tip}(\w) - f_{S}(\w) \right], \label{eq:second-order-inelastic-contribution}\end{aligned}$$ involves the transmission function $$\begin{aligned} \label{eq:rho2explicit} \tau^{(2)}_{\sigma}(\w) &= & \frac{1}{\pi} \lim_{\delta\to 0^+} \Im G^{(2)}_{d\sigma} (\w-i\delta) \nonumber \\ &=&\sum_{\mu\mu'}^M t_{\mu \sigma}t_{\mu' \sigma} \\ &&\times \sum_{\nu\nu'}^{N_\nu}\lambda^{\rm tip}_{\mu\nu}\lambda^{\rm tip}_{\mu'\nu'}\lim_{\delta\to 0^+} \frac{1}{\pi} \Im G_{\hat X_\nu d_{\mu\sigma}, \hat X_{\nu'}d^\dagger_{\mu'\sigma}} (\w-i\delta). \nonumber\end{aligned}$$ Up to second-order, the total inelastic contribution to the tunneling current is thus given by $I_{\rm inel} = I^{(1)}_{\rm inel} +I^{(2)}_{\rm inel}$. Again, the spectral sum rule of $G_{\hat X_\nu d_{\mu \sigma}, \hat X_{\nu'} d^\dagger_{\mu' \sigma}}(z)$ is related to the expectation value of the anticommutator, i. e.  $\langle \hat X_\nu \hat X_{\nu'} \rangle \delta_{\mu,\mu'}$. ### The limit of vanishing electron-phonon coupling in the system S {#sec-lambda-0} In order to make the connection to the literature and also to point out the major difference of our theory in comparison with earlier ones, we consider the limit of vanishing electron-phonon coupling in the system S, but maintain a small but non-zero $\lambda^{\rm tip}_{\mu\nu}$. Then, $\langle\hat X_\nu \rangle=0$. As argued in the previous section, as a consequence $G^{(1)}_{d\sigma}(z)=0$, $\tau^{(1)}_{\sigma}(\w)=0$ and $I^{(1)}_{\rm inel}=0$ hold, while $I^{(2)}_{\rm inel}$ is reduced to a simplified result [@MolecularVibrationTunnel1968], because the correlation function $G_{\hat X_\nu d_{\mu \sigma}, \hat X_{\nu'} d^\dagger_{\mu' \sigma}}(t)$ in Eq.  factorizes in the time domain into the product of the electronic Green’s function and the phonon propagator $G_{\hat X_\nu, \hat X_\nu}(t)$, $$\begin{aligned} \label{eqn:26} G_{\hat X_\nu d_{\mu \sigma}, \hat X_{\nu'} d^\dagger_{\mu' \sigma}}(t) &=& G_{d_{\mu \sigma}, d^\dagger_{\mu' \sigma}}(t) G_{\hat X_\nu, \hat X_\nu}(t) \delta_{\nu\nu'}.\end{aligned}$$ Therefore, the spectral function is given by a convolution in the frequency domain. Summing over all free vibrational modes $\nu$ on a molecule with frequency $\w_{\nu}$, we obtain [@Mahan81] $$\begin{aligned} \label{eqn:27} \tau^{(2)}_{\sigma}(\w) &=& \sum_{\mu\mu'\nu} t_{\mu \sigma}t_{\mu' \sigma} \lambda^{\rm tip}_{\mu \nu} \lambda^{\rm tip}_{\mu' \nu} \left[ \rho_{d_{\mu \sigma},d^\dagger_{\mu'\sigma}}(\w-\w_\nu)(g(\w_\nu) +f_{S}(\w_\nu-\w)) + \rho_{d_{\mu \sigma},d^\dagger_{\mu'\sigma}}(\w+\w_\nu)(g(\w_\nu) +f_{S}(\w_\nu+\w)) \right] \nonumber \\ \label{eqn:freemode_inelastic}\end{aligned}$$ where $g(\w)$ denotes the Bose function. Using the approximation in Eq.  yields the identical inelastic current contribution as derived in Refs. [@Caroli72]. We briefly discuss two extreme cases for the electronic spectral function. For simplicity, we restrict ourselves to $M=1$ and a single vibrational mode with frequency $\w_0$. This excludes the possibility of a Fano resonance. In the first extreme, we assume a featureless density of states in the vicinity of the Fermi energy over an interval $[-2\w_0,2\w_0]$, i. e. $\rho_{\mu,\mu'\sigma}(\w)\to {\rm const.}$ in , and $\beta\w_0\gg 1$ so that the Bose function can be ignored. Then $\tau^{(2)}_{\sigma}(\w) $ will be dominated by the Fermi functions, introducing two threshold contributions in the overall differential conductance $dI/dV$ at $\pm \w_0$. These are the typical $dI/dV$ steps that are often encountered in inelastic tunnel spectroscopy as shown in Fig. 1 of Ref. [@REED2008]. In the second extreme, we consider an electronic DOS of the sample system S which possesses a sharp spectral peak located at $\w=0$ with a width $\Gamma\ll \w_0$. Then $\tau^{(2)}_{\sigma}(\w)$ exhibits again a sharp threshold behavior at $\pm \w_0$, but instead of a plateau the spectral function decreases with increasing $|\w|$ on a scale given by the peak broadening $\Gamma$. In this case, two “replicas” of the peak at $\w=0$ can be found at $\pm \w_0$ in the overall differential conductance $dI/dV$. However, the Fermi function $f_{\rm S}(\w_0\pm\w)$ cuts away the halves of the replicas on the low-$|\w|$ side and modifies them to a threshold behavior: a minimal energy transfer for $eV=\pm\w_0$ is required to generate an inelastic contribution replicating the standard picture [@MolecularVibrationTunnel1968]. These truncated replicas of the $\w=0$ peak are generated by the inelastic tunneling process due to the change of the distance between system S and STM tip. In case of a large electron-phonon coupling in S, the approximation is invalid and the proper Green’s function $G_{\hat X_\nu d_{\mu \sigma}, \hat X_{\nu'} d^\dagger_{\mu' \sigma}}(t)$ must be calculated, along with $G_{X_\nu d_{\mu\sigma}, d^\dagger_{\mu'\sigma}}$ and $G_{d_{\mu\sigma}, X_\nu d^\dagger_{\mu'\sigma}}$. ### Discussion The presented tunneling theory combines different limits [@MolecularVibrationTunnel1968; @Mahan81; @PerssonBaratoff1987; @LorentePersson2000; @PaakeFlensberg2005; @REED2008] discussed the literature. We stress again that no assumption is needed with respect to the nature and dynamics governing the system S. On the contrary, the theory can address arbitrary strengths of both the electron-electron and electron-phonon interactions in the system. The only input that is required are the Green’s functions of the participating orbitals and vibrational displacements in the absence of the STM tip. The theory is valid in the tunneling limit and we have restricted ourselves to the conventional single-particle electron transfer operator $\hat H_{T}$ [@Bardeen1961]. A further assumption that we have made concerns the relative distance changes between the tip and the system S that are induced by the relevant vibrational modes; they must be small enough such that a linear expansion of the tunneling matrix elements in the displacements suffices and higher-order terms can be neglected. Apart from allowing quantitative calculations of tunneling spectra for realistic systems, one of the most important benefits of the theory is that it allows a systematic separation of elastic contributions to the tunneling current (charge transfer does not involve an energy transfer) from inelastic ones (arising from correlated tunneling processes involving a fermionic hopping and a displacement operator). This differentiation in some cases deviates from the one given in the literature. In fact, the terminology elastic vs. inelastic current is not unambiguous throughout the literature. In some cases, certain contributions to the elastic and inelastic tunnel currents may have even the same analytic structure. This can be illustrated for a single orbital in the case of weak electron-phonon coupling in the system S, if moreover the free electron picture and the free phonon pictures are employed [@PerssonBaratoff1987; @LorentePersson2000; @Wehling2008]. We have seen in the previous section that in the case of vanishing electron-phonon coupling in S, but for finite coupling $\lambda^{\rm tip}$ in the tunneling matrix element, the general expression for $\tau^{(2)}_{\sigma}(\w)$ (Eq. ) in the inelastic current $I^{\rm (2)}_{\rm inel}$ takes on the shape as Eq. , leading to an inelastic current which for the special case of a flat DOS at the Fermi level leads to steps in the differential conductance at the vibrational energies $\pm \omega_0$. (\#1,\#2,\#3)\#4[(\#1,\#2)[(0,0)\[\#3\][\#4]{}]{}]{} (100,100)(0,0) (0,0)[ ![Second-order Feynman diagram of the generating Luttinger-Ward functional in the system S. The full line represents the full local electron Green’s function $G_d(z)$, the wiggled line the full phonon propagator. $i\w_n = i\pi (2n+1)/\beta$ denote the fermionic Matsubara frequencies and $i\w_n = i2\pi n/\beta$ the bosonic Matsubara frequencies [@LuttingerWard1960]. []{data-label="fig:self-electron-phonon-diagram"}](fig2-eph-functional-diagram "fig:") ]{} (60.22516,54.22514,t)[$i\omega _n+i\nu _n$]{}(60.22514,106.37524,b)[$i\nu _n$]{}(60.22514,14.07504,t)[$i\omega _n$]{} We now compare this result to the perturbatively calculated elastic current in the same limit. Under these circumstances, the Green’s function of the orbital with the single-particle energy $\e_a$ has the form $G_a(z)=[z- \e_a -\Sigma_{\rm el}(z) - \Sigma_{\rm el-ph}(z)]^{-1}$, where the self energy $\Sigma_{\rm el}(z)$ accounts for the purely electronic interactions and $\Sigma_{\rm el-ph}(z)$ arises from the additional electron-phonon coupling which is limited to the system S as assumed by Lorente et al. [@LorentePersson2000]. Introducing $G_{a}^{(0)} = [z- \e_a -\Sigma_{\rm el}(z)]^{-1}$ allows for a perturbation expansion in linear order of $\Sigma_{\rm el-ph}(z)$ in weak electron-phonon coupling, $$\begin{aligned} \label{equ:Gf-expansion} G_a(z) &=& G_{a}^{(0)}(z) + G_{a}^{(0)}(z) \Sigma_{\rm el-ph}(z)G_{a}^{(0)}(z) + \cdots\end{aligned}$$ If we substitute this expansion of $G_a(z)$ into the expression Eq.  for the elastic current, two contributions arise, first a purely electronic one generated by $ G_{a}^{(0)}(z)$, and second a contribution involving the self energy $\Sigma_{\rm el-ph}(z)$ in first order. This second term has been designated as an inelastic term in the literature [@PerssonBaratoff1987; @LorentePersson2000; @Wehling2008], but we include it in the elastic part of the current, since the electron energy is conserved during the tunneling process and the scattering process occurs in the system S. It is interesting to note that the second-order contribution to $\Sigma_{\rm el-ph}(z)$ in Eq.  is proportional to $G_{\hat X_\nu d_{\mu \sigma}, \hat X_\nu d^\dagger_{\mu' \sigma}}$ in weak coupling. This can be seen from Fig. \[fig:self-electron-phonon-diagram\], which depicts the generating functional [@LuttingerWard1960] for the conserving approximation of $\Sigma_{\rm el-ph}(z)$: by differentiating with respect to the electronic Green’s function $G_d(i\w_n)$ (equivalent to cutting the semi-circular full line) we obtain the diagram of the electronic self energy $\Sigma_{\rm el-ph}(z)$ due to the electron-phonon interaction [@AllenMitrovic; @JovchevAnders2013]. Evidently, this diagram has the structure of $G_{\hat X_\nu d_{\mu \sigma}, \hat X_\nu d^\dagger_{\mu' \sigma}}$. Therefore, the analytic structure of the *elastic* current calculated in second-order perturbation theory from Eq.  is identical to that of the *inelastic* current given by Eq. . But while the overall form of the two current contributions is identical, the physical mechanisms are different: In one case, an electron tunnels elastically from the tip into the system S, probing its density of states that includes the effects of vibration-induced electron scattering *within the system S*. In the other case, the electron loses energy *during the tunneling*, induced by the modulation of the distance between the tip and the system S, and transmits through the system S without further scattering on the vibration. Although both loss processes are governed by different coupling constants ($\lambda^{\rm tip}$ vs. $\lambda_c$ or $\lambda_d$, see section \[sec:modelling\_the\_system\_vibrations\]), in the limit of vanishing couplings these become of course indistinguishable. Thus, our approach incorporates the literature result in the limit of vanishing electron-phonon couplings. Once we leave the validity of the weak coupling limit, however, the two processes become distinguishable: On the one hand, the electron-phonon coupling in S leads to peaks in the density of states at the vibrational frequencies (see above), giving a distinct signature in the elastic current, and on the other hand the full composite Green’s function $G_{\hat X_\nu d_{\mu \sigma}, \hat X_\nu d^\dagger_{\mu' \sigma}}$ must be calculated in the appropriate higher order to obtain the correct inelastic current as given by Eq. ; this also properly accounts for the renormalization of the phonon frequency in the adiabatic regime which will affect the inelastic current profoundly. Returning to Fig. \[fig:self-electron-phonon-diagram\], we note that the phonon propagator obtains its self-energy by differentiation of the functional with respect to the phonon propagator (equivalent to cutting the wiggly phonon line that branches off from the particle-hole loop). In the spectral function and STS spectra, this correction to the phonon propagator has two consequences: First, it causes a renormalization of the phonon frequency itself [@HewsonMeyer02; @EidelsteinSchiller2013] and second, it induces multi-phonon processes. In the literature, however, the self-energy of the phonon propagator is often neglected [@Wehling2008] and the phonon is treated as a free excitation with an infinite lifetime, such that only the bare phonon frequency enters the final expression [@PerssonBaratoff1987; @LorentePersson2000]. Evidently, such approaches are limited to the case of a vanishing electron-phonon coupling in the system, i. e. the weak adiabatic limit, and cannot include renormalization effects stemming from multi-phonon processes. Already at moderate electron-phonon coupling corrections at $\w=2\w_0$ in the self-energy occur which also find their way into the tunneling spectra. Ref. [@JovchevAnders2013] discusses the deviations of the non-perturbatively calculated full electron-phonon self-energy from the lowest-order perturbative results. In conclusion, we maintain the terminology of the elastic current for all current contributions where the electrons travel ballistically between the tip and sample system S. Internal many-body scattering processes within the system S are all included the spectral functions within $\tau^{(0)}_{\sigma}(\w)$ and no assumption of the strength of the internal interactions are required. Therefore, $I_{\rm el}$ describes the current for a static distance between the system S and the STM tip. Modeling the system {#sec:Modeling the system} =================== In the previous section, we have presented a tunneling theory which relies on three spectral functions: one contains the information on the elastic tunneling current, the other two are connected linearly and quadratically to vibrational displacements. While this tunneling theory is completely general, for its application we need to specify the Hamiltonian of the system $\hat H_{S}$ and thus also the spectral functions which enter the tunneling theory. In the present section, we specify and discuss a $\hat H_{S}$ which turns out to be of sufficient generality to describe the physical sample system which we investigate experimentally in section \[sec:experiment-NTCDA\]. Electronic degrees of freedom ----------------------------- We employ a single-orbital single impurity Anderson model (SIAM) for the electronic degrees of freedom $$\begin{aligned} \label{eq:H-e} \hat H_{\rm e} &=& \sum_{\k\sigma} \e_{\k\sigma} c^\dagger_{\k\sigma} c_{\k\sigma} + \sum_\sigma \e_{d\sigma} n^d_\sigma + U n^d_\uparrow n^d_\downarrow \non && + \sum_{\k\sigma} V_{\k} ( c^\dagger_{\k\sigma} d_{0\sigma} + d^\dagger_{0\sigma} c_{\k\sigma} ) \label{eqn:SIAM}\end{aligned}$$ of the sample system S, thereby assuming that only one single molecular orbital is relevant for the energy spectral properties accessed by the STM. $c^\dagger_{\k\sigma}$ creates an effective substrate electron of energy $\e_{\k\sigma}$, momentum $\k$ and spin $\sigma$, while $ d^\dagger_{0\sigma}$ creates an electron in a local orbital, e.g. of a species adsorbed on the substrate surface, with the energy $\e_{d\sigma}$. The third term in the above equation specifies the Coulomb repulsion between electrons of opposite spin in the local orbital. The last term describes the hybridization between the substrate and the local orbital. We include the subscript 0 into the notation of the single active molecular orbital indicating that it will enter the tunnel Hamiltonian $H_T$, Eq.  , as $\mu=0$ orbital. For solving realistic systems with this ansatz, the SIAM needs to be mapped to the results of an atomistic simulation of the system in question. In this context, the projected density of states (PDOS) of the local orbital as calculated by a combination of DFT and many-body perturbation theory (MBPT) [@RevModPhys.74.601], plays a crucial role, because the mean-field parametrization of the local orbital’s Green’s function $$\begin{aligned} G^{\rm mf}_d(z) = [ z- \e_{d\sigma} -Un_{-\sigma} - \Delta(z)]^{-1} \label{eqn:MFGF}\end{aligned}$$ can be employed to extract $\e_{d\sigma}$ as well as the hybridization function $\Delta(z)$ [@PTCDAAgMove; @AU-PTCDA-monomer] as defined in the framework of the SIAM, $$\begin{aligned} \Delta(z) &=& \sum_{\k} \frac{|V_{\k}|^2}{z-\e_{\k\sigma}}. \label{eqn:HybFunc}\end{aligned}$$ Both serve as the input for a NRG calculation [@BullaCostiPruschke2008]. We note that in the absence of an electron-phonon coupling the influence of the substrate on the dynamics in the local orbital is completely determined by $\Delta(z)$, which justifies an effective single-band model [@BullaPruschkeHewson1997]. The spectral function required for the calculation of the elastic tunnel current through the system S as specified by the above Hamiltonian can be obtained by the standard approach [@PetersPruschkeAnders2006; @WeichselbaumDelft2007] that is based on the complete basis set of the NRG [@AndersSchiller2005; @AndersSchiller2006]. If the local orbital is close to integral filling, the exact solution of this model describes the Kondo effect [@Wilson75; @KrishWilWilson80a; @*KrishWilWilson80b]. Vibrational degrees of freedom and electron-phonon coupling ----------------------------------------------------------- ### Hamilton operator {#sec:modelling_the_system_vibrations} Naturally, we need to include a vibrational component into $\hat H_{S}$ if we want to calculate the two inelastic transmission functions $\tau^{(1)}_{\sigma}(\w)$ and $\tau^{(2)}_{\sigma}(\w)$ that play a role in the tunneling theory of section \[sec:tunnel-current\]. We divide the vibrational Hamiltonian into two parts, $\hat H_{\rm ph}$ and $\hat H_{\rm e-ph}$. We assume that there are $N_\nu$ phonon modes in the system S, and hence $\hat H_{\rm ph}$ is given by $$\begin{aligned} \label{eq:H-ph} \hat H_{\rm ph} &=& \sum_{\nu=0}^{N_\nu-1} \w_\nu b^\dagger_\nu b_\nu\end{aligned}$$ where a phonon of mode $\nu$ is created by $b^\dagger_\nu$. Even in the absence of an electron-phonon coupling in the system S, this term must be included in $\hat H_{S}$ when we evaluate the tunneling current as presented in section \[sec:tunnel-current\], because each of the modes $\nu$ can in principle modulate the tunneling matrix element between the tip and the system S. The second term in the vibrational Hamiltonian, $\hat H_{\rm e-ph}$, describes the electron-phonon coupling in S. While in principle all $N_\nu$ phonon modes may couple to the electrons in the system S, for simplicity we restrict the electron-phonon coupling in the present section to a single mode $b^\dagger_0$ of frequency $\omega_0$. All NRG calculations were performed with the restriction to a single phonon mode in order to keep the number of parameters in the model very small. Therefore $\omega_0$ always labels the eigenfrequency of the coupled vibrational mode that was included in the NRG, while $\omega_\nu (\nu>0)$ refers to eigenfrequencies of modes with no electron-phonon coupling in $H_S$ but contribute to the tunneling Hamiltonian $H_T$. Of course, the number of phonon modes that couple to the electrons can straightforwardly be extended to whatever number is required to explain the experimental observations. For example, it turns out that $N_\nu=2$ phonon modes, one with a non-zero electron-phonon coupling within S, the other without, are sufficient to reproduce the experimental spectra of NTCDA/Ag(111) in section \[sec:experiment-NTCDA\] with a minimal set of free parameters. We note that all molecular vibrations that do not have a finite or relevant electron-phonon coupling in the system S can be ignored in the calculation of the electronic properties of the system in absence of the STM tip. We assume that $\hat H_{\rm e-ph}$ is given by an extended Holstein Hamiltonian $$\begin{aligned} \label{eqn:AndersonHolstein} H_{\rm e-ph} &=& \lambda_d \hat X_0 (\sum_\sigma d^\dagger_{0\sigma} d_{0\sigma} - n_{d0}) \\ && \nonumber +\lambda_c \hat X_0 (\sum_\sigma c^\dagger_{0\sigma} c_{0\sigma} -n_{c0}),\end{aligned}$$ comprising *two* Holstein couplings $\lambda_d$ and $\lambda_c$ to two distinct orbitals. One of these orbitals is the local orbital $d_0$, the other an effective local substrate electron $c_{0\sigma}$ that hybridizes with the local orbital $d$ as described in Eq. . The annihilation operator of the effective local substrate electron is defined by $$\begin{aligned} c_{0\sigma} &=& \frac{1}{\bar V} \sum_{\k} V_{\k} c_{\k\sigma} \label{eqn:c_0}, \\ \bar V^2 &=& \sum_{\k} |V_{\k}|^2,\end{aligned}$$ and is entering the hybridization part in the SIAM, Eq. . This operator and the corresponding $c_{0\sigma}^\dagger$ obey the fermionic anticommutation relation by construction. $\hat X_0=b^\dagger_0 + b_0$ denotes the dimensionless vibrational displacement operator of the phonon mode $\omega_0$. The unconventional Holstein coupling $\lambda_c$ is included in $\hat H_{\rm e-ph}$ since it captures the fact that a vibrational excitation of the adsorbed molecule may couple to electrons in the substrate when parts of the molecule periodically beat onto the substrate surface. In particular, we will show below that this unconventional Holstein coupling can reduce the Kondo temperature [@PAM-E-PHON2013] of the system S. The additional constants $n_{d0}$ and $n_{c0}$ in Eq.  are often set to zero in the literature [@GalperinRatnerNitzan2007] when the polaronic energy shift in the single-particle energies is of primary interest, because they do not play a role then. Here, however, we focus on the quantum fluctuations with respect to some reference filling that are induced by the electron-phonon coupling [@EidelsteinSchiller2013; @JovchevAnders2013] and use these constants to ensure $\langle \hat X_\nu \rangle=0$. Typical values are $n_{d0}= n_{c0}=1$ at half filling. ### Interaction-driven displacement of the harmonic oscillator {#sec:e-phonon-displacement} Away from particle-hole symmetry, an electron-phonon coupling as the one in Eq.  generates a displacement of the equilibrium position of the corresponding harmonic oscillator. Since we are going to use an atomistic DFT calculation with relaxed atomic coordinates to generate the input parameters of the model Hamiltonian $\hat H_{\rm e}+ \hat H_{\rm ph} + \hat H_{\rm e-ph}$, such an additional displacement is not justified. We therefore include appropriately adjusted $n_{d0}$ and $n_{c0}$ added in Eq.  to ensure $\langle\hat X_0\rangle=0$. However, the perturbative derivation of the tunnel current does not rely on explicitly vanishing displacements $\langle \hat X_\nu \rangle$, and therefore the absorption of the equilibrium displacement in Eq.  is just a convention and must not alter the physics. The equilibrium displacement generated by the electron-phonon coupling also touches upon a more fundamental issue: Evidently the physical observables such as the total STS spectra must not depend on the precise definition of the operators $b_\nu$. We therefore need to analyze our theory in this respect. Specifically, we show in this section that the total current STS spectra calculated in our theory do not depend on the choice of the basis for the operators $b_\nu$. Interestingly, however, this choice of basis does determine the partitioning between elastic and the inelastic contributions to the total current. Inelastic and elastic currents are therefore not physical observables, but an interpretation based on a model-dependent partitioning of the total current. Let us assume that we have made a particular choice $\hat X$ of the oscillator basis and find a non-zero $\langle \hat X_0 \rangle=x_0$ for the mode $\w_0$ (for which $\hat H_{\rm S}$ foresees an electron-phonon coupling). For simplicity, we assume that the other $N_\nu-1$ vibrational modes $\langle \hat X_\nu \rangle=0$ holds. Then we can define a new bosonic operator $$\begin{aligned} \label{eq:XX} \bar b_0 = b_0 - \frac{1}{2}x_0 \end{aligned}$$ such that $\langle \hat{\bar{X_0}}=\bar b_0+\bar b_0^\dagger\rangle=0$. Substituting this expression into $\hat H_{\rm ph} + \hat H_{\rm e-ph}$ (Eqs. ,) and ignoring all vibrational modes except $\omega_0$ yields $$\begin{aligned} \label{eq:35} \hat H_{\rm ph} &+& \hat H_{\rm e-ph} = \sum_{\nu=1}^{N_\nu-1} \w_\nu b^\dagger_\nu b_\nu \\ &+& \w_0 \bar b^\dagger_0 \bar b_0 +\lambda_d x_0 N_d + \lambda_c x_ 0\hat N_{c} + E_0 \nonumber\\ &+& \hat{\bar{X_0}} \left( \lambda_d (\hat N_d - n_{d0}) +\lambda_c(\hat N_c -n_{c0}) +\frac{\w_0 x_0}{2}\right)\, \nonumber\end{aligned}$$ where we define $\hat N_d \equiv \sum_\sigma \hat n^d_\sigma$, $\hat N_c\equiv \sum_\sigma c^\dagger_{0\sigma} c_{0\sigma}$ and absorb all constants into $E_0$. To keep the Hamiltonian $\hat H_{S}=\hat H_{\rm e}+\hat H_{\rm ph} + \hat H_{\rm e-ph}$ invariant under the basis set change of the bosonic operator, we substitute $\e_d\to \e_d + \lambda_d x_0$ and define a single-particle energy $\e_c=\lambda_c x_0$ for the orbital $c_{0\sigma}$. In case of a non-zero $\lambda_c$, we also need to shift the constant $n_{c0}\to n_{c0} - \w_0 x_0/2\lambda_c$. Since under these conditions the Hamiltonian is unaltered, the dynamics of the fermion degrees of freedom remains identical and independent of this basis transformation $b_0\to \bar b_0$. In particular, this means that the Kondo temperature, if applicable, and other thermodynamic properties of the system S remain unchanged. We now analyze the effect of a non-zero expectation value $\langle \hat X_0 \rangle=x_0$ on the tunnel current. For simplicity we set $N_\nu=1$ and assume identical vibrational couplings in the tunneling Hamiltonian for all $M$ orbitals, i.e. $\lambda^{\rm tip}_{\mu \nu}=\lambda^{\rm tip}_{\mu' \nu}$. For the two inelastic density of states we require the Green’s functions involving the vibrational displacements either linearly or quadratically. The relation between these in the two bases $\hat{X}$ and $\hat{\bar{X}}$ follows from and is given by $$\begin{aligned} \label{eq:36} G_{\hat X d_{\mu \sigma}, d^\dagger_{\mu' \sigma}}(z) &=& G_{\hat{ \bar X} d_{\mu \sigma}, d^\dagger_{\mu' \sigma}}(z) + x_0 G_{d_{\mu \sigma}, d^\dagger_{\mu' \sigma}}(z)\\ \label{eq:37} G_{\hat Xd_{\mu \sigma}, \hat X d^\dagger_{\mu' \sigma}}(z) &=& G_{\hat {\bar X} d_{\mu \sigma}, \hat {\bar X} d^\dagger_{\mu' \sigma}}(z) \\ && + x_0\left( G_{ d_{\mu \sigma}, \hat {\bar X} d^\dagger_{\mu' \sigma}}(z) + G_{ \hat {\bar X} d_{\mu \sigma}, d^\dagger_{\mu' \sigma}}(z)\right) \non && + x_0^2 G_{d_{\mu \sigma}, d^\dagger_{\mu' \sigma}}(z) . \nonumber\end{aligned}$$ Substituting these expression into the formula for the total tunnel current and regrouping the different contributions, we obtain for the sum of the three transmission functions that enter the integral for the total tunnel current up to second order $$\begin{aligned} \label{basis change density of states} \tau^{(0)}_\sigma + \tau^{(1)}_\sigma &+&\tau^{(2)}_\sigma = \\ (&1& + \lambda^{\rm tip} x_0) ^2 \bar{\tau}^{(0)}_\sigma + (1+\lambda^{\rm tip} x_0)\bar{\tau}^{(1)}_\sigma +\bar{\tau}^{(2)}_\sigma \nonumber\end{aligned}$$ The purely fermionic Green’s function $G_{d_{\mu \sigma}, d^\dagger_{\mu' \sigma}}(z)$ picks up the factor $(1+\lambda^{\rm tip} x_0)^2$ which therefore appears as a prefactor in the elastic density of states and in the corresponding tunnel current. Remembering that our theory is accurate to quadratic order in $\lambda^{\rm tip}$, we may add corrections of order $O([\lambda^{\rm tip}]^3)$ and higher to the right hand side of Eq. . Since $\bar{\tau}^{1}_\sigma$ and $\bar{\tau}^{(2)}_\sigma$ are of orders $(\lambda^{\rm tip})$and $(\lambda^{\rm tip})^2$, respectively, we can thus write $$\begin{aligned} \label{basis change density of states 2} \tau^{(0)}_\sigma + \tau^{(1)}_\sigma + \tau^{(2)}_\sigma &\simeq&(1 + \lambda^{\rm tip} x_0)^2 (\bar{\tau}^{(0)}_\sigma + \bar{\tau}^{(1)}_\sigma +\bar{\tau}^{(2)}_\sigma)\end{aligned}$$ after adding the corresponding higher-order correction terms to the prefactors of $\bar{\tau}^{(1)}_\sigma $ and $\bar{\tau}^{(2)}_\sigma$. Therefore, a finite displacement $x_0$ generates an overall prefactor $(1+\lambda x_0)^2$ in the total tunnel current. This can be absorbed into the tunneling matrix element $t^2_{\mu \sigma}\to \bar t^2_{\mu \sigma}= t^2_{\mu \sigma}( 1+\lambda x_0)^2 \approx [t_{\mu \sigma}\exp(\lambda x_0)]^2$, leading to an identical total tunnel current for the two bases $\hat{X}$ and $\hat{\bar{X}}$, up to $O([\lambda^{\rm tip}]^3)$ corrections. However, while the total current is invariant under the basis change of the harmonic oscillator, the attribution of elastic and inelastic contributions remains basis-dependent, which becomes immediately obvious from the Eqs.  and : the inelastic current in the original oscillator basis contains an elastic part with respect to the shifted oscillator basis. We adopt the following strategy in order to ensure that all properties are discussed in the framework of a harmonic oscillator basis with vanishing displacements in the presence of the electron-phonon coupling: First, we calculate the displacement for a given $\hat H_{S}$, secondly we perform a basis set change of the harmonic oscillators to a basis $\hat{\bar{X}}_\nu$ with $\langle\hat{\bar{X}}_\nu\rangle=0$. This leads, thirdly, to a renormalization of the model parameters in $\hat H_{S}$, as outlined in Eq. . Fourthly, we calculate all spectral functions in the transformed basis. This implies that the effect of the displacement is absorbed into the definition of the prefactor via Eq.  and is consistent with the notion that an additional electron phonon coupling does not change the atomic equilibrium positions as determined by the LDA. Experiments on NTCDA/Ag(111) {#sec:experiment-NTCDA} ============================ Choice of system {#sec:experiment-NTCDA choice of system} ---------------- To meaningfully test our tunneling theory of section \[sec:tunnel-current\], we need a system S that exhibits both strong electron-electron interaction and electron-phonon interaction and can be investigated in very clean conditions with STM and STS. Specifically, building on the Hamilton operator introduced in section \[sec:Modeling the system\], a quantum impurity system which shows the Kondo effect appears prospective. Molecular adsorbates on metals are a good starting point to realize a quantum impurity system [@Kondo_molecule; @kondo_molecule2; @ptcda_kondo_cleavage; @vib_kondo_stm; @AU-PTCDA-monomer; @AU-PTCDA-dimer], since they have localized orbitals that may interact with the electrons of the metal substrate. At the same time, molecules display large number of vibrations, offering the possibility to find a sizable electron-phonon coupling at least for some of these modes. In fact, the combination of the Kondo effect and vibrational inelastic tunneling has been reported for a few molecule-on-metal systems [@kondo_vib_bjunc3; @vib_kondo_stm; @vib_kondo_stm2; @kondo_vib_bjunc2; @kondo_vib_bjunc]. For technical reasons, well-ordered, commensurate periodic layers have advantages, since in these layers the molecules are located at well-defined sites, enforced by both interactions with the substrate and interactions with the neighboring molecules. These considerations draw our attention to the system of 1,4,5,8-naphthalene-tetracarboxylic dianhydride (NTCDA) on Ag(111). For this system, the Kondo effect has been reported [@ziroff2012low]. An additional benefit is that NTCDA/Ag(111) bears similarity to PTCDA/Ag(111) and AuPTCDA/Au(111), for which the Hamiltonian in Eq.  has allowed a quantitative modeling of the Kondo effect. However, unlike PTCDA/Ag(111), NTCDA/Ag(111) displays the Kondo effect even in the native adsorbed state [@ziroff2012low], without artificially lifting the molecule from the surface, such that it can be studied in tunneling regime, a prerequisite for our theory. Moreover, it shows a rich vibrational signature [@C5CP06619K; @braatz2016switching]. This makes NTCDA/Ag(111) ideally suited to the present purpose. Experimental details {#sec:experiment-NTCDA experimental details} -------------------- The Ag(111) crystal was prepared by repeated cycles of $\mathrm{Ar}^+$ sputtering and annealing to $T \approx 800$K for $15$ minutes. A small coverage of NTCDA molecules (less than 15% of a monolayer) was deposited from a home-built Knudsen cell onto the clean Ag(111) surface held at $T \approx 80$K. After deposition the sample was annealed at $T \approx 350$K for $5$ minutes and afterwards cooled down to $T \approx 80$K within $2-3$ minutes. In order to minimize contaminations the sample was transferred into the STM immediately after the preparation. The scanning tunneling microscopy (STM) and spectroscopy (STS) experiments on NTCDA/Ag(111) were carried out in ultra-high vacuum (UHV) in a Createc STM with a base temperature of $T \approx 9.5$K and JT-STM (SPECS) with a base temperature of $T \approx 4.3$K. The JT-STM offers magnetic fields up to $3$T in the out-of-plane direction. Differential conductance $dI/dV(V)$ spectra were recorded with the lock-in technique with the current feedback loop switched off. Typical parameters were a modulation amplitude of $0.6 - 2$mV and modulation frequency of $833$Hz. Before experiments on NTCDA, a featureless tip density of states was ensured by measuring the surface state of clean Ag(111). After changing the temperature of the STM we waited for $20$h to obtain equilibrium conditions before measuring $dI/dV$ spectra. $dI/dV(V)$ spectra at different locations above the same molecule were measured as follows: First, the tip was positioned above the CH edge of a NTCDA molecule at tunneling current $I = 200$pA and bias voltage $V = 50$mV; then the feedback loop was switched off and the tip was moved at constant height to different locations above the molecule, followed by the measurement of $dI/dV(V)$ spectra at each of the desired positions. ![Constant-current STM image of the relaxed phase of NTCDA on Ag(111) ($I=200$pA, $V=50$mV). Graphical representations of (gas-phase) NTCDA molecules have been overlaid over the bright and dark molecules. The white, grey and red circles indicate hydrogen, carbon and oxygen atoms of NTCDA, respectively. The length of the scale bar is 10 Å.[]{data-label="fig:exp_fig1"}](fig3) ![Constant-current STM image of the rippled phase of NTCDA on Ag(111) ($I=200$pA, $V=50$mV). The arrow indicates a line along which the character of the molecules changes gradually from bright to dark and vice versa. The length of the scale bar is 20 Å.[]{data-label="fig:fig4"}](fig4) Structure {#sec:experiment-NTCDA structure} --------- The geometric structure and the electronic properties of NTCDA on Ag(111) have already been studied in previous works [@stahl1998coverage; @kilian2008commensurate; @ziroff2012low; @braatz2016switching; @C5CP06619K]. There are two phases, commonly referred to as the relaxed and the compressed ones [@stahl1998coverage; @kilian2008commensurate; @braatz2016switching]. Here, we focus on the relaxed phase of NTCDA/Ag(111). In Fig. \[fig:exp\_fig1\] an STM image of the relaxed phase of NTCDA is shown. The relaxed phase contains two molecules per unit cell, arranged in a brick-wall structure with a rectangular unit cell of area $11.57$Å $\times$ $15.04$Å. The structure is commensurate [@braatz2016switching]. Because of their different appearance in the STM image the two molecules in the unit cell will from now on be referred to as bright and dark molecules, respectively. Both molecules are aligned with their long axis along the $[01\bar1]$ direction of the substrate [@braatz2016switching]. The difference between the two molecules most probably arises from different adsorption sites on the surface. Because the arrangement of the molecules in the unit cell is consistent with two distinct high-symmetry sites, on-top and bridge [@braatz2016switching], it appears natural that the molecules are in fact located in these sites. However, it is not known whether bright molecules are in on-top and dark molecules in bridge sites or vice versa. From our PBE+vdW$^\text{surf}$ calculations (see below) we find that the NTCDA molecules at both sites are chemisorbed. The on-top molecule has an average distance of $\overline{z}=2.89$Å and a corrugation of $\Delta z = 0.35$Å, while for bridge molecule we observe $\overline{z}=2.86$Å and $\Delta z=0.40$Å. We report here also a phase that to the best of our knowledge has not been reported before, the rippled phase. An STM image of the rippled phase, a variant of the relaxed phase, is shown in Fig. \[fig:fig4\] in which over a distance of approximately six unit cells along the $[01\bar1]$ direction of the substrate the bright molecule turns into a dark one and vice versa. Kondo effect {#sec:experiment-NTCDA Kondo effect} ------------ Fig. \[fig:exp\_fig2\] displays STS spectra recorded above NTCDA/Ag(111) in four different positions, namely in the vicinity of the CH edges of the NTCDA molecule and in the center of the molecule, each for both the bright and the dark molecules. These positions were chosen because at the CH edges the lowest unoccupied molecular orbital (LUMO) of NTCDA exhibits an intense lobe (for the bright molecule this lobe is directly seen in the STM image of Fig. \[fig:exp\_fig1\]), while in the center of the molecule two nodal planes of the LUMO intersect. Note that the LUMO of NTCDA becomes partially occupied when the molecule adsorbs on the Ag(111) [@ziroff2012low; @braatz2016switching; @C5CP06619K]. An image of the probability amplitude of the LUMO is shown in Fig. \[fig:exp\_fig3\]. Fig. \[fig:exp\_fig2\] shows that at the CH edge both molecules exhibit a peak at zero bias (the precise peak position for the bright molecule is $+1.9$meV, while for the dark molecule it is $-0.6$meV), although with very different intensities. For the bright molecule this peak is much more intense. Ziroff et al. suggested that a corresponding peak observed in photoelectron spectroscopy at the Fermi energy is a manifestation of the Kondo effect with a Kondo temperature of $T_K \simeq 100$K  [@ziroff2012low], although its temperature evolution did not resemble the characteristic temperature dependence of a Kondo resonance. For the purpose of this paper we have to establish beyond doubt that the zero-bias peak for both molecules is indeed a Kondo resonance. To this end, we use a three-pronged approach, comprising measurements of the zero-bias peak as a function of temperature, magnetic field and hybridization, in each case looking for the dependence that is indicative of the Kondo effect. ![$dI/dV$ spectra of the bright and dark molecules acquired at the CH-edge (left) and at the center (right) of the bright (red line) and dark (black line) molecules, respectively. The spectra are plotted on the bias voltage axis as measured. A calibration of bias voltage scale to symmetrize the inelastic features would require shifting the spectra by $2.25$mV to the right. Fano fits according to Eq.  are indicated by blue lines, the fit parameters are: bright molecule, CH-edge: $\delta = 24.04$, $q = 6.44$. Bright molecule, center: $\delta = 32.50$, $q = 1.74$. Dark molecule, CH-edge: $\delta = 25.42$, $q = 6.95$. Dark molecule, center: $\delta = 29.84$, $q = 2.08$. Average fit parameters are listed in table \[tab:fwhm\_and\_q\].[]{data-label="fig:exp_fig2"}](fig5) We first analyze the temperature-dependence of the zero-bias peak for the bright molecule. In Fig. \[fig:fig2\](a) its full width at half maximum (FWHM) is displayed. The FWHM was extracted by fitting with a Fano line shape [@FanoResonance1961; @SchillerHershfield2000a]. Broadening effects due to temperature $T$ and modulation amplitude $V_\mathrm{mod}$ have been taken into account by subtracting them from the measured FWHM, using $\text{FWHM}= \sqrt{\text{FWHM}_\text{measured}^2-(1.7V_\mathrm{mod})^2 - (3.5k_{\mathrm{B}}T)^2}$ [@Kroger2005]. The such-determined intrinsic FWHM exhibits the expected temperature dependence of a Kondo resonance. Fitting the expression $\sqrt{(\alpha k_B T)^{2} + (2 k_{\mathrm{B}} T_K)^2}$ to the FWHM [@Nagaoka2002] we find a Kondo temperature of $T_K^\mathrm{bright} = 133$K and $\alpha = 4.53$. It should be noted that this is only a rough estimate, because the FWHM is related to $T_K$ by a non-universal scaling constant [@AU-PTCDA-monomer]. A more accurate analysis in of the Kondo temperature will be presented in section \[sec:application-to-NTCDA\]. Because of its low intensity and broad FWHM, the temperature dependence of the zero-bias peak of the dark molecule is difficult to study. A broad Kondo peak indicates that the system is in the weakly correlated regime, with a small ratio $U/\Gamma$, where $U$ is the intra-orbital Coulomb repulsion (Eq. ) and $\Gamma$ is an energy-averaged hybridization parameter (related to Eq. ), and a large Kondo temperature $T_{\rm K}$. To prove that the zero-bias peak of the dark molecule is also a Kondo resonance, we therefore apply a different strategy: Instead of decreasing the temperature to change its line shape, we decrease $\Gamma$, thus tuning the system further into the strong-coupling regime, in which the Kondo peak is sharper and more easy to pin down. The tuning of the hybridization is achieved by forming a contact (at $z \equiv 0$Å, where $z$ is the vertical tip coordinate) between the tip apex and one of the corner oxygen atoms of NTCDA. The corresponding part of the molecule can then either be pushed towards the surface ($z < 0$Å) or lifted up ($z > 0$Å) [@ptcda_kondo_cleavage; @gated_molecular_wire; @PTCDAAgMove; @gated_wire_spectral]. Fig. \[fig:fig2\](c) displays $dI/dV$ spectra recorded at different $z$ for both molecules, plotted as color maps. Both maps exhibit very similar behavior, albeit shifted with respect to each other by $\Delta z=0.9$Å  along the vertical axis. Based on the similarity of the maps, and the fact that for the bright molecule we have already shown, employing the temperature-dependence, that the zero-bias peak is a Kondo peak, we can conclude that the same is also true for the dark molecule. In fact, both maps in Fig. \[fig:fig2\](c) exhibit the expected dependence of a spin-$\frac{1}{2}$ Kondo effect, as the comparison with the well-studied case of lifting PTCDA molecules from Ag(111) shows [@ptcda_kondo_cleavage; @gated_molecular_wire; @PTCDAAgMove; @gated_wire_spectral]. ![(a) Temperature evolution of the FWHM of the zero-bias peak of the bright molecule. (b) $dI/dV(V,z)$ maps for the bright (left) and dark (right) molecules, recorded after the formation of the tip-molecule bond. The meaning of the $z$ coordinate is shown schematically in the illustrations. (c) $dI/dV$ spectra at various $z$ in a magnetic field of $B = 2.5$T, measured at $T = 4.3$K.[]{data-label="fig:fig2"}](fig6){width="85mm"} The maps clearly show the sharpening of the Kondo resonance that is expected if the hybridization is reduced and the Kondo effect is tuned from the weak- towards the strong-coupling regime. Note that in addition to reducing $\Gamma$, lifting the molecule may reduce the charge transfer to the molecule and also lead to a smaller dielectric screening due to the larger molecule-surface distance, both resulting in a increased Coulomb interaction $U$, thus further increasing $U/\Gamma$ with increasing $z$. We note furthermore that the FWHM of the zero-bias peak decreases by a factor of $\approx 2.4$ for the bright and dark molecules when the molecule is contacted by the tip (from $22$mV for the non-contacted bright molecule in Fig. \[fig:exp\_fig2\] to $9$mV for the contacted molecule at $z=0$, and from $45$mV to $19$mV for the dark molecule, see Fig. \[fig:fig7\]). This reduction of the FWHM can be explained by the partial dehybridization that occurs when the oxygen atom jumps into contact with the tip and lifts the surrounding parts of the molecule from the surface. It is well-known that the additional contact to the tip is electronically weak and does not lead to an appreciable hybridization with the LUMO [@Temirov2018; @Esat2018]. Since the FWHM of the zero-bias peak decreases by approximately the same factor for the bright and dark molecules when the molecule is contacted by the tip, we can conclude that the initial $T_K^\mathrm{dark}$ of the dark molecule, without the contact to the tip, must also be larger than $T_K^\mathrm{bright}$. This explains the broader Kondo peak of the dark molecule in Fig. \[fig:exp\_fig2\]. ![$dI/dV$ spectra of the different NTCDA molecules in the rippled phase, recorded directly after the formation of the bond of the tip to the molecules as indicated in the inset. The data was acquired at $T=4.3$K. Spectra are vertically offset by $5$$\mu$S for clarity.[]{data-label="fig:fig7"}](fig7){width="45.00000%"} In the strong-coupling regime moderate magnetic fields may split the Kondo resonance [@Zhang2013]. Applying a $B$ field of $2.5$T at an experimental temperature of $4.3$K, we indeed observe an incipient splitting of the Kondo resonance for the bright molecule at $z=0.5$Å. Assuming a Landé factor of $g = 2$, the Zeeman energy at $2.5$T is $g \mu_B B \approx 0.29$mV, slightly smaller than the thermal fluctuations $k_B T \approx 0.37$mV at $T = 4.3$K (at $z=0.5$Å, $T_K$ has dropped so far that $g \mu_B B\gg k_B T_K$). Therefore, the split is not well developed, but nevertheless clearly visible. Note that for the non-contacted bright molecule a $B_c \approx k_B T_K / (g \mu_B) \approx 49.5$T would be necessary to split the Kondo resonance, which is clearly out of reach. Another notable observation in Fig. \[fig:exp\_fig2\] is the fact that in the center of the molecule step-like structures at zero bias are observed instead of a Lorentzian peak. The spectra of the bright and dark molecules are almost identical and merely differ in the intensity of the step-like feature. Such features result from the quantum interference between two or more different tunneling paths [@FanoResonance1961; @SchillerHershfield2000a]. This interference leads to a zero-bias feature with a so-called Fano line shape. In the simplest case of two interfering channels, the differential conductance is approximated by $$\label{eq:Fano} \frac{dI}{dV} (V) \propto \rho_0 + \frac{(q+\epsilon)^2}{1+\epsilon^2},$$ with $$\epsilon = \frac{eV-E_K}{(\delta /2)}$$ and $$\label{NTCDA_q} q = \frac{t_2}{\pi \rho_0 \Gamma t_1}.$$ Here, $E_K$ describes the intrinsic position of the Kondo resonance, $\delta$ its FWHM, $\Gamma$ the hybridization between the local orbital and the substrate, $t_1$ and $t_2$ the tunneling probabilities from the tip directly into the substrate and into the local orbital, respectively, and $\rho_0$ the density of states. $q$ determines the line shape of the Kondo resonance. The blue lines in Fig. \[fig:exp\_fig2\] display fits of Eq.  to the experimental tunneling spectra. The fit parameters $q$ and $\delta$, averaged over fits for 10 data sets including the one shown in Fig. \[fig:exp\_fig2\], are summarized in Tab. \[tab:fwhm\_and\_q\]. As expected, $\delta$ is on average larger for the dark molecule than for the bright one (see above). However, it is noteworthy that the $\delta$ that is extracted from the spectra recorded at the CH edge is approximately the same as the one for the center of the molecule. This confirms that both the Lorentzian peak and the step-like feature indicate the same energy scale – we can thus conclude that the step is also a manifestation of the Kondo effect that leads to the peak recorded at the CH edge. Table \[tab:fwhm\_and\_q\] also reveals that $q$ is significantly smaller in the center of the molecule, indicating that there the probability to tunnel directly from the tip into the substrate ($t_1$) is larger than at the CH edge. The reason for the larger tunneling probability directly into the substate when the tip is located in the center of the molecule is a direct consequence of the spatial distribution of the LUMO wave function, which has a node in the center of the molecule and a pronounced lobe at the CH edge (Fig. \[fig:exp\_fig3\]b). This is reflected in Fig. \[fig:exp\_fig3\]a, which displays the LDOS of the LUMO $4$Å above the gas-phase NTCDA molecule. ![Local density of states (LDOS) of the LUMO of NTCDA calculated $4$Å above the gas-phase molecule (left panel). A graphical representation of the gas-phase NTCDA molecule has been overlaid for clarity. The right panel shows the top view of the LUMO of the gas-phase NTCDA molecule. The different colors indicate the positive ($\Psi (r) > 0$) and negative ($\Psi (r) < 0$) contributions of the wave function.[]{data-label="fig:exp_fig3"}](fig8){width="85mm"} We note that the fits displayed in Fig. \[fig:exp\_fig2\] and the derived parameters in Table \[tab:fwhm\_and\_q\] are merely heuristic and should only be used to ascertain that there are at least two tunneling paths present, and that the center of the molecule is more transparent to the tunneling current than the CH edge. More elaborate fits to the spectra, based on a more solid theoretical foundation, will be presented in section section \[sec:application-to-NTCDA\]. molecule location $\delta$ $q$ ---------- ---------- -------------------- ---------------- CH edge $ (28.5\pm 2.3)$mV $ 15.5\pm 6.9$ center $ (29.4\pm 6.8)$mV $ 1.2\pm 0.2$ CH edge $ (52.1\pm 4.7)$mV $ 21.4\pm 9.7$ center $ (48.5\pm 8.2)$mV $ 0.9\pm 0.4$ : $\delta$ and $q$ extracted from the fits of the Fano line shape (Eq. ) to the $dI/dV(V)$ spectra of the bright and dark molecules. The values are averages over the fit parameters for ten different data sets. One data set is shown in Fig. \[fig:exp\_fig2\]. []{data-label="tab:fwhm_and_q"} We conclude that there is overwhelming experimental evidence (from $T$-dependent data, junction stretching, magnetic field data and quantum interference) that both the dark and the bright molecule of the relaxed NTCDA/Ag(111) phase exhibit the Kondo effect. To explain the behavior of the present system quantitatively, it therefore appears natural to apply the theory that has been very successful for PTCDA/Ag(111) and AuPTCDA/Au(111) [@PTCDAAgMove; @gated_wire_spectral; @AU-PTCDA-monomer; @AU-PTCDA-dimer]. This is done in section \[sec:application-to-NTCDA\]. Vibrational features {#sec:experiment-NTCDA vibrational side bands} -------------------- ![The panel on the left shows a STM topography image measured at constant height above the molecules of the rippled phase ($V=47$mV). The panel on the right shows the corresponding $d^2I/dV^2$ image. The length of the scale bar is 10 Å.[]{data-label="fig:fig10"}](fig9){width="85mm"} ![ (left) $dI/dV$ spectra of the different NTCDA molecules in the rippled phase at $T=4.3$K. The spectra have been recorded at the CH edge of each molecule. Different colors correspond to the different molecules as indicated by the colored frames around the molecules. The color coding is the same as in Fig. \[fig:fig7\]. (right) $dI/dV$ spectra of the different NTCDA molecules in the rippled phase at $T=4.3$K, measured at the center of each molecule. Colors as in panel b. Spectra are vertically offset by $1$ nS for clarity.[]{data-label="fig:fig9"}](fig10){width="85mm"} In addition to the zero-bias features, the spectra in Fig. \[fig:exp\_fig2\] also show features at finite bias voltages. Most notable are peaks at approximately $+(47.0 \pm 0.3)$mV and $-(51.5 \pm 0.3)$mV in the spectra recorded at the CH edges of both the bright and the dark molecules. However, we also observe weak features around $\pm 30$mV. The (nearly) symmetric location of in particular the stronger features around zero bias (up to a shift of $2.25$mV towards negative energies) is suggestive of inelastic excitations, either during the tunneling process or within the sample system. Such excitations can occur as a result of, e.g., vibrational or magnetic degrees of freedom. We have not observed any change or shift of the side peaks in magnetic fields up to $3$T. It is therefore unlikely that the features are of magnetic origin and we conclude that they must be linked to vibrations. NTCDA indeed has a number of vibrational modes in the relevant frequency range [@C5CP06619K]. Some of them are listed in Table \[tab:vibrations\]. no. symmetry $\hbar\omega$ ----- -------------- -------------------------------- 1 B$_{\rm 3g}$ $ 41.6$meV ($335.35$cm$^{-1}$) 2 B$_{\rm 1u}$ $ 46.2$meV ($372.41$cm$^{-1}$) 3 B$_{\rm 3g}$ $ 50.4$meV ($406.62$cm$^{-1}$) 4 A$_{\rm g}$ $ 50.7$meV ($408.96$cm$^{-1}$) 5 B$_{\rm 1g}$ $ 52.8$meV ($525.53$cm$^{-1}$) : Vibrational modes of gas-phase NTCDA in the energy range $40$ to $50$meV, calculated by DFT (taken from reference [@C5CP06619K]). Gas-phase NTCDA has the symmetry group D$_{\rm 2h}$.[]{data-label="tab:vibrations"} When recorded in the center of the molecule, the features at $+(47.0 \pm 0.3)$mV and $-(51.5 \pm 0.3)$mV become much stronger. In Fig. \[fig:fig10\], the spatial distribution of the vibrational features is displayed, recorded as a $d^2I/dV^2$ image at $\simeq 50$mV, in comparison with a constant-height topographic image. One observes an image without nodal planes and a clear concentration of the intensity close to the center of the molecule. This is true for both molecules, although the bright molecule has a larger maximum intensity, which is consistent with Fig. \[fig:exp\_fig2\]. Moreover, for the spectra measured in the center of the molecules there is a clear difference in the line shape between the bright and dark molecules. In both cases they are asymmetric with a steep rise at the low-bias side, but for the dark molecule the drop on the high-bias side is more moderate than for the bright molecule, giving the vibrational feature a more step-like appearance for the dark molecule, in contrast to a “half-peak” for the bright molecule. ![Plot of the step height $\Delta\sigma_{\rm center}$ measured at the center of the molecules in the rippled phase as a function of the height $\sigma_{\rm CH}(0)$ of the Kondo resonance at the CH edge. The values are obtained from the spectra displayed in Fig. \[fig:fig9\]. Color coding as in Fig. \[fig:fig9\]. Data points have been fitted by a linear function of the form $\Delta\sigma_{\rm center}=c \times \sigma_{\rm CH}(0) +b$, fit parameters are $c = 0.13$, $b = 0.30$nS.[]{data-label="fig:fig11"}](fig11){width="85mm"} The dependence of line shape of the vibrational features at $+(47.0 \pm 0.3)$mV and $-(51.5 \pm 0.3)$mV on the shape of the spectrum at zero bias is also apparent in Fig. \[fig:fig9\], which displays the evolution of the Kondo peak and the vibrational side bands in the transition from the dark to the bright molecule in the rippled phase of Fig. \[fig:fig4\]. As the intensity of the Kondo peak (measured at the CH edge) increases, the vibrational features measured in the center of the molecule at $+(47.0 \pm 0.3)$mV and $-(51.5 \pm 0.3)$mV turn from a step for the dark molecule into an asymmetric peak for the bright molecule, in agreement with Fig. \[fig:exp\_fig2\]. For the feature at $+(47.0 \pm 0.3)$mV this behavior is also illustrated by Fig. \[fig:fig11\], in which $\Delta\sigma_{\rm center}$ is plotted versus $\sigma_{\rm CH}(0)$ and a linear correlation between the peak height of the vibrational feature and the Kondo peak is observed [@Esat2017]. NRG results {#sec:NRG-results} =========== Application of the tunneling theory to NTCDA {#sec:application-to-NTCDA} -------------------------------------------- ### General approach In this section we specify step by step the theoretical framework needed to reproduce and explain the experimentally measured differential conductance $dI/dV$ spectra in Fig. \[fig:exp\_fig2\], using Eqs. , and as a basis. We use an approach [@PTCDAAgMove; @gated_wire_spectral; @AU-PTCDA-monomer; @AU-PTCDA-dimer] in which we map the results of density functional theory (DFT) calculations, combined with many body perturbation theory (MBPT) to include quasi-particle corrections, onto the Hamiltonian $\hat H_\text{SIAM}$ of a single-orbital Anderson model (SIAM), here also including Holstein terms, which is then solved by NRG calculations. In particular, the NRG is used [@Wilson75; @BullaCostiPruschke2008] to exactly calculate all spectral functions that are required to calculate the transmission functions that enter Eqs. , and . As pointed out in Ref. [@AU-PTCDA-monomer], employing a fully energy-dependent hybridization function $\Gamma(\omega) = \Im \Delta(\w-i0^+)=\pi \Sigma |V_k|^2 \delta(\omega-\epsilon_k )$ in the NRG is crucial for an accurate description. Moreover, we do not impose particle-hole symmetry. Our theoretical framework is the same as discussed in Ref. [@PTCDAAgMove; @AU-PTCDA-monomer; @AU-PTCDA-dimer]. Structural optimization is performed within density-functional theory (DFT), using the SIESTA package [^1] [@Siesta1; @Siesta2], using ab-initio pseudopotentials and a double-zeta plus polarization basis (DZP). We employ the PBE exchange-correlation functional [@PBE]. Since the van der Waals interaction is crucial for weakly bound systems like organic molecules on metal surfaces, we include it in the formulation of Ruiz et al. [@vdWsurfPTCDA] (vdW$^\text{surf}$) for all structure optimizations. In addition to the structural data, the electronic mean-field spectrum of the adsorbed molecule (in particular its LUMO state) is required as input for the NRG. This cannot be calculated on the level of DFT, since DFT suffers from problems as long as electronic spectra are concerned [@PTCDAAgMove; @AU-PTCDA-monomer; @AU-PTCDA-dimer]. Instead, many-body perturbation theory (MBPT) provides a systematic approach to spectral features (except for the dynamical correlation to be treated by the NRG.) Here we employ the same approach as discussed in Ref. [@PTCDAAgMove; @AU-PTCDA-monomer; @AU-PTCDA-dimer]. Starting from a DFT-LDA calculation for a given geometry, we carry out a MBPT calculation within our LDA+$GdW$ approach. This yields realistic band-structure energies for all states of the adsorbate system, fully including all screening and broadening effects resulting from the metallic surface. After projecting on the LUMO state of the bare molecule, we arrive at a projected density of states (PDOS) as shown in Fig. \[NRG\_fig4aa\]. From this PDOS one can deduce the level position $\e_{d\sigma}$ of the LUMO state when adsorbed on the surface, as well as its hybridization function $\Delta(z)$ \[see Eq. (\[eqn:MFGF\])\]. In addition, the internal Coulomb interaction $U$ of the LUMO state is also obtained from MBPT [@PTCDAAgMove]. ### Input from ab-initio calculations and model without electron-phonon coupling {#sec:Input from ab-initio calculations and model without electron-phonon coupling} ![PDOS of the LUMO of NTCDA/Ag(111) as calculated by a combination of DFT and MBPT. The latter includes quasiparticle corrections. (a) on-top molecule, (b) bridge molecule. []{data-label="NRG_fig4aa"}](fig12-LDA-PDOS){width="50.00000%"} As suggested by experimental data (see Fig. \[fig:exp\_fig1\] and discussion in section \[sec:experiment-NTCDA structure\]), the molecules in the calculations are placed at on-top and bridge sites on the Ag(111) surface. According to the calculation, both molecules chemisorb stably at these sites. The *ab initio* calculation predicts well-separated NTCDA molecular orbitals, an energy broadening of the orbitals due to their hybridization with the substrate, a partial occupation of the lowest unoccupied molecular orbital (LUMO) due to charge transfer from the substrate, and a substantial intramolecular Coulomb interaction $U=1.25$eV for the LUMO. The specific local environments of the two adsorption sites lead to slightly different positions of the on-top and bridge LUMOs, as can be seen in the projected densities of state in Fig. \[NRG\_fig4aa\], with the weight of the PDOS spectrum of the bridge molecule appearing further to the left. Similarly, the value of the hybridization functions at the chemical potential differ slightly for the two molecules, being $\Gamma^\mathrm{bridge}(0) = 190$meV and $\Gamma^\mathrm{top}(0) = 165$meV. Because the Coulomb interaction is substantial compared to the hybridization strength, it enforces a single occupation of the LUMOs on both molecules, leading to a free spin of the radical that is ultimately screened by the Kondo effect for $T\to 0$. Since the *ab initio* PDOS spectra only contain the Coulomb interaction on a mean-field level, the half-filled orbitals are spin-degenerate in a paramagnetic calculation and the effective mean-field orbital energy must be pinned close to the Fermi energy, as is indeed apparent in Fig. \[NRG\_fig4aa\]. Both *ab initio* spectra in Fig. \[NRG\_fig4aa\] are much too wide compared to the STS data which exhibits a peak width of the order of $30$meV in the differential conductance at zero bias. This indicates that many-body correlations play an important role and must be taken into account for matching theory with experiment. Interpreting the *ab initio* PDOS as a mean-field solution [@PTCDAAgMove; @gated_wire_spectral; @AU-PTCDA-monomer; @AU-PTCDA-dimer] allows us to extract the single particle energies $\epsilon_{d\sigma}^\text{bridge}=-0.77\,\text{eV}$ and $\epsilon_{d\sigma}^\text{top}=-0.67\,\text{eV}$ for both types of molecules as well as the full complex hybridization function $\Delta(z)$. ![Comparison between the experimental $dI/dV(V)$ spectra recorded on NTCDA/Ag(111) (black lines) and the results of NRG calculations for the SIAM with the PDOS of Fig. \[NRG\_fig4aa\] as input (blue lines). The NRG spectra have been adjusted with a constant offset ($\rho_\text{offset}=0.2$nS in panel (a), $\rho_\text{offset}=1.1$nS in panel (b) to account for an experimental background signal such that the zero-bias peak heights and the high-frequency tails are matched to the experimental $dI/dV$ curve. []{data-label="NRG_fig4ab"}](fig13-NRG_withoutLc){width="50.00000%"} Next, we use these sets of *ab initio* parameters and functions as an input for NRG calculations to solve the SIAM for both the bridge and the on-top molecules in the absence of any electron-phonon coupling. For simplicity, we also set the tunneling matrix element $t_{c_{0\sigma}}$ from the tip to the local effective substrate orbital (defined in Eq. ) to zero and only include $\rho_{d_{0\sigma},d^\dagger_{0\sigma}}(\w)$ in the calculation of the theoretical $dI/dV$ curve (Eqs.  and ), where $d_{0\sigma}$ is the annihilation operator for an electron in the LUMO. The result for the on-top and bridge molecules are displayed in Fig. \[NRG\_fig4ab\], revealing a substantial narrowing of the zero-bias resonance relative to the *ab initio* PDOS. Thus, in the calculations as well as in experiments, both molecules exhibit a Kondo resonance. Driven by the differences in $\Gamma(0)$ the two molecules in the NTCDA/Ag(111) unit cell have different Kondo temperatures: The Kondo temperature of the bridge molecule is larger than that of the on-top molecule. If we associate the larger measured to the larger calculated Kondo temperature, we can identify the dark molecule with the bridge site and the bright molecule with the on-top site. This identification is consistent with expectations based on structural arguments, namely that the bridge molecule adsorbs slightly closer to the surface and exhibits a larger corrugation. This explains the larger $\Gamma$, the smaller coupling parameter $U/\Gamma$ and thus also the larger $T_K$. However, while the experimental trend $T_K^\mathrm{dark} > T_K^\mathrm{bright}$ is thus correctly predicted, the actual widths of the zero-bias anomalies, which is a measure of Kondo temperature, are too large compared to experiment, as indicated by the much narrower experimental curves in Fig. \[NRG\_fig4ab\]. ### Identification of the problem. The unconventional Holstein model as a possible solution {#sec:problem-and-strategy} The SIAM without electron-phonon coupling as discussed in the previous section has three shortcomings: (i) the Kondo temperature $T_{\rm K}$, determined from the full-width half maximum (FWHM) of the calculated spectra, is too large, (ii) the calculated spectra naturally lack the additional vibrational features that are observed in experiment, and (iii) the exclusive coupling of the STM tip to the LUMO orbital NTCDA cannot explain the marked differences between the $dI/dV$ spectra measured in the center and at the CH-edge of the molecules. While it is clear how shortcomings (ii) and (iii) can be addressed, namely by reverting to the general tunneling theory of section \[sec:tunnel-current\] that includes both inelastic tunneling and tunneling interference, we still need to identify a mechanism that is able to reduce the Kondo temperature below what is expected from standard theory without electron-phonon interaction as sketched out in the previous section. In section \[sec:experiment-NTCDA vibrational side bands\], we have reported the observation of two sets of vibrational features in our STS spectra. At the same time, in our theoretical analysis of sections \[sec:tunnel-current-general\] and \[sec:Modeling the system\] we have identified three distinct mechanisms in which vibrations may influence the differential conductance spectra measured in STS: a vibration-induced change of the tunnel coupling of the STM tip to an orbital of the system S, as well as an electron-phonon coupling purely within the system S, the latter either of conventional of unconventional Holstein type. The coincidence of observing in experiment both a reduced Kondo temperature and vibrational features that possibly result from an electron-phonon coupling in the system S suggests that the two observations might be connected. We therefore briefly explore the possibility that the electron-phonon coupling within the system S influences, and in particular reduces, its Kondo temperature. A finite electron-phonon coupling $|\lambda_d|>0$ (see Eq. \[eqn:AndersonHolstein\]) as in the conventional Holstein model [@HewsonMeyer02; @GalperinRatnerNitzan2007] generates a reduction of $U\to U^d_{\rm eff}=U-\lambda^2_d/\w_0$ [@Mahan81; @GalperinRatnerNitzan2007; @EidelsteinSchiller2013; @JovchevAnders2013] and would thus lead to an enhancement of the Kondo temperature. Only if $U^d_{\rm eff}<0$ the width of zero-frequency peak is rapidly reduced and the charge Kondo regime is entered [@HewsonMeyer02] – see the discussion in the literature [@HewsonMeyer02] or in Sec. \[sec:anti-adiabatic-regime-spectrum\]. However, for the case of NTCDA/Ag(111) the DFT calculation excludes a vibrational coupling of any local molecular vibrational mode to the molecular orbital, i. e. demands $\lambda_d\approx 0$ and, therefore, rules out the conventional Holstein model, both as a source of the sharpening of the Kondo peak and as a source of the vibrational features in the STS spectrum. Finally, a coupling of the phonon displacement to the hybridization $V_{\k}$ is known to lead to an enhancement of the effective hybridization [@Cornaglia2005], as well as reduce the local Coulomb interaction [@Grewe84] and thus an increase of the Kondo temperature, which can also be analytically derived by a employing a Lang-Firsov transformation [@LangFirsov1962]. In contrast, a possibility to reduce the Kondo temperature is provided by an unconventional Holstein term which linearly connects the charging energy of the effective local substrate orbital $c_{0\sigma}$ to one of the molecular vibrational modes. In section \[sec:modelling\_the\_system\_vibrations\] above, we have included such a term, parameterized by the coupling constant $\lambda_c$, in Eq. . The physical idea behind this term is that a molecular vibration perpendicular to the substrate can induce a local potential change that shifts the single particle energy of local substrate orbitals as function of the displacement. Such an unusual Holstein coupling has been investigated in the context of the periodic Anderson model to provide a microscopic mechanism for the Kondo volume collapse [@PAM-E-PHON2013], which is believed to be the origin of structural $\ensuremath{\gamma}\ensuremath{\rightarrow}\ensuremath{\alpha}$ phase transition in Cerium [@KondoVolumKollapse1982]. It offers the possibility to reduce the width of the equilibrium Kondo resonance in a straightforward manner: In analogy to the discussion before, the Holstein coupling induces an attractive (negative) contribution to the local Coulomb interaction $U^c$ which is initially zero in an uncorrelated free conduction band. Consequently, the singly occupied spin-degenerate states are energetically separated from the lower-lying empty and doubly occupied states of $c_{0\sigma}$. This energy separation reduced the charge fluctuations with the LUMO. For a large $\lambda_c$, this reduces the hybridization $\Gamma \to \Gamma_{\rm eff}$ between the local orbital $d$ and the substrate $c_{0\sigma}$, thus providing a mechanism for the reduction of the Kondo temperature by bipolaron formation [@PAM-E-PHON2013]. Before applying this model to the NTCDA/Ag(111) system, we investigate it in detail in the next two sections, first regarding its influence on the Kondo temperature and second regarding the spectral functions. To keep the analysis simple, we employ the assumption of particle-hole symmetry in the next two sections. ### Analysis of the Holstein coupling $\lambda_c$: The influence on the Kondo temperature {#sec:The influence on the Kondo temperature} To set the stage for the realistic description the NTCDA/Ag(111) system, we investigate the influence of the unusual Holstein coupling $\lambda_c$ between a vibrational mode of the molecule and a local substrate orbital on the particle-hole symmetric single impurity Anderson Hamiltonian Eq.  and its Kondo temperature. For clarity, we only include in Eq.  a featureless conduction band with a constant density of states $\rho_0=1/2D$, a spin-degenerate molecular orbital $d_{0\sigma}$ with single-particle energy $\epsilon_d=-U/2$ and a single vibrational mode $\omega_0$. The imaginary part of the hybridization function Eq. , $\Gamma_0=\Im(\Delta(\w-i0^+))=\pi V^2\rho_0$, is a constant over the whole band width $\pm D$ and serves as the natural unit for all model parameters. We also set all $\lambda^{\rm tip}_{\mu\nu}=0$ and hence only include the elastic contributions to the tunnel current. By fitting the differential conductance that is calculated by the NRG to the empirical formula [@GoldhaberGordon] introduced by Goldhaber-Gordon et al. [@GoldhaberGordon] $$\begin{aligned} \frac{d I}{d V}(V=0)=\frac{G_0}{\left[1+(2^{1/s}-1)(\frac{T}{T_{K}})^2\right]^s} , \label{eqn:Goldhaber}\end{aligned}$$ with $s=0.22$ for a spin-$\frac{1}{2}$ system, we obtain the Kondo temperature $T_{K}$ as function of $\lambda_c$. In Fig. \[fig:TK-Lc-a\] we plot the ratio $T_{K}(\lambda_c)/T_{K}(\lambda_c=0)$ as function of the Holstein coupling $\lambda_c$ for different phonon frequencies $\w_0$. The additional coupling of the vibrational mode to the surface orbital indeed leads to a reduction of the Kondo temperature with increasing $\lambda_c$. For a fixed coupling strength $\lambda_c$ the reduction of $T_{\rm K}$ decreases with increasing $\omega_0$. For a better understanding of the underlying mechanism, the inset shows the same data, but plotted as a function of the polaronic energy shift $E_{\rm p}= \lambda_c^2/\omega_0$. For phonon frequencies $\w_0 \simeq \Gamma_0$ and $\w_0>\Gamma_0$, the decrease of $T_{K}$ only depends on the polaron energy $E_{p}$. In this high-frequency or anti-adiabatic limit, the phonons can be integrated out and their main effect is to generate a negative effective $U$ in the local substrate orbital, $U^c_{\rm eff} \approx -2E_p$. For smaller frequencies, retardation effects play a role and we observe increasing deviations in the crossover regime to the adiabatic limit. ![Ratio $T_{K}(\lambda_c)/T_{K}(\lambda_c=0)$ calculated for the symmetric single impurity Anderson model plus unusual Holstein coupling $\lambda_c$ for a single vibrational mode $\omega_0$. The main panel shows the Kondo temperature as function of the coupling $\lambda_c$ for $U/\Gamma_0=10$ and different values for the vibrational frequency $\omega_0$, normalized at $\lambda_c=0$. The inset depicts the same quantity but plotted against the polaronic energy shift $E_p=\lambda_c^2/\omega_0$. []{data-label="fig:TK-Lc-a"}](fig14-TK-Lc_U-10_W0){width="45.00000%"} We have argued above in section \[sec:problem-and-strategy\] that the attractive $U^c_{\rm eff}$ acting on the local substrate electrons primarily suppresses the hybridization between the molecule and the substrate to an effective value $\Gamma_{\rm eff}$, which as a consequence reduces the Kondo temperature due to the increasing correlation measured by the ratio $U/\Gamma_{\rm eff}$. However, a more careful analysis reveals that the notion of an effective hybridization $\Gamma_{\rm eff}$ may be misleading, since the absolute height of the Kondo peak is usually pinned at $1/\pi \Gamma_0$. Additional correlations often only lead to a *narrowing* of the peak width, parametrized by a reduction of $T_{K}$, but not to a change of the height of the peak. In order to distinguish between between a peak narrowing and $\Gamma_{\rm eff}$, we define the latter by the orbital spectral function at zero frequency for a particle-hole symmetric Hamiltonian, $$\begin{aligned} \rho_{d_{0\sigma},d^\dagger_{0\sigma}}(0) &\equiv & \frac{1}{\pi \Gamma_\text{eff}}.\end{aligned}$$ Since the real part of the Green function must vanish at $\w=0$ in particle-hole symmetry, $$\begin{aligned} \label{equ:49-Gamma-eff} \Gamma_\text{eff} &=&\Gamma_0 + \Im[\Sigma_\sigma(-i0^+)] \end{aligned}$$ must hold using the general property $G_{d_{0\sigma}, d^\dagger_{0\sigma} }(z)=[ z-\e_{d \sigma} -\Delta(z) -\Sigma_\sigma(z)]^{-1}$, where we have divided the total self-energy of the molecular orbital $\Sigma_{\rm tot}(z)=\Delta_\sigma(z)+\Sigma_\sigma(z)$ into the hybridization-induced part $\Delta_\sigma(z)$ for the non-interacting problem and all correlation-induced and electron-phonon induced corrections $\Sigma_\sigma(z)$. Since the imaginary part of the self-energy $\Sigma_\sigma(z)$ vanishes for $T,\w\to 0$ in a local Fermi liquid in the standard case of a non-interacting conduction band, the Green’s function is pinned to a fixed value $\rho_{d_{0\sigma},d^\dagger_{0\sigma}}(0)=\frac{1}{\pi} (\pi V^2\rho_0)^{-1}=(\pi \Gamma_0)^{-1}$ independent of the model parameters [@Langreth1966; @YoshimoriZawadowski1982; @Anders1991]. The presence of the unusual Holstein coupling $\lambda_c$ however, leads to modifications of this picture. In appendix \[sec:EOM-GF\], we derive the exact analytic expression of the correlation-induced self-energy $\Sigma_\sigma(z)$ of the molecular orbital using the exact equation for motion (EOM) for the Green’s functions [@BullaHewsonPruschke98]. The result is $$\begin{aligned} \label{eq:self-energy} \Sigma_\sigma(z) &=& \frac{U F_\sigma(z) +\lambda_d M_\sigma(z) + \frac{ \lambda_c}{V_0}\Delta(z) N_\sigma(z) }{G_{d_{0\sigma},d^\dagger_{0\sigma}}(z)} ,\end{aligned}$$ with the definitions \[eq-def-corr-eom\] $$\begin{aligned} \label{equationa} F_\sigma(z) &= &G_{d_{0\sigma} n^d_{-\sigma},d^\dagger_{0\sigma}}(z),\\ M_\sigma(z) &= & G_{\hat X_0 d_{0\sigma},d^\dagger_{0\sigma}}(z), \label{equationb} \\ N_\sigma(z) &= & G_{\hat X_{0} c_{0\sigma} ,d^\dagger_{0\sigma}}(z). \label{equationc}\end{aligned}$$ We explicitly use Eq.  to obtain the Green’s function of the molecular orbital from the NRG solution which provides $F_\sigma(z)$, $M_\sigma(z)$, $N_\sigma(z)$ and $G_{d_{0\sigma},d^\dagger_{0\sigma}}(z)$ [@BullaHewsonPruschke98]. As shown by Hewson and Meyer [@HewsonMeyer02], the self-energy $\Sigma_\sigma(z)= [UF_\sigma(z) +\lambda_d M_\sigma(z)] /G_{d_{0\sigma},d^\dagger_{0\sigma}}(z)$ maintains Fermi liquid properties and its imaginary part vanishes for $T,\w\to 0$ for a coupling of the orbital to a free electron gas. This can be understood from the topology of a Feynman diagram expansion of these correlation functions independent of the analytic shape of $G_{d_{0\sigma},d^\dagger_{0\sigma}}(z)$. In the presence of a finite $\lambda_c$, this statement does not hold any longer: the imaginary part of the self-energy acquires a negative offset which we will quantify in the following. Applying the EOM to $F_\sigma(z)$ reveals that this composite Green’s function also contains additional self-energy corrections in the presence of a finite $\lambda_c$. Therefore, the self-energy contribution $\Sigma^U(z)= UF_\sigma(z) /G_{d_\sigma,d^\dagger_\sigma}(z)$ cannot be identified by the same skeleton expansion [@LuttingerWard1960] as for the $\lambda_c=0$ case. Another modification stems from the third term in the nominator of Eq.  , $$\begin{aligned} \Delta \Sigma(z) &=& \frac{ \lambda_c}{V_0 |G_{d_\sigma,d^\dagger_\sigma}(z)|^2}\Delta(z) N_\sigma(z) G_{d_\sigma,d^\dagger_\sigma}^*(z) .\end{aligned}$$ Assuming particle-hole symmetry and $T\to 0$, and using that the real part of $G_{d_\sigma,d^\dagger_\sigma}(z)$ as well as the real part of $N_\sigma(-i0^+)$ vanish yields $$\begin{aligned} \label{eq:27} \Delta \Sigma(-i0^+) &=& i \frac{ \lambda_c \Gamma_0}{V_0 \pi \rho_{d_{0\sigma},d^\dagger_{0\sigma}}(0)} \Im N_\sigma(-i0^+) .\end{aligned}$$ $N_\sigma(z)$ is an off-diagonal Green’s function and its spectral integral is zero. Therefore, spectrum has equal positive and negative spectral weight in different frequency regions. The NRG calculation shows that $\Im N_\sigma(-i0^+)<0$ for $\lambda_c>0$ and $\Im N_\sigma(-i0^+)\propto \lambda_c$ in leading order. ![Symmetric single impurity Anderson model plus unusual Holstein coupling $\lambda_c$ for a single vibrational mode $\omega_0$. Renormalized hybridization $\Gamma_0\rightarrow\Gamma_{\rm eff}(\lambda_c)$ as function of $\lambda_c$ for two different values for $\omega_0$ at $U/\Gamma_0=10$. The inset depicts the same quantity for $\omega_0/\Gamma_0=0.2$, $\lambda_c/\Gamma_0=0.35$ as function of the Coulomb interaction.[]{data-label="fig:TK-Lc-b"}](fig15-Geff-U-Lc){width="50.00000%"} Particle-hole symmetric demands $$\begin{aligned} G_{d_{0\sigma},d^\dagger_{0\sigma}}(-i0^+)&=& \frac{i}{\Gamma_\text{eff}} .\end{aligned}$$ We substitute Eq.  into Eq.  with $\lambda_d=0$, $$\begin{aligned} \Gamma_\text{eff} &=&\Gamma_0 \\ && \nonumber - \Gamma_\text{eff} \left[ U \Re F_\sigma(-i0^+) -\frac{ \lambda_c }{V_0} \Gamma_0\Im N_\sigma(-i0^+) \right] ,\end{aligned}$$ which we solve for the ratio $$\begin{aligned} \label{eq:gamma-eff} \frac{\Gamma_\text{eff}}{\Gamma_0} &=& \frac{1} { 1 + U \Re F(-i0^+)- \frac{ \lambda_c}{V_0}\Gamma_0 \Im N_\sigma(-i0^+) }\end{aligned}$$ A negative $\Im N_\sigma(-i0^+)$ in combination with a positive $\Re F(-i0^+)$ leads to a reduction of $\Gamma_{\rm eff}$ which is quadratic in $\lambda_c$ for small $\lambda_c$, since then $\Im N_\sigma(-i0^+)$ is proportional to $\lambda_c$. Clearly, the reduction $\Gamma_0 \to \Gamma_{\rm eff}$ is not only effected by $\lambda_c$, but also depends on $U$. Fig. \[fig:TK-Lc-b\] shows the dependence, as calculated by NRG, of the effective hybridization on the coupling $\lambda_c$ for two different vibrational frequencies $\omega_0$ and a fixed $U/\Gamma_0=10$. For a fixed coupling strength $\lambda_c$, the reduction of $\Gamma_{\rm eff}$ decreases with decreasing polaron energy $E_{p}$, as expected from the discussion in the context of Fig. \[fig:TK-Lc-a\], confirming the microscopic mechanism outlined above: the larger $E_{p}$, the more severe is the suppression of the hybridization and the stronger thus the reduction of the Kondo temperature. For weak electron-phonon coupling, we expect $\Im N_\sigma(-i0^+)\propto \lambda_c$, as confirmed by NRG calculations. Furthermore, the change in real part of correlation function $F_\sigma(z)$ must also depend quadratically on $\lambda_c$, because it scales with the polaron energy. Hence Eq.  predicts an analytic form $1/(1+\alpha \lambda^2_c)$ for $\Gamma_{\rm eff}/\Gamma_0$, which agrees well with the data presented in Fig. \[fig:TK-Lc-b\]. Moreover, $\Gamma_{\rm eff}/\Gamma_0$ should decrease linearly with increasing $U$ for small $U$, as an expansion of Eq.  in powers of $U$ shows. The inset in Fig. \[fig:TK-Lc-b\] confirms this prediction. ![Symmetric single impurity Anderson model plus unusual Holstein coupling $\lambda_c$ for a single vibrational mode $\omega_0$. $T_K$ plotted against the inverse of the renormalized hybridization. Different Coulomb interactions are indicated by different colors and different vibrational frequencies $\w_0$ by different points. For comparison we added the textbook expression for $T_K$ of the SIAM as the dashed line.[]{data-label="fig:TK-Lc-c"}](fig16-TK-Geff){width="50.00000%"} Finally, we address the question whether the change of $T_{K}$ could also be understood by using an effective SIAM *without* explicitly including the phonons, whose effect would then be accounted for summarily by a renormalized $\Gamma_{\rm eff}$. As we will show below, the answer is no, one also needs a renormalization $U\to U_{\rm eff}$ *in the molecular orbital*. Since it is possible to reproduce any $T_{\rm K}$ with an appropriate combination of $U$ and $\Gamma$, not much understanding would be gained if both parameters were left free. Therefore, we demand that $U_{\rm eff}=Uf(x)$ depends only via a universal function $f(x)$ on the ratio $x=\Gamma_0/\Gamma_\text{eff}$. Assuming the validity of the standard expression for the Kondo temperature [@KrishWilWilson80a], the ratio of the Kondo temperatures for fixed band widths but different hybridization strengths $\Gamma_\text{eff}$ is given by $$\frac{T_K^{\rm SIAM}(U_{\rm eff}=Uf(x),\Gamma_\text{eff})}{T_K^{\rm SIAM}(U,\Gamma_0)} =\frac{1}{\sqrt{x f(x)}} e^{ - \frac{\pi U}{8\Gamma_0}( x f(x)-1)} \label{eq:tk-ratio-gamma-eff}$$ which for a fixed initial value $U/\Gamma_0$ is only a function of $x$. In Fig. \[fig:TK-Lc-c\] we plot the NRG data of Fig. \[fig:TK-Lc-a\] as function of $x=\Gamma_0/\Gamma_\text{eff}$ and indeed observe universality: all data points for different phonon frequencies fall on top of a universal but $U$-dependent curve. However, a constant function $f(x)=1$, which would imply a $U_{\rm eff}=U$ that is not renormalized, yields a mismatch between the NRG results and Eq. , as shown for the case of $U/\Gamma=20$ by the dashed line in Fig. \[fig:TK-Lc-c\]. We obtain an excellent fit of the numerical data with a phenomenological universal function $f(x) = 1 + 0.21(x^{-2}-1)$ (thin dotted lines for the different values of $U$ in Fig. \[fig:TK-Lc-c\].) The inset shows $f(x)$ on the same interval. In conclusion, our NRG solution of a physical model comprising an electron-phonon coupling $\lambda_c$ between the molecular vibration and a local effective substrate orbital reveals a reduction of the Kondo temperature of the spin-$\frac{1}{2}$ degree of freedom in the molecular orbital. In the framework of a particle-hole symmetric single impurity Anderson model, this can be parametrized by a reduction of the hybridization between the molecular orbital and the substrate and a concurrent, but more moderate reduction of the intraorbital Coulomb repulsion $U$. The underlying mechanism is the generation of a negative $U^c$ in the local effective substrate orbital. In the antiadiabatic limit this negative $U^c$ is essentially given by the polaron energy. While it is intuitively clear that a negative $U^c$, by destabilizing the singly occupied state, will reduce the hybridization with the molecular orbital, the NRG shows that the screening of the intraorbital repulsion in the LUMO is also indirectly affected by a coupling of the vibrational mode to the local substrate orbital. We have established the following consequences of $\lambda_c$: (i) a reduction of $\Gamma_{\rm eff}$, corresponding to an increase of the Kondo peak height relative to $\Gamma_0$, (ii) a reduction of $T_K$ leading to a narrowing of the Kondo resonance as well as (iii) a parametrization of the Kondo temperature by replacing $\Gamma\to \Gamma_{\rm eff}$ as well as $U\to U_{\rm eff}=Uf(\Gamma_{\rm eff}/\Gamma_0)$ in the standard analytic expression for $T_K$. Although Eq.  does not hold in the particle-hole asymmetric case, the qualitative features will remain valid. ### Analysis of the Holstein coupling $\lambda_c$: Spectral and transmission functions Next, we investigate the influence of the unconventional Holstein coupling $\lambda_c$ on the various spectral functions $\rho(\w)$ that make up the elastic and inelastic transmission functions $\tau^{(0)}_\sigma$, $\tau^{(1)}_\sigma$ and $\tau^{(2)}_\sigma$. This prepares the comparison of the calculated to the experimental differential conductance spectra for the NTCDA/Ag(111) system. As in the previous section, we will consider a particle-hole symmetric scenario for simplicity. Then, the Holstein coupling $\lambda_c$ does not lead to a displacement of the harmonic oscillator $\nu=0$ with energy $\omega_0$, and $\langle \hat{X_0} \rangle=0$ is always fulfilled. Furthermore, we allow for tunneling into $M=2$ states $\mu$, $\mu'$, namely the molecular orbital (tunneling matrix element $t_{0\sigma}=t_{d}$) and the effective local substrate orbital ($t_{1\sigma}=t_{c}$) where we have dropped the spin dependency of the tunneling matrix elements assuming a non-magnetic tip. We thus explicitly include the possibility of a Fano interference in this section [@SchillerHershfield2000a]. ![Contributions to the elastic spectrum due to tunneling into the molecular orbital $d_{0\sigma}$ and the local surface orbital $c_{0\sigma}$, leading to the constituents (a) $\rho_{d_{0\sigma},d^\dagger_{0\sigma}}(\w)$, (b) $\rho_{c_{0\sigma},c^\dagger_{0\sigma}}(\w)$ and (c) $\rho_{d_{0\sigma},c^\dagger_{0\sigma}}(\w)$. The spectral functions have been calculated with NRG. We set $U/\Gamma_0=10$, $\omega_0/\Gamma_0=0.1$ and different colors indicate various unconventional Holstein couplings $\lambda_c$. []{data-label="NRG_fig2"}](fig17-Elastic-Lc){width="50.00000%"} The elastic part of the transmission function $\tau^{(0)}_\sigma(\omega)$ comprises three different contributions $$\begin{aligned} \label{eq:transmission_elastic-two_channel_model} \tau_\sigma^{(0)}(\omega)& =& t_{d}^2\, \rho_{d_{0\sigma},d^\dagger_{0\sigma}}(\w) +t_{c}^2 \rho_{c_{0\sigma}, c_{0\sigma}^\dagger}(\w) \nonumber\\ && +t_{d} t_{c}\big[ \rho_{c_{0\sigma},d^\dagger_{0\sigma}}(\w)+ \rho_{d_{0\sigma},c_{0\sigma}^\dagger}(\w)\big],\end{aligned}$$ stemming from the tunneling into the molecular orbital $d_{0}$ and into the effective local surface orbital $c_0$, introduced in Eq.  and implying $d_{1\sigma} =c_{0\sigma}$ in $H_T$, Eq. . The three relevant spectral functions are plotted versus frequency for two different values of $\lambda_c$ and a fixed $U$ in Fig. \[NRG\_fig2\]. For $ \rho_{d_{0\sigma},d^\dagger_{0\sigma}}(\w)$, displayed in panel (a), the narrowing of the Kondo resonance with increasing $\lambda_c$ is illustrated. The increase of the peak height is connected to the reduction of $\Gamma_{\rm eff}$, as discussed extensively in the previous section. The corresponding anti-resonance in $\rho_{c_{0\sigma}, c_{0\sigma}^\dagger}(\w)$ is clearly visible in panel (b) of Fig. \[NRG\_fig2\]. This anti-resonance can be associated with the contribution to the Kondo screening by the electrons in local substrate orbital. The mixed contribution $ \rho_{c_{0\sigma},d^\dagger_{0\sigma}}(\w)= \rho_{d_{0\sigma},c_{0\sigma}^\dagger}(\w)$ in panel (c) is an antisymmetric function and thus its integrated spectral weight vanishes. This contribution captures the interference between the two possible tunneling paths and generates Fano lineshapes in Eq. \[eq:transmission\_elastic-two\_channel\_model\]. We note that the low-frequency part of the all spectral functions in Fig. \[NRG\_fig2\] is governed by the same energy scale $T_{K}$ that is reduced with increasing $\lambda_c$. ![Contributions to the inelastic spectrum due to tunneling into the molecular orbital $d_{0\sigma}$, the local surface orbital $c_{0\sigma}$ and a coupling $\lambda^\text{tip}_{\mu\nu}$ of the vibrational mode $\omega_0$ to the STM tip, leading to the constituents (a) $\rho_{\hat{X}_0d_{0\sigma},\hat{X}_0d^\dagger_{0\sigma}}(\w)$, (b) $\rho_{\hat{X}_0d_{0\sigma},d^\dagger_{0\sigma}}(\w)$ and (c) $\rho_{\hat{X}_0d_{0\sigma},c^\dagger_{0\sigma}}(\w)$. The spectral functions have been calculated with NRG. We set $U/\Gamma_0=10$, $\omega_0/\Gamma_0=0.1$ and different colors indicate various Holstein couplings $\lambda_c$. []{data-label="NRG_fig3"}](fig18-Inelastic-Lc){width="50.00000%"} The constituents of the inelastic spectrum $$\begin{aligned} \label{eq:transmission_inelastic-two_channel_model} \tau^{(1)}_\sigma(\omega)&+&\tau^{(2)}_\sigma(\omega)= \lambda^{\rm tip} \Big\{\lambda^{\rm tip}t^2_{d} \rho_{\hat{X}_0 d_{0\sigma},\hat{X}_0d^\dagger_{0\sigma}} (\w)\nonumber \\ &+&t^2_{d} \big[ \rho_{ \hat{X}_0d_{0\sigma} ,d^\dagger_{0\sigma}}(\w) + \rho_{d_{0\sigma},\hat{X}_0 d^\dagger_{0\sigma}}(\w) \big] \\ &+&t_{d} t_{c}\big[ \rho_{ \hat{X}_0 d_{0\sigma} ,c_{0\sigma}^\dagger}(\w) + \rho_{c_{0\sigma},\hat{X}_0 d^\dagger_{0\sigma}}(\w) \big] \Big\} \nonumber\end{aligned}$$ are shown in Fig. \[NRG\_fig3\]. Here we have assumed that $\lambda^{\rm tip}_{\mu\nu}=0$ except for $\mu=0$ and $\nu=0$, i.e. only the molecular orbital $d_0$ (but not the local effective substrate orbital $c_0$) is coupled through the vibration $\hat X_0$ to the STM tip, with coupling constant $\lambda^{\rm tip}$. We note that $\rho_{\hat{X}_0d_{0\sigma},\hat{X}_0d^\dagger_{0\sigma}}(\w)$, displayed in panel (a) of Fig. \[NRG\_fig3\], can in principle also be obtained from Eq.  in the limit $\lambda_c=0$. It consists of two peaks at $\pm\omega_0$ that indicate the threshold for the excitation of a vibrational quantum by the tunneling electron. The smooth structure of the thresholds in the NRG spectrum is a consequence of broadening procedure in the NRG approach [@PetersPruschkeAnders2006; @WeichselbaumDelft2007; @BullaCostiPruschke2008]. Tracking the peak position of $\rho_{\hat{X}_0d_{0\sigma},\hat{X}_0d^\dagger_{0\sigma}}(\w)$ for increasing $\lambda_c$ reveals the well-understood renormalization of the phonon frequency $\omega_0^\prime(\lambda_c)$ in the adiabatic limit [@EidelsteinSchiller2013]. Furthermore, it is interesting to note that in the limit of a vanishing Holstein coupling $\lambda_c$, $\rho_{\hat{X}_0d_{0\sigma},\hat{X}_0d^\dagger_{0\sigma}} (\w)$ is the only non-zero contribution, because the inelastic terms that are linear in $\lambda^{\rm tip}$ require a non-zero electron-phonon coupling in the system S so the phonon number is not any longer a conserved quantity. ### Strategy for matching the experimental and theoretical spectra for NTCDA/Ag(111) We return to the NTCDA/Ag(111) system and show how the tunneling theory of section \[sec:tunnel-current-general\], the unconventional Holstein model (section \[sec:modelling\_the\_system\_vibrations\]) and the ab-initio input to the NRG (section \[sec:Input from ab-initio calculations and model without electron-phonon coupling\]) can be combined to match the experimental spectra. In the first step (section \[Holstein coupling for NTCDA/Ag(111)\]) we adjust the unconventional Holstein coupling $\lambda_c$ such that the zero-bias peak in the experimental differential conductance spectra is correctly reproduced, irrespective of the spectral signatures at finite voltages. This procedure can indeed reduce the simulated Kondo temperature sufficiently to achieve a match to the experimental Kondo temperature. In the second step (section \[sec:NRG-7\]), we focus on the inelastic parameters $\lambda_{\mu\nu}^{\rm tip}$ which parameterize the change of the tunnel coupling of the STM tip to the orbital $\mu$, induced by vibration $\nu$, and govern the differential conductance spectra at higher energies. Both coupling mechanisms together allow modeling the differential conductance spectra in excellent agreement with experiment, as we will demonstrate in section \[NRG results for NTCDA/Ag(111)\]. ### Holstein coupling $\lambda_c$ for NTCDA/Ag(111) {#Holstein coupling for NTCDA/Ag(111)} First, we need to identify the vibrational mode(s) which may couple to the local effective orbital $c_{0\sigma}$. To this end we use the free vibrational modes of the gas phase in the absence of the substrate as guidance. The energies $\omega_\nu$ of molecular eigenmodes in the relevant energy range are given in Table \[tab:vibrations\] of Sec. \[sec:experiment-NTCDA\], together with their irreducible representations. Two modes have B$_{\rm 3g}$ character, namely the ones at $\omega = 50.4\,\text{meV}$ and $\omega = 41.6\,\text{meV}$. The B$_{\rm 3g}$ eigenmodes describe specific out-of-molecular-plane vibrations, while the B$_{\rm 1u}$, A$_{\rm g}$ and B$_{\rm 1g}$ modes close in energy all correspond to in-plane vibrations. Since any mode that potentially contributes to the unconventional Holstein coupling $\lambda_c$ must change the local potential and thus shift the single particle energy of local substrate orbitals as function of the displacement, we can expect the relevant modes to involve displacements perpendicular to the surface. Thus, we can concentrate on the two B$_{\rm 3g}$ modes. Note that the symmetry considerations only provide guidance and are not strictly valid, since the $D_{2h}$ point group symmetry of the molecule in the gas phase is broken in the NTCDA/substrate system as well as the presence of the STM tip. Assuming that only one of the two possible B$_{\rm 3g}$ modes couples to the substrate, we calculate the Kondo temperature as function of the coupling strength $\lambda_c$ for both modes individually, using the electronic ab-initio parameters as input for the NRG calculations. The result is depicted in Fig. \[NRG\_fig4a\] for both types of molecules and both vibrational modes. A substantial narrowing is achieved for $\lambda_c>100\,\text{meV}$, whence the polaron energy $E_p$ exceeds the hybridization strength of $\Gamma^\text{dark}(0)=190\,\text{meV}$ and $\Gamma^\text{bright}(0)=165\,\text{meV}$ at the Fermi energy. Therefore, by fixing the appropriate value of $\lambda_c$ the NRG-calculated spectral width of the Kondo peak can be matched to the experimental findings, irrespective of which of the two B$_{\rm 3g}$ modes is used. ![NRG results for the Kondo temperature $T_K$ of the bright/on-top and dark/bridge molecules as a function of the unconventional Holstein coupling $\lambda_c$ for two different vibrational energies $\w_0=41.6,\text{meV}, 50.4,\text{meV}$. []{data-label="NRG_fig4a"}](fig19-TK_Lc_BridgeTop){width="50.00000%"} ### Tip-system coupling $\lambda^{\rm tip}_{\mu\nu}$ for NTCDA/Ag(111) {#sec:NRG-7} The unconventional Holstein coupling $\lambda_c$ substantially reduces the Kondo temperature and thus improves the overall agreement of the experimental differential conductance spectrum with the combined ab-initio and NRG spectrum, but it does not explain the additional features in the $dI/dV$ spectra at finite frequencies, most notably the ones at $+(47.0 \pm 0.3)$mV and $-(51.5 \pm 0.3)$mV which we have attributed to inelastic tunneling processes (section \[sec:experiment-NTCDA vibrational side bands\]). Since in experiment these features are linked to the evolution of the zero-bias anomaly (Fig. \[fig:fig11\]), we propose that they are incarnations of the so-called vibrational Kondo replica [@Kondo_vib_SM; @kondo_vib_bjunc; @kondo_vib_bjunc2; @kondo_vib_bjunc3; @vib_kondo_stm; @vib_kondo_stm2; @vib_kondo_stm3] and related to half of the Kondo peak shifted by $\pm \w_{\rm eff}$ as suggested by Eq.  (see the discussion of the second limiting case at the end of section \[sec-lambda-0\]). The features at $+(47.0 \pm 0.3)$mV and $-(51.5 \pm 0.3)$mV thus very likely appear in the differential conductance spectrum as a result of inelastic tunneling in the limit of a free phonon mode. The relevant coupling is the vibration-induced change of the tunneling matrix element, i.e. the tip-system coupling $\lambda^{\rm tip}_{\mu\nu}$, and no electron-phonon coupling in the system S is required. This raises the question which electronic states $\mu$ and which vibrational modes $\nu$ may take part in the tip-system coupling $\lambda^{\rm tip}_{\mu\nu}$. We note that the presence of the tip at a general position above the system, as well as its possibly axially non-symmetric shape, breaks the symmetry of the molecule completely and principally allows the observation of any mode (no selection rules). The only condition is that the excitation of the vibrational mode modulates the tunnel matrix elements, as shown in Eq. . Most likely, only the atomic displacements on the molecule will lead to a relevant change of the tunneling matrix elements and, therefore, $\lambda^{\rm tip}_{\mu\nu}$ is restricted to a tunneling into the molecular orbital $d_{0\sigma}$, i.e. $\lambda^{\rm tip}_{\mu\nu}=\lambda^{\rm tip}_{0\nu}$. Moreover, since electron densities vary exponentially with perpendicular distance from the surface of the system S, it is plausible that the most significant couplings $\lambda^{\rm tip}_{0\nu}$ will involve vibrational displacements perpendicular to the substrate — among the modes in the relevant energy window listed in Table \[tab:vibrations\] these are the B$_{\rm 3g}$ modes. Thus, it turns out that the *same* modes which induce a local potential change on the surface and thereby provide the unconventional Holstein coupling $\lambda_c$ are those which also change the tunneling matrix elements from the tip into the system strongly. Because a large $\lambda_c$ (which is needed for one vibrational mode to reduce the Kondo temperature, see section \[Holstein coupling for NTCDA/Ag(111)\]) yields a sizable downward renormalization of the energy $\w_{\rm eff}$ at which the vibrational feature is observed, i.e. $\w_{\rm eff}< \w_0$ where $\w_0$ is the bare vibrational energy, and also because it broadens the inelastic spectral functions substantially, a second mode which only couples weakly within the system S is needed to generate the sharp steps at $+(47.0 \pm 0.3)$meV and $-(51.5 \pm 0.3)$meV in the total spectrum. Therefore, we require two distinct vibrational modes in the tunneling Hamiltonian in $\hat H_T$ ($N_\nu=2$). It is interesting to note that the mode at $\omega_0=50.4$meV in particular changes the tunneling matrix element very effectively, as a DFT analysis reveals. Fig. \[Figure\_Rohlfing\_vibrational\_mode\] shows the elongation pattern of this mode, calculated for gas-phase NTCDA. The outer C atoms at the CH edge exhibit a small, the ones in line with the carboxylic C atoms the largest vibrational amplitude, while for symmetry reasons the C atoms located on the long axis of the molecule have zero amplitudes. If we estimate the contribution of each mode to the transmission functions in Eq.  and separately by calculating in DFT the change of the LUMO local density of states (LDOS) on excitation of a single quantum of each vibration, we observe that the mode at $\omega_0=50.4$meV produces a substantial modulation of the LUMO LDOS *above the center of the molecule*, although both the LUMO and the vibrational mode amplitude itself vanish there. The reason is that the positive and negative lobes of the LUMO (green and red in Fig. \[fig:exp\_fig3\]), being of equal size for the non-vibrating molecule, are distorted differently on excitation of this mode. This breaks the symmetry in the center of the molecule, leading to a non-vanishing LUMO density of states in the *center of the vibrationally distorted molecule*. Evidently, this gives rise to a large relative change of the LDOS on excitation of the vibration, and thus to a large $\lambda^{\rm tip}_{00}$. Apart from explaining the large value of $\lambda^{\rm tip}_{00}$, this also elucidates why the coupling is sharply focussed in the center of the molecule. ### NRG results for NTCDA/Ag(111) {#NRG results for NTCDA/Ag(111)} In summary, we arrive at the following model to calculate differential conductance spectra for NTCDA/Ag(111): In accordance with DFT, we set $\lambda_d= 0$ for all vibrational modes of the NTCDA molecule, while one vibrational mode ($\omega_0=50.4\,\text{meV}$) exhibits a nonzero coupling $\lambda_c$ to the substrate orbital $c_{0\sigma}$ and two modes ($\omega_0=50.4\,\text{meV}$ and $\omega_1=41.6\,\text{meV}$) exhibit finite couplings $\lambda^{\rm tip}_{00}$ and $\lambda^{\rm tip}_{01}$ to the tip. The inelastic contribution stemming from the mode $\omega_1$ is calculated via Eq. , using the NRG-calculated spectral function $\rho_{d_{0\sigma},d^\dagger_{0\sigma}}(\w)$, whereas the one stemming from the mode $\omega_0$ is calculated within the NRG using the full formalism of Eq. and . For the bright (on-top) molecule we set $\lambda_c=200\text{meV}$, while $\lambda_c=220\text{meV}$ is selected for the dark (bridge) molecule. Since the DFT predicts a larger hybridization $\Gamma^\text{bridge}(0)=190\,\text{meV}$ compared to $\Gamma^\text{on-top}(0)=165\,\text{meV}$, a 10% enhancement of the electron-phonon coupling for the bridge molecule appears justified. While the tunneling Hamiltonian may include an arbitrary number of orbitals in the system S, we focus on a minimal configuration $M=2$ to include (i) the Kondo effect, (ii) the feasibility of a Fano resonance, (iii) and the possibility to change the differential conductance spectra when moving from the CH edge to the center of the molecules by adjusting the tunneling matrix elements without altering the system S itself, i.e. using fixed spectral functions. For simplicity, we use the same two orbitals for the $M=2$ tunneling channels which have already been introduced in the framework of the two electron-phonon coupling mechanisms. As mentioned before, the hybridization functions $\Gamma^\text{on-top}(\omega)$, $\Gamma^\text{bridge}(\omega)$ and the intra-LUMO Coulomb repulsion $U$ are provided as input to the NRG calculation by a combination of DFT and MBPT, the latter in the shape of the GdW approximation. ![Individual contributions to the (a) elastic and (b) inelastic spectrum for the bright (on-top) molecule that are combined in Fig. \[NRG\_fig4\] to fit the experimental $dI/dV$ spectra. Note that at the CH-edge we only include the spectra displayed with solid lines, whereas the Fano effect in the center of the molecule leads to additional contributions which are plotted as dashed lines in Fig \[NRG\_fig4b\]. Results for the dark (bridge) molecule are qualitatively the same. Parameters as in Fig. \[NRG\_fig4\]. []{data-label="NRG_fig4b"}](fig21-Bright-contributions){width="50.00000%"} ![Comparison between theoretical (blue) and experimental (black) $dI/dV$ spectra for both types of molecules (bright and dark), each measured at the CH-edge and at the center of a molecule. Two vibrational energies have been used, $\omega_0=50.4\,\text{meV}$ and $\omega_1=41.6\,\text{meV}$. Parameters are: (a) bright/on-top molecule, CH-edge: $\lambda_c= 200$meV for $\omega_0$, $\lambda_c= 0$ for $\omega_1$, $\lambda^{\rm tip}_{00}= 0.25$, $\lambda^{\rm tip}_{01}= 0.35$, $t_c/t_d=0$, $\rho_{\rm offset}=0$. (b) bright/on-top molecule, center: $\lambda_c= 200$meV for $\omega_0$, $\lambda_c= 0$ for $\omega_1$, $\lambda^{\rm tip}_{00}= 0.45$, $\lambda^{\rm tip}_{01}= 1.0$, $t_c/t_d=0.5$, $\rho_{\rm offset}=1.85$nS. (c) dark/bridge molecule, CH-edge: $\lambda_c= 220$meV for $\omega_0$, $\lambda_c= 0$ for $\omega_1$, $\lambda^{\rm tip}_{00}= 0.28$, $\lambda^{\rm tip}_{01}= 0.40$, $t_c/t_d=0$, $\rho_{\rm offset}=0.52$nS. (d) dark/bridge molecule, center: $\lambda_c= 220$meV for $\omega_0$, $\lambda_c= 0$ for $\omega_1$, $\lambda^{\rm tip}_{00}= 0.35$, $\lambda^{\rm tip}_{01}= 0.75$, $t_c/t_d=0.5$, $\rho_{\rm offset}=2.18$nS. []{data-label="NRG_fig4"}](fig22a-CH-site "fig:"){width="50.00000%"} ![Comparison between theoretical (blue) and experimental (black) $dI/dV$ spectra for both types of molecules (bright and dark), each measured at the CH-edge and at the center of a molecule. Two vibrational energies have been used, $\omega_0=50.4\,\text{meV}$ and $\omega_1=41.6\,\text{meV}$. Parameters are: (a) bright/on-top molecule, CH-edge: $\lambda_c= 200$meV for $\omega_0$, $\lambda_c= 0$ for $\omega_1$, $\lambda^{\rm tip}_{00}= 0.25$, $\lambda^{\rm tip}_{01}= 0.35$, $t_c/t_d=0$, $\rho_{\rm offset}=0$. (b) bright/on-top molecule, center: $\lambda_c= 200$meV for $\omega_0$, $\lambda_c= 0$ for $\omega_1$, $\lambda^{\rm tip}_{00}= 0.45$, $\lambda^{\rm tip}_{01}= 1.0$, $t_c/t_d=0.5$, $\rho_{\rm offset}=1.85$nS. (c) dark/bridge molecule, CH-edge: $\lambda_c= 220$meV for $\omega_0$, $\lambda_c= 0$ for $\omega_1$, $\lambda^{\rm tip}_{00}= 0.28$, $\lambda^{\rm tip}_{01}= 0.40$, $t_c/t_d=0$, $\rho_{\rm offset}=0.52$nS. (d) dark/bridge molecule, center: $\lambda_c= 220$meV for $\omega_0$, $\lambda_c= 0$ for $\omega_1$, $\lambda^{\rm tip}_{00}= 0.35$, $\lambda^{\rm tip}_{01}= 0.75$, $t_c/t_d=0.5$, $\rho_{\rm offset}=2.18$nS. []{data-label="NRG_fig4"}](fig22b-Center "fig:"){width="50.00000%"} Combining all elastic and inelastic contributions for the outlined SIAM-Holstein model of the NTCDA/Ag(111) system as displayed in Fig. \[NRG\_fig4b\], the experiment/theory comparison of the $dI/dV$ curves is displayed in Fig. \[NRG\_fig4\] for two STM tip locations on both the dark (bridge) and bright (on-top) molecules. As the figure shows, our minimal model yields a remarkable agreement between NRG and the experiment, with calculated Kondo temperatures $T_K^\text{bright}=103.6$K and $T_K^\text{dark}=140.4$K. In the comparison in Fig. \[NRG\_fig4\], all other contributions beyond the $M=2$ tunnel paths that are explicitly contained in our model in section \[sec:tunnel-current-general\] are included into a constant background $\rho_{\rm offset}$ that is added to the NRG-calculated $dI/dV$ curves. Its value is uniquely fixed by the condition that the maximum $dI/dV$ value of the zero-bias peak agrees between theory and experiment. It is reassuring that the experimental and theoretical values at larger bias ($\pm 70$mV) are also comparable. We stress that the NRG curves in Fig. \[NRG\_fig4\] are not fits in the mathematical sense. Rather, we have chosen a set of electron-phonon input parameters for the NRG calculations (in addition to the electronic *ab initio* parameters) which illustrate that our formalism of section \[sec:tunnel-current-general\] and model of section \[sec:Modeling the system\] are general enough to predict the generic features that are observed in the experimental differential conductance spectra of NTCDA/Ag(111). We have also tried a model in which the role of the two modes $\w_0$ and $\w_1$ is reversed. However, the resulting fit to the experimental data is significantly worse than the one in Fig. \[NRG\_fig4\], the prime reason being that the vibrational mode renormalisation through $\lambda_c$ shifts the inelastic features substantially down to $\w_{\rm eff} < \omega_\nu$ and it is therefore preferential to start with a larger bare vibration energy. With regard to the comparison of the NRG-calculated spectra to the experiments in Fig. \[NRG\_fig4\] it should be noted that each experimental spectrum is inevitably measured with a slightly different tip, and different STM tips generally lead to different differential conductance spectra on the same molecule. Within our theory this can be accounted for by the modification of the fictitious STM tip orbital in Eq.  that also changes the individual matrix elements $t_{\mu \sigma\sigma'}(\{\vec{R}_i\})$ in the approximation Eq. . This modification of the tunneling matrix elements leads to a different background current and to a slightly different mixing of the different frequency components of the spectral functions. We therefore account for different tips by adjusting $\rho_{\rm offset}$. Experimental spectra may moreover contain a small offset in the voltage scale which is usually gauged away by a calibration. In the present case, the Kondo peak is a common feature in experiment and theory, and we have adjusted its precise location to coincide with the NRG calculation. To this end, we have shifted the experimental curve such that it coincides with the NRG curve at the Kondo peak. We have shifted the experimental curves rather than the theoretical ones because the precise calibration of the experimental energy axis has an uncertainty anyway and the location of NRG resonance is determined via the Friedel sum rule [@Langreth1966; @YoshimoriZawadowski1982; @Anders1991] by LUMO orbital filling and the hybridization function, both strongly constrained by the DFT+MBPT input. The offset of the Kondo peak from zero bias is indicative of a particle-hole asymmetry. As such, the NRG-calculated offsets $+7.5$mV for the bright/on-top molecule and $+3.8$mV for the dark/bridge molecule in Fig. \[NRG\_fig4\] stem directly from the DFT+MBPT-calculated mean-field PDOS in Fig. \[NRG\_fig4aa\]. The as-measured experimental spectra exhibit Kondo peak positions of $+1.9$mV for the bright/on-top molecule and $-0.6$mV for the dark/bridge molecule. However, since we know that the inelastic features in the $dI/dV$ spectra should be located symmetrically around zero bias, we may use them for a calibration of the experimental bias voltage scale. This results in Kondo peak positions in the calibrated experimental spectra at $+4.2$mV for the bright/on-top molecule and $+1.7$mV for the dark/bridge molecule. This reveals that the NRG-calculation agrees with experiment regarding the *direction* of the particle-hole asymmetry for both NTCDA molecules on Ag(111) (although the NRG predicts a larger particle-hole asymmetry than found in experiment), as well as regarding the fact that the particle-hole asymmetry is stronger for the bright/on-top molecule. In this respect, the absolute different between experiment and NRG is only $1.2$ mV (NRG predicts a difference of $3.7$mV between the on-top and bridge molecules, while in experiment the corresponding difference between the bright and dark molecules is $2.5$mV). In Fig. \[NRG\_fig4\] we have shifted the as measured CH-edge spectra $5.6$ / $4.4$mV (bright and dark molecules) to the right to achieve coincidence of the Kondo peaks with the NRG. A shift of $2.55$mV would have established a symmetric distribution of the inelastic features in the experimental curve. As a consequence, the inelastic features of the experimental spectrum appear off-center in Fig. \[NRG\_fig4\] with respect to their NRG counterparts (which are symmetric by construction). After what has been said it is clear that this difference is not an issue of the electron-phonon coupling in our model, but rather of the overestimated particle-hole asymmetry of the NRG calculation (and, more fundamentally of the DFT+MBPT calculation). In principle, a more correct comparison of the NRG and experimental curves in Fig. \[NRG\_fig4\] would require shifting the experimental Kondo peak by a larger value ($+5.6$ / $+4.4$meV for bright/dark molecules, to correct for the too large prediction of the particle-hole asymmetry) than the rest of the spectrum at the inelastic features ($+2.25$ meV for both molecules, to achieve the physically motivated symmetry of the inelastic features). Essentially, these shifts are small and also reveals the overall uncertainties in our procedure matching theory and experiment. In conclusion, Fig. \[NRG\_fig4\] shows that our model of the NTCDA/Ag(111) system explains all generic features of the experimental differential conductance spectra: (i) the different Kondo temperatures of the bright and dark molecules (by different adsorption heights and correspondingly different hybridizations with the substrate), (ii) the smaller-than-expected Kondo temperatures of both molecules including their absolute values (by electron-phonon coupling with an effective local substrate orbital through an unconventional Holstein term), (iii) the strong threshold features at approximately at $+(47.0 \pm 0.3)$mV and $-(51.5 \pm 0.3)$mV including their asymmetric peak shapes (by inelastic tunneling involving a free phonon including the replication of half of the Kondo peaks), (iv) the weak shoulders at lower bias (by inelastic tunneling involving the coupled vibration that is also responsible for the reduction of the Kondo temperatures), (v) the marked difference of the spectra at the CH edge and in the center of the molecules (by quantum interference between tunneling path into a molecular orbital and into the effective local substrate orbital which is also implicated in the unconventional Holstein coupling), (vi) the strong concentration of the inelastic tunneling in the center of the molecule (by the quantum interference and the symmetries of the involved modes), (vii) the offset of the Kondo peak to positive bias voltages in the calibrated spectra (by a particle-hole asymmetry in the PDOS of the NTCDA LUMO adsorbed on Ag(111)), (viii) the fact that the Kondo peak of the bright molecule appears at slightly larger bias voltages (by the stronger particle-hole asymmetry of the bright molecule). STS in the anti-adiabatic regime {#sec:STS-anti-adiabatic-regime} -------------------------------- ![Spectral function $\rho_{d_\sigma,d^\dagger_\sigma}(\w)$ of the molecular orbital for the particle-hole symmetric anti-adiabatic regime. (a) and (b) spectral evolution for increasing $\lambda_d> \lambda_{d,c}$. The $\lambda_d=0$ is added for comparison. (c) the spectral data of panel (b) plotted on a larger energy interval. Parameters: $\rho=const$, $D/\Gamma_0=10$, $U/\Gamma_0=-2\e_{d}/\Gamma_0=10$. []{data-label="NRG_fig15"}](fig23a-Kondo-Ld){width="50.00000%"} ![Spectral function $\rho_{d_\sigma,d^\dagger_\sigma}(\w)$ of the molecular orbital for the particle-hole symmetric anti-adiabatic regime. (a) and (b) spectral evolution for increasing $\lambda_d> \lambda_{d,c}$. The $\lambda_d=0$ is added for comparison. (c) the spectral data of panel (b) plotted on a larger energy interval. Parameters: $\rho=const$, $D/\Gamma_0=10$, $U/\Gamma_0=-2\e_{d}/\Gamma_0=10$. []{data-label="NRG_fig15"}](fig23b-Kondo-Ld_W0-1_Spektrum.eps){width="50.00000%"} In this section we focus on the conventional setup ($\lambda_d>0$) [@galperinNitzanRatner2006; @EidelsteinSchiller2013; @JovchevAnders2013] in anti-adiabatic regime and neglect the unconventional coupling of a local phonon mode to the substrate, i. e. $\lambda_c=0$. In the anti-adiabatic regime polaron energy $E_p$ exceeds the hybridization strength, i.e. $E_p=\lambda^2_d/\w_0 >\Gamma(0)$. For simplicity, we only consider a featureless symmetric conduction band with a constant density of states to separate the many-body effects from single-particle energy shifts induced by particle-hole asymmetric hybridization functions. ### Equilibrium electronic spectra {#sec:anti-adiabatic-regime-spectrum} We review the evolution of the molecular orbital equilibrium spectral properties [@HewsonMeyer02] with increasing electron-phonon coupling $\lambda_d$. The spectral function $\rho_{d_\sigma,d^\dagger_\sigma}(\w)$ for different values of $\lambda_d$ and a constant conduction band density of states is shown for a fixed $U/\Gamma_0=10$ and particle-hole symmetry in Fig. \[NRG\_fig15\]. In order to obtain sharp spectral features, we averaged over $N_z=30$ z-values in the NRG calculation and set the NRG broadening parameter to $b=0.2$ – see Ref. [@AndersSchiller2006; @PetersPruschkeAnders2006; @WeichselbaumDelft2007; @BullaCostiPruschke2008] for the technical details. The width of the zero-frequency resonance changes non-monotonically with $\lambda_d$. The initial width of the Kondo resonance increases (not shown here) and, after reaching a maximum, it decreases again with increasing $\lambda_d$. Small shoulders develop symmetrically around the zero-bias resonance that evolves into two separated peaks as clearly seen in Fig. \[NRG\_fig15\](b). Simultaneously, the width of the zero-frequency peak rapidly declines. The electron-phonon interaction generates an attractive contribution to electron-electron interaction [@Mahan81; @LangFirsov1962] that is related to the polaron energy and renormalizes the bare value of $U\to U_{\rm eff} = U -2 E_p= U-2\lambda^2_d/\w_0$ [@HewsonMeyer02; @Chowdhury2015; @KleineAnders2015]. $U_{\rm eff} $ vanishes at a critical value $\lambda_{d}^c= \sqrt{U\w_0/2}$ and changes its sign to an attractive interaction upon further increase of $\lambda_{d}$. The spectral properties can be understood in terms of an $U_{\rm eff}$ [@HewsonMeyer02]. Starting from the purely electronic problem at $\lambda_d=0$, added as a black line to Fig. \[NRG\_fig15\] as a reference spectrum, the decrease of $U_{\rm eff}$ with increasing $\lambda_d$ leads to an increasing Kondo temperature up to $U_{\rm eff}\approx \pi\Gamma_0$. The zero-frequency peak width monotonically grows up to this point. The zero-frequency peak width approached its largest values for $\lambda_d/\Gamma_0=2.24$ which is roughly a factor 2 larger than the value for $\lambda_d=0$ as shown in Fig. \[NRG\_fig15\](a). Once $\lambda_d$ exceeds $\lambda_{d}^{c}$, the system entered the attractive $U$ regime at low frequencies [@HewsonMeyer02; @Chowdhury2015; @KleineAnders2015] which is governed by a bi-polaron formation. The spectral properties for this regime are shown in Fig. \[NRG\_fig15\](b) and (c). There, the spin Kondo physics is replaced by a charge-Kondo effect with a rapidly decreasing low-temperature scale $T_K^c$ under further increasing of $\lambda_d> \lambda_{d}^{c}$. The spectra develop two shoulders when increasing $\lambda_d$ that are located approximately at $\pm U_{\rm eff}/2$. As depicted in Fig. \[NRG\_fig15\](b), this shoulders grow into symmetric side peaks once $|U_{\rm eff}|$ exceed the charge Kondo scale, i. e. $T_K^c < |U_{\rm eff}|$. Since the phonon frequency $\w_0$ is of the order of the charge fluctuation scale $\Gamma_0$ and smaller than $U$, the concept of an effective $U_{\rm eff}$ is only useful at low energies. In terms of the renormalization group approach [@Wilson75], $U$ becomes frequency dependent in the presence of the electron-phonon interaction and flows from its bare high energy value to $U_{\rm eff}$ for $|\w|\ll \omega_0$. Therefore the high energy features of the spectra depicted in Fig. \[NRG\_fig15\](c) are only moderately modified: the original charge excitations around $\pm U/2$ are renormalized slightly to smaller values which indicate that the renormalization of $U\to U_{\rm eff}$ has set in very moderately at high frequencies $\w \approx U/2$. Once the flow of $U_{\rm eff}$ has converged, the spectral developed additional new peaks: in addition to the slightly shifted high-energy charge fluctuation peaks located at $\w\approx \pm U/2$ additional low-frequency peaks located around $\pm U_{\rm eff}$ develop leading to a much richer spectrum as depicted in Fig. \[NRG\_fig15\]. As demonstrated in Fig. \[NRG\_fig15\](b), the low-frequency side peaks evolves with $U_{\rm eff}$. In order to avoid entering the negative $U_{\rm eff}$ regime, one could fix $U_{\rm eff} =const$ by adjusting the bare $U$ of the model upon increasing $\lambda_d$. However, with increasing $\lambda_d$ the renormalization of $\Gamma_0\to\Gamma_{\rm eff}\approx \Gamma_0\exp[-\lambda_d^2/\w_0^2 f(\lambda_d/\w_0)]$ reduces rapidly the charge fluctuation scale [@HewsonMeyer02] in the strong coupling regime. The reduction factor $\exp[-\lambda_d^2/\w_0^2]$ is generated by the local polaron formation and can be understood via the Lang-Firsov transformation [@Mahan81; @LangFirsov1962], while the scaling function $f(\lambda_d/\w_0)$ accounts for additional reduction of $\Gamma_{\rm eff}$ due to the softening of the phonon mode [@HewsonMeyer02]. Consequently, the Kondo temperature reduces rapidly once $E_p$ exceeds $\w_0$ suppressing the Kondo effect at finite temperature. We investigated this limit but since the spectral functions qualitatively do not differ much from those presented above, we spare the rather repetitive analysis. ![Spectral functions for local particle-hole asymmetry. (a) small $\Delta\e/\Gamma_0=0.004$ and (b) $\Delta\e/\Gamma_0=0.01$. NRG parameters as in Fig. \[NRG\_fig15\]. []{data-label="NRG_fig-ph-asymmetry"}](fig24-AsymSpectra_Ld){width="50.00000%"} A comment is in order with regards to particle-hole asymmetry. While for particle-hole symmetry, the resonance in the spectral function remains pinned to zero-frequency, a particle-hole asymmetry allows for a continuous change of the scattering phase [@Langreth1966; @YoshimoriZawadowski1982] of the low energy quasiparticles. In order to understand the spectra in this regime for $\lambda_d> \lambda_{d}^{c}$, we can perform a particle-hole transformation of one spin species to convert an attractive U back to a repulsive $U$ in the transformed model. Starting from the impurity Hamiltonian in the absence of an external magnetic field $ \e_{d\sigma} = \e_{d}$ and replacing $n_\uparrow = (1 - d_\uparrow d^\dagger_\uparrow) = 1 -\bar n_\uparrow$, where $\bar n_\uparrow$ is the number operator of the holes, we derive $$\begin{aligned} \sum_\sigma \e_{d\sigma} n^d_\sigma + U n^d_\uparrow n^d_\downarrow = \non \sum_\sigma \left(\frac{U}{2}-\sigma \Delta \e \right) \bar n^d_\sigma - U \bar n^d_\uparrow \bar n^d_\downarrow + \e_{d} \, ,\end{aligned}$$ where $\Delta \e = \e_d + U/2$ serves as a measure of the particle-hole asymmetry [@KrishWilWilson80b], and $\bar n^d_\downarrow= n^d_\downarrow$. A negative $U$ model describes the same physics as the positive $U$ model after the particle-hole transformation but with $\Delta \e$ acting as effective magnetic field. The increasing $(-U_{\rm eff})$ leads to a decreasing charge Kondo temperature $T^c_K$ since the hybridization $\Gamma_0\to\Gamma_{\rm eff}$ is also reduced [@Mahan81; @LangFirsov1962; @EidelsteinSchiller2013]. Consequently the dimensionless magnetic field $\Delta\e/T_K(\lambda_d)$ increasing with further increasing of $\lambda_d$. Therefore, we can understand the evolution of the spectra in Fig. \[NRG\_fig-ph-asymmetry\] for two values of $\Delta \e$ in terms of this analysis. The zero-bias Kondo resonance is shifted to a finite value $\Delta\e$, representing the effective magnetic field in the transformed model. Furthermore its peak high is increasingly reduced due to the destruction of the Kondo effect in a strong effective magnetic field. Therefore, the spectral properties shown in Fig. \[NRG\_fig-ph-asymmetry\] are consistent with those of an effective Anderson model in the attractive $U$ regime. ### Inelastic contributions After reviewing the present understanding of the electronic spectral function in the Anderson Holstein model [@HewsonMeyer02] in its anti-adiabatic, particle-hole symmetric as well as particle-hole anti-symmetric regime, we proceed by discussing the implications for a potential STS including elastic and inelastic contributions. We assume for simplicity that the STM tip only couples to the molecular orbital excluding Fano physics. In order to eliminate the coupling parameters that need to be adjusted for a specific experimental setup, we define the following two spectral functions $$\begin{aligned} \bar\rho^{(2)}(\w) &=& \frac{1}{(t_d \lambda^{\rm tip})^2} \tau^{(2)}(\w)\\ \bar\rho^{(1)}(\w)& = &\frac{1}{t^2_d \lambda^{\rm tip}} \tau^{(1)}(\w)\end{aligned}$$ that contain both inelastic terms. This eliminates the STM tip dependent prefactor and focuses only on the spectral features. ![All contributions to the STS spectra in the anti-adiabatic regime. (a) spectral function taken from Fig. \[NRG\_fig15\], (b) $\bar\rho^{(2)}(\w)$ for $\lambda_d$ stated in panel (a) and (c) $\bar\rho^{(1)}(\w)$ for $\lambda_d$ stated in panel (a). Parameters as in Fig. \[NRG\_fig15\]. []{data-label="NRG_fig16"}](fig25-Spectra_Ld){width="50.00000%"} The individual spectra contributing to the total STS are shown in Fig. \[NRG\_fig16\] for the different coupling constants $\lambda_d$. Panel (a) includes some of the data contained already in Fig. \[NRG\_fig15\] for comparison. Panel (b) of Fig. \[NRG\_fig16\] depicts the contribution to $\bar\rho^{(2)}(\w)$. We observe the same narrowing of the distance between the two peaks when increasing $\lambda_d$ as plotted in Fig. \[NRG\_fig3\]. Note however, that the data in Fig. \[NRG\_fig3\] were calculated for $\lambda_d=0$ and a finite $\lambda_c$ as well as a phonon frequency $\w_0 =0.1\Gamma_0$ which is ten times smaller than the charge fluctuation scale. While the peaks are located at $\pm \w_0$ in the weak coupling limit ($\lambda_d\to 0$), the energy difference between the two peak positions is significantly reduced in the anti-adiabatic regime. Common to both cases, the previously discussed limit, (i) $\lambda_d>0$ and $\lambda_c=0$ and (ii) the focus in this section $\lambda_d=0$ and $\lambda_c>0$, is the renormalization of the phonon propagator in the strong coupling limit. The charge susceptibility contributes to the phonon propagator as can be understood either in weak coupling derived from the Feynman diagram in Fig. \[fig:self-electron-phonon-diagram\] or in the atomic limit [@Mahan81; @LangFirsov1962; @HewsonMeyer02]. The softening of the phonon mode generates additional low frequency contributions to the correlated spectrum which is the origin of the peak narrowing observed in $\bar\rho^{(2)}(\w)$ as well as in the evolution of the inelastic spectrum $\bar\rho^{(1)}(\w)$ shown in panel (c) of Fig. \[NRG\_fig16\]. ![Combination of all three contributions for $\lambda_d/\Gamma_0=2.27$ and two values of $\lambda^{\rm tip}$. NRG parameters as in Fig. \[NRG\_fig15\]. []{data-label="NRG_fig18"}](fig26-Antiad_STS-Spectra){width="50.00000%"} After individually discussing the spectral contributions, we combine the results to a total STS spectrum in Fig. \[NRG\_fig18\]. We selected the spectra for the largest $\lambda_d$ in Fig. \[NRG\_fig16\]: $\lambda_d/\Gamma_0=2.27$. For this value $\rho_{d_\sigma,d^\dagger_\sigma}(\w)$ clearly shows side peaks associated with $U_{\rm eff}$ but not with $\pm \w_0$. With increasing $\lambda^{\rm tip}$, the STS becomes increasingly asymmetric due to the admixture of the odd function $\bar \rho^{(1)}(\w)$. The correlated spectral function $\bar\rho^{(2)}(\w)$ of a electron removal or addition with a simultaneous displacement $X$ of the harmonic oscillator only provides an incoherent background with a small gap at zero frequency remaining. In this case the elastic contributions stemming from the side peaks in $\rho_{d_\sigma,d^\dagger_\sigma}(\w)$ could be mistakenly attributed to inelastic contributions stemming from a fictitious phonon at frequency $\w_0' = |U_{\rm eff}|$. Summary and conclusion {#sec:conclusion} ====================== We extended the tunnel theory for STS that bridges between weak and strong electron phonon coupling in the system as well as includes the strong coupling limit of electronic degrees of freedom. It relies on an exact solution of all relevant system spectral functions of the system in the absence of the STM tip and requires only that the coupling to the tip remains weak so that the second order expansion in the tunneling matrix element is sufficient. Importantly, a Wick’s theorem is not required. One of the key incidence of our approach is the systematic derivation of the tunnel current operator from the charge conservation in the total system comprising the system S of interest and the STM tip. The analytic form of the current operator is determined by the tunneling Hamiltonian $H_T$ connecting the two parts of the total system. The strength of the approach is the treatment of all tunneling processes, elastic and inelastic contributions, on equal footing. In particular, our approach includes linear as well as quadratic contributions to the inelastic tunneling current. Neglecting the linear term is only applicable in the limit of vanishing electron-phonon coupling in the system. It becomes relevant in situations when the phonon mode not only enters $H_T$ but couples to the electronic degrees of freedom in the system S. We presented experimental STS for two different location of NTCDA molecules on Ag(111). Combining the LDA+MBPT with an NRG approach clarifies that the so-called bright molecule corresponds to the top molecule and the dark molecule corresponds to the bridge molecule in the ab-initio calculation. The projected LDA+MBPT spectrum of the LUMO as well as the calculated screened intraorbital Coulomb interaction enters the NRG as ab-initio parameters. Guided from the inelastic features of the experimental STS we added an extended Holstein term to the many-body calculation and were able to reproduce the zero-bias peaks seen in the experimental STS. The experimental differences in the peak width could be related to the different hybridzation strength of the LUMO orbital with the substrate in the bridge molecule and the top molecule as calculated by the LDA. In a second step, the different spectral functions calculated by the NRG were combined with sets of tunneling parameters to reproduce the experimental STS. Consistent with the influence of the LUMO electron density on the molecular motion of the B$_{3g}$ modes, the vibrational couplings in the center of the molecule are a bit larger than at the CH site. While at the CH the tunneling occurs mainly into the LUMO, a Fano mixing between the LUMO and the substrate orbital in the ratio of $2:1$ is found in accordance the phenomenological Fano fit of the experimental data. We presented a generalized tunneling theory of STM spectra what include inelastic vibrations processes. Calculating accurate spectral functions with an combined ab-inito DFT plus many-body approach including the NGR allows to reproduced the experimental STS in NTCDA molecules on Ag(111) and provide a deeper insight in this complex system. Our approach can also be extended to inelastic magnetic excitation processes during tunneling and opens new doors for our understanding of magnetic surfaces. F.B.A. acknowledges support from the Deutsche Forschungsgemeinschaft via project AN-275/8-1. F.S.T. and M.R. and acknowledge support from the Deutsche Forschungsgemeinschaft via the Collaborative Research Center SFB 1083, projects A12 and A13, respectively. Equation of motion for calculating the orbital Green function {#sec:EOM-GF} ============================================================= It is useful to derive a closed analytic expression for the self-energy of the molecular orbital GF [@Kolodzeiski2017] which is used to increase the precision of the NRG GF [@BullaHewsonPruschke98] as well as analyze the results. We consider the system Hamiltonian $H_S$ $$\begin{aligned} H_S&=& \sum_{\k\sigma} \e_{\k\sigma} c^\dagger_{\k\sigma} c_{\k\sigma} +\w_0 b^\dagger_0 b_0 + \sum_\sigma \e_{d\sigma} n^d_\sigma + U n^d_\uparrow n^d_\downarrow \non && + \sum_{\k\sigma} V_{\k} ( c^\dagger_{\k\sigma} d_\sigma + d^\dagger_\sigma c_{\k\sigma} ) \\ && \nonumber + \lambda_d \hat X_0 (\sum_\sigma n^d_\sigma - n_{d0}) +\lambda_c \hat X_0 (\sum_\sigma c^\dagger_{0\sigma} c_{0\sigma} -n_{c0})\end{aligned}$$ where we have defined $$\begin{aligned} c_{0\sigma} &=& \frac{1}{V_0} \sum_{\k} V_{\k} c_{\k\sigma} \\ V_0^2 &=& \sum_{\k} |V_{\k} |^2 \, .\end{aligned}$$ We start from the commutators $$\begin{aligned} \, [d_\sigma, H_S] &=& \e_{d\sigma}d_\sigma + U n^d_{-\sigma} d_\sigma +V_0 c_{0\sigma} +\lambda_d \hat X_0 d_\sigma \\ \, [c_{k\sigma}, H_S] &=& \e_{\k\sigma} c_{\k\sigma} +\lambda_c \hat X_0 \frac{V_k}{V_0} c_{0\sigma} + V_k d_\sigma \label{eq:commu-c0}\end{aligned}$$ and obtain the equation of motion (EOM) $$\begin{aligned} (z-\e_d) G_{d_\sigma,d^\dagger_\sigma}(z) &=& 1 + U F_\sigma(z) +\lambda_d M_\sigma(z) \\ && \nonumber + \sum_k V_k G_{c_{\k\sigma} ,d^\dagger_\sigma}(z)\end{aligned}$$ after introducing the notation $$\begin{aligned} F_\sigma(z) &=& G_{d_\sigma n_{-\sigma},d^\dagger_\sigma}(z) \\ M_\sigma(z) &=& G_{\hat X_0 d_\sigma,d^\dagger_\sigma}(z) .\end{aligned}$$ While the complex function $F_\sigma(z)$ contains the information about the local correlations between the electrons of different spins $\sigma$, the influence of the molecular vibration onto the equilibrium GF is account for by $M_\sigma(z)$ that also is relevant for the inelastic tunneling current – see Sec. \[sec:I-inelastic\]. In order to close the EOM, we use the commutator to derive $$\begin{aligned} (z- \e_{\k\sigma}) G_{c_{\k\sigma} ,d^\dagger_\sigma}(z) &=& V_k G_{d_\sigma,d^\dagger_\sigma}(z) \\ && \nonumber +\lambda_c \frac{V_k}{V_0} N_\sigma(z) .\end{aligned}$$ The off-diagonal composite correlation function $$\begin{aligned} N_\sigma(z) &=& G_{\hat X_0 c_{0\sigma} ,d^\dagger_\sigma}(z)\end{aligned}$$ accounts for the correlations between hybridization process and the vibrational displacement $\hat X_0$. We have seen its explicit importance for the renormalization of bare hybridization via Ref. . Defining $$\begin{aligned} \Delta_\sigma(z)&=& \sum_k \frac{V_k^2}{z- \e_{\k\sigma}}\end{aligned}$$ and using the standard parametrization of the GF in terms of self-energy corrections $\Sigma_\sigma$, $$\begin{aligned} G_{d_\sigma,d^\dagger_\sigma}(z) &=& \frac{1}{z-\e_d -\Delta_\sigma(z) - \Sigma_\sigma(z)}\end{aligned}$$ the self-energy can be expressed [@BullaHewsonPruschke98] as $$\begin{aligned} \label{eq:APP-self-energy} \Sigma_\sigma(z) &=& \frac{U F_\sigma(z) +\lambda_d M_\sigma(z) + \frac{ \lambda_c}{V_0}\Delta(z) N_\sigma(z) }{G_{d_\sigma,d^\dagger_\sigma}(z)}\end{aligned}$$ Since the NRG can calculate each individual Green function $F_\sigma(z), M_\sigma(z), N_\sigma(z)$ and $G_{d_\sigma,d^\dagger_\sigma}(z)$, Bulla et al. [@BullaHewsonPruschke98] have shown that replacing the GFs on the right side of by the NRG results yields a self-energy that becomes almost independent of the NRG discretization parameters and, therefore, is an accurate representation of the true self-energy for the continuum model. Eq.  is analytically exact and is also used in the main text to present an better analytical understanding of the numerical finding. 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--- abstract: | Let $n,k\in\mathbb{N}$ and let $p_{n}$ denote the $n$th prime number. We define $p_{n}^{(k)}$ recursively as $p_{n}^{(1)}:=p_{n}$ and $p_{n}^{(k)}=p_{p_{n}^{(k-1)}}$, that is, $p_{n}^{(k)}$ is the $p_{n}^{(k-1)}$th prime. In this note we give answers to some questions and prove a conjecture posed by Miska and Tóth in their recent paper concerning subsequences of the sequence of prime numbers. In particular, we establish explicit upper and lower bounds for $p_{n}^{(k)}$. We also study the behaviour of the counting functions of the sequences $(p_{n}^{(k)})_{k=1}^{\infty}$ and $(p_{k}^{(k)})_{k=1}^{\infty}$. author: - 'B[ł]{}ażej Żmija' title: A note on primes with prime indices --- [^1] Introduction ============ Let $(p_{n})_{n=1}^{\infty}$ be the sequence of consecutive prime numbers. In a recent paper [@MT] Miska and Tóth introduced the following subsequences of the sequence of prime numbers: $p_{n}^{(1)}:=p_{n}$ and for $k\geq 2$ $$\begin{aligned} p_{n}^{(k)}:=p_{p_{n}^{(k-1)}}.\end{aligned}$$ In other words, $p_{n}^{(k)}$ is the $p_{n}^{(k-1)}$th prime. They also defined $$\begin{aligned} {{\rm Diag}\mathbb{P}}:= & \{\ p_{k}^{(k)}\ |\ k\in\mathbb{N}\ \}, \\ \mathbb{P}_{n}^{T}:= & \{\ p_{n}^{(k)}\ |\ k\in\mathbb{N}\ \}\end{aligned}$$ for each positive integer $n$. The main motivation in [@MT] was the known result that the set of prime numbers is ($R$)-dense, that is, the set $\{\ \frac{p}{q}\ |\ p,q\in\mathbb{P}\ \}$ is dense in $\mathbb{R}_{+}$ (with respect to the natural topology on $\mathbb{R}_{+}$). It was proved in [@MT] that for each $k\in\mathbb{N}$ the sequence $\mathbb{P}_{k}:=(p_{n}^{(k)})_{n=1}^{\infty}$ is ($R$)-dense. This result might be surprising, because the sequences $\mathbb{P}_{k}$ are very sparse. In fact, for each $k$ set $\mathbb{P}_{k+1}$ is a zero asymptotic density subset of $\mathbb{P}_{k}$. On the other hand, it was showed, that the sequences $(p_{n}^{(k)})_{k=1}^{\infty}$ for each fixed $n\in\mathbb{N}$, and $(p_{k}^{(k)})_{k=1}^{\infty}$ are not ($R$)-dense. Results of another type that were proved in [@MT] concern the asymptotic behaviour of $p_{n}^{(k)}$ as $n\rightarrow\infty$, or as $k\rightarrow\infty$. In particular, as $n\rightarrow\infty$, we have for each $k\in\mathbb{N}$ $$\begin{aligned} p_{n}^{(k)}\sim n(\log n)^{k},\ \ \ \ p_{n+1}^{(k)}\sim p_{n}^{(k)}, \ \ \ \ \log p_{n}^{(k)}\sim \log n\end{aligned}$$ by [@MT Theorem 1]. Some results from [@MT] concerning $p_{n}^{(k)}$ as $k\rightarrow\infty$ are mentioned later. For a set $A\subseteq \mathbb{N}$ let $A(x)$ be its counting function, that is, $$\begin{aligned} A(x):=\# \left(A\cap [1,x]\right).\end{aligned}$$ Miska and Tóth posed four questions concerning the numbers $p_{n}^{(k)}$: 1. Is it true that $p_{k+1}^{(k)}\sim p_{k}^{(k)}$ as $k\rightarrow\infty$? 2. Are there real constants $c>0$ and $\beta$ such that $$\begin{aligned} \exp\mathbb{P}_{n}^{T}(x)\sim cx(\log x)^{\beta} \end{aligned}$$ for each $n\in\mathbb{N}$? 3. Are there real constants $c>0$ and $\beta$ such that $$\begin{aligned} \exp{{\rm Diag}\mathbb{P}}(x)\sim cx(\log x)^{\beta}? \end{aligned}$$ 4. Is it true that $$\begin{aligned} {{\rm Diag}\mathbb{P}}(x)\sim\mathbb{P}_{n}^{T}(x) \end{aligned}$$ for each $n\in\mathbb{N}$? The aim of this paper it to give answers to question B, C and D. The main ingredients of our proofs are the following inequalities: $$\begin{aligned} \label{ineqPrimes} n\log n<p_{n}<2n\log n.\end{aligned}$$ The first inequality holds for all $n\geq 2$, and the second one for all $n\geq 3$. For the proofs, see [@RS]. In Section \[Results\] we use (\[ineqPrimes\]) in order to show explicit bounds for $p_{n}^{(k)}$. In particular, for all $n>e^{4200}$ we have: $$\begin{aligned} \log p_{n}^{(k)}= & k(\log k+\log\log k+O_{n}(1)), \\ \log p_{k}^{(k)}= & k(\log k+\log\log k+O(\log\log\log k)),\end{aligned}$$ as $k\rightarrow\infty$, where the implied constant in the first line may depend on $n$, see Theorem \[MAIN\] below. In consequence, we improve the (in)equalities $$\begin{aligned} \lim_{k\rightarrow\infty}\frac{p_{n}^{(k)}}{k\log k}= & 1, \\ 1\leq \liminf_{k\rightarrow\infty}\frac{p_{k}^{(k)}}{k\log k}\leq & \limsup_{k\rightarrow\infty}\frac{p_{k}^{(k)}}{k\log k}\leq 2.\end{aligned}$$ that appeared in [@MT]. Then we show in Section \[Coro\] that the answers to questions B and C are negative (Corollary \[CoroBC\]), while the one for question D is affirmative (Theorem \[AsympEqThm\]). In fact, we find the following relation: $$\begin{aligned} \mathbb{P}_{n}^{T}(x)\sim{{\rm Diag}\mathbb{P}}(x)\sim\frac{\log x}{\log\log x}\end{aligned}$$ for all positive integers $n$. In their paper, Miska and Tóth also posed a conjecture, that we state here as a proposition, since it is in fact a consequence of a result that had already appeared in [@MT]. Let $n\in\mathbb{N}$ be fixed. Then $$\begin{aligned} \frac{p_{n}^{(k)}}{p_{k}^{(k)}}\longrightarrow 0\end{aligned}$$ as $k\longrightarrow\infty$. Let $k> p_{n}$. Then $$\begin{aligned} 0\leq \frac{p_{n}^{(k)}}{p_{k}^{(k)}}<\frac{p_{n}^{(k)}}{p_{p_{n}}^{(k)}}=\frac{p_{n}^{(k)}}{p_{n}^{(k+1)}}.\end{aligned}$$ The expression on the right goes to zero as $k$ goes to infinity, as was proved in [@MT Corollary 3]. It is worth to note, that primes with prime indices have already appeared in the literature, for example in [@BB] and [@BKO]. However, according to our best knowledge, our paper is the second one (after [@MT]), where the number of iterations of indices, that is, the number $k$ in $p_{n}^{(k)}$, is not fixed. Throughout the paper we use the following notation: $\log x$ denotes the natural logarithm of $x$, and for functions $f$ and $g$ we write $f\sim g$ if $\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=1$, $f=O(g)$ if there exists a positive constants $c$ such that $f(x)<cg(x)$, and $f=o(g)$ if $\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}=0$. Upper and lower bounds for $p_{n}^{(k)}$ {#Results} ======================================== In this Section, we find explicit upper and lower bounds for $p_{n}^{(k)}$. We start with the upper bound. \[lemUP\] Let $n\geq 9$. Then for each $k\in\mathbb{N}$ we have: $$\begin{aligned} p_{n}^{(k)}<2^{2k-1}\cdot n\cdot (k-1)!\cdot\big(\log(\max\{k,n\})\big)^{k}. \end{aligned}$$ In particular, $$\begin{aligned} p_{n}^{(k)}< \big(4\cdot k\log k\big)^{k}\end{aligned}$$ for $k\geq n$. We proceed by induction on $k$. For $k=1$ it is a simple consequence of (\[ineqPrimes\]). Then the second induction step goes as follows: let us denote $m:=\max\{n,k\}$. Observe that $(k-1)!<(k-1)^{k-1}<m^{k-1}$ and $4\log m<m$ for $m\geq 9$. Hence, $$\begin{aligned} p_{n}^{(k+1)}\leq & 2p_{n}^{k}\log p_{n}^{(k)}<2\cdot 2^{2k-1}\cdot n\cdot (k-1)!\cdot(\log m)^{k}\cdot \log\left[2^{2k-1}\cdot n\cdot (k-1)!\cdot (\log m)^{k}\right] \\ < & 2^{2k}\cdot n\cdot (k-1)!\cdot(\log m)^{k} \log\left[4^{k}\cdot m\cdot m^{k-1}\cdot (\log m)^{k}\right]=2^{2k}\cdot n\cdot (k-1)!\cdot(\log m)^{k} \log\left[4\cdot m\cdot \log m\right]^{k} \\ \leq & 2^{2k}\cdot n\cdot k!\cdot (\log m)^{k}\cdot \log[m]^{2}\leq 2^{2k+1}\cdot n\cdot k!\cdot (\log m)^{k+1}.\end{aligned}$$ The second part of the statement is an easy consequence of the first part and the inequalities $(k-1)!<k^{k-1}$ and $n\leq k$. In order to prove a lower bound for $p_{n}^{(k)}$ we will need the following fact. \[lemL\] Let $$\begin{aligned} L(x):=\left(\frac{x}{x+1}\right)^{x+1}\left(\frac{\log x}{\log (x+1)}\right)^{x+1}.\end{aligned}$$ Then we have $$\begin{aligned} L(x)>0.32627\end{aligned}$$ for all $x\geq 4200$. Observe, that the function $\left(\frac{x}{x+1}\right)^{x+1}$ is increasing. Indeed, if $$\begin{aligned} f(x):=\log\left(\frac{x}{x+1}\right)^{x+1}=(x+1)\left(\log x-\log (x+1)\right),\end{aligned}$$ then $$\begin{aligned} f'(x)=\log x-\log (x+1)+(x+1)\left(\frac{1}{x}-\frac{1}{x+1}\right)=\frac{1}{x}-\log \left(1+\frac{1}{x}\right)>0,\end{aligned}$$ where the last inequality follows from the well-known inequality $y>\log (1+y)$ used with $y=\frac{1}{x}$. Hence, we can bound $$\begin{aligned} \label{L1} \left(\frac{x}{x+1}\right)^{x+1}\geq \left(\frac{4200}{4201}\right)^{4201}\end{aligned}$$ for all $x\geq 4200$. Now we need to find a lower bound for $\left(\frac{\log x}{\log (x+1)}\right)^{x+1}$. Let us write $$\begin{aligned} \left(\frac{\log x}{\log (x+1)}\right)^{x+1}=\left[\left(1-\frac{1}{\frac{\log (x+1)}{\log (x+1)-\log x}}\right)^{\frac{\log (x+1)}{\log (x+1)-\log x}}\right]^{\log\left(1+\frac{1}{x}\right)^{x}\cdot\frac{1}{\log (x+1)}\cdot\left(1+\frac{1}{x}\right)}.\end{aligned}$$ At first, we prove that functions $g(t):=\left(1-\frac{1}{t}\right)^{t}$ and $h(x):=\frac{\log (x+1)}{\log (x+1)-\log x}$ are increasing. For the function $g(t)$ it is enough to observe, that $\log g(t)=f(t-1)$ and the function $f(x)$ is increasing. For the function $h(x)$ we have: $$\begin{aligned} h'(x)= & \frac{1}{(\log (x+1)-\log x)^{2}}\left[\frac{\log (x+1)-\log x}{x+1}-\log (x+1)\left(\frac{1}{x+1}-\frac{1}{x}\right)\right] \\ = & \frac{1}{(\log (x+1)-\log x)^{2}}\left[\frac{\log (x+1)}{x}-\frac{\log x}{x+1}\right] >0.\end{aligned}$$ The fact that the functions $g(t)$ and $h(x)$ are increasing, together with the properties $g(h(4200))\in (0,1)$ and $\log\left(1+\frac{1}{x}\right)^{x}<1$, give us $$\label{L2} \begin{aligned} \left(\frac{\log x}{\log (x+1)}\right)^{x+1}\geq & \big[g(h(4200))\big]^{\log\left(1+\frac{1}{x}\right)^{x}\cdot\frac{1}{\log (x+1)}\cdot\left(1+\frac{1}{x}\right)} > \big[g(h(4200))\big]^{\frac{1}{\log (x+1)}\cdot\left(1+\frac{1}{x}\right)} \\ \geq & \big[g(h(4200))\big]^{\frac{1}{\log (4200+1)}\cdot\left(1+\frac{1}{4200}\right)}=\left(\frac{\log 4200}{\log 4201}\right)^{\frac{4201}{4200}\cdot\frac{1}{\log 4201 -\log 4200}} \end{aligned}$$ for all $x\geq 4200$. Combining (\[L1\]) and (\[L2\]) we get the inequality $$\begin{aligned} L(x)\geq \left(\frac{4200}{4201}\right)^{4201}\left(\frac{\log 4200}{\log 4201}\right)^{\frac{4201}{4200}\cdot\frac{1}{\log 4201 -\log 4200}}\approx 0.3262768>0.32627.\end{aligned}$$ The proof is finished. In the next lemma we provide a lower bound for $p_{n}^{(k)}$. \[lemDOWN\] If $n>e^{4200}$, then for all $k\geq\lfloor\log n\rfloor$ we have $$\begin{aligned} p_{n}^{(k)}>\left(\frac{e\cdot k\log k}{\log\log n}\right)^{k}.\end{aligned}$$ First, let us observe that a simple induction argument on $k$ implies the inequality $$\begin{aligned} \label{ineqDOWN} p_{n}^{(k)}>n(\log n)^{k}.\end{aligned}$$ Indeed, for $k=1$ this follows from left inequality in (\[ineqPrimes\]). Using the same inequality we get also $$\begin{aligned} p_{n}^{(k+1)}>p_{n}^{(k)}\log p_{n}^{(k)}>n (\log n)^{k}\log(n (\log n)^{k})>n(\log n)^{k+1},\end{aligned}$$ and hence (\[ineqDOWN\]). Now we show that the inequality from the statement is true for $k=\lfloor \log n\rfloor$. Because of (\[ineqDOWN\]) it is enough to show: $$\begin{aligned} n(\log n)^{k}>\left(\frac{e\cdot k\log k}{\log\log n}\right)^{k},\end{aligned}$$ or equivalently, after taking logarithms we get $$\begin{aligned} \log n+\lfloor\log n\rfloor\log\log n>\lfloor\log n\rfloor+\lfloor\log n\rfloor\log\lfloor\log n\rfloor+\lfloor\log n\rfloor\log\log\lfloor\log n\rfloor -\lfloor\log n\rfloor\log\log\log n.\end{aligned}$$ This is equivalent to the inequality $$\begin{aligned} \big(\log n-\lfloor\log n\rfloor\big)+\lfloor\log n\rfloor\big(\log\log n-\log\lfloor\log n\rfloor\big)+\lfloor\log n\rfloor\big(\log\log\log n-\log\log\lfloor\log n\rfloor\big)>0,\end{aligned}$$ which is obviously true. In order to finish the proof, we again use the induction argument. The inequality from the statement of our lemma is true for $k=\lfloor \log n\rfloor$. Assume it holds for some $k\geq \lfloor\log n\rfloor$. Then by (\[ineqPrimes\]) and the induction hypothesis we get $$\begin{aligned} p_{n}^{(k+1)}> & p_{n}^{(k)}\log p_{n}^{(k)}>\left(\frac{e\cdot k\log k}{\log\log n}\right)^{k}\log \left(\frac{e\cdot k\log k}{\log\log n}\right)^{k}.\end{aligned}$$ It is enough to show that for all $n>e^{4200}$ and all $k\geq\lfloor\log n\rfloor$ we have $$\begin{aligned} \left(\frac{e\cdot k\log k}{\log\log n}\right)^{k}\log \left(\frac{e\cdot k\log k}{\log\log n}\right)^{k}>\left(\frac{e\cdot (k+1)\log (k+1)}{\log\log n}\right)^{k+1}.\end{aligned}$$ This is equivalent to $$\begin{aligned} k^{k+1}(\log k)^{k}\left[\log k+\log\left(\frac{e\log k}{\log\log n}\right)\right]>\frac{e}{\log\log n}(k+1)^{k+1}(\log(k+1))^{k+1}.\end{aligned}$$ Recall, that we assume that $k\geq \lfloor\log n\rfloor$. Thus $k^{e}>\log n$, that is, $e\log k>\log\log n$. Therefore, it is enough to show the following inequalities: $$\begin{aligned} k^{k+1}(\log k)^{k+1}>\frac{e}{\log\log n}(k+1)^{k+1}(\log(k+1))^{k+1},\end{aligned}$$ or equivalently $$\begin{aligned} \left(\frac{k}{k+1}\right)^{k+1}\left(\frac{\log k}{\log (k+1)}\right)^{k+1}>\frac{e}{\log\log n}.\end{aligned}$$ Notice that the left-hand side expression of the last inequality is equal to $L(k)$, where the function $L(x)$ is defined in the statement of Lemma \[lemL\]. If $n>e^{4200}$, then $k\geq \lfloor\log n\rfloor \geq 4200$, and Lemma \[lemL\] implies $L(k)>0.32627$. Therefore, if $N:=\max\left\{\lfloor e^{4200}\rfloor,\lceil e^{e^{e/0.32627}}\rceil\right\}=\lfloor e^{4200}\rfloor$, then for all $n>N$ and $k\geq \lfloor\log n\rfloor$ we have: $$\begin{aligned} L(k)>0.32627\geq \frac{e}{\log\log N}>\frac{e}{\log\log n}.\end{aligned}$$ This finishes the proof. Main results {#Coro} ============ We begin this section by a theorem that provides good information about asymptotic growth of $\log p_{n}^{(k)}$ for large fixed $n$, and for $\log p_{k}^{(k)}$ as $k\rightarrow\infty$. \[MAIN\] 1. Let $n>e^{4200}$. Then $$\begin{aligned} \log p_{n}^{(k)}=k(\log k+\log\log k+O_{n}(1)) \end{aligned}$$ as $k\rightarrow\infty$, where the implied constant may depend on $n$. 2. We have $$\begin{aligned} \log p_{k}^{(k)}=k(\log k+\log\log k+O(\log\log\log k)) \end{aligned}$$ as $k\rightarrow\infty$. If $n>e^{4200}$ and $k\geq n$, Lemmas \[lemUP\] and \[lemDOWN\] give us: $$\begin{aligned} \left(\frac{e\cdot k\log k}{\log\log n}\right)^{k}<p_{n}^{(k)}<\left(4\cdot k\log k\right)^{k}.\end{aligned}$$ After taking logarithms, we simply get the first part of our theorem. In order to get the second part, we need to put $n=k$ and repeat the reasoning. Now we give the answer to Question D. \[AsympEqThm\] For each $n\in\mathbb{N}$ we have $$\begin{aligned} {{\rm Diag}\mathbb{P}}(x)\sim \mathbb{P}_{n}^{T} (x)\end{aligned}$$ as $x\rightarrow\infty$. From [@MT Theorem 6] we know that $$\begin{aligned} \mathbb{P}_{m}^{T}(x)\sim \mathbb{P}_{n}^{T}(x)\end{aligned}$$ for each $m,n\in\mathbb{N}$. Therefore, it is enough to prove ${{\rm Diag}\mathbb{P}}(x)\sim \mathbb{P}_{n}^{T} (x)$ for some sufficiently large $n$. Let $n=\lfloor e^{4200}\rfloor +100$. We use the idea from the proof of [@MT Theorem 17]. Let $x$ be a large real number. Let $k$ be such that $p_{k}^{(k)}\leq x<p_{k+1}^{(k+1)}$. Then ${{\rm Diag}\mathbb{P}}(x)=k$. By [@MT Theorem 8] and Theorem \[MAIN\] above we have $$\begin{aligned} \frac{\mathbb{P}_{n}^{T}(x)}{{{\rm Diag}\mathbb{P}}(x)}\leq & 1+\frac{\log p_{k+1}^{(k+1)}}{k\log\log p_{n}^{(k)}}-\frac{\log p_{n}^{(k)}}{k\log\log p_{n}^{(k)}} \\ = & 1+\frac{(1+o(1))(k+1)\log (k+1)}{k\log \big[(1+o(1))k\log k\big]}-\frac{(1+o(1))k\log k}{k\log\big[(1+o(1))k\log k\big]} \\ = & 1+(1+o(1))\left(1+\frac{1}{k}\right)\frac{\log (k+1)}{\log k+\log\left[(1+o(1))\log k\right]}-(1+o(1))\frac{\log k}{\log k+\log\left[(1+o(1))\log k\right]}.\end{aligned}$$ The whole last expression goes to $1$ as $k$ goes to infinity. On the other hand, ${{\rm Diag}\mathbb{P}}(x)\leq \mathbb{P}_{n}^{T}(x)$ for $x\geq p_{n}^{(n)}$ and we get the result. The answers to Questions B and C will follow from our next result, which is of independent interest. \[AsympDiagP\] 1. Let $n\in\mathbb{N}$. Then $$\begin{aligned} \mathbb{P}_{n}^{T}(x)\sim\frac{\log x}{\log\log x}. \end{aligned}$$ 2. We have $$\begin{aligned} {{\rm Diag}\mathbb{P}}(x)\sim \frac{\log x}{\log\log x}. \end{aligned}$$ In view of Theorem \[AsympEqThm\], it is enough to show the statement for the function ${{\rm Diag}\mathbb{P}}(x)$. Let us fix an arbitrarily small number $\varepsilon >0$ and take a sufficiently large real number $x$ and find $k$ such that $p_{k}^{(k)}\leq x<p_{k+1}^{(k+1)}$. Then ${{\rm Diag}\mathbb{P}}(x)=k$ and by Lemmas \[lemUP\] and \[lemDOWN\] we have $$\begin{aligned} k^{k}<x<k^{(1+\varepsilon )k}.\end{aligned}$$ Let us write $x=e^{y}$. Then $$\begin{aligned} k\log k< y<(1+\varepsilon )k\log k.\end{aligned}$$ If $y$ is sufficiently large, this implies $$\begin{aligned} \label{ineqyk} (1-\varepsilon )\frac{y}{\log y}< k<(1+\varepsilon )\frac{y}{\log y}.\end{aligned}$$ Indeed, if $k\leq (1-\varepsilon )\frac{y}{\log y}$, then $$\begin{aligned} y<(1+\varepsilon)k\log k\leq (1-\varepsilon^{2})\frac{y}{\log y}\log\left((1-\varepsilon )\frac{y}{\log y}\right)<(1-\varepsilon^{2})\left(1-\frac{\log\log y}{\log y}\right)y,\end{aligned}$$ which is impossible. Similarly, if $k\geq (1+\varepsilon )\frac{y}{\log y}$, then $$\begin{aligned} y>k\log k\geq (1+\varepsilon )\frac{y}{\log y}\log\left((1+\varepsilon )\frac{y}{\log y}\right)>(1+\varepsilon )\left(1-\frac{\log\log y}{\log y}\right)y.\end{aligned}$$ The above inequality cannot hold if $y$ is sufficiently large. If we go back to $k={{\rm Diag}\mathbb{P}}(x)$ and $y=\log x$ in (\[ineqyk\]), we get $$\begin{aligned} (1-\varepsilon )\frac{\log x}{\log\log x}<{{\rm Diag}\mathbb{P}}(x)<(1+\varepsilon )\frac{\log x}{\log\log x}.\end{aligned}$$ The number $\varepsilon >0$ was arbitrary, so the result follows. \[CoroBC\] There do not exist constants $c>0$ and $\beta$ such that $$\begin{aligned} \exp\mathbb{P}_{n}^{T}(x)\sim cx(\log x)^{\beta}\end{aligned}$$ for some $n$, or $$\begin{aligned} \exp{{\rm Diag}\mathbb{P}}(x)\sim cx(\log x)^{\beta}.\end{aligned}$$ We prove the result only for ${{\rm Diag}\mathbb{P}}(x)$. The case of $\mathbb{P}_{n}^{T}(x)$ is analogous. Assume to the contrary, that $\exp{{\rm Diag}\mathbb{P}}(x)\sim cx(\log x)^{\beta}$ for some $c>0$ and $\beta$. Then $\exp{{\rm Diag}\mathbb{P}}(x)=(1+o(1))cx(\log x)^{\beta}$. This, after taking logarithms on both sides, implies ${{\rm Diag}\mathbb{P}}(x)=\log x+O(\log\log x)$, contradicting Theorem \[AsympDiagP\]. Acknowledgement {#acknowledgement .unnumbered} =============== I would like to express my gratitude to Piotr Miska, who informed me about the problem and helped to improve the quality of the paper. I would also like to thank Carlo Sanna, whose comments allowed me to simplify a part of the paper. [20]{} K. Broughan, R. Barnett, [*On the subsequence of primes having prime subscripts*]{}, Journal of Integer Sequences **12**, article 09.2.3, 2019. J. Bayless, D. Klyve, T. Oliveira e Silva, [*New bounds and computations on prime-indexed primes*]{}, Integers **13**: A43:1–-A43:21, 2013. P. Miska, J. Tóth, [*On interesting subsequences of the sequence of primes*]{}, preprint: arXiv:1908.10421. J. B. Rosser, L. Schoenfeld, [*Approximate formulas for some functions of prime numbers*]{}, Illinois J. Math. **6**, no. 1 , 64–94, 1962. doi:10.1215/ijm/1255631807. [^1]: During the preparation of the work, the author was a scholarship holder of the Kartezjusz program funded by the Polish National Center for Research and Development.

--- abstract: 'The effective theories for many quantum phase transitions can be mapped onto those of classical transitions. Here we show that the naive mapping fails for the sub-ohmic spin-boson model which describes a two-level system coupled to a bosonic bath with power-law spectral density, $J(\omega)\propto\omega^s$. Using an $\epsilon$ expansion we prove that this model has a quantum transition controlled by an [*interacting*]{} fixed point at small $s$, and support this by numerical calculations. In contrast, the corresponding classical long-range Ising model is known to display mean-field transition behavior for $0<s<1/2$, controlled by a [*non-interacting*]{} fixed point. The failure of the quantum–classical mapping is argued to arise from the long-ranged interaction in imaginary time in the quantum model.' author: - Matthias Vojta - 'Ning-Hua Tong' - Ralf Bulla date: 'Jan 20, 2005' title: | Quantum phase transitions in the sub-ohmic spin-boson model:\ Failure of the quantum–classical mapping --- Low-energy theories for certain classes of quantum phase transitions in clean systems with $d$ spatial dimensions are known to be equivalent to the ones of classical phase transitions in $(d+z)$ dimensions, where $z$ ist the dynamical exponent of the quantum transition [@book]. This mapping is usually established in a path integral formulation of the effective action for the order parameter, where imaginary time in the quantum problem takes the role of $z$ additional space dimensions in the classical counterpart. The tuning parameter for the phase transition, being the ratio of certain coupling constants in the quantum problem (where $T$ is fixed to zero), becomes temperature for the classical transition. For the quantum Ising model, where the transverse field can drive the system into a disordered phase at $T=0$, the quantum–classical equivalence in the scaling limit can be explicitly shown using transfer matrix techniques [@book]. While this formal proof is only applicable for [*short-range*]{} interactions in time direction, it is believed that it also holds for long-range interactions, which can arise upon integrating out gapless degrees of freedom coupled to the order parameter. (Counter-examples are phase transitions in itinerant magnets, where the elimination of low-energy fermions produces non-analyticities in the resulting order parameter field theory [@bkv].) A paradigmatic example is the spin-boson model [@Leggett; @Weiss], where an Ising spin (i.e. a generic two-level system) is coupled to a bath of harmonic oscillators: eliminating the bath variables leads to a retarded self-interaction for the local spin degree of freedom, which decays as $1/\tau^2$ in the well-studied case of ohmic damping. Interestingly, the same model is obtained as the low-energy limit of the anisotropic Kondo model which describes a spin-1/2 magnetic impurity coupled to a gas of conduction electrons [@yuval; @emery]. The purpose of this paper is to point out that the naive quantum–classical mapping can fail for long-ranged interactions in imaginary time even for the simplest case of $(0+1)$ dimensions and Ising symmetry. We shall explicitly prove this failure for the sub-ohmic spin-boson model, by showing that the phase transitions in the quantum problem and in the corresponding classical long-range Ising model fall in different universality classes. The spin-boson model is described by the Hamiltonian $${\cal H}_{\rm SB}=-\frac{\Delta}{2}\sigma_{x}+\frac{\epsilon}{2}\sigma_{z}+ \sum_{i} \omega_{i} a_{i}^{\dagger} a_{i} +\frac{\sigma_{z}}{2} \sum_{i} \lambda_{i}( a_{i} + a_{i}^{\dagger} ) \label{eq:sbm}$$ in standard notation. The coupling between spin $\sigma$ and the bosonic bath with oscillators $\{a_i\}$ is completely specified by the bath spectral function $$J\left( \omega \right)=\pi \sum_{i} \lambda_{i}^{2} \delta\left( \omega -\omega_{i} \right) \,,$$ conveniently parametrized as $$J(\omega) = 2\pi\, \alpha\, \omega_c^{1-s} \, \omega^s\,,~ 0<\omega<\omega_c\,,\ \ \ s>-1 \label{power}$$ where the dimensionless parameter $\alpha$ characterizes the dissipation strength, and $\omega_c$ is a cutoff energy. The value $s=1$ represents the case of ohmic dissipation, where a Kosterlitz-Thouless transition separates a delocalized phase at small $\alpha$ from a localized phase at large $\alpha$. These two phases asymptotically correspond to eigenstates of $\sigma_x$ and $\sigma_z$, respectively. In the following, we are interested in sub-ohmic damping, $0<s<1$ [@spohn; @KM]. The standard approach is to integrate out the bath, leading to an effective interaction $$\begin{aligned} {\cal S}_{\rm int} = \int d\tau d\tau' \sigma_z(\tau) g(\tau-\tau') \sigma_z(\tau')\end{aligned}$$ with $g(\tau) \propto 1/\tau^{1+s}$ at long times. Numerical renormalization group (NRG) calculations in Refs. , performed directly for the sub-ohmic spin-boson model, have established that a second-order quantum transition occurs for all $0<s<1$. Here we use an analytical renormalization group (RG) expansion, controlled by the small parameter $s$, to establish that the spin-boson transition at small $s$ is governed by an interacting fixed point with strong hyperscaling properties. This analytical result is supported by NRG calculations. In contrast, the transition in the classical Ising model is known to display mean-field behavior for $0<s<1/2$ [@fisher; @luijten]. [*Scaling and critical exponents.*]{} A scaling ansatz for the impurity part of the free energy takes the form $$F_{\rm imp} = T f(|\alpha-\alpha_c| T^{-1/\nu}, \epsilon T^{-y_\epsilon} ) \label{fscal}$$ where $|\alpha-\alpha_c|$ measures the distance to criticality. The bias $\epsilon$ takes the role of a local field (with scaling exponent $y_\epsilon$); and $\nu$ is the correlation length exponent which describes the vanishing of the energy scale $T^\ast$, above which critical behavior is observed [@book]: $T^\ast \propto |\alpha-\alpha_c|^{\nu}$. The ansatz (\[fscal\]) assumes the fixed point to be interacting; for a Gaussian fixed point the scaling function will also depend upon dangerously irrelevant variables. With the local magnetization $M_{\rm loc} = \langle\sigma_z\rangle = -\partial F_{\rm imp}/\partial\epsilon$ and the susceptibility $\chi_{\rm loc} = -\partial^2 F_{\rm imp}/(\partial\epsilon)^2$ we can define critical exponents (see also Ref. ): $$\begin{aligned} M_{\text{loc}}(\alpha > \alpha_c,T=0,\epsilon=0) &\propto& (\alpha-\alpha_c)^{\beta}, \nonumber\\ \chi_{\text{loc}}(\alpha < \alpha_c,T=0) &\propto& (\alpha_c-\alpha)^{-\gamma}, \nonumber\\[-1.75ex] \label{exponents} \\[-1.75ex] M_{\text{loc}}(\alpha=\alpha_c,T=0) &\propto& | \epsilon |^{1/\delta}, \nonumber\\ \chi_{\text{loc}}(\alpha=\alpha_c,T) &\propto& T^{-x}, \nonumber \\ \chi_{\text{loc}}''(\alpha=\alpha_c,T=0,\omega) &\propto& |\omega|^{-y} {\rm sgn}(\omega). \nonumber\end{aligned}$$ The last equation describes the dynamical scaling of $\chi_{\rm loc}$. In the absence of a dangerously irrelevant variable there are only two independent exponents, e.g., $\nu$ and $y_\epsilon$. The scaling form (\[fscal\]) yields hyperscaling relations: $$\begin{aligned} \beta = \gamma \frac{1-x}{2x},~~ 2\beta + \gamma = \nu, ~~ \gamma = \nu x,~~ \delta = \frac{1+x}{1-x} \,.\end{aligned}$$ Hyperscaling also implies $x=y$, which is equivalent to so-called $\omega/T$ scaling in the dynamical behavior. [*Long-range Ising model.*]{} The classical counterpart of the spin-boson model (\[eq:sbm\]) is the one-dimensional Ising model [@Leggett; @Weiss] $${\cal H}_{\rm cl} = - \sum_{\langle ij \rangle} J_{ij} S_i^z S_j^z + {\cal H}_{\rm SR} \label{hcl}$$ with interaction $J_{ij} = J/|i-j|^{1+s}$. ${\cal H}_{\rm SR}$ contains an additional generic short-range interaction which arises from the transverse field, but is believed to be irrelevant for the critical behavior [@fisher; @luijten]. As proven by Dyson [@dyson] this model displays a phase transition for $0<s\leq 1$. Both analytical arguments, based on the equivalence to a O(1) $\phi^4$ theory [@fisher], and extensive numerical simulations [@luijten] show that the upper-critical dimension for the $d$-dimensional long-range Ising model is $d_c^+ = 2s$, i.e., in $d=1$ the transition obeys non-trivial critical behavior for $1/2<s<1$. In contrast, mean-field behavior obtains for $0<s<1/2$, with exponents [@fisher; @luijten] $\beta = 1/2$, $\gamma = 1$, $\delta = 3$, $\nu = 1/s$, $y=s$ violating hyperscaling. As the exponent $s$ exclusively determines the power laws of spectra and correlations in a long-range model once spatial dimensionality ($d=1$) is fixed, $s$ takes the role of a “dimension”, i.e., we will refer to $s=1/2$ as upper-critical “dimension” of the classical Ising chain. Near $s=1$ the phase transition can be analyzed using a kink gas representation of the partition function, where the kinks represent Ising domain walls [@koster]. This expansion, controlled by the smallness of the kink fugacity, is done around the ordered phase of the Ising model, corresponding to the localized fixed point of the spin-boson quantum problem. For small $(1-s)$ the results obtained via the perturbative kink-gas RG are consistent with the NRG data for the spin-boson transition [@BTV], indicating that the quantum–classical mapping works in the asymptotic vicinity of the localized fixed point. [*Spin-boson model: perturbative RG.*]{} We now describe a novel RG expansion which is performed around the [*delocalized*]{} fixed point of the spin-boson model. NRG indicates that the critical fixed point merges with the delocalized one as $s\to 0^+$, thus we expect that the expansion will allow access to the quantum phase transition for small $s$. As shown below, the expansion is done about the [*lower-critical*]{} “dimension” $s=0$; it yields an interacting fixed point, and mean-field critical behavior for small $s$ does [*not*]{} obtain. For convenience we assume equal couplings, $\lambda_i \equiv \lambda$, then the energy dependence of $J(\omega)$ is contained in the density of states of the oscillator modes $\omega_i$, and we have $\alpha \propto \lambda^2$. How to set up a suitable RG expansion? Power counting about the free-spin fixed point, $\lambda=\Delta=0$, gives the scaling dimensions ${\rm dim}[\lambda] = (1-s)/2$, ${\rm dim}[\Delta] = 1$. Thus, both parameters are strongly relevant for small $s$. A better starting point is the delocalized fixed point, corresponding to [*finite*]{} $\Delta$. Eigenstates of the impurity are $|\!\rightarrow_x\rangle$ and $|\!\leftarrow_x\rangle$, with an energy splitting of $\Delta$. The low-energy Hilbert space contains the state $|\!\rightarrow_x\rangle$ only, and interaction processes with the bath arise in second-order perturbation theory, proportional to $\kappa_0 = \lambda^2/\Delta$. Power counting w.r.t. the $\lambda=0$ limit now gives ${\rm dim}[\kappa_0] = -s$, i.e., $\kappa_0$ is marginal at $s=0$, indicating that an $\epsilon$-type expansion for small $s$ is possible. =3.1in To lowest order, the RG can be performed within the low-energy sector, i.e., for the $\kappa_0$ vertex (Fig. \[fig:dgr\]b) and the propagator for the $|\!\rightarrow_x\rangle$ state. Consequently, our approach is valid as long as $\lambda \ll \Delta, \omega_c$. We introduce a renormalized coupling $\kappa$ according to $\kappa_0 = \Lambda^{-s} \kappa$ where $\Lambda$ is the running cutoff, $\Lambda=\omega_c$ initially. The one-loop beta function can be derived using the familiar momentum shell method, i.e., by successively eliminating high-energy bath bosons. To one-loop order only the coupling-constant renormalization in Fig. \[fig:dgr\]c enters, and we obtain $$\begin{aligned} \beta(\kappa) = -s \kappa + \kappa^2 \,.\end{aligned}$$ Besides the stable delocalized fixed point $\kappa=0$ this flow equation displays an infrared unstable fixed point at $$\kappa^\ast = s + {\cal O}(s^2) \label{fp}$$ which controls the transition between the delocalized and localized phases. No (dangerously) irrelevant variables are present in this theory, so we conclude that the critical fixed point (\[fp\]) is interacting [@potnote]. We proceed with the calculation of critical exponents. Expanding the RG beta function around the fixed point (\[fp\]) gives the correlation length exponent $$1 / \nu = s + {\cal O}(s^2) \,, \label{nu}$$ i.e., $\nu$ diverges as $s\to 0^+$, as characteristic for a lower-critical dimension. Parenthetically, we note that the RG structure for $s\to 1^-$ is also similar to that near a lower-critical dimension: The line of second-order transitions for $0<s<1$ terminates in a Kosterlitz-Thouless transition at $s=1$ and is thus bounded by [*two lower-critical “dimensions”*]{} – a similar situation was recently found in the pseudogap Kondo problem, which, however, is in a different universality class [@LFMV]. As usual for RG expansions around a lower-critical dimension the present RG can only capture one of the two phases (the delocalized one), whereas run-away flow occurs on the localized side. The exponents associated with the local field $\epsilon$ can be obtained in straightforward renormalized perturbation theory. To calculate observables, the diagrams are written down using the original model with couplings $\lambda$ and $\Delta$ and both $|\!\rightarrow_x\rangle$, $|\!\leftarrow_x\rangle$ states. The perturbation theory turns out to be organized in powers of $\lambda^2/\Delta$, as expected. Some of the relevant diagrams are displayed in Fig. \[fig:dgr\]d, details will appear elsewhere. Restricting ourselves to the lowest-order results for the disordered and quantum critical regimes we find $$\begin{aligned} \gamma &=& 1 + {\cal O}(s) ,~~ x = y = s + {\cal O}(s^2), \nonumber\\ 1/\delta &=& 1 - 2 s + {\cal O}(s^2) \,. \label{delta}\end{aligned}$$ Interestingly, we are able to derive an [*exact*]{} result for the exponents $x$, $y$, employing an argument along the lines of Refs. , based on the diagrammatic structure of $\chi_{\rm loc}$. We obtain $$\begin{aligned} x = y = s \label{x} \,.\end{aligned}$$ (Notably, $y=s$ was found to be the exact decay exponent of the critical spin correlations in the long-range Ising model for all $s$ [@fisher; @suzuki].) Hyperscaling yields $\delta = (1+s)/(1-s)$, consistent with the lowest-order result (\[delta\]). =3.3in As an aside, we note that at $s=0$ the bath coupling is marginally relevant. Therefore the impurity is always localized as $T\to 0$, with a localization temperature given by $T^\ast = \omega_c \exp(-\Delta\omega_c / \lambda^2)$. =3.3in [*Spin-boson model: Numerical results.*]{} We have performed extensive NRG calculations [@BLTV] to evaluate the critical exponents of the spin-boson transition, see Refs.  for numerical details [@footnrg]. Results are shown in Figs. \[fig:betadelta\] and \[fig:nux\]: these exponents obey hyperscaling including $x=y$ ($\omega/T$ scaling). They are in excellent agreement with the small-$s$ RG expansion, but at variance with the exponents of the long-range Ising model: the mean-field predictions are $\beta = 1/2$, $\delta = 3$ which are clearly violated by our results in Fig. \[fig:betadelta\]. Within error bars the mean-field exponents are realized [*at*]{} $s=1/2$. Further, the one-loop results for $\nu$ (\[nu\]) and $\gamma$ (\[delta\]) appear to be exact for all $0<s<1/2$, and logarithmic corrections to the power laws are observed at $s=1/2$. This suggests that the spin-boson transition [*does*]{} change its character at $s=1/2$, but the critical fixed points for both $0<s<1/2$ and $1/2<s<1$ are interacting and obey hyperscaling (the latter one being equivalent to that of the classical Ising model). [*Discussion.*]{} We have proven that the naive quantum–classical mapping fails for the sub-ohmic spin-boson model: Using a novel RG expansion around the delocalized fixed point, we have shown that the quantum transition at $0<s<1/2$ is controlled by an interacting fixed point, whereas the corresponding classical long-range Ising model shows mean-field behavior. Thus, the spin-boson problem for $s<1/2$ is equivalent neither to the classical Ising model nor to the corresponding (quantum or classical) O(1) $\phi^4$ theory [@luijten]. In physical terms the inequivalence can be traced back to the different disordered (delocalized) fixed points in the two situations (expansions around these fixed points are suitable to access the critical behavior for small $s$): In the quantum model the transverse field fully polarizes the spin in $x$ direction (which can be viewed as a “condensate” of spin flips), whereas the high-temperature limit of the classical Ising model is simply incoherently disordered. The inequivalence may come unexpected – so where does the quantum–classical mapping fail? Formal proofs of the mapping using transfer matrices rely on the short-ranged character of the interaction [@book]. For general interactions a Trotter decomposition of the quantum partition function is employed where the imaginary axis of length $\beta = 1/T$ is divided into $N$ slices of size $\Delta\tau=\beta/N$, leading to an Ising chain (\[hcl\]) with $N$ sites. This procedure is exact when the limits $\Delta\tau\to 0$ and $\beta\to\infty$ are taken in this order. However, the limit $\Delta\tau\to 0$ leads to a [*diverging*]{} near-neighbor coupling in the term ${\cal H}_{\rm SR}$ of the classical Ising model (\[hcl\]) [@emery]. This may in fact change the critical behavior of ${\cal H}_{\rm cl}$ (\[hcl\]), as a classical model with finite couplings arises upon taking $\beta\to\infty$ first. In other words, the quantum and classical problems are only equivalent if and only if the low-energy limit of ${\cal H}_{\rm cl}$ is independent of the order of the two limits $\Delta\tau\to 0$ and $\beta\to\infty$ [@emery]. As we have proven the inequivalence of ${\cal H}_{\rm SB}$ and ${\cal H}_{\rm cl}$ (with finite couplings) we conclude that the two limits cannot be interchanged for $s<1/2$. A recent paper [@stefan] investigated a SU($N$)-symmetric Bose-Fermi Kondo model in a certain large-$N$ limit and found a critical fixed point with $\omega/T$ scaling for all $s$. The authors argued that the apparent failure of the quantum–classical mapping is due to the quantum nature of the impurity spin. As discussed above, this alone is not sufficient: for short-range interactions in imaginary time the mapping can be proven to be asymptotically exact [@book]; [*long-range*]{} interactions are essential. Our results suggest that some conclusions drawn in the past for effectively long-range Ising systems on the basis of the quantum–classical mapping have to be re-examined. Further, we envision that our novel $\epsilon$ expansion will have applications for various quantum impurity problems, e.g., two-level systems coupled to multiple baths, Bose-Fermi Kondo models [@MVMK; @stefan], and also for quantum dissipative lattice models [@lattice]. We thank D. Grüneberg, H. Rieger, A. Rosch, S. Sachdev, Q. Si, M. Troyer, T. Vojta, and W. Zwerger for discussions. This research was supported by the DFG through SFB 484 (RB) and the CFN Karlsruhe (MV), and by the Alexander von Humboldt foundation (NT). S. Sachdev, [*Quantum Phase Transitions*]{}, Cambridge University Press, Cambridge (1999). D. Belitz [*et al.*]{}, Phys. Rev. B [**55**]{}, 9452 (1997); [*ibid.*]{} [**65**]{}, 165112 (2002); Ar. Abanov and A. V. Chubukov, cond-mat/0409601. A. J. Leggett [*et al.*]{}, Rev. Mod. Phys. [**59**]{}, 1 (1987). U. Weiss, [*Quantum dissipative systems*]{}, 2nd ed. (World Scientific, Singapore, 1999). G. Yuval and P. W. Anderson, Phys. Rev. B [**1**]{}, 1522 (1970); P. W. Anderson, G. Yuval, and D. R. Hamann, Phys. Rev. B [**1**]{}, 4464 (1970). V. J. Emery and A. Luther, , 215 (1974). R. Bulla, N. Tong, and M. Vojta, Phys. Rev. Lett. [**91**]{}, 170601 (2003). R. Bulla [*et al.*]{}, cond-mat/0407559. M. E. Fisher [*et al.*]{}, , 917 (1972). E. Luijten and H. W. J. Blöte, , 8945 (1997). M. Suzuki, Prog. Theor. Phys. [**49**]{}, 424, 1106, 1440 (1973). H. Spohn and R. Dümcke, J. Stat. Phys. [**41**]{}, 389 (1985). S. Kehrein and A. Mielke, Phys. Lett. A [**219**]{}, 313 (1996). F. J. Dyson, Commun. Math. Phys. [**12**]{}, 91 (1969). J. M. Kosterlitz, , 1577 (1976). K. Ingersent and Q. Si, Phys. Rev. Lett. [**89**]{}, 076403 (2002). In the low-energy sector, $|\!\rightarrow_x\rangle$, the effective theory reduces to a potential scattering problem for the bath bosons, which can be solved exactly, and indeed has a phase transition at $\kappa_0 G_{\rm loc}(\omega=0) = 1$. Corrections can arise from higher-order processes involving $|\!\leftarrow_x\rangle$. L. Fritz and M. Vojta, , 214427 (2005). M. Vojta [*et al.*]{}, Phys. Rev. B [**61**]{}, 15152 (2000). M. Vojta and M. Kirćan, , 157203 (2003). Within NRG the boson occupation number stays small for all $s$ in the vicinity of the critical fixed points (in contrast to the localized fixed points, where it diverges in the low-energy limit). Thus, the bosonic chain-NRG [@BLTV] can be used which has been proven to give reliable results in the quantum critical regime, e.g., for exponents. L. Zhu [*et al.*]{}, , 267201 (2004). S. Pankov [*et al.*]{}, , 054426 (2004), P. Werner [*et al.*]{}, , 047201 (2005)  \ [**Erratum: Quantum phase transitions in the sub-ohmic spin-boson model: Failure of the quantum–classical mapping, Phys. Rev. Lett. [**94**]{}, 070604 (2005)**]{}\  \ Matthias Vojta, Ning-Hua Tong, and Ralf Bulla\  \ We point out that the interpretation of data obtained by the Numerical Renormalization Group\ (NRG) method in Phys. Rev. Lett. [**94**]{}, 070604 (2005) was incorrect: Two different effects precluded\ the reliable determination of critical exponents. An analytical argument was as well incomplete.\ We discuss consequences for the quantum phase transition scenario.\  \ Ref. used results from NRG to argue that the critical properties of the spin-boson model with bath exponent $s<1/2$ are not those of mean-field theory. The difference is in the order-parameter exponents $\beta$, $\delta$ and $x$. By now, we have convinced ourselves that the exponents in question could not be reliably extracted from the NRG calculations, if the ordered phase is controlled by an operator which is dangerously irrelevant at criticality. The two relevant issues will be discussed in turn. \(A) [*Hilbert-space truncation*]{}: The bosonic NRG is presently unable to describe the physics of the localized fixed point for $s<1$, due to the truncation of the bosonic Hilbert space [@BLTV2]. From the absence of convergence problems at the critical fixed point we nevertheless concluded [@prl] that all critical exponents can be extracted from NRG. While correct for an interacting fixed point, this assertion is erroneous for exponents $\beta$ and $\delta$ of a Gaussian fixed point, because, in the presence of a dangerously irrelevant variable, those exponents are [*not*]{} properties of the critical fixed point and its vicinity, but instead properties of the flow towards the fixed point controlling the ordered phase. (This is what happens above the upper-critical dimension of a standard $\phi^4$ theory.) \(B) [*Mass flow:*]{} The NRG algorithm integrates out the bath iteratively. In the order-parameter language, this leads to a mass renormalization along the flow, arising from the bath (not from non-linear interactions), i.e., at any NRG step only a partial mass renormalization has been taken into account. As a result, the critical flow trajectory in NRG is that of a system with an additional mass set by the current NRG scale (i.e. by the temperature $T$), and, effectively, the system is not located at the critical coupling for any finite $T$. For an interacting fixed point, the missing mass correction from the bath and the interaction-generated mass have the same scaling, hence no qualitative error is introduced. In contrast, above the upper-critical dimension, the interaction-generated mass vanishes faster as $T\!\to\!0$, and the bath mass correction dominates all observables calculated along the critical flow trajectory, like the exponent $x$. Note that this conceptual “mass-flow” problem of NRG arises for all baths without low-energy particle-hole symmetry. Consequently, the NRG-extracted exponents $\beta$, $\delta$, and $x$ quoted in Ref.  were unreliable. The same applies to the analytical argument given in Ref. : The RG equations (9-11) were correct, but the following analysis relied on the absence of dangerously irrelevant variables as well. Analyzing (A) in detail, we have found that careful NRG calculations performed with a large number $N_b$ of bosonic states in principle allow to extract the correct exponents $\beta$ and $\delta$. While the truncation always dominates asymptotically close to criticality, the physical power laws can be seen on intermediate scales. As an example, we show in Fig. 1 the non-linear field response at criticality for $s=0.4$. For $N_b=50$ a crossover from $\delta\approx2.4$ to the mean-field value $\delta=3$ upon increasing the field is visible, before the quantum critical regime is left. ![ Magnetization curve of the critical spin-boson model for $s=0.4$. NRG parameters are $\Lambda=4$, $(N_s,N_b)=(40,12)$ and $(60,50)$ [@BLTV2]. The lines are power laws with exponents 1 (dotted), 1/2.4 (dashed), and 1/3 (dash-dot). ](delta.eps){width="3.5in"} Taking into account recent numerical work based on Quantum Monte Carlo and exact diagonalization techniques [@rieger; @fehske], which confirmed mean-field behavior in the sub-ohmic spin-boson model for $s<1/2$, we conclude that the quantum-to-classical mapping is valid here. [99]{} M. Vojta, N. Tong, and R. Bulla, Phys. Rev. Lett. [**94**]{}, 070604 (2005). R. Bulla [*et al.*]{}, Phys. Rev. B [**71**]{}, 045122 (2005). A. Winter [*et al.*]{}, Phys. Rev. Lett. [**102**]{}, 030601 (2009). A. Alvermann and H. Fehske, Phys. Rev. Lett. [**102**]{}, 150601 (2009).

--- abstract: 'We propose StartNet to address Online Detection of Action Start (ODAS) where action starts and their associated categories are detected in untrimmed, streaming videos. Previous methods aim to localize action starts by learning feature representations that can directly separate the start point from its preceding background. It is challenging due to the subtle appearance difference near the action starts and the lack of training data. Instead, StartNet decomposes ODAS into two stages: action classification (using ClsNet) and start point localization (using LocNet). ClsNet focuses on per-frame labeling and predicts action score distributions online. Based on the predicted action scores of the past and current frames, LocNet conducts class-agnostic start detection by optimizing long-term localization rewards using policy gradient methods. The proposed framework is validated on two large-scale datasets, THUMOS’14 and ActivityNet. The experimental results show that StartNet significantly outperforms the state-of-the-art by $15\%$-$30\%$ p-mAP under the offset tolerance of $1$-$10$ seconds on THUMOS’14, and achieves comparable performance on ActivityNet with $\times 10$ smaller time offset.' author: - | Mingfei Gao$^1$[^1] Mingze Xu$^2$ Larry S. Davis$^1$ Richard Socher$^3$ Caiming Xiong$^3$[^2]\ $^1$University of Maryland $^2$Indiana University $^3$Salesforce Research\ [{mgao,lsd}@umiacs.umd.edu, mx6@indiana.edu, {rsocher,cxiong}@salesforce.com]{} bibliography: - 'egbib.bib' title: 'StartNet: Online Detection of Action Start in Untrimmed Videos' --- [^1]: Work done when the author was an intern at Salesforce Research. [^2]: Corresponding author.

--- abstract: | Of the C$_{3}$H$_{x}$ hydrocarbons, propane (C${_3}$H$_{8}$) and propyne (methylacetylene, CH$_{3}$C$_{2}$H) were first detected in Titan’s atmosphere during the Voyager 1 flyby in 1980. Propene (propylene, C$_{3}$H$_{6}$) was first detected in 2013 with data from the Composite InfraRed Spectrometer (CIRS) instrument on Cassini. We present the first measured abundance profiles of propene on Titan from radiative transfer modeling, and compare our measurements to predictions derived from several photochemical models. Near the equator, propene is observed to have a peak abundance of 10 ppbv at a pressure of 0.2 mbar. Several photochemical models predict the amount at this pressure to be in the range 0.3 - 1 ppbv and also show a local minimum near 0.2 mbar which we do not see in our measurements. We also see that propene follows a different latitudinal trend than the other C$_{3}$ molecules. While propane and propyne concentrate near the winter pole, transported via a global convective cell, propene is most abundant above the equator. We retrieve vertical abundances profiles between 125 km and 375 km for these gases for latitude averages between 60$^{\circ}$S to 20$^{\circ}$S, 20$^{\circ}$S to 20$^{\circ}$N, and 20$^{\circ}$N to 60$^{\circ}$N over two time periods, 2004 through 2009 representing Titan’s atmosphere before the 2009 equinox, and 2012 through 2015 representing time after the equinox. Additionally, using newly corrected line data, we determined an updated upper limit for allene (propadiene, CH$_{2}$CCH$_{2}$, the isomer of propyne). We claim a 3-$\sigma$ upper limit mixing ratio of 2.5$\times$10$^{-9}$ within 30$^\circ$ of the equator. The measurements we present will further constrain photochemical models by refining reaction rates and the transport of these gases throughout Titan’s atmosphere. address: - 'Planetary Systems Laboratory, Solar System Exploration Division, NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD, USA' - 'Center for Space Science and Technology, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD, USA' - 'Department of Astronomy, University of Maryland College Park, College Park, MD, USA' - 'Laboratoire Interuniversitaire des Systémes Atmosphériques, Université Paris-Est, Creteil, France' - 'Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA, USA' - 'Atmospheric, Oceanic and Planetary Physics, Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, UK' author: - Nicholas A Lombardo - Conor A Nixon - Richard K Achterberg - Antoine Jolly - Keeyoon Sung - Patrick G J Irwin - F Michael Flasar bibliography: - 'c3.bib' title: 'Spatial and Seasonal Variations in C$_{3}$H$_{x}$ Hydrocarbon Abundance in Titan’s Stratosphere from Cassini CIRS Observations' --- Introduction ============ Titan, the largest moon of Saturn, has a CH$_{4}$ surface mixing ratio of about 5%, measured by the Huygens GCMS [@niemann:2010], and decreasing with altitude into the stratosphere where it remains constant with altitude at 1-1.5%, as measured in [@lellouch:2014]. Titan is thought to have many similarities to the Archean Earth, including an atmosphere abundant in in N$_{2}$ and significant quantities of CH$_{4}$ as well as global haze layers which continually shroud Titan and may have occurred intermittently on Earth. While factors like temperature, sources of atmospheric CH$_{4}$, and minor atmospheric constituents vary between the two bodies, Titan remains a good analog for studying the atmosphere of the Archean Earth [@arney:2016; @izon:2017]. The global haze on Titan is produced through photolysis of CH$_{4}$ as Saturn Magnetospheric Electrons and solar UV photons bombard the upper atmosphere. The products of this process -highly reactive CH$_{3}^{-}$, H$^{+}$, and N$^{+}$ ions, among others- may then react to form C$_{2}$H$_{6}$, C$_{2}$H$_{4}$, and other molecules. As this complex process continues, larger hydrocarbons (C$_{x}$H$_{y}$) and nitriles (C$_{x}$H$_{y}$(CN)$_{z}$) react further to give rise to the ’photochemical zoo’ of molecules present in Titan’s atmosphere [@yung:1984; @wilson:2004; @lavvas:2008; @loison:2015; @dobrijevic:2016; @willacy:2016]. Titan’s 26.7$^{\circ}$ obliquity (the axial tilt relative to the normal of the orbital plane), comparable to the Earth’s 23.5$^{\circ}$ obliquity, causes variations in the insolation of the moon over the course of a Titan year (about 29.5 Earth years). The resulting seasonal variations in the physical state of the atmosphere include molecule abundance [@vinatier:2015; @coustenis:2018], temperature [@achterberg:2011; @teanby:2017], and behavior of the haze layers [@jennings:2012], discussed more in the review by [@horst:17]. Noteworthy is the existence of a global circulation cell, which transports warm gases in the summer hemisphere towards the winter pole, where they subside lower into the stratosphere. This downward advection causes adiabatic warming in the winter stratosphere and entrains short-lived gases produced in the upper stratosphere, increasing their abundance lower in the atmosphere. As northern winter evolved to northern spring, this single circulation cell transformed into two circulation cells, upwelling near the equator and downwelling at both poles, as predicted in [@hourdin:2004] and observed in [@teanby:2012]. For additional explanation of Titan’s atmospheric dynamics and chemistry, the reader is directed to [@titan:2010] and [@titan:2014]. Regarding the C$_{3}$ hydrocarbons, propane (C$_{3}$H$_{8}$) and propyne (C$_{3}$H$_{4}$) were initially detected in Titan’s atmosphere after the 1980 Voyager 1 flyby [@hanel:1981] through spectra acquired by the IRIS instrument. Abundances for propyne were first estimated by [@maguire:1981] by comparing the strength of the 633 cm$^{-1}$ Q-branch of propyne to the 721 cm$^{-1}$ Q-branch of acetylene (also ethyne, C$_{2}$H$_{2}$), and estimated to be on the order of 3$\times$10$^{-8}$. Propane was modeled in the same paper using a synthetic spectrum constructed for its $\nu_{21}$ band, and a disk averaged value of 2$\times$10$^{-5}$ was reported. These values were updated by [@coustenis:1989] to 4.4$^{+1.7}_{-2.1}\times$10$^{-9}$ for propyne and (7$\pm$4)$\times$10$^{-7}$ for propane. Further weak bands of propane were detected by the Composite InfraRed Spectrometer (CIRS) aboard Cassini [@nixon:propane]. Over three decades later, CIRS spectra were used to make the first detection of C$_{3}$H$_{6}$ [@nixon:propene], however an exact abundance could not be retrieved from modeling the spectra due to the lack of a spectral line list, although an abundance estimate was made by comparing the intensities of propene and propane lines, discussed more in Section 4.2. Recent analyses have shown the abundance of propyne to vary strongly with season and latitude. [@vinatier:2015], using limb viewing observations, showed the vertical gradient of C$_{3}$H$_{4}$ increases dramatically over the mid northern latitudes as northern winter moves into northern spring and the polar vortex responds to the changing amount of sunlight. [@coustenis:2018], using nadir observations to probe abundance in a narrow altitude range in the middle stratosphere, show a similar trend at latitudes closer to the pole, between 60$^{\circ}$ and 90$^{\circ}$ either side of the equator. In the same studies, propane was shown to have a more constant abundance in latitude and time, remaining constant within error bars near 1$\times$10$^{-6}$ throughout the stratosphere, with the exception near the winter pole, where it increases with altitude. Two C$_{3}$ hydrocarbons have yet to be firmly detected on Titan, allene (CH$_{2}$CCH$_{2}$, isomer of propyne) and cyclopropane (CH$_{2}$CH$_{2}$CH$_{2}$, isomer of propene). There was a tentative detection of allene by [@roe:2011], however an accurate line list was not available at the time of the study, thus the authors were not able to model the potential allene feature and confirm its detection. In this paper, we discuss members of the C$_{3}$H$_{x}$ series known to be present in Titan’s atmosphere- propane (C$_{3}$H$_{8}$), propene (C$_{3}$H$_{6}$), and propyne (CH$_{3}$C$_{2}$H). We also searched for allene and provide an new upper limit for allene in regions close to the equator. This work was enabled by the creation of a propene pseudo-line list for Titan [@sung:18] and an updated line list for allene (see Section 2.2). We use spectra collected by the CIRS instrument to determine the abundance of propene in Titan’s stratosphere. We show latitudinal and seasonal variation in the distribution of propene, propane and propyne. The large number of CIRS observations used allows us to vertically resolve the profile of each gas. We compare the values determined to those predicted by photochemical models of [@hebrard:2013], [@kras:2014], [@li:2015], and [@loison:2015]. Additionally, we use a corrected line list for allene to determine an updated upper limit for the molecule. Methods ======= Dataset ------- CIRS is a Fourier Transform infrared spectrometer, with three focal planes spanning the 10 cm$^{-1}$ - 1500 cm$^{-1}$ spectral region [@jennings:17]. We use spectra acquired by Focal Plane 3 (FP3, 580 cm$^{-1}$-1100 cm$^{-1}$) and Focal Plane 4 (FP4, 1050 cm$^{-1}$-1500cm$^{-1}$ ), two parallel arrays of 10 detectors each. Limb observations were performed at a spectral resolution of 0.5 cm$^{-1}$ at distances between 10$^{5}$ km and 2$\times$10$^{5}$ km from Titan, during which time each focal plane was positioned normal to Titan’s surface, such that each detector sampled a different altitude. The arrays were centered at  125 km for between one and two hours and were then moved away from Titan’s surface to stare at a central altitude of  350 km for a similar amount of time. The footprint of each detector (the vertical resolution) on Titan’s atmosphere varied between 27 km and 54 km depending on the distance to the moon. The size of the footprint was comparable to Titan’s atmospheric scale height, and thus allows us to vertically resolve physical characteristics of the atmosphere. The C$_{3}$H$_{6}$ $\nu_{19}$ band detected in [@nixon:propene] at 912.67 cm$^{-1}$ sits between several C$_{2}$H$_{4}$ emissions, and on top of a broad C$_{3}$H$_{8}$ band. To increase S/N to the point where we can model this feature, we divide the CIRS dataset into six time-latitude bins, where in each bin the temperature varies less than 15 K. We make the assumption that over the time and latitudes covered in by each bin, Titan’s stratosphere has similar temperature and molecular abundance profiles. Spectra in each bin are averaged together to reduce random noise in the data before modeling. Each bin includes data from from between three and seven flybys (or between 187 and 728 spectra). We use two time periods - the pre equinox time from 2005 to 2009 (just before the northern vernal equinox of August 2009) and the post equinox time 2012 through 2015. During the time just after the northern vernal equinox, Cassini was in a very low inclination orbit relative to Saturn. Limb observations on Titan were focused on the polar regions, as Cassini was able to view these regions of the atmosphere continuously during a Titan flyby. We therefore do not include data from just after the equinox, as no limb data exists for the latitude regions we model in this work. In each time span, we combine observations representative of three latitude regions - northern, equatorial, and southern. In our averages, we include data from 20$^{\circ}$ to 60$^{\circ}$ for both hemispheres, and within 20$^{\circ}$ of the equator. The former two bins represent the mid-latitudes, and the third bin represent the equatorial atmosphere. While the latitude boundaries for observations we include are the same before and after equinox, the physical distribution of included observations varies. As an example, in the pre-equinox time, the northern bin contains observations from 24$^{\circ}$N through 54$^{\circ}$N, whereas post-equinox we have observations only between 25$^{\circ}$N and 48$^{\circ}$N. The distribution of observations used is shown in Fig. \[fig:map\] as black dots. The boundaries for each time-latitudinal bin are drawn as black boxes enclosing observations. We exclude observation centered at latitudes closer than 60$^{\circ}$ to either pole because temperature begins to vary strongly with latitude in these regions. Including these spectra in our averages would make the resultant spectra very difficult to model and obscure details of finer latitudinal variations in the retrieved profiles. A summary of the data used is in Table \[tab:data\]. [ccccccc|cccccc]{} Time Range & &\ Latitude Range & & & & & &\ Avg Lat& & & & & &\ \ Altitude (km) & spectra & NESR & spectra & NESR & spectra & NESR & spectra & NESR & spectra & NESR & spectra & NESR\ 100-150 & 322 & 1.58 & 330 & 1.68 & 412 & 1.53 & 299 & 2.30 & 440 & 1.82 & 288 & 2.41\ 150-200 & 451 & 1.37 & 428 & 1.48 & 537 & 1.37 & 416 & 1.96 & 608 & 1.67 & 395 & 2.19\ 200-250 & 515 & 1.32 & 460 & 1.50 & 568 & 1.30 & 441 & 1.85 & 601 & 1.61 & 464 & 1.89\ 250-300 & 496 & 1.34 & 440 & 1.45 & 634 & 1.23 & 463 & 1.99 & 680 & 1.60 & 446 & 1.74\ 300-350 & 271 & 1.67 & 299 & 1.79 & 294 & 1.60 & 229 & 2.60 & 297 & 2.16 & 341 & 1.92\ 350-375 & 235 & 1.96 & 242 & 1.89 & 258 & 1.59 & 218 & 2.33 & 287 & 2.21 & 191 & 2.70\ \ \ Altitude (km) & spectra & NESR & spectra & NESR & spectra & NESR & spectra & NESR & spectra & NESR & spectra & NESR\ 100-150 & 310 & 0.29 & 369 & 0.23 & 515 & 0.29 & 388 & 0.40 & 345 & 0.30 & 298 & 0.33\ 150-200 & 381 & 0.22 & 463 & 0.27 & 652 & 0.23& 530 & 0.27 & 488 & 0.30 & 397 & 0.32\ 200-250 & 372 & 0.24 & 573 & 0.24 & 691 & 0.17& 551 & 0.21 & 503 & 0.26 & 464 & 0.29\ 250-300 & 334 & 0.26& 559 & 0.19 & 728 & 0.18 & 583 & 0.20 & 556 & 0.23 & 450 & 0.24\ 300-350 & 187 &0.39 & 450 & 0.22& 345 & 0.28 & 305 & 0.29 & 230 & 0.33 & 336 & 0.28\ 350-400 & 187 & 0.29& 277 & 0.30 & 294 & 0.30 & 265 & 0.32 & 232 & 0.38 & 183 & 0.29\ ![Distribution of all Mid-IR Limb Integrations (MIRLMBINT) observations during the Cassini mission shown as black circles. Temperatures are shown as contours (colored) and correspond to those at 1 mbar, or about 175 km, and are updated from [@achterberg:2014]. The black boxes indicate the binning scheme with two time spans (2004-2009 and 2012-2015) further binned to three latitude ranges (60$^{\circ}$S-20$^{\circ}$S, 20$^{\circ}$S-20$^{\circ}$N, and 20$^{\circ}$N-60$^{\circ}$N). The dashed black line surrounded by gray is the solar latitude of Titan, where the Sun is directly overhead at 1200 local time.[]{data-label="fig:map"}](obs){width="\columnwidth"} Spectral Line Data ------------------ Molecular line lists used are described in Table \[tab:linedata\]. Molecule *a priori* VMR Source of Line Data ------------------- -------------------------------- ---------------------------- -- C$_{2}$H$_{4}$ (1.1$\pm$0.2)$\times$10$^{-7}$ [@geisa:2016] C$_{3}$H$_{8}$ (1$\pm$0.5)$\times$10$^{-6}$ [@sung:propane] C$_{3}$H$_{6}$ (3$\pm$1.5)$\times$10$^{-9}$ [@sung:18] CH$_{3}$C$_{2}$H (1$\pm$0.4) $\times$10$^{-8}$ same as [@coustenis:2007] C$_{4}$H$_{2}$ (5$\pm$2.5)$\times$10$^{-9}$ [@jolly:2010] CH$_{2}$CCH$_{2}$ (3$\pm$1.5$\times$10$^{-9}$) Discussed in Section 2.2.1 : The reference abundances of molecules retrieved in the model are set to be constant above saturation at the values listed here. Sources of the spectral line data are also listed.[]{data-label="tab:linedata"} ### Updated Allene Linedata No allene line lists are currently present in the HITRAN or GEISA databases. [@coustenis:1993] initially investigated the detectability of allene in Titan’s atmosphere using spectroscopic parameters by [@chazelas:1985] for the $\nu_{10}$ band centered at 845 cm$^{-1}$. Line intensities were obtained from band intensity measurements by [@koga:1979]. They concluded that the non-detection of allene implied an abundance below 5 ppbv. The same line list was used in [@coustenis:2003] and [@nixon:allene], with upper limits discussed in Section 4.4. As in [@jolly:2015], we notice that the locations of transitions and transition intensities listed in the older line list does not match those in room temperature laboratory spectra. In this search for allene, we make the necessary corrections to this older line list so the spectral data we use matches experimental laboratory spectra. In this work, we combine the existing spectroscopic data used previously with parameters obtained by high resolution studies [@hegelund:1993] that address the existence of hot bands. The hot band contribution is necessary at room temperature to allow comparisons between these calculated line lists with observed room temperature spectra. Calculations were compared and validated against room and high temperature spectra taken at 0.08 cm$^{-1}$ resolution in the $\nu_{10}$/$\nu_{9}$ wavenumber range [@es-sebbar:2014]. A comparison between the old line list and new line list are given against a laboratory spectrum in Fig. \[fig:allene\_ll\], where the misfit between the line list used in previous studies and the laboratory data can be clearly seen. Radiative Transfer Modeling --------------------------- Spectral fitting was achieved using the NEMESIS atmospheric modeling code [@irwin:nemesis]. NEMESIS operates on the method of optimal estimation, which involves the computation of a forward model and a retrieval process. The forward model was calculated using the correlated-k method of [@lacis:1991], and includes a Hamming apodization of Full Width at Half Maximum of 0.475 cm $^{-1}$ to recreate the instrument line shape. The retrieval process varies *a priori* profiles of chosen physical parameters to optimize the spectral fits. This is performed by minimizing a cost-function which includes the deviation of the retrieved profile from the *a priori* estimate, and the quality of fit to the spectra (similar to a $\chi^{2}$ goodness of fit test). NEMESIS has been extensively used to determine atmospheric abundances in the outer solar system using IR spectra, and application to Titan is described in [@teanby:2007], [@teanby:2009], and [@cottini:2012]. Spectral modeling for each bin proceeded in two steps: stratospheric temperatures during observations were extracted and then used in determining the abundances of trace gases. First, the 1275 cm$^{-1}$ - 1325 cm$^{-1}$ region of the $\nu_{4}$ band of methane was modeled to retrieve the stratospheric temperature profile. While we did not model the entire $\nu_{4}$ band that extends from 1250 cm$^{-1}$ - 1350 cm$^{-1}$, the region of the spectrum we modeled contains sufficient information to retrieve the temperatures in the altitude regions needed for modeling the trace gases. Temperature measurements proceeded by assuming a methane abundance of 1.41$\times$10$^{-2}$ above 140 km, consistent with previous measurements and models [@lellouch:2014], [@wilson:2004]) and an abundance below 140 km derived from measurements by the Huygens descent profile [@niemann:2010]. In the retrieval, the temperature profile, derived from the HASI temperature [@ful:2005] profile, and a non-gray aerosol haze were allowed to vary to fit the observed spectrum. This retrieved temperature was then fixed while molecules in the FP3 spectral region were allowed to vary. In the 900 cm$^{-1}$ - 930 cm$^{-1}$ region, the Q branch of the $\nu_{21}$ band of propane at 922 cm$^{-1}$, the $\nu_{19}$ propene band at 912.5 cm$^{-1}$, and $\nu_{7}$ lines of ethylene at 915.5 cm$^{-1}$ and 922.5 cm$^{-1}$ (among several weaker lines) were modeled. In the 620 cm$^{-1}$ - 640 cm$^{-1}$ region, the propyne $\nu_{9}$ band at 633 cm$^{-1}$ and diacetylene $\nu_{8}$ band at 628 cm$^{-1}$ were modeled. The *a priori* volume mixing ratios (VMRs) used were constant over our range of sensitive altitudes above 100 km and set to values comparable to those reported in previous literature including [@coustenis:2007], [@vinatier:2007], [@coustenis:2010], and [@vinatier:2015], and represent ’rough guesses’ for the abundance of each molecule in both time spans. Though some molecules have been shown to have vertical gradient that changes with time, we chose to use a constant profile above saturation that is the same for both time spans as to not influence the results of our comparisons across latitude and time. The *a priori* value for propene is comparable to predictions from [@hebrard:2013], [@nixon:propene], [@li:2015], and [@dobrijevic:2016]. Results ======= Example spectra from the altitude bins used in the pre-equinox southern temperature retrieval are shown in Fig \[fig:tempfit\]. Errors on the radiance were initially calculated as the standard deviation from the mean of all spectra included in the average, but were expanded to account for systematic uncertainties and prevent over-constraining the model. Normalized contribution functions - also called the inversion kernel - are the rate of change of radiance with respect to abundance, normalized to one at the maximum value. Altitudes with higher values ’contribute’ more to the calculated spectrum, and thus are the altitudes where the data give the most information - see Fig. \[fig:tempcf\]. Retrieved temperature profiles for the pre-equinox time span and the post-equinox time span are shown in Fig. \[fig:temps\]. We see variation in the stratospheric temperature and the shape and altitude of the stratopause as described in [@flasar:2009]. While the stratopause increases in altitude towards the north winter pole, we do not see the full extent of the stratopause, since our data is only sensitive to an altitude of 0.02 mbar. Contribution functions similar to Fig. \[fig:tempcf\] are shown in Fig. \[fig:propenecf\] for the propene retrieval. Figs. \[fig:fp3spec\] and \[fig:propynespec\] show spectral fits for the pre-equinox southern bin, similar to those shown for temperature. The retrieved profiles of C$_{3}$H$_{4}$, C$_{3}$H$_{6}$, C$_{3}$H$_{8}$ are given in Fig. \[fig:timecompare\] and Fig. \[fig:latcompare\], and are compared to profiles reported in [@bampasidis:2012] and [@vinatier:2015] in Fig. \[fig:latcompare\]. ![Normalized temperature contribution functions for each altitude bin, taken at 1311 cm$^{-1}$. Altitude bins are listed immediately to the right of each contribution function. The altitude where the contribution function peaks is listed in parentheses. In most cases, altitudes of peak contribution are not the center altitude for each bin.[]{data-label="fig:tempcf"}](2004_equator_fp4_cf){width="\columnwidth"} ![Comparison of retrieved temperatures in the 2012-2015 time span. 1-$\sigma$ error bars are given at altitudes where the contribution function peaks in each altitude bin. The solid gray line is the a priori profile with error envelope. Dot dashed lines are the retrieved profile where there are no spectra.[]{data-label="fig:temps"}](temps){width="\columnwidth"} ![Normalized propene contribution functions for each altitude bin, taken at 912.5 cm$^{-1}$. Altitude bins are listed immediately to the right of each contribution function. The altitude where the contribution function peaks is listed in parantheses. In most cases, altitudes of peak contribution are not the center altitude for each bin.[]{data-label="fig:propenecf"}](2004_equator_fp3_cf){width="\columnwidth"} ![Spectral fits for propene and propane for the 2004-2009 time span in the equatorial bin.[]{data-label="fig:fp3spec"}](2004_equator_propene_r2){height="\textheight"} ![Spectral fits used in the 2004-2009 south propyne retrieval.[]{data-label="fig:propynespec"}](2004_equator_propyne_r2){height="\textheight"} ![Vertical profiles C$_{3}$H$_{4} $(top), C$_{3}$H$_{6}$ (middle), C$_{3}$H$_{8}$ (bottom). Black is the northern bin, blue is the equatorial bin, and green is the southern bin. Volume mixing ratios are given as solid lines. 1-$\sigma$ errors are given as colored regions, dot-dashed lines, or horizontal error bars at altitudes of peak contribution.[]{data-label="fig:timecompare"}](timecompare.pdf){width="\columnwidth"} The C$_{3}$H$_{8}$ profile shows a peak stratospheric abundance at 0.5 mbar in the north during late winter, with the profile flattening out to a more vertically constant profile in the equatorial and southern bins. The post-equinox profiles show the peak altitude decreasing slightly in the north, and a similar peak at 0.7 mbar forming around the equator during early northern spring. C$_{3}$H$_{6}$ does not follow the same trend as other C$_{3}$ hydrocarbons, the maximum abundance of the gas in both time seasons is above 1 mbar near the equator. This differs from the general trend of the other C3 hydrocarbons and trace gases, which tend to increase above the mid to high winter latitudes. CH$_{3}$C$_{2}$H shows a nearly constant abundance within error bars in the northern pre-equinox bin, becoming more variable and decreasing in abundance as the latitude moves away from the winter pole. The CH$_{3}$C$_{2}$H abundance profiles for the post-equinox time span are comparable across latitudes. There appears to be some small but systematic error of the CH$_{3}$C$_{2}$H spectral fit that may require re-evaluation of the methylacetlyene line list. This misfit is also seen in Fig. 10 of [@vinatier:2007] and Fig. 12 of [@coustenis:2007]. Discussion ========== Propane ------- In the pre-equinox period, equatorial and southern propane increases with altitude from 5$\times$10$^{-7}$ at 10 mbar to 1$\times$10$^{-6}$ at 0.01 mbar. In the north (during winter at the time), propane achieves a maximum abundance of 1$\times$10$^{-6}$ at 0.3 mbar before it begins to decrease with altitude, returning to an abundance comparable to the southern and equatorial regions. In the post-equinox period, the propane distribution remains largely unchanged within error bars. The south shows the abundance around 1 mbar decrease from 1$\times$10$^{-6}$ to 8$\times$10$^{-7}$, just outside of the error bars on the retrieval. We are able to compare to [@vinatier:2007], who measure abundance from the first flyby of Titan by Cassini, Tb on 13 December 2004, at 15 $^{\circ}$S. They show propane increasing monotonically from 4$\times$10$^{-7}$ at 4 mbar to 1$\times$10$^{-6}$ at 0.01 mbar, which has a slightly greater slope than our pre-equinox values. In comparison, [@bampasidis:2012] probe the lower stratosphere using nadir observations between 2006 and 2012 and show propane generally at a southern and equatorial abundance of 5$\times$10$^{-7}$ between 2006 and 2009, in agreement with our results. At 50$^{\circ}$N, they retrieve an abundance near 2$\times$10$^{-6}$ , which is slightly higher than our values at the lowest altitudes, but agree with our limb sounding around 3 mbar. [@vinatier:2015] perform another analysis of chemical abundance during the northern spring, between 2009 and 2013. At 46$^{\circ}$N in 2012, they show propane increase from 4$\times$10$^{-7}$ at 5 mbar to 1$\times$10$^{-6}$ at 0.03 mbar. Because their analysis was done on a single observation, the error bars presented are large, so it is hard to compare variations in the vertical profile with our results, however they do seem compatible. Propyne ------- Propyne shows stronger seasonal variation than propane. In the pre-equinox period, our results show propyne having a weak vertical gradient from 8$\times$10$^{-9}$ at 8 mbar to just under 2$\times$10$^{-8}$ at 0.01 mbar in the south. The vertical gradient decreases near the equator, and becomes nearly vertical in the north, indicative of the effect of downward advection within the winter polar vortex on the abundance profiles of trace gases. In the post-equinox period, the propyne profile behaves similar across all latitudes, increasing from 1$\times$10$^{-8}$ at 8 mbar to 2$\times$10$^{-8}$ at 0.01 mbar. [@bampasidis:2012] report similar values to ours for the 2006-2009 period for 50$^{\circ}$N, 0$^{\circ}$N, and 50$^{\circ}$S. [@vinatier:2007] show a steeper vertical profile at 15$^{\circ}$S than our results indicate. Their modeled profile begins at a slightly lower abundance at 1 mbar and is in agreement with our results at higher altitudes. This could be explained by our larger dataset including observations later in the season and at slightly higher latitudes, where we would expect to see more propyne at lower altitudes. Propene ------- Photochemical models have included C$_{3}$H$_{6}$ beginning after the Voyager flyby [@yung:1984]. Since the start of the Cassini mission, several new models have also incorporated the molecule [@wilson:2004; @hebrard:2013; @kras:2014; @li:2015; @loison:2015; @dobrijevic:2016]. [@yung:1984] propose the main source of propene in Titan’s stratosphere (&lt;300 km) is from $$\label{} C_{2}H_{3} + CH_{3} + M \rightarrow C_{3}H_{6} + M$$ $$\label{} C_{2}H_{3} + CH_{3} + M \rightarrow C_{3}H_{6} + M$$ [@li:2015] follow with additional production pathways $$CH + C_{2}H_{6} \rightarrow C_{3}H_{6} + H$$ dominating the region between 600 km and 1000 km, where CH molecules are plentiful and $$C_{3}H_{8} + h\nu \rightarrow C_{3}H_{6} + H_{2}$$ dominating the region between 300 km and 600 km, where the pressure is not great enough for a termolecular reaction and CH is scarce. Alternatively, [@hebrard:2013] include $$H + C_{3}H_{5} \rightarrow C_{3}H_{6}$$ as the dominating production reactions at mid to high altitudes. The first detection of C$_{3}$H$_{6}$ in Titan’s atmosphere was made with the INMS instrument by [@magee:2009]. However, because mass spectrometers can generally not differentiate between isomers, it is unknown whether this detection is of propene or cyclopropane. The first definitive detection of propene was made by [@nixon:propene] using an average of CIRS spectra between 30$^{\circ}$S and 10$^{\circ}$N from 1 July 2004 through 1 July 2010, similar to our 20$^{\circ}$S-20$^{\circ}$N pre-equinox bin. However, a spectral line list for propene did not exist at the time, so the gas could not be included in their radiative transfer calculations. Instead, estimates of abundance were made by comparing the relative intensities of the propane and propene lines, and they claimed a 3-$\sigma$ abundance of (2.0-4.6)$\times$10$^{-9}$ between 100 and 200 km. In Fig. \[fig:licompare\], we compare the predicted propene profiles to the abundance we determined for the pre-quinox time span, centered on the equator. Of the models compared, [@loison:2015] shows the best agreement. The abundance below 6 mbar is within the error bars of our measured values. Above that, our measurements are within the 90th percentile of modeled profiles (90th percentile being the region which encloses 90% of modeled profiles.) In all other cases, the predictions have a peak abundance at or below 1 mbar, ranging from 1 to 6 ppbv. Our profiles display a peak abundance of 10 ppbv at 0.1 mbar, higher in abundance and altitude than all of the compared profiles. The profiles from [@li:2015] and [@kras:2014] show good agreement in abundance below 1 mbar, but also display an inversion above 0.2 mbar that we do not see in our measurements. We also retrieve about twice as much propene compared to the values inferred from relative line strengths in [@nixon:propene]. A modified version of the [@loison:2015] model was used as the [@dobrijevic:2016] model. Since the two models have different predicted profiles for propene, we can look at the differences between the two models in attempt to isolate the factors that caused the abundance of propene to change. [@dobrijevic:2016] list the changes from [@loison:2015] as: limiting the modeled hydrocarbons at C$_{4}$ species, excluding high mass nitriles, reducing the number of isomers included, and not considering reactions with ’very small fluxes’. The authors checked the effects of these changes on species included in the model, and while no major change was reported for significant molecules, we do see that the changes applied in [@dobrijevic:2016] decrease the predicted abundance of propene and worsen the agreement between the [@loison:2015] and our measurements. Due to the large number of reactions included in both models, we are unable to say which of these changes has the greatest effect on the modeled abundance of propene. ![Our 2004-2009 20$^{\circ}$S - 20$^{\circ}$N profile compared with published predictions for propene’s abundance. Errors on Hebrard 2013 correspond to 75th percentile, Loison 2015 correspond to 90th percentile, Dobrijevic 2016 correspond to 90the percentile. Nixon 2013 values are inferred abundances and uncertainties described in [@nixon:propene]. Of the models compared, only Loison 2015 has a similar shape and abundance to our measured values. Above 10$^{-4}$ bar, the signal from propene drops to near the NESR, so a the abundances presented here are most reliable below this altitude.[]{data-label="fig:licompare"}](licompare){width="\columnwidth"} Allene ------ The isomer of propyne, CH$_{2}$CCH$_{2}$ or allene, is also theorized to be produced in Titan’s atmosphere [@yung:1984; @li:2015]. Production pathways for allene (and propyne) include $$CH + C_{2}H_{4} \rightarrow C_{3}H_{4}$$ above 600 km, with $$H + C_{3}H_{5} \rightarrow C_{3}H_{4} + H_{2}$$ and $$CH_{3} + C_{3}H_{5} \rightarrow C_{3}H_{4} + CH_{4}$$ dominating throughout the rest of the atmosphere. The only potential detection of allene on Titan was made in [@roe:2011], however since the line list used in the authors’ analysis was in significant disagreement with the potential observed allene lines this remains only a tentative detection. Other analyses of Titan’s atmosphere to search for allene have resulted in many upper limits. [@coustenis:2003] claimed a 3-$\sigma$ upper limit of 2 ppbv, derived from disk averaged spectra from the Infrared Space Observatory (ISO). [@nixon:allene] estimate a 3-$\sigma$ upper limit of 0.3 ppbv at 25$^{\circ}$ N at 107 km, and 1.6 ppbv at 76$^{\circ}$ N, 224 km, using a method described in their paper. As discussed in Section 2.2.1, the line data used in these previous works is inaccurate. We update these upper limits using a corrected line list produced by us and included in our radiative transfer calculations. The spectrum used in the model is an average of nadir observations between 30$^{\circ}$S and 30$^{\circ}$N, from 2004-2015. Nadir observations were used to take advantage of the lower noise compared to limb viewing, however the abundance of allene is likely to be very low at the altitudes that nadir observations are sensitive to. Fig. \[fig:allenespec\] shows the best fit spectrum overlayed on the observed spectrum. The contributions of allene are visible as spikes in the dotted-line residual. We perform a $\Delta \chi ^{2}$ analysis similar to that described in [@teanby:2007] and [@nixon:allene] and estimate an updated 3-$\sigma$ upper limit of 2.5 ppbv at 150 km within 30$^{\circ}$ north and south of the equator. A retrieval of ethane, propane, and an aerosol haze was performed between 830 and 880 cm$^{-1}$, with allene line data not included. The $\chi^{2}$ value for this retrieval was considered $\chi_{0} ^{2}$. The best-fit abundances of ethane and propane were then fixed and varying amounts of allene were added to the atmosphere model. A forward model was run at each allene abundance to calculate a modified chi-squared value ($\chi_{m}$), the $\Delta \chi ^{2}$ value was calculated to be $\chi_{m} ^{2}$ - $\chi_{0} ^{2}$. We plot the values of $\Delta \chi ^{2}$ as a function of added allene in Fig. \[fig:allenechi\]. Where the $\Delta \chi ^{2}$ achieves a value of 9, we claim an upper limit with a confidence of $(\Delta \chi ^{2})^{1/2}$ = 3-$\sigma$. The results are shown in Fig. \[fig:allenespec\] and Fig. \[fig:allenechi\]. A major challenge to modeling this allene band is the overlap of the band with the band of ethane and the band of propane. The ethane band is very bright, as seen in Fig. \[fig:allenespec\]. Therefore, relatively small discrepancies in the modeling and line list of ethane propagate heavily through to measuring allene. The propane band also present in the region we modeled is very dim and broad, further increasing the difficulty of measuring allene in this region, since it contributes mostly to the continuum in this small spectral window. ![Comparison of modeled spectrum without allene (green) and forward modeled spectrum including allene (blue), against the observed spectrum (black). Differences in the spectra are most easily noted in the residuals, where contributions from allene are seen as excess emission in the dashed spectrum, noted by asterisks.[]{data-label="fig:allenespec"}](allenecompare){height="\textheight"} ![Plot of the $\Delta \chi^{2}$ against allene abundance. $\Delta \chi^{2}$ is calculated by subtracting the $\chi^{2}$ calculated from the forward model with allene included in the atmosphere from the $\chi^{2}$ value calculated from retrieved spectrum that does not include allene. Along these lines, a negative $\Delta \chi^{2}$ indicates the model-fit has improved, whereas a positive value indicates the fit is worsened. The $\Delta \chi^{2}$ value achieves a value of 9 near 2.5 $\times$10$^{-9}$, leading us to an estimated 3-$\sigma$ upper limit of 2.5 ppbv.[]{data-label="fig:allenechi"}](allene_chi2){width="\columnwidth"} Conclusion ========== In this work, we have determined the first abundance profiles of propene in Titan’s atmosphere, enabling us to compare the mixing ratio and spatial distribution of this gas with other trace gases, as well as compare to predictions from existing photochemical models. We’ve shown: 1. propene is present in Titan’s atmosphere at a mixing ratio between 4 and 10 ppbv in the stratosphere 2. the abundance of propene near the equator is consistent with predictions from the [@loison:2015] model, but other predictions show a local inversion centered around 0.1 mbar, and a generally lower abundance 3. contrary to other trace gases, propene does not show a poleward enhancement in the winter hemisphere equatorward of 60$^{\circ}$. Instead, propene shows an increased abundance above the equator relative to either pole. The results of our analysis will be useful in refining models of Titan’s atmospheric chemistry and dynamics as discrepancies between observations and predictions show that current photochemical models do not accurately predict the production and destruction rates of the molecule. The unique spatial trend exhibited by propene will make it a useful constraint in global circulation and transport models, as it may be a good tracer for horizontal transport. Additionally, the polymerization of propene and other $\pi$-bond molecules may lead to the formation of Titan’s haze[@teanby:haze; @trainer:2013]. We also provide a new 3-$\sigma$ upper limit for allene within 30$^\circ$ of the equator of 2.5$\times10^{-9}$. Because this value is calculated using corrected line data, it should be considered more reliable than values from previous calculations. Acknowledgments =============== N.A.L., C.A.N., R.K.A., and F.M.F. were supported by the NASA Cassini Project for the research work reported in this paper. C. A. N. also acknowledges support from the NASA Astrobiology Institute. Part of the research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. References ==========

--- abstract: 'The magnetic nature of Cs$_{2}$AgF$_{4}$, an isoelectronic and isostructural analogue of La$_2$CuO$_4$, is analyzed using density functional calculations. The ground state is found to be ferromagnetic and nearly half metallic. We find strong hybridization of Ag-$d$ and F-$p$ states. Substantial moments reside on the F atoms, which is unusual for the halides and reflects the chemistry of the Ag(II) ions in this compound. This provides the mechanism for ferromagnetism, which we find to be itinerant in character, a result of a Stoner instability enhanced by Hund’s coupling on the F.' author: - 'Deepa Kasinathan,$^1$ A. B. Kyker,$^1$ and D. J. Singh$^2$' title: 'Origin of ferromagnetism in Cs$_2$AgF$_4$: importance of Ag - F covalency' --- Cs$_2$AgF$_4$ is a member of a family of Ag(II) fluorides that form in perovskite and layered perovskite structures. The distinguishing feature is the presence of Ag(II), which is a powerful oxidizing agent. [@hoppe1; @hoffmann] This compound was first synthesized in 1974 by Odenthal and co-workers. [@hoppe] It occurs in the tetragonal K$_{2}$NiF$_{4}$ layered perovskite structure. This is the same structure as the parent of the high temperature superconducting cuprates, La$_2$CuO$_4$. Cs$_2$AgF$_4$ shows no tilts or rotations of the octahedra, which are common in oxide layered perovskites. Synthesis of isostructural Na$_2$AgF$_4$ and K$_2$AgF$_4$ was also reported and these compounds also have the K$_{2}$NiF$_{4}$ structure. All three compounds are reported as being blue or purple in appearance and ferromagnetic. While transport measurements have not been reported for these compounds, it is known that the related distorted perovskite compound KAgF$_3$ is metallic at high temperatures, and then has a metal insulator transition coincident with an antiferromagnetic ordering temperature. [@g-kagf3] In the doped high-T$_c$ cuprates, superconductivity develops from a paramagnetic metallic phase, with Fermi surfaces coming from hybridized Cu $d$ - O $p$ bands. These are formally antibonding bands of $d_{x^2-y^2}$ - $p_\sigma$ character. [@pickett] While the theory of high temperature cuprate superconductivity remains to be established, it is widely held that the phenomenon is associated with the physics of the undoped compounds, which are antiferromagnetic Mott insulators. Specifically, it is thought that there is a relationship between superconductivity and the antiferromagnetic fluctuations associated with the correlated $d$ electrons of cuprates. Cs$_2$AgF$_4$ has interesting similarities to the high-T$_c$ cuprates. As mentioned, it is isostructural, featuring AgF$_2$ sheets in place of CuO$_2$ sheets, it has a transition element with a $d^9$ configuration, and it is magnetic. Moreover, related compounds have been shown both in band structure calculations and X-ray photoelectron spectroscopy experiments to display significant Ag - F covalency, reminiscent of the Cu - O hybridization in the cuprates. [@hoffmann; @jaron; @g2] These similarities and other considerations have led to speculations about possible high temperature superconductivity in Ag(II) fluorides. [@hoffmann; @g3] One puzzling difference between the cuprates and the layered Ag(II) fluorides is that the undoped cuprates are antiferromagnetic, while the argentates are ferromagnetic. One possible explanation would be an orbital ordering that favors ferromagnetism within a superexchange framework, as was recently suggested. However, neutron measurements did not detect the symmetry lowering that would occur in this case. [@mclain] Here we use electronic structure calculations to elucidate the electronic structure of Cs$_2$AgF$_4$ and the origin of its magnetic properties. A previous density functional calculation for this material found it to be a covalent metal,[@hoffmann] with a substantial density of states (DOS) at the Fermi level (E$_{F}$) in the absence of magnetism. We did electronic structure calculations within the local spin density approximation (LSDA) and the generalized gradient approximation (GGA), [@pw; @pbe] using the general potential linearized augmented planewave method, with local orbitals, [@lapw; @lo] as implemented in the WIEN2K program. [@wien] The augmented planewave plus local orbital extension was used for the Ag $d$ and semicore levels. [@apw] The valence states were treated in a scalar relativistic approximation, while the core states were treated relativistically. Well converged basis set sizes and Brillouin zone samplings were employed. Except as noted otherwise, the LAPW sphere radii were 2.0 $a_0$ and 1.85 $a_0$ for the metal and fluorine atoms, respectively. The basis set cut-off was chosen to be $RK_{max}$=7.0, where $R$ is the radius of the F sphere. We tested the convergence by comparison of LSDA results with an independent code, employing the LAPW augmentation with local orbitals and with higher basis set cut-offs as well as different sphere radii. The structural data were obtained from the report[@hoppe] of Odenthal and co-workers: $a$ = 4.58Å, $c$ = 14.19Å, including the two internal parameters corresponding to the Cs and apical F heights above the AgF$_2$ square planar sheets. Minimization of the forces in the LDA approximation yielded a value of z$_{\rm Cs}$=0.361 and z$_{\rm F}$=0.147, in close agreement with the experimental values of z$_{\rm Cs}$=0.36 and z$_{\rm F}$=0.15. Within the LSDA we find a Cs$_2$AgF$_4$ to be a metal on the borderline of ferromagnetism. Fixed spin moment calculations showed a non-spin-polarized ground state, but with a 1 $\mu_B$ per formula unit fully polarized solution only 35 meV higher in energy. We also did LSDA calculations applying fields only inside the Ag LAPW spheres, which were chosen to be 2.1 $a_0$ in radius for this purpose. With 5 mRy fields of this type in a ferromagnetic pattern, moments of 0.35 $\mu_B$ were induced in the Ag spheres, and moments also appeared in the F spheres, for a total spin magnetization of 0.62 $\mu_B$. Application of the same field in an in-plane $c$(2x2) antiferromagnetic pattern yielded induced Ag moments in the spheres of only 0.17 $\mu_B$, with a small moment also appearing on the apical F, but no moments on the in-plane F, as is required by symmetry. This shows the system to be much closer to ferromagnetism than antiferromagnetism at the LSDA level, and suggests an important role for the in-plane F in the magnetism. Within the GGA, we obtain a ferromagnetic ground state, with spin magnetization, $M=0.9 \mu_B$ and energy 6 meV below the non-spin polarized solution. However, we do not find any metastable antiferromagnetic solution, implying itinerant magnetism, in particular, the absence of stable local moments. The calculated electronic density of states (DOS) for the ferromagnetic ground state is shown in Fig. \[dos\]. The band structure is shown in Fig. \[bands\], and the Fermi surface in Fig. \[fermi\]. The band structure is expected to be two dimensional, due to the bonding topology, which has 180$^\circ$ Ag-F-Ag links in the AgF$_2$ sheets, but no direct Ag-F-Ag connections in the $c$-axis direction. This in fact is the case. [@disp-note] As may be seen, Cs$_2$AgF$_4$ is close to a half metal, with the Fermi energy being near a band edge in the majority channel, but not in the minority channel. The minority spin Fermi surface consists of small hole cylinders running along the zone corner (from the $d_{x^2-y^2}$ band) and electron cylinders around the zone center i(from the $d_{z^2}$ band). The majority spin Fermi surface consists of a single large square cylindrical electron surface that almost fills the Brillouin zone, leaving a small region of holes around the zone boundary. Cs$_{2}$AgF$_{4}$ has two type of F sites forming distorted Ag centered octahedra; one is in the AgF$_2$ sheets (referred as F1 in this paper), and the other is the apical F along the $c$ - axis (referred as F2 in this paper). The apical Ag - F2 distance is slightly smaller than the in-plane Ag - F1 distance. A key point is that the F1 atoms bridge the Ag atoms in the sheets, with 180$^\circ$ bonds, while the apical, F2 atoms connect to only one Ag atom and therefore are not bridging. Examining the DOS and projections in more detail, one may note that the valence bands have substantially mixed Ag $d$ - F $p$ character, especially near the bottom and top of the manifold where $e_g$ - $p_\sigma$ bonding and antibonding combinations occur. This hybridization involves both F1 and F2 atoms, and is consistent with previous results for Ag(II) fluorides. [@hoffmann; @jaron; @g2] The result is a very stable metallic electronic structure, with substantial F character at the Fermi energy, $E_F$, and a valence band width of $\sim$ 5.5 eV. This strong hybridization can be understood in chemical terms considering the very strongly oxidizing character of Ag(II), which in this compound partially oxidizes F$^{-}$. Thus covalency in this compound is a consequence of the unusual valence state of Ag. Turning to the band structure, there are two bands crossing $E_F$ in the minority spin channel. These are the $d_{x^2-y^2}$, which hybridizes with the in-plane F, and the $d_{z^2}$ hybridized with the apical F. The $d_{x^2-y^2}$ - F1 combination has greater dispersion because of the in-plane topology, mentioned above. However, the $d_{z^2}$ - F2 combination is higher lying, with the result that the two band maxima nearly coincide. The higher lying position of the $d_{z^2}$ - F2 is readily explained by the fact that these bands are antibonding $e_g$ - $p_\sigma$ and the Ag - F2 bond is shorter. In the minority spin channel, the $d_{z^2}$ band extends from from -0.5 to 0.7 eV (relative to $E_F$), while the $d_{x^2-y^2}$ band extends from -2.1 to 0.5 eV. In the absence of the lighter $d_{x^2-y^2}$ band, one would have a half-filled $d_{z^2}$ band. Because the $d_{x^2-y^2}$ is in fact present, the $d_{z^2}$ is slightly electron doped away from half filling. This is in contrast to the cuprates, where only a $d_{x^2-y^2}$ band is active, and this band is hole doped away from half-filling in the highest $T_c$ compounds. The small size of F$^-$ relative to O$^{2-}$ emphasizes the effect of the perovskite bonding topology in the band structure. This is because direct F - F hopping is reduced by its small size, relative to O in oxides, and the strongly oxidizing nature of F and Ag(II) push the Cs conduction bands to high energy, reducing the assisted hopping via Cs for the valence bands. Thus, the hopping is dominated by nearest neighbor Ag-F channels, so for example, the $d_{xz} - p_\pi$ and $d_{yz} - p_\pi$ derived bands take strong one dimensional character and are seen to be almost perfectly flat along some directions as seen in Fig. \[bands\]. The mixed character of the bands is reflected in the distribution of the magnetic moments in the ferromagnetic ground state. Of the total spin moment of 0.9 $\mu_{B}$, only 0.5 $\mu_{B}$ lies within the Ag LAPW sphere, radius 2.0 $a_0$. The remaining $\sim$ 40% of the magnetization is F derived, approximately equally divided between the F1 and F2 sites. This is important for understanding the itinerant ferromagnetic ground state that we find. First of all, the large moments on the in-plane F1 atoms seen both in the GGA ferromagnetic ground state and in the LSDA calculations with ferromagnetic fields in the Ag (but not the F) spheres, mean that there is a contribution to the energy from the F polarization. F$^-$ is a relatively small anion, at the end of the first row of the periodic table. Thus, when magnetic, it can have a strong Hund’s coupling. This provides a generalized double exchange mechanism for favoring ferromagnetism, similar to the mechanism in SrRuO$_3$. [@singh-ru; @mazin-ru] In the ferromagnetic case, the F1 atoms take moments due to the hybridized character of the bands around $E_F$ and contribute to the Stoner instability through their Hund’s coupling. With antiferromagnetic ordering, no induced moments can be present on the F1 atoms by symmetry, and therefore the Hund’s coupling on these sites cannot stabilize the magnetism. Thus, the fact that the moments become unstable in an in-plane antiferromagnetic configuration supports this picture. Different from the ruthenates, the hybridized states in Cs$_2$AgF$_4$ involve $e_g$ instead of $t_{2g}$ states, and the F$^{-}$ ion is much smaller than O$^{2-}$. We studied the stability of this ferromagnetic, two-band electronic structure, using LDA+U calculations and treating the Coulomb $U$ as a parameter. We found, as expected, that a local moment, insulating state could be obtained. However this only happened when using a very high value $U$=7 eV. This is an unreasonably large value for a 4$d$ ion in a screening environment. The reason for the weak effect of more realistic values of $U$ is that the bands are strongly hybridized, and are really mixed F $p$ - Ag $d$ bands, and not narrow bands built from the Ag $d$ orbitals. Thus we conclude that on-site Coulomb correlations do not have a large effect on the electronic structure or magnetism of this compound. This is in contrast to the undoped cuprates, where the LSDA and GGA approximations incorrectly predict non-magnetic metallic ground states, and the Hubbard $U$ is crucial for obtaining moment formation and an insulating ground state. To summarize, density functional calculations of the electronic structure of Cs$_2$AgF$_4$ show strong covalency between Ag $d$ and F $p$ states. Within the GGA, the ground state is ferromagnetic, and is stabilized by Hund’s coupling on the in-plane F atoms which occurs due to F participation in the magnetism resulting from the $e_g$ - $p_\sigma$ hybridization. The electronic structure is nearly half-metallic, and not insulating. It would be of interest to experimentally test the prediction of a metallic electronic structure. The resulting picture of the electronic structure and magnetism is very different from the undoped cuprates. (1) Cs$_2$AgF$_4$ has two active bands: $d_{x^2-y^2}$ and $d_{z^2}$; neither is exactly half-filled; (2) moment formation in Cs$_2$AgF$_4$ is due to a Stoner type mechanism as opposed to on-site Coulomb repulsions that are crucial in the cuprates; (3) the magnetism has strong itinerant character due to F participation, as opposed to the local moment superexchange mediated character of cuprate antiferromagnetism; and (4) we find ferromagnetism with the ideal tetragonal structure; orbital ordering to obtain ferromagnetism within a superexchange mediated framework is not needed. We note that the predicted F contributions to the magnetism are large enough to be detected using neutron scattering. Finally, we note that the mechanism for ferromagnetism in Cs$_2$AgF$_4$ is quite robust, and would expected to occur in other Ag(II) fluorides with similar bond lengths and topologies. Since it does not rely on small structural effects, it provides a ready explanation for the observed ferromagnetism in the other $A_2$AgF$_4$ compounds. Furthermore, the above picture of itinerant magnetism may be more generally applicable to other Ag(II) fluoride compounds. For example, KAgF$_3$ shows a metal insulator transition coincident with an antiferromagnetic ordering. [@g-kagf3] This is much more natural in an itinerant system than in a strongly correlated local moment system, which would tend to be insulating on both sides of the ordering temperature at odd integer band fillings. The structure of that compound shows compressed octahedra and Ag-F-Ag chains along the $c$-axis direction with short bond lengths. Assuming that the above mechanism applies also to this compound, one may expect ferromagnetic chains along $c$. Considering that the ground state is known to be antiferromagnetic, one may anticipate a C-type ordering of antiferromagnetic $a-b$ planes, stacked ferromagnetically in that case. In any case, in perovskite derived Ag(II) fluorides, the mechanism that we propose would generally favor ferromagnetism or complex antiferromagnetic states, with ferromagnetic interactions along some bonding directions, as opposed to simple G-type ordering. We are grateful for helpful discussions with W.E. Pickett and J. Turner. Research at ORNL was sponsored by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, U.S. Department of Energy, under contract DE-AC05-00OR22725 with Oak Ridge National Laboratory, managed and operated by UT-Battelle, LLC. Work at UC Davis was supported by DOE contract DE-FG03-01ER45876. [100]{} R. Hoppe, Angew. Chem. Int. Ed. Engl. [**20**]{}, 63 (1981). W. Grochala and R. Hoffmann, Angew. Chem. Int. Ed. Engl. [**40**]{}, 2742 (2001). R.-H. Odenthal, D. Paus and R. Hoppe, Z. Anorg. Allg. Chem. [**407**]{}, 144 (1974). W. Grochala and P.P. Edwards, Phys. State Sol. (b) [**240**]{}, R11 (2003). W.E. Pickett, H. Krakauer, R.E. Cohen and D.J. Singh, Science [**255**]{}, 46 (1992). T. Jaron, W. Grochala and R. Hoffmann, Phys. State Sol. (b) [**242**]{}, R1 (2005). W. Grochala, R.G. Egdell, P.P. Edwards, Z. Mazej and B. Zemva, ChemPhysChem [**4**]{}, 997 (2003). W. Grochala, A. Porch and P.P. Edwards, Solid State Commun. [**130**]{}, 137 (2004). S.E. McLain, D. A. Tennant, J. F. C. Turner, T. Barnes, M. R. Dolgos, Th. Proffen, B. C. Sales and R. I. Bewley, cond-mat/0509194 (2005). J.P. Perdew and Y. Wang, Phys. Rev. B [**45**]{}, 13244 (1992). J.P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett. [**77**]{}, 3865 (1996). D.J. Singh and L. Nordstrom, Planewaves, Pseudopotentials and the LAPW Method, 2nd Ed. (Springer, Berlin, 2006). D. Singh, Phys. Rev. B [**43**]{}, 6388 (1991). P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, and J. Luitz, [WIEN2K]{}, 2002, an Augmented Plane Wave + Local Orbitals Program for calculating crystal properties (Karlheinz Schwarz, Technische Universitat Wien, Austria). E. Sjostedt, L. Nordstrom and D.J. Singh, Solid State Commun. [**114**]{}, 15 (2000). This can be seen for the valence bands in Fig. \[bands\] from the $\Gamma -M$ and $X - P$ lines, which are along $k_z$ in the body centered tetragonal zone. The two-dimensional character is also evident in the Fermi surfaces. The higher lying (mainly Cs) derived states above the gap are more three dimensional in character. D.J. Singh, J. Appl. Phys. [**79**]{}, 4818 (1996). I.I. Mazin and D.J. Singh, Phys. Rev. B [**56**]{}, 2556 (1997).

--- abstract: 'A wide range of stochastic processes that model the growth and decline of populations exhibit a curious dichotomy: with certainty either the population goes extinct or its size tends to infinity. There is a elegant and classical theorem that explains why this dichotomy must hold under certain assumptions concerning the process. In this note, I explore how these assumptions might be relaxed further in order to obtain the same, or a similar conclusion, and obtain both positive and negative results.' address: 'Biomathematics Research Centre, University of Canterbury, Christchurch, New Zealand' author: - Mike Steel title: 'Reflections on the extinction–explosion dichotomy' --- Extinction, Borel–Cantelli lemma, population size, coupling, Markov chain Introduction ============ The ‘merciless dichotomy’ (Section 5.2 of [@had]) concerning extinction refers to a very general property of stochastic processes that describes the long-term fate of populations. Roughly speaking, the result states that if there is always a strictly positive chance the population could become extinct in the future (depending, perhaps, on the current population size), then the population is guaranteed to either become extinct or to grow unboundedly large. More precisely, a formal version of this result, due to Jagers (Theorem 2 of [@jag]), applies to any sequence $X_1, X_2, \ldots, X_n \ldots $ of non-negative real-valued random variables that are defined on some probability space and which is absorbing at 0 (i.e. $X_n=0 \Rightarrow X_{n+1}=0$ for all $n$). It states that, provided: $$\label{strong} {{\mathbb P}}(\exists r: X_r=0|X_1, X_2, \ldots, X_n) \geq \delta_x>0 \mbox{ whenever $X_n \leq x$}$$ holds for all positive integers $n$, then, with probability 1, either $X_n \rightarrow \infty$ or a value of $n$ exists for which $X_k=0$ for all $k\geq n$ (notice that $\delta_x$ can tend towards 0 at any rate as $x$ grows). This result applies to a wide variety of stochastic processes studied in evolutionary and population biology (e.g. Yule birth-death models, branching processes etc) and the proof in [@jag] involves an elegant and short application of the martingale convergence theorem. Note that the processes in [@jag] (and here) need not be Markovian. Nevertheless, the lower-bound inequality condition in (\[strong\]) has a Markovian-like feature that it is required to hold for all values of $X_1, X_2, \ldots, X_{n-1}$ whenever $X_n$ is less than $x$. This raises the question of how much this uniform bounding across the previous history of the process might be relaxed without sacrificing the conclusion of certain extinction or explosion. In this short note, we consider possible extensions of Jagers’ theorem by weakening the assumption in (\[strong\]). Specifically, we will consider a lower bound that conditions just on the event that $0<X_n \leq x$, either alone or alongside another variable that is dependent on (but less complete than) the past history $X_1, \ldots, X_{n-1}$. First, we consider what happens if the probability in the lower bound (\[strong\]) were to condition just on $0<X_n \leq x$. In this case, we describe a positive result that delivers a slightly weaker conclusion than the original theorem of Jagers. We then show that the full conclusion cannot be obtained by lower bounds that condition solely on $0<X_n\leq x$ by exhibiting a specific counterexample. However, in the final section, we show that the full conclusion of Jagers’ theorem can be obtained by conditioning on $0<X_n\leq x$, together with some partial information concerning the past history of the process. A simple general lemma and its consequence for bounded populations ================================================================== We first present an elementary but general limit result, stated within the usual notation of a probability space $(\Omega, \Sigma, {{\mathbb P}})$ consisting of a sigma-algebra $\Sigma$ of ‘events’ (subsets of the sample space $\Omega$) and a probability measure ${{\mathbb P}}$ (for background on probability theory, see [@borel]). Suppose that $E_1, E_2,\ldots $ are [*increasing*]{} (i.e. $E_i \subseteq E_{i+1}$) and $E = \bigcup_{n=1}^{\infty}E_n$. For example, suppose that $E_n$ is the event that some particular ‘situation’ (e.g. extinction of the population) has arisen on or before a given time step $n$ (e.g. day, year). These events are increasing and their union $E$ is the event that the ‘situation’ eventually arises. We are interested in when ${{\mathbb P}}(E)=1$. A sufficient condition to guarantee this is to impose any non-zero lower bound on the probability that the ‘situation’ arises at time step $n$ given that it has not done so already; in other words, to require that the conditional probability ${{\mathbb P}}(E_n|\overline{E_{n-1}})$ is at least $\delta >0$ for all sufficiently large values of $n$ (throughout this paper an overline denotes the complementary event). On the other hand, it is equally easy to check that if $p_n={{\mathbb P}}(E_n|\overline{E_{n-1}})$ is allowed to converge to zero sufficiently quickly (so the probability of the ‘situation’ first arising on day $n$ goes to zero sufficiently fast that $\sum_n p_n < \infty$), then it is possible for ${{\mathbb P}}(E)<1$. For example, if accidents occur independently and the probability of a particular accident is reduced each year by $1\%$ of its current value, then there is a positive probability that no accident will ever occur; but if the probability reduces at the rate $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \cdots,$ then an accident is guaranteed to eventually occur (by the second Borel–Cantelli lemma). Rather than placing some lower bound on the probability that the situation arises at time step $n$, we can, following [@jag], make a weaker assumption that if the situation has not happened yet, there is always a non-vanishing chance that it will occur some time in the future (formally, requiring merely that ${{\mathbb P}}(E|\overline{E_n})$ is uniformly bounded away from $0$). For maximal generality, we also wish to avoid any Markovian or independence assumptions. The following lemma provides a sufficient condition for ${{\mathbb P}}(E)=1$ without any further assumptions, and uses an elementary argument that will be useful later. \[figure7\] \[mike\] Suppose $E_n$ is an increasing sequence with limit $E$ and suppose that for some $\epsilon > 0$, ${{\mathbb P}}(E|\overline{E_n}) \geq \epsilon$ holds for all $n \geq 1$. Then ${{\mathbb P}}(E)=1$. [*Proof:*]{} Let $p_n = P(E_n)$. Then, by the law of total probability: ${{\mathbb P}}(E) = {{\mathbb P}}(E|\overline{E_n})(1 - p_n) + {{\mathbb P}}(E|E_n)p_n$. Now, ${{\mathbb P}}(E|E_n) = 1$ and, by assumption, ${{\mathbb P}}(E|\overline{E_n}) \geq \epsilon$. Therefore: ${{\mathbb P}}(E) \geq \epsilon(1 - p_n) + p_n$. Since the events $E_n$ are increasing, a well known and elementary result in probability theory ensures that ${{\mathbb P}}(E) = \lim_{n \to \infty} p_n$. So, letting $n \to \infty$ in the previous inequality gives: ${{\mathbb P}}(E) \geq \epsilon (1 - {{\mathbb P}}(E)) + {{\mathbb P}}(E)$, which implies that ${{\mathbb P}}(E) = 1$, as claimed. $\Box$ Example 1 --------- Consider population of a species where $X_n$ denotes the size of the population at time step $n$. The event $E_n = \{X_n=0\}$ is the event that the population is extinct by time step $n$ and this increasing sequence has the limit $E$ equal to the event of eventual extinction. In this setting, Lemma  \[mike\] provides the following special case of Jagers’ theorem. \[coro1\] Suppose that $X_1, X_2, \ldots, X_n$ is a sequence of non-negative real-valued random variables that are absorbing at 0 and are constrained to lie between $0$ and $M$. Moreover, suppose that for some $\delta>0$ and all positive integers $n$ we have: ${{\mathbb P}}(\exists r: X_r=0|X_n \neq 0) \geq \delta.$ Then, with probability 1, a value $n$ exists for which $X_k=0$ for all $k \geq n$. $\Box$ Remarks ------- - One might view Lemma \[mike\] as a simple formulation of ‘Murphy’s Law’ – the idea that if something bad can happen, it will at some point (a popular claim often made in jest that has an interesting history [@murphy]). In that context, $E_n$ is simply the event that the ‘bad thing’ has happened on or before day $n$. - The proof of Proposition \[mike\] shows that $\lim_{n \rightarrow \infty} {{\mathbb P}}(E|\overline{E_n})>0 \Longrightarrow {{\mathbb P}}(E)=1.$ The converse also holds, provided that ${{\mathbb P}}(E_n)<1$ for all $n$; indeed under that restriction, a sharper limit can be stated: ${{\mathbb P}}(E)=1 \Longrightarrow \lim_{n \rightarrow \infty} {{\mathbb P}}(E|\overline{E_n})=1.$ With a view towards Borel–Cantelli type results, note also that one can have: $\sum_{n \geq 1} {{\mathbb P}}(E|\overline{E_n}) = \infty$ and ${{\mathbb P}}(E) <1$, if, for example, ${{\mathbb P}}(E_n) = q-\frac{1}{n}$, where $q<1$. - A general characterisation for when ${{\mathbb P}}(E)=1$ is the following result from [@bruss]. \[nice\] If $E_n$ is an increasing sequence of events with limit $E$, then ${{\mathbb P}}(E)=1$ if and only if either ${{\mathbb P}}(E_1)=1$ or ${{\mathbb P}}(E_i|\overline{E_{i-1}})=1$ for some $i$, or $\sum_{i=1}^\infty {{\mathbb P}}(E_{t_i}|\overline{E_{t_{i-1}}})= \infty$ for some strictly increasing sequence $t_i$. A convergence in probability result for $X_n$ ============================================= We now consider what happens if the population size is not bounded above by some maximal value $M$ as in Corollary \[coro1\]. In this case, by weakening the conditioning in Inequality (\[strong\]) to just $X\in (0,m]$, one can still derive a result a result concerning convergence in probability (rather than almost sure convergence) of the population size to 0 or infinity, as we now show. \[jag2\] Suppose that $X_1, X_2, \ldots, X_n$ is a sequence of non-negative real-valued random variables that are absorbing at 0, and that for each positive integer $m$, there is a value $\delta_m>0$ for which the following holds for all values of $n$: $$\label{boundful1} {{\mathbb P}}(\exists r: X_r=0|X_n \in (0,m]) \geq \delta_m.$$ Then, for every $m \geq 1$, we have $\lim_{n \rightarrow \infty} {{\mathbb P}}(X_n =0 \cup X_n >m) = 1.$ [*Proof:*]{} Throughout this proof we will let $E$ denote the event $\{\exists r: X_r=0\}$. The proof of Proposition \[jag2\] relies on the following result. > [**Claim:**]{} Both ${{\mathbb P}}(X_n=0|X_n \leq m)$ and ${{\mathbb P}}(E|X_n \leq m)$ converge to 1 as $n \rightarrow \infty$. Proposition \[jag2\] follows directly from this claim, since, for any $m \geq 1$: $${{\mathbb P}}(X_n =0 \cup X_n >m) = {{\mathbb P}}(X_n=0) + {{\mathbb P}}(X_n > m)$$ $$\geq {{\mathbb P}}(X_n=0|X_n \leq m){{\mathbb P}}(X_n\leq m)+ {{\mathbb P}}(X_n >m).$$ By the claim, $ {{\mathbb P}}(X_n=0|X_n \leq m)$ converges to 1 as $n$ grows, and so the previous inequality ensures that $\lim_{n \rightarrow \infty} {{\mathbb P}}(X_n =0 \cup X_n >m) = 1,$ as required. Thus it suffices to establish the claim. [*Proof of Claim:*]{} Consider any subsequence $n(k)$ of positive integers for which the bounded sequence ${{\mathbb P}}(X_{n(k)} \leq m)$ has a limit. Such subsequences exist (by the Bolzano–Weierstrass theorem), and since $\lim \inf_{n \rightarrow \infty} {{\mathbb P}}(X_n \leq m)>0$ (by (\[boundful1\])) for all $m\geq 1$, the limit of ${{\mathbb P}}(X_{n(k)} \leq m)$ for any such subsequence is strictly positive (this latter observation also ensures that some conditional probabilities below are well defined for large enough values of $k$). By the law of total probability: $${{\mathbb P}}(E|X_{n(k)} \leq m) = {{\mathbb P}}(E|X_{n(k)}=0) {{\mathbb P}}(X_{n(k)}=0|X_{n(k)} \leq m)$$ $$+ {{\mathbb P}}(E|X_{n(k)} \in (0, m]) {{\mathbb P}}(X_{n(k)}>0|X_{n(k)} \leq m).$$ Thus if we let $p_k = {{\mathbb P}}(X_{n(k)}=0|X_{n(k)} \leq m)$, then, by (\[boundful1\]): $$\label{pp1} {{\mathbb P}}(E|X_{n(k)} \leq m) \geq 1 \cdot p_k +\delta_m(1-p_k).$$ Now, $p_{k} = {{\mathbb P}}(X_{n(k)}=0)/{{\mathbb P}}(X_{n(k)} \leq m)$ and so $\lim_{k \rightarrow \infty} p_{k} = \frac{\lim_{k\rightarrow \infty} {{\mathbb P}}(X_{n(k)}=0)}{\lim_{k\rightarrow \infty} {{\mathbb P}}(X_{n(k)} \leq m)}$, since the numerator and denominator limits are non-zero. Moreover, we have ${{\mathbb P}}(E)=\lim_{k\rightarrow \infty} {{\mathbb P}}(X_{n(k)}=0)$ and so: $$\label{pp2} \lim_{k \rightarrow \infty} p_{k} = \lim_{k \rightarrow \infty} {{\mathbb P}}(E|X_{n(k)} \leq m).$$ Let $p$ denote the shared limit in Eqn. (\[pp2\]). Then, from Inequality (\[pp1\]) we have: $$p \geq p + \delta_m(1-p),$$ which implies that $p=1$. Thus, for [*all*]{} subsequences $n(k)$ of positive integers for which ${{\mathbb P}}(X_{n(k)} \leq m)$ has a limit, this limit takes the same value (namely 1). It follows from a well-known result in analysis (e.g. Theorem 11, p. 67 of [@mal92]) that the full sequence ${{\mathbb P}}(X_n \leq m)$ also converges to 1 as $n \rightarrow \infty$, and, therefore, so do the sequences ${{\mathbb P}}(X_n=0|X_n \leq m)$ and ${{\mathbb P}}(E|X_n \leq m)$. This establishes the two limit claims in the Claim, and so completes the proof of Proposition \[jag2\]. $\Box$ Notice that Proposition \[jag2\] also implies Corollary \[coro1\] by taking $m = M$ and $\delta_m = \delta$ in (\[boundful1\]). The conclusion of Proposition \[jag2\] cannot be strengthened to almost sure convergence. ----------------------------------------------------------------------------------------- Suppose $X_1, X_2, \ldots, X_n \ldots$ is a sequence of non-negative real-valued random variables that satisfy the conditions described in Proposition \[jag2\]. In this case, the proposition assures us that $X_n$ converges in probability either to 0 or to infinity. This is a weaker conclusion than the statement that, with probability $1$, either $X_n=0$ for all sufficiently large $n$, or $X_n \rightarrow \infty$. We now show, by an explicit example, that such a stronger conclusion (which holds under the stronger condition (\[strong\]) required for Jagers’ theorem) need not hold under just the conditions described in Proposition \[jag2\]. In other words, some additional conditioning on the past history of the process is required in order to secure the stronger conclusion (we describe this further in the next section). Example 2 --------- Consider the following process. Let $X_n^1, n \geq 1$ be a sequence of independent random variables with: $${{\mathbb P}}(X^1_n) = \begin{cases} 1, & \mbox{ with probability } \frac{1}{n};\\ n, & \mbox{otherwise}. \end{cases}$$ For each $k \geq 2$, let $X^k_n, n\geq 1$ be the (deterministic) random variables defined by: $${{\mathbb P}}(X^k_n) = \begin{cases} 1, & \mbox{ with probability 1 for all $n \in [1, \ldots, 2^k)$};\\ 0, & \mbox{with probability 1 for all $n \geq 2^k$}. \end{cases}$$ Now, let $X_n$ be the stochastic process which selects $K=k$ with probability $\frac{1}{2^k}$ (for $k=1, 2,\ldots$) and then takes $X_n$ to be the process $X^K_n$ for all $n\geq 1$. Firstly, note that this mixture process is well defined, since $\sum_{k \geq 1} {{\mathbb P}}(K=k) = 1$. Next, observe that since $X^1_n = 1$ infinitely often (with probability 1) by the Borel–Cantelli Lemma (for independent random variables) and since there is a probability of $\frac{1}{2}$ that $X_n = X^1_n$ for all $n$, then, with probability $\frac{1}{2}$, $X_n$ does not converge to infinity or hit zero (note that $X^1_n \neq 0$ for any $n$, and $X^1_n$ returns to 1 infinitely often and so does not tend to infinity). Thus, to establish the claim regarding our example it suffices to show that Inequality (\[boundful1\]) applies. This can be verified, and the details are provided in the Appendix. An extended extinction dichotomy theorem ======================================== The example in the previous section shows that in (\[boundful1\]) we need to supplement the condition $X_n \in ~(0, m]$ with some further information concerning the past history of the process, in order to guarantee eventual extinction or $X_n \rightarrow \infty$. Here, we provide a mild extension of Theorem 2 of [@jag] by conditioning on the number of times the process has dipped below each given value $m$ up to the present step of the process. \[jag3\] Suppose $X_1, X_2, \ldots, X_n \ldots$ is a sequence of non-negative real-valued random variables that are absorbing at 0. For each positive integer $m \geq 1$, let $\kappa_m(X_1, \ldots, X_{n-1})$ count the number of $X_1,X_2, \ldots, X_{n-1}$ that are less than or equal to $m$. Suppose that for each positive integer $m$, there exists $\delta_m>0$ for which the following holds for all $n$.: $$\label{boundful3} {{\mathbb P}}(\exists r: X_r=0| X_n \in (0, m], \kappa_m(X_1, \ldots, X_{n-1})) \geq \delta_m.$$ Then, with probability 1, either $X_n \rightarrow \infty$ or a value of $n$ exists for which $X_k=0$ for all $k\geq n$. [*Proof:*]{} For any strictly positive integers $n$ and $m$, let $E_n$ be the event that $X_n=0$ and let $J_m$ be the event that $X_k \leq m$ for infinitely many values of $k$. Notice that $E_n$ and $J_m$ are both increasing sequences. Moreover, if we let $E= \bigcup_{n \geq 1}E_n$, $J = \bigcup_{m \geq 1} J_m$ and $\overline{J} =\bigcap_{m \geq 1} \overline{J_m}$, then $E$ is the event that some $k$ exists such that $X_k=0$ and $\overline{J}$ is the event that $X_n \rightarrow \infty$. We wish to show the following: $$\label{ejeq} {{\mathbb P}}(E \cup \overline{J}) =1.$$ Notice that: $$\label{ej} E \subseteq J_m \mbox{ for each $m\geq 1$}.$$ Furthermore, ${{\mathbb P}}(E) >0$ by Inequality (\[boundful3\]) applied to $n=1$, and any value of $m\geq 1$ for which ${{\mathbb P}}(X_1\leq m)>0$. Thus, from (\[ej\]), ${{\mathbb P}}(J_m)>0$ (and so ${{\mathbb P}}(J)>0$ also), so the conditional probabilities ${{\mathbb P}}(E|J)$ and ${{\mathbb P}}(E|J_m)$ are well defined, and for each $m \geq 1$, the inclusion (\[ej\]) gives: $$\label{EJ2} {{\mathbb P}}(E) = {{\mathbb P}}(E|J_m){{\mathbb P}}(J_m).$$ We will show that: $$\label{basic} {{\mathbb P}}(E|J_m) = 1 \mbox{ for each } m \geq 1,$$ which, combined with Eqn. (\[EJ2\]), gives ${{\mathbb P}}(E) = {{\mathbb P}}(J_m)$ for each $m \geq 1$. Thus, since ${{\mathbb P}}(J) = \lim_{m \rightarrow \infty} {{\mathbb P}}(J_m)$ (recall $J_m$ are increasing), we have ${{\mathbb P}}(E) = {{\mathbb P}}(J)$, and consequently ${{\mathbb P}}(E)+{{\mathbb P}}(\overline{J})=1,$ since $E$ and $\overline{J}$ are mutually exclusive. In this way we obtain the required identity (\[ejeq\]) that establishes the theorem. Thus it suffices to establish Eqn. (\[basic\]). For this we employ a coupling-style argument. For each positive integer $m$, we will associate to $X_n$ a second sequence of random variables $Y_k, k\geq 1$ as follows. Let $O_m= \{n \geq 1: X_n \leq m\}$, and for each $k \leq |O_m|,$ let $Y_k = X_{\nu(k)}$ where the random variable $\nu(k)$ is the $k^{\rm th}$ element of $O_m$ under the natural ordering of the positive integers. If $O_m$ is finite, then set $Y_k =0$ for all $k>|O_m|$ (notice that this will not occur when we condition on $J_m$ below). We may assume that the joint probability ${{\mathbb P}}(Y_k \neq 0, J_m)$ is strictly positive; otherwise ${{\mathbb P}}(Y_k=0|J_m)=1$ and so (\[basic\]) holds, since ${{\mathbb P}}(Y_k=0|J_m) \leq {{\mathbb P}}(E|J_m)$. Consequently, the conditional probabilities are well defined in the following equation: $$\label{big} {{\mathbb P}}(E|Y_k \neq 0, J_m) = \sum_{n \geq 1} {{\mathbb P}}(E| X_n \in (0, m], \nu(k)=n, J_m)\cdot {{\mathbb P}}(\nu(k)=n|Y_k \neq 0, J_m).$$ From (\[ej\]) and (\[boundful3\]), we obtain the following equality and inequality, respectively: $$\label{one} {{\mathbb P}}(E| X_n \in (0, m], \nu(k) = n, J_m) \geq {{\mathbb P}}(E| X_n \in (0, m], \nu(k)=n) \geq \delta_m,$$ where the first inequality is from (\[ej\]) and the second inequality is from (\[boundful3\]), since conditioning on the conjunction $X_n \in (0, m], \nu(k) = n$ is equivalent to conditioning on the conjunction of $X_n \in (0,m]$ and $\kappa_m(X_1, \ldots, X_{n-1}) = k-1$. Substituting (\[one\]) into the right-hand side of (\[big\]) gives ${{\mathbb P}}(E|Y_k \neq 0, J_m) \geq \delta_m$. Thus, we have: $$\label{good} {{\mathbb P}}(E|J_m) ={{\mathbb P}}(E|Y_k \neq 0, J_m){{\mathbb P}}(Y_k \neq 0|J_m) + 1 \cdot {{\mathbb P}}(Y_k=0|J_m)$$ $$\geq \delta_m(1-p_k) + 1\cdot p_k,$$ where $p_k = {{\mathbb P}}(Y_k=0|J_m),$ and where the factor $1$ is because, conditional on $J_m$, the event $E$ occurs whenever $Y_k=0$. Now, $\{Y_k=0\}$ is an increasing sequence in $k$, so if we let $\mathcal{Y}:= \bigcup_{k\geq 1}\{Y_k=0\}$, then: $$\label{helps1} p:= \lim_{k \rightarrow \infty} p_k={{\mathbb P}}({\mathcal Y}|J_m) .$$ Moreover: $$\label{helps2} {{\mathbb P}}(E|J_m) = {{\mathbb P}}({\mathcal Y}|J_m).$$ Applying (\[helps1\]) and (\[helps2\]) into (\[good\]) gives: $p \geq \delta_m(1-p)+p,$ which, in turn, implies that $p=1$ (since $\delta_m>0$). Thus, ${{\mathbb P}}(E|J_m)=p=1$, which establishes (\[basic\]) and so completes the proof. $\Box$ Concluding remarks ================== Notice that Theorem \[jag3\] implies Theorem 2 of [@jag], since the lower bound (\[boundful3\]) involves conditioning on aggregates of values for $X_1, \ldots, X_{n}$, so it holds automatically under the lower bound (\[strong\]). Notice also that the proof of Theorem \[jag3\], though longer than the elegant martingale argument for Theorem 2 of [@jag], requires merely elementary notions in probability. It turns out that the collection of random variables $\kappa_m(X_1, \ldots, X_k)$ across all (real) values of $m$ and all integer values of $k$ between 1 and $n$ suffices to determine the sequence of random variables $X_1, \ldots, X_n$ (by induction on $k$), so it is not immediately clear that Theorem \[jag3\] really allows greater generality than Theorem 2 of [@jag]. Therefore we provide an example to show that this is indeed the case. Informally, the extra generality in Theorem \[jag3\], arises from imposing fewer inequalities: in (\[boundful3\]) there are $n$ inequalities corresponding to the $n$ possible values that $\kappa_m(X_1, \ldots, X_{n-1})$ can take, while in (\[strong\]), there are potentially infinitely many, corresponding to all possible values for $X_1, \ldots, X_{n-1}$ (and for $X_n \leq x$). Example 3 --------- Roughly speaking, the stochastic process we will construct becomes extinct unless it oscillates regularly within a fixed range for an initial period, and the longer that it oscillates the greater the chance that it will escape to infinity rather than become extinct. We show that such a process satisfies (\[boundful3\]) but not (\[strong\]). First, consider a simple Markov chain $Y_n$ on the three states $0,1,2$ that starts in state 2 (i.e. $Y_1=2$ with probability 1) and with transition probabilities described as follows: - 0 is an absorbing state; - from state 1 or state 2, the next state is chosen with equal probability ($\frac{1}{3}$) from 0,1,2. Thus, with probability 1, a value $n$ exists for which $Y_k =0$ for all $k\geq n$. We will say that a sequence of values $y_1, y_2, y_3, \ldots, y_k$ from $\{1,2\}$ is a [*terminated flip sequence*]{} (of length $k$) if $y_1 = 2$ and $y_{i} = y_{i-1}$ only for $i=k$. For example $(2,1,2,1,2,1,2,1,1)$ and $(2,1,2,1,2,2)$ are terminated flip sequences of lengths nine and six respectively. We use $Y_n$ to define our process $X_n$ which takes non-negative integer values as follows. If there is no value $N \geq 4$ for which $Y_1, Y_2, \ldots Y_N$ is a terminated flip sequence, then set $X_n=Y_n$ for all $n$; in which case $X_n$ absorbs at 0 with probability 1. On the other hand, if a value $N \geq 4$ exists for which $Y_1, Y_2, \ldots Y_N$ is a terminated flip sequence, then, conditional on this value of $N$, $X_n = Y_n$ for all $n \leq N$, and for $n> N$, $X_n = Z^N_{n-N}$, where $Z^N_1, Z^N_2, \ldots$ is a second Markov chain on the state space $\{0\} \cup \{ N-1, N, N+1, N+2,\ldots\}$. This second chain has $Z_1= N-1$ (with probability 1), and has transitions from each state $i \geq N-1$ to $0$ and to $i+1$ with probabilities of $2^{-i}$ and $1- 2^{-i}$, respectively. Notice that, although the process $X_n$ is absorbing at $0$, it fails to satisfy (\[strong\]) since, for an terminated flip sequence $(x_1,x_2, \ldots, x_n)$, of length 4 or more, we have $x_n \leq 2$ and yet: $${{\mathbb P}}(\exists r: X_r=0| \wedge_{i=1}^n \{X_i =x_i\}) = \sum_{j=n-1}^\infty \frac{1}{2^j} \rightarrow 0, \mbox{ as } n \rightarrow \infty.$$ To show that $X_n$ satisfies (\[boundful3\]), we consider the cases $m=1$, $m=2$ and $m>2$ separately. For $m=1$, (\[boundful3\]) is equivalent to the following inequality holding for all $n \geq 1$: $$\label{m1} {{\mathbb P}}(\exists r: X_r=0| X_n =1, \kappa_1(X_1, \ldots, X_{n-1})) \geq \delta_1>0.$$ Now, if $\kappa_1(X_1, \ldots, X_{n-1}) \neq \lfloor (n-1)/2\rfloor$ then $X_1, \ldots, X_n$ cannot be a terminated flip sequence, and so, with probability at least $\frac{1}{3}$, we have $X_{n+1} = 0$. On the other hand, if $\kappa_1(X_1, \ldots, X_{n-1}) = \lfloor (n-1)/2\rfloor$ then the probability that $Y_1, Y_2, \ldots, Y_n$ is a terminated flip sequence of length 4 or more is bounded away from 1 as $n$ grows, and so the event $\{\exists r: X_r=0\}$ has a probability that is bounded away from $0$ for all $n$ when we condition on $\kappa_1(X_1, \ldots, X_{n-1})$ and $X_n = 1$. Thus a value $\delta_1>0$ can be chosen to satisfy (\[m1\]) for all $n \geq 1$. For $m=2$, (\[boundful3\]) is equivalent to the following inequality holding for all $n \geq 1$: $$\label{m2} {{\mathbb P}}(\exists r: X_r=0| X_n \in (0,2])\geq \delta_2>0.$$ Notice that $\kappa_2$ has vanished, since $X_n \in (0,2]$ implies that $\kappa_2(X_1, \ldots, X_{n-1}) = n-1$ with probability 1. Now, conditional on $X_n \in (0,2]$, the probability that $Y_1, Y_2, \ldots, Y_n$ is a terminated flip sequence of length 4 or more is bounded away from 1 as $n$ grows, and so the event $\{\exists r: X_r=0\}$ has a probability that is bounded away from $0$ for all $n$ when we condition on$X_n \in \{1,2\}$. Thus a value $\delta_2>0$ can be chosen to satisfy (\[m2\]) for all $n \geq 1$. Finally, for each $m>2$, for all $n \geq 1$: $${{\mathbb P}}(\exists r: X_r=0| X_n \in (0,m], \kappa_m(X_1, \ldots, X_{n-1})) \geq 2^{-m}>0,$$ so we can set $\delta_m = 2^{-m}$ for all $m>2$. In summary, for all values of $m$, $X_n$ satisfies (\[boundful3\]) for all $n$, as claimed. Acknowledgments =============== I thank Elchanan Mossel for several helpful comments concerning an earlier version of this manuscript, and Elliott Sober for some motivating discussion. I also thank the Allan Wilson Centre for funding support for this work. References ========== [99]{} Bruss, F. T. (1980). A counterpart of the Borel–Cantelli Lemma, J. Appl. Probab. 17: 1094–1101. Grimmett, G. R. and Stirzaker, D. R. (2001). Probability and Random Processes, 3rd ed. Oxford University Press, New York, USA. Haccou, P., Jagers, P., and Vatutin, V.A., (2005). Branching processes: variation, growth, and extinction of populations. Cambridge Studies in Adaptive Dynamics, Cambridge University Press, Cambridge UK. Jagers, P. (1992). Stabilities and instabilities in population dynamics. J. Appl. Probab. 29(4): 770–780. Malik, S.C. and Arora, S. (1992). Mathematical Analysis. New Age International. New Delhi, India. Wikipedia (2014) http://en.wikipedia.org/wiki/Murphy’s law (accessed 15 September 2014). Appendix: Proof that Example 2 satisfies Inequality (\[boundful1\]) =================================================================== Firstly, if $m<1$, then conditioning on $X_n\leq m$ is equivalent to conditioning on $K>1$ and so we can take any positive value for $\delta_m$ (even $= 1$) and satisfy Inequality (\[boundful1\]). Next, suppose that $m \geq 1$, and, for any $n \geq 1$, write: $$\label{eq0} n = 2^q +r, \mbox{ where } 0\leq r < 2^q, q\geq 0.$$ SInce ${{\mathbb P}}(\exists r: X_r=0|X_n \in (0,m]) = {{\mathbb P}}(K>1|X_n\in (0,m])$ we have: $$\label{eq1} {{\mathbb P}}(\exists r: X_r=0|X_n \in (0,m]) = \sum_{k \geq 2} {{\mathbb P}}(K=k|X_n\in (0,m]),$$ and, from Bayes’ identity: $$\label{eq2} {{\mathbb P}}(K=k|X_n\in (0,m]) = \frac{{{\mathbb P}}(X_n \in (0,m]|K=k){{\mathbb P}}(K=k)}{{{\mathbb P}}(X_n \in (0,m])}.$$ Now, for any $k \geq 2$ (and still with $m>1$): $$\label{eqx} {{\mathbb P}}(X_n \in (0,m]|K=k) = \begin{cases} 1, & \mbox{ provided $k \geq q+1$};\\ 0, & \mbox{otherwise}; \end{cases}$$ since $X_n^k=1$ with probability 1 for all $n \in [1, \ldots, 2^k)$. Consequently, the numerator of (\[eq2\]) equals $\frac{1}{2^k}$ ( $={{\mathbb P}}(K=k)$) when $k \geq q+1$ and is zero otherwise. Now, the denominator of (\[eq2\]), namely ${{\mathbb P}}(X_n \in (0,m])$, can be written as: $$\label{eq3} \left[\sum_{k \geq 2} {{\mathbb P}}(X_n \in (0,m]|K=k){{\mathbb P}}(K=k)\right] + {{\mathbb P}}(X_n\in (0,m]|K=1){{\mathbb P}}(K=1).$$ From (\[eqx\]), the first term in (\[eq3\]) is: $$\label{eq4} \sum_{k \geq 2} {{\mathbb P}}(X_n \in (0,m]|K=k){{\mathbb P}}(K=k)= \sum_{k \geq q+1} \frac{1}{2^k} = \frac{1}{2^q}.$$ Regarding the second term in (\[eq3\]), observe that: $${{\mathbb P}}(X_n\in (0,m]|K=1) = \begin{cases} \frac{1}{n}, & \mbox{ provided $n > m$};\\ 1 , & \mbox{if $n \leq m$. } \end{cases}$$ Therefore, recalling (\[eq0\]), the second term in (\[eq3\]) is: $$\label{eq5} \begin{cases} \frac{1}{n} \times \frac{1}{2} = \frac{1}{2^{q+1}+2r} \leq \frac{1}{2^{q+1}} , & \mbox{ provided $n >m$};\\ 1 \times \frac{1}{2}, & \mbox{when $n \leq m$}. \end{cases}$$ Consequently, by combining (\[eq3\]), (\[eq4\]) and (\[eq5\]) into (\[eq2\]) (and noting again that $\sum_{k \geq q+1} \frac{1}{2^k} = \frac{1}{2^q}$) we have that if $n >m$, then $\sum_{k \geq 2} {{\mathbb P}}(K=k|X_n\in (0,m]) \geq \frac{\frac{1}{2^q}}{\frac{1}{2^q} + \frac{1}{2^{q+1}}} \geq \frac{1}{2},$ while if $n \leq m$, then $\sum_{k \geq 2} {{\mathbb P}}(K=k|X_n\in (0,m]) \geq \frac{\frac{1}{2^q}}{\frac{1}{2^q} + \frac{1}{2}} \geq \frac{\frac{1}{2^q}}{1+ \frac{1}{2}} \geq \frac{2}{3}\cdot \frac{1}{2^q} \geq \frac{2}{3m},$ where the last inequality is from $ m \geq n \geq 2^q$. Thus, if we take $\delta_1 = \frac{1}{2}$ and $\delta_m = \frac{2}{3m}$ for each $m \geq 2$, then, from (\[eq1\]), $ {{\mathbb P}}(\exists r: X_r=0|X_n \in (0,m])\geq \delta_m$ for all $n, m$, as claimed.

--- address: - 'Laboratoire de Mathématiques de l’Université Paris-Sud' - 'AgroParisTech/UMR INRA MIA 518' author: - 'A. Bonnet' - 'E. Gassiat' - 'C. Lévy-Leduc' bibliography: - 'biblio\_anna.bib' title: Heritability estimation in high dimensional linear mixed models --- Acknowledgments {#acknowledgments .unnumbered} =============== The authors would like to thank Edouard Maurel-Segala and Maxime Février for stimulating discussions on random matrix theory and Thomas Bourgeron and Roberto Toro for having led us to study this very interesting subject and for the discussions that we had together on genetic topics.

--- author: - | Kevin <span style="font-variant:small-caps;">Ebinger</span>$^{1}$, Sanjana <span style="font-variant:small-caps;">Sinha</span>$^{2}$, Carla <span style="font-variant:small-caps;">Fröhlich</span>$^{2}$, Albino <span style="font-variant:small-caps;">Perego</span>$^{3}$, Matthias <span style="font-variant:small-caps;">Hempel</span>$^{1}$,\ Marius <span style="font-variant:small-caps;">Eichler</span>$^{2}$, Jordi <span style="font-variant:small-caps;">Casanova</span>$^{2}$, Matthias <span style="font-variant:small-caps;">Liebendörfer</span>$^{1}$, and Friedrich-Karl <span style="font-variant:small-caps;">Thielemann</span>$^{1}$\ \ $^{1}$ Department für Physik, Universität Basel, CH-4056 Basel, Switzerland\ $^{2}$Department of Physics, North Carolina State University, Raleigh, NC, 29695-8202, USA\ $^{3}$Institut für Kernphysik, Technische Universität Darmstadt, D-64289 Darmstadt, Germany date: 'August 20, 2016' title: 'Explosion Dynamics of Parametrized Spherically Symmetric Core-Collapse Supernova Simulations' --- Introduction ============ Core-collapse supernovae (CCSNe) occur at the end of the evolution of massive stars and the ejecta of these violent events contribute to the chemical evolution of the universe. The explosion mechanism of CCSNe is still not fully understood and self consistent one-dimensional simulations of CCSNe, including general relativity and detailed neutrino transport, do not lead to explosions, with the exception of the lowest-mass CCSN progenitors[@lowmass]. Even though multi-dimensional simulations are promising to explode and well suited to investigate the explosion mechanism they are computationally too expensive to explore a large set of progenitors. The presented parametrized one-dimensional framework (PUSH, introduced in[@push]) is well suited to study explosive nucleosynthesis and remnant properties of a broad range of CCSN progenitors and with this also get a better understanding of phenomenon itself. Spherically symmetric simulations show a smaller heating efficiency of electron neutrinos behind the shock due to an absence of convective motion. PUSH provides extra energy deposition in the heating region by tapping the energy of $\mu$- and $\tau$- (anti)-neutrinos in otherwise consistent spherically symmetric simulations to mimic multi-dimensional effects (e.g., convection, SASI) that enhance neutrino heating. This enables a consistent evolution of the PNS and treatment of the electron fraction of the ejecta. Furthermore, after the onset of explosion the method also prevents a too strong decrease in $\nu$-heating behind the shock due to a drop in electron (anti)neutrino luminosity that occurs in 1D simulations due to drastic reduction of the mass accretion rate on the PNS (see Figure \[lum\]). Figure \[radii\] shows the temporal evolution of the shock, gain and PNS radius with and without PUSH of a CCSN simulation of a 20 M$_{\odot}$ star. ![Temporal evolution of the shock radius, the PNS radius and the gain radius of a 20 M$_{\odot}$ progenitor (WH07 [@prog1]).[]{data-label="radii"}](neutrino_analysis.eps){width="1.05\linewidth"} ![Temporal evolution of the shock radius, the PNS radius and the gain radius of a 20 M$_{\odot}$ progenitor (WH07 [@prog1]).[]{data-label="radii"}](rpns.eps){width="\linewidth"} Comparison with multi-dimensional simulations ============================================= Overall PUSH shows a behaviour more consistent with multi-dimensional models than older methods (e.g. pistons and thermal bombs [@piston],[@thiel96]). Figures \[pushentropy\] and \[flashentropy\] show the spherically averaged entropy per baryon as a function of radius obtained from a 2D Flash simulation (see [@kcpan] and references therein) and from a 1D simulation with PUSH for the same progenitor and electron (anti)neutrino transport [@lieb1]. The comparison of the two figures shows on average a similar heating pattern. Such a comparison can be used as a further fit requirement (besides explosion energy and nucleosynthesis yields [@push], see also proceeding of S. Sinha, this volume) for the free parameters of the PUSH method. ![Spherically averaged entropy per baryon as a function of radius obtained from a 2D Flash simulation of a 20 M$_{\odot}$ progenitor.[]{data-label="flashentropy"}](entropyset.eps){width="\linewidth"} ![Spherically averaged entropy per baryon as a function of radius obtained from a 2D Flash simulation of a 20 M$_{\odot}$ progenitor.[]{data-label="flashentropy"}](fig_s20_LS220_radial_averaged_entr_v1.eps){width="\linewidth"} Progenitor and Equation of State Dependence of Black Hole Formation =================================================================== To disentangle aspects - other than the explosion mechanism - that have an influence on black hole formation we investigate the effect that different choices of the equation of state and of the progenitor profiles can have in our 1D simulations. ![Black hole formation times for a collection of different progenitor ZAMS masses from two different progenitor sets (WH07[@prog1] in red and WHW02[@prog3] in blue).[]{data-label="fig:overviewcollapse"}](w02rhoc40.eps){width="\linewidth"} ![Black hole formation times for a collection of different progenitor ZAMS masses from two different progenitor sets (WH07[@prog1] in red and WHW02[@prog3] in blue).[]{data-label="fig:overviewcollapse"}](tdot.eps){width="\linewidth"} Figure \[fig:detailcollapse\] shows the temporal evolution of the central density of a 40 M$_{\odot}$ solar metallicity star for two progenitor models (WH07[@prog1] in red and WHW02[@prog3] in blue) and two equations of state (HS(DD2) solid lines, SFHO dashed lines, [@hempel],[@fischer],[@sfho]). The dependence of the black hole formation time on the equation of state (indicated by the colored areas) and the even stronger dependence on the progenitor model for this progenitor ZAMS mass (difference between red and blue lines) is evident. Baryonic PNS masses at collapse are given next to the corresponding central density curves. In Figure \[fig:overviewcollapse\] the black hole formation times for a set of different progenitor ZAMS masses are given. The differences for black hole formation time between the progenitors can be related to different accretion rates, which are correlated to compactness $\xi_{M}=\frac{M/M_{\odot}}{R(M)/1000km}$. Conclusions and Outlook ======================= In comparison to traditional effective methods, as pistons or thermal bombs, PUSH is better suited to study explosive nucleosynthesis, especially of the innermost ejecta, due to the inclusion of more neutrino physics and the preservation of charged current reactions. We have shown that the entorpy profiles obained with PUSH are similar to the spherical averages of multi-dimensional models and demonstrated the big effect the choice of progenitor and equation of state can have on black hole formation and thus on a study of explodability. It is planned to investigate the explodability of different progenitor sets with different equations of state with PUSH in the future. [9]{} A. Perego, M. Hempel, C. Fröhlich, K. Ebinger, M. Eichler, J. Casanova, M. Liebendörrfer, F.-K. Thielemann, Astrophys. J. **806**, 275 (2015) T. Fischer, S. C. Whitehouse, A. Mezzacappa, F.-K. Thielemann, M. Liebendörfer, A&A **517**, A80 (2010) S. E. Woosley & A. Heger, Phys. Rep. **442**, 269 (2007) S. E. Woosley & T. A. Weaver, Astrophys. J. S. **101**, 181 (1995) F.-K. Thielemann, K. Nomoto, M. A. Hashimoto, Astrophys. J. **460**, 408 (1996) K.-C. Pan et al., Astrophys. J. **817**, 72 (2016) M. Liebendörrfer, S. C. Whitehouse, T. Fischer, Astrophys. J. **698**, 1174 (2009) S. E. Woosley, A. Heger, T. A. Weaver, Rev. Mod. Phys. **74**, 1015 (2002) M. Hempel & J. Schaffner-Bielich, Nuc. Phys. A **837**, 210 (2010) T. Fischer, M. Hempel, I. Sagert, Y. Suwa, J. Schaffner-Bielich, Eur. Phys. J. A. **50**, 46 (2014) A. W. Steiner, M. Hempel, T. Fischer, Astrophys. J. **774**, 17 (2013)

--- abstract: 'We clarify certain important issues relevant for the geometric interpretation of a large class of $N = 2$ superconformal theories. By fully exploiting the phase structure of these theories (discovered in earlier works) we are able to clearly identify their geometric content. One application is to present a simple and natural resolution to the question of what constitutes the mirror of a rigid Calabi-Yau manifold. We also discuss some other models with unusual phase diagrams that highlight some subtle features regarding the geometric content of conformal theories.' author: - | Paul S. Aspinwall and Brian R. Greene\ F.R. Newman Lab. of Nuclear Studies,\ Cornell University,\ Ithaca, NY 14853\ title: | On the Geometric Interpretation of\ $N$ = 2 Superconformal Theories\ --- epsf \#1[date[*\#1*]{}]{} \#1[thefnmarkfootnotetext[1991 [*Mathematics Subject Classification.*]{} \#1]{}]{} \#1[thefnmarkfootnotetext[ [*Key words and phrases.*]{} \#1]{}]{} PS. @firstpage[PS. @empty oddhead evenheadoddhead ]{} ifundefined[reset@font]{}[@font]{} footnotetext\#1 [ ]{} [ ]{} \#1 Ø[[O]{}]{} ¶[[P]{}]{} \#1[\#1 \#1 \#1]{} \#1 \#1 \#1\#2 \#1\#2[\_[\#1]{}F\_[\#2]{}]{} 0 1.5cm Introduction and Summary {#s:intro} ======================== One of the most intriguing problems in string theory is to understand how space-time emerges naturally. Since the vacuum configuration for a critical string is given by a conformal field theory a question which arises in this context is the following. Given a conformal field theory, can one construct some corresponding geometrical interpretation? In this paper we will discuss this question for particularly troublesome conformal field theories. It is worthwhile to emphasize at the outset that in general when a conformal theory does have a geometrical interpretation it may not be unique. A perusal of even simple systems such as conformal theories with central charge $c = 1$ makes this clear. For instance, in this moduli space it is known that a string on the group manifold $SU(2)$ is equivalent to a string on a circle of radius $\sqrt{\alpha^\prime}$. Both target spaces have an equal right to be declared [*the*]{} geometrical interpretation of the conformal field theory. Similarly a circle of radius $R$ is equivalent to a circle of radius $\alpha^\prime/R$. Mirror symmetry, in which strings propagating on distinct Calabi-Yau spaces give identical physical models, is another substantial arena in which geometrical interpretations are not unique. These ambiguities are a reflection of the rich structure of quantum geometry; they arise because of the extended nature of the string. When there are multiple geometric interpretations of a given model, there is no reason why one should be forced to choose between the possibilities. Rather, one can exploit the geometric ambiguity as some interesting physical questions are more easily answered from one interpretation rather than another. In this paper we shall focus our investigation into the geometric content of certain of $N = 2$ conformal theories using the framework established in [@W:phase; @AGM:I; @AGM:II]. This approach has the virtue of giving us a physical and mathematical understanding of [*global*]{} properties of the moduli space of these theories as well as of the theories themselves. It also gives us the proper arena for understanding the global implications of mirror symmetry. We will apply this approach to study some theories whose geometrical content has been quite puzzling. For some of these theories, previous papers have proposed possible geometrical interpretations [@Drk:Z; @Schg:gen; @Set:sup]. We will see that when phrased in the language of [@W:phase; @AGM:I; @AGM:II], the previous puzzles are seen to disappear and the geometric status of these theories becomes apparent. Following our remarks above, there need not be one unique interpretation of a given model; however, we do feel that the approach provided here is especially enlightening and economical. We will also see that the less natural constructions of [@Drk:Z; @Schg:gen; @Set:sup] can give misleading results for properties of the corresponding physical model. We now recall some important background material which will naturally lead us to a summary of the problems we address and the solutions we offer. Our understanding of the geometric content of $N = 2, c = 3d$ superconformal theories has undergone impressive growth and revision over the last few years. The initial picture which emerged from numerous studies is schematically given in figure \[fig:1\]a. We have an abstract $N = 2, c = 3d$ conformal field theory moduli space that is geometrically interpretable in terms of complex structure and Kähler structure deformations of an associated Calabi-Yau manifold of $d$ complex dimensions and a fixed topological type. The space of Kähler forms naturally exists as a bounded domain (the complexification of the “Kähler cone”) which we denote as a cube. The moduli space of complex structures does not have this form and is more usually compactified to form a compact space. Observables in each of the conformal theories in the moduli space are related to geometrical constructs on the corresponding Calabi-Yau space, the latter being taken as the target space of a nonlinear sigma model. This picture was extended to that given in figure \[fig:1\]b after the discovery of mirror symmetry. Two Calabi-Yau spaces $X$ and $Y$ constitute a mirror pair if they yield isomorphic conformal theories when taken as the target space for a two-dimensional supersymmetric nonlinear , with the explicit isomorphism being a change in sign of the left moving $U(1)$ charges of all fields. Geometrically this implies that the Hodge numbers $h^{1,1}(X)$ and $h^{d-1,1}(X)$ are related to those of $Y$ by $h^{1,1}(X) = h^{d-1,1}(Y)$ and $h^{d-1,1}(X) = h^{1,1}(Y)$. Since the cohomology groups $H^{1,1}$ and $H^{d-1,1}$ correspond to Kähler and complex structure deformations, respectively, we see that the underlying conformal field theory moduli space has the two geometrical interpretations given in the figure. This immediately led to a problem since, as mentioned above, the geometric form of the moduli spaces of Kähler forms and complex structures appeared to be quite different. This was resolved by the works of [@W:phase; @AGM:I; @AGM:II] to that shown in figure \[fig:1\]c. Here we see that the appropriate interpretation of the conformal field theory moduli space has required that the Kähler moduli space of $X$ be replaced by its “enlarged Kähler moduli space” (and similarly for $Y$). The latter contains numerous regions in addition to the Kähler cone of the topological manifold $X$. For instance, it typically contains regions corresponding to the Kähler cones of Calabi-Yau spaces related to $X$ by the birational operation of flopping a rational curve, regions corresponding to the moduli space of singular blow-downs of $X$ and its birational partners, and regions interpretable in terms of the parameter space of (gauged or ungauged) Landau-Ginzburg models fibered over various compact spaces. The complex structure moduli space can also be equipped with a phase structure [@AGM:sd] — as must happen to preserve mirror symmetry. We note that from the  point of view the phase regions in the complex structure moduli space have a less pronounced physical interpretation. This is because in analyzing the we use perturbation theory in Kähler modes (which fix the size of the Calabi-Yau) and hence this approximation method is not mirror symmetric. However, the phase structure in the complex structure moduli space of $X$ [*is*]{} the phase structure in the enlarged Kähler moduli space of $Y$ and it is the latter interpretation where this phase structure is most manifest. For the purposes of this paper we may ignore the phase structure in the complex structure part of the moduli space and for this reason we have put parentheses around this in \[fig:1\]c. The results of the present paper all stem directly from a careful study of the phase diagrams of figure \[fig:1\]c. We shall review the quantitative construction of these phase spaces in section \[s:ph\]; for now we will content ourselves with the schematic description given and summarize our results with a similar level of informality. There are numerous ways of constructing $N = 2$ superconformal theories with $c = 3d$. Some constructions, such as the Calabi-Yau s described above, are manifestly geometric in character. Other constructions do not begin with a geometrical target space and hence their geometrical content, if any, can only be assessed after more detailed study. More generally and pragmatically, given an abstract conformal field theory in some presentation, how do we determine if it has a geometrical interpretation? We will not seek to answer this question in generality, but rather will focus attention on those theories for which we can construct the phase diagram illustrated in figure \[fig:1\]c. For theories of this sort, as we shall review, toric geometry supplies us with a geometric description of each theory. We hasten to emphasize, though, that Calabi-Yau s are but one kind of corresponding geometry. We will see, for instance, that Landau-Ginzburg orbifolds can be associated with noncompact, generally singular, configuration spaces. From our brief discussion here and also from [@W:phase; @AGM:I; @AGM:II] one might think that any theory with a phase diagram such as that in figure \[fig:1\]c, has regions interpretable in terms of Calabi-Yau s. After all, our progression from figures \[fig:1\]a through \[fig:1\]c has centered around Kähler cones of Calabi-Yau spaces. This conclusion, as we shall see in detail in section \[s:appl\], is false and comes to bear on a number of issues, including that of the generality of mirror symmetry. Namely, there are Calabi-Yau manifolds that are rigid, i.e. that have trivial $H^{d-1,1}$. The mirror to such a space, therefore, should have $h^{1,1} = 0$. This is troublesome, though, because Calabi-Yau spaces are Kähler and hence have at least one nontrivial element in $H^{1,1}$. With the above discussion, and explicit calculation in section \[s:appl\], the resolution to this puzzle becomes clear: the enlarged Kähler moduli space for the theory mirror to the one associated with the rigid space $X$ [*does not contain a region interpretable in terms of a Calabi-Yau* ]{}. In fact, the enlarged Kähler moduli space, in contrast to the generic case illustrated in figure \[fig:1\]c, is zero-dimensional and consists of a single point. By direct analysis, we show that the corresponding theory is a Landau-Ginzburg orbifold - not a Calabi-Yau  - and hence it is perfectly consistent for the theory to lack a Kähler modulus. We note at the outset that possible resolutions to the question of the identity of mirrors to rigid Calabi-Yau spaces have been previously presented in [@Drk:Z; @Schg:gen; @Set:sup]. These authors have invoked unexpected additional structures such as non-Calabi-Yau spaces of dimension greater than $d$ and supermanifolds in an attempt to resolve this issue. Contrary to these works, we see here that absolutely no additional structure is required. Rather, rigid Calabi-Yau manifolds fit perfectly into the general framework introduced in [@W:phase; @AGM:I; @AGM:II]. In addition to applying our analysis to the case of rigid Calabi-Yau manifolds and their mirrors, we also study two other interesting phenomena. First, we present an example of a theory with nonzero dimensional enlarged Kähler moduli space that does not contain a geometric region thus showing that the mere existence of a would-be Kähler form does not guarantee a Calabi-Yau interpretation. Second, we briefly discuss an example (first pointed out in [@W:phase]) whose enlarged Kähler moduli space has a phase whose target space has the desired dimension but is not of the Calabi-Yau type. Phase Diagrams: Supersymmetric Gauge Theory and Toric Geometry {#s:ph} ============================================================== The moduli space of $N = 2$ superconformal theories is most naturally interpretable in terms of a collection of regions within which the theory assumes a particular phase. Amongst the possibilities are smooth and singular geometric Calabi-Yau phases, gauged and ungauged Landau-Ginzburg phases, as well as orbifolds and hybrids thereof. This is the burden of figure \[fig:1\]c. The existence and quantitative construction of these phase diagrams has been approached from two distinct vantage points in the works of [@W:phase] and [@AGM:II]. In fact, a point which is not as fully appreciated as it might be is that these two approaches, although phrased in different languages, are [*isomorphic*]{}. Different questions, though, are often more easily answered from one of the two formalisms and hence it is important to fully understand both approaches and their precise relationship. It is the purpose of the present section to explain these issues. We note that the material in this section is implicit in [@W:phase] and [@AGM:II]; for our purposes we need to make the relation explicit. In brief, both [@W:phase] and [@AGM:II] build constrained $N = 2$ supersymmetric quantum field theories. In the physical approach of [@W:phase] these constraints are phrased in terms of [*symplectic*]{} quotients. In the mathematical approach of [@AGM:II] these constraints are phrased in terms of [*holomorphic*]{} quotients. The well-known equivalence [@Kirwan:; @Ness:] of these two approaches then implies that each constructs the same theory and hence also the same phase diagrams. The proper language for establishing these statements is that of [*toric geometry*]{} for which the reader can find a primer in [@AGM:II]. In the following we will try to convey the main points with a minimum of unnecessary technical detail. Complex projective space may be considered to be the prototypical toric variety. One constructs $\P^n$ by taking the $n+1$ homogeneous coordinates, $x_i$, spanning $\C^{n+1}$, removing the origin $x_i=0$ and modding out by the $\C^*$-action $x_i\to\lambda x_i$, $\lambda\neq0$. A toric variety is simply a generalization of this concept with perhaps more than one $\C^*$-action and a possibly more complicated point set removed prior to the modding out process. The most natural way of building a $N$=2  with a complex projective target space appears to be in terms of a $U(1)$-gauged field theory [@ADL:CPn]. In this construction, one begins with the homogeneous coordinates, $x_i$, denoting chiral superfields, each with the same $U(1)$ charge, $Q_i$, (which we may take to be 1). The classical vacuum of such a theory may be determined by finding the minimum of the classical potential energy. Solving the algebraic equations for the auxiliary D-component of the gauge multiplet and including the result in the scalar potential yields the familiar contribution $$( |x_1|^2 + |x_2|^2 +\ldots+|x_{n+1}|^2 - r )^2, \label{eq:ve1}$$ where $r$, a real number, is the coefficient of the familiar Fayet-Illiopoulos D-term. We take $r$ to be positive here to avoid naïvely breaking supersymmetry (see section 3.2 of [@W:phase] for a discussion on negative values of $r$). Minimizing the energy forces us to require that (\[eq:ve1\]) should vanish. This immediately removes the origin $x_i=0$ from consideration. It also forces the $x_i$ to lie on the sphere $S^{2n+1}$. We may now divide out by the $U(1)$ (i.e., $S^1$) action to form $S^{2n+1}/S^1\cong\P^n$. The process of dividing by $\C^*$, in the usual formulation of $\P^n$ may be viewed to having taken place in two stages. First we fix an $\R_+$ degree of freedom by imposing the vanishing of (\[eq:ve1\]) and then we divide out by $S^1$. The equivalence of these two constructions then follows from the fact that $\C^*\cong \R_+\times S^1$. Dividing by the former is a simple example of a holomorphic quotient; dividing by the latter is a simple example of a symplectic quotient. We have just seen, therefore, the essential reason why these two are equivalent. Let us now discuss how Witten generalized upon this quantum field theory approach of generating symplectic quotients. We will then discuss their equivalent holomorphic quotient description as in [@AGM:II]. Witten [@W:phase] extended the above model to describe not a complex projective space but the “canonical” line bundle of complex projective space (see, for example, [@GH:alg] for the precise definition of this bundle). Let us reserve $n$ to denote the dimension of the toric variety in question so now we are looking at a line bundle over $\P^{n-1}$ and the variable $x_{n+1}$ will now be treated differently to the others. This space is then built from $\C^{n+1}$ by removing the point $x_1=x_2=\ldots=x_n=0$ and modding out by the action $$\eqalign{ x_i&\to \lambda x_1,\quad i=1\ldots n,\cr x_{n+1}&\to\lambda^{-n}x_{n+1}.\cr} \label{eq:Cs1}$$ To produce this from the gauged  point of view we put $Q_i=1$ for $i=1\ldots n$ and $Q_{n+1}=-n$. The vanishing of the classical potential now implies $$|x_1|^2 + |x_2|^2 +\ldots+|x_n|^2 -n|x_{n+1}|^2 = r. \label{eq:ve2}$$ We see that there are classical vacuum solutions for $r$ for either sign. If $r>0$, we thus recover the required target space as in the case of the projective space. If however $r<0$, we find that $x_{n+1}\neq0$ and we have no condition on $x_1,\ldots, x_n$. Let us consider this space more closely. Removing the point set $x_{n+1}=0$ from $\C^{n+1}$ and dividing by the action (\[eq:Cs1\]) produces another toric variety. $x_{n+1}$ may be fixed by choosing a value for $\lambda^n$ leaving the $n$th roots of unity to act on the space spanned by $x_1,\ldots,x_n$. Thus the toric variety is $\C^n/\Z_n$. Therefore we see that the geometry of the target space can change discontinuously as we vary $r$. This theory is said to have two [*phases*]{} where the relevant toric variety is either the canonical line bundle of $\P^{n-1}$ or $\C^n/\Z_n$. The construction of [@W:phase] doesn’t quite stop here. One may introduce a $U(1)$-invariant superpotential, $W$, i.e., a $\C^*$-invariant polynomial over the $x_i$’s. Minimizing the classical potential now also implies that we are at a critical point of $W$. Our toric “ambient” space will always turn out to be non-compact. This however will contain compact subspaces which may also be considered as toric varieties themselves. Clearly $\P^{n-1}$ is a toric subspace of the canonical line bundle over $\P^{n-1}$. The only compact toric subspace of $\C^n/\Z_n$ is the point at the origin. Assuming that $W$ is suitably generic, the effect of including the superpotential term is to force the classical vacuum to be equal to, or contained in some compact toric subspace of the ambient space. In our example, a suitable $W$ is $(x_1^n+x_2^n+\ldots+x_n^n)x_{n+1}$. In the canonical line bundle over $\P^{n-1}$ case, the critical point set of $W$ consists of the hypersurface $x_1^n+x_2^n+\ldots+x_n^n=0$ in $\P^{n-1}$. This is a compact  $(n-2)$-fold. This is thus named, the [**]{} phase. In the $\C^n/\Z_n$ case, the origin is the critical point set of $W$. Thus our classical vacuum is simply one point. The effective superpotential of this theory however allows for massless fluctuations around this point given by a Landau-Ginzburg superpotential $x_1^n+x_2^n+\ldots+x_n^n$. This is thus the [**]{} phase. Note that all fluctuations around the vacuum in the  phase are massive. Let us fix some notation.[^1] We will call the ambient non-compact toric space $\Vbig$. This contains a maximal compact toric subset $\Vlit$ (which may be reducible). Within $\Vlit$ we have the classical vacuum of the quantum field theory which we denote $X$. A simple generalization of the above construction is to consider a weighted projective space for $\Vlit$. Clearly this may be achieved by giving different charges to $x_1,\ldots,x_n$. Following the above formalism we would again obtain two phases depending on whether $r$ was less than or greater than zero. When we look at the associated conformal field theory it turns out that this does not capture the full moduli space, i.e., $h^{1,1}>1$ for many of these theories. It is not hard to generalize the present description to include at least some of these other degrees of freedom. For each such independent direction in the moduli space we are able to access in this formalism, we introduce a $U(1)$ gauge factor and a corresponding parameter $r_l$. Thus, the total gauge group is $G=U(1)^{s}$ where $s$ is the dimension of this subspace of the moduli space . The chiral fields will in general be charged under all of the $U(1)$ factors, and hence we write $Q_i^{(l)}$ to denote the charge of the $i^{th}$ chiral superfield under $U(1)_{(l)}$. The superpotential $W$ must now be a $G$-invariant combination of the chiral superfields. It turns out that the language of toric geometry is precisely suited for determining all of the data needed for building such a model. Namely, in the case of $s = 1$ (or more generally, $s$ is the number of distinct toric factors making up the ambient space) it is straightforward to figure out appropriate charges so that minimization of the scalar potential yields the desired model. When $s$ is not of this form, the problem requires a more systematic treatment; this is precisely what the formalism of toric geometry supplies. Furthermore, for these more general cases, it proves increasingly difficult to determine the phase diagram of the model by studying the minimum of the scalar potential for various values of the $r_1,...,r_{s}$. The formalism of toric geometry, as described in [@AGM:II], supplies us with a far more efficient means of determining the phase structure, as well. Hence, let us now recast the above formulation directly in terms of toric geometry. The homogeneous coordinates (in the sense of [@Cox:]) $x_1,\ldots, x_N$ form a natural representation of the group $(\C^*)^N$. Let us form a toric variety by removing some point set and dividing the resultant space by $(\C^*)^{N-n}$. Clearly the space formed, $\Vbig$, is acted upon non-trivially by $(\C^*)^n$. Let us introduce $\zeta_j$, $j=1,\ldots,n$, as the natural representation of this $(\C^*)^n$-action. That is, the $\zeta_j$ provide coordinates on a dense open subset of $\Vbig$. This follows since $\Vbig$ may be regarded itself as a compactification of $(\C^*)^n$. Let us relate these new “affine” coordinates to the homogeneous coordinates by $$\zeta_j = \prod_{i=1}^N x_i^{\alpha_{ij}}, \label{eq:aff}$$ where $\alpha_{ij}\in\Z$. We may represent the $N\times n$ matrix, $\alpha_{ij}$, by a collection of $N$ points, which we denote $\cA$, living in an $n$-dimensional real space where $\alpha_{ij}$ is the $j$th coordinate of the $i$th point. Let us demand that $\cA$ is such that there exists an $n$-dimensional lattice ${\bf N}$ within this same space (which we denote ${\bf N}_{\R}={\bf N}\otimes_{\Z}\R$) such that $$\cA = {\bf N} \cap (\hbox{Convex hull of $\cA$}\,). \label{eq:Acvx}$$ The notation $\alpha_i$ will denote the position vector of the $i$th point of $\cA$ in ${\bf N}$. Consider now the charges of the homogeneous coordinates under the $(\C^*)^{N-n}$ by which we modded out. Denote these $Q_i^{(l)}$ where $i=1,\ldots,N$ and $l=1,\ldots,N-n$. The obvious short exact sequence $$1\to(\C^*)^n\to(\C^*)^N\to(\C^*)^{N-n}\to1,$$ induces, $$\sum_{i=1}^N Q^{(l)}_i\alpha_{ij} = 0,\quad\forall l,j. \label{eq:krnl}$$ Thus, we see that the charges $Q^{(l)}_i$ are simply the [*kernel of the transpose of the matrix whose elements are*]{} $\alpha_{ij}$. The reader should check that in the simple case, say, of projective space discussed earlier, that the charge assignment posited can in fact be derived in this manner. Now define ${\bf M}$ as the dual lattice to ${\bf N}$. Let us demand that there is an element $\mu\in{\bf M}$ such that $$\langle\mu,\alpha_i\rangle=1,\quad\forall i. \label{eq:hypln}$$ This condition is similar to stating that $\Vbig$ be a space with vanishing canonical class, $K$, (or zero first Chern class). Actually $\Vbig$ need not be smooth so be need to be more careful about our language. The correct term from algebraic geometry is that $\Vbig$ is [*Gorenstein*]{} (see, for example, [@Reid:mm]). Applying (\[eq:hypln\]) to (\[eq:krnl\]) tells us that $$\sum_{i=1}^N Q^{(l)}_i =0,\quad\forall l. \label{eq:Q0}$$ This appears as an important condition in [@W:phase] ensuring freedom from anomalies in certain chiral currents which should be present if there is an infrared limit with $N$=2 superconformal invariance. It is curious to note that (\[eq:Q0\]) is not sufficient to guarantee (\[eq:hypln\]). We may have $\langle\mu,\alpha_i\rangle=k$ for example, for some integer $k$. $\Vbig$ would then be $\Q$-Gorenstein which is roughly saying that $kK=0$ but $K$ may be a non-trivial torsion element. The effect of this in terms of the two dimension quantum field theory has not been studied. This point set $\cA$ gives us all the information we require to build $\Vbig$ except which point set should be removed from $(\C^*)^N$ before performing the quotient. This is performed in toric geometry by building a fan, $\Delta$. A fan is a collection of tesselating cones in ${\bf N}_\R$ with apexes at the origin. The intersection of this fan with the hyperplane containing $\cA$ will be a set of tesselating polytopes. The convex hull of this set of polytopes must be the convex hull of $\cA$ and the vertices of the polytopes must be elements of $\cA$. Thus each cone, $\sigma$, in $\Delta$ is “generated” by a subset of $\cA$. We say $\alpha_i\in\sigma$ if $\alpha_i$ is one of the generators, i.e., $\alpha_i$ lies at a vertex of the intersection of $\sigma$ with the hyperplane in ${\bf N}_\R$ containing $\cA$. The point set $F_\Delta$ removed from $\C^N$ prior to quotienting is then specified by $$\bigcap_{\sigma\in\Delta} \Bigl\{ x\in\C^N; \prod_{{\alpha_i\in\cA,} \atop {\alpha_i\not\in\sigma}}\!\!x_i=0 \Bigr\}, \label{eq:Fset}$$ where $x$ is the point with coordinates $x_i$. The fact that different fans may be associated with the point-set $\cA$ gives rise to the phase structure. We need only consider the case where all the $\sigma$’s are simplicial based cones, i.e., we induce a simplicial decomposition of triangulation of $\cA$. To each such fan (satisfying in addition a certain “convexity” property, see [@AGM:II] for more details) we associate a phase. Other fans consistent with $\cA$ not satisfying these conditions may always be considered as models on the boundary between two or more phases. The parameters, $r$, in the linear  approach give us an identical fan structure. This is best understood from examining figure 11 of [@AGM:II]. The $r$ parameters, in essence, fix the heights of the points in this figure and hence following the discussion of section of 3.8 of [@AGM:II] their values determine a triangulation of the point set $\cA$. From a physical point of view we can group together those values for the $r$ parameters which yield the same phase for the model. In this way we partition the space of all possible $r$’s into a phase diagram. This phase diagram is the “secondary fan” for the moduli space as discussed in [@AGM:II]. We now have a dictionary between [@W:phase] and the toric approach: [*Specifying generic values of [“$r$”]{} parameters is equivalent to specifying a triangulation of $\cA$. The non-vanishing conditions on the fields $x_i$ specified by minimizing the $D$-term part of the classical potential is equivalent to removing the point set $F_\Delta$ given by*]{} (\[eq:Fset\]). Note that requiring $\cA$ to be “complete” in the sense of (\[eq:Acvx\]) is not necessary in the analysis of [@W:phase]. By imposing this condition we gain access to the largest subspace of the moduli space we can reach by this toric method. One point in the dictionary between [@W:phase] and the toric approach which we have not spelled out explicitly as yet is how we determine the superpotential $W$ from the toric data. This is straightforward as we now describe. Let us $G$ to denote the group $(\C^*)^{N-n}$. $W$ is a $G$-invariant polynomial in the chiral superfields. From (\[eq:krnl\]) we see that any monomial of the form $$\prod_{i=1}^N x_i^{\langle\alpha_i, v\rangle} \label{eq:prod}$$ for a fixed but arbitrary vector $v$ is $G$-invariant. However, we want all terms in $W$ to not only be $G$-invariant but also to have nonnegative integral exponents. Towards this end we are naturally led to introduce the cone $\Upsilon$ in ${\bf M}_\R$, dual to $\Sigma$ which is the cone over the convex hull of $\cA$ in ${\bf N}_\R$, defined by $$\Upsilon = \left\{ s\in{\bf M}_\R; \langle s,t\rangle\geq0, \forall t\in\Sigma \right\}.$$ The integral lattice points in $\Upsilon$, when substituted for the vector $v$ in (\[eq:prod\]), will then generate $G$-invariant monomials with nonnegative exponents. To systematize this, we now define $\cBp\subset \Upsilon$ by $$\cBp = {\bf M} \cap \Upsilon,$$ the integral lattice points contained in the dual cone. Any point in $\cBp$, if substituted for the vector $v$ in (\[eq:prod\]), yields a $G$-invariant nonnegative exponent monomial. Finally, we note that we would like $W$ to be a suitably “quasihomogeneous” polynomial of lowest nontrivial degree in the $x_i$. This will remove any “irrelevant” terms in the superpotential [@VW:] and may be achieved as follows. Let the monomials in this reduced superpotentials be labeled by elements of $\cB \subset \cBp$. Following [@BB:mir] let us put one last condition on $\cA$, namely that when we derive the point set $\cB$ exists a vector $\nu\in{\bf N}$ such that $$\langle \beta_v,\nu\rangle = 1, \quad\forall\beta_v\in\cB,$$ and that the vectors given by the elements of $\cB$ (or a subset of $\cB$) generate $\Upsilon$. We also impose the condition on $\cB$ paralleling our discussion for the point set $\cA$. Namely, we can say $$\cB = {\bf M} \cap (\hbox{Convex hull of $\cB$}\,),$$ with the elements of $\cB$ at the vertices of this convex hull generating $\Upsilon$. We denote by $M$ the number of points in $\cB$ so that $v=1,\ldots,M$. The superpotential $W$ is then constructed according to $$W = \sum_{v=1}^M a_v w_v,$$ for $a_v\in\C$ with $$w_v = \prod_{i=1}^N x_i^{\langle\beta_v,\alpha_i\rangle}. \label{eq:mon}$$ We may note at this point that mirror symmetry is conjectured to exchange the sets $\{{\bf M},\mu,M,\cA\,\}\leftrightarrow\{{\bf N},\nu,N,\cB\}$. This may be regarded as a generalization of the “monomial-divisor mirror map” of [@AGM:mdmm].[^2] The mirror pairs of [@GP:orb] (which is established at the conformal field theory level) are a subset of this general construction and the examples in sections \[ss:Z\] and \[ss:h1\] are in this subset. Thus statement concerning mirror symmetry with regards to these examples may be regarded as definitely true. Also note that our analysis of the phases of the moduli space does not depend on the mirror map and thus does not depend on this mirror conjecture. Now let us try to calculate the central charge $3d$ of the conformal field theory associated to this model. We may apply the same reasoning as was used in [@VW:] to determine this. Firstly we have $N$ chiral superfields each of which contributes $+1$ to $d$. This may be taken to correspond to the string propagating in $\C^N$. We also have $N-n$ vector superfields which we take to contribute $-1$ to $d$ since each removes one complex dimension from the target space. Thus, so far we have $d=n$. However, the string is further confined by the superpotential $W$ and we expect this to reduce the value of $d$ as we now show. Consider now rescaling by an element of $(\C^*)^N$, $$x_i\to\lambda^{\omega_i}x_i.$$ The monomial $w_v$ then scales to $\lambda^\chi w_v$ where $$\chi=\sum_{i=1}^N \langle\beta_v,\alpha_i\rangle\omega_i.$$ Consider choosing the weights $\omega_i$ such that $$\sum_{i=1}^N \omega_i\alpha_i = \nu. \label{eq:wcond}$$ Then all the monomials transform $w_v\to\lambda w_v$ and thus, declaring $a_v$ to be invariant, we have $W\to\lambda W$. Taking the inner product of (\[eq:wcond\]) with $\mu$ gives $$\sum_{i=1}^N\omega_i = \langle \mu,\nu\rangle.$$ It was shown in [@VW:] that the effect of the superpotential is to contribute $-2\sum\omega_i$ to $d$. Thus we have $$d=n-2\langle \mu,\nu\rangle, \label{eq:d}$$ in agreement with the conjecture in [@BB:mir].[^3] For the cases considered in [@AGM:II] based upon the construction of [@Bat:m] we had $\langle\mu,\nu\rangle=1$. This then is a generalization. It should be noted that this more generalized picture could have been deduced directly by applying the toric language to Witten’s formulation of [@W:phase] although historically it was first written in the form of [@Boris:m] where it was used specifically for conjecturing the mirror map for complete intersections in toric varieties. To summarize so far, all the data we require to build an abelian gauged linear  of the form studied in [@W:phase] is the matrix $\alpha_{ij}$. To provide a consistent model for a conformal field theory we demand that this matrix be compatible with $\mu$ and $\nu$ and be consistent with the existence of ${\bf N}$ in the form of (\[eq:Acvx\]). Once we have this information we may apply the technology of [@W:phase; @AGM:II] to determine the geometry of the various phases in the moduli space of Kähler forms. This is most easily determined in terms of triangulations of the point set $\cA$. There is one more piece of information we will need before moving on to some examples concerning orbifolding. The toric variety $\Vbig$ is acted upon by $(\C^*)^n$. It is simple in toric geometry to describe the orbifold of $\Vbig$ by a discrete subgroup of this $(\C^*)^n$. Consider the affine coordinates introduced by (\[eq:aff\]). Let us consider the element, $g\in(\C^*)^n$ which acts by $$g:(\zeta_1,\zeta_2,\ldots,\zeta_n)\mapsto(e^{2\pi ig_1}\zeta_1, e^{2\pi ig_2}\zeta_2,\ldots,e^{2\pi ig_n}\zeta_n) \label{eq:iden}$$ where $0\leq g_j<1$. We can see (for more details consult [@Reid:yp]) that dividing $\Vbig$ by the group generated by $g$ is equivalent to replacing the lattice ${\bf N}$ by a lattice generated by ${\bf N}$ and the vector $(g_1,g_2,\ldots,g_n)\in{\bf N}_\R$. The reason for this is that lattice points $p$ in ${\bf N}$ represent one (complex) parameter group actions on the toric variety $$p:(\zeta_1,\zeta_2,\ldots,\zeta_n)\mapsto(\lambda^{p_1}\zeta_1, \lambda^{p_2}\zeta_2,\ldots,\lambda^{p_n}\zeta_n).$$ For points $p$ whose components are non-integral, such a map is only well defined if certain global identifications are made on the $(\zeta_1,\zeta_2,\ldots,\zeta_n)$. In particular, one directly sees that taking $p$ to be $(g_1,g_2,\ldots,g_n)$ requires the desired identification of (\[eq:iden\]). Applications {#s:appl} ============ Let us now illustrate the general method of the previous section by applying it to various examples. The possibilities offered by this formulation appear to be very rich but we select here a few key examples to emphasize points relevant to our discussion. The Hypersurface Case {#ss:hyp} --------------------- Suppose that $\langle \mu,\nu\rangle=1$. In this case it is easy to see that $\nu\in\cA$ and that this point lies properly in the interior of the convex hull of $\cA$ (since $\langle\beta_v,\nu\rangle$ is strictly positive). One possible triangulation of the point set $\cA$ thus consists of drawing lines from $\nu$ to each point on the vertices of the convex hull and filling this skeleton in with a suitable set of simplices to form a triangulation. The resultant set forms a complete fan of dimension $n-1$ with center $\nu$. This fan $\delta$ corresponds to a compact $(n-1)$-dimensional toric sub-variety $\Vlit$ of $\Vbig$. Let us denote by $p$ the homogeneous coordinate corresponding to $\nu$. We see from (\[eq:mon\]) that every term in the superpotential appears linearly in $p$. Thus we may write $W=pG$ where $G$ is a function of the $N-1$ homogeneous coordinates describing $\Vlit$. Thus the condition $\partial W/\partial p=0$ implies $G=0$ — i.e., we are on a hypersurface within $\Vlit$. For a generic $G$, the other derivatives of $W$ imply that $p=0$. The target space, $X$, is now a hypersurface within $\Vlit$ which itself has dimension $n-1$. $X$ is thus of dimension $n-2$. The equation (\[eq:d\]) tells us that $d=n-2$. In fact $X$ is an anticanonical divisor of $\Vlit$ [@Bat:m] and is thus a  space of $d$ dimensions. Note that $X$ may not be smooth but these singularities can often be removed by further refinements of the fan $\delta$. Actually, in the case $d\leq3$ the singularities may always be removed in this way.[^4] For the case $\langle \mu,\nu\rangle=1$ we therefore always have a “” phase. That is, some limit in the moduli space where we may go to build some non-linear  of the conformal field theory (although in case $d>3$ we may have to include considerations such as terminal orbifold singularities in our model). The case considered here is basically of the type studied in [@Bat:m; @AGM:II] as shown in [@BB:mir]. It also includes the example of the model with the phase in $\C^n/\Z_n$ and the  hypersurface in $\P^{n-1}$ discussed above. The Mirror of the Z-orbifold {#ss:Z} ---------------------------- We now turn to the issue of rigid Calabi-Yau spaces and their mirrors. For concreteness we focus on the Z-orbifold of [@CHSW:]. Recall that this is the torus of six real dimensions divided by a diagonal $\Z_3$ action. It has 36 (1,1)-forms (9 from the original torus and 27 associated with blow up modes) and no (2,1)-forms. It is therefore rigid. Using the construction of [@GP:orb], it was shown in [@AL:geom] how to construct the Z-orbifold in terms of an orbifold of a Gepner model [@Gep:]. To phrase this more carefully allowing for the phase structure, one builds a conformal field theory as an orbifold of a Gepner model which may be deformed via marginal operators to a theory corresponding to a  whose target space is the blown-up Z-orbifold. It was also shown how to build a conformal field theory giving the mirror of the above theory, also as a orbifold of the Gepner model. The Gepner model itself is believed to be equivalent to an orbifold of a  theory. In the case under consideration (the ${\bf 1}^9$ model) the configuration space of this  orbifold theory is $\C^9/\Z_3$. The space $\C^9$ is a toric variety with $n=9$ described simply by the fan consisting of one cone, $\sigma$, isomorphic to the positive quadrant of $\R^9$. That is, $\cA$ consists of the points $(1,0,0,0,0,0,0,0,0),(0,1,0,0,0,0,0,0,0), \ldots,(0,0,0,0,0,0,0,0,1)$. The required $\Z_3$ quotient is performed by adding the generator $$g_1=(\ff13,\ff13,\ff13,\ff13,\ff13,\ff13,\ff13,\ff13,\ff13), \label{eq:g1Z}$$ to the integral lattice of $\R^9$. It was shown in [@AL:geom] that the mirror of the Z-orbifold was obtained by dividing by a further $\Z_3$ (i.e., taking a $\Z_3$ orbifold of the Gepner model) given by the vector $$g_2=(0,0,0,\ff13,\ff13,\ff13,\ff23,\ff23,\ff23),$$ to give the required ${\bf N}$-lattice. We may apply a $Gl(9,\R)$ transformation to ${\bf N}_\R$ to rotate ${\bf N}$ back into the standard integral lattice. This will act on $\sigma$ so that it is no longer the positive quadrant. One choice of transformation leaves $\sigma$ generated by $$\eqalignsq{ \alpha_1&=(3,0,0,1,1,1,-1,-1,-3)\cr \alpha_2&=(0,1,0,0,0,0,0,0,0)\cr \alpha_3&=(0,0,1,0,0,0,0,0,0)\cr \alpha_4&=(0,0,0,1,0,0,0,0,0)\cr \alpha_5&=(0,0,0,0,1,0,0,0,0)\cr \alpha_6&=(0,0,0,0,0,1,0,0,0)\cr \alpha_7&=(0,0,0,0,0,0,1,0,0)\cr \alpha_8&=(0,0,0,0,0,0,0,1,0)\cr \alpha_9&=(0,-1,-1,-2,-2,-2,0,0,3).\cr} \label{eq:AZ}$$ These 9 points lie in the hyperplane defined by $\mu=(3,1,1,1,1,1,1,1,3)$. It is a simple matter to show that the dual cone gives $\nu=(1,0,0,0,0,0,0,0,0)$ so that $\cA$ has the required properties. The important property of this model stems from the fact that the points $\alpha_1,\ldots,\alpha_9$ form the vertices of a simplex with [*no*]{} interior points lying on the lattice ${\bf N}$. That is, the set $\cA$ consists only of those points listed in (\[eq:AZ\]). Thus the only triangulation of $\cA$ consists of this simplex! This model has $\Vbig\cong\C^9/(\Z_3\times\Z_3)$ with superpotential $$W = a_1x_1^3+a_2x_2^3+\ldots+a_9x_9^3+a_{10}x_1x_2x_3+\ldots. \label{eq:WZ}$$ The critical point of $W$ is the origin. Thus we have an orbifold of a  theory as expected. Since there is no other triangulation of $\cA$ there is no other phase and, in particular, [*no  phase*]{}. Since $\langle \mu,\nu\rangle=3$ we are not in conflict with the section \[ss:hyp\]. As expected we see that $d=3$ in agreement with the fact that this theory is the mirror of a smooth  threefold (i.e., the blow-up of the Z-orbifold). Thus, by properly understanding the full content of mirror symmetry — as a symmetry between the moduli spaces of $N = 2$ superconformal theories — we see that there is no puzzle regarding the mirror of a rigid Calabi-Yau manifold. The mirror description simply does not have a Calabi-Yau phase and hence the absence of a Kähler form causes no conflict. It is important to realize that we have all the information we need to study this model without recourse to finding some other effective target space geometry. In particular, deformations of complex structure are achieved by deforming the $a_v$ parameters in (\[eq:WZ\]) in the usual way and one may then use mirror symmetry to study the moduli space of Kähler forms of the Z-orbifold as was done in [@Drk:Z].[^5] The lack of a  phase appears due to the existence of [*terminal*]{} singularities in algebraic geometry as we now discuss (see also [@Reid:yp] for a more thorough account). In section \[s:ph\] we discussed the case of a Landau-Ginzburg theory in $\C^n/\Z_n$. In this case, the $\Z_n$ symmetry is generated by $(\ff1n,\ff1n,\ldots)$. This singularity may be blown-up to give the canonical line bundle over $\P^{n-1}$. This smooth space has trivial canonical class. Thus the singularity $\C^n/\Z_n$ may be blown up without adding something non-trivial into the canonical class. Such a blow-up mode is always visible in the associated conformal field theory as a truly marginal operator since it may be regarded as a deformation of the Kähler form. A terminal singularity is a singularity which cannot be resolved (or even partially resolved) without adding something non-trivial to the canonical class. The singularity $\C^9/\Z_3$ generated by $g_1$ of (\[eq:g1Z\]) is precisely such a singularity. As such, from the conformal field theory point of view, it is “stuck”. This agrees with the fact that the Gepner model contains no marginal operators corresponding to deformations of the Kähler form. One can go ahead and blow-up the $\C^9/\Z_3$ singularity if one really desires some smooth manifold. There is no unique prescription for this but one may, for example, form the space $\O_{\P^8}(-3)$. This is a line bundle over $\P^8$ with $K<0$ (i.e., $c_1>0$). The homogeneous coordinates of this projective space may be given by the coordinates of the original $\C^9$. The superpotential of the Landau-Ginzburg theory is cubic in these fields and so one might try to associate this model to the cubic hypersurface in $\P^8$. This is the essence of the construction of [@Drk:Z; @Schg:gen]. Note that in the language of this paper, we no longer satisfy (\[eq:Q0\]) and so our field theory is expected to have undesirable properties in the infrared limit. When we try to describe the mirror of the Z-orbifold, the situation becomes even worse. The second $\Z_3$ quotient given by $g_2$ induces further terminal quotient singularities on $\P^8$ which require considerably more to be added to the canonical class. We hope the reader sees that this procedure of forcing a smooth geometrical interpretation when terminal singularities appear is completely unnatural when written in terms of the underlying conformal field theory and it is unnecessary when one adopts the phase picture. The mirror of the Z-orbifold need only be described as an orbifold of a Landau-Ginzburg theory in $\C^9$. We should add that the construction of [@Set:sup] should be expected to overcome the renormalization group flow problem inherent in the above hypersurface in $\P^8$ of [@Drk:Z; @Schg:gen]. In the construction of [@Set:sup] one adds ghost fields to reduce the effective dimension of the target space back down to that of $d$. Assuming this is the case, this target space with ghosts can be proposed as a good geometric interpretation of the conformal field theory. It should be pointed out however that such geometric interpretations are probably highly ambiguous. That is, one conformal field theory can be given many interpretations. This occurs in [@Set:sup] where constructions of K3 conformal field theories are given in terms of a 4 complex dimensional space with ghosts whereas the complete moduli space is already understood completely in terms of K3 surfaces [@AM:K3p]. In fact, it is probable that any geometric model may be blown-up to give $K<0$ and then nonzero contributions the the $\beta$-function be cancelled by adding suitable extra fields. Since there are an infinite number of such blow-ups for any model there is the possibility of ascribing an infinity of geometric interpretations of this form. Finally note that it might be possible to associate some geometry with the case discussed in this section by considering orbifolds with discrete torsion [@Berg:dt]. Since we do not understand precisely how to relate quotient singularities with discrete torsion to classical singularities we will not discuss this interpretation here. A Case with $h^{1,1}=1$ {#ss:h1} ----------------------- The above example may be considered rather trivial in that our phase space was zero dimensional, i.e., consisted of only one point. Let us now give a less trivial example which still has no  phase. Consider dividing $\C^9$ by the group $\Z_4\times\Z_4$ groups generated by $$\eqalign{g_1&=(\ff14,\ff14,\ff12,\ff14,\ff14,\ff12,\ff14,\ff14,\ff12)\cr g_2&=(\ff14,\ff34,0,0,0,0,0,0,0).\cr} \label{eq:gZ4}$$ Using the arguments of [@GP:orb; @AL:geom] one may show that the  theory in this space is the mirror of the orbifold $T^3/(\Z_4\times\Z_2)$ where $T$ is a complex torus, the $\Z_4$ group is generated by $(z_1,z_2,z_3)\mapsto(iz_1,-iz_2,z_3)$ and $\Z_2$ by $(z_1,z_2,z_3)\mapsto(z_1,-z_2,-z_3)$, where $z_i$ are the complex coordinates on the tori. The $(2,1)$-form $dz_1\wedge dz_2\wedge d\bar z_3$ is invariant under this group. Indeed $h^{2,1}$ for this orbifold is equal to 1. Thus we expect the case in question to have $h^{1,1}=1$. The point set $\cA$ corresponding to such a space is given by $n=9$ and $$\eqalignsq{ \alpha_1&=(4,-3,0,0,0,0,0,0,0)\cr \alpha_2&=(0,1,0,0,0,0,0,0,0)\cr \alpha_3&=(0,0,1,0,0,0,0,0,0)\cr \alpha_4&=(0,0,0,1,0,0,0,0,0)\cr \alpha_5&=(0,0,0,0,1,0,0,0,0)\cr \alpha_6&=(0,0,0,0,0,1,0,0,0)\cr \alpha_7&=(0,0,0,0,0,0,1,0,0)\cr \alpha_8&=(-4,2,-2,-1,-1,-2,-1,4,-2)\cr \alpha_9&=(0,0,0,0,0,0,0,0,1)\cr \alpha_{10}&=(3,-2,0,0,0,0,0,0,0)\cr \alpha_{11}&=(2,-1,0,0,0,0,0,0,0)\cr \alpha_{12}&=(1,0,0,0,0,0,0,0,0),\cr} \label{eq:Ao}$$ with $\mu=(1,1,1,1,1,1,1,3,1)$ and $\nu=(0,0,0,0,0,0,0,1,0)$. Therefore this theory has $d=3$ again. The points $\alpha_1,\ldots,\alpha_9$ form a simplex with $\alpha_{10},\alpha_{11},\alpha_{12}$ positioned along the edge joining $\alpha_1$ and $\alpha_2$. Thus all the interesting part of the point set as regards triangulations is contained in this line $\alpha_1\alpha_2$: $$\setlength{\unitlength}{0.007in}\begin{picture}(330,30)(155,635) \thinlines \put(400,660){\circle{10}} \put(160,660){\circle*{10}} \put(480,660){\circle*{10}} \put(320,660){\circle{10}} \put(240,660){\circle{10}} \put(160,660){\line( 1, 0){320}} \put(315,635){\makebox(0,0)[lb]{$\alpha_{11}$}} \put(155,635){\makebox(0,0)[lb]{$\alpha_1$}} \put(235,635){\makebox(0,0)[lb]{$\alpha_{10}$}} \put(395,635){\makebox(0,0)[lb]{$\alpha_{12}$}} \put(475,635){\makebox(0,0)[lb]{$\alpha_2$}} \end{picture}$$ The points $\alpha_{10},\alpha_{11},\alpha_{12}$ may, or may not be included in the triangulation (and are hence shown as circles rather than dots). If none of the points $\alpha_{10},\alpha_{11},\alpha_{12}$ are included in the triangulation, we have one simplex with vertices $\alpha_1,\ldots,\alpha_9$ and the associated toric variety is $\C^9/(\Z_4\times\Z_4)$ as expected. If all these points are included in the triangulation we have 4 simplices. The resulting space is a partial resolution of the $\C^9/(\Z_4\times\Z_4)$ space. The exceptional divisor introduced is a “plumb product” of three $\P^1$ spaces. Each of the points $\alpha_{10},\alpha_{11},\alpha_{12}$ may be taken to correspond to one of these $\P^1$ components. This is shown in figure \[fig:Z4b\]. The black dot on the left hand side shows the isolated singularity. On the right hand side the singularity (which is now terminal) covers the whole exceptional divisor. Clearly, other triangulations represent intermediate steps in this blow-up. Let us now analyze the critical point set of $W$. Finding $\cB$ we determine from (\[eq:mon\]) that $$W = a_1x_1^4x_{10}^3x_{11}^2x_{12}+a_2x_2^4x_{10}x_{11}^2x_{12}^3+ a_3x_3^2+a_4x_4^4+a_5x_5^4+a_6x_6^2+a_7x_7^4+a_8x_8^4+a_9x_9^2+ \ldots \label{eq:WZ4}$$ In total there are 87 points in $\cB$ but we need only consider the above terms with nonzero $a_1,\ldots,a_9$ for a sufficiently generic $W$. Consider the maximal triangulation. This includes all three points $\alpha_{10},\alpha_{11},\alpha_{12}$. Since $N=12$ we need to remove the set $F_\Delta$ given by (\[eq:Fset\]) from $\C_{12}$. This amounts in imposing $x_2x_{11}x_{12}\neq0$ or $x_1x_2x_{12}\neq0$ or $x_1x_2x_{10}\neq0$ or $x_1x_{10}x_{11}\neq0$. We also wish to impose $\partial W/\partial x_i=0$ for $i=1,\ldots,12$. It is straight-forward to show that these conditions require $$\eqalign{x_3=x_4=x_5=x_6=x_7&=x_8=x_9=x_{11}=0\cr x_1&\neq0\cr x_2&\neq0\cr}$$ and that $x_{10}$ and $x_{12}$ cannot both be zero simultaneously. As $N-n=3$ we have three $\C^*$ actions to divide this subspace of $\C^{12}$ by. Two may be used to fix $x_1$ and $x_2$ to specific values. The other $\C^*$ may be used to turn $x_{10}$ and $x_{12}$ into homogeneous coordinates parametrizing $\P^1$. The vacuum is thus $\P^1$. One may also determine the superpotential in this vacuum to show that we have a  theory fibered over this $\P^1$ to obtain the familiar hybrid-type models of [@W:phase; @AGM:II]. One may also show that the fiber has a $\Z_4$-quotient singularity at the zero section. In terms of the ambient toric variety $V_\Delta$, what we have just described in the previous paragraph is the $\P^1$ that appears in the middle of the chain of three $\P^1$’s on the right in figure \[fig:Z4b\]. Thus although $\Vbig$ appears to have three degrees of freedom for the Kähler form — giving the three independent sizes of the three $\P^1$’s, only one makes it down to $X$, the critical point set of $W$. Therefore $X$ only has one Kähler-type deformation. Sometimes additional modes appear in the fibre for these hybrid models but in this case the fibre contains no twist fields with the correct charges to be considered a (1,1)-form. We will therefore assert that $h^{1,1}(X)=1$. Thus we are in agreement with the assertions concerning the mirror space at the start of this section. Analyzing the other possible triangulations we find that we reproduce one of the two phases we know about — either the  orbifold in $\C^9/(\Z_4\times\Z_4)$ or the hybrid model over $\P^1$. The points $\alpha_{10}$ and $\alpha_{12}$ may be ignored when considering $X$. Thus we have constructed a model with a non-trivial phase diagram — there are 2 phases — but neither is a  space. In general there is a homomorphism: $$\kappa:H^{1,1}(V_\Delta)\to H^{1,1}(X). \label{eq:kappa}$$ In general however $\kappa$ is neither injective nor surjective. The example in this section shows a failure of injectivity since $h^{1,1}(V_\Delta)=3$ and $h^{1,1}(X)=1$. In the more simple case of $\langle\mu,\nu\rangle=1$ it was shown in [@AGM:mdmm] that the kernel of $\kappa$ was described by points in the interior of co-dimension one faces of the convex hull of $\cA$. In the case described in this section we see that such a simple criterion cannot be used — all the points $\alpha_{10},\alpha_{11},\alpha_{12}$ lie in a co-dimension 7 face and yet $\alpha_{10}$ and $\alpha_{12}$ contribute to the kernel and $\alpha_{11}$ survives through to $H^{1,1}(X)$. At this point in we know of no simple method of determining the image of $\kappa$ except to explicitly calculate the critical point set of $W$ on a case by case basis. Let us conclude this section by discussing the short-comings of analyzing this model in terms of the “generalized  manifolds” of [@Schg:gen]. The $\Z_4$ singularity in $\C^9$ generated by $g_1$ of (\[eq:gZ4\]) may be partially resolved by a line bundle over the weighted projective space $\P^8_{\{1,1,2,1,1,2,1,1,2\}}$. The resultant space has $K<0$. The “generalized  manifold”, $R$, would be identified as the hypersurface in this weighted projective space given by the vanishing of (\[eq:WZ4\]) with $x_1,\ldots,x_9$ taken to be the quasi-homogeneous coordinates and $x_{10}=x_{11}=x_{12}=1$. The $\Z_4$-action of $g_2$ acts on $R$ to induce $\Z_4$-quotient singularities over some subspace of codimension two. These latter singularities are not terminal and may be resolved without adding anything further to $K$. In fact, resolving these latter singularities may be achieved by introducing the points $\alpha_{10},\alpha_{11},\alpha_{12}$ into the toric fan. It is easy to see that something similar will happen in general. That is, any $K=0$ toric resolutions we may perform in $\Vbig$ may also be performed after blowing up any terminal singularities in $\Vbig$. It follows that the points in the interior of the convex hull of $\cA$ may be counted by analyzing singularities which may be locally resolved with $K=0$ in $R$. (Note that this is a rather inefficient way of proceeding in our picture — one may as well just analyze the singularities in $\Vbig$ without any destroying the $K=0$ condition.) This observation sheds light on a conjecture in [@Schg:fno] that $h^{1,1}(X)$ could be determined by counting the contribution to $h^{1,1}$ of any resolutions of singularities within the $R$. We see now that this will count $h^{1,1}(V_\Delta)$ which is, in general, not equal to $h^{1,1}(X)$. Thus this conjecture is false. In the example above, counting this way would imply that $h^{1,1}(X)=3$. With regards to determining $h^{1,1}(X)$, it appears hard to save the construction of [@Set:sup] from a similar fate. The problem is that the divisors associated with $\alpha_{10}$, $\alpha_{11}$ and $\alpha_{12}$ appear on equal footing in $R$. Thus unless some unsymmetric rules are devised for resolving canonical singularities in superspace one cannot obtain the correct answer $h^{1,1}(X)=1$. A Case with $X=\hbox{\bigbbbfont P}^3$ {#ss:P3} -------------------------------------- One might be led to suspect the following to be the general picture for the geometric interpretation of an $N$=2 superconformal field theory. Either $X$ is a  space and the string is free to move within $X$ and there are no massless modes normal to $X$, or $X$ is a space of dimension $<d$ in which the string is free to move and there are massless modes governed by some superpotential normal to $X$ inside some bigger ambient space $\Vbig$ containing $X$. We now show an example (which also appeared in [@W:phase]) which is an exception to this. Consider the following point set for $\cA\,$: $$\eqalignsq{ \alpha_1&=(1,0,0,0,0,0,0,1,0,0,0)\cr \alpha_2&=(0,1,0,0,0,0,0,1,0,0,0)\cr \alpha_3&=(0,0,1,0,0,0,0,0,1,0,0)\cr \alpha_4&=(0,0,0,1,0,0,0,0,1,0,0)\cr \alpha_5&=(0,0,0,0,1,0,0,0,0,1,0)\cr \alpha_6&=(0,0,0,0,0,1,0,0,0,1,0)\cr \alpha_7&=(0,0,0,0,0,0,1,0,0,0,1)\cr \alpha_8&=(-1,-1,-1,-1,-1,-1,-1,0,0,0,1)\cr \alpha_9&=(0,0,0,0,0,0,0,1,0,0,0)\cr \alpha_{10}&=(0,0,0,0,0,0,0,0,1,0,0)\cr \alpha_{11}&=(0,0,0,0,0,0,0,0,0,1,0)\cr \alpha_{12}&=(0,0,0,0,0,0,0,0,0,0,1).\cr} \label{eq:AP3}$$ Thus $\mu=(0,0,0,0,0,0,0,1,1,1,1)$. One can also determine $\cB$ with a little effort and find $\nu=(0,0,0,0,0,0,0,1,1,1,1)$. Thus $d=3$ again. The superpotential $W$ may be written $$W = x_9G_1 + x_{10}G_2 + x_{11}G_3 + x_{12}G_4,$$ where the $G_k$ are generic homogeneous polynomials of total degree two in $x_1,\ldots,x_8$. There are two triangulations of the point set $\cA$. The first consists of taking 8 simplices each of which has $\alpha_9,\ldots,\alpha_{12}$ as 4 of its vertices with the other 7 vertices taken from the set $\{\alpha_1,\ldots,\alpha_8\}$. In terms of $F_\Delta$ this amounts to removing the point $x_1=\ldots=x_8=0$ from consideration. Restricting to the critical point set of $W$ forces $x_9=\ldots=x_{12}=0$ and $G_1=\ldots=G_4=0$. Dividing out by the single required $\C^*$-action forms $\P^7$ with homogeneous coordinates $x_1,\ldots,x_8$. Thus $X$ is the intersection of 4 quadric equations $G_k=0$ in $\P^7$. This is a known  space dating back to [@CHSW:]. The other triangulation consists of 4 simplices with 8 vertices given by $\alpha_1,\ldots,\alpha_8$ with the other 3 taken from the set $\{\alpha_9,\ldots,\alpha_{12}\}$. This amounts to removing $x_9=\ldots=x_{12}=0$ from consideration. Restricting to the critical point set of $W$ forces $x_1=\ldots=x_8=0$. Now the $\C^*$-action may be used to form $\P^3$ with homogeneous coordinates $x_9,\ldots,x_{12}$. Thus $X$ in this phase is $\P^3$. Our phase diagram consists of two phases — both of which have the dimension of $X$ equal to $d$. One phase is a  manifold with $h^{1,1}=1$ and $h^{2,1}=65$ which we understand. The other phase is $\P^3$. The reader might be alarmed at the appearance of the latter since $\P^3$ is not a  space and lacks a nonvanishing holomorphic 3-form for example. The resolution is as follows. Although we have correctly identified the vacuum of the field theory as $X$ we have to be a little careful in declaring it to be the effective target space of a conformal field theory. Let us consider the variables $\alpha_1,\ldots,\alpha_8$ which we forced to zero. The superpotential is quadratic in these variables so we certainly haven’t missed any massless degrees of freedom (which would only add to our troubles by increasing $d$ anyway). The point is that there is actually a $\Z_2$-quotient singularity coming from the identification of homogeneous coordinates in $\Vbig$ to affine ones. Thus we have a fibration of a  orbifold theory over $\P^3$ which may appear trivial in that the superpotential is quadratic but we may expect twist fields is add to our spectrum. In particular we expect to have an analog to a Calabi-Yau $H^{3,0}$ mode (i.e. a field of charge (3,0) under the $U(1)_L \times U(1)_R$ of the superconformal algebra) coming from such twisted sectors. Of course, such a mode cannot be given a literal geometric interpretation in terms of a (3,0)-form. It is interesting also to ask how literally we can take this $\P^3$ to be a target space for the conformal field theory. To find the actual size of truly conformally invariant  target space one needs to solve the Picard-Fuchs system as described in [@AGM:sd]. We will not present the details here since they are rather lengthy but we may quickly summarize as follows. One solves equations (42) of [@AGM:sd] where the “$\beta$” vector of this system is set equal to $-\nu$. (One could then count rational curves on this  space if one so desired.) The complexified Kähler form $B+iJ$ of the phase can then be analytically continued into $\P^3$ phase (which is most easily done by the method of [@me:min-d]). Taking $z$ to be the local coordinate on the moduli space where $z=0$ corresponds to the limit point in the $\P^3$ phase we obtain $$B+iJ=-\ff12 -\frac{3\pi i}{2\log(z)} +O(\log(z)^{-2}).$$ Thus $J\geq 0$ in the region near the limit point $|z|\ll1$. The effective size of target space is very small as $|z|\to0$. In other words, the effect of integrating out the massive modes in the linear  of [@W:phase] has caused an infinite renormalization of the “$r$” parameter (unlike what is believed to happen for the  phase). To summarize we see that the phase picture can produce phases with dimension equal to $d$ which do not correspond to  non-linear s. To understand these phases more completely will require a better understanding of the hybrid models. Conclusions {#s:conc} =========== Geometrical methods have proven themselves to be a powerful conceptual and calculational tool in understanding the physical content of certain conformal theories and their associated string models. As such, it is a worthwhile task to gain as complete an understanding as possible of the geometrical status of conformal field theories, especially for the case of $N = 2$ worldsheet supersymmetry relevant for spacetime supersymmetric string models. The phase structure of such $N = 2$ models, as found in [@W:phase; @AGM:II], goes a long way towards capturing the full geometric content of these theories, and, in particular, certainly provides the correct framework for discussing mirror symmetry. In this paper we have used this phase structure analysis to address certain previously puzzling issues regarding the geometrical content of certain theories. In particular, at first sight mirror symmetry seems to come upon the puzzle regarding the identity of the mirror of a rigid manifold. We have seen, though, that this appears to be a puzzle only because the question itself is not phrased in the correct context. That is, mirror symmetry tells us that certain [*a priori*]{} distinct pairs of families of conformal theories actually are composed of isomorphic members. When the phase structure of each family in such a pair contains a Calabi-Yau  region, then these Calabi-Yau’s form a mirror pair. However, in certain cases, at least one of the families does [*not*]{} have a Calabi-Yau  region. In such cases mirror symmetry will simply not yield a mirror pair of Calabi-Yau manifolds. A family which has a rigid Calabi-Yau phase, as we have seen, provides one such example — the mirror family does not have a Calabi-Yau region. Thus the absence of a Kähler form for the mirror is not an issue. The mirror moduli space has no Calabi-Yau phase and hence does not require a Kähler form. The previous puzzle disappears, therefore, when the question is phrased in the correct context. Beyond the rigid case, we have also seen that even when there is a conformal mode that can play the part of a Kähler form, there need not be a Calabi-Yau phase on which it can realize this potential. So, whereas in the previous problem we resolved the issue of a “Calabi-Yau in search of a Kähler form” here we have “a Kähler form in search of a Calabi-Yau”. We have established that there are examples in which there simply is no Calabi-Yau to be found. It is worth noting that the map $\kappa$ in (\[eq:kappa\]) is neither injective or surjective. We saw the effect on this map not being injective in section \[ss:h1\]. The failure of surjectivity shows that some (1,1)-forms on $X$ do not come from the toric ambient space $\Vbig$. In the case of models of the form discussed in section \[ss:hyp\] it is still possible to count the number $h^{1,1}(X)$ because of the properties of hypersurfaces [@Bat:m; @AGM:mdmm]. In the cases considered here however we deal with more general complete intersections. The problem of counting $h^{1,1}(X)$ in this context is the mirror of the problem of counting $h^{2,1}$ for the mirror model. The analogue of the fact that $\kappa$ is not an isomorphism is the fact that deforming the polynomial giving the superpotential is not the same as deformation the complex structure. It would be interesting to see if methods along the lines of [@GH:poly] could be applied in this context to determine the Hodge numbers of $X$. An interesting question, to which we do not know the answer, is whether there are examples in which [*neither*]{} family in a mirror pair has a Calabi-Yau region. Such an example would establish that there are $N = 2, c = 3d$ conformal theories which are not interpretable in terms of Calabi-Yau compactifications (or analytic continuations thereof). To answer this question is difficult because of the failure of the surjectivity of $\kappa$. One may find mirror pairs of orbifolds of the Gepner model, ${\bf 1}^9$, for which neither has an obvious  interpretation. An example with $h^{1,1}$ and $h^{2,1}$ equal to 4 and 40 was mentioned in [@AL:geom]. This is a good candidate for a situation where neither of the mirror partners have a  phase (despite the assertions of [@AL:geom]). Unfortunately if $X$ is the model with $h^{1,1}=4$ then the image of $\kappa$ is trivial, i.e., none of the (1,1)-forms come from $\Vbig$. Because of this the methods of this paper cannot be used to draw any conclusions regarding the lack of  phase for this example. Acknowledgements {#acknowledgements .unnumbered} ================ It is a pleasure to thank D. Morrison and R. Plesser for useful conversations. The work of P.S.A. is supported by a grant from the National Science Foundation. The work of B.R.G. is supported by a National Young Investigator Award, the Alfred P. Sloan Foundation and the National Science Foundation. Note Added {#note-added .unnumbered} ========== The mirror of the example of section \[ss:P3\] was studied in [@lots:per] where it was discovered that there is an extra $\Z_2$ symmetry in the moduli space. This should act on the moduli space in section \[ss:P3\] to identify the two phases with each other. This may be viewed as a new kind of $R\leftrightarrow1/R$ symmetry. [10]{} E. Witten, , Nucl. Phys. [**B403**]{} (1993) 159–222. P. S. Aspinwall, B. R. Greene, and D. R. Morrison, , Phys. Lett. [**303B**]{} (1993) 249–259. P. S. Aspinwall, B. R. Greene, and D. R. Morrison, , Nucl. Phys. [**B416**]{} (1994) 414–480. P. Candelas, E. Derrick, and L. Parkes, , Nucl.Phys. [**B407**]{} (1993) 115–154. R. Schimmrigk, , Phys. Rev. Lett. [**70**]{} (1993) 3688–3691. S. Sethi, , Nucl. Phys. [**B430**]{} (1994) 31–50. P. S. Aspinwall, B. R. Greene, and D. R. Morrison, , Nucl. Phys. [**B420**]{} (1994) 184–242. F. C. Kirwan, , Number 31 in Mathematical Notes, Princeton University Press, Princeton, 1984. L. Ness, , Amer. J. Math. [**106**]{} (1984) 1281–3129. A. D’Adda, P. D. Vecchia, and M. L[ü]{}scher, , Nucl. Phys. [**B152**]{} (1979) 125–144. P. Griffiths and J. Harris, , Wiley-Interscience, 1978. D. A. Cox, , Amherst 1992 preprint, alg-geom/9210008, to appear in J. Algebraic Geom. M. Reid, , in S. Iitaka, editor, “Algebraic Varieties and Analytic Varieties”, volume 1 of Adv. Studies in Pure Math., pages 131–180, Kinokuniya, Tokyo, 1983. C. Vafa and N. Warner, , Phys. Lett [**218B**]{} (1989) 51–58. V. V. Batyrev and L. A. Borisov, , Essen and Michigan 1994 preprint, alg-geom/9402002. P. S. Aspinwall, B. R. Greene, and D. R. Morrison, , Internat. Math. Res. Notices [**1993**]{} 319–338. S. Katz and D. Morrison, , to appear. B. R. Greene and M. R. Plesser, , Nucl. Phys. [**B338**]{} (1990) 15–37. V. V. Batyrev, , J. Alg. Geom. [**3**]{} (1994) 493–535. L. Borisov, , Michigan 1993 preprint, alg-geom/9310001. M. Reid, , Proc. Symp. Pure Math. [**46**]{} (1987) 345–414. P. Candelas, G. Horowitz, A. Strominger, and E. Witten, , Nucl. Phys. [**B258**]{} (1985) 46–74. P. S. Aspinwall and C. A. L[ü]{}tken, , Nucl. Phys. [**B353**]{} (1991) 427–461. D. Gepner, , Phys. Lett. [**199B**]{} (1987) 380–388. P. S. Aspinwall and D. R. Morrison, , Duke and IAS 1994 preprint DUK-TH-94-68, IASSNS-HEP-94/23, hep-th/9404151, to appear in “Essays on Mirror Manifolds 2”. P. Berglund, , Phys. Lett. [**319B**]{} (1993) 117–124. R. Schimmrigk, , Bonn 1994 preprint BONN-TH-94-007, hep-th/9405086. P. S. Aspinwall, , Nucl. Phys. [**B431**]{} (1994) 78–96. P. Green and T. H[ü]{}bsch, , Commun. Math. Phys. [**113**]{} (1987) 505–528. P. Berglund et al., , Nucl. Phys. [**B419**]{} (1994) 352–403. [^1]: This notation is not entirely consistent with [@AGM:II]. For example the $\Delta$ of this paper is the $\Delta^+$ of [@AGM:II]. [^2]: This has also been studied independently by S. Katz and D. Morrison [@KM:mir]. [^3]: Except that there appears to be a typographical error in conjecture (2.17) of [@BB:mir]. [^4]: This is because Gorenstein singularities can only be terminal in more than 3 dimensions [@Reid:mm]. [^5]: Note that the periods deduced in [@Drk:Z] can be determined from the analysis of the Picard-Fuchs equation as we mention briefly later.

--- abstract: 'Mean-reverting assets are one of the holy grails of financial markets: if such assets existed, they would provide trivially profitable investment strategies for any investor able to trade them, thanks to the knowledge that such assets oscillate predictably around their long term mean. The modus operandi of cointegration-based trading strategies [@tsay2005analysis §8] is to create first a portfolio of assets whose aggregate value mean-reverts, to exploit that knowledge by selling short or buying that portfolio when its value deviates from its long-term mean. Such portfolios are typically selected using tools from cointegration theory [@granger; @johansen], whose aim is to detect combinations of assets that are stationary, and therefore mean-reverting. We argue in this work that focusing on stationarity only may not suffice to ensure profitability of cointegration-based strategies. While it might be possible to create synthetically, using a large array of financial assets, a portfolio whose aggregate value is stationary and therefore mean-reverting, trading such a large portfolio incurs in practice important trade or borrow costs. Looking for stationary portfolios formed by many assets may also result in portfolios that have a very small volatility and which require significant leverage to be profitable. We study in this work algorithmic approaches that can take mitigate these effects by searching for maximally mean-reverting portfolios which are sufficiently sparse and/or volatile.' author: - | Marco Cuturi\ Graduate School of Informatics\ Kyoto University\ `mcuturi@i.kyoto-u.ac.jp`\ \ Alexandre d’Aspremont\ D.I., UMR CNRS 8548\ Ecole Normale Supérieure, `aspremon@ens.fr` title: | Mean-Reverting Portfolios:\ Tradeoffs Between Sparsity and Volatility --- Introduction ============ Mean-reverting assets, namely assets whose price oscillates predictably around a long term mean, provide investors with an ideal investment opportunity. Because of their tendency to pull back to a given price level, a naive contrarian strategy of buying the asset when its price lies below that mean, or selling short the asset when it lies above that mean can be profitable. Unsurprisingly, assets that exhibit significant mean-reversion are very hard to find in efficient markets. Whenever mean-reversion is observed in a single asset, it is almost always impossible to profit from it: the asset may typically have very low volatility, be illiquid, hard to short-sell, or its mean-reversion may occur at a time-scale (months, years) for which the borrow-cost of holding or shorting the asset may well exceed any profit expected from such a contrarian strategy. ### Synthetic Mean-Reverting Baskets Since mean-reverting assets rarely appear in liquid markets, investors have focused instead on creating synthetic assets that can mimic the properties of a single mean-reverting asset, and trading such synthetic assets as if they were a single asset. Such a synthetic asset is typically designed by combining long and short positions in various liquid assets to form a *mean-reverting portfolio*, whose aggregate value exhibits significant mean-reversion. Constructing such synthetic portfolios is, however, challenging. Whereas simple descriptive statistics and unit-root test procedures can be used to test whether a single asset is mean-reverting, building mean-reverting portfolios requires finding a proper vector of algebraic weights (long and short positions) that describes a portfolio which has a mean-reverting aggregate value. In that sense, mean-reverting portfolios are made by the investor, and cannot be simply chosen among tradable assets. A mean-reverting portfolio is characterized both by the pool of assets the investor has selected (starting with the dimension of the vector), and by the fixed nominal quantities (or weights) of each of these assets in the portfolio, which the investor also needs to set. When only two assets are considered, such baskets are usually known as long-short trading pairs. We consider in this paper baskets that are constituted by more than two assets. ### Mean-Reverting Baskets with Sufficient Volatility and Sparsity A mean-reverting portfolio must exhibit sufficient mean-reversion to ensure that a contrarian strategy is profitable. To meet this requirement, investors have relied on cointegration theory [@granger; @maddala1998urc; @johansen2005cointegration] to estimate linear combinations of assets which exhibit stationarity (and therefore mean-reversion) using historical data. We argue in this work, as we did in earlier references [@alex; @cuturi2013mean], that mean-reverting strategies cannot, however, only rely on this approach to be profitable. Arbitrage opportunities can only exist if they are large enough to be traded without using too much leverage or incurring too many transaction costs. For mean-reverting baskets, this condition translates naturally into a first requirement that the gap between the basket valuation and its long term mean is large enough on average, namely that the basket price has sufficient variance or volatility. A second desirable property is that mean-reverting portfolios require trading as few assets as possible to minimize costs, namely that the weights vector of that portfolio is sparse. We propose in this work methods that maximize a proxy for mean reversion, and which can take into account at the same time constraints on variance and sparsity.\ \ We propose first in Section \[s:crit\] three proxies for mean reversion. Section \[s:opt\] defines the basket optimization problems corresponding to these quantities. We show in Section \[s:sdp\] that each of these problems translate naturally into semidefinite relaxations which produce either exact or approximate solutions using sparse PCA techniques. Finally, we present numerical evidence in Section \[s:numres\] that taking into account sparsity and volatility can significantly boost the performance of mean-reverting trading strategies in trading environments where trading costs are not negligible. Proxies for Mean-Reversion {#s:crit} ========================== Isolating stable linear combinations of variables of multivariate time series is a fundamental problem in econometrics. A classical formulation of the problem reads as follows: given a vector valued process $x=(x_t)_t$ taking values in $\RR^n$ and indexed by time $t\in\NN$, and making no assumptions on the stationarity of each individual component of $x$, can we estimate one or many directions $y\in\RR^n$ such that the univariate process $(y^Tx_t)$ is stationary? When such a vector $y$ exists, the process $x$ is said to be cointegrated. The goal of cointegration techniques is to detect and estimate such directions $y$. Taken for granted that such techniques can efficiently isolate sparse mean reverting baskets, their financial application can be either straightforward using simple event triggers to buy, sell or simply hold the basket [@tsay2005analysis §8.6], or more elaborate optimal trading strategies if one assumes that the mean-reverting basket value is a Ohrstein-Ullenbeck process, as discussed in [@jurek; @liu2010optimal; @elie:hal-00573429]. Related Work and Problem Setting -------------------------------- @granger provided in their seminal work a first approach to compare two non-stationary univariate time series $(x_t,y_t)$, and test for the existence of a term $\alpha$ such that $y_t-\alpha x_t$ becomes stationary. Following this seminal work, several techniques have been proposed to generalize that idea to multivariate time series. As detailed in the survey by @maddala1998urc [§5], cointegration techniques differ in the modeling assumptions they require on the time series themselves. Some are designed to identify only one cointegrated relationship, whereas others are designed to detect many or all of them. Among these references, @johansen proposed a popular approach that builds upon a VAR model, as surveyed in [@johansen2005cointegration; @johansen2009cointegration]. These approaches all discuss issues that are relevant to econometrics, such as de-trending and seasonal adjustments. Some of them focus more specifically on testing procedures designed to check whether such cointegrated relationships exist or not, rather than on the robustness of the estimation of that relationship itself. We follow in this work a simpler approach proposed by @alex, which is to trade-off interpretability, testing and modeling assumptions for a simpler optimization framework which can be tailored to include other aspects than only stationarity. @alex did so by adding regularizers to the predictability criterion proposed by @box1977cam. We follow in this paper the approach we proposed in [@cuturi2013mean] to design mean-reversion proxies that do not rely on any modeling assumption. Throughout this paper, we write $\symm_n$ for the $n\times n$ cone of positive definite matrices. We consider in the following a multivariate stochastic process $x=(x_t)_{t\in\NN}$ taking values in $\RR^n$. We write $\Acal_k= \Expect[x_t x_{t+k}^T], k\geq 0$ for the lag-$k$ autocovariance matrix of $x_t$ if it is finite. Using a sample path $\bx$ of $(x_t)$, where $\bx=(\bx_1,\ldots,\bx_T)$ and each $\bx_t\in\RR^n$, we write $A_k$ for the *empirical* counterpart of $\Acal_k$ computed from $\bx$, $$\label{eq:autos} A_k\defeq \frac{1}{T-k-1}\sum_{t=1}^{T-k} \tilde{\bx}_t \tilde{\bx}_{t+k}^T,\; \tilde{\bx}_t\defeq \bx_t-\frac{1}{T}\sum_{t=1}^T \bx_t.$$ Given $y\in\RR^n$, we now define three measures which can all be interpreted as proxies for the mean reversion of $y^Tx_t$. **Predictability** – defined for stationary processes by @box1977cam and generalized for non-stationary processes by @Bewl94 – measures how close to noise the series is. The **portmanteau** statistic [@Ljun78] is used to test whether a process is white noise. Finally, the **crossing statistic** [@ylvisaker1965expected] measures the probability that a process crosses its mean per unit of time. In all three cases, low values for these criteria imply a fast mean-reversion. Predictability {#subsec:pred} -------------- We briefly recall the canonical decomposition derived in [@box1977cam]. Suppose that $x_t$ follows the recursion: \[eq:ar1\] x\_t= \_[t-1]{} + \_t, where $\hat{x}_{t-1}$ is a predictor of $x_t$ built upon past values of the process recorded up to $t-1$, and $\varepsilon_t$ is a vector of i.i.d. Gaussian noise with zero mean and covariance $\Sigma \in \symm_n$ independent of all variables $(x_{r})_{r<t}$. The canonical analysis in [@box1977cam] starts as follows. ### Univariate case Suppose $n=1$ and thus $\Sigma\in\RR_+$, Equation (\[eq:ar1\]) leads thus to $$\Expect[x_t^2]=\Expect[\hat{x}_{t-1}^2]+\Expect{[\varepsilon_t^2]}, \text{ thus } 1=\frac{\hat{\sigma}^2}{\sigma^2}+\frac{\Sigma}{\sigma^2},$$ by introducing the variances $\sigma^2$ and $\hat{\sigma}^2$ of $x_t$ and $\hat{x}_t$ respectively. @box1977cam measure the *predictability* of $x_t$ by the ratio $$\lambda\defeq\frac{\hat{\sigma}^2}{\sigma^2}.$$ The intuition behind this variance ratio is simple: when it is small the variance of the noise dominates that of $\hat{x}_{t-1}$ and $x_t$ is dominated by the noise term; when it is large, $\hat{x}_{t-1}$ dominates the noise and $x_t$ can be accurately predicted on average. ### Multivariate case Suppose $n>1$ and consider now the univariate process $(y^Tx_t)_{t}$ with weights $y\in\RR^{n}$. Using (\[eq:ar1\]) we know that $y^Tx_t =y^T\hat{x}_{t-1}+y^T\varepsilon_t$, and we can measure its predicability as \[eq:pred\] (y), where $\hat{\Acal}_0$ and $\Acal_0$ are the covariance matrices of $x_t$ and $\hat{x}_{t-1}$ respectively. Minimizing predictability $\lambda(y)$ is then equivalent to finding the minimum generalized eigenvalue $\lambda$ solving \[eq:pred2\] (\_0 - \_0) =0. Assuming that $\Acal_0$ is positive definite, the basket with minimum predictability will be given by $y=\Acal_0^{-1/2}y_0$, where $y_0$ is the eigenvector corresponding to the smallest eigenvalue of the matrix $\Acal_0^{-1/2} \hat{\Acal}_0 \Acal_0^{-1/2}$. ### Estimation of $\lambda(y)$ All of the quantities used to define $\lambda$ above need to be estimated from sample paths. $\Acal_0$ can be estimated by $A_0$ following Equation . All other quantities depend on the predictor $\hat{x}_{t-1}$. @box1977cam assume that $x_t$ follows a vector autoregressive model of order $p$ – VAR(p) in short – and therefore $\hat{x}_{t-1}$ takes the form, $$\hat{x}_{t-1}=\sum_{k=1}^p \Hca_k x_{t-k},$$ where the $p$ matrices $(\Hca_k)$ contain each $n\times n$ autoregressive coefficients. Estimating $\Hca_k$ from the sample path $\bx$, @box1977cam solve for the optimal basket by inserting these estimates in the generalized eigenvalue problem displayed in Equation . If one assumes that $p=1$ (the case $p>1$ can be trivially reformulated as a VAR(1) model with adequate reparameterization), then $$\hat{\Acal}_0=\Hca_1 \Acal_0 \Hca_1^T \text{ and }\Acal_1=\Acal_0 \Hca_1,$$ and thus the Yule-Walker estimator [@lutkepohl2005nim §3.3] of $\Hca_1$ would be $H_1=A_0^{-1} A_1$. Minimizing predictability boils down to solving in that case $$\min_{y} \hat{\lambda}(y), \; \hat{\lambda}(y)\defeq \frac{y^T \left( H_1 A_0 H_1^T\right) y}{y^T A_0 y}=\frac{y^T \left( A_1 A_0^{-1} A_1^T\right) y}{y^T A_0 y},$$ which is equivalent to computing the smallest eigenvector of the matrix $A_0^{-1/2}A_1 A_0^{-1} A_1^T A_0^{-1/2}$ if the covariance matrix $A_0$ is invertible. The machinery of @box1977cam to quantify mean-reversion requires defining a model to form $\hat{x}_{t-1}$, the conditional expectation of $x_t$ given previous observations. We consider in the following two criteria that do without such modeling assumptions. Portmanteau Criterion {#ss:portm} --------------------- Recall that the [*portmanteau*]{} statistic of order $p$ [@Ljun78] of a centered univariate stationary process $x$ (with $n=1$) is given by $$\por_p(x)=\frac{1}{p}\sum_{i=1}^p \left(\frac{\Expect[x_t x_{t+i}]}{\Expect[x_t^2]}\right)^2$$ where ${\Expect[x_t x_{t+i}]}/{\Expect[x_t^2]}$ is the $i$th order autocorrelation of $x_t$. The portmanteau statistic of a white noise process is by definition $0$ for any $p$. Given a multivariate $(n>1)$ process $x$ we write $$\phi_p(y)=\por_p(y^T x)=\frac{1}{p}\sum_{i=1}^p\left(\frac{y^T \Acal_i y}{y^T \Acal_0 y}\right)^2,$$ for a coefficient vector $y\in\RR^n$. By construction, $\phi_p(y)=\phi_p(ty)$ for any $t\ne 0$ and in what follows, we will impose $\|y\|_2=1$. The quantities $\phi_p(y)$ are computed using the following estimates [@Hami94 p.110]: \[eq:portm\] \_p(y)=\_[i=1]{}\^p()\^2. Crossing Statistics {#ss:cross} ------------------- @Kede94 [§4.1] define the [*zero crossing rate*]{} of a univariate $(n=1)$ process $x$ (its expected number of crosses around $0$ per unit of time) as \[eq:cross-rate\] (x)=, A result known as the cosine formula states that if $x_t$ is an autoregressive process of order one AR(1), namely if $|a|<1$, $\varepsilon_t$ is i.i.d. standard Gaussian noise and $x_t=a x_{t-1} + \varepsilon_t$, then [@Kede94 §4.2.2]: $$\gamma(x)=\frac{\arccos(a)}{\pi}.$$ Hence, for AR(1) processes, minimizing the first order autocorrelation $a$ also directly maximizes the crossing rate of the process $x$. For $n>1$, since the first order autocorrelation of $y^Tx_t$ is equal to $y^T\Acal_1y$, we propose to minimize $y^T\Acal_1y$ and ensure that all other absolute autocorrelations $\abs{y^T\Acal_ky}$, $k>1$ are small. Optimal Baskets {#s:opt} =============== Given a centered multivariate process $\bx$, we form its covariance matrix $A_0$ and its $p$ autocovariances $(A_1,\ldots,A_p)$. Because $y^TAy=y^T(A+A^T)y/2$, we symmetrize all autocovariance matrices $A_i$. We investigate in this section the problem of estimating baskets that have maximal mean reversion (as measured by the proxies proposed in Section\[s:crit\]), while being at the same time sufficiently volatile and supported by as few assets as possible. The latter will be achieved by selecting portfolios $y$ that have a small “0-norm”, namely that the number of non-zero components in $y$, $$\|y\|_0\defeq \#\{1\leq i\leq d | y_i\ne 0\},$$ is small. The former will be achieved by selecting portfolios whose aggregated value exhibits a variance over time that exceeds a given threshold $\nu>0$. Note that for the variance of $(y^Tx_t)$ to exceed a level $\nu$, the largest eigenvalue of $A_0$ must necessarily be larger than $\nu$, which we always assume in what follows. Combining these two constraints, we propose three different mathematical programs that reflect these trade-offs. Minimizing Predictability {#ss:opt-pred} ------------------------- Minimizing Box-Tiao’s predictability $\hat{\lambda}$ defined in §\[subsec:pred\] while ensuring that both the variance of the resulting process exceeds $\nu$ and that the vector of loadings is sparse with a 0-norm equal to $k$, means solving the following program: \[eq:P1\] & y\^T M y\ & y\^T A\_0y,\ & y\_2=1,\ & y\_0=k, in the variable $y\in\RR^n$ with $M\defeq A_1 A_0^{-1} A_1^T$, where $M,A_0\in\symm_n$. Without the normalization constraint $\|y\|_2=1$ and the sparsity constraint $\|y\|_0=k$, problem  is equivalent to a generalized eigenvalue problem in the pair $(M,A_0)$. That problem quickly becomes unstable when $A_0$ is ill-conditioned or $M$ is singular. Adding the normalization constraint $\|y\|_2=1$ solves these numerical problems. Minimizing the Portmanteau Statistic {#ss:opt-portm} ------------------------------------ Using a similar formulation, we can also minimize the order $p$ portmanteau statistic defined in §\[ss:portm\] while ensuring a minimal variance level $\nu$ by solving: \[eq:P2\] &\_[i=1]{}\^[p]{}(y\^T A\_i y)\^2\ & y\^T A\_0y ,\ & y\_2=1,\ & y\_0=k, in the variable $y\in\RR^n$, for some parameter $\nu>0$. Problem  has a natural interpretation: the objective function directly minimizes the portmanteau statistic, while the constraints normalize the norm of the basket weights to one, impose a variance larger than $\nu$ and impose a sparsity constraint on $y$. Minimizing the Crossing Statistic {#ss:opt-portm2} --------------------------------- Following the results in §\[ss:cross\], maximizing the crossing rate while keeping the rest of the autocorrelogram low, \[eq:P3\] & y\^TA\_1y + \_[k=2]{}\^[p]{}(y\^T A\_k y)\^2\ & y\^T A\_0y ,\ & y\_2=1,\ & y\_0=k, in the variable $y\in\RR^n$, for some parameters $\mu,\nu>0$, will produce processes that are close to being AR(1), while having a high crossing rate. Semidefinite Relaxations and Sparse Components {#s:sdp} ============================================== Problems ,  and   are not convex, and can be in practice extremely difficult to solve, since they involve a sparse selection of variables. We detail in this section convex relaxations to these problems which can be used to derive relevant sub-optimal solutions. A Semidefinite Programming Approach to Basket Estimation {#subsec:asemidefinite} -------------------------------------------------------- We propose to relax problems ,  and   into Semidefinite Programs (SDP) [@vandenberghe1996semidefinite]. We show that these semidefinite programs can handle naturally sparsity and volatility constraints while still aiming at mean-reversion. In some restricted cases, one can show that these relaxations are tight, in the sense that they solve exactly the programs described above. In such cases, the true solution $y^\star$ of some of the programs above can be recovered using their corresponding SDP solution $Y^\star$. However, in most of the cases we will be interested in, such a correspondence is not guaranteed and these SDP relaxations can only serve as a guide to propose solutions to these hard non-convex problems when considered with respect to vector $y$. To do so, the optimal solution $Y^\star$ needs to be *deflated* from a large rank $d\times d$ matrix to a rank one matrix $yy^T$, where $y$ can be considered a good candidate for basket weights. A typical approach to deflate a positive definite matrix into a vector is to consider its eigenvector with the leading eigenvalue. Having sparsity constraints in mind, we propose to apply a heuristic grounded on sparse-PCA [@zou2006sparse; @d2007direct]. Instead of considering the lead eigenvector, we recover the leading *sparse* eigenvector of $Y^\star$ (with a $0$-norm constrained to be equal to $k$). Several efficient algorithmic approaches have been proposed to solve approximately that problem; we use the SPASM toolbox [@sjostrand2012spasm] in our experiments. Predictability {#predictability} -------------- We can form a convex relaxation of the predictability optimization problem  over the variable $y\in\RR^n$, $$\BA{ll} \mbox{minimize} & y^T M y\\ \mbox{subject to} & y^T A_0y\geq \nu\\ & \|y\|_2=1,\\ & \|y\|_0=k, \EA$$ by using the lifting argument of @Lova91, writing $Y=yy^T$, to solve now the problem using a semidefinite variable $Y$, and by introducing a sparsity-inducing regularizer on $Y$ which considers the $L_1$ norm of $Y$, $$\norm{Y}_1\defeq \sum_{ij}\abs{Y_{ij}},$$ so that Problem  becomes (here $\rho>0$), $$\BA{ll} \mbox{minimize} & \Tr(MY) + \rho \norm{Y}_1\\ \mbox{subject to} & \Tr(A_0Y)\geq\nu\\ & \Tr(Y)=1,~\Rank(Y)=1,~Y\succeq 0. \EA$$ We relax this last problem further by dropping the rank constraint, to get \[eq:SDP1\] & (MY) + \_1\ & (A\_0Y)\ & (Y)=1, Y0 which is a convex semidefinite program in $Y\in\symm_n$. Portmanteau ----------- Using the same lifting argument and writing $Y=yy^T$, we can relax problem  by solving \[eq:SDP2\] & \_[i=1]{}\^p (A\_iY)\^2 + \_1\ & (BY)\ & (Y)=1, Y0, a semidefinite program in $Y\in\symm_n$. Crossing Stats -------------- As above, we can write a semidefinite relaxation for problem : \[eq:SDP3\] & (A\_1Y)+ \_[i=2]{}\^p (A\_iY)\^2 + \_1\ & (BY)\ & (Y)=1, Y0 ### Tightness of the SDP Relaxation in the Absence of Sparsity Constraints Note that for the crossing stats criterion (with $p=1$ and no quadratic term in $Y$) criteria, the original problem \[eq:P3\] and its relaxation \[eq:SDP3\] are equivalent, taken for granted that no sparsity constraint is considered in the original problems and $\mu$ set to $0$ in the relaxations. This relaxations boil down to an SDP’s that only has a linear objective, a linear constraint and a constraint on the trace of $Y$. In that case, @Bric61 showed that the range of two quadratic forms over the unit sphere is a convex set when the ambient dimension $n\geq 3$, which means in particular that for any two square matrices $A,B$ of dimension $n$ &{(y\^TAy,y\^TBy): y\^n, y\_2=1}=&\ &{((AY),(BY)): Y\_n, Y=1, Y0}& We refer the reader to [@Barv02 §II.13] for a more complete discussion of this result. As remarked in [@cuturi2013mean], the same equivalence holds for \[eq:P1\] and \[eq:SDP1\]. This means that, in the case where $\rho,\mu=0$ and the 0-norm of $y$ is *not* constrained, for any solution $Y^\star$ of the relaxation  there exists a vector $y^\star$ which satisfies $\norm{y}_2^2=\Tr(Y^\star)=1$, $y^{\star T} A_0 y^\star=\Tr(BY^\star)$ and $y^{\star T}My^\star=\Tr(MY^\star)$ which means that $y^\star$ is an optimal solution of the original problem . @Boyd:1072 [App.B] show how to explicitly extract such a solution $y^\star$ from a matrix $Y^\star$ solving . This result is however mostly anecdotical in the context of this paper, in which we look for sparse and volatile baskets: using these two regularizers breaks the tightness result between the original problems in $\RR^d$ and their SDP counterparts. Numerical Experiments {#s:numres} ===================== ![**Option implied volatility** for Apple between January 4 2004 and December 30 2010.[]{data-label="fig:vol"}](aapl.pdf){width=".7\textwidth"} In this section, we evaluate the ability of our techniques to extract mean-reverting baskets with sufficient variance and small 0-norm from a universe of tradable assets. We measure performance by applying to these baskets a trading strategy designed specifically for mean-reverting processes. We show that, under realistic trading costs assumptions, selecting sparse and volatile mean-reverting baskets translates into lower incurred costs and thus improves the performance of trading strategies. Historical Data --------------- We consider daily time series of option implied volatilities for 210 stocks from January 4 2004 to December 30 2010. A key advantage of using option implied volatility data is that these numbers vary in a somewhat limited range. Volatility also tends to exhibit regime switching, hence can be considered piecewise stationary, which helps in extracting structural relationships. We plot a sample time series from this dataset in Figure \[fig:vol\] that corresponds to the implicit volatility of Apple’s stock. In what follows, we mean by asset the implied volatility of any of these stocks, whose value can be efficiently replicated using option portfolios. Mean-reverting Basket Estimators -------------------------------- We compare the three basket selection techniques detailed here, **predictability**, **portmanteau** and **crossing statistic**, implemented with varying targets for both sparsity and volatility, with two cointegration estimators that build upon principal component analysis [@maddala1998urc §5.5.4]. By the label ‘PCA’ we mean in what follows the eigenvector with smallest eigenvalue of the covariance matrix $A_0$ of the process [@stock1988tct]. By ‘sPCA’ we mean the sparse eigenvector of $A_0$ with 0-norm $k$ that has the smallest eigenvalue, which can be simply estimated by computing the leading sparse eigenvector of $\lambda I-A_0$ where $\lambda$ is bigger than the leading eigenvalue of $A_0$. This sparse principal component of the covariance matrix $A_0$ should not be confused with our utilization of sparse PCA in Section \[subsec:asemidefinite\] as a way to recover a vector solution from the solution of a positive semidefinite problem. Note also that techniques based on principal components do not take explicitly variance levels into account when estimating the weights of a co-integrated relationship. @jurek Trading Strategy ----------------------- While option implied volatility is not directly tradable, it can be synthesized using baskets of call options, and we assimilate it to a tradable asset with (significant) transaction costs in what follows. For baskets of volatilities isolated by the techniques listed above, we apply the [@jurek] strategy for log utilities to the basket process recording out of sample performance. @jurek propose to trade a stationary autoregressive process $(x_t)_{t}$ of order $1$ and mean $\mu$ governed by the equation $x_{t+1} = \rho x_t +\sigma \varepsilon_t$, where $\abs{\rho}<1$, by taking a position $N_t$ in the asset $x_t$ which is proportional to $$\label{eq:jurek} N_t = \frac{\rho (\mu-x_t)}{\sigma^2}W_t$$ In effect, the strategy advocates taking a long (resp. short) position in the asset whenever it is below (resp. above) its long-term mean, and adjust the position size to account for the volatility of $x_t$ and its mean reversion speed $\rho$. Given basket weights $y$, we apply standard AR estimation procedures on the in-sample portion of $y^T\bx$ to recover estimates for $\hat{\rho}$ and $\hat{\sigma}$ and plug them directly in Equation . This approach is illustrated for two baskets in Figure \[fig:syn\]. -2.5cm![**Three sample trading experiments, using the PCA, sparse PCA and the Crossing Statistics estimators**. \[a\] Pool of 9 volatility time-series selected using our fast PCA selection procedure. \[b\] Basket weights estimated with in-sample data using either the eigenvector of the covariance matrix with smallest eigenvalue, the smallest eigenvector with a sparsity constraint of $k=\lfloor 0.5 \times 9\rfloor=4$ and the Crossing Statistics estimator with a volatility threshold of $\nu=0.2$, a constraint on the basket’s variance to be larger than $0.2 \times$ the median variance of all $8$ assets. \[c\] Using these 3 procedures, the time series of the resulting basket price in the in-sample part \[c\] and out-sample parts \[d\] are displayed. \[e\] Using the [@jurek] trading strategy results in varying positions (expressed as units of baskets) during the out-sample testing phase. \[f\] Transaction costs that result from trading the assets to achieve such positions accumulate over time. \[g\] Taking both trading gains and transaction costs into account, the net wealth of the investor for each strategy can be computed (the Sharpe over the test period is displayed in the legend). Note how both sparsity and volatility constraints translate into portfolios composed of less assets, but with a higher variance.[]{data-label="fig:syn"}](example_3basks2.pdf "fig:"){width="140.00000%"} Transaction Costs ----------------- We assume that fixed transaction costs are negligible, but that transaction costs per contract unit are incurred at each trading date. We vary the size of these costs across experiments to show the robustness of the approaches tested here to trading costs fluctuations. We let the transaction cost per contract unit vary between 0.03 and 0.17 cents by increments of 0.02 cents. Since the average value of a contract over our dataset is about 40 cents, this is akin to considering trading costs ranging from about 7 to about 40 Base Points (BP), that is 0.07 to 0.4%. Experimental Setup ------------------ We consider 20 sliding windows of one year (255 trading days) taken in the history, and consider each of these windows independently. Each window is split between 85% of days to estimate and 15% of days to test-trade our models, resulting in 38 test-trading days. We do not recompute the weights of the baskets during the test phase. The 210 stock volatilities (assets) we consider are grouped into 13 subgroups, depending on the economic sector of their stock. This results in 13 sector pools whose size varies between 3 assets and 43 assets. We look for mean-reverting baskets in each of these 13 sector pools. Because all combinations of stocks in each of the 13 sector pools may not necessarily mean-reverting, we select smaller candidate pools of $n$ assets through a greedy backward-forward minimization scheme, where $8\leq n\leq 12$. To do so, we start with an exhaustive search of all pools of size 3 within the sector pool, and proceed by adding or removing an asset using the PCA estimator (the smallest eigenvalue of the covariance matrix of a set of assets). We use the PCA estimator in that backward-forward search because it is the fastest to compute. We score each pool using that PCA statistic, the smaller meaning the better. We generate up to 200 candidate pools per each of the 13 sector pools. Out of all these candidate pools, we keep the best 50 in each window, and use then our cointegration estimation approaches separately on these candidates. One such pool was, for instance, composed of the stocks `{BBY,COST,DIS,GCI,MCD,VOD,VZ,WAG,T}` observed during the year 2006. Figure \[fig:syn\] provides a closeup on that universe of stocks, and shows the results of three trading experiments using either PCA, sparse PCA or the Crossing Stats estimator to build trading strategies. Results ------- ### Robustness of Sharpe Ratios to Costs In Figure \[fig:sharpe\], we plot the average of the Sharpe ratio over the $922$ baskets estimated in our experimental set versus transaction costs. We consider different PCA settings as well as our three estimators using, in all 3 cases, the variance bound $\nu$ to be $0.3$ times the median of all variances of assets available in a given asset pool, and the 0-norm to be equal to 0.3 times the size of the universe (itself between 8 and 12). We observe that Sharpe ratios decrease the fastest for the naive PCA based method, this decrease being somewhat mitigated when adding a constraint on the 0-norm of the basket weights obtained with sparse PCA. Our methods require, in addition to sparsity, enough volatily to secure sufficient gains. These empirical observations agree with the intuition of this paper: simple cointegration techniques can produce synthetic baskets with high mean-reversion, large support, low variance. Trading a portfolio with low variance which is supported by multiple assets translates in practice into high trading costs which can damage the overall performance of the strategy. Both sparse PCA and our techniques manage instead to achieve a trade-off between desirable mean-reversion properties and, at the same time, control for sufficient variance and small basket size to allow for lower overall transaction costs. ### Tradeoffs Between Mean Reversion, Sparsity, and Volatility In the plots of Figure \[fig:sharpeCrossing\] and \[fig:sharpeCrossing2\], this analysis is further detailed by considering various settings for $\nu$ (volatility threshold) and $k$. To improve the legibility of these results we summarize, following the observation in Figure \[fig:sharpe\] that the relationship between Sharpes and transactions costs seems almost linear, each of these curves by two numbers: an intercept level (Sharpe ratio when costs are low) and a slope (degradtion of Sharpe as costs increase). Using these two numbers, we locate all considered strategies in the intercept/slope plane. We first show the spectral techniques, PCA and sPCA with different levels of sparsity, meaning that $k$ is set to $\lfloor u \times d\rfloor$ where $u\in\{0.3,0.5,0.7\}$ and $d$ is the size of the original basket. Each of the three estimators we propose is studied in a separate plot. For each we present various results characterized by two numbers: a volatility threshold $\nu\in\{0,0.1,0.2,0.3,0.4,0.5\}$ and a sparsity level $u\in\{0.3,0.5,0.7\}$. To avoid cumbersome labels, we attach an arrow to each point: the arrow’s length in the vertical direction is equal to $u$ and characterizes the size of the basket, the horizontal length is equal to $\nu$ and characterizes the volatility level. As can be seen in these 3 plots, an interesting interplay between these two factors allows for a continuum of strategies that trade mean-reversion (and thus Sharpe levels) for robustness to cost level. ![Average Sharpe ratio for the @jurek trading strategy captured over about 922 trading episodes, using different basket estimation approaches. These 922 trading episodes were obtained by considering 7 disjoint time-windows in our market sample, each of a length of about one year. Each time-window was divided into 85% in-sample data to estimate baskets, and 15% outsample to test strategies. On each time-window , the set of 210 tradable assets during that period was clustered using sectorial information, and each cluster screened (in the in-sample part of the time-window) to look for the most promising baskets of size between 8 and 12 in terms of mean reversion, by choosing greedily subsets of stocks that exhibited the smallest minimal eigenvalues in their covariance matrices. For each trading episode, the same universe of stocks was fed to different mean-reversion algorithms. Because volatility time-series are bounded and quite stationary, we consider the PCA approach, which uses the eigenvector with the smallest eigenvalue of the covariance matrix of the time-series to define a cointegrated relationship. Besides standard PCA, we have also consider sparse PCA eigenvectors with minimal eigenvalue, with the size $k$ of the support of the eigenvector (the size of the resulting basket) constrained to be 30%, 50% or 70% of the total number of considered assets. We consider also the portmanteau, predictability and crossing stats estimation techniques with variance thresholds of $\nu=0.2$ and a support whose size $k$ (the number of assets effectively traded) is targeted to be about $30\%$ of the size of the considered universe (itself between 8 and 12). As can be seen in the figure, the sharpe ratios of all trading approaches decrease with an increase in transaction costs. One expects sparse baskets to perform better under the assumption that costs are high, and this is indeed observed here. Because the relationship between sharpe ratios and transaction costs can be efficiently summarized as being a linear one, we propose in the plots displayed in Figures \[fig:sharpeCrossing\] and  \[fig:sharpeCrossing2\] a way to summarize the lines above with two numbers each: their intercept (Sharpe level in the quasi-absence of costs) and slope (degradation of Sharpe as costs increase). This visualization is useful to observe how sparsity (basket size) and volatility thresholds influence the robustness to costs of the strategies we propose. This visualization allows us to observe how performance is influenced by these parameter settings.\[fig:sharpe\]](ex2-eps-converted-to.pdf){width=".8\textwidth"} ![Relationships between Sharpe in a low cost setting (intercept) in the $x$-axis and robustness of Sharpe to costs (slope of Sharpe/costs curve) of a different estimators implemented with varying volatility levels $\nu$ and sparsity levels $k$ parameterized as a multiple of the universe size. Each colored square in the figures above corresponds to the performance of a given estimator (Portmanteau in subfigure $(a)$, Predictability in subfigure $(b)$) using different parameters for $\nu\in\{0,0.1,0.2,0.3,0.4,0.5\}$ and $u\in\{0.3,0.5,0.7\}$. The parameters used for each experiment are displayed using an arrow whose vertical length is proportional to $\nu$ and horizontal length is proportional to $u$.\[fig:sharpeCrossing\]](Portmanteau___-eps-converted-to.pdf "fig:"){width="\textwidth"}\ (a) ![Relationships between Sharpe in a low cost setting (intercept) in the $x$-axis and robustness of Sharpe to costs (slope of Sharpe/costs curve) of a different estimators implemented with varying volatility levels $\nu$ and sparsity levels $k$ parameterized as a multiple of the universe size. Each colored square in the figures above corresponds to the performance of a given estimator (Portmanteau in subfigure $(a)$, Predictability in subfigure $(b)$) using different parameters for $\nu\in\{0,0.1,0.2,0.3,0.4,0.5\}$ and $u\in\{0.3,0.5,0.7\}$. The parameters used for each experiment are displayed using an arrow whose vertical length is proportional to $\nu$ and horizontal length is proportional to $u$.\[fig:sharpeCrossing\]](Predictability___-eps-converted-to.pdf "fig:"){width="\textwidth"}\ (b) ![Same setting as Figure \[fig:sharpeCrossing\], using the crossing statistics (c).\[fig:sharpeCrossing2\]](Crossing___-eps-converted-to.pdf "fig:"){width="\textwidth"}\ (c) Conclusion ========== We have described three different criteria to quantify the amount of mean reversion in a time series. For each of these criteria, we have detailed a tractable algorithm to isolate a vector of weights that has optimal mean reversion, while constraining both the variance (or signal strength) of the resulting univariate series to be above a certain level and its 0-norm to be at a certain level. We show that these bounds on variance and support size, together with our new criteria for mean reversion, can significantly improve the performance of mean reversion statistical arbitrage strategies and provide useful controls to adjust mean-reverting strategies to varying trading conditions, notably liquidity risk and cost environment.

--- abstract: 'A measurement of the top quark pair production cross section in proton anti-proton collisions at an interaction energy of $\sqrt{s}=1.96~{\rm TeV}$ is presented. This analysis uses 405 pb$^{-1}$ of data collected with the DØ detector at the Fermilab Tevatron Collider. Fully hadronic $t\bar{t}$ decays with final states of six or more jets are separated from the multijet background using secondary vertex tagging and a neural network. The $t\bar{t}$ cross section is measured as $\sigma_{t\bar{t}}=4.5_{-1.9}^{+2.0}({\rm stat}) _{-1.1}^{+1.4}({\rm syst}) \pm 0.3 ({\rm lumi})~{\rm pb }$ for a top quark mass of $m_{t} = 175~{\rm GeV/c^2}$.' date: 'December 13, 2006' title: 'Measurement of the $p\bar{p} \to t\bar{t}$ production cross section at $\sqrt{s}=1.96$ TeV in the fully hadronic decay channel ' --- list\_of\_authors\_r2.tex The standard model (SM) predicts that the top quark decays primarily into a $W$ boson and a $b$ quark. The measurement presented here tests the prediction of the SM in the dominant decay mode of the $t\bar{t}$ system: when both $W$ bosons decay to quarks, the so-called fully hadronic decay channel. This topology occurs in 46% of $t\bar{t}$ events. The theoretical signature for fully hadronic $t\bar{t}$ events is six or more jets originating from the hadronization of the six quarks. Of the six jets, two originate from $b$ quark decays. Fully hadronic $t\bar{t}$ events are difficult to identify at hadron colliders because the background rate is many orders of magnitude larger than that of the $t\bar{t}$ signal. We report a measurement of the production cross-section of top quark pairs, $\sigma_{t\bar{t}}$, using data collected with DØ in the fully hadronic channel, that exploits the long lifetime of the $b$ hadrons in identifying $b$ jets. To increase the sensitivity for $t\bar{t}$ events, we used a neural network to distinguish signal from the overwhelming background of multijet production through Quantum Chromodynamic processes (QCD). The DØ detector [@d0det] has a central tracking system consisting of a silicon micro strip tracker (SMT) and a central fiber tracker (CFT), both located within a 2 T superconducting solenoidal magnet, with designs optimized for tracking and vertexing at pseudorapidities $|\eta|<3$ and $|\eta|<2.5$, respectively. Rapidity $y$ and pseudorapidity $\eta$ are defined as functions of the polar angle $\theta$ and parameter $\beta$ as $y(\theta,\beta)= \frac{1}{2} \ln [ (1+\beta \cos \theta)/(1-\beta \cos \theta )]$ and $\eta(\theta)=y(\theta,1)$, where $\beta$ is the ratio of the particle’s momentum to its energy. The liquid-argon and uranium calorimeter has a central section (CC) covering pseudorapidities $|\eta|$ up to $\approx 1.1$ and two end calorimeters (EC) that extend coverage to $|\eta| \approx 4.2$, with all three housed in separate cryostats. Each calorimeter cryostat contains a multilayer electromagnetic calorimeter, a finely segmented hadronic calorimeter and a third hadronic calorimeter that is more coarsely segmented, providing both segmentation in depth and in projective towers of size $0.1 \times 0.1$ in $\eta$-$\phi$ space, where $\phi$ is the azimuthal angle in radians. An outer muon system, covering $|\eta|<2$, consists of a layer of tracking detectors and scintillation trigger counters in front of 1.8 T iron toroids, followed by two similar layers after the toroids. The luminosity is measured using plastic scintillator arrays placed in front of the EC cryostats. The data set was collected between 2002 and 2004, and corresponds to an integrated luminosity $\mathcal{L}=405~\pm~25~{\rm pb}^{-1}$ [@newlumi]. To isolate events with six jets, we used a dedicated multijet trigger. The requirements on the trigger, particularly on jet and trigger tower energy thresholds, were tightened during the collection of the data set to manage the increasing instantaneous luminosities delivered by the Fermilab Tevatron Collider. The change in trigger requirements had little effect on the efficiency for signal, while removing an increasing number of background events [@footnote1]. The trigger was tuned for the fully hadronic $t\bar{t}$ channel and was optimized to remain as efficient possible while using limited bandwidth. The collection rate after all trigger levels was fixed to a few Hz, which was completely dominated by QCD multijet events as the hadronic $t\bar{t}$ event production rate is expected to be a few events per day. We required three or four trigger towers above an energy threshold of 5 GeV at the first trigger level, three reconstructed jets with transverse energies ($E_T$) above 8 GeV at the second trigger level, combined with a requirement on the sum of the transverse momenta ($p_{T}$) of the jets, and four or five reconstructed jets at transverse energy thresholds between 10 and 30 GeV at the highest trigger level [@d0det]. We simulated $t\bar{t}$ production using [alpgen 1.3]{} to generate the parton-level processes, and [pythia 6.2]{} to model hadronization [@alpgen; @pythia]. We used a top quark invariant mass of $m_{t}=175~{\rm GeV/c^2}$. The decay of hadrons carrying bottom quarks was modeled using [evtgen]{} [@evtgen]. The simulated $t\bar{t}$ events were processed with the full [geant]{}-based DØ detector simulation, after which the Monte Carlo (MC) events were passed through the same reconstruction program as was used for data. The small differences between the MC model and the data were corrected by matching the properties of the reconstructed objects. The residual differences were very small and were corrected using factors derived from detailed comparisons between the MC model and the data for well understood SM processes such as the jets in $Z$ boson and QCD dijet production. In the offline analysis, jets were defined with an iterative cone algorithm [@jetsdef]. Before the jet algorithm was applied, calorimeter noise was suppressed by removing isolated cells whose measured energy was lower than four standard deviations above cell pedestal. In the case that a cell above this threshold was found to be adjacent to one with an energy less than four standard deviations above pedestal, the latter was retained if its signal exceeded 2.5 standard deviations above pedestal. Cells that were reconstructed with negative energies were always removed. The elements for cone jet reconstruction consisted of projective towers of calorimeter cells. First, seeds were defined using a preclustering algorithm, using calorimeter towers above an energy threshold of 0.5 GeV. The cone jet reconstruction, an iterative clustering process where the jet axis was required to match the axis of a projective cone, was then run using all preclusters above 1.0 GeV as seeds. As jets from $t\bar{t}$ production are relatively narrow due to relatively high jet $p_{T}$, the jets were defined using a cone with radius $R_{{\rm cone}}=0.5$, where $\Delta R = \sqrt{(\Delta y)^2+(\Delta \phi)^2}$ . The resulting jets (proto-jets) took into account all energy deposits contained in the jet cone. If two proto-jets were within $1<\Delta R / R_{{\rm cone}} <2$, an additional midpoint clustering was applied, where the combination of the two proto-jets was used as a seed for a possible additional proto-jet. At this stage, the proto-jets that share transverse momentum were examined with a splitting and merging algorithm, after which each calorimeter tower was assigned to one proto-jet at most. The proto-jets were merged if the shared $p_T$ exceeded 50% of the $p_T$ of the proto-jet with the lowest transverse momentum and the towers were added to the most energetic proto-jet while the other candidate was rejected. If the proto-jets shared less than half of their $p_{T}$, the shared towers were assigned to the proto-jet which was closest in $\Delta R$ space. The collection of stable proto-jets remaining was then referred to as the [*reconstructed*]{} jets in the event. The minimal $p_{T}$ of a reconstructed jet was required to be 8 GeV/$c$ before any energy corrections were applied. We removed jets caused by electromagnetic particles and jets resulting from noise in hadronic sections of the calorimeter by requiring that the fraction of the jet energy deposited in the calorimeter ($EMF$) was $0.05 < EMF < 0.95$ and the fraction of energy in the coarse hadronic calorimeter was less than 0.4. Jets formed from clusters of calorimeter cells known to be affected by noise were also rejected. The remaining noise contribution was removed by requiring that the jet also fired the first level trigger. To correct the calorimeter jet energies back to the level of particle jets, a jet energy scale (JES) correction $C^{JES}$ was applied. The same procedure has to be applied to Monte Carlo jets to ensure an identical calorimeter response in data and simulation. The particle level or true jet energy $E^{true}$ was obtained from the measured jet energy $E^m$ and the detector pseudorapidity, measured from the center of the detector ($\eta_{det}$), using the relation $$E^{true}= \frac{E^m - E_0 ( \eta_{det}, \mathcal{L})}{\mathcal{R}(\eta_{det}, E^m) S(\eta_{det},E^m)} = C^{JES} (E^m,\eta_{det},\mathcal{L}) \cdot E^m .$$ In data and MC the total correction was applied to the measured energy $E^m$ as a multiplicative factor $C^{JES}$. $E_{0}(\eta_{det},\mathcal{L})$ was the offset energy created by electronics noise and noise signal caused by the uranium in the calorimeter, pile-up energy from previous collisions and the additional energy from the underlying physics event. The dependence on the luminosity $\mathcal{L}$ was caused by the fact that the number of additional interactions was dependent on the instantaneous luminosity, while the dependence on $y$ was caused by variations in the calorimeter occupancy as a function of the jet rapidity. $\mathcal{R}(\eta_{det},E^m)$ parameterized the energy response of the calorimeter, while $S(\eta_{det},E^m)$ represents the fraction of the true partonic jet energy that was deposited inside the jet cone. This out-of-cone showering correction depended on the energy of the jet and its location in the calorimeter. The JES was measured directly using $p_T$ conservation in photon + jet events. The method was identical for data and simulation and used transverse momentum balancing between the jet and the photon. As the energy scale of the photon was directly and precisely measured (the electromagnetic calorimeter response was derived from measurements of resonances in the $e^+ e^-$ spectrum like the $Z$ boson), the true jet energy could be derived from the difference between the photon and jet energy. $E_0$, $\mathcal{R}$ and $S$ were fit as a function of jet rapidity and measured energy, which lead to uncertainties coming from the fit (statistical) and the method (systematic). The total correction $C^{JES}$ was approximately 1.4 for data jets in the energy range expected for jets associated with top quark events. The uncertainties on $C^{JES}$, which were dominated by the systematic uncertainty of the out-of-cone showering correction $S(\eta_{det},E^m)$, were a few percent and were dependent on the jet energy and rapidity. The jet energy resolution was measured in photon + jet data for low jet energies and dijet data for higher jet energy values. Fits to the transverse energy asymmetry $[p_T(1) - p_T(2)]/[p_T(1)+p_T(2)]$ between the transverse momenta of the back-to-back jets and/or photon ($p_T(1)$ and $p_T(2)$) were then used to obtain the jet energy resolution as a function of jet rapidity and transverse energy. The uncertainties on the jet energy resolution were dominated by limited statistics in the samples used. In this analysis, we considered a data set consisting of events with four or more reconstructed jets, in which the scalar sum of the uncorrected transverse momenta $H_T^{uncorr}$ of all the jets in the event was greater than 90 GeV/$c$. The final analysis sample was a subset of this sample, where at least six jets with corrected transverse momentum greater than 15 GeV/$c$ and $|y|<2.5$ were required. Events with isolated high transverse momentum electron or muon candidates were vetoed to ensure that the all-hadronic and leptonic $t\bar{t}$ samples were disjoint [@ttbarlepjetsrun2; @ttbarlepjetsvtxtag]. In addition, we rejected events where two distinct $p\bar{p}$ interactions with separate primary vertices were observed and the jets in the event were not assigned to only one of the two primary vertices. The primary vertex requirement did not affect minimum bias interactions or $t\bar{t}$ events. Table \[effmostcuts\] lists the efficiencies after the first set of selection cuts, commonly referred to as preselection, which includes the requirements on the primary vertex, the number of reconstructed jets and the presence of isolated leptons, and the efficiency after preselection and after preselection and the trigger. Besides selecting all hadronic $t\bar{t}$ events, the analysis was also expected to accept a small contribution from the semi-leptonic (lepton+jets) $t\bar{t}$ decay channel. The combined efficiency included the fully hadronic and semi-leptonic $W$-boson branching fractions of $0.4619\pm0.0048$ and $0.4349\pm0.0027$ respectively [@pdg]. We used a secondary vertex tagging algorithm (SVT) to identify $b$-quark jets. The algorithm was the same as used in previously published DØ $t\bar{t}$ production cross section measurements [@ttbarlepjetsrun2; @ttbarlepjetsvtxtag]. Secondary vertex candidates were reconstructed from two or more tracks in the jet, removing vertices consistent with originating from long-lived light hadrons as for example $K_S^0$ and $\Lambda$. Two configurations of the secondary vertex algorithm were used; these were labeled “loose” and “tight” respectively. If a reconstructed secondary vertex in the jet had a transverse decay length $L_{xy}$ significance ($L_{xy}/\sigma_{L_{xy}}$) $>$ 5 (7), the jet was tagged as a loose (tight) $b$-quark jet. The loose SVT was chosen to efficiently identify $b$-quark jets, while the tight SVT was configured to accept only very few light quark jets while sacrificing a small reduction in the efficiency for $b$-quark jets. Events with two or more loosely tagged jets were called double-tag events. The sample that did not contain two loosely tagged jets was inspected for events with one tight tag. Events thus isolated were labeled single-tag events. The fully exclusive samples of single-tag and double-tag events were treated separately because of their different signal-to-background ratios. The use of the tight SVT selection for single tagged events optimized the rejection of mistags, the main background in the single-tag analysis. When two tags were required, the background sample started to be dominated by direct $b\bar{b}$ production. The choice to use the loose SVT optimized the double-tag analysis for signal efficiency instead of background rejection. cut $t\bar{t} \to {\rm hadrons}$ $t\bar{t} \to \ell + {\rm jets}$ any $t\bar{t}$ ------------------ ------------------------------ ---------------------------------- ------------------------- [preselection]{} [$0.2706 \pm 0.0016$]{} [$0.0311\pm0.0008$]{} [$0.1385\pm 0.0011$]{} [trigger]{} [$0.2527\pm 0.0015$]{} [$0.0268 \pm 0.0007$]{} [$0.1284 \pm 0.0010$]{} : \[effmostcuts\] Efficiency for selection criteria applied before $b$-jet identification. Efficiencies listed include the efficiency for all previous selection criteria. The trigger efficiency is quoted for events that have passed the preselection. The uncertainties are due to Monte Carlo statistics. Listed are the selection efficiencies as determined for $t\bar{t}$ in the hadronic decay channel, the lepton+jets decay channel and the efficiency for all different decay channels corrected for $W$ boson branching fractions. Compared to light-quark QCD multijet events, $t\bar{t}$ events on average have more jets of higher energy and with less boost in the beam direction, resulting in events with many central jets that all have similar and relatively high energies. Moreover, the fully hadronic decay makes it possible to reconstruct the $W$ boson and $t$ quark four-momenta. To distinguish between signal and background, we used the following event characteristics [@run1alljetsxsec1]: ![\[fig1\] The $H_T$ distribution for single-tag events (a) and double-tag events (b). Shown are the data (points), the background (solid line) and the expected $t\bar{t}$ distribution (filled histogram) multiplied by 140 (60) for the single (double)-tag analysis.](./fig1.eps){width="\linewidth"} \(1) $H_T$: The scalar sum of the corrected transverse momenta of the jets (Fig. \[fig1\]). ![\[fig2\] The $E_{T}^{56}$ distribution for single-tag events (a) and double-tag events (b). Shown are the data (points), the background (solid line) and the expected $t\bar{t}$ distribution (filled histogram) multiplied by 140 (60) for the single (double)-tag analysis.](./fig2.eps){width="\linewidth"} \(2) $E_{T}^{56}$: The square root of the product of the transverse momenta of the fifth and sixth leading jet (Fig. \[fig2\]). ![\[fig3\] The $\mathcal{A}$ distribution for single-tag events (a) and double-tag events (b). Shown are the data (points), the background (solid line) and the expected $t\bar{t}$ distribution (filled histogram) multiplied by 140 (60) for the single (double)-tag analysis.](./fig3.eps){width="\linewidth"} \(3) $\mathcal{A}$: The aplanarity as calculated from the normalized momentum tensor (Fig. \[fig3\]) [@ttbarlepjetsrun2; @ttbarlepjetsvtxtag; @run1alljetsxsec1]. ![\[fig4\] The $\langle \eta^2 \rangle$ distribution for single-tag events (a) and double-tag events (b). Shown are the data (points), the background (solid line) and the expected $t\bar{t}$ distribution (filled histogram) multiplied by 140 (60) for the single (double)-tag analysis.](./fig4.eps){width="\linewidth"} \(4) $\langle \eta^2 \rangle$: The $p_T$-weighted mean square of the $y$ of the jets in an event (Fig. \[fig4\]), see also Ref.  [@run1alljetsxsec1]. ![\[fig5\] The $\mathcal{M}$ distribution for single-tag events (a) and double-tag events (b). Shown are the data (points), the background (solid line) and the expected $t\bar{t}$ distribution (filled histogram) multiplied by 140 (60) for the single (double)-tag analysis.](./fig5.eps){width="\linewidth"} \(5) $\mathcal{M}$: The mass-$\chi^2$ variable, which was defined as $\mathcal{M} = (M_{W_1}- M_W)^2/\sigma_{M_W}^2 +(M_{W_2}- M_W)^2/\sigma_{M_W}^2 +(m_{t_1}-m_{t_2})^2/\sigma_{m_t}^2$, where the parameters $M_W$, $\sigma_{M_W}$ and $\sigma_{m_t}$ were the invariant mass and mass resolution from the jet four-momenta calculated as observed in all-hadronic $t\bar{t}$ MC, respectively 79, 11 and 21 GeV/$c^2$ after all corrections and resolutions were included [@footnote2]. $M_{W_i}$ and $m_{t_i}$ were calculated for every possible permutation of the jets in the event. We did not distinguish between tagged and untagged jets. The combination of jets that yielded the lowest value of $\mathcal{M}$ is used (Fig. \[fig5\]). ![\[fig6\] The $M_{min}^{34}$ distribution for single-tag events (a) and double-tag events (b). Shown are the data (points), the background (solid line) and the expected $t\bar{t}$ distribution (filled histogram) multiplied by 140 (60) for the single (double)-tag analysis.](./fig6.eps){width="\linewidth"} \(6) $M_{min}^{34}$: The second-smallest dijet mass in the event. First, all possible dijet masses were considered and the jets that yield the smallest mass were rejected. $M_{min}^{34}$ was the smallest dijet mass as found from the remaining jets (Fig. \[fig6\]). ![\[fig7\] The output discriminant of an artificial neural network ($NN$) with six input nodes. All distributions are normalized to area. $NN$ is optimized to distinguish between fully hadronic $t\bar{t}$ Monte Carlo events (signal) and the background from multijet production (background) as predicted by the tag rate functions. ](./fig7.eps){width="\linewidth"} The top quark production cross section was calculated from the output of [*NN*]{}, an artificial neural network trained to force its output near 1 for $t\bar{t}$ events and near $-1$ for QCD multijet events, using the multilayer perceptron in the [root]{} analysis program [@root]. The six parameters illustrated in Figs. \[fig1\]-\[fig6\] were used as input for the neural net. The very large background-to-signal ratio in the untagged data allowed us to use untagged data as background input for the training of [*NN*]{}, while $t\bar{t}$ MC was used for the signal. Fig. \[fig7\] shows the [*NN*]{} discriminant for $t\bar{t}$ signal and multijet background. Although the distributions for single- and double-tag events were different due to increased heavy flavor content in the double-tag sample, both samples showed a clear discrimination between signal and background. The overwhelming background also made it possible to use the entire (tagged and untagged) sample to estimate the background. For the loose and tight SVT, we derived a tag rate function ([trf]{} — the probability for any individual jet to have a secondary vertex tag ) from the data with $N_{tags} \leq 1$. The [trf]{} was parameterized in terms of the $p_T$, $\phi$ and $y$ of the jet and the coordinate along the beam axis ($z$) of the primary vertex of the event, $z_{PV}$, in four different $H_T$ bins. To predict the number of tagged jets in the event, it was necessary to correct for a possible correlation between tagged jets. In the single-tag analysis the correlation factor was negligible, unlike in the double-tag analysis, where the presence of $b\bar{b}$+jets events in the sample enhanced the correlation correction. We corrected for correlations caused by $b\bar{b}$ background by applying a correlation factor $C_{ij}$, that was parameterized as a function of the cone distance between the tagged jets, $\Delta R$. Figure \[fig8\] shows the number of double-tagged events versus $\Delta R$ as observed in data, and the distribution as modeled by the [trf]{} with and without including $C_{ij}$. We considered significantly different functional forms for the parameterization of $C_{ij}$ and found that the choice of parameterization had little effect on the shape of the modeled background distribution. ![\[fig8\] The performance of the [trf]{} prediction on double-tag events (points), without including the correlation factor $C_{ij}$ (dashed histogram), and including $C_{ij}$ for two different functional parameterizations (solid histograms).](./fig8.eps){width="\linewidth"} The probabilities $p_i$ were used to assign a weight, the probability that the event could have a given number of tags, to every tagged and untagged event in the sample. To ensure the [trf]{} prediction was accurate in the region of phase space outside the “background” peak of the neural network, we used the region $-0.7<NN<0.5$ to determine a normalization. In this region of phase space, the $t\bar{t}$ content was negligible. A possible dependence on $t\bar{t}$ content was studied by the addition and/or subtraction of simulated $t\bar{t}$ events, as was the variation of the interval used for the normalization. Outside the background peak, the [trf]{} predictions were corrected by: [*SF*]{}$_{1} = 1.000 \pm 0.009$ for the single-tag analysis, and [*SF*]{}$_{2} = 0.969 \pm 0.014 $ for the double-tag analysis. The errors on the normalization were taken into account as a systematic uncertainty on the number of background events. ![\[fig9\] The distribution of the $NN$ output variable for single-tag events. Shown are the data (points), background (hashed band), signal (filled histogram) and signal+background (dashed histogram). The vertical line represents the used cut of $NN>0.81$.](./fig9.eps){width="\linewidth"} Both the single-tag and the double-tag analysis were expected to be dominated by background, even at large values of $NN$. Figures \[fig9\] and \[fig10\] show the distribution for data (points), the Monte Carlo simulation prediction for $\sigma_{t\bar{t}}=6.5~{\rm pb}$ (filled histogram), the background prediction (line histogram) and the signal+background distribution (dashed histogram) [@ttbarlepjetsvtxtag; @theoryxsec]. The cross section was calculated from the number of $t\bar{t}$ and background candidates above a cut value of the $NN$ discriminant. The cut value was chosen to maximize the expected statistical significance $s/\sqrt{s+b}$, where $s$ and $b$ were the number of expected signal and background events. The signal and background distributions were estimated using the [trf]{} prediction and $t\bar{t}$ Monte Carlo events [@footnote3]. For both analyses, the expected statistical significance was about two standard deviations. The optimal cut for the single (double)-tag analysis was $NN\geq 0.81~(0.78)$ shown by a vertical line in Figs. \[fig9\] and \[fig10\]. Table \[tabres1\] gives the observed numbers of events ($N_{obs}^i$), the background prediction ($N_{bg}^i$) and the efficiency for signal ($\varepsilon_{t\bar{t}}$) that can be used to calculate the $t\bar{t}$ production cross section via: $$\sigma_{t\bar{t}}=\frac{N_{obs}^i - N_{bg}^i}{ \varepsilon_{t\bar{t}}^i \mathcal{L} (1 - \varepsilon_{TRF}^i) } ,$$ where $i$ was “$=1$” for the single-tag analysis and “$\geq 2$” for the double-tag analysis. The number of background events is predicted using the [trf]{} method. It was likely that at values of $NN$ close to unity a certain fraction of the sample used to predict the background actually consists of tagged or untagged $t\bar{t}$ events, resulting in an increased background prediction. The expected $t\bar{t}$ contamination of the background sample was corrected by a factor $\varepsilon_{TRF}^i$. In the higher value bins of $NN$, the contribution from untagged $t\bar{t}$ events was significant. $\varepsilon_{TRF}^i$ was estimated by applying the [trf]{} on $t\bar{t}$ MC, and comparing the predicted tagging probability for signal to what was expected from background. The size of the Monte Carlo sample dominates the uncertainty on $\varepsilon_{TRF}^i$. Table \[tabres1\] lists the systematic uncertainties on the estimate of the number of background events, the selection efficiency and the background contamination. The first was uncorrelated between the two analyses, while the latter two were correlated as they were derived from the same Monte Carlo samples. For the single-tag analysis, the systematic uncertainty on the selection efficiency was dominated by the uncertainty in the jet calibration and identification, which were estimated by varying the parameterizations used by one standard deviation. The uncertainty on the background prediction was dominated by the uncertainty on the [trf]{} method and the uncertainty on $\varepsilon_{TRF}$ was due to limited Monte Carlo statistics. The uncertainty of the [trf]{} prediction was comprised from the uncertainties coming from the fits of the probability density functions at the jet level, the statistics of the background sample and the uncertainty on the normalization and correlation factors $SF$ and $C_{ij}$. For the double-tag analysis, the contribution from the uncertainties due to calibration of the $b$ quark jet identification efficiency was an additional systematic uncertainty on $\varepsilon_{t\bar{t}}$. These uncertainties were derived by varying the parameterizations used within their known uncertainties.   symbol value -------------------------- ----------------------------------- --------------------------------------------- -- observed events $N_{obs}^{=1}$ 495 background events $N_{bg}^{=1}$ $464.3 \pm 4.6 ({\rm syst})$ $t\bar{t}$ efficiency $\varepsilon_{t\bar{t}}^{=1}$ $0.0242 _{-0.0058}^{+0.0049}({\rm syst})$ $t\bar{t}$ contamination $\varepsilon_{TRF}^{=1}$ $0.245 \pm 0.031 ({\rm syst})$ observed events $N_{obs}^{\geq 2}$ 439 background events $N_{bg}^{\geq 2}$ $400.2_{-6.2}^{+7.3} ({\rm syst})$ $t\bar{t}$ efficiency $\varepsilon_{t\bar{t}}^{\geq 2}$ $0.0254 _{- 0.0070}^{+0.0065} ({\rm syst})$ $t\bar{t}$ contamination $\varepsilon_{TRF}^{\geq 2}$ $0.194 \pm 0.048({\rm syst})$ : \[tabres1\]Overview of observed events, background predictions and efficiencies. ![\[fig10\] The distribution of the $NN$ output variable for double-tag events. Shown are the data (points), background (hashed band), signal (filled histogram) and signal+background (dashed histogram). The vertical line represents the used cut of $NN>0.78$. ](./fig10.eps){width="\linewidth"} The single-tag analysis yielded a cross section of $$\sigma_{t\bar{t}}=4.1_{-3.0}^{+3.0}({\rm stat}) _{-0.9}^{+1.3}({\rm syst}) \pm 0.3 ({\rm lumi})~{\rm pb}.$$ For the double-tag analysis the measured cross section was $$\sigma_{t\bar{t}}=4.7_{-2.5}^{+2.6}({\rm stat}) _{-1.4}^{+1.7}({\rm syst}) \pm 0.3 ({\rm lumi})~{\rm pb}.$$ As the single-tag and double-tag analysis were measured on independent samples, the statistical uncertainties were uncorrelated. The uncertainties on the selection efficiency were completely correlated. Taking all uncertainties into account, a combined cross section measurement of $$\sigma_{t\bar{t}}=4.5_{-1.9}^{+2.0}({\rm stat}) _{-1.1}^{+1.4}({\rm syst}) \pm 0.3 ({\rm lumi})~{\rm pb}$$ was obtained, for a top quark mass of $m_t=175~{\rm GeV}/c^2$. For a top quark mass of $m_t=165~{\rm GeV}/c^2$, the cross section is $\sigma_{t\bar{t}}(165)=6.2_{-2.7}^{+2.8}({\rm stat}) _{-1.5}^{+2.0}({\rm syst}) \pm 0.4 ({\rm lumi})$ pb, while for a top quark mass of $m_t=185~{\rm GeV}/c^2$ the value shifted down to $\sigma_{t\bar{t}}(185)=4.3_{-1.8}^{+1.9}({\rm stat}) _{-1.0}^{+1.4}({\rm syst}) \pm 0.3 ({\rm lumi})$ pb. In summary, we have measured the $t\bar{t}$ production cross section in $p\bar{p}$ interactions at $\sqrt{s}=1.96$ TeV in the fully hadronic decay channel. We used lifetime $b$-tagging and an artificial neural network to distinguish $t\bar{t}$ from background. Our measurement yields a value consistent with SM predictions and previous measurements. [99]{} list\_of\_visitor\_addresses\_r2.tex DØ Collaboration, V.M. Abazov [*et al.*]{}, Nucl. Instrum. Methods Phys. Res. A [**565**]{}, 463 (2006). T. Andeen [*et. al.*]{}, FERMILAB-TM-2365-E (2006), in preparation. The efficiency for signal remained between 85 and 90% throughout the data collection period. Efficiencies were was measured both on $t\bar{t}$ Monte Carlo and derived from parameterizations determined from data. M.L. Mangano [*et al.*]{}, J. High Energy Phys. [**07**]{}, 001 (2003). T. Sjöstrand [*et al.*]{}, Comput. Phys. Commun. [**135**]{}, 238 (2001). D. Lange, Nucl. Instrum. Methods Phys. Res. A [**462**]{}, 152 (2001). G.C. Blazey [*et al.*]{}, in [*Proceedings of the Workshop: QCD and Weak Boson Physics in Run II*]{}, U. Baur, R.K. Ellis and D. Zeppenfeld (ed.), Fermilab, Batavia, IL (2000). DØ Collaboration, V.M. Abazov [*et al.*]{}, Phys. Lett. B [**626**]{}, [ 35]{} (2005). DØ Collaboration, V.M. Abazov [*et al.*]{}, Submitted to Phys. Rev. D, FERMILAB-PUB-06-386-E (2006). W.-M. Yao [*et al.*]{}, Journal of Physics G [**33**]{}, 1 (2006). DØ Collaboration, B. Abbott [*et al.*]{}, Phys. Rev. Lett. [**83**]{} 1908 (1999). The possibility that the wrong permutations of jets could be chosen was taken into account in the determination of the values of the values of 79, 11, and 21 GeV/$c^2$ for $M_W$, $\sigma_{M_W}$ and $\sigma_{m_t}$. R. Brun and F. Rademakers, Nucl. Inst. Meth. in Phys. Res. A [**389**]{} (1997) 81-86. See also http://root.cern.ch/. N. Kidonakis and R. Vogt, Phys. Rev. D [**68**]{}, 114014 (2003). The expected $t\bar{t}$ content used to optimize the $NN$ cut was equivalent with a hypothetical cross section of $\sigma_{t\bar{t}}=6.5~{\rm pb}$. The chosen cuts are stable under variation of the value assumed for the optimization.

--- abstract: 'We present a complete atlas of the Cygnus Loop supernova remnant in the light of ø3 ($\lambda 5007$), , and 2 ($\lambda\lambda 6717, 6731$). We include low-resolution ($25\arcsec$) global maps and smaller fields at $6\arcsec$ resolution from observations using the Prime Focus Corrector on the 0.8-m telescope at McDonald Observatory. Despite its shell-like appearance, the Cygnus Loop is not a current example of a Sedov-Taylor blast wave. Rather, the optical emission traces interactions of the supernova blast wave with clumps of gas. The surrounding interstellar medium forms the walls of a cavity through which the blast wave now propagates, including a nearly complete shell in which non-radiative filaments are detected. We identify non-radiative shocks around half the perimeter of the Cygnus Loop, and they trace a circle of radius $R = 1\fdg 4$ (19 pc) in the spherical cavity walls. The Cygnus Loop blast wave is not breaking out of a dense cloud, but is instead running into confining walls. Modification of the shock velocity and gas temperature due to interaction of the blast wave with the surrounding medium introduces errors in estimates of the age of this supernova remnant. The optical emission of radiative shocks arises only where the blast wave encounters inhomogeneities in the ambient medium; it is not a consequence of gradual evolution to a global radiative phase. Distance measurements that rely on this uniform blast wave evolution are uncertain, but the radiative shocks can be used as distance indicators because of the spherical symmetry of the surrounding medium. The interstellar medium dominates not only the appearance of the Cygnus Loop but also the continued evolution of the blast wave. If this is a typical example of a supernova remnant, then global models of the interstellar medium must account for such significant blast wave deceleration.' author: - 'N. A. Levenson and James R. Graham' - 'Luke D. Keller and Matthew J. Richter' nocite: - '[@Oort46]' - '[@Fes82]' - '[@Min58]' - '[@Kir76]' - '[@Gre91]' - '[@Shu91]' - '[@Fes82]' - '[@McC79]' - '[@Cha85]' - '[@McK75]' - '[@Lev97]' - '[@Hes83]' - '[@Hes94]' - '[@Fes92]' - '[@Fes92]' - '[@Hes94]' - '[@Hes94]' - '[@Bla91]' - '[@Mor96e]' - '[@Cha85]' - '[@Lev97]' - '[@McK84]' - '[@Fal82]' - '[@Shu91]' - '[@Lev97]' - '[@PShu85]' - '[@Van92]' - '[@Vin97]' - '[@Lev97]' - '[@Min58]' - '[@Hub37]' - '[@Hes86]' - '[@Che80]' - '[@McK77]' - '[@Shu87]' title: Panoramic Views of the Cygnus Loop --- ø3[\[\]]{} 2[\[\]]{} \#1\#2 \#1\#2\#3[ ]{} Introduction\[secintro\] ======================== Supernova remnants greatly determine the large-scale structure of the interstellar medium. The energy of supernova remnants heats and ionizes the interstellar medium (ISM), and their blast waves govern mass exchange between various phases of the ISM. In doing so, supernova remnants (SNRs) influence subsequent star formation and the recycling of heavy elements in galaxies. Global models of the interstellar medium that include a hot ionized component ([@Cox74]; [@McK77]) are sensitive to the supernova rate, the persistence of their remnants, and the sizes they attain. A simple calculation of the last of these assumes that the blast wave expands adiabatically in a uniform medium once the blast wave has swept up mass comparable to the mass of the ejecta. During this Sedov-Taylor phase, the radius of the SNR as a function of $E_{51}$, the initial energy in units of $10^{51}$ erg, $n_o$, the ambient number density in units of ${\rm cm^{-3}}$, and $t_4$, time in units of $10^4$ yr, is $R=13 (E_{51}/n_o)^{1/5} t_4^{2/5} {\rm \,pc}$ in a medium where the mean mass per particle is $2.0 \times 10^{-24} {\rm \, g}$. This phase will last until radiative losses become important. The beginning of this subsequent phase, marked by the initial loss of pressure behind the blast wave, occurs at $t=1.9\times 10^4 E_{51}^{3/14} n_o^{-4/7} {\rm \,yr}$, when the radius is $R=16.2 E_{51}^{2/7} n_o^{-3/7} {\rm \,pc}$ ([@Shu87]), although the radiating shell is not fully formed yet. We approach these large-scale questions with analysis of complete images of a particular supernova remnant, the Cygnus Loop, in three optical emission lines. This supernova remnant appears to be a limb-brightened shell at radio ([@Keen73]), infrared ([@Bra86]), optical ([@Fes82]), and X-ray ([@Ku84]; [@Lev97]) energies, which at first glance suggests that it is presently in the transition to the radiative stage. The Cygnus Loop has the advantages of being nearby, bright, and relatively unobscured by dust. This allows us to examine in detail the evolution of various portions of the shock front and to determine physical parameters, such as shock velocity and local ambient density, as they vary throughout the remnant. Despite its appearance, the Cygnus Loop is not a current example of blast wave propagation in a uniform medium at any stage. Instead, its evolution is governed by the inhomogeneous interstellar medium, which we map using the shock as a probe. Many of the features we discuss have been noted by others. Oort (1946) first suggested that the Cygnus Loop is an expanding supernova shell. Spectroscopy of radiative shocks in selected locations (e.g., [@Mil74], [@Ray80a], and Fesen et al. 1982) combined with theoretical models of these shocks (e.g., [@Cox72], [@Dop77], [@Ray79], and [@Shu79]) has been used to derive the physical conditions of the observed shocks. We utilize the radial velocity measures of Minkowski (1958), Kirshner & Taylor (1976), Greidanus & Strom (1991), and Shull & Hippelein (1991) to discern some of the three-dimensional structure that is ambiguous from the data we present. Many non-radiative or Balmer-dominated shocks in the Cygnus Loop have been identified (e.g., [@Kir76], [@Ray80b], [@Tref81], Fesen et al. 1982, [@Fes92], and [@Han92]). Our observations qualitatively match these, and we rely on these works and others ([@Ray83]; [@Long92]; [@Hes94]) for quantitative measures of parameters such as shock velocity and preshock density. McCray & Snow (1979) and Charles, Kahn, & McKee (1985) have suggested that the Cygnus Loop is the result of a cavity explosion, and we adapt this global model to interpret the surrounding interstellar medium, as well. This paper is a companion to the soft X-ray survey presently in progress with the [*ROSAT*]{} High Resolution Imager ([@Gra96]; [@Lev97]). With these two surveys, we examine the Cygnus Loop as a whole, not restricting our investigation only to those regions that are exceptionally bright or that appear to be particularly interesting. We hope to understand both the global processes and the specific variations that are responsible for the emission we detect. The X-rays probe hot (temperature $T\sim 10^6$ K) gas that shocks with velocities $v_s \sim 400 \kms$ heat. The optical emission is expected from slower shocks ($v_s \lesssim 200 \kms$) in which the post-shock region cools to temperatures $T\sim 10^4$ K, yet the most prominent regions at optical wavelengths are also bright in X-rays. McKee & Cowie (1975) suggested that the broad correlation of X-ray and optical emission is the result of a blast wave propagating in an inhomogeneous medium. In this scenario, the shock is significantly decelerated in dense clumps of gas, while portions of it proceed unimpeded through the lower-density intercloud medium. We apply the principles of this basic cloud–blast-wave interaction to a range of locations in the Cygnus Loop. In particular, we refine the cavity model introduced in Levenson et al. (1997), using these optical data to constrain the current ISM in the vicinity of the Cygnus Loop and to determine how the stellar progenitor modified it in the past. We present the observations in §2. We describe them in detail, noting individual regions of interest, and we use these data to measure the physical conditions of the blast wave and the ambient medium in particular locations in §3. The purpose of the detailed examination is to combine the results in a complete map of the surrounding ISM. We present this three-dimensional model while providing a coherent explanation of the history that accounts for it in §4. We predict the future of this SNR and relate its fate to more general theories of supernova modification of the interstellar medium in §5 and summarize our conclusions in §6. Observations ============ We obtained narrow-band images of the Cygnus Loop in 1995 August and September and 1995 September on the 0.8-m telescope at McDonald Observatory with the Prime Focus Corrector (PFC; [@Cla92]). The PFC system is a 5-element catadioptric corrector located at the F/3 prime focus of the telescope and projects a flat focal plane onto a Loral-Fairchild $2048 \times 2048$, $15\micron$-pixel charge-coupled device. The camera and corrector system has a $0\farcs3$ spot size in the V-band and projects each pixel to $1\farcs35$ on the sky, producing a $46\arcmin \times 46\arcmin$ field of view. The interference filters each have a rectangular transmission profile and full width at half maximum $FWHM = 40$Å  in the converging beam of the PFC. The filter centers in the F/3 beam are 5007Å, 6723Å, and 6563Å  to detect line emission from ø3 ($\lambda5007$), 2 ($\lambda\lambda6717, 6731$), and +\[\], which we refer to throughout as . Each individual integration of approximately $600$ s duration was bias-subtracted, flat-fielded, and scaled by its median sky value before being combined with other images. Every region of the Cygnus Loop was observed at least twice through each filter in order to remove cosmic rays and exclude bad regions of the detector, and most regions were observed three times in . We obtained a total of 62 observations in ø3, 89 in , and 54 and 2. Over the large spatial scale of the Cygnus Loop ($3^\circ \times 3\fdg 5$), each observation is a distinct projection of the sky onto the plane of the detector, so images must be remapped to a common projection before being co-added. An astrometric solution for each integration was calculated based on the [*Hubble Space Telescope*]{} Guide Star Catalog. Alignment does not rely on overlapping observations of a particular region but depends instead on the astrometry of individual integrations. The final images are the result of pixel-by-pixel averaging in the combined image plane using nearest-neighbor matching and rejecting bad regions of the detector. The global maps of the Cygnus Loop with $25\arcsec$ pixels are shown with linear scaling in , ø3, and 2, and in a false-color combination with logarithmic scaling (Figure \[figwhole\]). While the observations at different wavelengths resemble each other broadly, their variation is apparent in the combined image. In this display, blends of  and 2 alone are yellow, ø3 and 2  together appear as magenta, and  and ø3  combine to make cyan. Only the regions that are displayed as white exhibit strong emission at all three wavelengths. In addition to the global maps, we present 5 smaller fields, each $95\arcmin$ across, at $6\arcsec$ resolution (Figs. \[figsubv\]–\[figsubz\]). Although we first describe the current physical conditions in specific regions of interest, our ultimate goal is to combine the understanding of the particular sites into a complete account of the surrounding ISM, its modification by the progenitor star, and the passage of the blast wave through it. Emission Morphology and Structure of the ISM ============================================ Observed Morphologies --------------------- The radiative shocks that are responsible for most of the optical emission from the Cygnus Loop result in two characteristic morphologies: filaments and diffuse emission. Figure \[figann\] identifies examples of these features and other structures to which we will refer by name. The morphological distinction is the result of different viewing geometry. As the “wavy sheet” model (Hester et al. 1983; [@Hes87]) describes, bright, sharp filaments are the result of long lines of sight through tangencies of the shock front. When the shock front is viewed face-on, the emission appears more diffuse. Combined with the X-ray data, these observations provide more specific information on the evolutionary state of the Cygnus Loop and the distribution of inhomogeneities of the ISM around it. The widespread X-ray emission indicates that the global shock velocity is several hundred , so the optical emission comes from regions of denser gas that have decelerated the blast wave. The correlation of the X-rays with optically-bright regions make clear that this SNR is not globally in the gradual transition to the radiative phase that is the result of blast wave propagation in a uniform medium. The blast wave has recently and suddenly decelerated where radiative shocks are detected. This rapid, environmentally-imposed evolution is distinct from the simple model prediction for incremental change in a homogeneous medium. These complete data sets demonstrate that many characteristics that have been investigated in small regions of the Cygnus Loop are widespread, and we adapt the physical explanations of these limited regions in previous work to understand the Cygnus Loop globally. For example, the bright emission from radiative shocks is found to be stratified at select observed locations ([@Fes82]), and we show this stratification to be typical of all prominent radiative shocks. The ø3 extends toward the projected exterior, and the  and 2 are located toward the projected interior. The 2 is farthest to the interior and is particularly diffuse. The stratification is a function of the column density that the shock has swept up. The incomplete ø3 emission is concentrated behind shocks that have encountered relatively little gas, while where the shock lags because it has swept up a greater column of gas,  and then 2 are prominent. The strong ø3 emission requires that the shock has progressed through a column density $N \approx 3\times 10^{17} {\rm \ cm^{-2}}$, while the complete recombination zone develops after the swept-up column $N \gtrsim 10^{18} {\rm \ cm^{-2}}$ for $v_s < 150 \kms$ ([@Ray88]). The apparent thickness of these zones and the extensive regions of bright emission in a combination of all the observed lines is a consequence of projection effects. The curved surface of the blast wave encounters the cloud at different projected radii, so some lines of sight intersect multiple stages of shock evolution. Around most of the perimeter of the Cygnus Loop, filaments are detected in  alone. These Balmer-dominated filaments are the result of non-radiative shocks ([@Che78]; [@Che80]). Unlike the more common radiative shocks, the  emission here does not come from the extended cooling zone behind the shock. Instead, it is the result of collisional ionization before excitation in the immediate post-shock region, or the downward cascade toward the ground state following charge exchange. In either case, neutral atoms are required in the post-shock region. Because these do not survive long, the emission originates at the shock front. Thus, we use the Balmer filaments to trace the shock front directly. This  emission is intrinsically weak, so it is most obvious when the line of sight through the shock front is long. This is the case at the projected edge of the SNR, where the line of sight is tangent to the spherical blast wave. Projected Edges\[subsecedge\] ----------------------------- The projected edges of the Cygnus Loop are the simplest regions to understand because the geometric complications are minimized there. The observed structures may be slightly—but not significantly—in the foreground or background and still appear near the projected edge. The prominent regions, NGC 6992 at the northeast ($\alpha=\hms{20}{56}{24},\delta=31^\circ 43\arcmin$; Fig. \[figsubv\]) and NGC 6960 at the west ($\alpha=\hms{20}{45}{42},\delta=30^\circ 43\arcmin$; Fig. \[figsubw\]), are examples of interactions of the blast wave with large clouds ([@HesCox]; Hester et al. 1994; [@Gra95]; [@Lev96]), and the corresponding X-ray enhancement arises from reflected shocks that propagate through the hot, shocked SNR interior. The observed length scales of optical and X-ray emission from NGC 6992 and NGC 6960 reveal the sizes of these large clouds. At the 770 pc distance of the Cygnus Loop, the coherent networks of filaments over $20\arcmin$ imply that the clouds have lengths of about $10^{19}$ cm. Balmer filaments, high-resolution X-ray observations or both constrain the location of clouds along the line of sight. X-ray emission extends beyond both of these bright optical edges, where some portion of the three-dimensional blast wave is not yet impeded by the clouds. Thus, the clouds do not extend over the complete edge of the blast wave along the line of sight. In contrast, the southeast knot at $\alpha=\hms{20}{56}{20},\delta=30^\circ 25\arcmin$ (Fesen et al. 1992; [@Gra95]) is an example of a cloud that is clearly extended along the line of sight but is not necessarily large across the plane of the sky. The nearly-circular edge of the X-ray SNR and the Balmer filaments are distinctly concave at the southeast knot (Fig. \[figsubx\]a), and there is no X-ray emission to the exterior. Thus, this cloud must be at least 11 pc long along the line of sight in order to impede the entire projected edge of the blast wave at this location. Again, the presence of the complete optical cooling structure indicates that the shock has progressed through a column $N \gtrsim 10^{18} {\rm cm^{-2}}$ here. The breakout to the south, centered on $\alpha=\hm{20}{50},\delta=29^\circ 10\arcmin$ and extending approximately $1^\circ$ across (Fig. \[figsubz\]), is the most significant departure from circularity in the remnant. The greater advance of the blast wave in the breakout compared with the circular loop north of $\delta=29^\circ40\arcmin$ indicates that the ambient density is lower in this southern region. Balmer-dominated filaments surround much of the breakout, where the blast wave has now encountered atomic gas. There are sections of incomplete shocks to the east and west (e.g., $\alpha=\hm{20}{53},\delta=29^\circ 20\arcmin$ and $\alpha=\hm{20}{48},\delta=29^\circ 15\arcmin$) and fully radiative shocks at the east ($\alpha=\hm{20}{52},\delta=29^\circ 15\arcmin$). Emission is absent across the face of this area. In the plane of the sky, the breakout is not very extended. Either the blast wave has only recently entered this low-density region, so it has not had enough time to advance significantly, or the extent of the low-density material is limited, so the blast wave no longer continues to progress rapidly. The stratification of emission around the edge of the breakout is apparent on smaller scales than those observed elsewhere in the Cygnus Loop, and the breakout exhibits smaller regions of common overlap. In the northeast, for comparison, the distinct regions of  and ø3 are typically $3\arcmin$ wide, and they are observed together over $10\arcmin$ scales, whereas in the breakout, the exclusive and common zones are both typically $1\arcmin$ wide. These characteristics are consequences of the more limited scale of interactions with clouds in the breakout. There is no evidence for indentation or blast wave progress beyond the optical-emitting regions, which could be due to either intrinsically small clouds or early stages of encounter. In either case, at least some dense material surrounds the SNR at its southern extremity. The exterior of the southwest quadrant exhibits many filaments that are nearly aligned with one another (Fig. \[figsuby\]). Some of these are part of the radiative shock structure related to the southern end of NGC 6960, while farther away from the center, these are non-radiative and incomplete shocks (Fesen et al. 1992). Along some of the filaments, the transition from non-radiative to incomplete shock structure is clear where ø3 emission arises. The multiple filaments are easily understood as tangencies of a wavy sheet. These filaments are shorter than those at the north and east of the Cygnus Loop. This indicates that the ISM is clumpier here, which would distort the blast wave in many places and account for the number of filaments that we observe. Shock diffraction through low-density ($\delta \rho/ \rho \gtrsim 1$) regions may be responsible for the large curvature of some of these filaments, as Fesen et al. (1992) suggest, although multiple interfering shock fronts are not necessary to produce the different filaments. They arise from multiple projections of the same shock front. ### Balmer Filaments and X-ray Emission The Balmer-dominated filaments around the periphery of the Cygnus Loop indicate that the fraction of atomic gas surrounding the SNR is high. The extreme circularity of these filaments requires that the blast wave remain nearly spherical with little deformation at the projected edge. With the exclusion of the breakout region, we fit them with a circle of radius $R=1\fdg 4$, which is equivalent to 19 pc at a distance of 770 pc. In Figure \[figcirc\], the model circle is drawn on the soft X-ray image obtained with the [*ROSAT*]{} High Resolution Imager ([@Lev97]). We calculate the correlation of radial variation as a function of angular separation and the corresponding Fourier coefficients for the Balmer filaments. Excluding the breakout, the amplitude of the $m = 2$ mode is less than 0.02, which implies that the surroundings of the Cygnus Loop are homogeneous on large scales. The individual filaments tend to be very long, typically extending over more than $40\arcmin$. The continuity and smoothness of the Balmer filaments indicate the uniformity of the medium in which these shocks propagate. For constant ram pressure $\rho v_s^2$ behind the shock, variations in density and velocity are related $\delta \rho / \rho = -2 \delta v_s / v_s$. In terms of the radius, $R$, and time, $t$, $v=\eta R/t$, where $\eta$ is of order unity; for free expansion $\eta =1$, and for an adiabatic supernova remnant in a uniform medium, $\eta = 2/5$. Thus, $\delta \rho / \rho = (-2/\eta) (\delta R/ R)(t/\delta t)$. The observed radial variations of a particular filament are $\delta R/ R \sim 0.01$. An approximate timescale of these shocks is 1000 yr, and the age of the Cygnus Loop is about 14,000 yr (cf. §\[secage\]), so the overall density variations in the present medium of the Balmer-dominated filaments are of order $\delta \rho / \rho \sim 0.3 \eta$. We expect $ 2/5 < \eta < 1 $, so $0.1 \lesssim \delta \rho / \rho \lesssim 0.3$. The Balmer-dominated filaments define the current location of the blast wave and mark the presence of neutral material. Detailed studies of particular locations at the northeast have used these non-radiative shocks as density probes ([@Ray83]; [@Long92]; Hester et al. 1994) and derive densities $n \sim 1 {\rm \ cm^{-3}}$. The total observed Balmer line intensity is sensitive to density, shock velocity, and geometric perspective. All of the filaments around the periphery share the same edge-on geometry. Because of the remnant’s circularity, it is reasonable to assume that the shock velocity is nearly constant around the extreme projected edge except at the breakout. The filaments that have been studied in detail to determine shock velocity and ambient density are the brightest ones, however, so these regions are most likely in the transition to incomplete radiative emission, and these densities are expected to be higher than elsewhere. Thus, we conclude that where typical non-radiative filaments are observed around the periphery, the density is constrained $n \lesssim 1 {\rm \, cm^{-3}}$, and most likely $n \sim 0.1 {\rm \ cm^{-3}}$. We roughly calculate the  surface brightness of a non-radiative shock in the Cygnus Loop. We examine a portion of the blast wave in the southeast, near $\alpha=\hm{20}{56}, \delta=30^\circ 00\arcmin$. This is not an exceptionally bright filament, and there is no obvious interaction with a large cloud here. The whole filament extends over more than $20\arcmin$, so the blast wave edge is smooth here. We estimate a surface brightness of $4.2\times 10^{-6} {\rm \,erg\,s^{-1}\,cm^{-2}\,sr^{-1}}$ from this filament. Assuming that this line of sight passes through two folds of the spherical blast wave and the ISM does not distort it, the intensity of a single shock surface is $2.1\times 10^{-6} {\rm \,erg\,s^{-1}\,cm^{-2}\,sr^{-1}}$. The line intensity, $I$, depends on the energy of the transition, $h\nu$, and is given by $$I = {{h \nu}\over{4\pi}}n_H v_s{q_{ex}\over{q_i}} {\rm \,erg\,s^{-1}\,cm^{-2}\,sr^{-1}},$$ where $n_H$ is the preshock neutral density and the ratio of excitation and ionization rates $q_{ex}/q_i \approx 0.2$ for  ([@Ray91]). If we reasonably constrain the shock velocity $200 < v_s < 400 \kms$, these observations limit the density $0.3 > n_H > 0.15 {\rm \,cm^{-3}}$ at the location of the non-radiative shock. The lower velocity limit is based on observations of Balmer filaments ([@Ray83]; [@Long92]; Hester et al. 1994), which are likely to represent the slowest non-radiative shocks, as noted above, and the upper bound assumes that the forward shock does not heat the X-ray-emitting medium to temperatures greater than $2 \times 10^6 {\rm \, K}$. The Balmer filaments are related to adjacent X-ray emission: the optical filaments form the exterior boundary of low surface brightness, limb-brightened X-rays ([@Lev97]). This characteristic morphology reveals where the blast wave is decelerated in the cavity walls. We combine the density determined from the Balmer observations with X-ray temperature measurements to calculate the unperturbed blast wave velocity and the original density of the cavity, and we compare this with the cavity density based on the observed X-ray surface brightness of the interior of the SNR. A bimodal temperature distribution has been detected in the Cygnus Loop, with peaks at $1.5 \times 10^6$ and $5 \times 10^6$ K in EXOSAT observations with the channel multiplier array (sensitive over 0.05–2 keV) and the medium energy experiment (sensitive over 1–50 keV; [@Bal89]). We suggest that the higher temperature component is due to the reflected shock and that previous low single-temperature measures are contaminated by the slow, forward shock in the cavity walls. We model these simply as a reflected shock and ignore non-equilibrium effects, which are probably important. The physical parameters temperature, pressure, and density, are related by the equations of conservation of mass, momentum, and energy flux across shock fronts. Assuming the temperature of $1.5 \times 10^6 {\rm \,K}$ corresponds to the decelerated shock in the cavity wall and $T=5 \times 10^6 {\rm \,K}$ corresponds to the reflected shock, the current blast wave velocity is around $300 \kms$, and the original, unperturbed blast wave velocity was around $500 \kms$. The reflected shock model also predicts that the original cavity density, $n_c$, is related to the shell density $n_s/n_c = 5$. Combined with the surface brightness measurement of the Balmer filaments, this yields an atomic shell density $n_s = 0.2 {\rm \,cm^{-3}}$ and a cavity density $n_c = 0.04{\rm \,cm^{-3}}$ if the shell is entirely neutral. As described below (§\[subsecmodel\]), the shell is partially ionized, so the original cavity density was $n_c = 0.08 {\rm \,cm^{-3}}$. In addition to neglect of non-equilibrium effects, other sources of error include confusion of the reflected shock and the unperturbed shock in the cavity, uncertainty of the derived X-ray temperature, and uncertainty of the Balmer intensity of a single shock surface derived from observation through the spherical edge of the blast wave. The predicted density is consistent with that derived from X-ray observations of the projected interior. In a featureless region of the southeast near $\alpha=\hm{20}{55},\delta=30^\circ 12\arcmin$, the median HRI surface brightness is $0.01 {\rm \, counts \, s^{-1} \, arcmin^{-2}}$. We compare this with a model of a Raymond-Smith plasma at $T_e = 3.5 \times 10^6$, correcting for absorbing column of $N_H = 6\times 10^{20} {\rm \, cm^{-2}}$. Assuming the emitting region is 22 pc deep (the predicted line of sight through the SNR at this radius), the observed count rate corresponds to a density $n_e = 0.2 {\rm \, cm^{-3}}$. This is the density of the shocked material; the original cavity density would have been $n_c = 0.05 {\rm \, cm^{-3}}$. ### Photoionization One unusual feature at the east is a region of very smooth emission adjacent to the bright structure of NGC 6995 at $\alpha=\hms{20}{57}{30},\delta=31^\circ 00\arcmin$. This area is obvious in  and detectable in 2 but absent in ø3. There are no X-rays here ([@Lev97]), so a fast shock wave has not passed through. The X-ray map shows a clear indentation in this region, where dense gas has significantly impeded the blast wave over large scales, similar to the southeast cloud. There is no other direct evidence, such as a Balmer filament, that the blast wave is presently located toward the exterior of this region. We propose that this gas is photoionized by a shock precursor, and it has not been shocked. Radiation from the shock-heated gas of the interior of the Cygnus Loop serves as the ionization source. Typically, when the shock velocity $v_s \gtrsim 100 {\rm \ km \ s^{-1}}$, much of the subsequent radiation is at UV wavelengths ([@Ray79]; [@Shu79]; [@Dop84]), which can ionize the surrounding gas, both ahead of and behind the shock front. The hardness of the ionizing spectrum increases with shock velocity, and the resulting emission from photoionized gas is a function of shock velocity and preshock density ([@Sut93]; [@Dop96]; [@Mor96]). The absence of ø3  in the ionized precursor indicates that the shock has intermediate velocity ($v_s \lesssim 200 {\rm \ km \ s^{-1}}$), which is consistent with UV spectroscopy of Blair et al. (1991) who find $v_s \sim 170 \kms$ is typical in this region. The bright emission ahead of the shock front is probably a consequence of relatively high density; the Balmer luminosity of the precursor scales linearly with density in the steady case ([@Ray79]; [@Shu79]; [@Dop96]). A second important factor is that the shock is fully radiative, having been decelerated in the dense cloud, so the shock front does not move through the ionized region before the ions have time to radiate (cf. [@Ray88]). Finally, the column density ahead of the shock must be great for the precursor emission to be observable. The images do allow us to estimate crudely the density of the photoionized material. We assume that the length of the photoionized region along the line of sight is equal to its extent north-south in the plane of the sky and use the observed surface brightness to predict a density $n\sim 60 {\rm \, cm^{-3}}$. Spectral data would allow a more exact determination of the characteristics of the emitting region and quantitative measurements of the photoionizing source. The blast wave history causes the ambient medium to be photoionized here and not elsewhere. The penetrating X-rays of very fast shocks ($v_s \gtrsim 400 \kms$) do not efficiently ionize the surrounding medium; the optical depth to 200 eV photons $\tau_{\rm 200 eV} = N/(1.1 \times 10^{20}{\rm \, cm^{-2}})$. Only the softer UV-dominated spectra of slower shocks are effective ionizing sources. In other regions of the Cygnus Loop, the recent shock velocity has been too great to produce much photoionization, and there has not been enough time for the ionizing field of the post-shock cooling and recombination zone to develop. Thus, most of the surrounding medium has remained sufficiently neutral to exhibit Balmer-dominated filaments. The exceptional photoionization at the eastern edge is consistent with the cavity model of this SNR. The blast wave has not slowed sufficiently elsewhere for most of the shocked gas to be a useful source of ionizing photons. Over nearly all of the Cygnus Loop, the deceleration has been recent, as the blast wave has run into the walls of the cavity. Only in this region has the softer spectrum of a slower shock existed long enough to ionize the preshock medium so that its radiative transitions can be detected. Photoionized regions have been observed around other SNRs, including N132D ([@Mor95]) and N49 ([@Van92]) in the Large Magellanic Cloud, for example. One difference between the Cygnus Loop and the former SNR, however, is that while in the Cygnus Loop the photoionized precursor is the exception, detected only in a limited region, in N132D it is the rule, occurring over the majority of the younger remnant’s perimeter. Another difference is that the exceptional photoionization ahead of the Cygnus Loop blast wave is adjacent to the region that contains its brightest soft X-ray knots ([@Lev97]), while the photoionization and X-ray surface brightness variations are not correlated in N132D ([@Mor96e]). These differences are likely to be a consequence of the different sources of ionizing photons and the density of the surrounding medium. First, in N132D, the X-ray–emitting shock can account for only half of the ionization ([@Mor96e]), so another significant source must also contribute to the ionization in this case. The ubiquitous radiative shocks in clouds and O-rich filaments that Morse et al. (1996) propose as this additional source are not preferentially associated with the X-ray emission. Second, a large molecular cloud extends over the entire southern hemisphere of this SNR ([@Ban97]). While the present blast wave is not in the dense core that is observed in CO, the outer regions of the cloud are expected to be over-dense with respect to the typical ambient ISM. The higher densities that make the ionizing precursor visible are therefore characteristic of the majority of the perimeter of N132D. In contrast, we suggest that it is the unusual conditions of extreme blast wave deceleration and related high density that result in both the bright X-ray emission and the photoionized precursor in this small area at the eastern edge of the Cygnus Loop. These conditions make the Cygnus Loop precursor similar to that of N49. This middle-aged SNR is associated with a small molecular cloud on its eastern limb ([@Ban97]). The blast wave interaction with the cloud results in local optical emission and X-ray enhancement, blast wave deceleration observed in a range of shock velocities ([@Van92]), and a prominent photoionized precursor on the eastern limb of N49. Projected Interior ------------------ The interior of the Cygnus Loop contains diffuse emission and filaments. Both morphologies share some characteristics, such as exclusive ø3 emission (e.g., $\alpha =\hm{20}{48}, \delta=30^\circ 30\arcmin$ in diffuse regions and $\alpha =\hm{20}{52}, \delta=30^\circ 30\arcmin$ in filaments), and blends of  and 2 (e.g., $\alpha =\hm{20}{50}, \delta=29^\circ 45\arcmin$ and $\alpha =\hm{20}{50}, \delta=29^\circ 45\arcmin$). A combination of emission in all three lines is observed only in diffuse emission (e.g., $\alpha =\hm{20}{52}, \delta=31^\circ 10\arcmin$), and  alone is observed only in filaments (e.g., $\alpha =\hm{20}{50}, \delta=30^\circ 30\arcmin$). Projection effects are more severe in the interior, but these optical images and high-resolution spectroscopy ([@Kir76]; [@Gre91]; [@Shu91]) together help unravel the tangled Cygnus Loop. In all cases, the optical emission is the result of the interaction of the three-dimensional blast wave surface with dense structures in the ISM. The emitting regions are truly at the outside of the SNR, but some appear projected to the interior. The brighter filaments tend to have lower absolute radial velocities and narrower velocity dispersions than the fainter diffuse emission ([@Kir76]; [@Shu91]). This is consistent with the model in which filaments are long lines of sight through the edge of a wavy shock front, and diffuse emission results when the blast wave is observed nearly face-on. The shock velocity is primarily perpendicular to the line of sight in the former case and along the line of sight in the latter case. In some instances, adjacent filamentary and diffuse emission are related, being tangential and normal views of the same portion of the blast wave. Elsewhere, spatially separated shock regions are projected onto the same location in the observer’s view, causing significant confusion. The extensive diffuse emission at the western interior near $\alpha=\hm{20}{47}$—$49^{\rm m}$ and $\delta=30^\circ 40\arcmin$—$31^\circ 20\arcmin$ suggests that a large wall of dense gas is present across the entire face of the SNR (Figs. \[figsubw\] and \[figsuby\]). The irregularity of the diffuse emission is due to the clumpiness of the cloud. This emission is blueshifted with respect to nearby gas ([@Gre91]; [@Shu91]) so it is on the front face. The bright filaments of the “carrot” (at $\alpha =\hm{20}{49}, \delta=31^\circ 40\arcmin$), having positive radial velocities, are on the rear face. The filaments at $\alpha =\hm{20}{49}, \delta=31^\circ 00\arcmin$ (Fig. \[figsubw\]) are distinct on small scales, similar to the edges of the breakout, although here the region of common overlap is absent. The origin of the stratification is the same: the limited physical scale of the interaction with the intervening cloud, either because the cloud is intrinsically small or because the encounter is recent and the blast wave has not progressed far into the cloud. Along a particular line of sight, only a single coherent section of the blast wave is detected. Unlike the areas where strong emission is observed at all wavelengths, here there are not many projected overlapping regions of various stages of shock evolution. The filaments that are bright in  and 2 illustrate an earlier section of the cloud shock than do the ø3 filaments. In this area on the western half of the Cygnus Loop, there is no planar blast wave that actually moves to the east, however. Rather, the [*interaction*]{} with a cloud is progressing from west to east as the blast wave surface expands radially. The intervening cloud must be oriented along the line of sight to some degree, as shown in Figure \[figcartc\]. Another region of diffuse emission and fainter filaments runs north-south across the center of the remnant, around $\alpha=\hm{20}{50}$—$52^{{\rm m}}, \delta=30^\circ 20\arcmin$—$50\arcmin$ (Figs. \[figsubx\] and \[figsuby\]). The high-resolution spectra in the vicinity are of the diffuse component alone, which includes very bright central knots. These clumps are blueshifted, so these clouds are on the near face. The westernmost of these filaments, at $\alpha=20^{{\rm h}} 50^{{\rm m}} 20^{{\rm s}}$, is detected only in . The blast wave has run into an extended wall of gas, enabling us to look through a long edge of the shock front. These filaments are broader than most of the other pure Balmer filaments, which is expected in this scenario. The broad filaments appear where the blast wave is viewed through a point of inflection, whereas the sharper, narrow ones are viewed at a tangency. For these non-radiative filaments to be observable, the wall of gas must be extremely smooth at the present projected location of the shock front. The wall of gas is likely to be related to the source of the rest of the optical emission at the center of the SNR, not the clouds that are illuminated as the “carrot,” although this is uncertain. History of the Cygnus Loop ========================== Modelling the ISM\[subsecmodel\] -------------------------------- Except in the single photoionized section, the optical emission traces dense shocked regions. Using some knowledge of shock evolution and geometry, we map the surrounding ISM. The basic principles applicable to these observations are: (1) optical emission due to radiative shocks requires that the blast wave has swept up a significant column density ($N \gtrsim 10^{18} {\rm \, cm^{-2}}$), which is likely in regions of increased density ($n\gtrsim 1 {\rm \, cm^{-3}}$); (2) sharp filaments are edge-on views of the blast wave surface, and diffuse emission is the result of face-on views of this surface; (3) the optical line emission from a radiative shock is progressively dominated by ø3, , then 2; and (4) filaments of  alone are the result of non-radiative shocks in material that has a significant neutral fraction. With these data, we construct a three-dimensional model of the interstellar medium around the Cygnus Loop . Over much of its surface, the blast wave is propagating through large clouds. Long filaments, detected in  around the perimeter and in other lines across the interior, indicate that a spherical shell of atomic gas surrounds the SNR. The dense clumps of gas and the atomic shell together form the walls of a cavity in which the supernova occurred. Slices through the Cygnus Loop parallel to the plane of the sky from the near side to the far side (Fig. \[figcartmap\]) show some of the three-dimensional structure of the ISM before the supernova event. The center of the Cygnus Loop is at latitude $b = -8^\circ$. At the rear of the cavity, toward the Galactic plane, is a large wall of gas that is detected in neutral hydrogen ([@DeN75]) and in the IRAS sky survey ([@Bra86]). The cloud that causes the bright northeastern emission is also on the far side. Clumps at the edge of the breakout and at the northwest are on the near side. Some clouds, including the southeast knot and the molecular gas at the west, extend along the line of sight and appear in all slices. The Cygnus Loop is not the result of a supernova explosion in an arbitrary region of the interstellar medium. Rather, the blast wave propagates through a medium that the progenitor star has processed. Several characteristics imply a massive progenitor, the sort most capable of modifying the surrounding ISM, which Charles et al. (1985) and Levenson et al. (1997) discuss. One strong piece of evidence is the paradoxical morphology of the Cygnus Loop. At optical and X-ray wavelengths, it appears to be nearly circular, yet it exhibits tremendous structure on smaller scales. The optical data alone could be interpreted as a SNR in transition from the adiabatic to the radiative phase, but the presence and correlation of X-rays require that overall the blast wave still has (or until very recently had) a high velocity, $v_s \gtrsim 400 {\rm \ km \ s^{-1}}$. A model of blast wave interaction with large clouds accounts for this correlation ([@HesCox]; [@Gra95]; [@Lev96]), but such large clouds cannot be typical of the pre-supernova interstellar medium. If they were, the SNR would not appear so circular, and X-ray emission due to clouds evaporating in the interior would be observed. Thus, although large clouds are found around the periphery of the Cygnus Loop, they were not located within the current blast wave prior to the supernova event. This structure is expected in the vicinity of an O or B star. During its main-sequence lifetime, such a massive star homogenizes the surrounding ISM, creating a uniform region at constant density, temperature, and pressure. The star’s UV emission removes clumps of denser gas within the H II region, either destroying the clouds through photoevaporation or relocating them to the exterior by the rocket effect ([@McK84]). The result is a homogeneous spherical region surrounded by dense clouds. A star of type earlier than B0 will clear a cavity with a radius of about 50 pc (McKee et al. 1984 ). The Cygnus Loop cavity is smaller, so its progenitor must be a later-type star ([@Cha85]), yet one still able to create the photoionized cavity. Thus, we suggest that a B0, $M\approx 15M_{\sun}$ star was the progenitor of the Cygnus Loop. The observer’s line of sight through shocked clouds at the projected edge is preferentially longer than through those that appear at the projected center. Photoevaporation of clouds progresses radially outward from the central star, so lines of sight tangent to the cavity are biased to be through the longer dimensions of surviving clumps. Thus, optical emission on the projected interior is not as bright as on the projected edge of the SNR, despite their common cause. The same effect is even more pronounced in the X-ray data. The atomic shell in which the non-radiative filaments appear is also a consequence of the progenitor’s evolution. While material is eroded from clouds during the star’s main sequence lifetime, increasing the density of the H II region, the radius of the ionized region decreases as the outer portions recombine, creating the shell ([@Elm76]). The atomic gas of this recombined region together with the cloud remnants on the periphery form the walls of the cavity (Fig. \[figcartmap\]). The Cygnus Loop blast wave has expanded through the uniform cavity and is now hitting the surrounding walls ([@Lev97]). The atomic shell is not actually a sharp density discontinuity. It is more likely a gradual transition from the cavity interior to the maximum shell density. At the end of the progenitor’s main sequence lifetime, the pressure in the former H II region declines as the gas cools and recombines. (See [@PShu85] for a discussion of this phenomenon applied to N49.) In the extreme case of a sudden loss of pressure at the boundary, a rarefaction wave would move back through the dense shell resulting in a density profile of the form $n \propto (1 + {3r \over {at_{RG}}})^3$ ([@Zel66]), where the distance, $r$, from the original shell edge is constrained $-3at_{RG} \le r \le 0$ in terms of sound speed, $a$, and red giant lifetime, $t_{RG}$. The drop in pressure is not so sudden, however. Over $t_{RG} \sim 10^6$ for $M\approx 15M_{\sun}$ ([@Mae89]), about half the atoms in the former H II region have recombined. Thus, the density profile of the shell edge is likely to be intermediate between a sharp discontinuity and the offset power law above. The Cygnus Loop has been suggested previously to be the result of an explosion in an incomplete cavity. Both Falle & Garlick (1982) and Shull & Hippelein (1991), for example, argue for a partial cavity at a density discontinuity, such as the edge of a molecular cloud. In these models, only half of the blast wave runs into the cavity boundary. Consequently, the bright radiative emission of NGC 6992 and NGC 6960 and the corresponding X-ray emission are physically separated by great distances along the line of sight: the former are the result of radiative shocks in the cavity wall, and the opposite hemisphere of the blast wave propagates through the low-density medium outside the cloud to cause the latter. In contrast, the model presented here and in Levenson et al. (1997) has a completely defined cavity, the atomic shell and dense clouds together forming its spherical boundary. The blast wave does not break [*out*]{} of the edge of the molecular cloud into lower-density material; instead, it slows down when it hits a wall of [*denser*]{} gas. Although NGC 6992 and NGC 6960 and the X-rays that appear to the exterior of these radiative regions are two slightly different portions of the blast wave, they are not significantly separated. Furthermore, some of the large clouds are obviously extended along the line of sight, not restricted to the rear boundary (Fig. \[figcartmap\]). These optical and previous X-ray data ([@Lev97]) demonstrate that the cavity wall is complete. All around the projected edges of the Cygnus Loop, the Balmer filaments and X-ray enhancements mark the smooth component of the cavity wall. [*Even the region called the breakout illustrates where the blast wave runs into a boundary of dense gas, not continued expansion in a low-density medium.*]{} Only the eastern third of the projected interior of the Cygnus Loop lacks direct evidence for a cavity wall, although the smooth atomic shell that is apparent in Balmer-dominated filaments elsewhere would not be detected across the face of the SNR. Only clumps of gas could be observed, and their distribution depends on the arrangement that existed before the ISM was processed by the progenitor. The southeast of the Cygnus Loop is farthest from the galactic plane, so we expect the original filling factor of clouds to have been lowest there. Other SNRs, including RCW 86 ([@Vin97]) and N132D ([@Mor96e]), have been identified as cavity explosions. \[Shull (1985) identifies N49 as a cavity supernova, but Vancura et al. (1992) argue that the dense ambient material associated with the single observed CO cloud accounts for the data better.\] In these cases, interaction of the blast wave with the cavity wall enhances the observable properties of the SNRs. RCW 86 shares many features of the Cygnus Loop, including similar size, a limb-brightened (though very incomplete) X-ray shell (Vink et al. 1997), and Balmer-dominated filaments to the exterior of the X-ray emission ([@Smi97]). N132D is a younger remnant, and the apparent cavity is smaller. The progenitor’s wind during its main sequence lifetime may form the cavity boundary, or the scale of photoevaporation may have been limited in this dense environment adjacent to a molecular cloud ([@Ban97]). The bright interior X-ray emission ([@Mor96e]) does imply that the cavity interior was not uniform. Age and Distance\[secage\] -------------------------- Although the shell-like visual morphology of the Cygnus Loop and its X-ray limb-brightening broadly suggest that this is an example of a supernova remnant in the Sedov-Taylor or adiabatic expansion phase, in detail this model is inconsistent with the data presented here and in Levenson et al. (1997). The observed X-ray and optical limb-brightening are a consequence of interaction with the inhomogeneous interstellar medium, not the result of evolution of the blast wave in a uniform medium. Assuming that the blast wave has slowed sufficiently during its progress through a uniform medium to produce optical line emission (i.e. to $v_s \lesssim 200 \kms$) yields an age estimate of about 50,000 yr ([@Min58]; [@Fal82]). This is much too great because the age is inversely proportional to the blast wave velocity, and only in propagating through dense clouds has the shock been decelerated to $v_s \lesssim 200 \kms$. The Balmer filaments mark the location of the unimpeded, spherical blast wave, and the X-ray emission immediately behind these filaments demonstrates that the typical shock velocity is $v_s \gtrsim 400 \kms$, which is still to great for radiative cooling of the Sedov-Taylor shell to be significant. This SNR has not made the global transition to the radiative evolutionary phase. The X-ray emission alone has been used to determine an age of around 18,000 yr ([@Rap74]; [@Ku84]), assuming that this is presently an adiabatic blast wave expanding in a uniform medium; $t = 2R/5v_s$, where $R=19{\rm \ pc}$ and $v_s = 400 \kms$. The X-ray data provide a more accurate measurement of the age, but it is still likely to be an upper limit because these calculations contain uncertainties in the shock temperature and the velocity that is derived from it. Only the current unperturbed shock velocity is relevant to the age in this model, but any observation of the SNR includes contributions from a variety of temperatures, heated by different stages of the blast wave’s history. Observing the projected edge of the SNR minimizes confusion due to overlapping projections of different temperature components, but the interaction of the blast wave with the inhomogeneous ISM greatly alters the shock velocity and the corresponding postshock gas temperature. When the blast wave hits a density discontinuity, as it does over most of the surface of the Cygnus Loop, the continuing shock is suddenly decelerated and a reflected shock propagates back toward the interior, further heating the dense, hot, shocked material. At a density contrast of 10, for example, the twice-shocked matter will have a temperature approximately 1.5 times the singly-shocked material at the same radius, which leads to an overestimate of the unperturbed shock velocity by about $20\%$. Both the reflected shock and the forward shock through the dense cloud enhance the X-ray surface brightness and thus bias the mean measured temperature to favor blast-wave–cloud interactions rather than the undisturbed blast wave. If the X-ray measurement of two temperature components ([@Bal89]) due to the decelerated cloud shock and the reflected shock is correct and the original velocity of the unperturbed blast wave was closer to $500 \kms$ (§\[subsecedge\]), the corresponding age $t = 14,000$ yr. A reliable determination of the age of the Cygnus Loop is required to determine the initial energy, $E_o$, accurately (§\[secintro\]). Adopting the parameters $n_o = 0.08 {\rm \, cm^{-3}}$, $R = 19$ pc, and $t = 14,000$ yr, and a partially-ionized cavity interior, we find $E_o=2\times10^{50} {\rm\, erg}$. Because the radiative emission shows where the blast wave has run into clumpy material and decelerated, care is required to use it to measure the distance to the Cygnus Loop. Minkowski (1958) determined the expansion velocity to be $116 \kms$, fitting an ellipse to radial velocity measurements as a function of radius. This, combined with Hubble’s (1937) observed proper motion of filaments of $0.03\arcsec {\rm \ year^{-1}}$ yields the frequently-cited distance of 770 pc. This calculation assumes that all the optical filaments are part of a common expanding surface. In fact, the optical emission reveals different regions of inhomogeneity in the ISM. A second complication is the distinct geometry of the diffuse and filamentary emission on the face of the SNR. One is the result of a face-on shock, and the other is an example of a shock viewed edge-on. The latter does not have the radial velocity structure as a function of projected radius that is assumed in the model calculation. A third source of confusion arises because the observed filaments are the cooling zone behind the shock front, so their apparent motion may be the result of further development of the postshock region, not motion of the same clump of gas. The cooling zone may be propagating, without any physical motion of a parcel of gas in the ISM. Despite these difficulties, however, the radiative shocked gas can be used to measure distance to the Cygnus Loop or other supernova remnants when the distribution of clumps in the surrounding medium is spherically symmetric. In this case, although the emission comes from distinct regions of the ISM, not a single expanding blast wave surface, the different clouds acquire the same velocity when the blast wave passes through them. This approach is not appropriate where the surrounding medium is asymmetric, however. IC 443, for example, is asymmetrical, and its corresponding distance determinations are highly uncertain ([@Gre84]). The non-radiative filaments can also be used to determine distance, as Hester, Danielson, & Raymond (1986) have done, finding a distance of about 700 pc. These structures are advantageous because proper motion and spatial velocity can be calculated from the same filament. There is some uncertainty in the shock velocity because the derived value depends on the amount of ion-electron temperature equilibration (Chevalier et al. 1980). Measuring proper motion of filaments also introduces some uncertainty. All filaments are detected where the observer’s line of sight through the shock front is long, which may be the tangent surface of a spherical bubble. If it is instead the deformed surface of the blast wave that appears as a filament, however, the apparent filament will not necessarily persist or be detectable over the timescales required to obtain proper motion measurements. Long baselines over which to measure proper motion are desirable, but the oldest observations are sensitive to the brightest filaments. These bright filaments are exactly the locations where the shock is likely slowed in denser gas, so the present velocity differs from the spatial velocity that corresponds to the proper motion. With these caveats in mind, we adopt a distance of 770 pc to the Cygnus Loop. Consequences ============ The Cygnus Loop blast wave is now in the walls of the cavity, the boundary between the low-density former H II region and the ambient ISM. Blast wave propagation in the atomic shell will be a long-lived phenomenon. The smooth material of the shell recombined as clumps were photoevaporated in the interior. We crudely calculate the thickness of the shell assuming a constant number of ions in the H II region in both the initial Stromgren sphere, in which only the less-dense intercloud material was ionized, and in the final Stromgren sphere of radius $R_f$, in which all material, having mean density $n_m$, is ionized. The typical, unprocessed ISM determines the initial conditions; we assume the intercloud density $n_{ic}=0.1 {\rm \, cm^{-3}}$, and a volume filling factor of 0.03 of clouds of density $n_{cl}=1 {\rm \, cm^{-3}}$, so $n_{m}=0.13 {\rm \, cm^{-3}}$. Conserving ion number and neglecting increased recombination as clouds are evaporated, we obtain for the shell thickness $$R_{sh}=R_f ((n_m/n_{ic})^{1/3} - 1),$$ so using the observed radius $R_f = 19{\rm \,pc}$ yields $R_{sh}=1.5 {\rm \, pc}$. A blast wave of velocity $v_s= 300\kms$ will propagate through the shell for 5000 years. Eventually the blast wave will propagate through the denser medium of the cavity walls and clouds everywhere. It will rapidly decelerate, not only in the select locations where it has already encountered dense clumps that protrude into the cavity. Some of the surrounding material consists of large dense clumps like the ones observed presently in the very extended regions of radiative shocks. These, too, will light up and will be detectable at optical wavelengths. The southeast cloud may be an example of the earliest stages of interaction with a large cloud, and it may later resemble the northeast region ([@Gra95]). We also expect the entire northern limb, which in the optical regime is now detected primarily in bright Balmer filaments, to turn into an extended radiative region. The incomplete shocks along the northern limb are the early stages of interaction with a larger structure that is perhaps directly associated with the bright northeastern region. The entire western face of the Cygnus Loop will soon be bright when the blast wave has fully encountered the wall of gas that we have noted there. If the long filaments across the center of the SNR are the result of another wall of gas, the middle will also light up with optical emission in several hundred years. Large clouds of gas will significantly affect blast wave evolution. The shock will no longer be adiabatic when the cooling timescale becomes comparable with the dynamical timescale. In large clouds, where density $n_{cl}=1 {\rm \, cm^{-3}}$, radius $R_{cl}= 5 {\rm \, pc}$, and in which the shock velocity $v_{cl}=200\kms$, the dynamical time $R_{cl}/v_{cl} \approx 2\times 10^4 {\rm \, yr}$. In these clouds, the cooling time $3kT/n_{cl}\Lambda(T)\approx 4 \times 10^4 {\rm \, yr}$, in terms of the cooling function, $\Lambda$, having units ${\rm erg\, cm^3\, s^{-1}}$. If the covering fraction of clumps in the surrounding medium is sufficiently great, they will force the Cygnus Loop into the radiative phase, and energy losses will further decelerate the blast wave. The time and radius at which this occurs are distinct from self-similar calculations, such as those of McKee & Ostriker (1977) and Shull & Draine (1987) in which the gradual transition to the radiative phase happens as the blast wave propagates through a uniform medium. We use the pressure-driven snowplow model ([@McK77]) to set an upper limit on the final size of the SNR. In this model, work of the blast wave pushing into the ambient material and radiative losses of the hot interior are taken into account. In the Cygnus Loop, the radiative losses of the outer shell also will be significant, slowing the blast wave and making the final radius smaller. During this phase, $PV^\gamma$ is constant, where $P$ is the pressure inside the SNR, $V$ is its volume, and $\gamma$ is the ratio of specific heats. For $\gamma = 5/3$, this phase ends when $R = R_{PDS}(P_{PDS}/P_{ISM})^{1/5}$, in terms of the external pressure of the interstellar medium, $P_{ISM}$, and the radius and pressure inside the supernova remnant at the onset of the pressure-driven snowplow stage, $R_{PDS}$ and $P_{PDS}$, respectively. At this final time, the pressure inside the supernova remnant is equal to the ambient pressure. For the Cygnus Loop, we take $P_{ISM} = 10^{-12} {\rm \, dyne\, cm^{-2}}$, $R_{PDS}=20 {\rm \, pc}$, and $P_{PDS} = 4\times 10^{-9} {\rm \, dyne\, cm^{-2}}$, which places an upper limit on the final radius $R < 105 {\rm \, pc}$. Assuming that blast waves propagate through uniform media over great distances yields misleading conclusions about the size of SNRs and how they distribute energy through the ISM. Global models of the interstellar medium that include the hot contribution of SNRs ([@Cox74]; [@McK77]) are sensitive to the supernova rate, the final sizes of supernova remnants, and their persistence as hot bubbles. These factors determine whether the dominant component of the ISM is hot and in which smaller regions of warm and cold gas are embedded, or whether the warm and cold components form the substrate in which the distinct hot bubbles of SNRs are located. For the former to occur, the onset of the radiative phase of the SNR must be postponed until $R > 100 {\rm \, pc}$, independent of its prior evolution ([@Cox86]). (This calculation assumes that the supernova rate is 1 per 30 yr in a disk of radius 15 kpc, yet the conclusion holds for a range of reasonable supernova rates.) Although the cavity of the Cygnus Loop is small because its progenitor is one of the least massive stars to undergo core collapse, the initial mass function favors these lower mass (and smaller cavity) progenitors. Furthermore, even more massive stars, where cavities of 50 pc radius are expected, will be forced into the radiative phase and not expand to overlap if the density of the cavity walls is sufficient. The cavity walls are indeed likely to be denser on average than the typical ISM because the walls contain much of the material that was cleared out of the cavity interior. The conditions of the ISM in general, however, constrain the mass that will be commonly available to form clouds and cavity walls around pre-supernova stars. If the hot, low-density medium is pervasive, cavity walls typically would not be sufficiently massive to induce the radiative phase early, and SNRs would then normally attain larger radii before their blast waves decelerate. In any case, the typical environments of supernova progenitors must be considered. If dense, molecular material normally surrounds these stars, the radiative phase will be induced early, even if globally the ISM has low density. The cavity structure is significant also in determination of the mass of gas that a supernova remnant heats. The onset of shell formation for a supernova blast wave expanding in a uniform medium occurs at $R \propto n^{-3/7}$, so the shocked mass is then $M \propto n^{-2/7}$. If the ambient density varies by three orders of magnitude, the heated mass changes by only a factor of 7, for example. In contrast, when the supernova environment is a cavity with dense walls, the initial size of the cavity constrains the mass of the hot interior. The cavity interior is rarefied, so the heated cavity mass is significantly lower. In the Cygnus Loop, assuming a cavity density $n_c = 0.08 {\rm \, cm^{-3}}$, the heated mass is $20 M_{\sun}$, whereas in a uniform medium of this density, the heated mass would be greater than $400 M_{\sun}$. Conclusions =========== This optical emission line atlas of the Cygnus Loop supernova remnant provides the information to render a portrait of the surrounding ISM. The ambient medium, which the massive stellar progenitor shaped as it evolved, consists of many large, dense regions. These will significantly affect the subsequent development of the blast wave. The blast wave will decelerate, and it will no longer have sufficient velocity to excite the material through which it propagates to X-ray-emitting temperatures. Although this is a study of a particular object, it has far-reaching consequences for the ISM as a whole. Supernova remnants provide the energy to heat the ISM, affect the velocity dispersion of interstellar clouds, and set the stage for future generations of star formation. Thus, any global model of the gas in the Galaxy critically depends on the evolution of SNR blast waves in their inhomogeneous environments, of which this example is typical. While optical emission-line characteristics identify supernova remnants, they may preferentially select those whose evolution the extant ISM determines. Other SNRs need to be examined over a broad range of energies in the same careful way to draw more certain conclusions about their net effect on the interstellar medium. This research was supported in part by the Packard Foundation. Ballet, J., & Rothenflug, R. 1989, , 218, 277 Banas, K. R., Hughes, J. P., Bronfman, L., & Nyman, L.-Å. 1997, , 480, 607 Blair, W. P., et al. 1991, , 379, L33 Braun, R., & Strom, R. G. 1986, , 164, 208 Charles, P. A., Kahn, S. M., & McKee, C. F. 1985, , 295, 456 Chevalier, R. A., Kirshner, R. P., & Raymond, J. C. 1980, , 235, 186 Chevalier, R. A., & Raymond, J. C. 1978, , 225, L27 Claver, C. F. 1997, , 24, 1282 Cox, D. P. 1972, , 178, 143 Cox, D. P. 1986, , 334, 252 Cox, D. P., & Smith, B. W. 1974, , 189, L105 DeNoyer, L. K. 1975, , 196, 479 Dopita, M. A. 1977, , 33, 437 Dopita, M. A., Binette, L., D’Odorico, S., & Benvenuti, P. 1984, , 276, 653 Dopita, M. A., & Sutherland, R. S. 1996, , 102, 161 Elmegreen, B. G. 1976, , 205, 405 Falle, S. A. E. G., & Garlick, A. R. 1982, , 201, 635 Fesen, R. A., Blair, W. P., & Kirshner, R. P. 1982, , 262, 171 Fesen, R. A., Kwitter, K. B., & Downes, R. A. 1992, , 104, 719 Graham, J. R., & Aschenbach, B. 1996, in ROSAT Newsletter \#13, Ed. S. L. Snowden, 6 Graham, J. R., Levenson, N. A., Hester, J. J., Raymond, J. C., & Petre, R. 1995, , 444, 787 Green D. A. 1984, , 209, 449 Greidanus, H., & Strom, R. G. 1991, , 89, 15 Hanson, G. J., & Winkler, P. F. 1992, , 24, 1231 Hester, J. J. 1987, , 314, 187 Hester, J. J., & Cox, D. P. 1986, , 300, 675 Hester, J. J., Raymond, J. C., & Danielson, G. E. 1986, , 303, L17 Hester, J. J., Parker, R. A. R., & Dufour, R. J. 1983, , 273, 219 Hester, J. J., Raymond, J. C., & Blair, W. P. 1994, , 420, 721 Hubble, E. P. 1937, Carnegie Institution Yearbook No. 36, p. 189 Keen, N. J., Wilson, W. E., Haslam, C. G. T., Graham, D. A., & Thomasson, P. 1973, , 28, 197 Kirshner, R. P., & Taylor, K. 1976, , 208, L83 Ku, W. H.-M., Kahn, S. M., Pisarski, R. & Long, K. S. 1984, , 278, 615 Levenson, N. A., Graham, J. R., Hester, J. J., & Petre, R. 1996, , 468, 323 Levenson, N. A., et al. 1997, , 483, 304 Long, K. S., Blair, W. P., Vancura, O., Bowers, C. W., Davidsen, A. F., & Raymond, J. C. 1992, , 400, 214 Maeder, A., & Meynet, G. 1989, , 210, 155 McCray, R., & Snow, T. P., Jr. 1979, , 17, 213 McKee, C. F., & Cowie, L. L. 1975, , 195, 715 McKee, C. F., & Ostriker, J. P. 1977, , 218, 148 McKee, C. F., Van Buren, D. & Lazareff, B. 1984, , 278, L115 Miller, J. S. 1974, , 189, 239 Minkowski, R. 1958, Rev. Mod. Phys, 30, 1048 Morse, J. A. et al. 1996, , 112, 509 Morse, J. A., Winkler, P. F., & Kirshner, R. P. 1995, , 109, 2104 Morse, J. A., Raymond, J. C., & Wilson, A. S. 1996, , 108, 426 Oort, J. H. 1946, , 106, 159 Rappaport, S., Doxsey, R., Solinger, A., &  Borken, R. 1974, , 194, 329 Raymond, J. C. 1979, , 39, 1 Raymond, J. C. 1991, , 103, 781 Raymond, J. C., Black, J. H., Dupree, A. H., Hartmann, L., & Wolff, R. S. 1980a, , 238, 881 Raymond, J. C., Blair, W. P., Fesen, R. A., & Gull, T. R. 1983, , 275, 636 Raymond, J. C., Davis, M., Gull, T. R., & Parker, R. A. R. 1980b, , 238, L21 Raymond, J. C., Hester, J. J., Cox, D., Blair, W. P., Fesen, R. A., & Gull, T. R., Jr. 1988, , 324, 869 Scoville, N. Z., Irvine, W. M., Wannier, P. G., & Predmore, C. R. 1977, , 216, 320 Shull, J. M., & Draine, B. T. 1987, in [Interstellar Processes]{}, ed. D. J. Hollenbach and H. A. Thronson, Jr. (Dordrecht: Reidel), p. 283 Shull, J. M., & McKee, C. F. 1979, , 227, 131 Shull, P., Jr., Dyson, J. E., Kahn, F. D., and West, K. A. 1985, , 212, 799 Shull, P., Jr., & Hippelein, H. 1991, , 383 714 Smith, R. C. 1997, , 114, 2664 Sutherland, R. S., Bicknell, G. V., & Dopita, M. A. 1993, , 414, 510 Treffers, R. R. 1981, , 250, 213 Vancura, O., Blair, W. P., Long, K. S., & Raymond, J. C. 1992, , 394, 158 Vink, J., Kaastra, J. S., & Bleeker, J. A. M. 1997, , 328, 628 Zeldovich, Ya. B., & Raizer, Yu. P. 1966, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, Vol. 1, ed. W. D. Hayes and R. F. Probstein, (New York: Academic Press)

--- abstract: 'Employing two state-of-the-art methods, second-order many-body perturbation theory and multiconfiguration Dirac-Fock, highly accurate calculations are performed for the lowest 318 fine-structure levels arising from the $2s^{2} 2p^{4}$, $2s 2p^{5}$, $2p^{6}$, $2s^{2} 2p^{3} 3l$, $2s 2p^{4} 3l$, $2p^{5} 3l$, and $2s^{2} 2p^{3} 4l$ configurations in O-like . Complete and consistent atomic data, including excitation energies, lifetimes, wavelengths, and E1, E2, M1 line strengths, oscillator strengths, and transition rates among these 318 levels are provided. Comparisons are made between the present two data sets, as well as with other available experimental and theoretical values. The present data are accurate enough for identification and deblending of emission lines involving the $n=3,4$ levels and are also useful for modeling and diagnosing fusion plasmas be considered as a benchmark for other calculations.' address: - 'Hebei Key Lab of Optic-electronic Information and Materials, The College of Physics Science and Technology, Hebei University, Baoding 071002, China' - 'Shanghai EBIT Lab, Key Laboratory of Nuclear Physics and Ion-beam Application, Institute of Modern Physics, Department of Nuclear Science and Technology, Fudan University, Shanghai 200433, China' - 'School of Science, Hunan University of Technology, Zhuzhou, 412007, China' - 'Institute of Applied Physics and Computational Mathematics, Beijing 100088, China' author: - Kai Wang - Wei Zheng - Xiao Hui Zhao - Zhan Bin Chen - Chong Yang Chen - Jun Yan bibliography: - 'ref.bib' title: 'Extended calculations of energy levels, radiative properties, and lifetimes for oxygen-like ' --- atomic data; O-like ; many-body perturbation theory; multiconfiguration Dirac-Fock method Introduction ============ Walls of fusion reactors often contain alloys of molybdenum, and ions of Mo are present in the plasmas due to sputtering from the walls [@Mansfield.1978.V11.p1521; @Reader.2015.V48.p144001]. Therefore, in order to simulate and diagnose plasmas that contain Mo as a constituent, accurate atomic data for different Mo ions are required. In view of this, soft X-ray emission lines from molybdenum plasmas generated by dual laser pulses were measured for different Mo ions [@Lokasani.2016.V109.p194103].  @Feldman.1991.V8.p531 identified 13 lines of the $(1s^2)2s^22p^4$ - $2s2p^5$ transitions in a laser-produced plasma. the theoretical side, excitation energies and radiative transition data for the low-lying states of the $2s^22p^4$, $2s2p^5$, and $2p^6$ configurations were provided by different calculations [@Fontes.2015.V101.p143; @Hu.2011.V9.p1228; @Gu.2005.V89.p267; @Zhang.2002.V82.p357; @Vilkas.1999.V60.p2808]. Atomic parameters of the $n > 2$ levels are also needed for applications in plasma physics [@Rice.2000.V33.p5435; @Kink.2001.V63.p46409]. The present study is to provide a complete accurate data set of energy levels, radiative transition data, and lifetimes involving high-lying levels in O-like . of our previous work O-like ions [@Wang.2017.V229.p37; @Wang.2017.V194.p108]. Excitation energies, wavelengths, line strengths, oscillator strengths, transition rates, and lifetimes are provided the lowest 318 levels of the $2s^{2} 2p^{4}$, $2s 2p^{5}$, $2p^{6}$, $2s^{2} 2p^{3} 3l$ ($l=s, p, d$), $2s 2p^{4} 3l$ ($l=s, p, d$), $2p^{5} 3l$ ($l=s, p, d$), and $2s^{2} 2p^{3} 4l$ ($l=s, p, d, f$) configurations using the many-body perturbation theory (MBPT) method [@Lindgren.1974.V7.p2441; @Safronova.1996.V53.p4036; @Vilkas.1999.V60.p2808; @Gu.2005.V156.p105; @Gu.2007.V169.p154]. In order to assess the accuracy of our MBPT calculations, the multiconfiguration Dirac-Fock (MCDF) and relativistic configuration interaction (RCI) method [@Grant.2007.V.p; @FroeseFischer.2016.V49.p182004] is used to calculate the corresponding data. The present study significantly increases the amount of accurate data for the $n = 3, 4$ levels of O-like Mo to directly aid and confirm experimental identifications. Calculations ============ The MBPT method integrated in the FAC code [@Gu.2008.V86.p675] and the MCDF method implemented in the GRASP2K code [@Jonsson.2007.V177.p597; @Jonsson.2013.V184.p2197] are used to perform the calculations. Both of the methods have been successfully used to calculate atomic parameters for L- and M-shells systems with high accuracy [@Wang.2014.V215.p26; @Wang.2015.V218.p16; @Wang.2016.V223.p3; @Wang.2016.V226.p14; @Wang.2017.V119.p189301; @Wang.2017.V194.p108; @Wang.2017.V187.p375; @Wang.2017.V229.p37; @Wang.2018.V235.p27; @Wang.2018.V239.p30; @Wang.2018.V234.p40; @Wang.2018.V208.p134; @Wang.2018.V864.p127; @Wang.2018.V220.p5; @Chen.2017.V113.p258; @Chen.2018.V206.p213; @Chen.2019.V234.p90; @Chen.2019.V225.p76; @Guo.2015.V48.p144020; @Guo.2016.V93.p12513; @Si.2016.V227.p16; @Si.2017.V189.p249; @Si.2018.V239.p3; @Zhao.2018.V119.p314]. We only give an outline of the MBPT and MCDF calculations, since these two methods are described in our earlier work in detail. MBPT ---- In the MBPT method, the Hilbert space of the system is divided into two subspaces, including a model space $M$ and an orthogonal space $N$. By means of solving the eigenvalue problem of a non–Hermitian effective Hamiltonian in the space $M$, we can get the true eigenvalues of the Dirac–Coulomb–Breit Hamiltonian. The configuration interaction effects in the $M$ space is exactly considered, and the interaction of the spaces $M$ and $N$ is accounted for with the many-body perturbation theory up to the second order. In the calculations, we include all states of the $2s^{2} 2p^{4}$, $2s 2p^{5}$, $2p^{6}$, $2s^{2} 2p^{3} 3l$ ($l=s, p, d$), $2s 2p^{4} 3l$ ($l=s, p, d$), $2p^{5} 3l$ ($l=s, p, d$), $2s^{2} 2p^{3} 4l$ ($l=s, p, d, f$), and $2s 2p^{4} 4s$ configurations in the model space $M$, Through the single and double (SD) virtual excitations of the states spanning the $M$ space, are contained in the space $N$. The maximum $n$ values for the single/double excitations are 200/65, respectively, while the maximum $l$ value is 20. The leading quantum electrodynamic (QED) effects are also considered in our work. MCDF ---- The MCDF method has been described by @FroeseFischer.2016.V49.p182004. Based on the active space (AS) approach [@Sturesson.2007.V177.p539] for the generation of the configuration state function (CSF) expansions, separate MCDF calculations are done for the even and odd parity states. We start the calculation without any excitation from the reference configurations which is usually referred to as the Dirac-Fock (DF) calculation. The reference configurations are . Subsequently, the CSFs expansions are obtained through the SD excitations from the shells of the reference configurations up to a $ n_{max}l_{max} $ orbital, with $n_{max} \leq 8$ and $l_{max} \leq 5$. To reduce the number of CSFs, the $1s^2$ core is closed during the relativistic self-consistent field (RSCF) calculations, but is opened during the RCI calculations, where the Breit interaction and QED effects , i.e., vacuum polarization and self- energy, are included in the Hamiltonian. The number of CSFs in the final even and odd state expansions are approximately 5 060 000 and 3 930 000, respectively, distributed over the different $J$ symmetries. To provide the $LSJ$ labeling system used in databases such as the Atomic Spectra Database (ASD) of the National Institute of Standards and Technology (NIST) [@Kramida.2018.V.p], the wave functions in both the MCDF and MBPT calculations are transformed from a $jj$-coupled CSF basis into a $LSJ$-coupled CSF basis using the methods developed by Gaigalas [@Gaigalas.2004.V157.p239; @Gaigalas.2017.V5.p6]. Results and Discussions ======================= The computed excitation energies for the lowest 318 levels of the $2s^{2} 2p^{4}$, $2s 2p^{5}$, $2p^{6}$, $2s^{2} 2p^{3} 3l$ ($l=s, p, d$), $2s 2p^{4} 3l$ ($l=s, p, d$), $2p^{5} 3l$ ($l=s, p, d$), and $2s^{2} 2p^{3} 4l$ ($l=s, p, d, f$) configurations from our MBPT and MCDF calculations are listed in Table \[table1\], along with the radiative lifetimes estimated from E1, E2, and M1 transition rates, and the $LSJ$-coupled and $jj$-coupled labels obtained from our calculations. Table \[table2\] lists wavelengths $\lambda_{ij}$, and E1, E2, M1 line strengths $S_{ji}$, oscillator strengths $g_{i}f_{ji}$, and radiative rates $A_{ji}$ among the 318 energy levels along with branching fractions (${\rm BF}_{ji} = \frac{A_{ji}}{\sum \limits_{k=1}^{j-1} A_{jk}}$), obtained from both the MBPT and MCDF methods. All the E1 and E2 values are computed in the Babushkin gauge (equivalent to the non-relativistic length form), which is considered to be more accurate than the Coulomb gauge (equivalent to the non-relativistic velocity form). Excitation energies ------------------- Since excitation energies for O-like Mo are only available for the $n=2$ levels in the previous experimental and theoretical studies, the MBPT and MCDF excitation energies are compared with the experimental values from the NIST ASD, and the other theoretical results calculated by @Gu.2005.V89.p267 and @Vilkas.1999.V60.p2808 in Table \[table3\]. The average difference with the standard deviation the NIST values are $-249 \pm 748$ cm$^{-1}$ for MBPT, $-68 \pm 471$ cm$^{-1}$ for MCDF, $1053 \pm 707$ cm$^{-1}$ for Gu, and $1955 \pm 1758$ cm$^{-1}$ for , respectively. Comparing with the previous calculations, there is generally a better agreement between the NIST values and our MBPT (MCDF) results due to electron correlation effects included in our work. For the remaining levels belonging to the $n = 3$ and $n = 4$ configurations, the relative difference between our two calculations for each level is shown in Table \[table1\]. The average absolute difference with standard deviation of the present MBPT and MCDF energy values are $296\pm646$ cm$^{-1}$, corresponding to the average relative difference with the standard deviation of $0.001~\%\pm0.002~\%$, which are satisfactory. Wavelengths, transition rates, and lifetimes {#sec_tr} --------------------------------------------  @Feldman.1991.V8.p531 identified 13 lines of the $2s^22p^4$ - $2s2p^5$ transitions in a laser-produced plasma, and these lines have been compiled in the NIST ASD. In Table \[table4\], the experimental wavelengths for these 13 lines are compared with the present MBPT and MCDF results. Transition rates from our MBPT and MCDF calculations are also compared with each other in the table. As in Table \[table4\], the difference between the NIST and MBPT(MCDF) wavelength for each transition is within 0.1 %, which is satisfactory. Our MBPT and MCDF transition rates show good agreement with each other, the within 5 %. the uncertainty estimation method suggested by @Kramida.2013.V63.p313 [@Kramida.2014.V2.p22; @Kramida.2014.V212.p11], the difference $\delta S$ of line strengths $S$ from the MBPT and MCDF calculations for each E1 transition is defined as $\delta S$ = $\left|S_{\rm MCDF} - S_{\rm MBPT} \right|$/max($S_{\rm MCDF}$, $S_{\rm MBPT}$). The averaged uncertainties $\delta S_{av}$ for line strengths $S$ from the present MBPT and MCDF calculations for E1 transitions in various ranges of $S$ are assessed to 1.7 % for $S \geq 10^{-1}$; 4.1 % for $10^{-1} > S \geq 10^{-2}$; 6.1 % for $10^{-2} > S \geq 10^{-3}$; 8.7 % for $10^{-3} > S \geq 10^{-4}$; 18 % for $10^{-4} > S \geq 10^{-5}$, and 35 % for $10^{-5} > S \geq 10^{-6}$. Then, the larger of $\delta S_{ij}$ and $\delta S_{av}$ is accepted as the uncertainty of each particular line strength. About 1.4 % have uncertainties of $\leq$ 2 % (the category A+), 1.0 % have uncertainties of $\leq$ 3 % (the category A), 20.0 % have uncertainties of $\leq$ 7 % (the category B+), 23.3 % have uncertainties of $\leq$ 10 % (the category B), 8.9 % have uncertainties of $\leq$ 18 % (the category C+), 19.6 % have uncertainties of $\leq$ 25 % (the category C), and 13.5 % have uncertainties of $\leq$ 40 % (the category D+), while 12.3 % have uncertainties of $>$ 40 % (categories D and E). Again, using the method suggested in [@Kramida.2014.V2.p22; @Kramida.2014.V212.p11], the uncertainties of the $S$ values for M1 and E2 transitions are estimated. The estimated uncertainties for all E1, M1, and E2 transitions with BF $\geq 10^{-5}$ are listed in Table \[table2\]. Our MBPT and MCDF lifetimes for the lowest 318 levels of the $n \leq 4$ configurations in O-like , which are calculated by considering all possible E1, M1, and E2 transitions, are listed in Table \[table1\]. For the lowest 10 levels of the $n = 2$ complex, the MCDF lifetimes agree very well with the MBPT results, and the differences are within 3 %. For the remaining 308 levels belonging to the $n = 3$ and $n = 4$ configurations, the average difference with the standard deviation $0.5~\% \pm 4.2~\%$. Conclusions =========== Employing two state-of-the-art methods (MBPT and MCDF), excitation energies and lifetimes of the lowest 318 levels for the $n \leq 4$ configurations have been calculated for O-like . Wavelengths, E1, M1, and E2 transition rates, line strengths, and oscillator strengths for the transitions among these 318 levels are also reported. The accuracy of energy levels and transition probabilities is estimated by comparing the MBPT and MCDF results with available experimental and theoretical data. The average difference with the standard deviation with the NIST values for the levels are $-249 \pm 748$ cm$^{-1}$ for MBPT $-68 \pm 471$ cm$^{-1}$ which indicates the high accuracy of the present two calculations. For the $n=3, 4$ levels, the average absolute difference of the present MBPT and MCDF energy values $296\pm646$ cm$^{-1}$, corresponding to the average relative difference of $0.001~\%\pm0.002~\%$, The uncertainty of line strength is assessed for each transition, and is available in Table \[table2\]. The present calculations provide a consistent and accurate data set for line identification and modeling purposes, which can also be considered as a benchmark for other calculations. Acknowledges ============ We acknowledge the support from the National Key Research and Development Program of China under Grant No. 2017YFA0403200, the Science Challenge Project of China Academy of Engineering Physics (CAEP) under Grant No. TZ2016005, the National Natural Science Foundation of China (Grant No. 11703004, No. 11674066, No. 11504421, and No. 11474034), the Natural Science Foundation of Hebei Province, China (A2019201300 and A2017201165), and the Natural Science Foundation of Educational Department of Hebei Province, China (BJ2018058). Kai Wang expresses his gratitude to the support from the visiting researcher program of Fudan University. [cllrrrccl]{} \ Key & $LSJ$-coupled CSF& $J\pi$ &$E_{\rm MCDF}$ & $E_{\rm MBPT}$& $\Delta E$ & $\tau_{\rm MCDF}$ & $\tau_{\rm MBPT}$ & $jj$-coupled CSF$^{a,b,c}$\ \ Key & $LSJ$-coupled CSF& $J\pi$ &$E_{\rm MCDF}$ & $E_{\rm MBPT}$& $\Delta E$ & $\tau_{\rm MCDF}$ & $\tau_{\rm MBPT}$ & $jj$-coupled CSF$^{a,b,c}$\ 1 &$2s^{2}\,2p^{4}(^{3}P)~^{3}P_{2}$ &2+ &0 &0 &0 & & &2p+2(4)4\ 2 &$2s^{2}\,2p^{4}(^{1}S)~^{1}S_{0}$ &0+ &207772 &207830 &57 &7.987E-02 &7.987E-02 &2p+2(0)0\ 3 &$2s^{2}\,2p^{4}(^{3}P)~^{3}P_{1}$ &1+ &863686 &863215 &-471 &7.767E-08 &7.767E-08 &2p-1(1)1.2p+3(3)2\ 4 &$2s^{2}\,2p^{4}(^{1}D)~^{1}D_{2}$ &2+ &976768 &976432 &-337 &1.414E-07 &1.414E-07 &2p-1(1)1.2p+3(3)4\ 5 &$2s^{2}\,2p^{4}(^{3}P)~^{3}P_{0}$ &0+ &1921492 &1920667 &-825 &2.842E-08 &2.842E-08 &2p+4(0)0\ 6 &$2s~^{2}S\,2p^{5}~^{3}P_{2}^{\circ}$ &2- &2379704 &2380355 &651 &4.938E-12 &4.938E-12 &2s+1(1)1.2p+3(3)4\ 7 &$2s~^{2}S\,2p^{5}~^{3}P_{1}^{\circ}$ &1- &2654158 &2655091 &933 &2.294E-12 &2.294E-12 &2s+1(1)1.2p+3(3)2\ 8 &$2s~^{2}S\,2p^{5}~^{3}P_{0}^{\circ}$ &0- &3289601 &3289830 &228 &3.061E-12 &3.061E-12 &2s+1(1)1.2p-1(1)0\ 9 &$2s~^{2}S\,2p^{5}~^{1}P_{1}^{\circ}$ &1- &3529720 &3530442 &722 &1.845E-12 &1.845E-12 &2s+1(1)1.2p-1(1)2\ 10 &$2p^{6}~^{1}S_{0}$ &0+ &5271717 &5272387 &670 &1.321E-12 &1.321E-12 &2p+4(0)0\ 11 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3s~^{3}D_{2}^{\circ}$ &2- &20650637 &20651158 &521 &1.486E-13 &1.486E-13 &2p+1(3)3.3s+1(1)4\ 12 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3s~^{3}D_{1}^{\circ}$ &1- &20696082 &20696833 &752 &4.980E-14 &4.980E-14 &2p+1(3)3.3s+1(1)2\ 13 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3p~^{3}D_{1}$ &1+ &21088106 &21087880 &-226 &1.209E-11 &1.209E-11 &2p+1(3)3.3p-1(1)2\ 14 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,3p~^{3}P_{2}$ &2+ &21100632 &21100445 &-188 &1.121E-11 &1.121E-11 &2p+1(3)3.3p-1(1)4\ 15 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3p~^{1}F_{3}$ &3+ &21339859 &21339494 &-365 &1.264E-11 &1.264E-11 &2p+1(3)3.3p+1(3)6\ 16 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,3p~^{3}P_{1}$ &1+ &21342916 &21342630 &-286 &7.800E-12 &7.800E-12 &2p+1(3)3.3p+1(3)2\ 17 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,3s~^{5}S_{2}^{\circ}$ &2- &21442137 &21442500 &363 &1.304E-13 &1.304E-13 &2p-1(1)1.2p+2(4)3.3s+1(1)4\ 18 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3p~^{3}D_{2}$ &2+ &21443360 &21443353 &-7 &1.076E-11 &1.076E-11 &2p+1(3)3.3p+1(3)4\ 19 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,3s~^{3}S_{1}^{\circ}$ &1- &21486265 &21486867 &603 &4.736E-14 &4.736E-14 &2p-1(1)1.2p+2(4)3.3s+1(1)2\ 20 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,3p~^{3}P_{0}$ &0+ &21539884 &21540069 &186 &2.594E-12 &2.594E-12 &2p+1(3)3.3p+1(3)0\ 21 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3s~^{3}D_{3}^{\circ}$ &3- &21572314 &21572756 &443 &7.952E-14 &7.952E-14 &2p-1(1)1.2p+2(4)5.3s+1(1)6\ 22 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3s~^{1}D_{2}^{\circ}$ &2- &21606547 &21607149 &602 &5.000E-14 &5.000E-14 &2p-1(1)1.2p+2(4)5.3s+1(1)4\ 23 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3s~^{3}P_{0}^{\circ}$ &0- &21744917 &21745043 &126 &1.148E-13 &1.148E-13 &2p-1(1)1.2p+2(0)1.3s+1(1)0\ 24 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3s~^{3}P_{1}^{\circ}$ &1- &21759710 &21759924 &214 &5.902E-14 &5.902E-14 &2p-1(1)1.2p+2(0)1.3s+1(1)2\ 25 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3d~^{3}D_{2}^{\circ}$ &2- &21839491 &21840372 &881 &2.556E-13 &2.556E-13 &2p+1(3)3.3d-1(3)4\ 26 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3d~^{3}G_{3}^{\circ}$ &3- &21862744 &21863661 &917 &9.965E-13 &9.965E-13 &2p+1(3)3.3d-1(3)6\ 27 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,3p~^{5}P_{1}$ &1+ &21870635 &21870247 &-388 &1.675E-11 &1.675E-11 &2p-1(1)1.2p+2(4)3.3p-1(1)2\ 28 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,3d~^{5}D_{1}^{\circ}$ &1- &21879333 &21880156 &823 &3.845E-14 &3.845E-14 &2p+1(3)3.3d-1(3)2\ 29 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3d~^{1}S_{0}^{\circ}$ &0- &21881341 &21882166 &825 &1.763E-12 &1.763E-12 &2p+1(3)3.3d-1(3)0\ 30 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3d~^{1}G_{4}^{\circ}$ &4- &21916177 &21917059 &882 &7.989E-11 &7.989E-11 &2p+1(3)3.3d+1(5)8\ 31 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3p~^{3}F_{2}$ &2+ &21918698 &21918410 &-288 &2.298E-11 &2.298E-11 &2p-1(1)1.2p+2(4)3.3p-1(1)4\ 32 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,3d~^{5}D_{2}^{\circ}$ &2- &21962255 &21963172 &916 &1.412E-14 &1.412E-14 &2p+1(3)3.3d+1(5)4\ 33 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3p~^{3}F_{3}$ &3+ &22015588 &22015303 &-285 &1.645E-11 &1.645E-11 &2p-1(1)1.2p+2(4)5.3p-1(1)6\ 34 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3d~^{3}F_{3}^{\circ}$ &3- &22044284 &22045384 &1101 &6.373E-15 &6.373E-15 &2p+1(3)3.3d+1(5)6\ 35 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,3d~^{3}D_{1}^{\circ}$ &1- &22071986 &22073049 &1063 &9.081E-15 &9.081E-15 &2p+1(3)3.3d+1(5)2\ 36 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,3p~^{5}P_{3}$ &3+ &22126012 &22125475 &-537 &1.060E-11 &1.060E-11 &2p-1(1)1.2p+2(4)3.3p+1(3)6\ 37 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,3p~^{5}P_{2}$ &2+ &22127625 &22127143 &-482 &5.112E-12 &5.112E-12 &2p-1(1)1.2p+2(4)3.3p+1(3)4\ 38 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3p~^{3}P_{0}$ &0+ &22138613 &22138217 &-395 &6.649E-12 &6.649E-12 &2p-1(1)1.2p+2(4)3.3p+1(3)0\ 39 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3p~^{3}P_{2}$ &2+ &22148113 &22148354 &241 &2.477E-12 &2.477E-12 &2p-1(1)1.2p+2(4)5.3p-1(1)4\ 40 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3p~^{3}D_{1}$ &1+ &22149046 &22148482 &-565 &9.339E-12 &9.339E-12 &2p-1(1)1.2p+2(0)1.3p-1(1)2\ 41 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3p~^{1}P_{1}$ &1+ &22244154 &22243774 &-380 &6.644E-12 &6.644E-12 &2p-1(1)1.2p+2(4)3.3p+1(3)2\ 42 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3p~^{3}D_{3}$ &3+ &22252714 &22252265 &-450 &1.188E-11 &1.188E-11 &2p-1(1)1.2p+2(4)5.3p+1(3)6\ 43 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3p~^{3}F_{4}$ &4+ &22263500 &22263072 &-427 &1.520E-11 &1.520E-11 &2p-1(1)1.2p+2(4)5.3p+1(3)8\ 44 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3p~^{3}P_{1}$ &1+ &22350764 &22350673 &-91 &1.665E-12 &1.665E-12 &2p-1(1)1.2p+2(4)5.3p+1(3)2\ 45 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3p~^{3}P_{0}$ &0+ &22358146 &22358194 &47 &2.632E-12 &2.632E-12 &2p-1(1)1.2p+2(0)1.3p-1(1)0\ 46 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3p~^{1}D_{2}$ &2+ &22421438 &22421218 &-220 &1.940E-12 &1.940E-12 &2p-1(1)1.2p+2(4)5.3p+1(3)4\ 47 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3p~^{3}P_{1}$ &1+ &22462553 &22462031 &-522 &2.220E-12 &2.220E-12 &2p-1(1)1.2p+2(0)1.3p+1(3)2\ 48 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3p~^{3}D_{2}$ &2+ &22486651 &22486365 &-286 &2.754E-12 &2.754E-12 &2p-1(1)1.2p+2(0)1.3p+1(3)4\ 49 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3s~^{3}P_{2}^{\circ}$ &2- &22538636 &22538645 &9 &5.859E-14 &5.859E-14 &2p+3(3)3.3s+1(1)4\ 50 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3s~^{1}P_{1}^{\circ}$ &1- &22568066 &22568222 &156 &4.521E-14 &4.521E-14 &2p+3(3)3.3s+1(1)2\ 51 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3d~^{3}D_{1}^{\circ}$ &1- &22648460 &22649188 &727 &6.105E-14 &6.105E-14 &2p-1(1)1.2p+2(4)3.3d-1(3)2\ 52 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3d~^{3}F_{2}^{\circ}$ &2- &22655526 &22656275 &749 &5.151E-14 &5.151E-14 &2p-1(1)1.2p+2(4)3.3d-1(3)4\ 53 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,3d~^{5}D_{0}^{\circ}$ &0- &22659553 &22660287 &734 &7.713E-14 &7.713E-14 &2p-1(1)1.2p+2(4)3.3d-1(3)0\ 54 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,3d~^{5}D_{3}^{\circ}$ &3- &22685184 &22685956 &772 &3.907E-14 &3.907E-14 &2p-1(1)1.2p+2(4)3.3d-1(3)6\ 55 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,3d~^{5}D_{4}^{\circ}$ &4- &22704297 &22705079 &782 &8.306E-11 &8.306E-11 &2p-1(1)1.2p+2(4)3.3d+1(5)8\ 56 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,3d~^{3}D_{3}^{\circ}$ &3- &22712027 &22712898 &871 &9.071E-14 &9.071E-14 &2p-1(1)1.2p+2(4)3.3d+1(5)6\ 57 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3s~^{5}P_{3}$ &3+ &22715626 &22714630 &-996 &5.899E-13 &5.899E-13 &2s+1(1)1.2p+2(4)5.3s+1(1)6\ 58 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3d~^{3}G_{4}^{\circ}$ &4- &22783461 &22784420 &959 &4.242E-11 &4.242E-11 &2p-1(1)1.2p+2(4)5.3d-1(3)8\ 59 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3d~^{1}P_{1}^{\circ}$ &1- &22790735 &22791579 &844 &1.430E-14 &1.430E-14 &2p-1(1)1.2p+2(4)3.3d+1(5)2\ 60 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3d~^{3}P_{2}^{\circ}$ &2- &22812793 &22813601 &807 &8.916E-15 &8.916E-15 &2p-1(1)1.2p+2(4)3.3d+1(5)4\ 61 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3s~^{3}P_{2}$ &2+ &22829810 &22829389 &-421 &6.458E-14 &6.458E-14 &2s+1(1)1.2p+2(4)5.3s+1(1)4\ 62 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3d~^{3}F_{4}^{\circ}$ &4- &22830158 &22831043 &885 &7.617E-11 &7.617E-11 &2p-1(1)1.2p+2(4)5.3d+1(5)8\ 63 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3d~^{3}D_{3}^{\circ}$ &3- &22837163 &22838068 &905 &6.548E-15 &6.548E-15 &2p-1(1)1.2p+2(4)5.3d-1(3)6\ 64 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3d~^{3}G_{5}^{\circ}$ &5- &22840848 &22841781 &933 &8.437E-11 &8.437E-11 &2p-1(1)1.2p+2(4)5.3d+1(5)10\ 65 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3d~^{3}P_{0}^{\circ}$ &0- &22878419 &22879263 &844 &1.369E-14 &1.369E-14 &2p-1(1)1.2p+2(4)5.3d+1(5)0\ 66 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3d~^{3}P_{1}^{\circ}$ &1- &22881477 &22882356 &879 &6.387E-15 &6.387E-15 &2p-1(1)1.2p+2(4)5.3d-1(3)2\ 67 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3d~^{3}D_{2}^{\circ}$ &2- &22885462 &22886390 &928 &6.273E-15 &6.273E-15 &2p-1(1)1.2p+2(4)5.3d-1(3)4\ 68 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3d~^{1}D_{2}^{\circ}$ &2- &22930758 &22931410 &652 &2.118E-14 &2.118E-14 &2p-1(1)1.2p+2(0)1.3d-1(3)4\ 69 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3d~^{3}S_{1}^{\circ}$ &1- &22940955 &22941641 &686 &6.400E-15 &6.400E-15 &2p-1(1)1.2p+2(4)5.3d+1(5)2\ 70 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3d~^{1}F_{3}^{\circ}$ &3- &22977582 &22978659 &1077 &9.410E-15 &9.410E-15 &2p-1(1)1.2p+2(4)5.3d+1(5)6\ 71 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3d~^{3}F_{2}^{\circ}$ &2- &23012320 &23013008 &688 &1.064E-14 &1.064E-14 &2p-1(1)1.2p+2(4)5.3d+1(5)4\ 72 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3p~^{3}S_{1}$ &1+ &23036093 &23035602 &-491 &1.174E-12 &1.174E-12 &2p+3(3)3.3p-1(1)2\ 73 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3d~^{3}P_{2}^{\circ}$ &2- &23063727 &23064263 &536 &6.713E-15 &6.713E-15 &2p-1(1)1.2p+2(0)1.3d+1(5)4\ 74 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3p~^{1}D_{2}$ &2+ &23069439 &23069193 &-246 &8.834E-12 &8.834E-12 &2p+3(3)3.3p-1(1)4\ 75 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3d~^{3}F_{3}^{\circ}$ &3- &23069895 &23070507 &613 &9.205E-15 &9.205E-15 &2p-1(1)1.2p+2(0)1.3d+1(5)6\ 76 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3d~^{3}D_{1}^{\circ}$ &1- &23081697 &23082323 &626 &5.017E-15 &5.017E-15 &2p-1(1)1.2p+2(0)1.3d-1(3)2\ 77 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3s~^{5}P_{1}$ &1+ &23094509 &23093648 &-861 &1.684E-13 &1.684E-13 &2s+1(1)1.2p+2(0)1.3s+1(1)2\ 78 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3s~^{3}D_{2}$ &2+ &23128999 &23128516 &-484 &3.537E-13 &3.537E-13 &2s+1(1)1.2p+2(4)3.3s+1(1)4\ 79 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3s~^{3}D_{1}$ &1+ &23157234 &23156969 &-265 &5.432E-14 &5.432E-14 &2s+1(1)1.2p+2(4)3.3s+1(1)2\ 80 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3p~^{5}P_{2}^{\circ}$ &2- &23165854 &23166004 &150 &2.510E-13 &2.510E-13 &2s+1(1)1.2p+2(4)5.3p-1(1)4\ 81 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3p~^{5}D_{3}^{\circ}$ &3- &23186328 &23186536 &208 &1.043E-13 &1.043E-13 &2s+1(1)1.2p+2(4)5.3p-1(1)6\ 82 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3p~^{3}D_{3}$ &3+ &23213386 &23212496 &-891 &9.651E-12 &9.651E-12 &2p+3(3)3.3p+1(3)6\ 83 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3s~^{3}P_{0}$ &0+ &23218954 &23218728 &-227 &8.762E-14 &8.762E-14 &2s+1(1)1.2p+2(0)1.3s+1(1)0\ 84 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3p~^{1}P_{1}$ &1+ &23222776 &23222019 &-758 &1.549E-12 &1.549E-12 &2p+3(3)3.3p+1(3)2\ 85 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3p~^{3}P_{2}$ &2+ &23274896 &23274181 &-715 &1.733E-12 &1.733E-12 &2p+3(3)3.3p+1(3)4\ 86 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3p~^{5}D_{4}^{\circ}$ &4- &23407091 &23407083 &-8 &1.195E-11 &1.195E-11 &2s+1(1)1.2p+2(4)5.3p+1(3)8\ 87 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3p~^{3}D_{3}^{\circ}$ &3- &23413650 &23413685 &35 &4.956E-14 &4.956E-14 &2s+1(1)1.2p+2(4)5.3p+1(3)6\ 88 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3p~^{3}P_{1}^{\circ}$ &1- &23444458 &23444653 &195 &4.504E-14 &4.504E-14 &2s+1(1)1.2p+2(4)5.3p+1(3)2\ 89 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3p~^{1}S_{0}$ &0+ &23492292 &23492568 &276 &7.453E-13 &7.453E-13 &2p+3(3)3.3p+1(3)0\ 90 &$2s~^{2}S\,2p^{4}(^{1}S)~^{2}S\,3p~^{3}P_{0}^{\circ}$ &0- &23537382 &23537683 &301 &2.573E-13 &2.573E-13 &2s+1(1)1.2p+2(0)1.3p-1(1)0\ 91 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3p~^{3}D_{2}^{\circ}$ &2- &23541715 &23542198 &483 &1.184E-13 &1.184E-13 &2s+1(1)1.2p+2(4)5.3p+1(3)4\ 92 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3p~^{5}D_{1}^{\circ}$ &1- &23548591 &23549121 &531 &4.606E-14 &4.606E-14 &2s+1(1)1.2p+2(4)3.3p-1(1)2\ 93 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3p~^{3}S_{1}^{\circ}$ &1- &23564212 &23564591 &379 &3.439E-14 &3.439E-14 &2s+1(1)1.2p+2(0)1.3p-1(1)2\ 94 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3p~^{3}P_{2}^{\circ}$ &2- &23577362 &23578052 &690 &2.185E-14 &2.185E-14 &2s+1(1)1.2p+2(4)5.3p+1(3)4\ 95 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3s~^{5}P_{2}$ &2+ &23682878 &23681954 &-924 &1.228E-13 &1.228E-13 &2s+1(1)1.2p-1(1)0.2p+3(3)3.3s+1(1)4\ 96 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3s~^{3}P_{1}$ &1+ &23751261 &23750615 &-646 &6.640E-14 &6.640E-14 &2s+1(1)1.2p-1(1)0.2p+3(3)3.3s+1(1)2\ 97 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3d~^{3}P_{0}^{\circ}$ &0- &23753511 &23753847 &337 &7.562E-15 &7.562E-15 &2p+3(3)3.3d-1(3)0\ 98 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3p~^{1}S_{0}^{\circ}$ &0- &23779035 &23779438 &403 &2.892E-12 &2.892E-12 &2s+1(1)1.2p+2(4)3.3p+1(3)0\ 99 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3d~^{3}P_{1}^{\circ}$ &1- &23781592 &23781909 &316 &8.751E-15 &8.751E-15 &2p+3(3)3.3d-1(3)2\ 100 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3d~^{3}F_{4}^{\circ}$ &4- &23786871 &23787263 &392 &7.841E-11 &7.841E-11 &2p+3(3)3.3d+1(5)8\ 101 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3p~^{1}F_{3}^{\circ}$ &3- &23787902 &23788261 &358 &4.315E-14 &4.315E-14 &2s+1(1)1.2p+2(4)3.3p+1(3)6\ 102 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3p~^{5}S_{2}^{\circ}$ &2- &23791620 &23791878 &258 &1.948E-13 &1.948E-13 &2p+3(3)3.3d+1(5)4\ 103 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,3d~^{1}D_{2}^{\circ}$ &2- &23818232 &23818527 &295 &2.386E-13 &2.386E-13 &2p+3(3)3.3d+1(5)4\ 104 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3d~^{1}F_{3}^{\circ}$ &3- &23819697 &23820317 &620 &6.830E-15 &6.830E-15 &2p+3(3)3.3d-1(3)6\ 105 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3p~^{3}D_{1}^{\circ}$ &1- &23837218 &23837547 &329 &2.559E-14 &2.559E-14 &2s+1(1)1.2p+2(0)1.3p+1(3)2\ 106 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3p~^{5}D_{2}^{\circ}$ &2- &23843955 &23844431 &476 &1.309E-14 &1.309E-14 &2p+3(3)3.3d-1(3)4\ 107 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,3d~^{3}D_{2}^{\circ}$ &2- &23854364 &23854973 &608 &1.751E-14 &1.751E-14 &2p+3(3)3.3d-1(3)4\ 108 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3d~^{3}D_{3}^{\circ}$ &3- &23861693 &23862126 &433 &2.050E-13 &2.050E-13 &2p+3(3)3.3d+1(5)6\ 109 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3s~^{3}D_{3}$ &3+ &23862007 &23861188 &-820 &6.398E-14 &6.398E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)5.3s+1(1)6\ 110 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{5}D_{3}$ &3+ &23899088 &23898277 &-811 &2.026E-12 &2.026E-12 &2s+1(1)1.2p+2(4)5.3d-1(3)6\ 111 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{5}D_{2}$ &2+ &23903361 &23902537 &-824 &9.447E-13 &9.447E-13 &2s+1(1)1.2p+2(4)5.3d-1(3)4\ 112 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{5}D_{4}$ &4+ &23916369 &23915549 &-820 &1.326E-11 &1.326E-11 &2s+1(1)1.2p+2(4)5.3d-1(3)8\ 113 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{5}P_{1}$ &1+ &23924709 &23923905 &-804 &1.037E-13 &1.037E-13 &2s+1(1)1.2p+2(4)5.3d-1(3)2\ 114 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3p~^{3}D_{1}^{\circ}$ &1- &23936560 &23937388 &828 &3.786E-14 &3.786E-14 &2s+1(1)1.2p+2(4)3.3p+1(3)2\ 115 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3s~^{1}D_{2}$ &2+ &23941001 &23940596 &-405 &7.081E-14 &7.081E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)5.3s+1(1)4\ 116 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{5}F_{5}$ &5+ &23959870 &23959106 &-764 &1.837E-11 &1.837E-11 &2s+1(1)1.2p+2(4)5.3d+1(5)10\ 117 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{3}F_{4}$ &4+ &24002699 &24002145 &-554 &4.271E-12 &4.271E-12 &2s+1(1)1.2p+2(4)5.3d-1(3)8\ 118 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,3d~^{1}P_{1}^{\circ}$ &1- &24007125 &24007800 &676 &3.648E-15 &3.648E-15 &2p+3(3)3.3d+1(5)2\ 119 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{3}P_{0}$ &0+ &24043050 &24042440 &-611 &4.455E-14 &4.455E-14 &2s+1(1)1.2p+2(4)5.3d+1(5)0\ 120 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3s~^{3}P_{1}$ &1+ &24080223 &24079741 &-482 &9.645E-14 &9.645E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)1.3s+1(1)2\ 121 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{3}P_{1}$ &1+ &24098456 &24097870 &-586 &1.475E-14 &1.475E-14 &2s+1(1)1.2p+2(4)5.3d+1(5)2\ 122 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3s~^{3}P_{0}$ &0+ &24102121 &24101864 &-258 &6.571E-14 &6.571E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)1.3s+1(1)0\ 123 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3p~^{5}P_{1}^{\circ}$ &1- &24114366 &24114531 &164 &6.908E-14 &6.908E-14 &2s+1(1)1.2p-1(1)0.2p+3(3)3.3p-1(1)2\ 124 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{3}F_{3}$ &3+ &24120725 &24120266 &-459 &1.295E-14 &1.295E-14 &2s+1(1)1.2p+2(4)5.3d+1(5)6\ 125 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{3}D_{2}$ &2+ &24142448 &24141982 &-467 &9.583E-15 &9.583E-15 &2s+1(1)1.2p+2(4)5.3d+1(5)4\ 126 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3p~^{3}F_{2}^{\circ}$ &2- &24164804 &24165094 &290 &5.559E-14 &5.559E-14 &2s+1(1)1.2p-1(1)0.2p+3(3)3.3p-1(1)4\ 127 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3s~^{1}P_{1}$ &1+ &24179558 &24179348 &-210 &5.853E-14 &5.853E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)3.3s+1(1)2\ 128 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3s~^{3}P_{2}$ &2+ &24180325 &24180133 &-192 &9.200E-14 &9.200E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)3.3s+1(1)4\ 129 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3d~^{3}D_{1}$ &1+ &24278875 &24278397 &-478 &1.091E-13 &1.091E-13 &2s+1(1)1.2p+2(0)1.3d-1(3)2\ 130 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{5}F_{2}$ &2+ &24300148 &24299642 &-506 &5.128E-14 &5.128E-14 &2s+1(1)1.2p+2(0)1.3d-1(3)4\ 131 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3d~^{3}G_{3}$ &3+ &24309230 &24308888 &-342 &5.839E-13 &5.839E-13 &2s+1(1)1.2p+2(4)3.3d-1(3)6\ 132 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3p~^{3}F_{3}^{\circ}$ &3- &24313266 &24313602 &336 &7.613E-14 &7.613E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)5.3p-1(1)6\ 133 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{5}F_{1}$ &1+ &24324592 &24324195 &-397 &6.513E-13 &6.513E-13 &2s+1(1)1.2p+2(4)3.3d-1(3)2\ 134 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{5}P_{2}$ &2+ &24327534 &24327238 &-296 &7.392E-14 &7.392E-14 &2s+1(1)1.2p+2(4)3.3d-1(3)4\ 135 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3d~^{3}P_{0}$ &0+ &24360229 &24359906 &-323 &1.321E-14 &1.321E-14 &2s+1(1)1.2p+2(4)3.3d-1(3)0\ 136 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3p~^{5}P_{3}^{\circ}$ &3- &24360230 &24360243 &13 &8.943E-14 &8.943E-14 &2s+1(1)1.2p-1(1)0.2p+3(3)3.3p+1(3)6\ 137 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3d~^{1}G_{4}$ &4+ &24366985 &24366677 &-308 &2.316E-12 &2.316E-12 &2s+1(1)1.2p+2(4)3.3d+1(5)8\ 138 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{5}P_{3}$ &3+ &24379840 &24379443 &-397 &2.081E-13 &2.081E-13 &2s+1(1)1.2p+2(4)3.3d+1(5)6\ 139 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3p~^{3}P_{1}^{\circ}$ &1- &24386256 &24386326 &70 &4.343E-14 &4.343E-14 &2s+1(1)1.2p-1(1)0.2p+3(3)3.3p+1(3)2\ 140 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3p~^{3}D_{2}^{\circ}$ &2- &24394336 &24394538 &202 &5.661E-14 &5.661E-14 &2s+1(1)1.2p-1(1)0.2p+3(3)3.3p+1(3)4\ 141 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{3}D_{3}$ &3+ &24416056 &24415861 &-195 &8.853E-15 &8.853E-15 &2s+1(1)1.2p+2(4)3.3d+1(5)6\ 142 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{3}D_{1}$ &1+ &24434711 &24434324 &-387 &9.120E-15 &9.120E-15 &2s+1(1)1.2p+2(4)3.3d+1(5)2\ 143 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{3}F_{2}$ &2+ &24444562 &24444222 &-340 &1.619E-13 &1.619E-13 &2s+1(1)1.2p+2(0)1.3d+1(5)4\ 144 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3p~^{3}P_{2}^{\circ}$ &2- &24445502 &24446359 &857 &7.606E-14 &7.606E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)5.3p-1(1)4\ 145 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3p~^{3}P_{0}^{\circ}$ &0- &24489635 &24490187 &551 &3.792E-14 &3.792E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)1.3p-1(1)0\ 146 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3p~^{3}D_{1}^{\circ}$ &1- &24515929 &24516564 &635 &3.625E-14 &3.625E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)1.3p-1(1)2\ 147 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3d~^{3}D_{2}$ &2+ &24528295 &24528295 &0 &5.996E-15 &5.996E-15 &2s+1(1)1.2p+2(4)3.3d+1(5)4\ 148 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3p~^{3}F_{4}^{\circ}$ &4- &24538583 &24538751 &168 &4.262E-12 &4.262E-12 &2s+1(1)1.2p-1(1)2.2p+3(3)5.3p+1(3)8\ 149 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3p~^{3}P_{0}^{\circ}$ &0- &24545023 &24545705 &682 &3.004E-14 &3.004E-14 &2s+1(1)1.2p-1(1)0.2p+3(3)3.3p+1(3)0\ 150 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3p~^{1}D_{2}^{\circ}$ &2- &24562245 &24562530 &285 &5.495E-14 &5.495E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)5.3p+1(3)4\ 151 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3p~^{3}D_{3}^{\circ}$ &3- &24593856 &24594131 &275 &5.987E-14 &5.987E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)5.3p+1(3)6\ 152 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3p~^{1}D_{2}^{\circ}$ &2- &24606863 &24607631 &767 &3.471E-14 &3.471E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)3.3p-1(1)4\ 153 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3p~^{1}P_{1}^{\circ}$ &1- &24622010 &24622435 &425 &2.334E-14 &2.334E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)5.3p+1(3)2\ 154 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3p~^{3}P_{1}^{\circ}$ &1- &24737223 &24738642 &1420 &9.350E-14 &9.350E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)3.3p-1(1)2\ 155 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3p~^{3}D_{2}^{\circ}$ &2- &24747317 &24747707 &390 &5.444E-14 &5.444E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)1.3p+1(3)4\ 156 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3p~^{3}S_{1}^{\circ}$ &1- &24789575 &24790042 &467 &6.047E-14 &6.047E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)1.3p+1(3)2\ 157 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{5}D_{0}$ &0+ &24839947 &24839227 &-720 &3.581E-14 &3.581E-14 &2s+1(1)1.2p-1(1)0.2p+3(3)3.3d-1(3)0\ 158 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3p~^{3}D_{3}^{\circ}$ &3- &24845306 &24846013 &708 &4.998E-14 &4.998E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)3.3p+1(3)6\ 159 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3p~^{3}P_{2}^{\circ}$ &2- &24845945 &24846550 &605 &8.525E-14 &8.525E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)3.3p+1(3)4\ 160 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{5}D_{1}$ &1+ &24865894 &24865164 &-730 &4.577E-14 &4.577E-14 &2s+1(1)1.2p-1(1)0.2p+3(3)3.3d-1(3)2\ 161 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3d~^{3}F_{2}$ &2+ &24910092 &24909473 &-620 &4.024E-14 &4.024E-14 &2s+1(1)1.2p-1(1)0.2p+3(3)3.3d-1(3)4\ 162 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{5}F_{4}$ &4+ &24910538 &24909840 &-698 &4.430E-12 &4.430E-12 &2s+1(1)1.2p-1(1)0.2p+3(3)3.3d+1(5)8\ 163 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3p~^{3}P_{0}^{\circ}$ &0- &24919786 &24920725 &939 &1.501E-12 &1.501E-12 &2s+1(1)1.2p-1(1)2.2p+3(3)3.3p+1(3)0\ 164 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{5}F_{3}$ &3+ &24929143 &24928562 &-580 &1.671E-14 &1.671E-14 &2s+1(1)1.2p-1(1)0.2p+3(3)3.3d-1(3)6\ 165 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3d~^{3}P_{2}$ &2+ &24974082 &24973434 &-648 &6.415E-14 &6.415E-14 &2s+1(1)1.2p-1(1)0.2p+3(3)3.3d+1(5)4\ 166 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3p~^{1}P_{1}^{\circ}$ &1- &24975613 &24976824 &1211 &1.530E-13 &1.530E-13 &2s+1(1)1.2p-1(1)2.2p+3(3)3.3p+1(3)2\ 167 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3d~^{3}F_{3}$ &3+ &24981008 &24980378 &-629 &1.460E-12 &1.460E-12 &2s+1(1)1.2p-1(1)0.2p+3(3)3.3d+1(5)6\ 168 &$2s~^{2}S\,2p^{4}(^{1}S)~^{2}S\,3s~^{3}S_{1}$ &1+ &24991737 &24990877 &-860 &2.985E-14 &2.985E-14 &2s+1(1)1.3s+1(1)2\ 169 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3d~^{3}G_{4}$ &4+ &25065955 &25065520 &-435 &3.539E-12 &3.539E-12 &2s+1(1)1.2p-1(1)2.2p+3(3)5.3d-1(3)8\ 170 &$2s~^{2}S\,2p^{4}(^{1}S)~^{2}S\,3s~^{1}S_{0}$ &0+ &25074897 &25074460 &-437 &7.952E-14 &7.952E-14 &2s+1(1)1.3s+1(1)0\ 171 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3d~^{3}G_{5}$ &5+ &25084575 &25084065 &-510 &5.984E-12 &5.984E-12 &2s+1(1)1.2p-1(1)2.2p+3(3)5.3d+1(5)10\ 172 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3d~^{3}S_{1}$ &1+ &25091782 &25091097 &-685 &7.707E-15 &7.707E-15 &2s+1(1)1.2p-1(1)2.2p+3(3)5.3d-1(3)2\ 173 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3d~^{3}P_{1}$ &1+ &25116378 &25115856 &-523 &5.358E-15 &5.358E-15 &2s+1(1)1.2p-1(1)2.2p+3(3)5.3d-1(3)2\ 174 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3d~^{3}P_{2}$ &2+ &25117330 &25116815 &-515 &7.269E-15 &7.269E-15 &2s+1(1)1.2p-1(1)2.2p+3(3)5.3d-1(3)4\ 175 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3d~^{3}D_{3}$ &3+ &25126659 &25126183 &-476 &1.076E-14 &1.076E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)5.3d-1(3)6\ 176 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3d~^{3}F_{4}$ &4+ &25169919 &25169489 &-430 &3.166E-12 &3.166E-12 &2s+1(1)1.2p-1(1)2.2p+3(3)5.3d+1(5)8\ 177 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3d~^{1}F_{3}$ &3+ &25178112 &25177723 &-389 &3.888E-14 &3.888E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)5.3d+1(5)6\ 178 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3d~^{1}D_{2}$ &2+ &25217779 &25217430 &-349 &1.462E-14 &1.462E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)5.3d+1(5)4\ 179 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3d~^{1}P_{1}$ &1+ &25237223 &25236908 &-314 &1.374E-14 &1.374E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)1.3d-1(3)2\ 180 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3d~^{3}D_{1}$ &1+ &25289570 &25289238 &-331 &1.204E-14 &1.204E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)5.3d+1(5)2\ 181 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3d~^{3}F_{2}$ &2+ &25299701 &25299519 &-182 &3.130E-14 &3.130E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)1.3d-1(3)4\ 182 &$2s~^{2}S\,2p^{4}(^{1}D)~^{2}D\,3d~^{1}S_{0}$ &0+ &25304627 &25304509 &-118 &4.820E-15 &4.820E-15 &2s+1(1)1.2p-1(1)2.2p+3(3)5.3d+1(5)0\ 183 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3d~^{1}F_{3}$ &3+ &25320879 &25320621 &-258 &1.797E-13 &1.797E-13 &2s+1(1)1.2p-1(1)2.2p+3(3)1.3d+1(5)6\ 184 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3d~^{3}F_{3}$ &3+ &25361522 &25361601 &79 &5.420E-13 &5.420E-13 &2s+1(1)1.2p-1(1)2.2p+3(3)3.3d-1(3)6\ 185 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3d~^{3}P_{2}$ &2+ &25361770 &25361604 &-166 &2.257E-14 &2.257E-14 &2s+1(1)1.2p-1(1)2.2p+3(3)1.3d+1(5)4\ 186 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3d~^{3}D_{2}$ &2+ &25421749 &25421715 &-34 &8.859E-15 &8.859E-15 &2s+1(1)1.2p-1(1)2.2p+3(3)3.3d-1(3)4\ 187 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3d~^{3}D_{3}$ &3+ &25422745 &25422670 &-75 &8.206E-13 &8.206E-13 &2s+1(1)1.2p-1(1)2.2p+3(3)3.3d+1(5)6\ 188 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3d~^{3}F_{4}$ &4+ &25425477 &25425543 &65 &1.442E-12 &1.442E-12 &2s+1(1)1.2p-1(1)2.2p+3(3)3.3d+1(5)8\ 189 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3d~^{3}P_{1}$ &1+ &25442114 &25442087 &-28 &7.542E-15 &7.542E-15 &2s+1(1)1.2p-1(1)2.2p+3(3)3.3d-1(3)2\ 190 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3d~^{3}P_{0}$ &0+ &25450323 &25450523 &199 &6.246E-15 &6.246E-15 &2s+1(1)1.2p-1(1)2.2p+3(3)3.3d-1(3)0\ 191 &$2s~^{2}S\,2p^{4}(^{3}P)~^{4}P\,3p~^{5}D_{0}^{\circ}$ &0- &25502918 &25503446 &528 &2.076E-12 &2.076E-12 &2s+1(1)1.3p-1(1)0\ 192 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3d~^{1}D_{2}$ &2+ &25515508 &25515873 &365 &5.021E-15 &5.021E-15 &2s+1(1)1.2p-1(1)2.2p+3(3)3.3d+1(5)4\ 193 &$2s~^{2}S\,2p^{4}(^{3}P)~^{2}P\,3d~^{1}P_{1}$ &1+ &25521091 &25521081 &-10 &4.531E-15 &4.531E-15 &2s+1(1)1.2p-1(1)2.2p+3(3)3.3d+1(5)2\ 194 &$2s~^{2}S\,2p^{4}(^{1}S)~^{2}S\,3p~^{3}P_{1}^{\circ}$ &1- &25524464 &25525228 &764 &3.352E-14 &3.352E-14 &2s+1(1)1.3p-1(1)2\ 195 &$2p^{5}(^{(}5)~^{2}P\,3s~^{3}P_{2}^{\circ}$ &2- &25603295 &25604088 &793 &1.042E-12 &1.042E-12 &2p+3(3)3.3s+1(1)4\ 196 &$2p^{5}(^{(}5)~^{2}P\,3s~^{1}P_{1}^{\circ}$ &1- &25646363 &25647503 &1140 &7.074E-14 &7.074E-14 &2p+3(3)3.3s+1(1)2\ 197 &$2s~^{2}S\,2p^{4}(^{1}S)~^{2}S\,3p~^{3}P_{2}^{\circ}$ &2- &25691236 &25691517 &280 &4.416E-13 &4.416E-13 &2s+1(1)1.3p+1(3)4\ 198 &$2s~^{2}S\,2p^{4}(^{1}S)~^{2}S\,3p~^{1}P_{1}^{\circ}$ &1- &25700199 &25700387 &189 &3.201E-14 &3.201E-14 &2s+1(1)1.3p+1(3)2\ 199 &$2p^{5}(^{(}5)~^{2}P\,3p~^{3}S_{1}$ &1+ &26015423 &26015552 &129 &3.997E-14 &3.997E-14 &2p+3(3)3.3p-1(1)2\ 200 &$2p^{5}(^{(}5)~^{2}P\,3p~^{3}D_{2}$ &2+ &26034919 &26035134 &214 &4.112E-14 &4.112E-14 &2p+3(3)3.3p-1(1)4\ 201 &$2s~^{2}S\,2p^{4}(^{1}S)~^{2}S\,3d~^{3}D_{3}$ &3+ &26235340 &26234920 &-420 &1.138E-13 &1.138E-13 &2s+1(1)1.3d+1(5)6\ 202 &$2s~^{2}S\,2p^{4}(^{1}S)~^{2}S\,3d~^{3}D_{1}$ &1+ &26248454 &26247964 &-490 &6.585E-15 &6.585E-15 &2s+1(1)1.3d-1(3)2\ 203 &$2p^{5}(^{(}5)~^{2}P\,3p~^{1}P_{1}$ &1+ &26258920 &26258969 &49 &3.247E-14 &3.247E-14 &2p+3(3)3.3p+1(3)2\ 204 &$2s~^{2}S\,2p^{4}(^{1}S)~^{2}S\,3d~^{3}D_{2}$ &2+ &26265339 &26265073 &-266 &8.657E-15 &8.657E-15 &2s+1(1)1.3d-1(3)4\ 205 &$2p^{5}(^{(}5)~^{2}P\,3p~^{3}D_{3}$ &3+ &26265848 &26265747 &-100 &5.467E-14 &5.467E-14 &2p+3(3)3.3p+1(3)6\ 206 &$2p^{5}(^{(}5)~^{2}P\,3p~^{3}P_{2}$ &2+ &26297747 &26297612 &-135 &2.772E-14 &2.772E-14 &2p+3(3)3.3p+1(3)4\ 207 &$2s~^{2}S\,2p^{4}(^{1}S)~^{2}S\,3d~^{1}D_{2}$ &2+ &26317458 &26317333 &-125 &5.496E-14 &5.496E-14 &2p+3(3)3.3p+1(3)4\ 208 &$2p^{5}(^{(}5)~^{2}P\,3p~^{3}P_{0}$ &0+ &26486871 &26488001 &1130 &3.340E-14 &3.340E-14 &2p+3(3)3.3p+1(3)0\ 209 &$2p^{5}(^{(}5)~^{2}P\,3s~^{3}P_{0}^{\circ}$ &0- &26510588 &26511171 &584 &9.894E-13 &9.894E-13 &2p-1(1)1.3s+1(1)0\ 210 &$2p^{5}(^{(}5)~^{2}P\,3s~^{3}P_{1}^{\circ}$ &1- &26533438 &26534202 &764 &8.035E-14 &8.035E-14 &2p-1(1)1.3s+1(1)2\ 211 &$2p^{5}(^{(}5)~^{2}P\,3d~^{3}P_{0}^{\circ}$ &0- &26716298 &26717081 &783 &7.803E-13 &7.803E-13 &2p+3(3)3.3d-1(3)0\ 212 &$2p^{5}(^{(}5)~^{2}P\,3d~^{3}P_{1}^{\circ}$ &1- &26744101 &26744956 &854 &7.735E-13 &7.735E-13 &2p+3(3)3.3d-1(3)2\ 213 &$2p^{5}(^{(}5)~^{2}P\,3d~^{3}F_{3}^{\circ}$ &3- &26769316 &26770513 &1197 &7.744E-13 &7.744E-13 &2p+3(3)3.3d-1(3)6\ 214 &$2p^{5}(^{(}5)~^{2}P\,3d~^{3}D_{2}^{\circ}$ &2- &26791462 &26792495 &1034 &7.673E-13 &7.673E-13 &2p+3(3)3.3d-1(3)4\ 215 &$2p^{5}(^{(}5)~^{2}P\,3d~^{3}F_{4}^{\circ}$ &4- &26802700 &26803792 &1092 &1.437E-12 &1.437E-12 &2p+3(3)3.3d+1(5)8\ 216 &$2p^{5}(^{(}5)~^{2}P\,3d~^{1}D_{2}^{\circ}$ &2- &26832698 &26833736 &1038 &1.122E-12 &1.122E-12 &2p+3(3)3.3d+1(5)4\ 217 &$2p^{5}(^{(}5)~^{2}P\,3d~^{3}D_{3}^{\circ}$ &3- &26872028 &26873194 &1167 &9.005E-13 &9.005E-13 &2p+3(3)3.3d+1(5)6\ 218 &$2p^{5}(^{(}5)~^{2}P\,3p~^{3}D_{1}$ &1+ &26926428 &26926304 &-124 &3.609E-14 &3.609E-14 &2p-1(1)1.3p-1(1)2\ 219 &$2p^{5}(^{(}5)~^{2}P\,3d~^{1}P_{1}^{\circ}$ &1- &27001375 &27002884 &1509 &5.012E-15 &5.012E-15 &2p+3(3)3.3d+1(5)2\ 220 &$2p^{5}(^{(}5)~^{2}P\,3p~^{1}S_{0}$ &0+ &27126800 &27128336 &1536 &2.454E-14 &2.454E-14 &2p-1(1)1.3p-1(1)0\ 221 &$2p^{5}(^{(}5)~^{2}P\,3p~^{3}P_{1}$ &1+ &27174101 &27173903 &-199 &3.414E-14 &3.414E-14 &2p-1(1)1.3p+1(3)2\ 222 &$2p^{5}(^{(}5)~^{2}P\,3p~^{1}D_{2}$ &2+ &27184333 &27184154 &-178 &3.657E-14 &3.657E-14 &2p-1(1)1.3p+1(3)4\ 223 &$2p^{5}(^{(}5)~^{2}P\,3d~^{3}F_{2}^{\circ}$ &2- &27671201 &27672066 &865 &1.098E-12 &1.098E-12 &2p-1(1)1.3d-1(3)4\ 224 &$2p^{5}(^{(}5)~^{2}P\,3d~^{3}P_{2}^{\circ}$ &2- &27737195 &27737913 &718 &9.126E-13 &9.126E-13 &2p-1(1)1.3d+1(5)4\ 225 &$2p^{5}(^{(}5)~^{2}P\,3d~^{1}F_{3}^{\circ}$ &3- &27748433 &27749343 &910 &8.899E-13 &8.899E-13 &2p-1(1)1.3d+1(5)6\ 226 &$2p^{5}(^{(}5)~^{2}P\,3d~^{3}D_{1}^{\circ}$ &1- &27799974 &27801251 &1277 &5.219E-15 &5.219E-15 &2p-1(1)1.3d-1(3)2\ 227 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4s~^{3}D_{2}^{\circ}$ &2- &28153615 &28154924 &1309 &1.604E-13 &1.604E-13 &2p+1(3)3.4s+1(1)4\ 228 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4s~^{3}D_{1}^{\circ}$ &1- &28168810 &28170171 &1361 &9.480E-14 &9.480E-14 &2p+1(3)3.4s+1(1)2\ 229 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4p~^{3}D_{1}$ &1+ &28336449 &28336902 &453 &1.793E-13 &1.793E-13 &2p+1(3)3.4p-1(1)2\ 230 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4p~^{5}P_{2}$ &2+ &28340331 &28340699 &368 &1.792E-13 &1.792E-13 &2p+1(3)3.4p-1(1)4\ 231 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4p~^{1}F_{3}$ &3+ &28438542 &28438722 &180 &2.046E-13 &2.046E-13 &2p+1(3)3.4p+1(3)6\ 232 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4p~^{3}P_{1}$ &1+ &28441026 &28441362 &336 &1.962E-13 &1.962E-13 &2p+1(3)3.4p+1(3)2\ 233 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4p~^{3}P_{2}$ &2+ &28476873 &28477132 &259 &2.047E-13 &2.047E-13 &2p+1(3)3.4p+1(3)4\ 234 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4p~^{3}P_{0}$ &0+ &28510339 &28510933 &593 &2.111E-13 &2.111E-13 &2p+1(3)3.4p+1(3)0\ 235 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,4d~^{3}D_{2}^{\circ}$ &2- &28629526 &28630935 &1409 &8.362E-14 &8.362E-14 &2p+1(3)3.4d-1(3)4\ 236 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4d~^{3}G_{3}^{\circ}$ &3- &28638159 &28639415 &1256 &9.288E-14 &9.288E-14 &2p+1(3)3.4d-1(3)6\ 237 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4d~^{3}D_{1}^{\circ}$ &1- &28643060 &28644207 &1147 &5.376E-14 &5.376E-14 &2p+1(3)3.4d-1(3)2\ 238 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4d~^{3}P_{0}^{\circ}$ &0- &28644326 &28645366 &1040 &9.697E-14 &9.697E-14 &2p+1(3)3.4d-1(3)0\ 239 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,4d~^{3}F_{4}^{\circ}$ &4- &28660718 &28661851 &1133 &9.592E-14 &9.592E-14 &2p+1(3)3.4d+1(5)8\ 240 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,4d~^{1}D_{2}^{\circ}$ &2- &28678042 &28679238 &1197 &2.516E-14 &2.516E-14 &2p+1(3)3.4d+1(5)4\ 241 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,4d~^{1}F_{3}^{\circ}$ &3- &28706336 &28707602 &1265 &1.272E-14 &1.272E-14 &2p+1(3)3.4d+1(5)6\ 242 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,4d~^{1}P_{1}^{\circ}$ &1- &28717020 &28718330 &1310 &1.780E-14 &1.780E-14 &2p+1(3)3.4d+1(5)2\ 243 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4f~^{3}G_{3}$ &3+ &28780086 &28780227 &141 &4.518E-14 &4.518E-14 &2p+1(3)3.4f-1(5)6\ 244 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4f~^{3}F_{2}$ &2+ &28786709 &28786770 &61 &4.524E-14 &4.524E-14 &2p+1(3)3.4f-1(5)4\ 245 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4f~^{5}F_{4}$ &4+ &28786803 &28786933 &130 &4.559E-14 &4.559E-14 &2p+1(3)3.4f-1(5)8\ 246 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,4f~^{3}D_{1}$ &1+ &28795346 &28795286 &-60 &4.525E-14 &4.525E-14 &2p+1(3)3.4f-1(5)2\ 247 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4f~^{3}F_{4}$ &4+ &28796888 &28797126 &238 &4.642E-14 &4.642E-14 &2p+1(3)3.4f+1(7)8\ 248 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4f~^{3}D_{3}$ &3+ &28797759 &28797875 &116 &4.566E-14 &4.566E-14 &2p+1(3)3.4f+1(7)6\ 249 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4f~^{1}H_{5}$ &5+ &28798780 &28798893 &113 &4.597E-14 &4.597E-14 &2p+1(3)3.4f+1(7)10\ 250 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,4f~^{1}D_{2}$ &2+ &28809948 &28810014 &66 &4.614E-14 &4.614E-14 &2p+1(3)3.4f+1(7)4\ 251 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4s~^{5}S_{2}^{\circ}$ &2- &28948743 &28949863 &1120 &1.665E-13 &1.665E-13 &2p-1(1)1.2p+2(4)3.4s+1(1)4\ 252 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4s~^{3}S_{1}^{\circ}$ &1- &28962444 &28963605 &1162 &9.330E-14 &9.330E-14 &2p-1(1)1.2p+2(4)3.4s+1(1)2\ 253 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4s~^{3}D_{3}^{\circ}$ &3- &29077316 &29078502 &1187 &1.656E-13 &1.656E-13 &2p-1(1)1.2p+2(4)5.4s+1(1)6\ 254 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4s~^{1}D_{2}^{\circ}$ &2- &29087497 &29088746 &1248 &9.912E-14 &9.912E-14 &2p-1(1)1.2p+2(4)5.4s+1(1)4\ 255 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4p~^{5}P_{1}$ &1+ &29127260 &29127537 &276 &1.762E-13 &1.762E-13 &2p-1(1)1.2p+2(4)3.4p-1(1)2\ 256 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4p~^{3}F_{2}$ &2+ &29149379 &29149613 &234 &1.785E-13 &1.785E-13 &2p-1(1)1.2p+2(4)3.4p-1(1)4\ 257 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4p~^{5}P_{3}$ &3+ &29231835 &29231824 &-10 &2.044E-13 &2.044E-13 &2p-1(1)1.2p+2(4)3.4p+1(3)6\ 258 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4p~^{3}D_{2}$ &2+ &29236428 &29236784 &356 &2.007E-13 &2.007E-13 &2p-1(1)1.2p+2(4)3.4p+1(3)4\ 259 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4p~^{3}P_{0}$ &0+ &29236687 &29236879 &192 &1.948E-13 &1.948E-13 &2p-1(1)1.2p+2(4)3.4p+1(3)0\ 260 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,4s~^{3}P_{0}^{\circ}$ &0- &29246189 &29248331 &2142 &1.447E-13 &1.447E-13 &2p-1(1)1.2p+2(0)1.4s+1(1)0\ 261 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,4s~^{3}P_{1}^{\circ}$ &1- &29251428 &29253504 &2076 &1.204E-13 &1.204E-13 &2p-1(1)1.2p+2(0)1.4s+1(1)2\ 262 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4p~^{3}F_{3}$ &3+ &29260375 &29260486 &110 &1.763E-13 &1.763E-13 &2p-1(1)1.2p+2(4)5.4p-1(1)6\ 263 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4p~^{3}P_{1}$ &1+ &29279033 &29279325 &293 &2.033E-13 &2.033E-13 &2p-1(1)1.2p+2(4)3.4p+1(3)2\ 264 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4p~^{3}P_{2}$ &2+ &29293177 &29293525 &348 &1.838E-13 &1.838E-13 &2p-1(1)1.2p+2(4)5.4p-1(1)4\ 265 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4p~^{3}D_{3}$ &3+ &29359017 &29359139 &121 &2.000E-13 &2.000E-13 &2p-1(1)1.2p+2(4)5.4p+1(3)6\ 266 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4p~^{3}F_{4}$ &4+ &29361214 &29361255 &42 &2.046E-13 &2.046E-13 &2p-1(1)1.2p+2(4)5.4p+1(3)8\ 267 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4p~^{1}P_{1}$ &1+ &29373041 &29373924 &883 &2.004E-13 &2.004E-13 &2p-1(1)1.2p+2(4)5.4p+1(3)2\ 268 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4p~^{1}D_{2}$ &2+ &29418608 &29418956 &348 &2.106E-13 &2.106E-13 &2p-1(1)1.2p+2(4)5.4p+1(3)4\ 269 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4d~^{5}D_{1}^{\circ}$ &1- &29430093 &29431197 &1104 &6.090E-14 &6.090E-14 &2p-1(1)1.2p+2(4)3.4d-1(3)2\ 270 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,4p~^{3}D_{1}$ &1+ &29430710 &29431558 &847 &1.780E-13 &1.780E-13 &2p-1(1)1.2p+2(0)1.4p-1(1)2\ 271 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4d~^{5}D_{0}^{\circ}$ &0- &29434998 &29436071 &1072 &5.723E-14 &5.723E-14 &2p-1(1)1.2p+2(4)3.4d-1(3)0\ 272 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4d~^{5}D_{2}^{\circ}$ &2- &29436242 &29437390 &1149 &4.302E-14 &4.302E-14 &2p-1(1)1.2p+2(4)3.4d-1(3)4\ 273 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4d~^{5}D_{3}^{\circ}$ &3- &29449321 &29450457 &1136 &3.313E-14 &3.313E-14 &2p-1(1)1.2p+2(4)3.4d-1(3)6\ 274 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4d~^{5}D_{4}^{\circ}$ &4- &29453993 &29454976 &983 &9.547E-14 &9.547E-14 &2p-1(1)1.2p+2(4)3.4d+1(5)8\ 275 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4d~^{3}D_{3}^{\circ}$ &3- &29460370 &29461519 &1148 &4.980E-14 &4.980E-14 &2p-1(1)1.2p+2(4)3.4d+1(5)6\ 276 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,4p~^{3}P_{0}$ &0+ &29482229 &29483516 &1286 &1.863E-13 &1.863E-13 &2p-1(1)1.2p+2(0)1.4p-1(1)0\ 277 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4d~^{3}D_{1}^{\circ}$ &1- &29490006 &29491077 &1070 &2.294E-14 &2.294E-14 &2p-1(1)1.2p+2(4)3.4d+1(5)2\ 278 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4d~^{3}D_{2}^{\circ}$ &2- &29505708 &29506933 &1225 &1.538E-14 &1.538E-14 &2p-1(1)1.2p+2(4)3.4d+1(5)4\ 279 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,4p~^{3}P_{1}$ &1+ &29539393 &29540355 &961 &2.040E-13 &2.040E-13 &2p-1(1)1.2p+2(0)1.4p+1(3)2\ 280 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,4p~^{3}D_{2}$ &2+ &29544345 &29545162 &817 &2.026E-13 &2.026E-13 &2p-1(1)1.2p+2(0)1.4p+1(3)4\ 281 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4d~^{3}G_{4}^{\circ}$ &4- &29559189 &29560168 &979 &9.638E-14 &9.638E-14 &2p-1(1)1.2p+2(4)5.4d-1(3)8\ 282 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4d~^{3}D_{3}^{\circ}$ &3- &29571520 &29572646 &1126 &2.633E-14 &2.633E-14 &2p-1(1)1.2p+2(4)5.4d-1(3)6\ 283 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4d~^{3}F_{4}^{\circ}$ &4- &29580866 &29581848 &982 &9.532E-14 &9.532E-14 &2p-1(1)1.2p+2(4)5.4d+1(5)8\ 284 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4f~^{5}F_{3}$ &3+ &29581384 &29581408 &24 &4.531E-14 &4.531E-14 &2p-1(1)1.2p+2(4)3.4f-1(5)6\ 285 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4f~^{5}F_{2}$ &2+ &29582534 &29582488 &-46 &4.493E-14 &4.493E-14 &2p-1(1)1.2p+2(4)3.4f-1(5)4\ 286 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4d~^{3}D_{2}^{\circ}$ &2- &29583203 &29584426 &1223 &1.759E-14 &1.759E-14 &2p-1(1)1.2p+2(4)5.4d-1(3)4\ 287 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4f~^{3}H_{4}$ &4+ &29583489 &29583560 &71 &4.579E-14 &4.579E-14 &2p-1(1)1.2p+2(4)3.4f-1(5)8\ 288 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4d~^{3}G_{5}^{\circ}$ &5- &29584086 &29585038 &952 &9.558E-14 &9.558E-14 &2p-1(1)1.2p+2(4)5.4d+1(5)10\ 289 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4f~^{5}F_{1}$ &1+ &29586615 &29586524 &-91 &4.489E-14 &4.489E-14 &2p-1(1)1.2p+2(4)3.4f-1(5)2\ 290 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4d~^{3}P_{1}^{\circ}$ &1- &29591177 &29592433 &1256 &1.725E-14 &1.725E-14 &2p-1(1)1.2p+2(4)5.4d-1(3)2\ 291 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4f~^{1}G_{4}$ &4+ &29591930 &29591957 &26 &4.556E-14 &4.556E-14 &2p-1(1)1.2p+2(4)3.4f+1(7)8\ 292 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4f~^{5}F_{5}$ &5+ &29592653 &29592677 &24 &4.574E-14 &4.574E-14 &2p-1(1)1.2p+2(4)3.4f+1(7)10\ 293 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4d~^{1}S_{0}^{\circ}$ &0- &29594969 &29596178 &1209 &3.003E-14 &3.003E-14 &2p-1(1)1.2p+2(4)5.4d+1(5)0\ 294 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4f~^{3}F_{3}$ &3+ &29597550 &29597606 &55 &4.586E-14 &4.586E-14 &2p-1(1)1.2p+2(4)3.4f+1(7)6\ 295 &$2s^{2}\,2p^{3}(^{4}S)~^{4}S\,4f~^{3}F_{2}$ &2+ &29600693 &29600687 &-6 &4.569E-14 &4.569E-14 &2p-1(1)1.2p+2(4)3.4f+1(7)4\ 296 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4d~^{3}S_{1}^{\circ}$ &1- &29617671 &29618902 &1230 &1.601E-14 &1.601E-14 &2p-1(1)1.2p+2(4)5.4d+1(5)2\ 297 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4d~^{1}D_{2}^{\circ}$ &2- &29629837 &29631117 &1280 &1.557E-14 &1.557E-14 &2p-1(1)1.2p+2(4)5.4d+1(5)4\ 298 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4d~^{1}F_{3}^{\circ}$ &3- &29630987 &29632140 &1152 &1.504E-14 &1.504E-14 &2p-1(1)1.2p+2(4)5.4d+1(5)6\ 299 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4f~^{3}H_{5}$ &5+ &29707410 &29707484 &74 &4.578E-14 &4.578E-14 &2p-1(1)1.2p+2(4)5.4f-1(5)10\ 300 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4f~^{3}G_{4}$ &4+ &29710182 &29710158 &-25 &4.539E-14 &4.539E-14 &2p-1(1)1.2p+2(4)5.4f-1(5)8\ 301 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4f~^{3}F_{3}$ &3+ &29711323 &29711234 &-89 &4.505E-14 &4.505E-14 &2p-1(1)1.2p+2(4)5.4f-1(5)6\ 302 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4f~^{3}D_{2}$ &2+ &29713264 &29713141 &-124 &4.483E-14 &4.483E-14 &2p-1(1)1.2p+2(4)5.4f-1(5)4\ 303 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4f~^{3}D_{1}$ &1+ &29715228 &29715083 &-145 &4.476E-14 &4.476E-14 &2p-1(1)1.2p+2(4)5.4f-1(5)2\ 304 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4f~^{3}P_{0}$ &0+ &29716529 &29716377 &-152 &4.480E-14 &4.480E-14 &2p-1(1)1.2p+2(4)5.4f-1(5)0\ 305 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4f~^{3}H_{6}$ &6+ &29718692 &29718763 &71 &4.603E-14 &4.603E-14 &2p-1(1)1.2p+2(4)5.4f+1(7)12\ 306 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4f~^{3}G_{5}$ &5+ &29719620 &29719596 &-23 &4.557E-14 &4.557E-14 &2p-1(1)1.2p+2(4)5.4f+1(7)10\ 307 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4f~^{3}F_{4}$ &4+ &29723846 &29723835 &-11 &4.578E-14 &4.578E-14 &2p-1(1)1.2p+2(4)5.4f+1(7)8\ 308 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4f~^{1}P_{1}$ &1+ &29725929 &29725774 &-155 &4.489E-14 &4.489E-14 &2p-1(1)1.2p+2(4)5.4f+1(7)2\ 309 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4f~^{1}F_{3}$ &3+ &29726441 &29726399 &-41 &4.562E-14 &4.562E-14 &2p-1(1)1.2p+2(4)5.4f+1(7)6\ 310 &$2s^{2}\,2p^{3}(^{2}D)~^{2}D\,4f~^{1}D_{2}$ &2+ &29726863 &29726771 &-92 &4.526E-14 &4.526E-14 &2p-1(1)1.2p+2(4)5.4f+1(7)4\ 311 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,4d~^{3}F_{2}^{\circ}$ &2- &29731200 &29732936 &1736 &5.399E-14 &5.399E-14 &2p-1(1)1.2p+2(0)1.4d-1(3)4\ 312 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,4d~^{3}P_{2}^{\circ}$ &2- &29765678 &29767493 &1815 &2.792E-14 &2.792E-14 &2p-1(1)1.2p+2(0)1.4d+1(5)4\ 313 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,4d~^{3}D_{1}^{\circ}$ &1- &29768117 &29770185 &2068 &1.492E-14 &1.492E-14 &2p-1(1)1.2p+2(0)1.4d-1(3)2\ 314 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,4d~^{3}F_{3}^{\circ}$ &3- &29771761 &29773468 &1707 &2.829E-14 &2.829E-14 &2p-1(1)1.2p+2(0)1.4d+1(5)6\ 315 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,4f~^{3}G_{3}$ &3+ &29878769 &29879390 &620 &4.609E-14 &4.609E-14 &2p-1(1)1.2p+2(0)1.4f-1(5)6\ 316 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,4f~^{3}F_{2}$ &2+ &29882953 &29883528 &575 &4.610E-14 &4.610E-14 &2p-1(1)1.2p+2(0)1.4f-1(5)4\ 317 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,4f~^{3}G_{4}$ &4+ &29891679 &29892359 &680 &4.660E-14 &4.660E-14 &2p-1(1)1.2p+2(0)1.4f+1(7)8\ 318 &$2s^{2}\,2p^{3}(^{2}P)~^{2}P\,4f~^{3}D_{3}$ &3+ &29892128 &29892669 &541 &4.609E-14 &4.609E-14 &2p-1(1)1.2p+2(0)1.4f+1(7)6\ 1. $^a$ The number at the end or is 2$J$. 2. $^b$ $s+ = s_{1/2}$, $p- = p_{1/2}$, $p+ = p_{3/2}$, $d- = d_{3/2}$, $d+ = d_{5/2}$, $f- = f_{5/2}$, and $f+ = f_{7/2}$. 3. $^c$ The number after $\pm$ is the occupation number of the corresponding sub-shell. For example, the $jj$-coupled CSF of level 9 is $2s_{1/2} 2p_{1/2}2p_{3/2}^{4}$. The full sub-shell $2p_{3/2}$ is omitted. [llcccccccccccc]{} \ $i$ & $j$& Type & $\lambda_{\rm MBPT}$ & $A_{\rm MBPT}$ & $gf_{\rm MBPT}$ & $S_{\rm MBPT}$ & $\rm BF_{\rm MBPT}$& $\lambda_{\rm MCDF}$ & $A_{\rm MCDF}$ & $gf_{\rm MCDF}$ & $S_{\rm MCDF}$ & $\rm BF_{\rm MCDF}$& Acc.\ \ $i$ & $j$ Type & $\lambda_{\rm MBPT}$ & $A_{\rm MBPT}$ & $gf_{\rm MBPT}$ & $S_{\rm MBPT}$ & $\rm BF_{\rm MBPT}$& $\lambda_{\rm MCDF}$ & $A_{\rm MCDF}$ & $gf_{\rm MCDF}$ & $S_{\rm MCDF}$ & $\rm BF_{\rm MCDF}$& Acc.\ 1 & 2& E2& 4.81296E+02& 1.252E+01& 4.348E-10& 2.887E-04& 1.000E+00& 4.81163E+02& 1.243E+01& 4.314E-10& 2.862E-04 & 1.000E+00& B+\ 1 & 3& M1& 1.15783E+02& 1.066E+07& 6.430E-05& 1.841E+00& 8.279E-01& 1.15846E+02& 1.066E+07& 6.435E-05& 1.844E+00 & 8.280E-01& AA\ 1 & 3& E2& 1.15783E+02& 4.675E+03& 2.819E-08& 2.605E-04& 3.631E-04& 1.15846E+02& 4.610E+03& 2.782E-08& 2.576E-04 & 3.581E-04& B+\ 2 & 3& M1& 1.52459E+02& 2.211E+06& 2.311E-05& 8.713E-01& 1.717E-01& 1.52582E+02& 2.209E+06& 2.313E-05& 8.728E-01 & 1.716E-01& AA\ 1 & 4& M1& 1.02378E+02& 7.057E+06& 5.545E-05& 1.404E+00& 9.979E-01& 1.02414E+02& 7.056E+06& 5.548E-05& 1.405E+00 & 9.977E-01& AA\ 1 & 4& E2& 1.02378E+02& 9.136E+03& 7.180E-08& 4.587E-04& 1.292E-03& 1.02414E+02& 9.036E+03& 7.105E-08& 4.545E-04 & 1.278E-03& B+\ 2 & 4& E2& 1.30040E+02& 8.323E+02& 1.055E-08& 1.382E-04& 1.177E-04& 1.30106E+02& 8.266E+02& 1.049E-08& 1.376E-04 & 1.169E-04& B+\ 3 & 4& M1& 8.84311E+02& 5.022E+03& 2.944E-06& 6.438E-01& 7.101E-04& 8.83264E+02& 5.043E+03& 2.949E-06& 6.442E-01 & 7.131E-04& AA\ 3 & 5& M1& 9.45353E+01& 3.516E+07& 4.710E-05& 1.101E+00& 9.992E-01& 9.45670E+01& 3.517E+07& 4.715E-05& 1.103E+00 & 9.992E-01& AA\ 4 & 5& E2& 1.05851E+02& 2.954E+04& 4.962E-08& 3.505E-04& 8.394E-04& 1.05906E+02& 2.923E+04& 4.915E-08& 3.477E-04 & 8.304E-04& B+\ 1 & 6& E1& 4.20220E+01& 1.720E+11& 2.277E-01& 3.150E-02& 8.493E-01& 4.20105E+01& 1.677E+11& 2.219E-01& 3.069E-02 & 8.491E-01& B+\ 3 & 6& E1& 6.59623E+01& 1.905E+10& 6.213E-02& 1.349E-02& 9.407E-02& 6.59135E+01& 1.861E+10& 6.062E-02& 1.315E-02 & 9.422E-02& B+\ 4 & 6& E1& 7.12791E+01& 1.146E+10& 4.365E-02& 1.024E-02& 5.659E-02& 7.12289E+01& 1.119E+10& 4.256E-02& 9.981E-03 & 5.665E-02& B+\ 1 & 7& E1& 3.76767E+01& 3.046E+11& 1.945E-01& 2.412E-02& 6.987E-01& 3.76635E+01& 2.972E+11& 1.896E-01& 2.351E-02 & 6.978E-01& B+\ 2 & 7& E1& 4.08766E+01& 9.258E+10& 6.957E-02& 9.362E-03& 2.124E-01& 4.08620E+01& 9.073E+10& 6.814E-02& 9.166E-03 & 2.130E-01& B+\ 3 & 7& E1& 5.58512E+01& 1.985E+10& 2.785E-02& 5.120E-03& 4.553E-02& 5.58074E+01& 1.940E+10& 2.717E-02& 4.993E-03 & 4.555E-02& B+\ 4 & 7& E1& 5.96164E+01& 1.868E+10& 2.986E-02& 5.861E-03& 4.285E-02& 5.95714E+01& 1.828E+10& 2.918E-02& 5.722E-03 & 4.292E-02& B+\ 5 & 7& E1& 1.36488E+02& 2.250E+08& 1.885E-03& 8.470E-04& 5.161E-04& 1.36161E+02& 2.178E+08& 1.816E-03& 8.142E-04 & 5.114E-04& B\ 3 & 8& E1& 4.12216E+01& 3.267E+11& 8.322E-02& 1.129E-02& 1.000E+00& 4.12097E+01& 3.183E+11& 8.104E-02& 1.099E-02 & 9.998E-01& B+\ 7 & 8& M1& 1.57370E+02& 8.553E+06& 3.177E-05& 1.236E+00& 2.618E-05& 1.57545E+02& 8.545E+06& 3.180E-05& 1.239E+00 & 2.684E-05& AA\ 1 & 9& E1& 2.83309E+01& 3.138E+10& 1.133E-02& 1.057E-03& 5.789E-02& 2.83251E+01& 3.091E+10& 1.115E-02& 1.040E-03 & 5.830E-02& B+\ 2 & 9& E1& 3.01028E+01& 2.491E+09& 1.015E-03& 1.006E-04& 4.595E-03& 3.00968E+01& 2.605E+09& 1.061E-03& 1.052E-04 & 4.913E-03& B\ 3 & 9& E1& 3.75089E+01& 4.121E+10& 2.608E-02& 3.220E-03& 7.602E-02& 3.74921E+01& 4.009E+10& 2.535E-02& 3.129E-03 & 7.561E-02& B+\ 4 & 9& E1& 3.91703E+01& 4.447E+11& 3.069E-01& 3.957E-02& 8.204E-01& 3.91541E+01& 4.346E+11& 2.997E-01& 3.863E-02 & 8.197E-01& B+\ 5 & 9& E1& 6.21802E+01& 2.227E+10& 3.872E-02& 7.927E-03& 4.108E-02& 6.21205E+01& 2.190E+10& 3.801E-02& 7.773E-03 & 4.130E-02& B+\ 6 & 9& M1& 8.69553E+01& 1.268E+07& 4.311E-05& 9.270E-01& 2.339E-05& 8.69500E+01& 1.268E+07& 4.313E-05& 9.274E-01 & 2.391E-05& AA\ 7 & 10& E1& 3.82035E+01& 5.076E+11& 1.111E-01& 1.397E-02& 6.704E-01& 3.82074E+01& 4.917E+11& 1.076E-01& 1.354E-02 & 6.697E-01& B+\ 9 & 10& E1& 5.74054E+01& 2.496E+11& 1.233E-01& 2.330E-02& 3.296E-01& 5.74071E+01& 2.425E+11& 1.198E-01& 2.264E-02 & 3.303E-01& B+\ 1. Only transitions among the lowest 10 levels of the $n=2$ configurations are shown here. Table \[table2\] is available online in its entirety the *JQSRT* website. [llccccccccc]{} \ Key & State & NIST & & & &\ (r)[4-5]{}(r)[6-7]{}(r)[8-9]{}(r)[10-11]{} & & $E$ & $E$ & $\Delta E$ & $E$ & $\Delta E$ & $E$ & $\Delta E$& $E$ & $\Delta E$\ \ Key & State & NIST & & & &\ (r)[4-5]{}(r)[6-7]{}(r)[8-9]{}(r)[10-11]{} & & $E$ & $E$ & $\Delta E$ & $E$ & $\Delta E$ & $E$ & $\Delta E$& $E$ & $\Delta E$\ 1 & $2s^{2}\,2p^{4}(^{3}P)~^{3}P_{2}$ & 0 & 0 & & 0 & & 0 & & 0 &\ 2 & $2s^{2}\,2p^{4}(^{1}S)~^{1}S_{0}$ & 208080 & 207772 & -308 & 207830 & -250 & 207878 & -202 & 207956 & -124\ 3 & $2s^{2}\,2p^{4}(^{3}P)~^{3}P_{1}$ & 863620 & 863686 & 66 & 863215 & -405 & 864158 & 538 & 864147 & 527\ 4 & $2s^{2}\,2p^{4}(^{1}D)~^{1}D_{2}$ & 976560 & 976768 & 208 & 976432 & -128 & 977254 & 694 & 977113 & 553\ 5 & $2s^{2}\,2p^{4}(^{3}P)~^{3}P_{0}$ & 1920420 & 1921492 & 1072 & 1920667 & 247 & 1922438 & 2018 & 1922041 & 1621\ 6 & $2s~^{2}S\,2p^{5}~^{3}P_{2}^{\circ}$ & 2380360 & 2379704 & -656 & 2380355 & -5 & 2381134 & 774 & 2382023 & 1663\ 7 & $2s~^{2}S\,2p^{5}~^{3}P_{1}^{\circ}$ & 2654560 & 2654158 & -402 & 2655091 & 531 & 2655814 & 1254 & 2656837 & 2277\ 8 & $2s~^{2}S\,2p^{5}~^{3}P_{0}^{\circ}$ & 3289690 & 3289601 & -89 & 3289830 & 140 & 3291540 & 1850 & 3292303 & 2613\ 9 & $2s~^{2}S\,2p^{5}~^{1}P_{1}^{\circ}$ & 3530130 & 3529720 & -410 & 3530442 & 312 & 3531756 & 1626 & 3532737 & 2607\ 10 & $2p^{6}~^{1}S_{0}$ & 5273440 & 5271717 & -1723 & 5272387 & -1053 & 5274365 & 925 & 5279298 & 5858\ [lllllcccccccc]{} \ $i$ & $j$ & Lower level & Upper level & & & &\ (r)[5-7]{}(r)[8-9]{}(r)[10-11]{}(r)[12-12]{} & & & & NIST & MBPT & MCDF & NIST & MBPT & NIST& MCDF & MBPT & MCDF& MBPT& MCDF\ \ 3 &9 & $2s^{2}\,2p^{4}(^{3}P)~^{3}P_{1}$ & $2s~^{2}S\,2p^{5}~^{1}P_{1}^{\circ}$ & 37.483 & 37.5089 & 37.4921 & 0.07 & 0.02 & 4.121E+10 & 4.009E+10 & -3\ 1 &7 & $2s^{2}\,2p^{4}(^{3}P)~^{3}P_{2}$ & $2s~^{2}S\,2p^{5}~^{3}P_{1}^{\circ}$ & 37.661 & 37.6767 & 37.6635 & 0.04 & 0.01 & 3.046E+11 & 2.972E+11 & -2\ 7 &10 &$2s~^{2}S\,2p^{5}~^{3}P_{1}^{\circ}$ & $2p^{6}~^{1}S_{0}$ & 38.187 & 38.2035 & 38.2074 & 0.04 & 0.05 & 5.076E+11 & 4.917E+11 & -3\ 4 &9 & $2s^{2}\,2p^{4}(^{1}D)~^{1}D_{2}$ & $2s~^{2}S\,2p^{5}~^{1}P_{1}^{\circ}$ & 39.161 & 39.1703 & 39.1541 & 0.02 & -0.02 & 4.447E+11 & 4.346E+11 & -2\ 2 &7 & $2s^{2}\,2p^{4}(^{1}S)~^{1}S_{0}$ & $2s~^{2}S\,2p^{5}~^{3}P_{1}^{\circ}$ & 40.866 & 40.8766 & 40.8620 & 0.03 & -0.01 & 9.258E+10 & 9.073E+10 & -2\ 3 &8 & $2s^{2}\,2p^{4}(^{3}P)~^{3}P_{1}$ & $2s~^{2}S\,2p^{5}~^{3}P_{0}^{\circ}$ & 41.221 & 41.2216 & 41.2097 & 0.00 & -0.03 & 3.267E+11 & 3.183E+11 & -3\ 1 &6 & $2s^{2}\,2p^{4}(^{3}P)~^{3}P_{2}$ & $2s~^{2}S\,2p^{5}~^{3}P_{2}^{\circ}$ & 42.014 & 42.0220 & 42.0105 & 0.02 & -0.01 & 1.720E+11 & 1.677E+11 & -3\ 3 &7 & $2s^{2}\,2p^{4}(^{3}P)~^{3}P_{1}$ & $2s~^{2}S\,2p^{5}~^{3}P_{1}^{\circ}$ & 55.842 & 55.8512 & 55.8074 & 0.02 & -0.06 & 1.985E+10 & 1.940E+10 & -2\ 9 &10 &$2s~^{2}S\,2p^{5}~^{1}P_{1}^{\circ}$ & $2p^{6}~^{1}S_{0}$ & 57.362 & 57.4054 & 57.4071 & 0.08 & 0.08 & 2.496E+11 & 2.425E+11 & -3\ 4 &7 &$2s^{2}\,2p^{4}(^{1}D)~^{1}D_{2}$ & $2s~^{2}S\,2p^{5}~^{3}P_{1}^{\circ}$ & 59.590 & 59.6164 & 59.5714 & 0.04 & -0.03 & 1.868E+10 & 1.828E+10 & -2\ 5 &9 & $2s^{2}\,2p^{4}(^{3}P)~^{3}P_{0}$ & $2s~^{2}S\,2p^{5}~^{1}P_{1}^{\circ}$ & 62.135 & 62.1802 & 62.1205 & 0.07 & -0.02 & 2.227E+10 & 2.190E+10 & -2\ 3 &6 & $2s^{2}\,2p^{4}(^{3}P)~^{3}P_{1}$ & $2s~^{2}S\,2p^{5}~^{3}P_{2}^{\circ}$ & 65.933 & 65.9623 & 65.9135 & 0.04 & -0.03 & 1.905E+10 & 1.861E+10 & -2\ 4 &6 & $2s^{2}\,2p^{4}(^{1}D)~^{1}D_{2}$ & $2s~^{2}S\,2p^{5}~^{3}P_{2}^{\circ}$ & 71.223 & 71.2791 & 71.2289 & 0.08 & 0.01 & 1.146E+10 & 1.119E+10 & -2\

--- abstract: 'We prove the existence of primitive sets (sets of integers in which no element divides another) in which the gap between any two consecutive terms is substantially smaller than the best known upper bound for the gaps in the sequence of prime numbers. The proof uses the probabilistic method. Using the same techniques we improve the bounds obtained by He for gaps in geometric-progression-free sets.' address: 'Department of Mathematics, Towson University, 8000 York Road, Towson, MD 21252' author: - Nathan McNew bibliography: - 'bibliography.bib' title: 'Primitive and Geometric-Progression-Free Sets without large gaps' --- Introduction ============ Despite the rich history of research on the gaps in the sequence of prime numbers, including many recent breakthroughs, the magnitudes of the largest gaps in this sequence are still poorly understood. Denoting by $p_1, p_2, \ldots$ the sequence of prime numbers, it has been known since 2001, due to Baker, Harman, and Pintz [@bhp], that $$p_n -p_{n-1} \ll p_n^{0.525}.$$ Assuming the Riemann Hypothesis gives a small improvement. Cramér [@cra] shows $$p_n -p_{n-1} \ll \sqrt{p_n}\log p_n.$$ Cramér [@Cra2] conjectures, however, that the bound $p_n -p_{n-1} \ll \log^2 p_n$ gives the true order of magnitude of the largest gaps. As for lower bounds, it follows immediately from the prime number theorem that there must exist gaps where $p_n -p_{n-1} \geq \log p_n$. This can be improved upon slightly. It has recently been shown by Ford, Green, Konyagin, Maynard and Tao [@fgkmt] that, for some positive constant $c$, the innequality $$p_n-p_{n-1} > \frac{c\log p_n \log \log p_n \log_4p_n}{\log_3 p_n}$$ holds infinitely often, improving on the previous result of Rankin [@rankinprimes] which included an additional triple $\log $ factor in the denominator. Here, and throughout the paper, $\log_i x$ will be used to denote the $i$-fold iterated logarithm when $i\geq 3$. Since $\log \log x$ is commonly used it will be used for readability when $i=2$. Generalizing from the set of primes, one can consider any primitive set of integers. We say a set is primitive if no integer in the set divides another integer in the set. The study of primitive sets also has a rich history. For example, it is known that primitive sets can have counting function substantially larger than the prime numbers. Ahlswede, Khachatrian, and Sárközy [@AKS] showed there exists a primitive sequence $s_1<s_2<\cdots$ with $$n \asymp \frac{s_n}{\log \log s_n (\log_3 s_n)^{1+\epsilon}}$$ for sufficiently large $n$. Martin and Pomerance [@mp] show that this can be improved slightly, in fact there exists such a sequence where $$n \asymp \frac{s_n}{\log \log s_n \log_3 s_n \cdots \log_k s_n (\log_{k+1} s_n)^{1+\epsilon}}$$ for sufficiently large $n$ and any $k\geq 2$. This is, in a sense, best possible, as Erdős [@erdosprimitive] shows that any primitive sequence $s_1, s_2, \ldots$ must satisfy $$\sum_{n=1}^\infty \frac{1}{s_n \log s_n} < \infty.$$ Compared to the sequence of prime numbers, where the average gap grows like $\log x$, we see from these results that primitive sets can have substantially smaller gaps on average, on the order of $\log \log x \log_3 x \cdots \log_k x (\log_{k+1} x)^{1+\epsilon}$ for any $k\geq 2$. Nevertheless, it has not yet been possible to show that the largest gaps among these sequences is any smaller than what is known for the prime numbers. We show here that there exist primitive sequences in which the gap between consecutive terms is substantially smaller than has been previously shown for the primes or any other primitive sequence. In particular, we get the following upper bound. \[thm:primitive\] For any $\epsilon>0$ there exists a primitive sequence $q_1< q_2 < \cdots$ of integers in which the gap between any two consecutive terms is bounded above by $$q_n-q_{n-1} \leq \exp \left(\sqrt{2\log q_n \log \log q_n + (2+\epsilon)\log q_n \log_3 q_n}\right). \label{primitive bound}$$ The proof utilizes the probabilistic method, and so it is not constructive. It generalizes, however, to the related problem of geometric-progression-free sets, where the analogous problem has recently attracted attention. If $r>1$ is rational (sometimes we insist it be integral), then a geometric progression of length $k$ with ratio $r$ is a progression of integers $(g_1,g_2,\ldots g_k$) in which $g_i=rg_{i-1}$. We say $S$ avoids geometric progressions of length $k$ if it is not possible to find $k$ integers from $S$ in a geometric progression. Note that primitive sets can be described as sets avoiding geometric progressions of length 2 in which we insist that the ratio $r$ must be an integer. For the remainder of the paper we will assume that our geometric progressions have length at least 3, and, unless otherwise stated, are allowed to have rational ratio. In the case of geometric-progression-free sets, unlike primitive sets, there exist such sets with positive density. In particular, the squarefree numbers avoid geometric progressions and have density $\frac{6}{\pi^2}$, though this density isn’t best possible. (See [@mcnewgpf; @NO; @Rankin] for results on the maximum density of such a set.) Because of this it is not clear, a priori, that there cannot exist such sets in which all of the gaps are bounded above by a fixed constant. In ergodic theory a set in which every gap is bounded by a constant is known as a *syndetic* set. Bieglböck, Bergelsen, Hindman and Strauss [@BBHS] first posed the question of whether there exists a syndetic set that is geometric-progression-free. This problem has become well-known as a good example of the difficulty inherent in studying problems that mix the additive and multiplicative structure of the integers, and remains open. There has been partial progress toward this question for 2-syndetic sets (sets in which the difference between any two consecutive terms is at most two). He [@He] shows by a computer search that any subset of the range \[1,640\] containing at least one of any pair of consecutive numbers must contain three term geometric progressions. Recently Patil [@Patil] shows that any sequence of integers $s_1<s_2<\cdots$ with $s_n-s_{n-1} \leq 2$ must contain infinitely pairs $\{a,ar^2\}$ with $r$ an integer. In general, one can avoid geometric progressions of length $k{+}1$ by taking the sequence of $k$-free numbers. Denoting by $s_1<s_2<\cdots$ the sequence of $k$-free numbers, the best known bound on the gaps, due to Trifonov [@trifonov] is that $$s_n-s_{n-1} \ll s_n^{\frac{1}{2k+1}} \log s_n.$$ Though this, again, is likely far greater than the truth. He [@He] considers the existence of geometric-progression-free sets with gaps provably smaller than the bounds for $k$-free numbers. He shows the following. For each $\epsilon>0$ there exists a sequence $b_1<b_2<\cdots$ avoiding 6-term geometric progressions satisfying $$b_n-b_{n-1} \ll_\epsilon \exp\left( \left(\frac{5\log 2}6 +\epsilon \right) \frac{\log b_n}{\log \log b_n}\right).$$ Furthermore, there exists a sequence $c_1<c_2<\cdots$ avoiding 5-term geometric progressions satisfying $$c_n-c_{n-1} \ll_\epsilon c_n^\epsilon$$ and a sequence $d_1<d_2<\cdots$ that avoids 3-term geometric progressions with integral ratio in which $$d_n-d_{n-1} \ll_\epsilon d_n^\epsilon.$$ The technique developed here allows us to treat 3-term geometric progressions with rational ratio and obtain a substantially smaller bound on the size of the gaps. In particular we prove the following in section \[sec:gpf\] . \[thm:gpf\] For any $\epsilon>0$ there exists a sequence of integers $t_1< t_2< \cdots$ free of 3-term-geometric-progressions, such that $$t_n-t_{n-1} \leq \exp \left(2\sqrt{\log 2\log t_n + \tfrac{3+\epsilon}{2}\sqrt{\log 2\log t_n}\log \log t_n}\right). \label{gp bound}$$ Coprime subsets of intervals ============================ We first prove that in any short interval we can find a relatively large subset of integers that are pairwise coprime. Using the linear sieve of Rosser and Iwaniec (see for example Theorem 12.14 and Corollary 12.15 of [@cribro]) one can sieve an interval of length $y$ by primes up to nearly $\sqrt{y}$. The result can be stated as follows. \[lem:sieve\] There exist positive constants $c_1$ and $c_2$ so that every interval of length $c_1y$ with $y\geq 2$ contains at least $\frac{y}{ \log^2 y}$ integers free of prime factors smaller than $\sqrt{y}$, and at most $\frac{c_2 y}{\log y}$ such integers. Using this we can show that the short interval $[x-y,x]$ contains a reasonably large subset of pairwise coprime integers. Erdős and Selfridge [@erdself] (see also [@erdric]) prove that for sufficiently large $y$ and any $\epsilon>0$ any such interval has a pairwise coprime subset of size at least $y^{1/2-\epsilon}$, though their proof is not correct as written. We correct and refine the argument, using Lemma \[lem:sieve\] to show the following. \[thm:coprimeset\] For sufficiently large $y$ and $x\geq y+1$, any interval $[x-y,x]$ contains a subset of pairwise coprime integers of size at least $ \frac{c_3 \sqrt{y}}{\log y}$ for some positive constant $c_3$. Let $y'=y/c_1$, where $c_1$ is the constant from Lemma \[lem:sieve\]. That lemma then implies that the set $S \subset [x-y,x]$ consisting of integers in this interval free of prime factors smaller than $\sqrt{y'}$ contains at least $\frac{y'}{ \log^2 y'}$ integers. Now, let $p \geq \sqrt{y'}$ be prime, and suppose $p|n$ for some $n \in S$. Then $n=pm$ where $m \in \left[\frac{x}{p} - \frac{y}{p},\frac{x}{p}\right]$ (an interval of length $\frac{c_1y'}{p}$). Since $n$ is free of prime factors smaller than $\sqrt{y'}$, $m$ will be free of such primes as well. While we can’t sieve this shorter interval of primes as large as $\sqrt{y'}$, we can use Lemma \[lem:sieve\] to sieve this interval of primes up to $\sqrt{\frac{y'}{p}}$, at least so long as $\frac{y'}{p}$ is at least two. Thus for each prime $\sqrt{y'}\leq p < \frac{y'}{2}$, the number of integers in $S$ divisible by the prime $p$ is at most $$\frac{c_2 y'}{p \log \frac{y'}{p}}.$$ For those primes $\frac{y'}{2}\leq p<y$, we can bound the number of integers in $S$ divisible by $p$ trivially by $\left \lceil \frac{y}{p}\right \rceil =O(1)$. We now use Turan’s graph theorem to prove that a large subset of $S$ is pairwise coprime. Construct a graph in which the vertices are the elements of $S$ and the edges connect vertices corresponding to integers which share a prime factor. Adding together the total number of edges produced by each prime, we find that the total number of edges in the graph is at most $$\begin{aligned} \frac{1}{2}\sum_{\sqrt{y'}\leq p < \frac{y'}{2}} \left(\frac{c_2 y'}{p \log \frac{y'}{ p}}\times \left(\frac{ c_2 y'}{p \log \frac{y'}{ p}}-1\right) \right) &+\frac{1}{2}\sum_{\frac{y'}{2}\leq p <y} \left \lceil \frac{y}{p}\right \rceil\left(\left \lceil \frac{y}{p}\right \rceil-1 \right) \\ &\leq \sum_{\sqrt{y'}\leq p < \frac{y'}{2}} \frac{c_2^2y'^2}{p^2 \log^2 \frac{y'}{p}} + \sum_{\frac{y'}{2} \leq p < y } O(1).\end{aligned}$$ By partial summation this expression is at most $\frac{c' y'^{3/2}}{\log^3 y'}$ for some constant $c'$. Turan’s graph theorem states that any graph with $v$ vertices and $e$ edges has an independent set of vertices of size at least $\frac{v^2}{v+2e}$. Applying this to our graph we find there must be an independent set of vertices (corresponding to a set of pairwise coprime integers) of size at least $$\frac{\frac{y'^2}{\log^4 y'}}{\frac{y'}{\log^2 y'} + \frac{2c' y'^{3/2}}{\log^3 y'}} \gg \frac{\sqrt{y'}}{\log y'}$$ and the result follows. \[rem:pfactors\] Note that in the construction above, the integers in the set were free of prime factors less than $\sqrt{y'}$, and thus have at most $$\frac{\log x }{\log \sqrt{y'}} = \frac{2 \log x}{\log y -\log c_1} = \frac{2 \log x}{\log y} + O\left(\frac{\log x}{\log^2 y}\right)$$ prime factors. Primitive sets without large gaps ================================= Using these results we are now able to give a proof of Theorem \[thm:primitive\] using the probabilistic method. We construct a primitive set according to the following probabilistic construction and then show that, with high probability, the set we constructed does not have any gaps greater than the bound . Fix $\epsilon>0$. For each prime number $p_i$ we choose a corresponding positive-integer-valued random variable $X_i$ with distribution $$P(X_i = n) = \frac{C_{\epsilon}}{n\log^{1+\tfrac{\epsilon}{8}} (n+2)},$$ with $C_{\epsilon}$ chosen to normalize the distribution. (Note that the sum of these terms converges since the power on the logarithm is greater than 1. The purpose of adding two inside the logarithm is just to make the probability positive when $n$ is either 1 or 2.) We then construct the set of integers $Q = \{n \geq 2 : p_i|n \rightarrow \Omega(n) = X_i\}$, consisting of only those integers $n$ for which the total number of prime factors dividing $n$ agrees with the random variable $X_i$ corresponding to every single one of its prime divisors, $p_i$. It is readily seen that this construction always produces a primitive set, since if $a \in Q$, and $a|b$ with $b>a$, then $\Omega(a) < \Omega(b)$, but any prime dividing $a$ also divides $b$, and so $b$ cannot be in $Q$. We now show that we expect every interval of size to contain an element of this set. Let $$y = \exp \left(\sqrt{2\log x \log \log x + (2+\epsilon)\log x \log_3 x }\right), \label{primitivey}$$ and consider the interval $[x-y,x]$. Using Theorem \[thm:coprimeset\], along with the observation of Remark \[rem:pfactors\], there exists a subset $S$ of the integers in this interval containing at least $\frac{c_3 \sqrt{y}}{\log y}$ integers from this interval which are pairwise coprime. Furthermore, the integers in $S$ have no more than $\frac{2 \log x}{\log y} + O\left(\frac{\log x}{\log^2 y}\right)$ prime factors. Suppose $n \in S$, then the probability that $n \in Q$ is $$\begin{aligned} P&(n \in Q) = \prod_{p_i|n} P(X_i = \Omega(n)) = \prod_{p_i|n} \frac{C_{\epsilon}}{\Omega(n)\log^{1+\tfrac{\epsilon}{8}} (\Omega(n)+2)} \\ &\geq \left(\frac{C_{\epsilon}}{\left(\frac{2 \log x}{\log y} + O\left(\frac{\log x}{\log^2 y}\right)\right)\log^{1+\tfrac{\epsilon}{8}}\left(\frac{2 \log x}{\log y} +O(1)\right)}\right)^{\frac{2 \log x}{\log y} + O\left(\frac{\log x}{\log^2 y}\right)} \notag \\ &= \exp\left(\left(-\frac{2 \log x}{\log y} {+} O\left(\frac{\log x}{\log^{2} y}\right)\right){\times} \left(\log\left(\frac{\log x}{\log y}\right){+}\left(1{+}\tfrac{\epsilon}{8}\right)\log_3 x {+}O_\epsilon\left(1 \right)\right)\right)\notag \\ &= \exp\left(-\frac{2 \log x}{ \log y}\left(\log\left(\frac{\log x}{\log y}\right) +\left(1+\frac{\epsilon}{8}\right) \log_3 x+ O_\epsilon(1)\right)\right). \notag\end{aligned}$$ Since the elements of $S$ are pairwise coprime, the probability that any one element of $S$ is included in $Q$ is independent of the probability of any other element is included. Thus the probability that no integer from the interval $[x~-~y,x]$ is included in $Q$ can be bounded as follows. $$\begin{aligned} &P([x-y,y]\cap Q = \varnothing) \leq P(S\cap Q = \varnothing) = \prod_{n \in S} \left(1-P(n \in Q)\right) \\ & \leq \prod_{n \in S}\left(1{-} \exp\left(-\frac{2 \log x}{ \log y}\left(\log\left(\frac{\log x}{\log y}\right) +\left(1+\tfrac{\epsilon}{8}\right) \log_3 x+ O_\epsilon(1)\right)\right)\right) \\ &\leq \left(1- \exp\left(-\frac{2 \log x}{ \log y}\left(\log\left(\frac{\log x}{\log y}\right) +\left(1{+}\tfrac{\epsilon}{8}\right) \log_3 x+ O_\epsilon(1)\right)\right)\right)^{\frac{c_3 \sqrt{y}}{\log y}} \\ & \leq \exp\left( - \frac{c_3 \sqrt{y}}{\log y} {\times} \exp\left(-\frac{2 \log x}{ \log y}\left(\log\left(\frac{\log x}{\log y}\right) {+}\left(1{+}\tfrac{\epsilon}{8}\right) \log_3 x{+} O_\epsilon(1)\right)\right)\right) \\ & = \exp\left(\hspace{-.5mm} {-}\exp \left(\tfrac{1}{2}\log y {-}\log \log y {-} \frac{2\log x}{\log y}\left( \log \left(\frac{\log x}{\log y}\right)\hspace{-.5mm}{+}\left(1{+}\tfrac{\epsilon}{8}\right) \log_3 x {+} O_\epsilon\hspace{-.5mm}(1)\hspace{-.2mm}\right)\hspace{-.2mm}\right)\hspace{-.2mm}\right).\end{aligned}$$ Now, inserting our choice for the length $y$ of the interval, the innermost exponent above becomes $$\begin{aligned} \tfrac{1}{2}&\sqrt{2\log x (\log \log x {+} \left(1{+}\tfrac{\epsilon}{2}\right) \log_3 x)} {-} \frac{2\log x\left(\log \left(\frac{\sqrt{\log x}}{\sqrt{\log \log x }}\right) \hspace{-0.5mm}{+}\left(1{+}\frac{\epsilon}{8}\right) \log_3 x{+}O_\epsilon(1)\right)}{\sqrt{2\log x (\log \log x + \left(1+\frac{\epsilon}{2}\right) \log_3 x)}} \\ &=\sqrt{\tfrac{1}{2}\log x}\left(\sqrt{\log \log x {+} \left(1{+}\tfrac{\epsilon}{2}\right) \log_3 x}- \frac{\log \log x {+} \left(1{+}\tfrac{\epsilon}{2}{-}\frac{\epsilon}{4}\right)\log_3 x+O_\epsilon(1)}{\sqrt{\log \log x + \left(1{+}\frac{\epsilon}{2}\right) \log_3 x}}\right) \\ &= \frac{\epsilon}{8}\frac{\sqrt{2\log x}}{\sqrt{\log \log x}}\left(\log_3 x+O_\epsilon(1)\right).\end{aligned}$$ Therefore the probability that none of the integers from the interval $[x-y,x]$ are included in $Q$, which is less than the probability that no integer in $S$ is included in $Q$ since $S \subset [x-y,x]$, is at most $\exp\left(-\exp\left(\frac{\epsilon}{8}\frac{\sqrt{2\log x}}{\sqrt{\log \log x}}\left(\log_3 x+O_\epsilon(1)\right)\right)\right)$. Using linearity of expectation, and by starting the sequence at a sufficiently high initial value $N$, we can ensure that the expected number of intervals of the form $[x-y,x]$ which do not contain an integer in $Q$ is at most $$\begin{aligned} \sum_{x>N} P(&[x {-} y,x] \cap Q = \varnothing) \leq \sum_{x>N} \exp\left(-\exp\left(\frac{\epsilon\sqrt{2\log x}}{8\sqrt{\log \log x}}\left(\log_3 x{+}O_\epsilon(1)\right)\right)\right) <1\end{aligned}$$ since this series converges. Because the expected number of intervals that do not contain an integer in $Q$ is less than 1, there must exist a sequence $Q$ which intersects every such interval, and thus satisfies the properties of the theorem. Geometric-Progression-Free sets without large gaps {#sec:gpf} ================================================== A very similar construction can be used to prove Theorem \[thm:gpf\], producing a set free of 3-term geometric progressions with gaps smaller than those obtained by He. Following the method of proof of Theorem \[thm:primitive\], we construct a set similar to the squarefree numbers, in the sense that each prime number is only allowed to appear (if it appears at all) to one fixed power in any element of the set. As before, we construct this set probabilistically and then bound the probability that it omits any interval of the size given in . For each prime $p_i$ choose a positive-integer-valued random variable $X_i$ with distribution $$P(X_i = n) = \frac{1}{2^n}.$$ Now construct the set of integers $T = \{n \geq 2 : p_i|n \rightarrow p_i^{X_i}||n\}$ consisting of those integers $n$ where the exponent on each of its prime divisors $p_i$ is equal to the random variable $X_i$. (If $p_i$ divides $n$ then $p_i^{X_i}$ is the largest power of $p_i$ that divides $n$.) This set $T$ is free of 3-term geometric progressions of integers for essentially the same reason that the squarefree integers avoid such progressions. If $\{a,ar,ar^2\}$ is any geometric progression with $r\in \mathbb{Q}$, $r>1$ and $p$ divides the numerator of $r$ but not the denominator, then $p$ appears to different, positive, powers in $ar$ and $ar^2$, and hence both cannot be in $T$. We now show that we expect every interval of size to contain an element of this set. Let $$y = \exp \left(2\sqrt{\log 2\log x +\tfrac{3+\epsilon}{2}\sqrt{\log 2 \log x}\log \log x }\right) \label{eq:gpfy}$$ and consider the interval $[x-y,x]$. We again use Theorem \[thm:coprimeset\] to obtain a pairwise coprime subset $S$ of this interval of size at least $\frac{c_3 \sqrt{y}}{\log y}$ consisting of integers having at most $\frac{2 \log x}{\log y} + O\left(\frac{\log x}{\log^2 y}\right)$ prime factors. The probability an integer $n$ from this set is contained in $T$ is $$\begin{aligned} P(n \in T) = \prod_{p_i^\alpha||n} P(X_i = \alpha) &= \prod_{p_i^\alpha||n} \frac{1}{2^\alpha} = \left(\frac{1}{2}\right)^{\Omega(n)} \geq \left(\frac{1}{2}\right)^{\frac{2 \log x}{\log y} + O\left(\frac{\log x}{\log^2 y}\right)} \\ &= \exp\left(-\left(\frac{2 \log 2 \log x}{\log y} + O\left(\frac{\log x}{\log^2 y}\right)\right) \right).\end{aligned}$$ Exploiting the fact that elements of $S$ are pairwise coprime, the probability that none of the elements of $S$ are included in $T$ is $$\begin{aligned} \prod_{n \in S} \left(1-P(n \in T)\right) & \leq \prod_{n \in S}\left(1-\exp\left(-\left(\frac{2 \log 2 \log x}{\log y} + O\left(\frac{\log x}{\log^2 y}\right)\right) \right)\right) \\ &\leq \left(1-\exp\left(-\left(\frac{2 \log 2 \log x}{\log y} + O\left(\frac{\log x}{\log^2 y}\right)\right) \right)\right)^{\frac{c_3 \sqrt{y}}{\log y}} \\ & \leq \exp\left( - \frac{c_3 \sqrt{y}}{\log y} \times \exp\left(-\left(\frac{2 \log 2 \log x}{\log y} + O\left(\frac{\log x}{\log^2 y}\right)\right) \right)\right) \\ & = \exp\left( -\exp \left(\tfrac{1}{2}\log y - \frac{2\log 2 \log x}{\log y} - \log \log y + O(1) \right)\right) .\end{aligned}$$ Inserting here for $y$ the innermost exponent above becomes $$\begin{aligned} &\sqrt{\log 2\log x +\tfrac{3+\epsilon}2\sqrt{\log 2 \log x}\log \log x } - \frac{ \log 2 \log x }{\sqrt{ \log 2 \log x +\frac{3+\epsilon}{2}\sqrt{\log 2 \log x}\log \log x}} \\ &\hspace{3cm} - \frac{1}{2} \log \log x +O(1)\\ & =\sqrt{\log 2 \log x}\left(\sqrt{1 {+}\frac{(3{+}\epsilon)\log \log x}{2\sqrt{\log 2\log x}}} - \frac{1}{\sqrt{1 {+}\frac{(3+\epsilon)\log \log x}{2\sqrt{\log 2\log x}}}}\right) {-} \frac{1}{2} \log \log x +O(1)\\ &=\sqrt{\log 2 \log x}\left(\frac{(3+\epsilon)\log \log x}{2\sqrt{\log 2\log x}}+ O\left(\frac{(\log \log x)^2}{\log x}\right)\right) - \frac{1}{2} \log \log x +O(1)\\ &=\left(1+\frac{\epsilon}{2}\right) \log \log x +O\left(1\right).\end{aligned}$$ The fourth line above was obtained using the Taylor expansion $$\sqrt{1+x}-\frac{1}{\sqrt{1+x}} = x +O(x^2)$$ around $x=0$. Therefore, the probability that no such integer is included in $T$ is at most $\exp\left(-\exp\left(\left(1+\frac{\epsilon}{2}\right) \log \log x +O\left(1\right)\right)\right)$. As before, by linearity of expectation, we may choose a sufficiently high initial value $N$ so that the expected number of intervals of the form $[x-y,x]$ which do not contain an integer in $T$ is at most $$\sum_{x>N} P([x - y,x] \cap T = \varnothing) \leq \sum_{x>N} \exp\left(-\exp\left(\left(1+\frac{\epsilon}{2}\right) \log \log x +O\left(1\right)\right)\right) <1$$ since this series is convergent. Thus there exists a geometric-progression-free sequence $T$ satisfying the properties of the theorem. Final Remarks ============= While we were able to show that there exist primitive sets in which the gap between consecutive terms was much smaller than what is known to be true, even conditionally for the primes, the method developed doesn’t seem to generalize to sequences of pairwise coprime integers. Can one prove that there exists a sequence $v_1<v_2<\cdots$ of pairwise coprime integers in which the difference between consecutive terms $v_n{-}v_{n-1}$ is smaller than the best known upper bound for the gaps between primes? aknowledgements {#aknowledgements .unnumbered} =============== The author is grateful to Angel Kumchev, Greg Martin and Carl Pomerance for helpful discussions during the development of this paper, and to the anonymous referee for useful feedback.

--- abstract: | We investigate the nonlinear current-voltage characteristic of mesoscopic conductors and the current generated through rectification of an alternating external bias. To leading order in applied voltages both the nonlinear and the rectified current are quadratic. This current response can be described in terms of second order conductance coefficients and for a generic mesoscopic conductor they fluctuate randomly from sample to sample. Due to Coulomb interactions the symmetry of transport under magnetic field inversion is broken in a two-terminal setup. Therefore, we consider both the symmetric and antisymmetric nonlinear conductances separately. We treat interactions self-consistently taking into account nearby gates. The nonlinear current is determined by different combinations of second order conductances depending on the way external voltages are varied away from an equilibrium reference point (bias mode). We discuss the role of the bias mode and circuit asymmetry in recent experiments. In a photovoltaic experiment the alternating perturbations are rectified, and the fluctuations of the nonlinear conductance are shown to decrease with frequency. Their asymptotical behavior strongly depends on the bias mode and in general the antisymmetric conductance is suppressed stronger then the symmetric conductance. We next investigate nonlinear transport and rectification in chaotic rings. To this extent we develop a model which combines a chaotic quantum dot and a ballistic arm to enclose an Aharonov-Bohm flux. In the linear two-probe conductance the phase of the Aharonov-Bohm oscillation is pinned while in nonlinear transport phase rigidity is lost. We discuss the shape of the mesoscopic distribution of the phase and determine the phase fluctuations. author: - 'M. L. Polianski' - 'M. Büttiker' title: Rectification and nonlinear transport in chaotic dots and rings --- Introduction {#sec:intro} ============ A large part of modern physics is devoted to nonlinear classical and quantum phenomena in various systems. Such effects as the generation of the second harmonic or optical rectification are known from classical physics, while quantum electron pumping through a small sample due to interference of wave functions is a quantum nonlinear effect. Experiments on nonlinear electrical transport often combine classical and quantum contributions. A macroscopic sample without inversion center [@UFN] exhibits a current-voltage characteristic which with increasing voltage departs from linearity due to terms proportional to the square of the applied voltage. If now an oscillating (AC) voltage is applied, a zero-frequency current (DC) is generated. If the sample is sufficiently small, quantum effects can appear due to the wave nature of electrons. The uncontrollable distribution of impurities or small variations in the shape of the sample result in quantum contributions to the DC which are random. For a mesoscopic conductor with terminals $\a ,\b, ... $ we can describe the quadratic current response in terms of second order conductances $\G_{\a\b\g}$. They relate voltages $V_{\b,\oo}$ applied at contacts or neighboring gates $\b$ at frequency $\oo$ to the current at zero frequency at contact $\a$, $$\begin{aligned} \label{eq:IV} I_\a &=&\sum_{\b\g} \G_{\a\b\g} |V_{\b,\oo}-V_{\g,\oo}|^2.\end{aligned}$$ The second order conductances include in detail the role of the shape and the nearby conductors (gates). They depend on external parameters like the frequency of the perturbation, temperature, magnetic field or the connection of the sample to the environment. We concentrate here on the quantum properties of nonlinear conductance through coherent chaotic samples. Chaos could result from the presence of impurities (disorder) or random scattering at the boundaries (ballistic billiard). Due to electronic interference the sign of this effect is generically random even for samples of macroscopically similar shape. [@WW; @AK; @KL] When averaged over an ensemble, the second order conductances vanish. As a consequence, for a fully chaotic sample there is no classical contribution to the DC and the nonlinear response is the result of the sample-specific quantum fluctuations. Interestingly enough, from a fundamental point of view these fluctuations of nonlinear conductance are sensitive to the presence of Coulomb interactions and magnetic field. While interactions strongly affect the fluctuations’ amplitude, their sign is easily changed by a small variation of magnetic flux $\Phi$, similarly to universal conductance fluctuations (UCF) in linear transport. More importantly, without interactions the current (\[eq:IV\]) through a two-terminal sample is a symmetric function of magnetic field, just like linear conductance. However, the idea that Coulomb interactions are responsible for magnetic-field asymmetry in nonlinear current was recently proposed theoretically [@SB; @SZ] and demonstrated experimentally in different mesoscopic systems. [@wei; @Zumbuhl; @marlow; @ensslin; @Bouchiat; @Bouchiat_preprint] (Various aspects of nonlinear quantum [@PB; @Coulomb; @Tsvelik; @PhysicaE] and classical [@AG] charge and spin transport [@Feldman] have been discussed later on.) It is useful to consider (anti) symmetric second order conductance $\Ga,\Gs$ defined as $$\begin{aligned} \label{eq:IVfield} {\genfrac{\{}{\}}{0pt}{}{{\mathcal G}_{s}(\Phi)}{{\mathcal G}_{a}(\Phi)}} &= &\frac{h}{\nu_s e^3}\frac{\DD^2}{2\DD \tilde V^2}\left(\frac{I(\Phi)\pm I(-\Phi)}{2}\right)_{\tilde V\to 0},\end{aligned}$$ where $\tilde V$ is a combination of voltages at the gates and contacts varied in the experiment and $\nu_s$ accounts for the spin degeneracy. We emphasize that, depending on the way voltages are varied, experiments probe different linear combinations of second order conductance elements $\G_{\a\b\g}$ of Eq. (\[eq:IV\]). From now on we will simply call $\Gs ,\Ga$ conductances and if no confusion is possible leave out the expression “second order”. In the presence of a DC perturbation the mesoscopic averages of antisymmetric [@SB; @SZ] and symmetric [@PB; @PhysicaE] conductances vanish, and it is their sample-to-sample fluctuations that are measured. Experiments are usually performed for strongly interacting samples and the magnetic-field components $\Gs,\Ga$ allow one to evaluate the strength of interactions. [@Zumbuhl; @Bouchiat] In previous theoretical works on nonlinear transport through chaotic dots several important issues have been discussed using Random Matrix Theory (RMT). [@SB; @PB; @PhysicaE] Sánchez and Büttiker [@SB] found the fluctuations of $\Ga$ in a dot with arbitrary interaction strength at zero temperature and broken time-reversal symmetry due to magnetic field. Polianski and Büttiker considered the statistics of both $\Ga$ and $\Gs$ for arbitrary flux $\Phi$, the temperature $T$, and the dephasing rate. [@PB] The fluctuations of relative asymmetry $\A=\Ga/\Gs$ and the role of the contact asymmetry on this quantity were discussed in Ref.. The results of RMT approach were compared with experimental data of Zumbühl [@Zumbuhl] and Angers [@Bouchiat] Previously we considered statistics of $\Ga,\Gs$ for the dots where only one DC voltage was varied. However, to avoid parasitic circuit effects some experiments are performed varying several voltages simultaneously. Surprisingly, the importance of the chosen combination of varied voltages (bias mode) was not addressed before in the literature. It turns out that an experiment where only one of the voltages is varied [@marlow; @Lofgren; @Bouchiat] or two voltages are asymmetrically shifted [@Zumbuhl; @ensslin] measure different combinations of nonlinear conductances $\G_{\a\b\g}$. For example, in a weakly interacting dot in the first mode we found that $\Gs\gg \Ga$, [@PhysicaE] but in the second bias mode the fluctuations of nonlinear current are strongly reduced, so that $\Gs\sim \Ga$. It is also important to generalize the previous treatment of the nonlinear current to mesoscopic systems biased by an AC-voltage at [*finite*]{} frequency. The resulting DC is sometimes called “photovoltaic current”. We expect that in such mesoscopic AC/DC converters the interactions lead to significant magnetic field-asymmetry in the DC-signal. The rectification effect of mesoscopic diffusive metallic microjunctions was theoretically considered by Falko and Khmelnitskii [@FK] assuming that electrons do not interact. Therefore, a magnetic-field asymmetry was not predicted and was also not observed in subsequent experiments. [@Bykov; @BykovAB; @Bartolo; @Lin; @Liu] The fact that the interactions induce a magnetic field-asymmetry of the photovoltaic current when the size of the sample is strongly reduced was recently demonstrated in Aharonov-Bohm rings by Angers [@Bouchiat_preprint] However, it turns out that for an AC perturbation another quantum interference phenomenon, also quadratic in voltage, random in sign and magnetic field-asymmetric, contributes to the DC. Due to [*internal*]{} AC- perturbations of the sample, the energy levels are randomly shifted and a phenomenon commonly referred to as “quantum pumping” [@pump; @SAA] appears. Brouwer demonstrated that two voltages applied out of phase generate pumped current linear in frequency, while a single voltage pumps current quadratic in frequency $\oo$. [@pump] Although theory usually considers small (adiabatic) frequencies, a photovoltaic current could be induced by voltages applied at arbitrary frequency. At small $\oo$ the pumping contribution vanishes and only the rectification effect survives. In contrast, it is not clear what the ratio of pumping current to rectification current is at large $\oo$. To distinguish between different mechanisms it is therefore important to consider rectification in a wide range of frequencies in detail. [![Top: Quantum pumping sources include oscillating voltage $V_{\rm p}(\oo)$ at the locally applied gate, which slightly changes the shape of the dot (shown dashed), or microwave antenna emitting photons with energy $\hbar\oo$ into the dot. Bottom: Rectification sources include external bias $V_{1,2}(\oo)$, top gate voltage $V_{0}(\oo)$ with capacitance $C_{g0}$, and parasitic coupling of $V_{\rm p}(\oo)$ due to stray capacitances $C_{\rm stray}$. Microwave antenna can emit photons to the contacts and lead not only to photon-assisted AC transport but also to a rectified DC.[]{data-label="fig:dot"}](3dot3d.eps "fig:"){width="9.cm"}]{} We point here to a crucial difference between rectification and pumping contributions to the photovoltaic effect. Rectification results from external perturbations or the perturbations that can be reduced to the exterior by a gauge transformation. Typical examples are external AC-bias, or gate voltage which shifts all levels uniformly, [@pedersen] or a bias induced by parasitic (stray) capacitance which connects sources of possible internal perturbations to macroscopic reservoirs, [@pump_rectif] see the bottom panel in Fig. \[fig:dot\]. Pumping, on the other hand, is due to internal perturbations like those of a microwave antenna [@VAA] or a locally applied gate voltage, [@pump] see the top panel in Fig. \[fig:dot\]. Internal and external sources affect the Schrödinger equation and its boundary conditions, respectively. In experiment pumping and rectification, often considered together under the name of photovoltaic effect, [@Bykov; @Liu; @Lin; @Bartolo; @BykovAB; @Kvon] are hard to distinguish. Can one clearly separate quantum pumping from rectification effects? To distinguish them it was proposed to use magnetic field asymmetry of DC as a signature of a true quantum pump effect. In Refs. and rectification by (non-interacting) quantum dot was due to stray capacitances of reservoirs with pumping sources. The rectified current was found to be symmetric with respect to $\Phi\to -\Phi$.[@pump_rectif] While such field-symmetric rectification dominated in the experiments of Switkes [@Switkes] and DiCarlo [@DiCarlo] at MHz frequencies, an asymmetry $\Phi\to -\Phi$ observed at larger GHz frequencies seemed to signify a quantum pump effect. [@DiCarlo] It was noted that the Coulomb interactions treated self-consistently do not lead to any drastic changes in the mesoscopic distribution of a pumped current.[@pump] Probably, that is why the effect of interactions on the rectification have not been considered yet, even though the Coulomb interaction in such dots is known to be strong.[@Zumbuhl] However, as it turned out later, Coulomb interactions are responsible for magnetic-field asymmetry in nonlinear transport through quantum dots. [@SB] Similarly this could be expected for rectification as well. Then the magnetic field asymmetry alone can not safely distinguish pumping from rectification. Therefore we thoroughly examine the frequency dependence of the magnetic-field (anti)symmetric conductances $\Ga,\Gs$. Here we neglect any quantum pumping effects and their interference with rectification. [@Vavilov05; @Moskalets_AC] While the role of Coulomb interactions and the full frequency dependence in quantum pumping are yet to be explored, here we answer two important questions concerning a competing mechanism, rectification: (1) In the DC limit $\oo\to 0$ for a strongly interacting quantum dot $\Ga$ and $\Gs$ are of the same order. Is this also the case at finite frequencies? (2) How are the experimental data affected by the bias mode for alternating voltages? A number of very recent experiments on nonlinear DC transport [@ensslin; @Bouchiat] and AC rectification [@Bouchiat_preprint] have used submicron ring-shaped samples with a relatively large aspect ratio. In this work we develop a model of a ring which includes chaotic dynamics due to possible roughness of its boundary and/or the presence of impurities. Similarly to quantum dots, the two-terminal nonlinear conductance of such a ring is field-asymmetric because field-asymmetry exists in each arm. In particular, this leads to deviations of the phase in AB oscillations from $0\mbox{(mod) }\pi$ which characterizes linear conductance obeying Onsager symmetry relations. Experiments find that the amplitude and phase of AB oscillations exhibit rather curious properties. For example, the DC experiment of Leturcq [@ensslin] finds that during many AB oscillations with period $hc/e$ the phase is well-defined. The experiment demonstrates that a nearby gate can vary the phase of the AB oscillations over the full circle. The amplitude of the second harmonic $hc/2e$ is strongly suppressed. On the other hand, the DC experiment [@Bouchiat] and AC experiment [@Bouchiat_preprint] of Angers find that the phase can be defined only for few oscillations at low magnetic fields. For high frequencies, the phase fluctuates strongly as function of frequency. Both in the nonlinear and the rectified current the amplitude of the second harmonic $hc/2e$ in AB oscillations is always comparable with the first harmonic $hc/e$. This is in contrast with the experiments in Ref.. Although we do not fully address all these questions here, our model of a chaotic ring allows us to consider them at least on a qualitative level. Principal results ================= To introduce the reader to the problem of nonlinear transport in Sec. \[sec:bias\] we first qualitatively discuss the Coulomb interaction effect in the simplest DC problem. In reality the statistical properties of conductances $\G_{\a\b\g}$ in Eq. (\[eq:IV\]) are sensitive to electronic interference but to assess the role of Coulomb interactions we can consider a specific sample. In contrast to linear transport, it turns out that the nonlinear current strongly depends on the way voltages at the contacts and/or nearby conductors are varied from their equilibrium values (bias mode). For example, we find that the experiments when only one voltage at the contact is varied [@Lofgren; @marlow; @Bouchiat] or when two contact voltages are shifted oppositely [@Zumbuhl; @ensslin] measure different nonlinear currents. Indeed, for a current $I(\{V_i\})$, bilinear in voltages, its second derivative should depend on the chosen direction in the space of voltages $\{V_i\}$. Interestingly, a sample with weak interactions is very sensitive to the choice of the bias mode, which we attribute to the strong effect of capacitive coupling of the sample with nearby conductors. To make our arguments quantitative and consider the role of magnetic flux $\Phi$ for a quantum dot which is (generally) AC-biased at arbitrary frequency $\oo$, in Sec. \[sec:DC\] we take electronic interference into account. Having done that, we illustrate the interplay between interactions and interference on several important examples. First, we consider nonlinear transport due to a constant applied voltage and then consider rectification of AC voltages. For a two-terminal dot, in a generally asymmetric circuit (capacitive couplings included), in Sec. \[sec:2terminal\] we find the statistics of (anti) symmetric conductances $\Ga,\Gs$ defined in Eq. (\[eq:IVfield\]). Both $\Ga$ and $\Gs$ vanish on average. Quantum fluctuations of $\Gs$ strongly depend on the interaction strength, circuit asymmetry and bias mode. This is in accordance with our qualitative picture. On the other hand, the antisymmetric component $\Ga$ depends only on interactions. Our arguments agree with recent experiments in quantum dots:[@Lofgren; @Zumbuhl] depending on the bias mode different features of the nonlinear conductance tensor are probed. The fluctuations of nonlinear current can be minimized or maximized (on average), which becomes important for weakly interacting electrons. Curiously, for symmetric coupling (transmission and capacitance) of contacts and dot the bias mode in which the voltages at the contacts are changed in opposite directions generally [*minimizes*]{} fluctuations of $\Gs$. Consequently, such a mode is more advantageous for the observation of $\Ga$ or a cleaner linear signal. Near the end of Sec. \[sec:2terminal\] we also demonstrate how to take into account possible classical circuit-induced asymmetry [@Lofgren] due to the finite classical resistance of the wires. In Sec. \[sec:rectify\] we present results elucidating the role of interaction in rectification through two-terminal dots. Usually there are two important time-scales: the dwell time $\dwell$ an electron spends inside the dot and the charge relaxation time $\RC\leq\dwell $ of the dot. For a given geometry, the dwell time depends on the coupling of the dot with reservoirs, but the charge relaxation time is also sensitive to the interaction strength. We have $\RC\ll\dwell$ for strong interactions and $\RC=\dwell$ in the weak interaction limit. Our results for fluctuations of $\Ga(\oo),\Gs(\oo)$ are obtained for arbitrary frequency $\oo$. Although the fluctuations of both $\Ga(\oo)$ and $\Gs(\oo)$ monotonically decrease when $\oo\to\infty$, as functions of frequency $\oo$ they behave differently. At nonadiabatic frequencies $\oo\dwell\gg 1$ the nearby gate short-circuits currents. This effect is even in magnetic field and thus affects only $\Gs$. As a result, for a high-frequency voltage the asymptotes of $\Ga$ and $\Gs$ are generally different and strongly depend on the bias mode. Since the regime of parameters is quite realistic, we expect that the predicted difference of $\Gs(\oo)$ and $\Ga(\oo)$ should be experimentally observable. In the noninteracting limit our results qualitatively agree with those in diffusive metallic junctions. Our model of a ring consisting of a chaotic dot with a ballistic arm which encloses an AB flux is presented in Section \[sec:phase\]. Although it is impossible to find the full mesoscopic distribution of the AB phase $\delta$, its shape can be discussed qualitatively. Since $\tan\d$ is similar to the asymmetry parameter $\A=\Ga/\Gs$ in quantum dots, its distribution can become very wide for a particular choice of the bias mode. On average $\la\d\mbox{(mod)}\pi\ra=0$ in our model, and we find the dependence of the fluctuations of $\d$ on temperature, interactions, and number of channels of the contacts and the arm. Our treatment allows a straightforward generalization to treat AC voltages applied to the ring. The technical calculations are presented in the Appendix. \[\]\[\]\[0.8\][$C_{g0}$]{}\[\]\[\]\[0.8\][$C_{g1}$]{} [![(Left) Rectified current is measured through a coherent quantum dot biased by voltages with (AC) amplitude $V_{i,\oo},i=1,2$ at reservoirs connected by $N_{i}$ ballistic channels and capacitances $C_{i}$ and by voltages $V_{gi,\oo}$ applied at additional gates with capacitances $C_{gi}$. Transport through the dot is sensitive to the total magnetic flux $\Phi$ through the area of the dot. (Center and right) Forward and reverse connection of Ref. exchange voltages at the contacts and classical resistors $r_{1,2}$.[]{data-label="fig:3dot"}](3dotnew.eps "fig:"){width="9.cm"}]{} Model {#sec:model} ===== The 2D quantum dot, see the left panel in Fig. \[fig:3dot\], is biased with several voltages $\{V_{i}\}$ at $M$ ballistic quantum point contacts (QPCs) with $N_{i}, i=1,...,M$ orbital channels. The reservoirs can be capacitively coupled to the dot via capacitances $C_i$. An additional set of voltages $\{V_{gi}\}$ is applied to (several) gates with capacitances $C_{gi}$. All perturbations are assumed to be at the same frequency $\oo$, which is not necessarily small (adiabatic). The dot is in the universal regime, [@Beenakker] when the Thouless energy $E_{\rm Th}=\hbar/\erg$ is large. The dots with area $A=\pi L^2$ (taken circular) are either diffusive with mean free path $l\ll L$, or ballistic, with $l\gg L$ and chaotic classical dynamics (in the latter case the substitution $l\to \pi L/4$ should be used). The mean level spacing (per spin direction) $\Delta=2\pi\h^2/(m^*A)$ and the total number of ballistic channels $N$ together define the dwell time $\dwell=h/(N\Delta)\gg \erg$. We also require that $eV\ll N\Delta$ when we can treat the nonlinearity only to $(eV)^2$. Scattering is spin-independent and this spin degeneracy is accounted for by the coefficient $\nu_s$. The noninteracting electrons are treated using the scattering matrix approach and Random Matrix Theory (RMT) for the energy-dependent scattering matrix $\S(\e)$. For details we refer the reader to reviews. [@Beenakker; @ABG] In this approach the fundamental property of a dot is its scattering matrix $\S$ distributed over circular ensembles of proper symmetry, see Ref. (An alternative method is the Hamiltonian approach based on the properties of the dot’s Hamiltonian $\cal H$ taken from a Gaussian Ensemble. [@ABG]) Transport properties of chaotic dots in RMT for matrices $\S$ or $\cal H$ are usually expressed in terms of an effective, magnetic field-dependent number of channels. Predictions based on this approach are in good agreement with experiment. For multichannel samples with $N\gg 1$ we use the diagrammatic technique described in Refs. and . However, when interactions are present, this treatment should be modified. The approach which assumes that in a pointlike scatterer the interactions appear in the form of a self-consistent potential was introduced by Büttiker and co-authors [@buttiker1] on the basis of gauge-invariance and charge conservation. This (Hartree) approach neglects contributions leading to Coulomb blockade (Fock terms), but is a good approximation for open systems. If the screening in the dot inside the medium with dielectric constant $\e$ is strong, $r_s=(k_{\rm F}a_B)^{-1}=e^2/(\e\hbar v_{\rm F})\lesssim 1$, an RPA treatment of Coulomb interactions is sufficient. For large dots, $L\gg a_B$, the details of screening potential on the scale $\sim a_B$ are not important and we can assign an electric potential $U(\vec r,t)$ defined by excess electrons at $\vec r,t$ at any point $\vec r$ of the sample. If additionally the number of ballistic channels $N$ is much smaller than the dimensionless conductance of a closed sample, $g_{\rm dot}=\Thou/\Delta\gg N$, the potential drops over the contacts and therefore in the interior of the dot it can be taken uniform (“zero-mode approximation”). [@ABG] This potential shifts the bottom of the energy band in the dot and thus modifies the $\S$-matrix. As a consequence, electrons with kinetic energy $E$ have an electro-chemical potential $\tilde E_\a=E-eV_\a$ in the contact $\a$ and $\tilde E=E-eU$ in the dot. (We point out that we neglect the quantum pumping in the dot and consequently the $\S$-matrix depends only on one energy.) Recently, Brouwer, Lamacraft, and Flensberg demonstrated that this self-consistent approach gives the leading order in an expansion in the inverse number of channels $1/N\ll 1$. [@BLF] Therefore, our analytical results present the leading order effect, valid for $1/N\ll 1$. In the self-consistent approach the influx of charge changes the internal electrical potential of the dot $U(t)$, which in turn affects the currents incoming through each conducting lead and/or redistributes charges among the nearby conductors (gates). Such capacitive coupling can often be estimated simply from the geometrical configuration. For example, the capacitance of a dot covered by a top gate at short distance $d\ll L$ is $C\sim \e L^2/d$ and a single quantum dot has $C\sim \e L$. The ratio of charging energy $E_c\sim e^2/C$ to mean level spacing $\Delta$ characterizes the interaction strength. It is proportional to the ratio of the smallest geometrical scale to the effective Bohr’s radius, $E_c/\Delta\sim \mbox{min }\{d,L\}/a_B$. We refer to interactions as strong if $E_c\gg \Delta$ and weak if $E_c\ll \Delta$. Importance of bias mode {#sec:bias} ======================= [![Depending on the bias mode, the experiment probes different transport properties. Plots present (left) linear and (right) nonlinear components of the current as functions of $x=V_1-V_0$ and $y=V_2-V_0$, the dashed curves correspond to equal currents. Thin line shows fixed $V_2$ and $I(\tilde V)$ is a function of source voltage $\tilde V=V_1$. Thick line corresponds to fixed $V_1+V_2$, such that $I(\tilde V)$ depends only on $\tilde V=(V_1-V_2)/\sqrt{2}$. Full and empty dots on the right figure correspond to the forward or reverse configurations shown in Fig. \[fig:dot\].[]{data-label="fig:Vaxes"}](Vaxes.eps "fig:"){width="9.cm"} ]{} We suppose for simplicity that at equilibrium the voltages $V_1=V_2=V_0$ are set. In the following we consider the situation when the (single) gate voltage $V_0$ is held fixed at its equilibrium value. Experiments can be performed in different [*bias modes*]{}, usually either (i) with fixed drain voltage $V_2$ or (ii) at fixed $V_1+V_2$ (the variations of the voltages at the contacts are equal in magnitude but opposite in sign). These different modes correspond to straight lines in the $\{V_1 , V_2\}$ plane shown in Fig. \[fig:Vaxes\]. Let us consider the nonlinear current as a function $I(x,y)$, where $x=V_1-V_0$ and $y=V_2-V_0$ are deviations of contacts voltages from equilibrium. For generality we consider below a situation when the linear combination $-x\sin(\eta-\pi/4)+y\cos(\eta-\pi/4)=0$ is held fixed and the only variable is $$\begin{aligned} \label{eq:tildeV} \tilde V=x\cos(\eta-\pi/4)+y\sin(\eta-\pi/4).$$ This corresponds to a rotation of the original $x , y$ axes such that the new coordinate axis $\tilde V$ makes an angle $\eta$ with the $y=-x$ line, as illustrated in Fig. \[fig:Vaxes\]. The value of $\eta$ fully characterizes the bias mode. Now the two modes introduced above are simply (i) $\eta=\pi/4$ which implies $\tilde V=x$; and (ii) $\eta=0$, which implies $\tilde V=(x-y)/\sqrt{2}$ and corresponds to an asymmetric variation of the voltages. The linear current depends only on $x-y$ (dashed lines on the left panel in Fig. \[fig:Vaxes\] correspond to the lines of equal currents) and in any bias mode the measured linear current $I_{\rm lin}$ is the same for a given $x-y$. If we consider the nonlinear current $I$ as a function of $x,y$, it is by construction a bilinear function of $x,y$. As in the linear case the current must vanish if the voltages are the same and thus $I=0$ for $x-y=0$. Therefore, the bilinear function must be of the form $$\begin{aligned} \label{eq:simpleI} I=I_0 \,\left[ (x+y)\cos\phi+(x-y)\sin\phi\right]\, (x - y)\end{aligned}$$ with unknown (generally fluctuating) parameters $I_0$ and $\phi\in(-\pi/2,\pi/2]$. It is important that the qualitative behavior of $I(x,y)$ depends on the interaction strength: one could expect that transport depends not only on voltages in the leads, but also on the internal nonequilibrium potential $U$ of the sample. This potential can be found if potentials in all reservoirs and the nearby gate are known. In the limit of weak interactions the equilibrium point $V_0$ is important, and if we reverse the bias voltage, $(V,0)\to (0,V)$ the current is fully reversed, that is $\DD^2_{xx}I=-\DD^2_{yy}I$. For the current defined in Eq. (\[eq:simpleI\]) it is possible only when $I\propto (x-y)(x+y)\Rightarrow \phi=0$. Another way to see this is to use the usual expression for the total current in terms of scattering matrices. In this formula the current depends on the difference between Fermi distributions in the leads $\propto f(\e-ex)-f(\e-ey)$, and its expansion up to the second order yields $f''(\e)(x^2-y^2)$. The lines of equal current are curved and directions $\eta=0,\pm \pi/2$ correspond to zero current directions. Thus the dependence of current on the angle $\eta$ is strong. In addition this approach predicts that the current through a two-terminal sample is symmetric with respect to the magnetic flux inversion. In contrast, for strong interactions, the value of $V_0$ is irrelevant and the nonequilibrium electrical potential $U$ is independent of $V_0$. In this case current depends only on the voltage difference $x-y$ and thus $I\propto (x-y)^2\Rightarrow |\phi|=\pi/2$. The equal-current lines are straight and the picture is similar to the left plot in Fig. \[fig:Vaxes\] for linear transport. Therefore we do not expect any nontrivial dependence of the nonlinear current on the choice of the bias mode. It is noteworthy that qualitative considerations can predict neither the sign, nor the magnitude of $I_0$. The only general conclusion which we can make for a weakly or strongly interacting dot is $I(x,y)\propto x^2-y^2$ and $I(x,y)\propto (x-y)^2$, respectively. Experiments extract derivatives of $I$ with respect to the applied voltages. Importantly, this derivative depends on the chosen direction $\eta$. The nonlinear current measured in this bias mode is $$\begin{aligned} \label{eq:IVangle} \label{eq:Iexample} I(\eta)&=& I_0 \tilde V^2\cos\eta\sin (\phi+\eta).\end{aligned}$$ The current is zero when $\eta=-\phi$ and $\eta=\pm\pi/2$, and the bisectrix of the angle between the two zero-current directions at $\eta=-\phi/2+(\pi/4) \mbox{sgn }\phi$ maximizes $\DD^2 I/\DD\tilde V^2$. Sometimes experiments extract information on nonlinearity from measurements in different connections schematically shown in the central and right panels in Fig. \[fig:3dot\]: “Forward” connection corresponds to $x=\pm V,y=0$, while “reverse” connection for the same voltage configuration corresponds to $x=0,y=\pm V$. The gate voltages $V_g$ are kept fixed. In Fig. \[fig:Vaxes\] these forward and reverse points are indicated by black and white dots, respectively. To find the nonlinear conductance Marlow [@marlow] and Löfgren [@Lofgren] determine the difference of conductance at these measurement points. Löfgren [@Lofgren_2004; @Lofgren] use the term “rigidity” for samples for which $G_f(V)=G_r(-V)$ in the points $f^+ =(V,0)$ and $r^- =(0,-V)$. [@Lofgren_2004] Equation (\[eq:simpleI\]) gives the nonlinear contribution $\G$ to the full conductance $G_{f,r}(\pm V)$: $$\begin{aligned} \G\propto I_0\left[(x-y)\sin\phi+(x+y)\cos\phi\right]\,.\end{aligned}$$ Thus for a sample which is called rigid this implies $I_0\cos\phi\to 0$. Since $I_0 = 0$ would mean that there is no second-order response, we must have $\cos\phi\to 0$ which is the case for samples with strong interaction. In other words, “rigidity” in samples which exhibit $O(V^2)$ current is equivalent to strong Coulomb interactions. On the other hand, comparison of data at another pair of points $f^+=(V,0)$ and $r^+=(0,V)$ gives $G_f(V)-G_r(V)\propto I_0\sin\phi$ and provides [*additional*]{} information about the two fluctuating quantities $I_0,\phi$. Reference expects that a Left-Right (LR)-symmetric system has $G_f(V) = G_r(V)$. Therefore rigid and LR-symmetric sample should necessarily have $I_0 \to 0$ and thus could not exhibit a second-order current $O(V^2)$. This point is discussed more quantitatively in Sec. \[sec:2terminal\]. It is important to note that to find the linear DC current one needs to know only $x-y=V_1-V_2$, while for the nonlinear current in general one needs two variables $x=V_1-V_0,y=V_2-V_0$ or any independent pair of their linear combinations. The projection of the vector $(V_1,V_2,V_0)$ on the $V_1+V_2+V_0=$const plane uniquely defines the nonlinear current. This projection can be parametrized by the pair of Cartesian $(x,y)$ or axial coordinates $(\tilde V,\eta)$. However, if in the experiment the voltages $V_{1,2}$ were fixed, this would not be enough to define $(x,y)$ uniquely. In this case Ref. points to the importance of the reference point $V_0$. Indeed, one could arrive at the point with a given $(V_1,V_2)$ from any equilibrium point and the measured current would depend on $V_0$. We prefer to characterize the measurement by the pair $(\tilde V,\eta)$ instead of three variables $(V_1,V_2,V_0)$ because of the simplicity of the final results. The weaker the interaction (or the stronger the capacitive coupling of the sample to the nearby gate) the more important the role of $\eta$ chosen in experiment. We illustrate this important conclusion by quantitative results for nonlinear conductance $\G\propto\DD^2 I/\DD\tilde V^2$ in the following sections. We point out that conductance with respect to the voltage difference $V=V_1-V_2$ is often used, even when a linear combination $\tilde V$ is actually varied in experiment. Voltages $\tilde V$ and $V$ are related, $\tilde V=V/\sqrt{2}\cos\eta$, and one can straightforwardly find $\DD^2 I/\DD V^2$. Generation of DC in quantum dots {#sec:DC} ================================ Now we quantify the qualitative arguments of Sec. \[sec:bias\] and consider the more general situation of a DC current generated by an AC bias. If at first we neglect Coulomb interactions, the nonlinear DC current $I_\a$ in response to the Fourier components $V_{\b,\oo}=V_\b e^{i\phi_\b}$ of the AC voltages applied at the contacts $\b=1,...,M$, can be expressed with the help of the DC-conductance matrix $g_{\a\b}(\e)$ of the dot at the energy $\e$ [@pedersen] $$\begin{aligned} \label{eq:Pedersen} I_\a &=&\frac{\nu_s e^3}{h}\int d\e \frac{f(\e+\hbar\oo)+f(\e-\hbar\oo)-2f(\e)}{(\hbar\oo)^2} \nonumber \\ &\times& \sum_{\b=1}^M g_{\a\b}(\e)|V_{\b,\oo}|^2,\\ \label{eq:gDC} g_{\a\b}(\e)&=&\Tr [{1\!\! 1}_\a\d_{\a\b}-\S^\dagger(\e) {1\!\! 1}_\a\S(\e){1\!\! 1}_\b].$$ If we now include interactions using a self-consistent potential $U_{\oo}$ this formula is modified: [@pedersen] in Eq. (\[eq:Pedersen\]) the Fourier components of the voltages at [*all*]{} contacts are shifted down by the Fourier component of the internal potential $-U_\oo$ $$\begin{aligned} U_\oo &=&\sum_\g u_\g V_{\g,\oo}, \,\,u_\g= \frac{\sum_{\b}G_{\b\g}(\oo)-i\oo C_\g}{\sum_{\b\g}G_{\b\g}(\oo)-i\oo C_\Sigma}\label{eq:uomega},\\ G_{\b\g}(\oo)&=& \frac {\nu_s e^2}{h}\int d\e\,\Tr\left[{1\!\! 1}_\b{1\!\! 1}_\g - {1\!\! 1}_\g{\cal S}^\dagger(\e){1\!\! 1}_\b{\cal S}(\e+\hbar\oo)\right]\nonumber \\ &\times& \frac{f(\e)-f(\e+\hbar\oo)}{\hbar\oo}.\label{eq:sumG}\end{aligned}$$ In Eq. (\[eq:uomega\]) the index $\g$ runs not only over real leads $1,...,M$, but also over all gates $gi$. However, when $\g\in\{gi\}$ the AC conductance $G_{\b\g}(\oo)$ is absent and only capacitive coupling $i\oo C_\g$ remains in the numerator. We point out that the matrix $ G(\oo)$ of dynamical AC conductance at frequency $\oo$ given in Eq. (\[eq:sumG\]) should not be confused with the degenerate matrix $g(\e)$ of energy-dependent DC conductances of electrons with kinetic energy $\e$ given in Eq. (\[eq:gDC\]). The results of Ref. can be expressed in terms of the DC conductances $g_{\a\b}$ and frequency-dependent characteristic potentials $u_\g$, $$\begin{aligned} \label{eq:current4omega} I_\a &=&\frac{\nu_s e^3}{h}\int d\e \frac{f(\e+\hbar\oo)+f(\e-\hbar\oo)-2f(\e)}{(\hbar\oo)^2} \nonumber \\ &\times &\sum_{\b\g} g_{\a\b}(\e)\mbox{Re }\, u_\g |V_{\b,\oo}-V_{\g,\oo}|^2.\end{aligned}$$ Here $\mbox{Re }u_\g$ stands for the real part of $u_\g$, which is in general a complex quantity. In contrast to Eq. (\[eq:Pedersen\]), Eq. (\[eq:current4omega\]) is expressed via differences of voltages applied to all present conductors. Therefore, the current is gauge-invariant. The charge conservation, $\sum_\a I_\a=0$, is obvious from Eq. (\[eq:gDC\]). From this point on we consider Eq. (\[eq:current4omega\]), a specific expression of Eq. (\[eq:IV\]), in detail for several regimes. In Sec. \[sec:2terminal\] we discuss the nonlinear current due to DC applied voltages (previously considered in Ref. ) and the importance of different bias modes in experiments in two-terminal quantum dots. In Sec. \[sec:rectify\] we consider the frequency dependence of $\Gs(\oo)$ and $\Ga(\oo)$. Nonlinearity in quantum dots {#sec:2terminal} ---------------------------- In the static limit [@ChristenButtiker] $\hbar\oo/T\to 0$ the integrand in the first line of Eq. (\[eq:current4omega\]) simplifies to $f''(\e)$ and for $\hbar\oo/N\Delta\to 0$ the derivatives $u_\g$ are real and expressed via subtraces of the Hermitian Wigner-Smith matrix $\S^\dagger \DD_\e\S/(2\pi i)$ [@WignerSmith; @BP] $$\begin{aligned} \label{eq:current4} I_\a &=&\frac {-\nu_s e^3}{h}\sum_{\b\g}\int f'(\e)d\e g'_{\a\b}(\e) u_{\g}(V_\b-V_\g)^2,\\ \label{eq:u} u_\g &=&\frac{ C_\g/\nu_s e^2-\int d\e f'(\e)\Tr {1\!\! 1}_\g\S^\dagger \DD_\e\S/(2\pi i) }{C_\Sigma/\nu_s e^2 -\int d\e f'(\e)\Tr \S^\dagger \DD_\e\S/(2\pi i)}\label{eq:u0}.\end{aligned}$$ For a two-terminal sample the nonlinear current through the first lead is $$\begin{aligned} \label{eq:current} I_1 &=&\frac {-\nu_s e^3}{h}\int f'(\e)g'_{11}(\e)d\e\left[\sum_i u_{gi}\left[(V_1-V_{gi})^2 \right.\right.\nonumber \\ && \mbox{} \left.\left. -(V_2-V_{gi})^2\right]+(u_2-u_1)(V_1-V_2)^2\right].\end{aligned}$$ The characteristic potentials in the last term of Eq. (\[eq:current\]) are sensitive to the asymmetry of the contacts. Indeed, in a strongly interacting dot $u_{gi}=0$ and $u_2-u_1\approx (N_2-N_1)/N$. The current magnitude grows with asymmetry due to the last term in Eq. (\[eq:current\]). On the other hand, the sign of $I_1$ is random because of quantum fluctuations of $g'_{11}$ around zero. [@deriv] As a consequence, if in an experiment the Fermi level is shifted by $\d\m_{\rm F}\sim N\Delta/2\pi$ (or the shape of the dot is changed) the sign of nonlinearity can be inverted. Different modes of bias having been discussed in Sec. \[sec:bias\], we concentrate here on the (anti)symmetric conductances through the quantum dot at fixed gate voltages. When the reservoir voltages are varied in the $\eta$ direction, the nonlinear current is given by the expression $$\begin{aligned} \label{eq:derivIV} I &=&\frac {-2\nu_s e^3}{h}\int f'(\e)g'_{11}(\e)d\e\left[(1-u_1-u_2)\sin \eta\right.\nonumber \\ &&\left.+(u_2-u_1)\cos\eta\right]\cos\eta\tilde V^2,\end{aligned}$$ and one can define exactly the unknown parameters $I_0,\phi$ which we introduced in the qualitative argument leading to Eq. (\[eq:Iexample\]). Depending on $\eta$ one measures different linear combinations of conductances. If we consider conductances $\DD^2 I/2\DD \tilde V^2$ in units of $\nu_s e^3/h$, Eqs. (\[eq:IVfield\]) and (\[eq:derivIV\]) yield $$\begin{aligned} \label{eq:defG} {\cal G}_{a,s}=\frac{2\pi\cos^2\eta\int d\e d{\tilde \e}f'(\e)f'(\tilde \e)\chi_1(\e)\chi_{2,a(s)}(\tilde \e)}{\Delta^2[C_\Sigma/(e^2\nu_s)-\int d\e f'(\e)\Tr\S^\dagger \DD_\e\S/(2\pi i)]}\end{aligned}$$ expressed in terms of fluctuating functions $\chi$ and a traceless matrix $\Lambda= (N_2/N){1\!\! 1}_1-(N_1/N){1\!\! 1}_2$: $$\begin{aligned} \label{eq:chi1} \chi_1(\e) &=& (\Delta/2\pi)\DD_\e \Tr\Lambda {\cal S}^\dagger\Lambda{\cal S}, \\ \label{eq:chi2a}\chi_{2,a}(\e) &=&(i\Delta/2\pi) \Tr\Lambda[\S^\dagger,\DD_\e\S],\\ \label{eq:chi2s} \chi_{2,s}(\e) &=&\Delta\left(\frac{C_0\tan\eta+C_2-C_1}{e^2\nu_s} +\frac{N_2-N_1}{N}\frac{\Tr \S^\dagger\DD_\e\S}{2\pi i }\right.\nonumber \\ &&\left. +\frac{1}{2\pi i} \Tr\Lambda\{\S^\dagger,\DD_\e\S\}\right).\end{aligned}$$ Standard calculations using the Wigner-Smith and/or $\S$-matrix averaging [@waves; @PietBeen; @iop] yield $\la\Ga\ra=\la\Gs\ra=0$. This result signifies that the nonlinear current through a quantum dot is indeed a quantum effect. As a consequence the size of the measured nonlinearity must be evaluated from correlations of $\Ga,\Gs$. The functions $\chi_1(\e,\Phi)$ and $\chi_{2,a/s}(\e',\Phi')$ are uncorrelated, and their autocorrelations [@PhysicaE] readily allow one to find statistical properties of ${\cal G}_{a,s}$. Our results can be expressed in terms of diffuson $\Diff$ or cooperon $\Coop$ in a time representation, $\exp(-\tau/\tau_{\Diff})$ and $\exp(-\tau/\tau_{\Coop})$. Both can be introduced using the $\S$-matrix correlators [@PVB] (correlations of retarded and advanced Green functions lead to the same expression up to a normalization constant [@ABG]). We have $$\begin{aligned} {\cal S}(\tau,\Phi) &=& \int\frac{d\e}{2\pi\hbar} \, {\cal S}(\e,\Phi)e^{i \e\tau/\hbar},\nonumber \\ \langle {\cal S}_{i j} (\tau,\Phi) {\cal S}^{*}_{k l} (\tau',\Phi')\rangle &=& (e^{-\tau/\tau_\Diff}\delta_{ik} \delta_{jl} + e^{-\tau/\tau_\Coop} \delta_{il} \delta_{jk})\nonumber \\ &\times &\frac{\Delta}{2\pi\hbar}\delta(\tau-\tau') \theta(\tau), \label{eq:cum1t}\\ \label{eq:channels} \tau_{\Coop,\Diff}=\frac{h}{N_{\Coop,\Diff}\Delta},\, {\genfrac{\{}{\}}{0pt}{}{N_{\cal C}}{N_{\cal D}}} &=& N+\frac{(\Phi\pm\Phi')^2}{4\Phi_0^2}\frac{h v_F l}{L^2\Delta}.\end{aligned}$$ We also introduce the electrochemical capacitance $C_\m$ [@PietMarkus] which relates the non-quantized mesoscopically averaged excess charge $\la Q\ra$ in the dot in response to small shift of the voltages $\d V$ at all gates. In addition the charge relaxation time $\RC$ of the dot is conveniently introduced by this electrochemical capacitance and the total contact resistance, $$\begin{aligned} \label{eq:excess} C_\m= \frac{\la \d Q\ra}{\d V}=\frac{C_\Sigma}{1+C_\Sigma\Delta/(\nu_s e^2)},\,\,\,\RC= \frac{hC_\m}{\nu_s Ne^2}.\end{aligned}$$ The denominator of Eq. (\[eq:defG\]) is a self-averaging quantity, $\la(...)^2\ra=\la (...)\ra^2=\Delta^2(C_\Sigma/C_\m)^2$. Using the diffusons and cooperons defined in Eq. (\[eq:cum1t\]) we find the following correlations of $\Ga$ and $\Gs$: $$\begin{aligned} \label{eq:main} &&{\genfrac{\{}{\}}{0pt}{}{\la\Ga(\Phi)\Ga(\Phi')\ra}{\la\Gs(\Phi)\Gs(\Phi')\ra}} = {\genfrac{\{}{\}}{0pt}{}{{\mathcal F}_\Diff-{\mathcal F}_\Coop}{{\mathcal F}_\Diff+{\mathcal F}_\Coop+X}}({\mathcal F}_\Diff+{\mathcal F}_\Coop)\nonumber \\ &&\times \left(2\cos^2\eta\frac{2\pi} {\Delta}\frac{C_\m}{C_\Sigma}\right)^2\frac{N_1^3 N_2^3}{N^6},\\ && {\cal F}_{\l}=\left(\frac{\Delta T }{2\hbar^2}\right)^2\int\frac {\tau_\l\tau^2 e^{-\tau/\tau_\l}}{\sinh^2 \pi T \tau/\hbar}d\tau, \label{eq:Fraw}\\ &&\label{eq:X}X= \frac{N^2}{2N_1N_2} \left(\frac{C_0\tan\eta+C_2-C_1}{\nu_s e^2/\Delta} +\frac{N_2-N_1}{N}\right)^2.\end{aligned}$$ There are two very different contributions to Eq. (\[eq:main\]), ${\mathcal F}_{\Coop,\Diff}$ due to quantum interference and $X$ defined by the classical response of the internal potential to external voltage. The terms denoted by ${\cal F}_{\Coop,\Diff}$ are sensitive to temperature, magnetic field, and decoherence. Asymptotical values of ${\cal F}$ in the low temperature, $T\ll \hbar/\tau_{\l}$, or high temperature limits, $T\gg \hbar/\tau_{\l}$, are ${\cal F}_\l \to 1/N_\l^2=(\tau_\l\Delta/h)^2$ and ${\cal F}_\l \to \Delta/(12 T N_\l)=\tau_\l\Delta^2/ (12 hT)$, respectively. The term denoted by $X$ and given by Eq. (\[eq:X\]) contains only quantities specifying the geometry of the sample and gates and the bias mode. In a real experiment the coupling due to capacitances $C_{1,2}$ is usually stronger then that of the external gates, $C_{1,2}\gg C_0$. Symmetrization of the circuit $C_1=C_2$ can diminish the value of $X$. If in addition $N_1=N_2$ and $\eta=0$ (used in the experiments [@Zumbuhl; @ensslin]) we have $X\to 0$. Thus such a symmetric setup and bias mode minimize the fluctuations of the nonlinear current and actually would be best for an accurate measurement of [*linear*]{} transport. Indeed, this regime is not affected by the fluctuations of capacitive coupling $u_0$ of the dot with the nearby gate and thus minimizes fluctuations of $\Gs$ around 0. Fluctuations of $\Ga,\Gs$ are given by different expressions, see the first line of Eq. (\[eq:main\]), where the first term is due to $\la\chi_{2,a}^2\ra$ or $\la\chi_{2,s}^2\ra$. Importantly, $\la\chi_{2,s}^2\ra$ contains both quantum ${\mathcal F}_{\l}\lesssim 1/N_\l^2$ and classical $X$ contributions. If the classical term dominates, $X\gg 1/N^2$, the current is mostly symmetric, $\Gs^2\gg\Ga^2$. This could be expected either for a weakly interacting dot or a very asymmetric setup, $N_1\neq N_2$ .[@PhysicaE] However, if the classical term is reduced due to, e.g., the bias mode, the fluctuations of $\Ga$ and $\Gs$ become comparable. This experimentally important conclusion remains valid for [*any interaction strength*]{}. (Particularly, it leads to a very wide distribution of the Aharonov-Bohm phase considered in Sec. \[sec:phase\].) Experiments of Zumbühl [@Zumbuhl] and Leturcq [@ensslin] are performed in this regime when $\eta=0$ and $X\to 0$. Data in Ref. demonstrate that the part of the total current symmetrized with respect to magnetic field is by far dominated by linear conductance. From Eq. (\[eq:main\]) we expect mesoscopic fluctuations in linear conductance to be $\sim N^2$ times larger then those of $\Gs\Delta$. Thus only when the number of channels is decreased will the nonlinear $\Gs$ become noticeable. A clear observation of $\Gs$ without linear transport contribution was performed in a DC Aharonov-Bohm experiment by Angers [@Bouchiat] in the mode $\eta=\pm \pi/4$ (only one contact voltage was varied). This allowed to evaluate the interaction strength from the ratio of $\Gs/\Ga$. Experiments of Marlow [@marlow] and Löfgren [@Lofgren] measure the full two-terminal conductance and extract nonlinear conductance properties related to various spatial symmetries of the dot. Although the current through a weakly interacting sample is field-symmetric, this is not true in general. Samples of Ref. differ in “rigidity” and degree of symmetry. Rigid samples, $u_0\to 0$, with Left-Right(LR) and Up-Down (UD)-symmetry should have $(u_2-u_1)_s=0$ and $(u_2-u_1)_a=0$ respectively, according to the expectations of Löfgren [@Lofgren] (indices $s$ and $a$ mean the symmetric and antisymmetric part in magnetic field). Due to quantum fluctuations, in experiment none of these symmetry-relations can be exactly fulfilled, see Eq. (\[eq:main\]). According to Eq. (\[eq:derivIV\]), the difference in the full conductances $g=(h/\nu_s e^2)I/V$ measured between different points probes different characteristic potentials. Reference defines three differences $g_{\rm i,ii,iii}$ for three pairs of points in the forward and reverse connection discussed after Eq. (\[eq:IVangle\]). Using Eq. (\[eq:main\]) we find (i) $g_{\rm i}\equiv g_f(V,B)-g_r(-V,B)\propto u_0$, (ii) $g_{\rm ii}\equiv g_f(V,B)-g_f(V,-B)\propto (u_2-u_1)_a$, and (iii) $g_{\rm iii}\equiv g_f(V,B)-g_f(-V,-B)\propto (u_0+u_2-u_1)_s$. The ensemble average of these differences vanishes and their fluctuations for $C_{1,2}=0,N_1=N_2$ are given by $$\begin{aligned} \left\{\begin{array}{c} g_{\rm i}^2 \\ g_{\rm ii}^2 \\ g_{\rm iii}^2\\ \end{array}\right\}=\left\{\begin{array}{c} X \\ {\mathcal F}_\Diff-{\mathcal F}_\Coop\\ X+{\mathcal F}_\Diff+{\mathcal F}_\Coop\\ \end{array}\right\}({\mathcal F}_\Diff+{\mathcal F}_\Coop) \left(\frac{\pi e V} {2\Delta}\frac{C_\m}{C}\right)^2,\nonumber\end{aligned}$$ where $X=2(C/C_\m-1)^2$ is found from Eq. (\[eq:X\]) at $\eta=\pm \pi/4$. In weakly interacting dots $C_\m/C\to 0$ and only magnetic-field symmetric signals $g_{\rm i}$ and $g_{\rm iii}$ survive. In strongly interacting (“rigid”) dots $C_\m/C\to 1$ and $g_{\rm ii}$ becomes similar to $g_{\rm iii}$. We point out that even if the rigid samples are made symmetric with respect to Left-Right inversion, the quantum fluctuations of the sample properties are unavoidable and $g_{\rm ii}^2\neq 0$ at $\Phi\neq 0$. For high magnetic fields and arbitrary interactions ${\mathcal F}_\Coop\to 0$ and experiment should observe $g_{\rm i}^2+g_{\rm ii}^2=g_{\rm iii}^2$. Clearly fluctuations exist also for large magnetic fields beyond the range of applicabilty of RMT. Experimental data (see inset of Fig. 6 in Ref. ) show that $g_{\rm i}^2+g_{\rm ii}^2\sim g_{\rm iii}^2$. It is hard to make a quantitative comparison with Refs. and , since the quantum fluctuations in the nonlinear conductance exist possibly on the background of classical effects due to macroscopic symmetries. We expect that quantum effects become more pronounced as contacts are narrowed. To conclude this subsection we briefly discuss here the case of a macroscopically asymmetric setup. If the experiment were aimed to measure large $\Ga$ compared to $\Gs$, one would try to minimize $\Gs$ by adjusting the setup. Such a procedure minimizes the value of $X$ in Eq. (\[eq:X\]). For $C_{1,2}=0, \eta=\pi/4$ the role of asymmetric contacts $N_1\neq N_2$ was discussed in Ref.. Analogously, one could consider a more general case of $C_{1,2}$ and an arbitrary bias mode $\eta$. This is especially important if the difference $C_1\neq C_2$ can not be neglected due to occasional loss of contact symmetry. The results of an experiment could also be affected by the presence of classical resistance loads $r_{1,2}$ between macroscopic reservoirs and the dot (shown in Fig. \[fig:3dot\]). Swapping of such resistances in the experiment, when connection is switched between “forward” and “reverse” [@Lofgren] affects the voltage division between loads. If we assume the capacitive connection of the dot and reservoirs is still the same, the modification of the expression for $u_\g$ in Eq. (\[eq:uomega\]) is straightforward, $\sum_{\b}G_{\b\g}\to \sum_{\b}G_{\b\g}/(1+r_\g \sum_{\b}G_{\b\g})$. Naturally, at large $r_\g$, $(2 e^2 N_\g/h)r_\g \gg 1$, the main drop of the voltage occurs over the resistor $r_\g$ and not over the QPCs. As a consequence, if $r_{1,2}\neq 0$, values of $u_{1,2}$ can become unequal due to $r_1\neq r_2$ and this leads to the classical circuit asymmetry which we do not consider here. Rectification in quantum dots {#sec:rectify} ----------------------------- Here we consider the DC generated by a quantum dot subject to an AC bias at the frequency $\oo$. In experiment at high bias frequency $\oo\dwell\gtrsim 1$ current is usually measured at zero frequency. In contrast, at small bias frequency $\oo\dwell\ll 1$ higher harmonics (for instance the second harmonic $2\oo$) can be measured. However, up to corrections small due to $\oo\dwell\ll 1$, the second harmonic is just equal to the rectified current, $I_{2\oo}\approx I_0$. Therefore, to leading order, our results for the rectified current describe both experiments. Generally, there are several important time-scales characteristic for time-dependent problems in chaotic quantum dots. To see how they appear let us first consider frequency-dependent linear transport of noninteracting electrons. Its statistics usually depend only on the flux-dependent time scales $\tau_{\Coop,\Diff}$, see Eq. (\[eq:channels\]). If we consider an analog of UCF $\la G^2(\Phi)\ra$ for the frequency-dependent conductances introduced in Eq. (\[eq:sumG\]), we find $$\begin{aligned} \label{eq:linear} \la G(\oo,\Phi)G(\oo',\Phi')\ra=\left(\frac{\nu_s e^2}{h}\frac{N_1N_2}{N}\right)^2\sum_{\l=\Coop,\Diff}&& \nonumber \\ \left(\frac{\Delta T}{2\hbar^2}\right)^2\int \frac{\tau_\l d\tau}{\oo\oo'}\frac{ e^{-\tau/\tau_\l}(e^{i\oo\tau}-1)(1-e^{i\oo'\tau})} {[1-i(\oo+\oo')\tau_\l/2]\sinh^2 \pi T\tau/\hbar}.&&\end{aligned}$$ The presence of $i\oo\tau_\l$ in the diffuson and cooperon contributions in the second line of Eq. (\[eq:linear\]) is due to the energy dependence of the scattering matrix $\S(\e)$, which usually brings up imaginary corrections to the matrix-element correlators. In a DC-problem $\oo\to 0$ it is usually useful to introduce a dimensionless number of channels $N_{\Coop,\Diff}$ modified by the magnetic field, see Eq. (\[eq:channels\]). In this limit at $T\to 0$ the integration in Eq. (\[eq:linear\]) becomes straightforward and summation is then performed over $N_\l^{-2}$. For equal magnetic fields, $\Phi=\Phi'$, we have $N_\Diff=N$, but $N_\Coop$ is strongly modified by large fields, $N_\Coop\to \infty$, which suppresses the weak localization correction and diminishes UCF. However, for an AC problem (especially for $\oo\dwell\gtrsim 1$) it is more convenient to express results in terms of dimensionless quantities $\oo\tau_{\Coop},\oo\tau_{\Diff}$. For example, from Eq. (\[eq:linear\]) the statistics of conductance can be easily evaluated: $\la |G(\oo,\Phi)|^2\ra/\la G(0,\Phi)^2\ra\sim 1/(\oo\dwell)$ and the real and imaginary parts of conductance are similar and uncorrelated at high frequency $\oo\dwell\gg 1$. Inclusion of interactions introduces an (additional) dependence on $\RC$, the charge-relaxation time defined in Eq. (\[eq:excess\]). To leading order in $1/N\ll 1$ the effect of interactions is often to substitute $\dwell\to \RC$ in the noninteracting results, e.g., for the linear conductance [@PietMarkus] or shot noise. [@BP; @Hekking] Interestingly, the subleading corrections depend on both $\RC$ and $\dwell$, e.g., in the weak localization correction in the absence of magnetic field.[@PietMarkus] When the magnetic field is increased to values which finally break time-reversal symmetry, the appearance of different time scales $\tau_{\Coop,\Diff}$ is expected, see e.g., Eq. (\[eq:linear\]). Therefore at intermediate magnetic fields, when $\tau_\Diff\neq \tau_\Coop$, and the interactions taken into account, $\RC\neq \dwell$, the solution of an AC problem is expected to show a complicated dependence on all these time scales. Indeed, if we consider the rectified current such an interplay between $\tau_{\Coop,\Diff}$ and $\RC$ does appear. We find $\la\Ga\ra=\la\Gs\ra=0$ and present below results for correlations of $\Ga$ and $\Gs$: $$\begin{aligned} {\genfrac{\{}{\}}{0pt}{}{\la\Ga(\Phi)\Ga(\Phi')\ra}{\la\Gs(\Phi)\Gs(\Phi')\ra}}= {\genfrac{\{}{\}}{0pt}{}{{\cal F}_{U,\Diff}(\oo)-{\cal F}_{U,\Coop}(\oo)}{{\cal F}_{U,\Diff}(\oo)+{\cal F}_{U,\Coop}(\oo)+X(\oo)}}\nonumber \\ \times [{\cal F}_{G,\Diff}(\oo)+{\cal F}_{G,\Coop}(\oo)]\left( \frac{4\pi\cos^2\eta} {\Delta}\frac{C_\m}{C_\Sigma}\right)^2\frac{N_1^3 N_2^3}{N^6} \label{eq:varGs}.\end{aligned}$$ Here the functions ${\cal F}_{U}(\oo),{\cal F}_{G}(\oo)$ are finite-frequency generalizations of Eq. (\[eq:Fraw\]) $$\begin{aligned} \label{eq:FU}{\cal F}_{U,\l}(\oo)&=&\left(\frac{\Delta T}{\hbar^2 \oo}\right)^2\int \tau_\l d\tau\frac{e^{- \tau/\tau_{\l}}\sin^2 \oo\tau/2 }{2\sinh^2 \pi T\tau/\hbar}\frac{1}{1+\oo^2\RC^2} \nonumber \\&\times&\left(1+ \mbox{Re }\frac{1+i\oo\RC}{1-i\oo\RC} \frac{e^{i\oo\tau}}{1-i\oo\tau_\l}\right) ,\\ {\cal F}_{G,\l}(\oo)&=& \left(\frac{\Delta T}{\hbar^2 \oo^2}\right)^2\int \frac{2d\tau}{\tau_\l}\frac{e^{- \tau/\tau_{\l}}\sin^4 \oo\tau/2}{\sinh^2 \pi T\tau/\hbar} \label{eq:FG}.\end{aligned}$$ The subscripts $U(G)$ of ${\mathcal F}_{U(G)}(\oo)$ illustrate the origin of these functions: they result from averaging of different scattering properties over the energy band defined by $\mbox{max }\{\hbar\oo, T,\hbar/\tau_{\Coop(\Diff)}\}$. The function ${\mathcal F}_{U}(\oo)$ is a characteristic of the internal potential $U_\oo$, see Eq. (\[eq:uomega\]). The function ${\mathcal F}_{G}(\oo)$ results from the energy averaging of the DC conductance $g(\e)$. Such averaging appears because both $G(\oo)$ defined in Eq. (\[eq:sumG\]) and $g(\e)$ in Eq. (\[eq:current4omega\]) are coupled to the Fermi distribution. The function $X(\oo)$ is $$\begin{aligned} \label{eq:Xgeneral} X(\oo) &=&\frac{N^2}{2N_1N_2} \left(\frac{(C_0\tan\eta+C_2-C_1)(1+\oo^2\dwell\RC)}{(1+\oo^2\RC^2)\nu_s e^2/\Delta}\right.\nonumber \\ && \mbox{} \left. +\frac{N_2-N_1}{N(1+\oo^2\RC^2)}\right)^2,\end{aligned}$$ and in the static limit $\oo\to 0$ it is given by Eq. (\[eq:X\]). We point out that when the interactions are negligible, $E_c\sim e^2/C\ll\Delta$, the role of the bias mode is significant. A quantum dot with (fully broken) time-reversal symmetry can be labeled by Dyson symmetry parameter ($\b=2$) $\b=1$. When the setup is ideal, $C_{1,2}=0$, and $\eta\neq 0$, the fluctuations of $\Ga,\Gs$ at large frequencies $\oo\dwell\gg 1$ are $$\begin{aligned} \label{eq:FK} \d\Gs &=& \la\Gs^2\ra^{1/2}=\frac{N_1 N_2}{N^2} \left(\frac 2\beta\frac{\pi}{\oo\dwell}\right)^{1/2}\frac{2\sin 2\eta}{\hbar\oo},\\ \d\Ga &=&\left(\frac{N_1 N_2}{N^2}\right)^{3/2}\frac{\nu_s e^2}{2C}\frac{\cos^2\eta}{\hbar^2\oo^3\dwell},\,\,\b=2.\label{eq:FKasym}\end{aligned}$$ In chaotic quantum dots the role of the Thouless energy $\Thou$ of the open systems is often played by the escape rate $\hbar/\dwell$. If we take this into account, our result (\[eq:FK\]) qualitatively agrees with that of Falko and Khmelnitskii [@FK] obtained for open diffusive metallic junctions. However, when $\eta\to 0$, the fluctuations of $\Gs$ are much smaller and for $|N_1-N_2|\ll\oo\dwell$ they become comparable with those of the antisymmetric conductance (\[eq:FKasym\]). However, very often experiments are performed in samples, where the interaction is not weak. Since $\Ga$ and $\Gs$ are comparable for strong Coulomb interactions in the static limit $\oo\to 0$,[@PhysicaE] we concentrate here on this experimentally relevant regime of $\Delta/E_c\sim \RC/\dwell\ll 1$ and take an ideal symmetric setup, $N_1=N_2$ and $C_{1,2}=0$. Then we have $$\begin{aligned} \label{eq:Xomega} X(\oo)\approx 2\tan^2\eta\left(\frac{\dwell^{-1}+\oo^2\RC}{\RC^{-1}+\oo^2\RC}\right)^2.\end{aligned}$$ Below we consider in detail the case $\eta\neq 0$ and how this bias mode affects the behavior of $\Gs^2(\oo)$. Several frequency regimes can be separated: adiabatic $\oo\tau_\l\ll 1$, intermediate, where $1/\tau_\l\ll \oo\ll 1/\RC$, and high frequencies $\oo\RC\gtrsim 1$. The asymptotes of the functions defined in Eqs. (\[eq:FU\]), (\[eq:FG\]), and (\[eq:Xomega\]) in these regimes are presented in Table \[tab:table\] for reference. For adiabatic frequencies $\oo\tau_\l\ll 1$ the integrands in Eqs. (\[eq:FU\]) and (\[eq:FG\]) do not oscillate on the short time scale $\tau_\l$. At such small frequencies ${\cal F}_U(\oo)={\cal F}_G(\oo)$ are equal to ${\cal F}$ of Eq. (\[eq:Fraw\]) and $X(\oo)\propto (\RC/\dwell)^2\ll 1$ can be neglected. This is essentially the zero frequency regime considered before for nonlinear DC transport. As the frequency grows, an intermediate regime is reached when max $\{T,\hbar/\tau_\l\}\ll \hbar\oo\ll \hbar/\RC$ and ${\cal F}_{U}(\oo), {\cal F}_{G}(\oo)$ start to differ. The scattering properties at large energy difference $\hbar\oo\gg\hbar/\tau_\l$ are uncorrelated and the response of the dot is randomized. Therefore both the conductance averaged over a large energy window $\hbar \oo$ and the response of the internal potential $U_\oo$ to the AC voltage at $\oo\dwell \gg 1$ are strongly suppressed, see Table \[tab:table\]. As a result, if $X(\oo)$ is still negligible, both $\Ga^2$ and $\Gs^2$ decrease with growing frequency as $1/\oo^4$. ------------------------------------ ---------------------- ---------------------------------- ---------------------------- Adiabatic Intermediate High $\mbox{Function}$ $\oo\ll\tau_\l^{-1}$ $\tau_\l^{-1}\ll\oo\ll \RC^{-1}$ $\oo\RC\gg 1$ ${\mathcal F}_U(\oo)\times N_\l^2$ 1 $\pi/(4\oo\tau_\l)$ $\pi/(4\oo^3\tau_\l\RC^2)$ ${\mathcal F}_G(\oo)\times N_\l^2$ 1 $X(\oo)$ $2\tan^2\eta$ ------------------------------------ ---------------------- ---------------------------------- ---------------------------- : Asymptotes of ${\mathcal F}_{U,\l}(\oo),{\mathcal F}_{G,\l}(\oo),X(\oo)$ at $T\to 0$ \[tab:table\] One could expect that interactions qualitatively change the behavior of $\Ga,\Gs$ when the frequencies become comparable to $1/\RC\sim NE_c/\hbar$, the scale defined by the interaction strength. At such frequencies the response of a dot to the potentials at the contacts is not resistive as occurs at low frequencies, but mostly capacitive. If the frequency is high, $\oo\RC\gtrsim 1$, we have $\mbox{Re }u_{1,2}\to 0$ and the function ${\mathcal F}_{U}$ in Eq. (\[eq:FU\]) is suppressed $\sim 1/(\oo\RC)^{2}$. As a consequence, $\Ga^2$ is suppressed stronger then $1/\oo^4$ and goes as $1/\oo^6$ at $\oo\RC\gtrsim 1$. However, a more important signature of this capacitive coupling is the growth of $X(\oo)$ in Eq. (\[eq:Xomega\]), which affects $\Gs^2$. To see the role of this growth we consider now sufficiently large fields $\Phi=\Phi'$ when only the diffuson contribution survives. The growth of $X(\oo)$ in Eq. (\[eq:Xomega\]) reflects enhanced sensitivity of the internal potential $U_\oo$ to the gate voltage, $X(\oo)\propto (\tan\eta \mbox{Re }u_{0})^2$. At high frequencies the impedance of the capacitor $C$ becomes negligible and therefore the internal potential follows the gate voltage and not the reservoir voltages, $u_0\to 1,u_{1,2}\to 0$. Enhanced from its small static value $\RC/\dwell$ to 1 at large frequencies, such coupling affects the fluctuations of $\Gs(\oo)$ if $\eta\neq 0$. The situation is somewhat similar to the weak interaction limit, when the coupling with nearby gates was strong, $u_0\to 1,u_{1,2}\ll 1$, and lead to $\Gs\gg\Ga$. [ ![Zero-temperature large-field fluctuations of $\Ga(\oo)$ (dashed) and $\Gs(\oo)$ (solid curve) in units of $(\pi/4\Delta N^2)^2$ for the bias mode $\eta=\pi/4$. Data are presented in the log-log scale at $N_{1,2}=5$ and $\RC/\dwell=0.05$. The asymptotes $\Ga^2\propto \oo^{-6}$ and $\Gs^2\propto \oo^{-3}$ are different due to $\eta\neq 0$, see Eqs. (\[eq:Gainter\]) and (\[eq:Gsinter\]). []{data-label="fig:GsGaMath"}](GsGaMath.eps "fig:"){width="9cm"}]{}\ The fluctuations of $\Ga(\oo),\Gs(\oo)$ for $\oo\dwell\gg 1$ can be evaluated: $$\begin{aligned} \label{eq:Gainter} &&\Ga^2(\oo)\sim \frac{\Delta^2}{(\hbar\oo)^4(1+\oo^2\RC^2)},\\ &&\Gs^2(\oo)\sim\Ga^2(\oo)+ \frac{(\RC\tan\eta[1+\oo^2\dwell\RC])^2}{\hbar^2\oo\dwell(1+\oo^2\RC^2)^3}. \label{eq:Gsinter}\end{aligned}$$ Fluctuations of $\Ga^2(\oo)$ and $\Gs^2(\oo)$ demonstrate qualitatively different behavior, which we illustrate in Fig. \[fig:GsGaMath\]. Indeed, at sufficiently high frequencies, the dependence of $X(\oo)$ on $\omega$ makes the last term in Eq. (\[eq:Gsinter\]) dominant. At $\oo\RC\gg 1$ the asymptotes of $\Ga^2\propto 1/\oo^6$ and $\Gs^2\propto 1/\oo^3$ become different due to the presence of the second term in Eq. (\[eq:Gsinter\]). These results show that for nonadiabatic frequencies of the external bias the DC current strongly depends on the bias mode $\eta$. We predict that the magnetic field asymmetry of the rectified current, noticeable at small frequencies, might become suppressed for large frequencies, when the symmetrized component dominates due to the presence of capacitive coupling. For convenience, the low-temperature estimates for $\la\Ga^2\ra$ and $\la\Gs^2\ra$ for $\eta\neq 0$, $\Phi\gg \Phi_c$ are collected in Table \[tab:GaGs\]. ----------------------------------------- ----------- ------------------------------------ ------------------------------------- Adiabatic Intermediate High $\mbox{Function}$ $z\ll 1$ $1\ll z\ll \dwell/\RC$ $z\gg \dwell/\RC$ $\frac{\hbar^4}{\Delta^2\dwell^4}\Ga^2$ $1$ $z^{-4}$ $ \frac{\dwell^{2}}{\RC^{2}}z^{-6}$ $\frac{\hbar^2}{\RC^2}(\Gs^2-\Ga^2)$ $1$ $(1+\frac\RC\dwell z^2)^2 z^{-1} $ $\frac{\dwell^4}{\RC^4}z^{-3}$ ----------------------------------------- ----------- ------------------------------------ ------------------------------------- : Estimates for $\la\Ga^2\ra$ and $\la\Gs^2\ra$, ($z=\oo\dwell$) \[tab:GaGs\] It is noteworthy that a recent experiment in AB rings [@Bouchiat_preprint] finds that $\G(\oo,\Phi=0)$ grows with frequency until $\oo\sim 2\Thou$ and then decreases $\sim 1/\oo^{3/2},\oo\to\infty$. While we predict a monotonic decrease of $\la\Gs^2(\oo)\ra$, this growth could be the result of quantum pumping or an interference of the pumping and rectification (both effects were neglected here). Phase of Aharonov-Bohm oscillations {#sec:phase} =================================== In this section we consider nonlinear transport through a chaotic Aharonov-Bohm (AB) ring. The nonlinear conductance $\G$ exhibits periodic AB oscillations and non-periodic fluctuations, similarly to the linear conductance $G$. However, since Coulomb interactions produce asymmetry of $\G$ with respect to magnetic field inversion, the phase of these oscillations is not pinned to $0\mbox{ (mod)}\pi$. As a quantum effect this AB phase is characterized by a mesoscopic distribution. The width of this distribution represents a typical fluctuation. We first discuss what kind of distribution could be expected in a chaotic AB ring and then calculate the fluctuation of the AB phase. Let us assume that $\G$ as a function of magnetic flux $\Phi$ can be expanded into the series of well-defined Fourier harmonics similarly to the linear conductance $G$: $$\begin{aligned} \label{eq:expansion} {\genfrac{\{}{\}}{0pt}{}{ G(\Phi)}{\G(\Phi)}}&=&\sum_{n=0}^\infty {\genfrac{\{}{\}}{0pt}{}{G_n}{\G_n}}\cos\left( \frac{2\pi n\Phi}{\Phi_0}+{\genfrac{\{}{\}}{0pt}{}{0}{\delta_n}}\right).\end{aligned}$$ The phase $\d$ of the main (first) harmonic $\Phi_0=hc/e$ is obtained from the ratio of the (anti) symmetrized conductances defined in Eq. (\[eq:IVfield\]) $$\begin{aligned} \label{eq:tan} \tan\delta=\frac {\int d\Phi \exp(2\pi i \Phi/\Phi_0)\Ga(\Phi)}{\int d\Phi \exp(2\pi i \Phi/\Phi_0)\Gs(\Phi)}.\end{aligned}$$ We can not find the full mesoscopic distribution of the phase $P(\d)$. We can gain some insight in the behavior of this phase by investigating a similar quantity, namely, the asymmetry parameter $\A=\Ga/\Gs$ considered previously for chaotic dots. [@PhysicaE] Based on Eq. (\[eq:tan\]) we argue that the statistical properties of $\arctan \A$ and the AB phase $\d$ should be similar. In quantum dots the parameter $\A$ is given by the ratio $\A=\Ga/\Gs=\chi_{2a}/\chi_{2s}$, see Eqs. (\[eq:defG\]), (\[eq:chi2a\]), and (\[eq:chi2s\]). The functions $\chi_{2a,2s}$ at $T\neq 0$ are convolved separately with $f'(\e)$, and at $T=0$ (which we consider below) they are evaluated at the Fermi energy. The properties of $\chi_{2a,2s}$ and the dependence of $\chi_{2s}$ on the bias mode were described after Eq. (\[eq:X\]). The function $\chi_{2s}$ can have a nonzero (classical) average $\la\chi_{2s}\ra\sim X^{1/2}$ defined by the interaction strength, geometry of the setup, and the bias mode $\eta$. Since $\la\chi_{2,a}\ra=0$ and the fluctuations of $\chi_{2a,2s}$ are small as $1/N^2$, the mesoscopic distribution of $\arctan\A$ is narrow and concentrated close to 0. However, $\la\chi_{2s}\ra=0$ is possible if $X\to 0$, e.g., for symmetric contacts and the bias mode $\eta=0$. In this case, the distribution of $\arctan \A$ becomes wide regardless of the interaction strength. [\[t\][$\phi$]{}\[l\][$P(\phi)$]{} \[l\]\[\]\[0.8\][$N=16,N_L=8$]{} \[l\]\[\]\[0.8\][$N=16,N_L=4$]{} \[l\]\[\]\[.7\][$N=24$]{} \[l\]\[\]\[.7\][$N=2$]{} ![Mesoscopic distribution $P(\phi)$ of $\phi=\arctan \Ga/\Gs$. (Main plot) If the contacts are asymmetric (bold curve, $N=16,N_L=4$) the distribution is narrow, while for symmetric contacts (dashed, $N=16,N_L=8$) it is almost uniform. As shown in the inset, for symmetric contacts at large $N$ the distribution becomes uniform, compare bold curve for $N=2$ and dashed for $N=24$.[]{data-label="fig:delta"}](Angle.eps "fig:"){width="8cm"}]{}\ The role of the classical contribution on the shape of $P(\arctan\A)$ is demonstrated in the main plot in Fig. \[fig:delta\] for $\eta=0$, where the distributions for asymmetric, $N_L=4,N=16$, and symmetric contacts, $N_L=8,N=16$, are presented. While the distribution is almost uniform, when the classical contribution $X$ is absent, it is highly peaked near zero when $X$ dominates. If $X$ is absent, the correlations between $\Ga$ and $\Gs$ are significant at small $N$. This leads to a nonuniform distribution of $P(\arctan\A)$, which is peaked at $0$ and $\pi/2$ when $N=2$, see the inset in Fig. \[fig:delta\]. When $N$ grows, the correlations between $\Ga$ and $\Gs$ vanish and therefore the distribution becomes uniform. Such a distribution could be easily obtained if we make the natural assumption that $\Ga,\Gs$ are independent and distributed by the Gaussian law with the same width. These numerics were performed for $\eta=0$, when the mesoscopic distribution of $\A=\Ga/\Gs$ becomes insensitive to the interaction strength. The role of interactions appears only if $\eta\neq 0$, when the classical contribution $X$ becomes dominant. Similarly, we expect that the distribution of the phase of AB oscillations is also strongly affected by the bias mode. If the bias mode is chosen such that the classical contribution $X$ vanishes, the phase $\d$ strongly fluctuates [*even for weak interactions*]{}. It would be very interesting to check this surprising conclusion experimentally. Let us now consider the fluctuations of the AB phase. Since the scattering theory turned out to be very useful for the discussion of the nonlinear/ rectified current through a chaotic quantum dot, we extend this theory to rings. We make two key assumptions (discussed in the Appendix in more detail) that the magnetic flux through the annulus of the ring is smaller then the flux quantum $\Phi_0$ and that the mean free path $l$, the radius $R$, the width of the ring $W$ and the contacts $W_c$ satisfy the condition $\pi^2 l W\gg 2R W_c$. In this case the RMT can be applied to such chaotic rings as well. Unlike the experiments on large open rings with high aspect ratio $R/W\gg 1$,[@WW; @Liu; @Lin; @Bartolo; @BykovAB] the recent experiments [@ensslin; @Bouchiat; @Bouchiat_preprint] are performed in rings of submicron size, which are effectively zero-dimensional. The treatment of such rings is similar to chaotic quantum dots, and the fluctuations of $\Gs,\Ga$ can be expressed in terms of the diffuson $\Diff$ and the cooperon $\Coop$, see Eq. (\[eq:main\]). The only problem is to find the expression for the effective number of channels as a function of magnetic field, similar to Eq. (\[eq:channels\]). [ ![Model of a chaotic Aharonov-Bohm ring with $N=N_1+N_2$ channels. The model consists of a quantum dot with $M$ channels combined with a ballistic arm with $N_3=N_4=(M-N)/2$ channels.[]{data-label="fig:model"}](ABdotmodel.eps "fig:"){width="6cm"}]{}\ The model we propose for a chaotic AB ring combines chaos and a ring geometry: a chaotic dot is attached to a long ballistic arm which serves to include an AB flux large compared to the fraction of the flux through the sample. This model is shown in Fig. \[fig:model\], where the ring with $N=N_1+N_2$ ballistic channels in the contacts 1,2 is modeled by a dot with $M>N$ channels and a ballistic arm with $N_3=N_4=(M-N)/2$ channels in contacts 3,4. The parameter $\rho=1-N/M$, the ratio of $N_3+N_4$ to the total number of channels $M$, can vary between 0 when the arm is much narrower then the contacts and 1 in the opposite limit. The electronic phase is randomized in the quantum dot, but when electrons propagate along the arm their phase is determined by the geometry and applied magnetic field. This model is a reasonable approximation for the real experiment, it takes into account the long time spent by electron inside the ring and the randomness of its motion. The discussion of the model and the details of calculation of $\Coop,\Diff$ are presented in the Appendix. In experiment the Fourier transform is often taken over the total flux (or applied magnetic field) and the flux through the hole $\Phi_{h}$ cannot be separated from the flux through the dot $\Phi_{d}$. Then the dependence of the diffuson and cooperon on magnetic field is non-periodic, which is indeed observed in the form of nonperiodic fluctuations in the (non-)linear conductance and phase slips of AB oscillations. A possible weakness of this model is in its spatial separation of chaotic scattering and the main part of magnetic field, but in the limit when the arm is much wider then the contacts $1$ and $2$ such a separation is not important and the averaged properties of AB oscillation phase become independent of the arm’s width. If the flux $\Phi_d$ through the dot is much smaller then the flux $\Phi_h$ through the hole, the nonperiodic fluctuations and the periodic AB oscillations are well-separated, which is usually the case in experiment.[@ensslin; @Bouchiat] In view of this separation we can neglect the flux through the chaotic dot, $\Phi_d\ll\Phi_h$, to find the statistics of the AB phase. We assume that the averaging is taken over a magnetic field range containing many AB oscillations but still small compared to the characteristic field of the nonperiodic fluctuations. In such a simplified model of a chaotic AB ring $N_{\Coop,\Diff}$ are given by Eq. (\[eq:DCfinal\]) with $\Phi_d=0,\Phi_h=\Phi$ and the parameter $\rho=(M-N)/M$. The effective number of channels is $$\begin{aligned} \label{eq:DCsimple} {\genfrac{\{}{\}}{0pt}{}{N_{\cal C}}{N_{\cal D}}} &=& M\left(1-\rho\cos \frac{2\pi(\Phi\pm\Phi')}{\Phi_0}\right).\end{aligned}$$ Using this expression for the $N_{\Coop,\Diff}$ we can evaluate the quantum fluctuations of linear conductance in AB rings. At low temperature a typical fluctuation of $G$ at $\Phi=0$ is $\d G=\sqrt{2}(N_1 N_2/N^2)(\nu_s e^2/h)$ and the amplitude of AB oscillations is $\d G_{\rm AB}^2\sim\d G^2 \rho(1-\rho)^{1/2}/(1+\rho)^{3/2}$, which reaches maximum when the widths of the arm and the contacts are equal, $\rho=1/2$. The first two moments of $\tan\d$ can be found analytically. It is zero on average, $\la\tan\d\ra=0$, since the numerator and denominator in Eq. (\[eq:tan\]) are independent random quantities. Equation (\[eq:DCsimple\]) show that $\Diff,\Coop$ are the same functions of $\Phi\pm\Phi'$, so that all necessary ingredients can be expressed in terms of the functions ${\mathcal F}_{U,\Diff}$ and ${\mathcal F}_{G,\Diff}$ of $\Phi-\Phi'$ defined in Eqs. (\[eq:FU\]) and (\[eq:FG\]). We omit now the index $\Diff$ for brevity and denote the average over magnetic field by $\overline{ (...)}=\int_0^{\Phi_0}(...) d\Phi/\Phi_0 $, to find $$\begin{aligned} \label{eq:tandelta} &&\la\tan ^2\delta\ra = \left[\overline{{\cal F}_{U }(\Phi){\cal F}_{G }(\Phi)\cos\Phi} +\overline{ {\cal F}_{G }}\cdot\overline{{\cal F}_{U }(\Phi)\cos\Phi}\right.\nonumber \\ && \mbox{} \left.-\overline{ {\cal F}_{U }}\cdot\overline{{\cal F}_{G }(\Phi)\cos\Phi}\right]/ \left[\overline{({\cal F}_{U }(\Phi)+X(\oo)){\cal F}_{G }(\Phi)\cos\Phi} \right.\nonumber \\ && \mbox{} \left.+ \overline{\frac{{\cal F}_{U }(\Phi)}{2}}\overline{{\cal F}_{G }(\Phi)\cos\Phi}+ \overline{\frac{{\cal F}_{G }(\Phi)}{2}}\overline{{\cal F}_{U }(\Phi)\cos\Phi}\right],\end{aligned}$$ where the function $X(\oo)$ is defined by the setup geometry in Eq. (\[eq:Xomega\]). Again, in the static limit $\oo\to 0$ we have ${\cal F}_{U}={\cal F}_{G}={\cal F}$ and $X$ defined by Eqs. (\[eq:Fraw\]) and (\[eq:X\]). In this case Eq. (\[eq:tandelta\]) can be rewritten as $$\begin{aligned} \label{eq:tan^2result} \frac{1}{\la\tan ^2\delta\ra} &=&1+\frac{[\overline{{\cal F}(\Phi)}+X]\overline{\cos\Phi{\cal F}(\Phi)}}{\overline{\cos\Phi{\cal F}^2(\Phi)}}.\end{aligned}$$ In the limits of high, $T\gg N\Delta/2\pi$, and low temperature, $T\ll N\Delta/2\pi$ the asymptotical values of $\la\tan^2\d\ra$ are $$\begin{aligned} \label{eq:tan^2asymp} \frac{1}{\la\tan ^2\delta\ra}&=&1+\left\{\begin{array}{cc} \frac{\sqrt{1-\rho^2}}{1+\sqrt{1-\rho^2}}+ \frac{12 T}{ \Delta}\frac{XM(1-\rho^2)}{1+\sqrt{1-\rho^2}}, & T\gg \frac{N\Delta}{2\pi}\\ \frac{2\sqrt{1-\rho^2}}{4+\rho^2} + \frac{2XM^2(1-\rho^2)^2}{4+\rho^2}, &T\ll\frac{N\Delta}{2\pi}\end{array}\right.\nonumber\end{aligned}$$ Very important is the case of symmetric contacts, $N_1=N_2$, and antisymmetric bias mode, $\eta=0$, which is used in Ref.. Then $X$ vanishes and the average $\tan^2\d$ becomes independent of interaction strength and as a function of $T$ it is very weak. That is not the case if $\eta\neq 0$, for example, when only one of the voltages changes, $\eta=\pm \pi/4$.[@Bouchiat; @Bouchiat_preprint] Then the statistics of the AB phase becomes temperature and interaction dependent due to the presence of $X$. The limit $M\gg N$ corresponds to a uniformly chaotic ring, which we suppose to be closer to the experimental situation. Then the dependence on $M$ drops out and the high/low temperature asymptotic read $$\begin{aligned} \label{eq:ANGLEwidearm} \frac{1}{\la\tan ^2\delta\ra}&=&1+8X\left\{\begin{array}{cc} 3NT/\Delta, & T\gg N\Delta/2\pi,\\ N^2/5, &T\ll N\Delta/2\pi\end{array}\right. .\end{aligned}$$ This result clearly demonstrates that the phase of the oscillations is expected to deviate strongly from 0, especially if the temperature is low and the number of channels in the contacts is diminished. The temperature is taken into account only in the form of temperature-averaging and the dephasing (previously considered for nonlinear transport of noninteracting electrons in Refs.) is not included. We expect our model for chaotic AB rings to work both for experiments at small frequencies [@ensslin; @Bouchiat] and for large frequencies.[@Bouchiat_preprint] Similarly to quantum dots, the generalization on the finite-frequency case is obvious, if we use Eq. (\[eq:Xomega\]). Even in cases where RMT cannot be assured to be valid for open diffusive rings, the dependence of the AB phase on interaction strength, temperature, and number of external channels given by Eq. (\[eq:ANGLEwidearm\]) should be correct qualitatively. The experiment of Leturcq [@ensslin] is performed in a bias mode $\eta=0$ when $X=0$. Then Eq. (\[eq:ANGLEwidearm\]) gives $\la\tan^2\d\ra=1$. The phase of the oscillations is evaluated from data according to Eq. (\[eq:tan\]) over a large range of fields. In experiment the AB phase is varied continuously as a function of the gate voltage at one of the arms of the ring. The data demonstrate that the phase $\d$ indeed changes in a wide range and is usually far from 0. This substantiates our conclusion that in the mode when the classical contribution is minimized, $X\to 0$, the mesoscopic distribution of $\d$ is very wide. Experiment of Angers [@Bouchiat] varies voltage in a different way, $\eta=\pi/4$, and therefore has $X\neq 0$. We would expect the phase $\d\mbox{(mod)}\pi$ to take values closer to $0$ and the antisymmetric component of the oscillations be relatively smaller even for large fields. Although phases close to 0 are indeed observed, the field averaging is taken only over first few oscillations. In this range $\Ga$, the numerator in Eq. (\[eq:tan\]) is still small and grows linearly with magnetic field. Averaging over a larger field-range similar to Ref. could not be performed because of the phase slips. Another interesting question is a difference in data [@ensslin; @Bouchiat] for the relative magnitudes $\G_2/\G_1$ of the second $hc/2e$ and main harmonic $hc/e$, see Eq. (\[eq:expansion\]). In the nonlinear transport regime this harmonic is small compared to its contribution in the linear transport, $\G_2/\G_1\ll G_2/G_1$,[@ensslin] while in Ref. they were comparable, $\G_2/\G_1\approx G_2/G_1$. Our model also predicts the mesoscopically averaged contribution of $hc/2e$ into linear and nonlinear conductance to be comparable with that of $hc/e$. Our approach assumes full quantum coherence of the ring, and probably the difference in data is due to decoherence. Conclusions {#sec:conclusions} =========== In this paper, we consider mesoscopic chaotic samples (quantum dots or rings) and find the statistics of their nonlinear conductance $\G$. This transport coefficient characterizes nonlinear DC current due to DC-bias or a rectified current due to AC bias or photon-assisted transport. For chaotic samples, the nonlinear effect is of quantum origin, which is clear from the fact that its ensemble average over similar samples vanishes. The linear response of the sample in two-terminal measurements is always symmetric with respect to magnetic field inversion. However, the Coulomb interactions lead to magnetic field asymmetry of the nonlinear DC response, which fluctuates due to the electronic interference. For the quantum dots we consider the fluctuations of (anti) symmetrized components $\Ga,\Gs$ of the nonlinear conductance. In chaotic rings the statistics of the phase of AB oscillations in the nonlinear transport regime, closely related to the ratio $\Ga/\Gs$, is of interest. Unlike the linear conductance measurements, in mesoscopic nonlinear transport experiments the way voltages are varied (“bias mode”) turns out to be important, especially for a weakly interacting sample. We demonstrate this fact qualitatively and discuss the role of Coulomb interactions. Quantitative self-consistent treatment of interactions allows us to consider magnetic-field asymmetry in chaotic quantum dots with many channels. Using Eqs. (\[eq:main\])–(\[eq:X\]) we show that the fluctuations of $\Gs$ are strongly affected by the geometry of the setup and discuss how the bias mode influences data of recent DC experiments. Another important issue is rectification of AC bias, which is quadratic in applied voltage, random, and asymmetric with respect to the magnetic flux inversion. The photovoltaic DC current can be due to rectification of external perturbations or quantum pumping by internal perturbations. Both rectification and quantum pumping share the aforementioned properties, and it is important to clearly separate them especially when the frequency of perturbations is high (nonadiabatic). We consider here only the effects of the external perturbations and discuss the dependence of the fluctuations of $\Ga,\Gs$ on frequency $\oo$. We show that the fluctuations of both $\Ga$ and $\Gs$, presented in Eqs. (\[eq:varGs\])–(\[eq:Xgeneral\]), decrease monotonically as $\oo\to\infty$. However, contrary to naive expectations, their asymptotical behavior can be very different. Since at high frequencies the response of the dot to the external bias becomes rather capacitive then resistive, the coupling to the nearby gates can be strongly enhanced. If the experiment is performed in a bias mode where such coupling contributes, the symmetrized $\Gs^2(\oo)\propto 1/\oo^3$ can become much larger then $\Ga^2(\oo)\propto 1/\oo^6$ valid for a strongly interacting quantum dot. The same conclusion holds in the weakly interacting limit, when $\Gs^2\propto 1/\oo^{3/2}$ and $\Ga^2\propto 1/\oo^{3}$. In addition, we show that recent experiments in chaotic Aharonov-Bohm rings might be considered similarly to quantum dots. The multiply connected geometry alone leads to AB oscillations, yet the mesoscopic distribution of their phase is expected to be qualitatively similar to that of $\arctan \Ga/\Gs$ in quantum dots. Therefore, the bias mode should strongly affect the shape of mesoscopic distribution of the AB phase. The model of an AB ring, which we develop, consists of a dot and a long ballistic arm and takes into account both chaos and a ring geometry. As an application of our model we consider fluctuations of the AB phase. Unlike the AB phase in the linear conductance, pinned to $0\mbox{(mod)}\pi$ by the Onsager symmetry relations, the fluctuations of the AB phase in nonlinear transport are shown to depend on the bias mode, interaction strength, and temperature. Acknowledgements ================ We thank Hélène Bouchiat, Piet Brouwer, Renaud Leturcq, David Sánchez, Maxim Vavilov, and Dominik Zumbühl for valuable discussions. We also thank the authors of Ref. for sharing their results with us before publication. This work was supported by the Swiss National Science Foundation, the Swiss Center for Excellence MaNEP, and the STREP project SUBTLE. diffuson and cooperon for chaotic ring {#sec:appendix} ====================================== In this Appendix we determine the diffuson and cooperon contributions to the $\S$-matrix correlators of the random scattering matrix of a chaotic Aharonov-Bohm (AB) ring. This calculation is performed using Random Matrix Theory (RMT). First we explain what approximations should be made to ensure validity of RMT. Our starting point is the assumption that the $\S$-matrix of the ring is uniformly distributed over the unitary group. This means that the ring is essentially zero-dimensional, similarly to quantum dots. RMT is applicable if all energy-scales are much smaller then the Thouless energy $\Thou$ and the total flux through the annulus of the ring is much smaller then $\Phi_0$. Assume the ring of radius $R$ and width $W\ll R$ to be diffusive with diffusion coefficient $D=l v_{\rm F}/2$. To evaluate $\Thou$ we neglect with transversal motion of an electron and find $\Thou=\hbar/\erg=(\hbar l v_{\rm F})/2R^2$ as a solution to Laplace equation along the circumference of the ring. RMT can be applied to a closed ring if the dimensionless conductance is large, $g=\Thou/\Delta=k_{\rm F}l W/2R\gg 1$, which is usually satisfied for a weak disorder even if $W\ll R$. ![Chaotic dot combined with long ballistic multichannel arm.[]{data-label="fig:Abdot"}](ABdot.eps "fig:"){width="4cm"}\ An open ring with ballistic contacts of the width $W_c$ gains a new energy parameter, the escape rate $\hbar/\dwell=N\Delta/2\pi$, where $N$ is the total number of ballistic channels. The scattering matrix $\S$ is uniformly distributed and independent of the exact positions of the contacts (and therefore the length of the arms) if $\hbar/\dwell\ll\Thou\Rightarrow \pi^2 l W\gg 2R W_c$. In this case the main drop of the potential occurs in the contacts. If a magnetic field is applied, the RMT is valid if the total flux through the annulus of the ring is much less then the flux quantum, $\Phi\ll\Phi_0$. Due to narrow contacts the time-reversal symmetry (TRS) of the $\S$-matrix can be broken at a much smaller scale, $\Phi\sim \Phi_0\sqrt{\erg/\dwell}$. Since in our rings $\erg\ll\dwell$, a full crossover to the broken TRS can be considered. How well are these conditions fulfilled in the experiment? In Ref. chaos was mainly due to diffusive scattering on the boundary and $l\approx R$. The width of the arm is 2-4 channels, while the number of channels in the contacts is $N\sim 2$, estimated from the linear conductance measurements, so $\dwell/\erg\sim 5\div 10\gg 1$. In semiballistic samples of Ref. (obtained by etching, and therefore having diffusive boundary scattering) $W=W_c$ and the mean free path is estimated $l\sim 1\div 2\mu$m$\sim L= 1.2 \mu$m, the side length. Therefore, we have a similar estimate for the ratio $\dwell/\erg$. Although this ratio is not parametrically large due to, e.g., weak disorder $k_{\rm F}l\gg 1$, we believe that such AB rings still can be assumed zero-dimensional due to their good conducting properties together with relatively narrow contacts. In our calculations we make a further simplification by spatially separating chaotic scattering which randomizes the electronic phase and the long ballistic arm attached to it. To find the correlators of the $\S$-matrix elements we use a simplified model, see Fig. \[fig:Abdot\], which combines chaos and a ring geometry. A chaotic $M$-channel dot is attached to a long multi-channel ballistic arm with $(M-N)/2$ orbital channels. We assume that the size of the dot $L$ and the length of the arm $L_a$ are such that $L_a\gg L\gg \sqrt {(M-N)\lambda_{\rm F} L_a}$ to ensure that in the hierarchy of different fluxes the main flux $\Phi_h$ is concentrated in the region embraced by the arm, the flux through the dot $\Phi_d$ is much smaller, but still much larger then the flux through the cross section of the arm. The amplitude of AB oscillations depends on the width of the arm $\propto (M-N)$. The wider the arm (relatively to the contacts) the closer the results should be to a uniformly chaotic ring. For the case when $M\gg N$ we expect it to be valid for the chaos uniformly distributed over the ring. Indeed, in this case an electron makes $\sim M/N\gg 1$ windings around the arm before exiting. In this appendix it is more convenient for us to work with an energy-dependent matrix $\S(\e)$, and the final transformation to time-representation is rather obvious. The total scattering matrix $\S$ is of size $N\times N$ due to scattering channels in the contacts 1 and 2. Chaotic scattering in the $M$-channel quantum dot is characterized by the $M\times M$ matrix $\U$. The scattered electron can either exit the sample through the $N=N_1+N_2$ channels (projection operator $P_0=1_1\oplus 1_2$) or propagate into the arm with $N_3=N_4=(M-N)/2$ channels. Electrons propagate through this arm ballistically and gain phases which depend on the flux through the hole. In the absence of backscattering the electronic amplitudes at energy $\e$ are related to the path length $L_a$ and magnetic field phase $\phi$: $$\begin{aligned} \label{eq:Pmatrix} \binom{b_3}{b_4}=e^{-ik(\e)L_a}\left(\begin{array}{cc} 0 & e^{-i\phi} \\ e^{i\phi}& 0 \\ \end{array}\right)\binom{a_3}{a_4}=P\binom{a_3}{a_4}.\end{aligned}$$ The scattering matrix of the arm is $\P(\e)=0_1\oplus 0_2\oplus P$. Each time an electrons enters the arm either through the third or fourth lead, the matrix $\P$ contributes to the scattering amplitude of the process. The total scattering matrix $\S$ is determined from the following equation: $$\begin{aligned} \S=P_0\sum_{n=0}^\infty\U(\P \U)^n P_0=P_0\U\frac{1}{1-\P\U}P_0,\end{aligned}$$ where multiple $n\geq 0$ windings are taken into account. Both $\U(\e,B)$ and $\P(\e,B)$ are field and energy dependent. Once we are interested only in pair correlators of $\S(\e),\S^\dagger(\e')$ for $N,M\gg 1$, the diffuson $\Diff$ and cooperon $\Coop$ of our scattering matrix are expressed via correlators of the dot, $\Diff_{\cal U}$,$\Coop_{\cal U}$, and tr $\P(\e)\P^{*(\dagger)}(\e')$. The correlators of the $\U$-matrix are known, see Eq. (\[eq:cum1t\]) for their time representation, and for $\Diff$ and $\Coop$ we derive $$\begin{aligned} \label{eq:DC} {\genfrac{\{}{\}}{0pt}{}{\Coop}{\Diff}}^{-1}&=&{\genfrac{\{}{\}}{0pt}{}{{\cal C}_{\cal U}^{-1}-\mbox{tr }\P(\e)\P^*(\e')}{{\cal D}_{\cal U}^{-1}-\mbox{tr }\P(\e)\P^\dagger(\e')}}, \\ {\genfrac{\{}{\}}{0pt}{}{\Coop}{\Diff}}_{\cal U}^{-1}&=& M-2\pi i\frac{\e-\e'}{\Delta}+\frac{h v_F l}{L^2\Delta}\left(\frac{\Phi_d\pm \Phi_d'}{2\Phi_0}\right)^2.\label{eq:DCU}\end{aligned}$$ The flux penetrating the dot is denoted as $\Phi_d$ and the phase $\phi\approx 2\pi\Phi_h/\Phi_0$ gained in the arm depends on the flux $\Phi_h$ through the hole. The traces read $$\begin{aligned} \label{eq:tr} \mbox{ tr }{\genfrac{\{}{\}}{0pt}{}{\P(\e,\Phi)\P^*(\e',\Phi')}{\P(\e,\Phi)\P^\dagger(\e',\Phi')}}&=&(M-N) \cos{\genfrac{\{}{\}}{0pt}{}{\phi+\phi'}{\phi-\phi'}}\nonumber \\ &\times &e^{iL_a [k(\e)-k(\e')]}.\end{aligned}$$ Since we assumed that the area of the arm is small compared to that of the dot, the energy-dependence of Eq. (\[eq:tr\]) can be neglected compared to that of $\Diff_{\cal U},\Coop_{\cal U}$ in Eq. (\[eq:DCU\]). We also assumed that since the arm is much longer than the size of the dot, $L_a\gg L$, the phases $\phi,\phi'$ of open trajectories in the arm correspond to the flux $\Phi_h,\Phi_h'$ through the hole. Therefore, the effective number of channels $N_{\Coop,\Diff}$, similar to Eq. (\[eq:channels\]) for quantum dots is $$\begin{aligned} \label{eq:DCfinal} {\genfrac{\{}{\}}{0pt}{}{N_{\cal C}}{N_{\cal D}}} &=& M-(M-N)\cos \frac{2\pi(\Phi_h\pm\Phi'_h)}{\Phi_0}\nonumber \\ &&+\frac{h v_F l}{ L^2\Delta}\left(\frac{\Phi_d\pm \Phi_d'}{2\Phi_0}\right)^2.\end{aligned}$$ The energy-dependent cooperon and diffuson in energy representation are given by $X(\e,\e')=1/[N_X-2\pi i(\e-\e')/\Delta], X=\Coop,\Diff$. Notice that when $\Phi=\Phi'$ the cooperon $\Coop$ is nonperiodic in the total flux $\Phi=\Phi_h+\Phi_d$ due to finite flux through the material of the sample, $\Phi_d\neq 0$. [MMM]{} V. I. Belinicher and B. I. Sturman, Usp. Fiz. Nauk [**130**]{}, 415 (1980) \[Sov. Phys. Usp. [**23**]{}, 199 (1980)\]. S. Washburn and R. A. Webb, Rep. Prog. Phys. [**55**]{}, 1311 (1992). B. L. Altshuler and D. E. Khmelnitskii, Pis’ma Zh. Eksp. Teor. Fiz. [**42**]{}, 291 (1985) \[JETP Lett. [**42**]{}, 359 (1985)\]. D. E. Khmelnitskii and A. I. Larkin, Phys. Scr., T [**T14**]{}, 4 (1986); Zh. Eksp. Teor. Phys. [**91**]{}, 1815 (1986) \[Sov. Phys. JETP [**64**]{}, 1075 (1986)\]. D. Sánchez and M. Büttiker, Phys. Rev. Lett. [**93**]{}, 106802 (2004). B. Spivak and A. Zyuzin, Phys. Rev. Lett. [**93**]{}, 226801 (2004). J. Wei, M. Shimogawa, Z. H. Wang, I. Radu, R. Dormaier, and D. H. Cobden , Phys. Rev. Lett. [**95**]{}, 256601 (2005). C. A. Marlow, R. P. Taylor, M. Fairbanks, I. Shorubalko, and H. Linke , Phys. Rev. Lett. [**96**]{}, 116801 (2006). R. Leturcq, D. Sánchez, G. Götz, T. Ihn, K. Ensslin, D. C. Driscoll, and A. C. Gossard, Phys. Rev. Lett. [**96**]{}, 126801 (2006). D. M. Zumbühl, C. M. Marcus, M. P. Hanson, A. C. Gossard, Phys. Rev. Lett. [**96**]{}, 206802 (2006). L. Angers, E. Zakka-Bajjani, R. Deblock, S. Guéron, H. Bouchiat, A. Cavanna, U. Gennser, and M. Polianski, Phys. Rev. B [**75**]{}, 115309 (2007). L. Angers, A. Chepelianskii, R. Deblock, B. Reulet, and H.  Bouchiat, Phys. Rev. B [**76**]{}, 075331 (2007). D.  Sánchez and M. Büttiker, Phys. Rev. B [**72**]{}, 201308(R) (2005). M. L. Polianski and M. Büttiker, Phys. Rev.  Lett. [**96**]{}, 156804 (2006). A. De Martino, R. Egger, and A. M. Tsvelik, Phys. Rev. Lett. [**97**]{}, 076402 (2006). M. L. Polianski and M. Büttiker, Physica E (Amsterdam) [**40**]{}, 67 (2007). A. V. Andreev and L. I. Glazman, Phys. Rev. Lett. [**97**]{}, 266806 (2006). B. Braunecker, D. E. Feldman, and F. Li, Phys. Rev. B [**76**]{}, 085119 (2007). A. Löfgren, C. A. Marlow, T. E. Humphrey, I. Shorubalko, R. P. Taylor, P. Omling, R. Newbury, P. E. Lindelof, and H. Linke, Phys. Rev. B [**73**]{}, 235321 (2006). V. I. Falko and D. E. Khmelnitsky, Zh. Eksp. Teor. Fiz. [**95**]{}, 328 (1989) \[Sov. Phys.  JETP [**68**]{}, 186 (1989)\]. A. A. Bykov, G. M. Gusev, Z. D. Kvon, D. I. Lubyshev, and V. P. Migal, JETP Lett. [**49**]{}, 13 (1989), J. Liu, M. A. Pennington, and N. Giordano, Phys. Rev. B [**45**]{}, 1267 (1992). J. J. Lin, R. E. Bartolo, and N. Giordano, Phys. Rev. B [**45**]{}, 14231 (1992). R. E. Bartolo, N. Giordano, X. Huang, and G. H. Bernstein, Phys. Rev. B [**55**]{}, 2384 (1997). A. A. Bykov, Z. D. Kvon, L. V. Litvin, Yu. V. Nastaushev, V. G. Mansurov, V. P. Migal, and S. P. Moschenko, Pis’ma Zh. Eksp. Teor. Fiz. [**58**]{}, 538 (1993) \[JETP Lett. [**58**]{}, 543 (1993)\]; A. A. Bykov, L.  V. Litvin, N. T. Moshegov, and A. I. Toropov, Superlattices Microstruct. [**23**]{}, 1285 (1998). P. W. Brouwer, Phys. Rev. B [**58**]{}, R10135 (1998). T. A. Shutenko, I. L. Aleiner, and B. L. Altshuler Phys. Rev. B [**61**]{}, 10366 (2000). M. H. Pedersen and M. Büttiker, Phys. Rev. B [**58**]{}, 12993 (1998). P. W. Brouwer, Phys. Rev. B [**63**]{}, 121303(R) (2001). M. G. Vavilov, V. Ambegaokar, and I. L. Aleiner, Phys. Rev. B [**63**]{}, 195313 (2001). J. Q. Zhang, S. Vitkalov, Z. D. Kvon, J. C. Portal, and A. Wieck, Phys. Rev. Lett. [**97**]{}, 226807 (2006). L. DiCarlo, C. M. Marcus, and J. S. Harris, Phys. Rev. Lett. [**91**]{}, 246804 (2003). M. Switkes, C. M. Marcus, K. Campman, and A. C. Gossard, Science [**283**]{}, 1905 (1999). M. Moskalets and M. Büttiker, Phys. Rev. B [**69**]{}, 205316 (2004); [**72**]{}, 035324 (2005). M. G. Vavilov, L. DiCarlo, and C. M. Marcus, Phys. Rev. B [**71**]{}, 241309(R) (2005) showed that the interference between parasitic rectification and quantum pumping produces current asymmetric to $\Phi\to-\Phi$ even in noninteracting dot. C. W. J. Beenakker, Rev. Mod. Phys. [**69**]{}, 731 (1997). I. L. Aleiner, P. W. Brouwer, and L. I. Glazman, Phys. Rep. [**358**]{}, 309 (2002). P. W. Brouwer and C. W. J. Beenakker, J. Math. Phys. [**37**]{}, 4904 (1996). M. L. Polianski and P. W. Brouwer, J. Phys. A [**36**]{}, 3215 (2003). M. Büttiker, A. Prêtre, and H. Thomas, Phys. Rev. Lett. [**70**]{}, 4114 (1993); M. Büttiker, J. Phys.: Condens. Matter [**5**]{}, 9361 (1993); M. Büttiker, H. Thomas, and A. Prêtre, Z. Phys. B [**94**]{}, 133 (1994). P. W. Brouwer, A Lamacraft, and K. Flensberg, Phys. Rev. B [**72**]{}, 075316 (2005). A. Löfgren, C. A. Marlow, I. Shorubalko, R. P. Taylor, P. Omling, L. Samuelson, and H. Linke, Phys. Rev. Lett. [**92**]{}, 046803 (2004). T. Christen and M. Büttiker, Europhys. Lett. [**35**]{}, 523 (1996). E. P. Wigner, Phys. Rev. [**98**]{}, 145 (1955); F. T. Smith, [*ibid*]{} [**118**]{}, 349 (1960). A number of transport problems where the Wigner-Smith matrix plays an important role are reviewed in M. Büttiker and M. L. Polianski, J. Phys. A [**38**]{}, 10559 (2005). P. W. Brouwer, S. A. van Langen, K. M. Frahm, M. Büttiker, and C. W. J. Beenakker, Phys. Rev. Lett. [**79**]{}, 913 (1997). P. W. Brouwer, K. M. Frahm, and C. W. J. Beenakker, Waves Random Media [**9**]{}, 91 (1999). M. L. Polianski, M. G. Vavilov, and P. W. Brouwer, Phys. Rev. B [**65**]{}, 245314 (2002). P. W. Brouwer and M. Büttiker, Europhys. Lett. [**37**]{}, 441 (1997). F. W. J. Hekking and J. P. Pekola, Phys. Rev. Lett. [**96**]{}, 056603 (2006). V. E. Kravtsov and V. I. Yudson, Phys. Rev. Lett. [**70**]{}, 210 (1993). A. G. Aronov and V. E. Kravtsov, Phys. Rev. B [**47**]{}, 13409 (1993). V. I. Yudson, Phys. Rev. B [**65**]{}, 115309 (2002).

--- author: - 'Lydie Koch-Miramond' - Péter Ábrahám - Yaël Fuchs - | \ Jean-Marc Bonnet-Bidaud - Arnaud Claret bibliography: - 'biblio.bib' date: 'Received 5 June 2002; Accepted 28 June 2002' title: 'A 2.4 - 12 $\mu$m spectrophotometric study with ISO of CygnusX-3 in quiescence [^1]' --- Introduction ============ CygnusX-3 has been known as a binary system since its discovery by @gia67, but there is still debate about the masses of the two stars and the morphology of the system (for a review see @bon88). The distance of the object is 8-12.5kpc with an absorption on the line of sight A$_V$ $\sim$ 20 mag [@ker96]. The flux modulation at a period of 4.8 hours, first discovered in X-rays [@par72], then at near infrared wavelengths [@bec73], and observed simultaneously at X-ray and near-IR wavelengths by @mas86, is believed to be the orbital period of the binary system. Following infrared spectroscopic measurements [@ker92], where WR-like features have been detected in I and K band spectra, the nature of the mass-donating star is suggested to be a Wolf-Rayet-like star, but an unambiguous classification, similar to the other WR stars, is still lacking. @mit96 and @van98 pointed out that it is not possible to find a model that meets all the observed properties of CygnusX-3 where the companion star is a normal Population I Wolf-Rayet star with a spherically symmetric stellar wind. In the evolution model originally proposed by @heu73 a final period of the order of 4.8 h may result from a system with initial masses M${_1^0}$=15[[M$_{\odot}$]{}]{}, M${_2^0}$=1[[M$_{\odot}$]{}]{}, P$^0$=5d, the final system being a neutron star accreting at a limited rate of $\sim$ 10$^{-7}$ [[M$_{\odot}$]{}]{}.yr$^{-1}$, from the wind of a core He burning star of about 3.8[[M$_{\odot}$]{}]{}. @van98 proposed that the progenitor of Cygnus X-3 is a 50[[M$_{\odot}$]{}]{}+10[[M$_{\odot}$]{}]{} system with P$^0$=6d; after spiral-in of the black hole into the envelope of the companion, the hydrogen reach layers are removed, and a 2-2.5 [[M$_{\odot}$]{}]{} Wolf-Rayet like star remains with P=0.2d. A system containing a black hole and an He core burning star is also favored by @erg98. In addition, CygnusX-3 undergoes giant radio bursts and there is evidence of jet-like structures moving away from CygnusX-3 at 0.3-0.9 c [@mio98; @mio01; @mar01].\ The main objective of the Infrared Space Observatory (ISO) spectrophotometric measurements in the 2.4-12 $\mu$m range was to constrain further the nature of the companion star to the compact object: the expected strong He lines as well as the metallic lines in different ionization states are important clues, together with the spectral shape of the continuum in a wavelength range as large as possible. An additional motivation for the imaging photometry with ISOCAM was to provide spatial resolution to a possible extended emission feature as a remnant of the expected high mass loss from the system. The paper is laid out as follows. In Section 2 observational aspects are reviewed. Section 3 summarizes the results on the continuum and line emissions from Cygnus X-3 and four Wolf-Rayet stars of WN6, 7 and 8 types. Section 4 reviews the constraints set by the present observations on the wind and on the nature of the companion to the compact object in Cygnux X-3. Finally, Section 5 summarizes the conclusions of this paper. -------------- ---------- ---------------- ---------- ---------- ------------ Instrument TDTNUM Wavelength Aperture range ($\mu$m) arcsec time (s) TU (start) ISOCAM-LW10 14200701 8-15 1.5 1134 6:57:09 ISOPHOT-SS 2.4-4.9 24x24 ISOPHOT-SL 5.9-11.7 24x24 ISOPHOT-P3.6 2.9-4.1 10 ISOPHOT-P10 9-10.9 23 ISOPHOT-P25 20-29 52 ISOPHOT-P60 48-73 99 -------------- ---------- ---------------- ---------- ---------- ------------ Observations and data reduction\[sobserv\] ========================================== We observed CygnusX-3 with the Infrared Space Observatory (ISO, see @kes96) on April7, 1996 corresponding to JD 2450180.8033 to 2450180.8519. The subsequent observing modes were: ISOCAM imaging photometry at 11.5$\mu$m (LW10 filter, bandwith 8 to 15$\mu$m), ISOPHOT-S spectrophotometry in the range 2.4-12$\mu$m, for 4096s, covering the orbital phases 0.83 to 1.04 (according to the parabolic ephemeris of @kit92); ISOPHOT multi-filter photometry at central wavelengths 3.6, 10, 25 and 60$\mu$m. Observing modes and observation times are summarized in Table 1. Preliminary results were presented in @koc01.\ ISOPHOT-S data reduction ------------------------ A low resolution mid-infrared spectrum of CygnusX-3 was obtained with the ISOPHOT-S sub-instrument. The spectrum covered the 2.4-4.9 and 5.9-11.7$\mu$m wavelength ranges simultaneously with a spectral resolution of about 0.04 and 0.1$\mu$m, respectively. The observation was performed in the triangular chopped mode with two background positions located at $\pm$120$''$, and with a dwelling time of 128s per chopper position. The field of view is 24[”]{}x24[”]{}. The whole measurement consisted of 8 OFF1–ON–OFF2–ON cycles and lasted 4096s. The ISOPHOT-S data were reduced in three steps. We first used the Phot Interactive Analysis (PIA[^2], @gab97) software (Version 8.2) to filter out cosmic glitches in the raw data and to determine signals by performing linear fits to the integration ramps. After a second deglitching step, performed on the signals, a dark current value appropriate to the satellite orbital position of the individual signal was subtracted. Finally we averaged all non-discarded (typically 3) signals in order to derive a signal per chopper step. Due to detector transient effects, at the beginning of the observation the derived signals were systematically lower than those in the consolidated part of the measurement. We then discarded the first $\sim$800sec (3 OFF-ON transitions), and determined an average \[ON–OFF\] signal for the whole measurement by applying a 1-dimensional Fast Fourier Transformation algorithm (for the application of FFT methods for ISOPHOT data reduction see @haa00). The \[ON–OFF\] difference signals were finally calibrated by applying a signal-dependent spectral response function dedicated to chopped ISOPHOT-S observations [@aco01], also implemented in PIA. In order to verify our data reduction scheme (which is not completely standard due to the application of the FFT algorithm) and to estimate the level of calibration uncertainties, we reduced HD184400, an ISOPHOT standard star observed in a similar way as CygnusX-3. The results were very consistent with the model prediction of the star, and we estimate that the systematic uncertainty of our calibration is less than 10$\%$. ISOPHOT spectral energy distribution ------------------------------------ {width="13.cm"} The observed spectral energy distribution is shown on Fig. 1. The observed (not dereddened) continuum flux in the range 2.4-7$\mu$m is 20$\pm10$mJy in good agreement with that observed by @ogl01 with ISOCAM on the same day (the dereddened fluxes are shown on Fig. 2) ; the observed flux decreases to about 10$\pm8$mJy around 9$\mu$m. An unresolved line is observed at about 4.3 $\mu$m peaking at 57$\pm$10 mJy. The linewidth is 0.04$\mu$m, consistent with the instrumental response and corresponding to $\sim$2500 km.s$^{-1}$. Note that the measured line flux might be underestimated because the ISOPHOT-S pixels are separated by small gaps, and a narrow line might falls into a gap. ISOPHOT-P data analysis and results ----------------------------------- The data reduction in the multi-filter mode was performed using the Phot Interactive Analysis [@gab97] software. After corrections for non-linearities of the integration ramps, the signal was transformed to a standard reset interval. Then an orbital dependent dark current was subtracted and cosmic ray hits were removed. In case the signal did not fully stabilize during the measurement time due to the detector transients, only the last part of the data stream was used. The derived flux densities were corrected for the finite size of the aperture by using the standard correction values as stated in the ISOPHOT Observer’s Manual [@kla94]. The flux detected at 3.6$\mu$m (bandwidth 1$\mu$m) in the $10''$ diameter aperture is 8.1$\pm$3.3 mJy at a confidence level of 2.4$\sigma$. No detection above the galactic noise was obtained at 10, 25 and 60$\mu$m with $23''$, $52''$ and $99''$ diameter aperture, respectively. ISOCAM data reduction and results --------------------------------- The LW10 filter centered at 11.5$\mu$m was used with the highest spatial resolution of $1.5''\times1.5''$ per pixel. The ISOCAM data were reduced with the Cam Interactive Analysis software (CIA[^3]) version 3.0, following the standard processing outlined in @sta99. First a dark correction was applied, then a de-glitching to remove cosmic ray hits, followed by a transient correction to take into account memory effects, using the inversion algorithm of @abe96, and a flat-field correction. Then individual images were combined into the final raster map, whose pixel values were converted into milli-Jansky flux densities. No colour correction was applied. A point source is clearly visible at the Cygnus X-3 position on the ISOCAM map at 11.5$\mu$m. The measured flux is 7.0$\pm$2.0mJy above a uniform background at a level of about 1.2mJy, in good agreement with our ISOPHOT result. This flux is lower than the 15.2$\pm$1.6mJy (at 11.5 $\mu$m) measured by @ogl01, on the same day, using ISOCAM/LW10 with a $6''\times 6''$ aperture. The high resolution configuration of the ISOCAM camera has been used to constrain the spatial extension of the infrared source. The measured FWHM for the source is 3.90$\pm$0.45arcsec (mean value of the four individual images composing the final raster map). This can be compared to the ISOCAM catalogued point spread functions at these energy and configuration which show a FWHM mean value of 3.44$\pm$0.45 arcsec, including the effects of the satellite jitter and of the pixel sampling. The slightly larger value for Cygnus X-3, though only marginally significant, might therefore indicate an extended source. The deconvolved extension would be 1.84 $\pm$ 0.64 arcsec which at a distance of 10 kpc corresponds to a linear extension of $\sim$ $2.7\times10^{17}$cm. The extended infrared source may be the result of the heating of the surrounding medium by the radio jets whose existence have been now clearly demonstrated both at arcsec [@mar00; @mar01] and sub-arcsec [@mio01] scales, but it clearly deserves confirmation. Results and discussion\[sresult\] ================================= Continuum spectral energy distribution:model fitting and comparison with four Wolf-Rayet stars ---------------------------------------------------------------------------------------------- {height="5.5cm"} {height="5.5cm"} {width="8cm"} {width="8cm"} The dereddening of the Cygnus X-3 spectrum is made using either @lut96 or @dra89 laws, and using an absorption value of A$_V$= 20 mag [@ker96]; the two dereddened spectra are shown in Fig. 2. They have clearly different shapes, but since the molecular composition of the absorbing material on the line of sight to CygnusX-3 is unknown, we cannot choose between these two laws.\ The spectral fitting of the dereddened spectra is shown in Fig. 3 and the results are given in Table 2. With the @lut96 law the best fit is obtained with a unique power law $S_\nu$$\propto$$\lambda$$^{-0.6\pm}$$^{0.3}_{0.4}$ with a reduced $\chi^2$=4.3 in the 2.4-12$\mu$m range, in good agreement with the Ogley et al (2001) result. With the @dra89 law the best fit is obtained with the sum of two components: a power law with slope $\lambda$$^{-1.6\pm}$$^{0.2}$ and a black body at T=250 K with a radius of 5000[[R$_{\odot}$]{}]{} at a distance of 10kpc, (reduced $\chi^2$=4.2 between 2.4 and 12$\mu$m), a hint of the presence of circumstellar dust. The power law part of the continuum spectrum ($S_\nu$$\propto$$\lambda$$^{-\alpha}$) can be explained by free-free emission of an expanding wind in the intermediate case between optically thick ($\alpha$=2) and optically thin ($\alpha$$\sim$0) regimes [@wri75]. ----------- -------------------- -------------------------- ------------------------ Object PL slope[^4] PL slope[^5] Flux at 4.7$\mu$m (Jy) CygnusX-3 1.6 $\pm$ 0.2 [^6] 0.6 $\pm$ $^{0.3}_{0.4}$ 0.079 $\pm$ 0.011 WR78 1.4 $\pm$ 0.2 [^7] 1.4 $\pm$ 0.2 0.114 $\pm$ 0.008 WR134 0.2 $\pm$ 0.2 $^d$ 0.2 $\pm$ 0.2 0.081 $\pm$ 0.006 WR136 1.0 $\pm$ 0.2 [^8] 1.0 $\pm$ 0.2 0.076 $\pm$ 0.006 WR147 1.6 $\pm$ 0.1 $^c$ 1.0 $\pm$ 0.1 0.085 $\pm$ 0.005 ----------- -------------------- -------------------------- ------------------------ : Comparison of the infrared continuum spectra of CygnusX-3 and four WR stars: power law slopes $\alpha$ such as S$_\nu$ $\propto$ $\lambda$$^{-\alpha}$ and 4.7 $\mu$m flux densities rescaled at 10 kpc ------- -------------------- -------------------- --------------- ----------- ------------------- ---------- Star Type Binarity Distance A$_V$[^9] Reference TDTNUM WR78 WN7h$^a$ WNL No 2.0kpc 1.48-1.87 @cro95II 45800705 WR134 WN6 possible $\sim$ 2.1kpc 1.22-1.99 @more99 17601108 WR136 WN6b(h) WNE-s[^10] possible 1.8kpc 1.35-2.25 @ste99 38102211 WR147 WN8(h) WNL B0.5V at 0.554$''$ 630 $\pm70$pc 11.2 @morr99 [@morr00] 33800415 ------- -------------------- -------------------- --------------- ----------- ------------------- ---------- {width="10cm"} Using the ISO archive data we have analysed the SWS spectra of four Wolf-Rayet stars: WR147 (WN8+B0.5), WR136 (WN6b), WR134 (WN6) and WR78 (WN7) whose main characteristics are given in Table3. We compare them to the Cygnus X-3 spectrum, after smoothing the SWS spectra to the resolution of the ISOPHOT-S instrument (using an IDL routine of B. Schulz dowloaded from the Home Page of the ISO Data Centre at Vilspa). The observed WR spectra are shown on Fig. 4 on top of the observed CygnusX-3 spectrum; the identification of the emission lines is from @morr00. The dereddened spectra of the Wolf-Rayet stars, using either the @dra89 law or the @lut96 law with the A$_V$ shown in Table3, have been fitted with power law slopes given in Table 2. Wolf-Rayet stars emit free-free continuum radiation from their extended ionized stellar wind envelopes and the different slopes reflect different conditions in the wind [@wil97]. It is noticeable that the mean continuum flux density of is the same (within a factor 1.5 at 4.7$\mu$m as seen in Table 2) as that of the four WR stars when their flux density is rescaled to a CygnusX-3 distance of 10kpc. The comparison between the CygnusX-3 spectrum and that of the Wolf-Rayet WR147 at 10kpc is shown in Fig. 5 after dereddening with the @dra89 law (left) and with the @lut96 law (right). The WR147 spectrum appears as the closest WR one to the CygnusX-3 spectrum, with almost the same mean flux density at 10kpc and the same power law slope (whithin the statistical errors). {width="8cm"} {width="8cm"} We note that WR147 is known as a colliding-wind binary that has been spatially resolved [@wil97; @ski99], with a separation on the sky large enough for the wind-wind collision zone between the stars to be resolved at near-infrared and radio [@wil97], and X-ray energies [@pit02]. The spectral energy distribution of WR147 in the 0.5$\mu$m to 2mm wavelength range (including all components) shown by @wil97 is dominated by the free-free emission from the stellar wind of the WN8 star; in the 2 to 10$\mu$m range these authors find $\alpha$=1.0, in good agreement with our ISOPHOT-S measurement (when dereddened with the @lut96 law); and in the mid-infrared to radio range they find $\alpha$=0.66. Emission lines; comparison with four Wolf-Rayet stars ----------------------------------------------------- The measured 4.3$\mu$m line flux above the continuum in the CygnusX-3 spectrum is 58$\pm$11mJy (dereddening with the @dra89 law), and 126$\pm$25mJy (dereddening with the @lut96 law), using respectively $\alpha$=1.6 and $\alpha$=0.6, the best fitted continuum slopes as given in Table 3, both detections being at more than 4.3$\sigma$. This line is interpreted as the HeI (3p-3s) line at 4.295$\mu$m, a prominent line in the WR147 (WN8+B0.5) spectrum as seen in @morr00 and in Fig.4. Again WR 147 appears as the closest WR to Cygnus X-3 as being the only WR in our sample with a HeI emission line at 4.3$\mu$m, the only line clearly seen in our CygnusX-3 data. The other expected He lines at 2.62, 3.73, 4.05, 7.46 and 10.5$\mu$m are not detected, probably due to the faintness of the object, at the limits of the instrument’s sensitivity. We note (Fig. 4) that the second highest peak in the SS-part of the CygX-3 spectrum is at 3.73$\mu$m, and there are also local maxima at 4.05 and 10.5$\mu$m. These expected lines are all blended with H lines and the absence of H observed by @ker96 in the I and K band spectra of CygnusX-3 could explain the weakness or absence of these lines in our data. We note that the Br$\alpha$+HeI-II line at 4.05$\mu$m is not detected in CygnusX-3 in quiescence, but is present in the four Wolf-Rayet stars. Strong HeI and HeII lines have been previously observed in the K-range in during quiescence [@ker92; @fen96; @fen99]. These lines have been interpreted [@ker96; @che94], as emission from the wind of a massive companion star to the compact object, and @fen99 suggest that the best candidate is probably an early WN Wolf-Rayet star. We note that the close match we have found between the mid-infrared luminosity and the spectral energy distribution, the HeI emission line in CygnusX-3 in quiescence, and that of the WR147, is consistent with a Wolf-Rayetlike companion of WN8 type to the compact object in CygnusX-3, a later type than suggested by earlier works [@ker96; @fen99; @han00]. Mass loss rate evaluation ------------------------- As @ogl01, we evaluated the mass loss rate of this free-free emitting wind, following the @wri75 formula (8) giving the emitted flux density (in Jansky) by a stellar wind assumed to be spherical, homogeneous and at a constant velocity : $$\centering \mathrm{S}_\nu \, =\, 23.2 \, \left(\frac{\dot{\mathrm{M}}}{\mu \mathrm{v}_\infty}\right)^{4/3} \ \frac{\nu^{2/3}}{ \mathrm{D}_ \mathrm{kpc}^2} \ \gamma^{2/3}\,\mathrm{g}^{2/3}\,\mathrm{Z}^{4/3} \quad \mathrm{Jy}$$ where D is the distance to the source in kpc. It gives : $$\centering \dot{\mathrm{M}}\, =\, 5.35.10^{-7} \, \mathrm{S}_\nu^{3/4}\, (\nu_ \mathrm{GHz}\, \gamma\, \mathrm{Z^2})^{-1/2} \, \mu\,\mathrm{v}_\infty \quad \mathrm{M_\odot .yr^{-1}}$$ (where $\nu$ is in GHz). With an assumed distance D=10kpc, a Gaunt factor g=1, a flux density deduced from the continuum fitting (@lut96 law, Fig. 3) of S$_\nu$=63 mJy at 6.75$\mu$m (4.44$\times 10^{4}$GHz), and for a WN-type wind (where the mean atomic weight per nucleon $\mu$=1.5, the number of free electrons per nucleon $\gamma_e$=1 and the mean ionic charge Z=1), and with a velocity of v$_\infty$= 1500km.s$^{-1}$ (van Kerkwijk 1996), one obtains M=1.2$\times$$10^{-4}$[M$_{\odot}$]{}.yr$^{-1}$. This is in agreement with the mass transfer rate estimated by @ker96 and (within a factor of 2) by @ogl01. This result is in good agreement with the recent revised WN mass-loss rate estimates, which have been lowered by a factor of 2 or 3 due to clumping in the wind [@morr99]. Note that if we assume a flattened disc-like wind as reported by @fen99, who detected double peaked emission lines, the mass loss rate decreases but remains within less than a factor of 2 of that obtained in the spherical case, in all but extreme cases when the ratio of structural length scales exceeds about 10, as shown by @sch82. It is noticeable that @chu92, considering the radio flux from the southern, stellar wind component of WR147, derived a mass-loss rate of M=4.2$\times$$10^{-5}$[M$_{\odot}$]{}.yr$^{-1}$ and @wil97 found a non spherically symetric stellar wind with a mass-loss rate of M=4.6$\times$$10^{-5}$[M$_{\odot}$]{}.yr$^{-1}$. Orbital modulation ------------------ Since the length of this spectrophotometric measurement was comparable to the 4.8h modulation period seen in the K-band at the level of 5$\%$ [@fen95], we attempted to detect this modulation in our data set. The points in Fig.6 refer to the average for the whole (2-12$\mu$m) spectrum. Although, as shown in Fig.6, the measurement uncertainty of the orbitally phase-resolved spectra was relatively high, the data clearly exclude periodic variations of amplitude higher than 15$\%$. ![Flux density in the 2.4-12 $\mu$m range versus Cygnus X-3 orbital phase []{data-label="Fig6"}](figCyg_lightcurve.ps){width="5.5cm"} Radio and X-ray fluxes ---------------------- Figure 7 shows the mean flux density of CygnusX-3 on MJD 50180.3 from radio to hard X-rays. The quiescent state observed in the mid-infrared range with ISO was also seen during monitoring radio observations of CygnusX-3 with the Ryle telescope (Mullard Radio Observatory, Cambridge) shortly before and after the ISO observations, the mean flux density at 15GHz being about 120mJy [@poo99], and with the Green Bank Interferometer (GBI) monitoring program [@mcc99] during a quiescent period before the ISO observations, the mean flux densities being 80$\pm$30mJy at 2.25GHz and 125$\pm$60mJy at 8.3GHz and the spectral index $\alpha$=0.3$\pm$0.1. These flux densities are at least one order of magnitude higher than that observed during the quench periods of very low radio emission preceeding the major flares of Cygnus X-3 [@mcc99]. In fact, this quiescent state was still present in 1996 May, June and July [@fen99]. ![Quasi-simultaneous observations of CygnusX-3 in the radio, infrared, soft and hard X-rays on April7, 1996, averaged over the orbital phase; the ISOPHOT-S spectrum is dereddened with the @lut96 law and rebinned to the resolution of 0.3$\mu$m; note that the GBI was dormant after 1996 April 1 till November 1996, and that the given flux densities (in dash lines) are mean values during the quiescent period March 17 to April 1, 1996 []{data-label="Fig7"}](figCygX3radioXnouv.ps){width="8.7cm"} In the X-ray range, at the same epoch, the $\it{Rossi}$ XTE/All Sky Monitor count rate was $\sim$7.5count.s$^{-1}$ corresponding to a mean flux of $\sim$1 mJy from 2 to 12keV (see XTE archive and @lev96), and the BATSE instrument on board the Compton Gamma Ray Observatory observed a mean photon flux of 0.039count.s$^{-1}$ corresponding to a flux density of 0.04mJy in the 20-100keV range. Thus the mid-infrared continuum spectrum whose shape is explained by thermal free-free emission in an expanding wind has a different origin than the non-thermal radio emission and the hard X-ray emission which are closely coupled [@mio01; @cho02]. Conclusions \[Sconclusion\] =========================== We have shown that the mid-infrared continuum (between 2.4-12$\mu$m) of CygnusX-3 in quiescence can be explained by the free-free emission of an expanding wind in the intermediate case between optically thick and optically thin regimes. The low quiescent luminosity of the object in the mid-infrared allows only detection of an upper limit of 15 percent on the possible 4.8h orbital modulation. A line at 4.3 $\mu$m is detected at a confidence level of more than 4.3$\sigma$, and is interpreted as the expected HeI (3p-3s) emission line. The close match between the mid-infrared brightness and spectral energy distribution of CygnusX-3 in quiescence, the HeI emission line, the high mass loss rate in the wind and that of the colliding-wind Wolf-Rayet system WR147, is consistent with a Wolf-Rayetlike companion of WN8 type to the compact object in CygnusX-3, a later type than suggested by previous works [@ker96; @fen99; @han00]. We warmly thank the ISO project and the ISOCAM and ISOPHOT Teams in Villafranca, Saclay and Heidelberg. We express our gratitude to R. Ogley and to R. Fender for helpful comments, to G. Pooley for giving us the Ryle telescope data and to J.L. Starck for very useful discussions on data analysis. We thank the referee J. Martí for helpful comments on the manuscript. This research has made use of data from the Green Bank Interferometer, a facility of the National Science Foundation operated by the NRAO in support of NASA High Energy Astrophysics programs, of data which were generated by the CGRO BATSE Instrument Team at the Marshall Space Flight Center (MSFC) using the Earth occultation technique, and of quick-look results provided by the ASM/RXTE team. P.A. acknowledges the support of a Hungarian science grant. + [^1]: Based on observations with ISO, an ESA project with instruments funded by ESA Member States (especially the PI countries: France, Germany, the Netherlands and the United Kingdom) and with the participation of ISAS and NASA. [^2]: PIA is a joint development by the ESA Astrophysics Division and the ISOPHOT consortium led by the Max–Planck–Institut für Astronomie, Heidelberg. [^3]: CIA is a joint development by the ESA Astrophysics Division and the ISOCAM consortium. The ISOCAM consortium was led by the ISOCAM PI, C. Cesarsky, Direction des Sciences de la Matière, C.E.A, France. [^4]: dereddening with @dra89 law [^5]: dereddening with @lut96 law; fit between [^6]: fit between 2.4-6.5 $\mu$m [^7]: fit between 2.4-12 $\mu$m [^8]: fit between 2.4-8 $\mu$m [^9]: from @huc01 except for WR 147 [^10]: from @cro95IV

--- author: - | \ Physics Department, National Technical University, GR-15780 Athens, Greece\ E-mail: - | P. Manousselis\ Physics Department, National Technical University, GR-15780 Athens, Greece\ E-mail: - | G. Zoupanos\ Physics Department, National Technical University, GR-15780 Athens, Greece\ Institute of Theoretical Physics, D-69120 Heidelberg, Germany\ Max-Planck Institut für Physik, Fohringer Ring 6, D-80805 Munchen, Germany\ Laboratoire d’ Annecy de Physique Theorique, Annecy, France\ E-mail: title: Noncommutative Gauge Theories and Gravity --- Introduction ============ Three out of four interactions of nature are grouped together under a common description by the Standard Model in which they are described by gauge theories. However, the gravitational interaction is not part of this picture, admitting a separate, geometric formulation, that is the theory of General relativity. In order to make contact among the two different pictures, there has been an undertaking in which gravity admits a gauge-theoretic approach, besides the geometric one [@Utiyama:1956sy]-[@Witten:1988hc]. Pioneer in this field was Utiyama, whose work was focused on describing 4-d gravity of General Relativity as a gauge theory, localizing the Lorentz symmetry, SO(1,3) [@Utiyama:1956sy]. However, the results were not considered to be successful, since the inclusion of the vielbein did not happen in a convincing way. A few years later, it was Kibble [@Kibble:1961ba] who modified the above consideration, adopting the inhomogeneous Lorentz group (Poincaré group), ISO(1,3), as the gauge group in which, along with the spin connection, the vierbein were also identified as gauge fields of the theory. Nevertheless, the dynamics of General Relativity remained unretrieved, since there was no action of gauge-theoretic origin of the Poincaré gauge group that would be identified as the Einstein-Hilbert action. Solution to this problem was given with the consideration of an SO(1,4) gauge invariant Yang-Mills action (instead of the Poincaré one) along with the involvement of a scalar field in the fundamental representation of the gauge group, SO(1,4) [@Stelle:1979aj] (see also [@MacDowell:1977jt; @Ivanov:1980tw; @Kibble:1985sn]). The gauge fixing of this scalar field led to a spontaneous symmetry breaking, recovering the Einstein-Hilbert action. Therefore, the 4-d gravitational theory of General Relativity was successfully described as a gauge theory with the presence of a scalar field. Moreover, in the absence of cosmological constant, 3-d Einstein gravity can be also described as a gauge theory of the 3-d Poincaré group, ISO(1,2). In turn, the 3-d de Sitter and Anti de Sitter groups, SO(1,3) and SO(2,2), respectively are employed, in case a cosmological constant is present [@Witten:1988hc]. The first part of the construction, that is the calculation of the transformation of the gauge fields (dreibein and spin connection) and the curvature tensors is similar to the 4-d case. However, the dynamic part is less tedious than that of the 4-d case. The 3-d Einstein-Hilbert action is recovered after the consideration of a Chern-Simons action functional, which is, in fact, identical to the 3-d Einstein-Hilbert’s action. Thus, 3-d Einstein gravity is precisely equivalent to an ISO(1,2) Chern-Simons gauge theory. Another contribution in this aspect is related with the gauge-theoretic approach of Weyl gravity (and supergravity) as a gauge theory of the 4-d conformal group [@Kaku:1977pa; @Fradkin:1985am][^1]. Proceeding in the same spirit as in the previous cases, the transformations of the gauge fields and the expressions of the various curvature tensors are obtained. The action is determined to be SO(2,4) gauge invariant of Yang-Mills type, as it is expected. Then, constraints are imposed on the curvature tensors and along with gauge fixing of the fields, the final action is actually the Weyl action. Therefore, it is understood that Weyl gravity admits a gauge-theoretic interpretation of the conformal group. An appropriate framework for the construction of physical theories at the high-energy regime (Planck scale), in which commutativity of the coordinates cannot be naturally assumed, is that of noncommutative geometry [@connes] - [@Gavriil:2015lka]. A very improtant feature of this framework is the potential regularization of quantum field theories and the construction of finite theories. Nevertheless, building quantum field theories on noncommutative spaces is a tedious task and, moreover, problematic ultraviolet features have been encountered [@filk] (see also [@grosse-wulkenhaar] and [@grosse-steinacker]). Despite that, the framework of noncommutative geometry is considered to be a suitable background for accommodating particle physics models, formulated as noncommutative gauge theories [@connes-lott] (see also [@martin-bondia; @dubois-madore-kerner; @madorejz]). Also, taking into account the above correspondence between gravity and (ordinary) gauge theories, the well-established formulation of noncommutative gauge theories [@Madore:2000en] allows one to use it as methodology for the construction of models of noncommutative gravity. Such approaches have been considered before, see for example refs. [@Chamseddine:2000si]-[@Ciric:2016isg] and, specifically, for 3-d models, employing the Chern-Simons gauge theory formulation, see [@Cacciatori:2002gq]-[@Banados:2001xw]. The authors of the above works to which we refered make use of constant noncommutativity (Moyal-Weyl) and also use the formulation of the $\star$-product and the Seiberg-Witten map [@Seiberg:1999vs]. However, besides the $\star$-product formulation, noncommutative gravitational models can be constructed using the noncommutative realization of matrix geometries [@Banks:1996vh; @Ishibashi:1996xs]. Such approaches, specifically for Yang-Mills matrix models, were proposed in the past few years, see refs. [@Aoki:1998vn]-[@Nair:2006qg]. Also, for alternative approaches on the subject see [@Buric:2006di; @Buric:2007zx; @Buric:2007hb], but also [@Aschieri:2003vyAschieri:2004vhAschieri:2005wm]. In general, formulation of noncommutative gravity implies that the noncommutative deformations break the Lorentz invariance. However, there exist specific noncommutative deformations which preserve the Lorentz invariance and the corresponding background spaces are called covariant noncommutative spaces [@Snyder:1946qz; @Yang:1947ud]. Along these lines, in ref.[@Heckman:2014xha], a noncommutative deformation of a general conformal field theory defined on 4-d dS or AdS spacetime has been employed, see also [@Buric:2015wta]-[@Steinacker:2016vgf]. In this proceedings contribution, our recent contributions in the above field of noncommutative gravity are included. First, we briefly review our proposition for a matrix model of 3-d noncommutative gravity [@Chatzistavrakidis:2018vfi] (see also [@Manolakos:2018isw; @Manolakos:2018hvn]), in which the corresponding background space is the $\mathbb{R}_\lambda^3$, introduced in ref. [@Hammou:2001cc] (see also ref. [@Vitale:2014hca] for field theories on this space), which is actually the 3-d Euclidean space foliated by multiple fuzzy spheres of different radii. As explained in ref.[@Kovacik:2013yca], the above fuzzy space admits an SO(4) symmetry, which is in fact the gauge group we considered. Noncommutativity implies the enlargement of the SO(4) to the U(2)$\times$U(2) gauge group, in a fixed representation, in order that the anticommutators of the generators close. In the same spirit, the Lorentz analogue of the above construction was also explored, in which the corresponding noncommutative space is the $\mathbb{R}_\lambda^{1,2}$, that is the 3-d Minkowski spacetime foliated by fuzzy hyperboloids [@Jurman:2013ota]. In this case too, the initial gauge group, SO(1,3), is eventually extended to GL(2,$\mathbb{C}$) in a fixed representation, for the same reasons as in the Euclidean case. In both signatures, the action proposed is a functional of Chern-Simons type and its variation produces the equations of motion. In addition, the commutative limit is considered, retrieving the expressions of the 3-d Einstein gravity. Second, a 4-d gravity model as a noncommutative gauge theory is constructed [@Manolakos:2019fle]. Motivated by Heckman-Verlinde [@Heckman:2014xha] who were based on Yang’s early work [@Yang:1947ud], we considered a noncommutative version of the 4-d de Sitter space, which is in fact a covariant fuzzy space, preserving Lorentz invariance. Some of the generators of the algebra of the final symmetry group of this space, SO(1,5), are identified as its noncommutative coordinates. As in the previous 3-d case, the final gauge group is a minimal extension of the initial, SO(1,4), specifically the SO(1,5)$\times$U(1), for the same reasons. According to the standard procedure, the corresponding gauge fields are determined and their gauge transformations and their corresponding component curvature tensors are obtained. Eventually, an action of Yang-Mills type is employed and its initial gauge symmetry breaks imposing certain conditions (constraints) on the curvature tensors and the gauge fields. The commutative limit of the model reduces the obtained expressions of gauge transformations of the fields and the ones of the tensors to those of the conformal gravity. The outline of the present contribution is as follows: First we briefly recall the gauge-theoretic approaches of gravitational theories mentioned above. Then, we include the necessary information about the construction of gauge theories in the noncommutative framework. Next, we review our suggestions for noncommutative gravity models in three and four dimensions. Eventually, we write down our conclusions and comment on the results. Gauge-theoretic approach of gravity =================================== In this section we briefly review the gauge-theoretic approach of various gravity theories [@Utiyama:1956sy]-[@Witten:1988hc], which consists the basis of our works in which the whole scheme is translated to the framework of noncommutativity. 3-d Einstein gravity {#3dgravity} -------------------- Let us begin with the 3-d case, in which Einstein gravity is precisely described by a Chern-Simons gauge theory of the ISO(1,2), Poincaré group [@Witten:1988hc]. Specifically for the dynamic part, the 3-d Einstein-Hilbert action is written down as: $$\label{eh3} S_{\text{EH3}}=\frac{1}{2}\int {\mathrm{d}}^3x \,\epsilon^{\mu\nu\rho}\epsilon_{a b c}\, e_{ \mu}{}^{a} R_{ \nu \rho}{}^{ b c}(\omega)~,$$ which, as it will be explained, is identical to a Chern-Simons functional of ISO(1,2). The ISO(1,2) algebra is generated by six operators, i.e. the three local translations, $P_{ a}$ and the three Lorentz transformations, $M^{ a}=\epsilon^{ a b c}M_{ b c}$, with $ a=1,2,3$ and determine the algebra through the following commutation relations: $$[M_{ a},M_{ b}]=\epsilon_{ a b c}M^{ c}~,\,\,\,\,\,\, {[}P_{ a},M_{ b}]= \epsilon_{ a b c}P^{ c}~,\,\,\,\,\, {[}P_{ a},P_{ b}]=0~.$$ According to the gauging procedure, the gauge connection and gauge parameter are written down as: $$A_{ \mu}=e_{ \mu}{}^{ a}P_{ a}+\omega_{ \mu}{}^{ a}M_{ a}~,\,\,\,\,\, \epsilon=\xi^{ a}P_{ a}+\lambda^{ a}M_{ a}~,$$ where the vielbein and the spin connection are identified as the gauge fields of the theory. The above expressions along with the transformation rule of the gauge connection: $$\delta A_\mu=D_\mu \epsilon=\partial_\mu+[A_\mu,\epsilon]~,\label{poincaretransformation}$$ lead to the calculation of the gauge transformations of the component fields: $$\begin{aligned} \delta e_{\mu}^{~a}&=\partial_{\mu}\xi^a+\omega_{\mu}^{~ab}\xi_b-\lambda^a_{~b}e_{\mu}^{~b}~, \\ \delta \omega_{\mu}^{~ab}&=\partial_{\mu}\lambda^{ab}+\lambda^a_{~c}\omega_\mu^{~bc}-\lambda^b_{~c}\omega_\mu^{~ac}~\end{aligned}$$ and the component curvature tensors: $$\begin{aligned} T_{\mu\nu}^{~~a}&=\partial_\mu e_\nu^{~a}-\partial_\nu e_\mu^{~a}-\omega_\mu^{~ab}e_{\nu b}+\omega_\nu^{~ab}e_{\mu b}~,\nonumber \\ R_{\mu\nu}^{~~ab}&=\partial_\mu\omega_\nu^{~ab}-\partial_\nu\omega_\mu^{~ab}-\omega_\mu^{~ac}\omega_{\nu c}^{~~b}+\omega_\nu^{~ac}\omega_{\mu c}^{~~b}~,\label{poincarecomponenttensors}\end{aligned}$$ starting from the defining relation of the curvature two-form, $R_{\mu\nu}$: $$R_{\mu\nu}=[D_\mu,D_\nu]=\partial_\mu A_\nu - \partial_\nu A_\mu+[A_{\mu},A_{\nu}]\label{poincarecurvature}$$ and expanding it on the generators of the algebra as: $$R_{\mu\nu}=T_{\mu\nu}^{~~a}P_a+\frac{1}{2} R_{\mu\nu}^{~~ab}M_{ab}~.\label{poincarecurvatureexpansion}$$ Also, consideration of a Chern-Simons action functional leads to the Einstein-Hilbert action, . In addition, 3-d gravitational theory with cosmological constant is also described in a gauge-theoretic approach. In this case, the gauge groups considered are the 3-d dS or AdS groups, SO(1,3) and SO(2,2), respectively, depending on the sign of the constant. The procedure of the construction of the gauge theory is the same, obtaining results that generalize the above of the ISO(1,2) case, because of the difference in the right hand side of the commutator of the translations, which is now non-zero: $$[P_{ a},P_{ b}]=\lambda M_{ a b}~.$$ 4-d Einstein gravity {#einsteingravity} -------------------- Now, gravitational interaction in four dimensions is described by General Relativity, which consists a solid and successful theory, having passed many tests since its early days. Its formulation is geometric, differentiating it from the rest interactions, which are described as gauge theories. Aiming at a connection between gravitational and the other interactions, the undertaking of the gauge-theoretic approach of gravity took place [@Utiyama:1956sy]-[@Kibble:1985sn]. Here we mention briefly the main features of this gauge-theoretic approach of the 4-d Einstein’s gravity. First, for this gauge-theoretic approach of 4-d gravity, the vierbein formulation of General Relativity has to be considered. The gauge group is reasonably chosen to be the ISO(1,3) (Poincaré group), since it is the isometry group of the 4-d Minkowski spacetime. The generators of the corresponding algebra satisfy the following commutation relations, which in fact determine the algebra: $$[M_{ab},M_{cd}]=4\eta_{[a[c}M_{d]b]}~,\,\,\,\,\, [P_a,M_{bc}]=2\eta_{a[b}P_{c]}~,\,\,\,\,\, [P_a,P_b]=0~,$$ where $\eta_{ab}=\mathrm{diag}(-1,1,1,1)$ is the metric tensor of the 4-d Minkowski spacetime, $M_{ab}$ are the generators of the Lorentz group (Lorentz transformations) and $P_a$ are the generators of the local translations. According to the standard gauging procedure, that is the one followed in the 3-d case described in the previous section, the gauge potential, $A_\mu$, is defined and it is expressed as a decomposition on the generators of the Poincaré algebra, as: $$A_{\mu}(x)=e_{\mu}{}^a(x)P_a+\frac{1}{2}\omega_{\mu}{}^{ab}(x)M_{ab}~.\label{connection}$$ The functions accompanying the generators of the algebra in the above decomposition are identified as the gauge fields of the theory and, specifically in this case, they are identified as the vierbein, $e_\mu^{~a}$ and the spin connection, $\omega_\mu^{~ab}$, which correspond to the translations, $P_a$, and Lorentz generators, $M_{ab}$, respectively. The consideration of the vierbein as a gauge field implies the mixing between the internal symmetry and spacetime making this kind of construction special, compared to the ordinary gauge theories. The gauge connection transforms according to the following rule: $$\delta A_{\mu}=\partial_{\mu}\epsilon+[A_{\mu},\epsilon]~,\label{transformconnection}$$ where $\epsilon=\epsilon(x)$ is the gauge transformation parameter which is also expanded on the generators of the algebra: $$\epsilon(x)=\xi^a(x) P_a+\frac{1}{2} \lambda^{ab}(x)M_{ab}~.\label{parameter}$$ Combination of the equations and with lead to the expressions of the transformations of the gauge fields: $$\begin{aligned} \delta e_{\mu}{}^a&=\partial_{\mu}\xi^a+\omega_{\mu}{}^{ab}\xi_b-\lambda^a{}_be_{\mu}{}^b~, \\ \delta \omega_{\mu}{}^{ab}&=\partial_{\mu}\lambda^{ab}-2\lambda^{[a}{}_c\omega_{\mu}{}^{cb]}~.\end{aligned}$$ The corresponding field strength tensor of the gauge theory is defined as: $$R_{\mu\nu}(A)=2\partial_{[\mu}A_{\nu]}+[A_{\mu},A_{\nu}]~\label{usualformula}$$ and is expanded on the generators as it is valued in the algebra: $$R_{\mu\nu}(A)=R_{\mu\nu}{}^a(e)P_a+\frac{1}{2} R_{\mu\nu}{}^{ab}(\omega)M_{ab}~,\label{expansion_of_curvature}$$ where $R_{\mu\nu}{}^a(e)$ and $R_{\mu\nu}{}^{ab}(\omega)$ are the curvatures associated to the component gauge fields, identified as the torsion and curvature, respectively. Replacement of the equations and in the leads to their explicit expressions: $$\begin{aligned} R_{\mu\nu}{}^a(e)&=2\partial_{[\mu}e_{\nu]}{}^a-2\omega_{[\mu}{}^{ab}e_{\nu]b}~,\\ R_{\mu\nu}{}^{ab}(\omega)&=2\partial_{[\mu}\omega_{\nu]}{}^{ab}-2\omega_{[\mu}{}^{ac}\omega_{\nu]c}{}^b~.\label{curvtwoform}\end{aligned}$$ Moving on with the dynamic part of the theory, the most reasonable choice is an action of Yang-Mills type, being invariant under the ISO(1,3) gauge group. However, the aim is to result with the Einstein-Hilbert action, which is Lorentz invariant and, therefore, the Poincaré symmetry of the initial action has to be broken to the Lorentz. This can be carried out through a spontaneous symmetry breaking, induced by a scalar field which belongs to the fundamental representation of the SO(1,4) [@Stelle:1979aj; @Ivanov:1980tw], that is also included in the theory. The choice of the 4-d de Sitter group is an alternative and preferred choice to that of the Poincaré group, since all generators of the algebra can be considered on equal footing. The spontaneous symmetry breaking leads to the breaking of the translational generators, resulting to a theory with vanishing torsion constraint and a Lorentz invariant action involving the Ricci scalar (and a topological Gauss-Bonnet term), that is Einstein-Hilbert action. Concluding, Einstein’s 4-d gravity theory is not equivalent to a pure Poincaré gauge theory but to an SO(1,4) gauge theory with the inclusion of a scalar field and the addition of an appropriate potential term in the Lagrangian, which leads to a spontaneous symmetry breaking. An alternative way to obtain an action with Lorentz symmetry, is to impose that the action is invariant only under the Lorentz symmetry and not under the total Poincaré symmetry with which one begins. This means that the curvature tensor related to the translations has to be zero, that is the imposition of the torsionless condition, that is a constraint that is necessary for resulting with an action with Lorentz symmetry. Solution of this constraint leads to a relation of the spin connection with the vielbein: $$\omega_\mu^{~ab}=\frac{1}{2}e^{\nu a}(\partial_\mu e_\nu^{~b}-\partial_\nu e_\mu^{~b})-\frac{1}{2}e^{\nu b}(\partial_\mu e_\nu^{~a}-\partial_\nu e_\mu^{~a})-\frac{1}{2}e^{\rho a}e^{\sigma b}(\partial_\rho e_{\sigma c}-\partial_\sigma e_{\rho c})e_\mu^{~c}\,. \label{spin-viel}$$ However, straightforward consideration of an action of Yang-Mills type with Lorentz symmetry, would lead to an action involving the $R(M)^2$ term, which is not the correct one, since the target is the Einstein-Hilbert action. Also, such an action would imply the wrong dimensionality (zero) of the coupling constant of gravity. In order to result with the Einstein-Hilbert action, which includes a dimensionful coupling constant, the action has to be considered in an alternative, non-straightforward way, that is the construction of Lorentz invariants out of the quantities (curvature tensor) of the theory. The one that is built by certain contractions of the curvature tensor is the correct one, ensuring the correct dimensionality of the coupling constant, and is identified as the Ricci scalar and the corresponding action is eventually the Einstein-Hilbert action. 4-d Conformal gravity leading to Weyl or Einstein gravity {#weylgravity} --------------------------------------------------------- Besides the above, also Weyl gravity admits a gauge-theoretic formulation, specifically of the 4-d conformal group, SO(2,4). In this case, too, the transformations of the fields and the expressions of the curvature tensors are obtained in a straightforward way. The action that is considered initially is an SO(2,4) invariant action of Yang-Mills type which breaks by the imposition of constraints on the curvature tensors. After the consideration of the constraints, the resulting action of the theory is identical to the scale invariant Weyl action [@Kaku:1977pa; @Fradkin:1985am; @vanproeyen] (see also [@cham-thesis; @Chamseddine:1976bf]). Let us start with the identification of the generators of the conformal algebra of SO(2,4). The fifteen generators are the local translations, ($P_a$), the Lorentz transformations, ($M_{ab}$), the conformal boosts, ($K_a$) and the dilatations, ($D$). The algebra of SO(2,4) is determined by the commutation relations of the above generators: $$\begin{aligned} [M_{ab},M^{cd}]&=4M_{[a}^{~[d}\delta_{b]}^{c]}\,,\quad [M_{ab},P_c]=2P_{[a}\delta_{b]c}\,,\quad [M_{ab},K_c]=2K_{[a}\delta_{b]c}\,,\nonumber \\ [P_a,D]&=P_a\,,\quad [K_a,D]=-K_a\,,\quad [P_a,K_b]=2(\delta_{ab}D-M_{ab})\,,\end{aligned}$$ where $a,b,c,d=1\ldots 4$. According to the gauging procedure, the gauge potential of the theory is defined and is expanded on the various generators as: $$A_{\mu} = e_{\mu}^{~a}P_{a} + \frac{1}{2}\omega_{\mu}^{~ab}M_{ab} + b_{\mu}D + f_{\mu}^{~a}K_{a}\,,\label{connection-conformal}$$ in which a gauge field has been associated with each generator. In this case, too, the vierbein and the spin connection are identified as gauge fields of the theory. The transformation rule of the gauge connection, , is: $$\delta_{\epsilon}A_{\mu} = D_{\mu}\epsilon = \partial_{\mu}\epsilon+ [A_{\mu},\epsilon]\,,\label{commrule}$$ where $\epsilon$ is a gauge transformation parameter valued in the Lie algebra of the SO(2,4) group and for this reason it can be written as: $$\epsilon = \epsilon_{P}^{~a} P_{a} + \frac{1}{2}\epsilon_{M}^{~~ab} M_{ab} + \epsilon_{D}D+ \epsilon_{K}^{~a}K_{a}\,.\label{parameter-conformal}$$ Combination of the equations , and , leads to the expressions of the transformations of the gauge fields of the theory: $$\begin{aligned} \delta e_{\mu}^{~a} &= \partial_\mu\epsilon_P^{~a}+2ie_{\mu b}\epsilon_M^{~ab}-i\omega_\mu^{~ab}\epsilon_{Pb}-b_\mu\epsilon_K^{~a}+f_\mu^{~a}\epsilon_D\,, \nonumber \\ \delta \omega_{\mu}^{~ab} &= \frac{1}{2}\partial_\mu\epsilon_M^{~ab}+4ie_\mu^{~a}\epsilon_P^{~b}+\frac{i}{4}\omega_\mu^{~ac}\epsilon_{M~c}^{~~b}+if_\mu^{~a}\epsilon_K^{~b}\,, \nonumber \\ \delta b_{\mu} &= \partial_\mu\epsilon_D-e_\mu^{~a}\epsilon_{Ka}+f_\mu^{~a}\epsilon_{Pa}\,, \nonumber \\ \delta f_{\mu}^{~a} &= \partial_\mu\epsilon_K^{~a}+4ie_\mu^{~a}\epsilon_D-i\omega_\mu^{~ab}\epsilon_{Kb}-4ib_\mu\epsilon_P^{~a}+if_\mu^{~b}\epsilon_{M~b}^{~~a}\,. \label{conformaltrans}\end{aligned}$$ The field strength tensor is defined by the following relation: $$R_{\mu\nu} = 2 \partial_{[\mu} A_{\nu]} - i [ A_{\mu} ,A_{\nu} ]\label{fieldstrengthconformal}$$ and is expanded on the generators as: $$R_{\mu\nu}=\tilde{R}_{\mu\nu}^{~~a}P_a+\frac{1}{2}R_{\mu\nu}^{~~ab}M_{ab}+R_{\mu\nu}+R_{\mu\nu}^{~~a}K_a~.\label{conformalexpansion}$$ Combining the equation and , the expressions of the component curvature tensors are obtained: $$\begin{aligned} R_{\mu\nu}^{~~~a}(P) &= 2 \partial_{[\mu}e_{\nu]}^{~~a} + f_{[\mu}^{~~a}b_{\nu]} + e^{~~b}_{[\mu} \omega_{\nu]}^{~~ac} \delta_{bc}, \nonumber \\ R_{\mu\nu}^{~~~ab}(M) &= \partial_{[\mu} \omega_{\nu]}^{~~ab} + \omega_{[\mu}^{~~ca} \omega_{\nu]}^{~~db} \delta_{cd} + e_{[\mu}^{~~a}e_{\nu]}^{~~b} + f_{[\mu}^{~~a}f_{\nu]}^{~~b}, \nonumber \\ R_{\mu\nu}(D) &= 2 \partial_{[\mu}b_{\nu]} + f_{[\mu}^{~~a}e_{\nu]}^{~~b}\delta_{ab}, \nonumber \\ R_{\mu\nu}^{~~~a}(K) &= 2 \partial_{[\mu}f_{\nu]}^{~~a} + e_{[\mu}^{~~a}b_{\nu]} + f_{[\mu}^{~~b}\omega_{\nu]}^{~~ac}\delta_{bc}\,. \label{conformalcurvatures}\end{aligned}$$ Regarding the action, at first it is taken to be an SO(2,4) invariant of Yang-Mills type. The initial, SO(2,4), symmetry gets broken by the imposition of certain constraints, [@Kaku:1977pa; @Fradkin:1985am; @vanproeyen], that is the torsionless condition, $R(P)=0$ and an additional constraint on $R(M)$. The two constraints admit an algebraic solution leading to expressions of the fields $\omega_\mu^{~ab}$ and $f_\mu^{~a}$ in terms of the independent fields $e_\mu^{~a}$ and $b_\mu$. Also, the gauge fixing $b_\mu=0$ can be employed and, inclusion of all constraints in the initial action lead to the well-known Weyl action. Besides the above breaking of the conformal symmetry which led to the Weyl action, it is possible to employ an alternative breaking route, this time leading to an action with Lorentz symmetry, explicitly the Einstein-Hilbert action [@Chamseddine:2002fd]. From our prespective, the latter can be achieved through an alternative symmetry breaking mechanism, specifically with the inclusion of two scalar fields in the fundamental representation of the conformal group [@Li:1973mq], being a generalization of the case of the breaking of the 4-d de Sitter group down to the Lorentz group by the inclusion of a scalar in the fundamental representation of SO(1,4), as explained in section \[einsteingravity\]. The inclusion of two scalars could trigger a spontaneous symmetry breaking in a theory with matter fields and the resulting action would be the Einstein-Hilbert one, respecting Lorentz symmetry. Calculations and details on this issue will be included in a future work. Moreover, the argument used in the previous section in the four-dimensional Poincaré gravity case, that is an alternative way to break the initial symmetry to the Lorentz, can be generalized for this case of conformal gravity. Since it is desired to result with the Lorentz symmetry out of the initial SO(2,4), the vacuum of the theory is considered to be directly SO(4) invariant, which means that every other tensor, except for the $R(M)$, has to be vanishing. Setting these tensors to zero will produce the constraints of the theory leading to expressions that relate the gauge fields. In particular, in [@Chamseddine:2002fd], it is argued that if both tensors $R(P)$ and $R(K)$ are simultaneously set to zero, then from the constraints of the theory it is understood that the corresponding gauge fields, $f_\mu^{~a}, e_\mu^{~a}$ are equal - up to a rescaling factor - and $b_\mu=0$. Noncommutative gauge theories ============================= Let us now briefy recall the basic concepts of the formulation of gauge theories on noncommutative spaces, in order to use them later for the construction of the noncommutative gravity models. For convenience, the following methodology is performed on the most typical noncommutative (fuzzy) space, the fuzzy sphere [@Madore:1991bw]. Obviously, the results can be easily generalized in the cases of other noncommutative spaces, too. Let a field $\phi(X_a)$ of the fuzzy sphere, written in terms of powers of $X_a$ [@Madore:2000en], and a gauge group, G. An infinitesimal gauge transformation of $\phi(X_a)$ is: $$\delta\phi(X)=\lambda(X)\phi(X)\,,\label{gaugetransffuzzy}$$ where $\lambda(X)$ is the gauge transformation parameter. If $\lambda(X)$ is a function of the coordinates, $X_a$, then it is an infinitesimal Abelian transformation and G=U(1), while if $\lambda(X)$ is a P$\times$P Hermitian matrix, then the transformation is non-Abelian and the gauge group is G=U(P). The coordinates are invariant under an infinitesimal transformation of the the gauge group, G, namely $\delta X_a=0$. Now, the gauge transformation of the product of the field and a coordinate is not covariant: $$\delta(X_a\phi)=X_a\lambda(X)\phi\,,$$ since, in general, it holds: $$X_a\lambda(X)\phi\neq\lambda(X)X_a\phi\,.$$ Drawing lessons by the construction of ordinary gauge theories, in which a covariant derivative is defined, in the noncommutative case, the covariant coordinate, $\phi_a$, is introduced by its transformation property: $$\delta(\phi_a\phi)=\lambda\phi_a\phi\,,$$ which is satisfied when: $$\delta(\phi_a)=[\lambda,\phi_a]\,.\label{transformationofphia}$$ Therefore, the covariant coordinate is defined as: $$\phi_a\equiv X_a+A_a\,,\label{covariantfield}$$ where $A_a$ is identified as the gauge connection of the noncommutative gauge theory. Combining equations , , the gauge transformation of the connection, $A_a$, is obtained: $$\delta A_a=-[X_a,\lambda]+[\lambda,A_a]\,.$$ In the above relation, the role of $A_a$ as a gauge field is demonstrated. Next, the field strength tensor, $F_{ab}$, is defined on the fuzzy sphere as: $$F_{ab}\equiv[X_a,A_b]-[X_b,A_a]+[A_a,A_b]-C^c_{ab}A_c=[\phi_a,\phi_b]-C^c_{ab}\phi_c\,,$$ which is covariant under a gauge transformation: $$\delta F_{ab}=[\lambda,F_{ab}]\,.$$ In the next sections, the above methodology is applied on the construction of noncommutative gravity models as gauge theories. A 3-d noncommutative gravity model {#noncommutativegravitythreedims} ================================== The solid framework for constructing noncommutative gauge theories, as described in the previous section, combined with the description of 3-d gravity as a gauge theory, section \[3dgravity\], gives rise to the construction of a 3-d model of noncommutative gravity. First, one has to identify an appropriate noncommutative space, on which the noncommutative gauge theory is constructed on. A suitable 3-d fuzzy space is constructed by the foliation of the 3-d Euclidean space by multiple fuzzy spheres of different radii, called $\mathbb{R}_\lambda^3$, first considered in ref.[@Hammou:2001cc] (see also [@Vitale:2014hca]). The coordinates of $\mathbb{R}_\lambda^3$, satisfy the commutation relation of the SU(2) algebra, just like the coordinates of a fuzzy sphere do. However, unlike the case of the fuzzy sphere, the generators of SU(2), to which coordinate operators are related, are not accommodated in an irreducible (higher-dimensional) representation, but in a reducible one. The employment of a reducible representation of SU(2) means that the coordinates can be expressed as matrices in a block-diagonal form, with each block being some irreducible representation, corresponded to a fuzzy sphere of certain radius. Therefore, the Hilbert space would be: $${\mathcal{H}}=\oplus [\ell],\quad \ell=0,1/2,1,\ldots~.$$ The three coordinates of $\mathbb{R}_\lambda^3$, $X_i$, are the operators which satisfy the following commutation relation: $$[X_i,X_j]=i\lambda \epsilon_{ijk}X_k~,$$ and are described by matrices in reducible representations of the algebra of SU(2) (cf. [@Hammou:2001cc]). Therefore, the coordinates, $X_i$, are allowed to be set in a reducible representation and this is equivalent to a sum of fuzzy 2-spheres of different radii. Thus, the noncommutative space can be viewed as a discrete foliation of 3-d Euclidean space by fuzzy 2-spheres, each fuzzy sphere being a leaf of the foliation[^2] (cf. [@DeBellis:2010sy]). The gauge group that is adopted for the construction of the theory is the SO(4), that is the parametrization of the symmetry of the $\mathbb{R}_\lambda^3$, as it is explained in ref.[@Kovacik:2013yca]. During the gauging procedure, the typical problem of the non-closure of the anticommutators of the generators of the algebra is encountered. The indicated solution for this problem is to pick a specific representation of the group to accommodate the generators and enlarge the algebra to the minimal extend, including the operators that are produced by the anticommutators. Accordingly, in the specific SO(4) case of interest, the final gauge group is the U(2)$\times$U(2) in a fixed representation, due to the inclusion of two more operators $\one$ and $\gamma_5$ to the SO(4) set of generators. After the determination of the fuzzy space and the gauge group, the procedure that is followed is a modification of the one of the continuous case, adjusted in the framework of noncommutative geometry. At first, the commutation and anticommutation relations of the generators of the gauge group are obtained: $$\begin{aligned} && [P_a,P_b]=i\epsilon_{abc}M_c~, \quad [P_a,M_b]=i\epsilon_{abc}P_c~, \quad [M_a,M_b]=i\epsilon_{abc}M_c~, \\ && \{P_a,P_b\}=\tfrac{1}{2} \delta_{ab}\one~,\quad \{P_a,M_b\}=\tfrac{1}{2} \delta_{ab}\gamma_5~, \quad \{M_a,M_b\}=\tfrac{1}{2} \delta_{ab}\one~.\end{aligned}$$ Next, in order to write down the expression of the gauge connection, a gauge field has to be introduced for each generator. Therefore, the covariant coordinate is defined as: $${\cal X}_{\mu}= X_{\mu}\otimes i\one +e_{\mu}{}â\otimes P_a+\omega_{\mu}{}â\otimes M_a+A_{\mu}\otimes i\one+{\widetilde{A}}_{\mu}\otimes \gamma_5~~.$$ Also, a gauge transformation parameter is introduced and it is expanded on the generators of the algebra as: $$\epsilon=\xi^a\otimes P_a+\lambda^a\otimes M_a+\epsilon_0\otimes i\one+\widetilde{\epsilon}_0\otimes\gamma_5~.$$ The above relations combined with the transformation rule of the covariant coordinate, produce the following transformations of the gauge fields: $$\begin{aligned} \delta e_\mu^{~a}&=-i[X_\mu+A_\mu,\xi^a]+2\{\omega_{\mu b},\xi_c\}\epsilon^{abc}+2\{e_{\mu b},\lambda^c\}\epsilon^{abc}+2i[\lambda_a,\tilde{A}_\mu]+2i[\tilde{\epsilon}_0,\omega_{\mu a}]+i[\epsilon_0,e_{\mu a}]~,\nonumber\\ \delta\omega_\mu^{~a}&=-i[X_\mu+A_\mu,\lambda^a]+2\{\omega_{\mu b},\lambda_c\}\epsilon^{abc}-\frac{1}{2}\{e_{\mu b},\xi_c\}\epsilon^{abc}+\frac{i}{2}[\xi^a,\tilde{A}_\mu]+i[\epsilon_0,\omega_\mu^{~a}]+\frac{i}{2}[\tilde{\epsilon}_0,e_\mu^{~a}]~,\nonumber\\ \delta A_\mu&=-i[X_\mu+A_\mu,\epsilon_0]-i[\xi^a,e_{\mu a}]+4i[\lambda^a,\omega_{\mu a}]-i[\tilde{\epsilon}_0,\tilde{A}_\mu]~,\nonumber\\ \delta\tilde{A}_\mu&=-i[X_\mu+A_\mu,\tilde{\epsilon}_0]+2i[\xi^a,\omega_{\mu a}]+2i[\lambda^a,e_{\mu a}]+i[\epsilon_0,\tilde{A}_\mu]~.\label{transformationsofthefieldsnoncomm3-dimgravity}\end{aligned}$$ Also, making use of the definition of the covariant coordinate in the following defining relation of the field strength tensor: $$\mathcal{R}_{\mu\nu}=[\mathcal{X}_\mu,\mathcal{X}_\nu]-i\lambda\epsilon_{\mu\nu}^{~~~\rho}\mathcal{X}_\rho\,,$$ the corresponding component curvature tensors are obtained: $$\begin{aligned} T_{\mu\nu}^{~~a}&=i[X_\mu+A_\mu,e_\nu^{~a}]-i[X_\nu+A_\nu,e_\mu^{~a}]-2\epsilon^{abc}\left(\{e_{\mu b},\omega_{\nu c}\}+\{\omega_{\mu b},e_{\nu c}\}\right)\nonumber\\ &~~~+2i\left([\omega_\mu^{~a},\tilde{A}_\nu]-[\omega_\nu^{~a},\tilde{A}_\mu]\right)-i\lambda \epsilon_{\mu\nu}^{~~\rho}e_\rho^{~a}~,\\ R_{\mu\nu}^{~~a}&=i[X_\mu+A_\mu,\omega_\nu^{~a}]-i[X_\nu+A_\nu,\omega_\mu^{~a}]+\epsilon^{abc}\left(\tfrac{1}{2}\{e_{\mu b},e_{\nu c}\}-2\{\omega_{\mu b},\omega_{\nu c}\}\right)\nonumber\\ &~~~+\tfrac{i}{2}\left([e_\mu^{~a},\tilde{A}_\nu]-[e_\nu^{~a},\tilde{A}_\mu]\right)-i\lambda \epsilon_{\mu\nu}^{~~\rho}\omega_\rho^{~a}~,\\ F_{\mu\nu}&=i[X_\mu+A_\mu,X_\nu+A_\nu]-i[e_{\mu}^{~a},e_{\nu a}]+4i[\omega_\mu^{~a},\omega_{\nu a}]-i[\tilde{A}_\mu,\tilde{A}_\nu]-i\lambda \epsilon_{\mu\nu}^{~~\rho}(X_\rho+A_\rho)~,\\ \tilde{F}_{\mu\nu}&=i[X_\mu+A_\mu,\tilde{A}_\nu]-i[X_\nu+A_\nu,\tilde{A}_\mu]+2i\left([e_\mu^{~a},\omega_{\nu a}]+[\omega_\mu^{~a},e_{\nu a}]\right)-i\lambda \epsilon_{\mu\nu}^{~~\rho}\tilde{A}_\rho~.\label{componentcurvturetensorsrlammbdatothe1commatwo} \end{aligned}$$ Finally, the action that is proposed is of Chern-Simons type, specifically: $$S=\text{Tr}i\epsilon^{\mu\nu\rho}\mathcal{X}_\mu\mathcal{R}_{\nu\rho}\,.$$ The equations of motion are obtained after variation of the above action with respect to the various gauge fields: $$T_{\mu\nu}^{~~a}=0~,\quad R_{\mu\nu}^{~~a}=0~,\quad F_{\mu\nu}=0~\quad\tilde{F}_{\mu\nu}=0~.$$ It is worth-noting that the results of the above construction of noncommutative 3-d gravity reduce to the ones of the continuous case in section \[3dgravity\]. A 4-d noncommutative gravity model ================================== In this section, the construction of a 4-d gravity model as a noncommutative gauge theory is reviewed. First, the construction of an appropriate 4-d fuzzy space, on which the gravity model is constructed, is presented and then the features of the gravity model on this 4-d fuzzy space are explored [@Manolakos:2019fle]. Fuzzy de Sitter space {#fuzzydesitter} --------------------- The 4-d background space that is employed in this case is the fuzzy 4-d de Sitter space, $\mathrm{dS_4}$. The continuous $\mathrm{dS_4}$ is defined as a submanifold of the 5-d Minkowski spacetime and can be viewed as the Lorentzian analogue of the definition of the four-sphere as an embedding in the 5-d Euclidean space. Specifically, the defining, embedding equation of $\mathrm{dS_4}$ is: $$\eta^{MN}x_Mx_N=R^2\,,$$ where $M,N=0,\ldots, 4$ and $\eta^{MN}$ is the metric tensor of the 5-d Minkowski spacetime, $\eta^{MN}=\mathrm{diag}(-1,+1,+1,+1,+1)$. In order to obtain the fuzzy analogue of this space, one has to consider its coordinates, $X_m$, to be operators that do not commute with each other: $$[X_m,X_n]=i\theta_{mn}\,,\label{noncommgeneral}$$ where the spacetime indices are $m,n=1,\ldots,4$. Analogy to the fuzzy sphere case, in which the corresponding coordinates are identified as the three (rescaled) generators of SU(2) in an (large) N-dimensional representation, implies that the right hand side, , should be identified as a generator of the underlying algebra, ensuring covariance, that is $\theta_{mn}=C_{mn}^{~~~r}X_r$, where $C_{mnr}$ is a rescaled Levi-Civita symbol. However, in this fuzzy de Sitter case, such an identification cannot be achieved, in the sense that such an identification of the coordinate operators with generators of SO(1,4) would break Lorentz invariance, since the algebra would not be closing, i.e. $\theta_{mn}$ cannot be identified as generators into the algebra [@Heckman:2014xha][^3]. However, preservation of covariance is necessary for our purpose, therefore a group with larger symmetry, in which all operators identified as coordinates but also the noncommutativity tensor can be included in it, is considered. The enlargement of the symmetry leads to the consideration of the SO(1,5) group. Therefore, a fuzzy $\mathrm{dS}_4$ space, with its coordinates being operators represented by N-dimensional matrices, respecting covariance, too, is obtained after the enlargement of the symmetry to the SO(1,5) [@Manolakos:2019fle]. In order to facilitate the construction we make use of the Euclidean signature, therefore, instead of the SO(1,5), the resulting symmetry group is considered to be that of SO(6). For the explicit formulation of the above 4-d fuzzy space, let us consider the SO(6) generators, denoted as $\mathrm{J}_{AB} = - \mathrm{J}_{BA}$, with $A,B = 1,\ldots, 6$, which satisfy the following commutation relation: $$[J_{AB}, J_{CD} ] = i(\delta_{AC}J_{BD} + \delta_{BD}J_{AC} - \delta_{BC}J_{AD} - \delta_{AD}J_{BC} )\,.$$ The above generators can be written as a decomposition in an SO(4) notation, with the component generators being identified as various operators, including the coordinates: $$J_{mn} = \tfrac{1}{\hbar} \Theta_{mn}, \ \ J_{m5} = \tfrac{1}{\lambda} X_{m}, \ \ J_{m6} = \tfrac{\lambda}{2 \hbar}P_{m} , \ \ J_{56} = \tfrac{1}{2} \mathrm{h}\,,$$ with $m,n = 1,\ldots,4$. For dimensional reasons, an elementary length, $\lambda$, has been introduced in the above identifications, in which the coordinates, momenta and noncommutativity tensor are denoted as $X_{m}$, $P_{m}$ and $\Theta_{mn}$, respectively. The coordinate and momentum operators satisfy the following commutation relations: $$[ X_{m} , X_{n} ] = i \frac{\lambda^{2}}{\hbar} \Theta_{mn}, \ \ \ [P_{m}, P_{n} ] = 4i \frac{\hbar}{\lambda^{2}} \Theta_{mn}$$ $$[ X_{m}, P_{n} ] = i \hbar \delta_{mn}\mathrm{h}, \ \ \ [X_{m}, \mathrm{h} ] = i \frac{\lambda^{2}}{\hbar} P_{m}$$ $$[P_{m}, \mathrm{h} ] =4i \frac{\hbar}{\lambda^{2}} X_{m}\,.$$ The algebra of spacetime transformations is: $$[X_{m}, \Theta_{np} ] = i \hbar ( \delta_{mp} X_{n} - \delta_{mn} X_{p} )$$ $$[P_{m}, \Theta_{np} ] = i \hbar ( \delta_{mp} P_{n} - \delta_{mn} P_{p} )$$ $$[\Theta_{mn}, \Theta_{pq} ] = i \hbar ( \delta_{mp} \Theta_{nq} + \delta_{nq} \Theta_{mp} - \delta_{np} \Theta_{mq} - \delta_{mq} \Theta_{np} )$$ $$[\mathrm{h}, \Theta_{mn} ] = 0~.$$ In contrast to the Heisenberg algebra (see [@Singh:2018qzk]), the above algebra admits finite-dimensional matrices to represent the operators $X_{m}$, $P_{m}$ and $\Theta_{mn}$, therefore the kind of spacetime obtained above is a finite quantum system. Spaces like the above fuzzy $\mathrm{dS}_4$ fall into the general class of fuzzy spaces, that is the fuzzy covariant spaces [@Heckman:2014xha; @Buric:2015wta; @Barut]. A noncommutative gauge theory of 4-d gravity -------------------------------------------- In this section we review the construction of a noncommutative 4-d gravity model as a gauge theory on the fuzzy $\mathrm{dS}_4$ space of the previous section, \[fuzzydesitter\][^4]. In analogy to the 3-d translation of the gauge-theoretic description of gravity on the fuzzy space $\mathbb{R}_\lambda^3$, in section \[noncommutativegravitythreedims\], the same pattern is followed, this time translating the 4-d case presented in section \[weylgravity\]. Determination of the gauge group and representation by $4 \times 4$ matrices ---------------------------------------------------------------------------- In the previous section, the fuzzy $\mathrm{dS}_4$ space was constructed and its symmetry group was found to be the SO(6). Recalling the case of the construction of Einstein gravity as gauge theory in section \[einsteingravity\], in which the isometry group (the Poincaré group) was chosen to be the gauged, in this case the role of the gauge group will be given to the isometry group of the fuzzy $\mathrm{dS}_4$ space, namely the SO(5), viewed as a subgroup of the SO(6) group. However, the same problem related to the anticommutators of the generators of the algebra that emerged in the 3-d case of noncommutative gravity, section \[noncommutativegravitythreedims\], is encountered in this case, too [@Chatzistavrakidis:2018vfi; @Manolakos:2018isw; @Manolakos:2018hvn] (see also [@Aschieri1]). Specifically, the anticommutation relations of the generators of the gauge group, SO(5), produce operators that, in general, do not belong to the algebra. Therefore, to troubleshoot this problem, the representation of the generators has to be fixed and all operators produced by the anticommutators of the (fixed) generators have to be included into the algebra, identifying them as generators, too. Thus, the initial gauge group, SO(5) is extended to the SO(6)$\times$U(1) group with the generators being represented by 4$\times$4 matrices (in the 4 representation of SO(6)). In order to obtain the specific expressions of the matrices representing the generators, the four Euclidean $\Gamma$-matrices are employed, satisfying the following anticommutation relation: $$\{\Gamma_{a}, \Gamma_{b} \} = 2 \delta_{ab} \one\,,$$ where $a,b = 1,\ldots 4$. Also the $\Gamma_{5}$ matrix is defined as $ \Gamma_{5} = \Gamma_{1} \Gamma_{2} \Gamma_{3} \Gamma_{4} $. Therefore, the generators of the SO(6)$\times$U(1) gauge group are identified as:\ \ a) Six generators of the Lorentz transformations: $ \mathrm{M}_{ab} = - \tfrac{i}{4} [\Gamma_{a} , \Gamma_{b} ] = - \tfrac{i}{2} \Gamma_{a} \Gamma_{b}\,,a < b$,\ \ b) four generators of the conformal boosts: $ \mathrm{K}_{a} = \tfrac{1}{2} \Gamma_{a}$,\ \ c) four generators of the local translations: $ \mathrm{P}_{a} = -\tfrac{i}{2} \Gamma_{a} \Gamma_{5}$,\ \ d) one generator for special conformal transformations: $\mathrm{D} = -\tfrac{1}{2} \Gamma_{5}$ and\ \ e) one U(1) generator: $\one$.\ \ The $\Gamma$-matrices are determined as tensor products of the Pauli matrices, specifically: $$\Gamma_{1} = \sigma_{1} \otimes \sigma_{1}, \ \ \ \Gamma_{2} = \sigma_{1} \otimes \sigma_{2}, \ \ \ \Gamma_{3} = \sigma_{1} \otimes \sigma_{3}$$ $$\Gamma_{4} = \sigma_{2} \otimes \mathbf{1}, \ \ \ \Gamma_{5} = \sigma_{3} \otimes \mathbf{1}\,.$$ Therefore, the generators of the algebra are represented by the following 4$\times$4 matrices: $$M_{ij} = - \frac{i}{2}\Gamma_{i} \Gamma_{j} = \frac{1}{2} \one \otimes \sigma_{k}\,,$$ where $i,j,k = 1,2,3$ and: $$M_{4k} = - \frac{i}{2}\Gamma_{4} \Gamma_{k} = - \frac{1}{2} \sigma_{3} \otimes \sigma_{k}\,.$$ Straightforward calculations lead to the following commutation relations, which the operators satisfy: $$\begin{aligned} &&[ K_{a} , K_{b} ] = i M_{ab}, \ \ \ [P_{a}, P_{b} ] = i M_{ab} \nonumber \\ &&[ X_{a}, P_{b} ] = i \delta_{ab}D, \ \ \ [X_{a}, D ] = i P_{a} \nonumber \\ &&[P_{a}, D ] =i K_{a} , \ \ \ [K_a,P_b]=i\delta_{ab}D , \ \ \ [K_a,D]=-iP_a \nonumber \\ &&[K_{a}, M_{bc} ] = i( \delta_{ac} K_{b} - \delta_{ab} K_{c} ) \nonumber \\ &&[P_{a}, M_{bc} ] = i( \delta_{ac} P_{b} - \delta_{ab} P_{c} ) \nonumber \\ &&[M_{ab}, M_{cd} ] = i( \delta_{ac} M_{bd} + \delta_{bd} M_{ac} - \delta_{bc}M_{ad} - \delta_{ad}M_{bc} ) \nonumber \\ &&[D, M_{ab} ] = 0\,.\label{algebra}\end{aligned}$$ Noncommutative gauge theory of gravity -------------------------------------- Having determined the commutation relations of the generators of the algebra, the noncommutative gauging procedure can be initiated. First, the covariant coordinate is defined as: $$\hat{X}_m=X_m\otimes\one+A_m(X)\,.\label{covcoorddef}$$ The property of covariance of the coordinate $\hat{X}_m$, is expressed as: $$\delta\hat{X}_m=i[\epsilon,\hat{X}_m]\,,\label{covcoord}$$ where $\epsilon(X)$ is the gauge transformation parameter. It is a function of the coordinates (N$\times$N matrices), $X_m$, but also is valued in the SO(6)$\times$U(1) algebra. Therefore, it can be decomposed on the sixteen generators of the algebra: $$\epsilon=\epsilon_0(X)\otimes\one+\xi^a(X)\otimes K_a+\tilde{\epsilon}_0(X)\otimes D+\lambda_{ab}(X)\otimes\Sigma^{ab}+\tilde{\xi}^a(X)\otimes P_a\,.$$ Taking into account that a gauge transformation acts trivially on the coordinate $X_m$, namely $\delta X_m=0$, the transformation property of the $A_m$ is obtained by the combination of the equations and . In accordance to the corresponding procedure in the commutative case, the $A_m$ transforms in such a way admitting the interpretation of the connection of the gauge theory. Similarly to the case of the gauge transformation parameter, $\epsilon$, the $A_m$, is a function of the coordinates $X_m$ of the fuzzy space $\mathrm{dS}_4$, but also takes values in the SO(6)$\times$U(1) algebra, which means that it can be expanded on its sixteen generators as follows: $$A_m(X)=e_m^{~a}(X)\otimes P_a+\omega_{m}^{~ab}(X) \otimes \Sigma_{ab}(X) + b_{m}^{~a}(X) \otimes K_{a}(X) + \tilde{a}_{m}(X) \otimes D + a_{m}(X) \otimes \one\,,$$ where it becomes manifest that one gauge field has been corresponded to each generator. The component gauge fields are functions of the coordinates of the space, $X_m$, therefore they have the form of N$\times$N matrices, where N is the dimension of the representation in which the coordinates are accommodated. Therefore, instead of the ordinary product, between the gauge fields and their corresponding generators, the tensor product is used, since the factors are matrices of different dimensions, since the generators are represented by 4$\times$4 matrices. Therefore, it is concluded that each term in the expression of the gauge connection is a 4N$\times$4N matrix. After the decomposition of the gauge connection and the introduction of the gauge fields, the covariant coordinate is now written as: $$\hat{X}_{m} = X_{m} \otimes \one + e_{m}^{~a}(X) \otimes P_{a} + \omega_{m}^{~ab}(X) \otimes \Sigma_{ab} + b_{m}^{~a} \otimes K_{a} + \tilde{a}_{m} \otimes D + a_{m} \otimes \one\,.$$ The next step is to calculate the field strength tensor for this SO(6)$\times$U(1) noncommutative gauge theory, which, for the fuzzy de Sitter space, is defined as: $$\mathcal{R}_{mn} = [\hat{X}_{m}, \hat{X}_{n}] - \frac{i\lambda^2}{\hbar}\hat{\Theta}_{mn}\,,\label{fieldstrengtt}$$ where $\hat{\Theta}_{mn}=\Theta_{mn}\otimes\one+\mathcal{B}_{mn}$. The $\mathcal{B}_{mn}$ is a 2-form gauge field, which takes values in the SO(6)$\times$U(1) algebra. The $\mathcal{B}_{mn}$ field was introduced in order to make the field strength tensor covariant, since in its absence it does not transform covariantly[^5]. The $\mathcal{B}_{mn}$ field will contribute in the total action of the theory with a kinetic term of the following form: $$\mathcal{S}_{\mathcal{B}}=\text{Tr}\,\text{tr}\, \hat{\mathcal{H}}_{mnp}\hat{\mathcal{H}}^{mnp}~.$$ The $\hat{\mathcal{H}}_{mnp}$ field strength tensor transforms covariantly under a gauge transformation, therefore the above action is gauge invariant. The field strength tensor of the gauge connection, , can be expanded in terms of the component curvature tensors, since it is valued in the algebra: $$\mathcal{R}_{mn}(X) = R_{mn}^{~~~ab}(X) \otimes \Sigma_{ab} + \tilde{R}_{mn}^{~~a}(X) \otimes P_{a} + R_{mn}^{~~a}(X) \otimes K_{a} + \tilde{R}_{mn}(X) \otimes D + R_{mn}(X) \otimes \one\,.$$ All necessary information for the determination of the transformations of the gauge fields and the expressions of the component curvature tensors is obtained. The explicit expressions and calculations lie in ref.[@Manolakos:2019fle]. The constraints for the symmetry breaking and the action -------------------------------------------------------- The gauge symmetry of the resulting theory, with which we would like to end up, is the one described by the Lorentz group, in the Euclidean signature, the SO(4). In this direction, one could consider directly a constrained theory in which the only component curvature tensors that would not be imposed to vanish would be the ones that corresponds to the Lorentz and the U(1) generators of the algebra, achieving a breaking of the initial SO(6)$\times$U(1) symmetry to the SO(4)$\times$U(1). However, counting the degrees of freedom, adoption of the above breaking would lead to an over-constrained theory. Therefore, it is more efficient to follow a different procedure and perform the symmetry breaking in a less straightforward way [@Manolakos:2019fle]. Accordingly, the first constraint is the torsionless condition: $$\tilde{R}_{mn}^{~a}(P)=0\,, \label{ncconstraints}$$ which is also encountered in the cases in which the Einstein and conformal gravity theories are described as (ordinary) gauge theories. Next, the gauge field $b_m^{~a}$ would admit an interpretation of a second vielbein of the theory, which would lead to a bimetric theory, which is not our case of interest. Thus, the relation $e_m^{~a}={b}_m^{~a}$ in the solution of the constraint should be considered. Also, this leads to the expression of the spin connection $\omega_m^{~ab}$ in terms of the rest of the independent fields, ${e}_m^{~a}, {a}_m, {\tilde{a}}_m$. In order to obtain the explicit expression of the spin connection in terms of the other fields, the following two identities are employed: $$\delta^{abc}_{fgh}=\epsilon^{abcd}\epsilon_{fghd}\quad\quad\text{and}\quad\quad \frac{1}{3!}\delta^{abc}_{fgh}a^{fgh}=a^{[fgh]}\,.\label{ids}$$ Solving the constraint $\tilde{R}(P)=0$, one obtains: $$\epsilon^{abcd}[e_m^{~b},\omega_n^{~cd}]-i\{\omega_m^{~ab},e_{nb}\}=-[D_m,e_m^{~a}]-i\{e_m^{~a},\tilde{a}_m\}\,,$$ where $D_m=X_m+a_m$ being the covariant coordinate of an Abelian noncommutative gauge theory. The above equation leads to the following two: $$\epsilon^{abcd}[e_m^{~b},\omega_n^{~cd}]=-[D_m,e_m^{~a}]\quad\quad \text{and}\quad\quad \{\omega_m^{~ab},e_{nb}\}=\{e_m^{~a},\tilde{a}_n\}\,.$$ Taking into consideration the identities, , the above equations lead to the desired expression for the spin connection: $$\omega_n^{~ac}=-\frac{3}{4}e^m_{~b}(-\epsilon^{abcd}[D_m,e_{nd}]+\delta^{[bc}\{e_n^{~a]},\tilde{a}_m\})\,.\label{omegaintermsofe}$$ Next, according to ref.[@Green:1987mn], the vanishing of the field strength tensor in a gauge theory could lead to the vanishing of the associated gauge field. However, the vanishing of the torsion component tensor, $\tilde{R}(P)=0$, does not imply $e_\mu^{~a}=0$, because such a choice would lead to degeneracy of the metric tensor of the space [@Witten:1988hc]. The field that can be gauge-fixed to zero is the $\tilde{a}_m$. This fixing, $\tilde{a}_m=0$, will modify the expression of the spin connection, , leading to a more simplified expression: $$\omega_n^{~ac}=\frac{3}{4}e^m_{~b}\epsilon^{abcd}[D_m,e_{nd}]\,.\label{omegaintermsofeteliko}$$ It should be noted that the U(1) field strength tensor, $R_{mn}(\one)$, which is associated to the noncommutativity, is not considered to be vanishing. The symmetry breaking does not affect the U(1) part of the symmetry since the breaking takes place in the (high-energy) noncommutative regime. However, the corresponding field, $a_m$, decouples in the commutative limit of the broken theory. Therefore, in the commutative limit, the gauge symmetry would be just SO(4). Alternatively, another way to break the SO(6) gauge symmetry to the desired SO(4) is to induce a spontaneous symmetry breaking by including two scalar fields in the 6 representation of SO(6), extrapolating the argument developed for the case of the conformal gravity to the noncommutative framework. It is expected that the spontaneous symmetry breaking induced by the scalars would lead to a constrained theory as the one that was obtained above by the imposition of the constraints . The action and equations of motion ---------------------------------- Next, for the action of the theory, it is natural to consider one of Yang-Mills type[^6]: $$\mathcal{S}=\text{Tr}\text{tr}\{ \mathcal{R}_{mn},\mathcal{R}_{rs}\}\epsilon^{mnrs}\,,$$ where $\text{Tr}$ denotes the trace over the coordinates-N$\times$N matrices (it replaces the integration of the continuous case) and $\text{tr}$ denotes the trace over the generators of the algebra. After the symmetry breaking, that is including the constraints, the surviving terms of the action will be: $$\mathcal{S}=2\text{Tr}(R_{mn}^{~~~ab}R_{rs}^{~~cd}\epsilon_{abcd}\epsilon^{mnrs}+4\tilde{R}_{mn}R_{rs}\epsilon^{mnrs}+\frac{1}{3}H_{mnp}^{~~~~ab}H^{mnpcd}\epsilon_{abcd}+\frac{4}{3}\tilde{H}_{mnp}H^{mnp})\,.$$ Replacing with the explicit expressions of the component tensors and writing the $\omega$ gauge field in terms of the surviving gauge fields, , then variation with respect to the independent gauge fields would lead to the equations of motion. Conclusions =========== In this review the construction of noncommutative 3-d and 4-d gravity models as gauge theories were revisited. Although for both cases the main procedure is similar, since they are based on the corresponding works in the continuous regime, let us summarize and conclude separately for the two cases. In the 3-d case, the noncommutative background space we employed is the $\mathbb{R}_\lambda^3$ which is a foliation of the 3-d Euclidean space by multiple fuzzy spheres. This onion-like construction admits an SO(4) symmetry which is the one we chose as gauge group. The anticommutators of the generators of the gauge group would not close, that is why we promoted the symmetry to the U(2)$\times$U(2) and fixed its representation. Then, following the standard procedure, we defined the covariant coordinate and calculated the transformations of the fields and the expressions of the component curvature tensors. Naturally, the action we determined was of Chern-Simons type and the equations of motion were obtained after its variation. The results obtained in the above construction reduce to the ones of the commutative case. In the 4-d case, the noncommutative background space we employed is the fuzzy version of the 4-d de Sitter space. It is worth-mentioning that it consists a 4-d covariant noncommutative space, respecting Lorentz invariance, which is of major importance in our case. Next, we determined the gauge group, SO(5), which was enlarged for the same reasons as in the previous case, to the SO(6)$\times$U(1). Then, we went on following the standard procedure for the calculation of the transformations of the fields and the expressions of the component curvature tensors. Since we desired to result with a theory respecting the Lorentz symmetry, we imposed certain constraints in order to break the initial symmetry. After the symmetry breaking, the action takes its final form and its variation will lead to the equations of motion. The latter will be part of our future work. It should be noted that, before the symmetry breaking, the results of the above construction reduce to the ones of the conformal gravity in the commutative limit. Finally, it should be also stressed that the above is a matrix model giving insight into the gravitational interaction in the high-energy regime and also giving promises for improved UV properties as compared to ordinary gravity. Clearly, the latter, as well the inclusion of matter fields is going to be a subject of further study. Another possible direction of further investigation is to consider our construction embedded in the Lie superalgebra of SU(2,2/1) which is isomorphic to the algebra of the superconformal spacetime symmetry group [@Yates:2017zgl]. Gauging of the latter leads to $\mathcal{N}=1$ conformal supergravity [@Fradkin:1985am; @Kaku:1977pa; @Kaku:1978nz; @Kaku:1977rk; @Townsend:1979ki]. In the present noncommutative context it seems more natural pursuing the gauging of the supergroup of SU(2,2/1) algebra. This possibility appeared to be fruitful in relating the Connes-Lott model [@connes-lott] to those based on gauging the SU(2/1) superalgebra [@Neeman:1979wp]-[@Hussain:1991tw] (see also [@Batakis:1993cw]) and could be useful in our construction, too.\ \ [**Acknowledgements**]{}:\ We would like to thank Ali Chamseddine, Paolo Aschieri, Thanassis Chatzistavrakidis, Evgeny Ivanov, Larisa Jonke, Danijel Jurman, Alexander Kehagias, Dieter Lüst, Denjoe O’Connor, Emmanuel Saridakis, Harold Steinacker, Kelly Stelle, Patrizia Vitale and Christof Wetterich for useful discussions. The work of two of us (GM and GZ) was partially supported by the COST Action MP1405, while both would like to thank ESI - Vienna for the hospitality during their participation in the Workshop “Matrix Models for Noncommutative Geometry and String Theory”, Jul 09 - 13, 2018. One of us (GZ) has been supported within the Excellence Initiative funded by the German and States Governments, at the Institute for Theoretical Physics, Heidelberg University and from the Excellent Grant Enigmass of LAPTh. GZ would like to thank the ITP - Heidelberg, LAPTh - Annecy and MPI - Munich for their hospitality. [99]{} R. Utiyama, Phys. Rev.  [**101**]{} (1956) 1597. doi:10.1103/PhysRev.101.1597 T. W. B. Kibble, J. Math. Phys.  [**2**]{} (1961) 212. doi:10.1063/1.1703702 K. S. Stelle and P. C. West, Phys. Rev. D [**21**]{} (1980) 1466. doi:10.1103/PhysRevD.21.1466 S. W. MacDowell and F. Mansouri, Phys. Rev. Lett.  [**38**]{} (1977) 739 Erratum: \[Phys. Rev. Lett.  [**38**]{} (1977) 1376\]. doi:10.1103/PhysRevLett.38.1376, 10.1103/PhysRevLett.38.739 E. A. Ivanov and J. Niederle, Conference: C80-06-23.3, p.545-551, 1980; E. A. Ivanov and J. Niederle, Phys. Rev. D [**25**]{} (1982) 976. doi:10.1103/PhysRevD.25.976; E. A. Ivanov and J. Niederle, Phys. Rev. D [**25**]{} (1982) 988. doi:10.1103/PhysRevD.25.988 T. W. B. Kibble and K. S. Stelle, In Ezawa, H. ( Ed.), Kamefuchi, S. ( Ed.): Progress In Quantum Field Theory, 57-81. M. Kaku, P. K. Townsend and P. van Nieuwenhuizen, “Gauge Theory of the Conformal and Superconformal Group,” Phys. Lett.  [**69B**]{} (1977) 304. doi:10.1016/0370-2693(77)90552-4 E. S. Fradkin and A. A. Tseytlin, “Conformal Supergravity,” Phys. Rept.  [**119**]{} (1985) 233. doi:10.1016/0370-1573(85)90138-3 D. Z. Freedman and A. Van Proeyen “Supergravity,” Cambridge University Press, 2012 A. H. Chamseddine, “Supersymmetry and higher spin fields”, PhD Thesis, (1976) A. H. Chamseddine and P. C. West, Nucl. Phys. B [**129**]{} (1977) 39. doi:10.1016/0550-3213(77)90018-9 E. Witten, “(2+1)-Dimensional Gravity as an Exactly Soluble System,” Nucl. Phys. B [**311**]{} (1988) 46. Connes A., Academic Press, Inc., San Diego, CA, 1994. Madore J., London Mathematical Society Lecture Note Series, Vol. 257, Cambridge University Press, Cambridge, 1999. J. Madore, Class. Quant. Grav.  [**9**]{} (1992) 69. doi:10.1088/0264-9381/9/1/008 Buric M., Grammatikopoulos T., Madore J., Zoupanos G., JHEP 0604 (2006) 054; Buric M., Madore J., Zoupanos G., SIGMA 3:125,2007, arXiv:0712.4024 \[hep-th\]. T. Filk, Phys. Lett. B 376 (1996) 53; J. C. Várilly and J. M. Gracia-Bondía, Int. J. Mod. Phys. A 14 (1999) 1305 \[hep-th/9804001\]; M. Chaichian, A. Demichev and P. Presnajder, Nucl. Phys. B 567 (2000) 360, hepth/ 9812180; S. Minwalla, M. Van Raamsdonk and N. Seiberg, JHEP 0002 (2000) 020, hep-th/9912072. H. Grosse and R. Wulkenhaar, Lett. Math. Phys. 71 (2005) 13, hep-th/0403232. H. Grosse and H. Steinacker, Adv. Theor. Math. Phys. 12 (2008) 605, hep-th/0607235; H. Grosse and H. Steinacker, Nucl. Phys. B 707 (2005) 145, hep-th/0407089. Connes A., Lott J., Nuclear Phys. B Proc. Suppl. 18 (1991), 29-47; Chamseddine A.H., Connes A., Comm. Math. Phys. 186 (1997), 731-750, hep-th/9606001; Chamseddine A.H., Connes A., Phys. Rev. Lett. 99 (2007), 191601, arXiv:0706.3690. Martín C.P., Gracia-Bond[í]{}a M.J., V[á]{}rilly J.C., Phys. Rep. 294 (1998), 363-406, hep-th/9605001. Dubois-Violette M., Madore J., Kerner R., Phys. Lett. B217 (1989), 485-488; Dubois-Violette M., Madore J., Kerner R., Classical Quantum Gravity 6 (1989), 1709-1724; Dubois-Violette M., Kerner R., Madore J., J. Math. Phys. 31 (1990), 323-330. Madore J., Phys. Lett. B 305 (1993), 84-89; Madore J., (Sobotka Castle, 1992), Fund. Theories Phys., Vol. 52, Kluwer Acad. Publ., Dordrecht, 1993, 285-298. hep-ph/9209226. Connes A., Douglas M.R., Schwarz A., JHEP (1998), no.2, 003, hep-th/9711162. Seiberg N., Witten E., JHEP (1999), no.9, 032, hep-th/9908142. N.Ishibashi, H.Kawai, Y.Kitazawa and A.Tsuchiya, Nucl. Phys. B498 (1997) 467, arXiv:hep-th/9612115. Jurčo B., Schraml S., Schupp P., Wess J., Eur. Phys. J. C 17 (2000), 521-526, hep-th/0006246; Jurčo B., Schupp P., Wess J., Nuclear Phys. B 604 (2001), 148-180, hep-th/0102129; Jur[č]{}o B., Moller L., Schraml S., Schupp S., Wess J., Eur. Phys. J. C 21 (2001), 383-388, hep-th/0104153; Barnich G., Brandt F., Grigoriev M., JHEP (2002), no.8, 023, hep-th/0206003. Chaichian M., Pre[š]{}najder P., Sheikh-Jabbari M.M., Tureanu A., Eur. Phys. J. C 29 (2003), 413-432, hep-th/0107055. Calmet X., Jur[č]{}o B., Schupp P., Wess J., Wohlgenannt M., Eur. Phys. J. C 23 (2002), 363-376, hep-ph/0111115; Aschieri P., Jur[č]{}o B., Schupp P., Wess J., Nuclear Phys. B 651 (2003), 45-70, hep-th/0205214; Behr W., Deshpande N.G., Duplancic G., Schupp P., Trampetic J., Wess J., Eur.Phys.J.C29: 441-446, 2003. Aschieri P., Madore J., Manousselis P., Zoupanos G., JHEP (2004), no. 4, 034, hep-th/0310072; Aschieri P., Madore J., Manousselis P., Zoupanos G., Fortschr. Phys. 52 (2004), 718-723, hep-th/0401200; Aschieri P., Madore J., Manousselis P., Zoupanos G., Conference: C04-08-20.1 (2005) 135-146, hep-th/0503039. Aschieri P., Grammatikopoulos T., Steinacker H., Zoupanos G., JHEP (2006), no. 9, 026, hep-th/0606021; Aschieri P., Steinacker H., Madore J., Manousselis P., Zoupanos G., arXiv:0704.2880. Steinacker H., Zoupanos G., JHEP (2007), no. 9, 017, arXiv:0706.0398. A. Chatzistavrakidis, H. Steinacker and G. Zoupanos, Fortsch.Phys. 58 (2010) 537-552, arXiv:0909.5559 \[hep-th\]. A. Chatzistavrakidis, H. Steinacker and G. Zoupanos, JHEP 1005 (2010) 100, arXiv:hep-th/1002.2606 A. Chatzistavrakidis and G. Zoupanos, SIGMA 6 (2010) 063, arXiv:hep-th/1008.2049. D. Gavriil, G. Manolakos, G. Orfanidis and G. Zoupanos, Fortsch. Phys.  [**63**]{} (2015) 442 doi:10.1002/prop.201500022 \[arXiv:1504.07276 \[hep-th\]\]; G. Manolakos and G. Zoupanos, Phys. Part. Nucl. Lett.  [**14**]{} (2017) no.2, 322. doi:10.1134/S1547477117020194; G. Manolakos and G. Zoupanos, Springer Proc. Math. Stat.  [**191**]{} (2016) 203 doi:10.1007/978-981-10-2636-2-13 \[arXiv:1602.03673 \[hep-th\]\]. J. Madore, S. Schraml, P. Schupp and J. Wess, Eur. Phys. J. C [**16**]{} (2000) 161 doi:10.1007/s100520050012 \[hep-th/0001203\]. A. H. Chamseddine, “Deforming Einstein’s gravity,” Phys. Lett. B [**504**]{} (2001) 33 doi:10.1016/S0370-2693(01)00272-6 \[hep-th/0009153\]. A. H. Chamseddine, Phys. Rev. D [**69**]{} (2004) 024015 doi:10.1103/PhysRevD.69.024015 \[hep-th/0309166\]. P. Aschieri and L. Castellani, JHEP [**0906**]{} (2009) 086 doi:10.1088/1126-6708/2009/06/086 \[arXiv:0902.3817 \[hep-th\]\]. P. Aschieri and L. Castellani, JHEP [**0906**]{} (2009) 087 doi:10.1088/1126-6708/2009/06/087 \[arXiv:0902.3823 \[hep-th\]\]. M. Dimitrijević Ćirić, B. Nikolić and V. Radovanović, Phys. Rev. D [**96**]{} (2017) no.6, 064029 doi:10.1103/PhysRevD.96.064029 \[arXiv:1612.00768 \[hep-th\]\]. S. Cacciatori, D. Klemm, L. Martucci and D. Zanon, Phys. Lett. B [**536**]{} (2002) 101 doi:10.1016/S0370-2693(02)01823-3 \[hep-th/0201103\]. S. Cacciatori, A. H. Chamseddine, D. Klemm, L. Martucci, W. A. Sabra and D. Zanon, Class. Quant. Grav.  [**19**]{} (2002) 4029 doi:10.1088/0264-9381/19/15/310 \[hep-th/0203038\]. P. Aschieri and L. Castellani, JHEP [**1411**]{} (2014) 103 doi:10.1007/JHEP11(2014)103 \[arXiv:1406.4896 \[hep-th\]\]. M. Banados, O. Chandia, N. E. Grandi, F. A. Schaposnik and G. A. Silva, Phys. Rev. D [**64**]{} (2001) 084012 doi:10.1103/PhysRevD.64.084012 \[hep-th/0104264\]. N. Seiberg and E. Witten, JHEP [**9909**]{} (1999) 032 doi:10.1088/1126-6708/1999/09/032 \[hep-th/9908142\]. T. Banks, W. Fischler, S. H. Shenker and L. Susskind, Phys. Rev. D [**55**]{} (1997) 5112 doi:10.1103/PhysRevD.55.5112 \[hep-th/9610043\]. N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, Nucl. Phys. B [**498**]{} (1997) 467 doi:10.1016/S0550-3213(97)00290-3 \[hep-th/9612115\]. H. Aoki, S. Iso, H. Kawai, Y. Kitazawa and T. Tada, Prog. Theor. Phys.  [**99**]{} (1998) 713 doi:10.1143/PTP.99.713 \[hep-th/9802085\]. M. Hanada, H. Kawai and Y. Kimura, Prog. Theor. Phys.  [**114**]{} (2006) 1295 doi:10.1143/PTP.114.1295 \[hep-th/0508211\]. K. Furuta, M. Hanada, H. Kawai and Y. Kimura, Nucl. Phys. B [**767**]{} (2007) 82 doi:10.1016/j.nuclphysb.2007.01.003 \[hep-th/0611093\]. H. S. Yang, Int. J. Mod. Phys. A [**24**]{} (2009) 4473 doi:10.1142/S0217751X0904587X \[hep-th/0611174\]. H. Steinacker, Class. Quant. Grav.  [**27**]{} (2010) 133001 doi:10.1088/0264-9381/27/13/133001 \[arXiv:1003.4134 \[hep-th\]\]. S. W. Kim, J. Nishimura and A. Tsuchiya, Phys. Rev. Lett.  [**108**]{} (2012) 011601 doi:10.1103/PhysRevLett.108.011601 \[arXiv:1108.1540 \[hep-th\]\]. J. Nishimura, PTEP [**2012**]{} (2012) 01A101 doi:10.1093/ptep/pts004 \[arXiv:1205.6870 \[hep-lat\]\]. V. P. Nair, Nucl. Phys. B [**651**]{} (2003) 313 doi:10.1016/S0550-3213(02)01061-1 \[hep-th/0112114\]. Y. Abe and V. P. Nair, Phys. Rev. D [**68**]{} (2003) 025002 doi:10.1103/PhysRevD.68.025002 \[hep-th/0212270\]. P. Valtancoli, Int. J. Mod. Phys. A [**19**]{} (2004) 361 doi:10.1142/S0217751X04017598 \[hep-th/0306065\]. V. P. Nair, Nucl. Phys. B [**750**]{} (2006) 321 doi:10.1016/j.nuclphysb.2006.06.009 \[hep-th/0605008\]. M. Burić, T. Grammatikopoulos, J. Madore and G. Zoupanos, JHEP [**0604**]{} (2006) 054 doi:10.1088/1126-6708/2006/04/054 \[hep-th/0603044\]. M. Burić, J. Madore and G. Zoupanos, SIGMA [**3**]{} (2007) 125 doi:10.3842/SIGMA.2007.125 \[arXiv:0712.4024 \[hep-th\]\]. M. Burić, J. Madore and G. Zoupanos, Eur. Phys. J. C [**55**]{} (2008) 489 doi:10.1140/epjc/s10052-008-0602-x \[arXiv:0709.3159 \[hep-th\]\]. P. Aschieri, J. Madore, P. Manousselis and G. Zoupanos, JHEP [**0404**]{} (2004) 034 doi:10.1088/1126-6708/2004/04/034 \[hep-th/0310072\]; ibid, Fortsch. Phys.  [**52**]{} (2004) 718 doi:10.1002/prop.200410168 \[hep-th/0401200\]; ibid, hep-th/0503039. H. S. Snyder, Phys. Rev.  [**71**]{} (1947) 38. doi:10.1103/PhysRev.71.38 C. N. Yang, Phys. Rev.  [**72**]{} (1947) 874. doi:10.1103/PhysRev.72.874 J. Heckman and H. Verlinde, “Covariant non-commutative space–time,” Nucl. Phys. B [**894**]{} (2015) 58 \[arXiv:1401.1810 \[hep-th\]\]. M. Burić and J. Madore, Eur. Phys. J. C [**75**]{} (2015) no.10, 502 doi:10.1140/epjc/s10052-015-3729-6 \[arXiv:1508.06058 \[hep-th\]\]. M. Sperling and H. C. Steinacker, J. Phys. A [**50**]{} (2017) no.37, 375202 doi:10.1088/1751-8121/aa8295 \[arXiv:1704.02863 \[hep-th\]\]. M. Burić, D. Latas and L. Nenadović, arXiv:1709.05158 \[hep-th\]. H. C. Steinacker, JHEP [**1612**]{} (2016) 156 doi:10.1007/JHEP12(2016)156 \[arXiv:1606.00769 \[hep-th\]\]. A. Chatzistavrakidis, L. Jonke, D. Jurman, G. Manolakos, P. Manousselis and G. Zoupanos, Fortsch. Phys.  [**66**]{} (2018) no.8-9, 1800047 doi:10.1002/prop.201800047 \[arXiv:1802.07550 \[hep-th\]\]. D. Jurman, G. Manolakos, P. Manousselis and G. Zoupanos, PoS CORFU [**2017**]{} (2018) 162 doi:10.22323/1.318.0162 \[arXiv:1809.03879 \[gr-qc\]\]. G. Manolakos and G. Zoupanos, “Non-commutativity in Unified Theories and Gravity,” Springer Proc. Math. Stat.  [**263**]{} (2017) 177 doi:10.1007/978-981-13-2715-5-10 \[arXiv:1809.02954 \[hep-th\]\]. A. B. Hammou, M. Lagraa and M. M. Sheikh-Jabbari, Phys. Rev. D [**66**]{} (2002) 025025 doi:10.1103/PhysRevD.66.025025 \[hep-th/0110291\]. P. Vitale, Fortsch. Phys.  [**62**]{} (2014) 825 doi:10.1002/prop.201400037 \[arXiv:1406.1372 \[hep-th\]\]. S. Kováčik and P. Prešnajder, J. Math. Phys.  [**54**]{} (2013) 102103 doi:10.1063/1.4826355 \[arXiv:1309.4592 \[math-ph\]\]. D. Jurman and H. Steinacker, JHEP [**1401**]{} (2014) 100 doi:10.1007/JHEP01(2014)100 \[arXiv:1309.1598 \[hep-th\]\]. G. Manolakos, P. Manousselis and G. Zoupanos, arXiv:1902.10922 \[hep-th\]. A. H. Chamseddine, J. Math. Phys.  [**44**]{} (2003) 2534 doi:10.1063/1.1572199 \[hep-th/0202137\]. L. F. Li, “Group Theory of the Spontaneously Broken Gauge Symmetries,” Phys. Rev. D [**9**]{} (1974) 1723. doi:10.1103/PhysRevD.9.1723 A. H. Chamseddine, “Invariant actions for noncommutative gravity,” J. Math. Phys.  [**44**]{} (2003) 2534 doi:10.1063/1.1572199 \[hep-th/0202137\]. J. DeBellis, C. Saemann and R. J. Szabo, “Quantized Nambu-Poisson Manifolds in a 3-Lie Algebra Reduced Model,” JHEP [**1104**]{} (2011) 075 doi:10.1007/JHEP04(2011)075 \[arXiv:1012.2236 \[hep-th\]\]. A. Singh and S. M. Carroll, “Modeling Position and Momentum in Finite-Dimensional Hilbert Spaces via Generalized Clifford Algebra,” arXiv:1806.10134 \[quant-ph\]. A. Barut, “ From Heisenberg algebra to Conformal Dynamical Group ’ ’ in A. Barut, H. D. Doener (Eds) “Conformal Groups and related Symmetries.Physical Results and Mathematical Backgrounf ” Lecture Notes in Physics, Springer-Verlag 1985 L. Alvarez-Gaume, F. Meyer and M. A. Vazquez-Mozo, “Comments on noncommutative gravity,” Nucl. Phys. B [**753**]{} (2006) 92 doi:10.1016/j.nuclphysb.2006.07.009 \[hep-th/0605113\]. Y. Kimura, “Noncommutative gauge theory on fuzzy four sphere and matrix model,” Nucl. Phys. B [**637**]{} (2002) 177 doi:10.1016/S0550-3213(02)00469-8 \[hep-th/0204256\]. M. B. Green, J. H. Schwarz and E. Witten, “Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies And Phenomenology,” Cambridge, Uk: Univ. Pr. ( 1987) 596 P. ( Cambridge Monographs On Mathematical Physics) L. A. Yates and P. D. Jarvis, J. Phys. A [**51**]{} (2018) no.14, 145203 doi:10.1088/1751-8121/aab215 \[arXiv:1710.10533 \[hep-th\]\]. M. Kaku, P. K. Townsend and P. van Nieuwenhuizen, Phys. Rev. D [**17**]{} (1978) 3179. doi:10.1103/PhysRevD.17.3179 M. Kaku, P. K. Townsend and P. van Nieuwenhuizen, Phys. Rev. Lett.  [**39**]{} (1977) 1109. doi:10.1103/PhysRevLett.39.1109 P. K. Townsend and P. van Nieuwenhuizen, Phys. Rev. D [**19**]{} (1979) 3166. doi:10.1103/PhysRevD.19.3166 Y. Ne’eman, Phys. Lett. B [**81**]{} (1979) 190 \[Phys. Lett.  [**81B**]{} (1979) 190\]. doi:10.1016/0370-2693(79)90521-5 R. Coquereaux, G. Esposito-Farese and G. Vaillant, Nucl. Phys. B [**353**]{} (1991) 689. doi:10.1016/0550-3213(91)90323-P P. H. Dondi and P. D. Jarvis, Z. Phys. C [**4**]{} (1980) 201. doi:10.1007/BF01421797 F. Hussain and G. Thompson, Phys. Lett. B [**260**]{} (1991) 359. doi:10.1016/0370-2693(91)91625-6 F. Hussain and G. Thompson, Phys. Lett. B [**265**]{} (1991) 307. doi:10.1016/0370-2693(91)90058-X N. A. Batakis, A. A. Kehagias and G. Zoupanos, Phys. Lett. B [**315**]{} (1993) 319 Erratum: \[Phys. Lett. B [**318**]{} (1993) 650\]. doi:10.1016/0370-2693(93)90470-3, 10.1016/0370-2693(93)91619-X [^1]: See also [@vanproeyen]. [^2]: In the Lorentzian signature, an analogous construction is encountered, specifically the foliation of the 3-d Minkowski spacetime by fuzzy hyperboloids of different radii [@Jurman:2013ota]. [^3]: For more details on this issue, see [@Sperling:2017dts; @Kimura:2002nq], where the same problem emerges in the construction of the fuzzy four-sphere. [^4]: For a string theory approach on such a model, see [@AlvarezGaume:2006bn]. [^5]: Details on this generic issue on such spaces are given in Appendix A of [@Manolakos:2019fle]. [^6]: A Yang-Mills action $\text{tr}F^2$ defined on the fuzzy $\mathrm{dS}_4$ space is gauge invariant, for details see Appendix A of [@Manolakos:2019fle].

--- abstract: 'In several spectral sequences for (global and local) Iwasawa modules over (not necessarily commutative) Iwasawa algebras (mainly of $p$-adic Lie groups) over ${\mathbb Z}_p$ are established, which are very useful for determining certain properties of such modules in arithmetic applications. Slight generalizations of said results can be found in (for abelian groups and more general coefficient rings), (for products of not necessarily abelian groups, but with ${\mathbb Z}_p$-coefficients), and . Unfortunately, some of Jannsen’s spectral sequences for families of representations as coefficients for (local) Iwasawa cohomology are still missing. We explain and follow the philosophy that all these spectral sequences are consequences or analogues of local cohomology and duality à la Grothendieck (and Tate for duality groups).' address: | Mathematisches Institut\ Universität Heidelberg\ Im Neuenheimer Feld 205\ D-69120 Heidelberg author: - Oliver Thomas - Otmar Venjakob bibliography: - 'bib.bib' title: On Spectral Sequences for Iwasawa Adjoints à la Jannsen for Families --- Introduction {#sec:introduction} ============ Let $\mathcal O$ be a complete discrete valuation ring with uniformising element $\pi$ and finite residue field. Consider furthermore an $\mathcal O$-algebra $R$, which is a complete Noetherian local ring with maximal ideal $\mathfrak m$, of dimension $d$ and finite residue field. We are mainly interested in the case of a ring of formal power series $R=\mathcal O[[X_1,\dots, X_t]]$ in $t$ variables, which is a complete regular local ring of dimension $d=t+1$. We now have a number of dualities at hand. First, there is Matlis duality: Denote with $\mathcal E$ an injective hull of $R/\mathfrak m$ as an $R$-module. Then $T=\operatorname{Hom}_R(-,\mathcal E)$ induces a contravariant involutive equivalence between Noetherian and Artinian $R$-modules akin to Pontryagin duality. Second, there is local duality: If ${{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak m}}$ denotes the right derivation of $ M \mapsto \varinjlim_k \operatorname{Hom}_R(R/\mathfrak m^k, M)$ in the derived category of $R$-modules, then $${{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak m}} \cong [-d] \circ T \circ \operatorname{{\pmb{\mathrm{R}}}Hom}_R(-, R)$$ on coherent $R$-modules. Third, there is Koszul duality: The complex ${{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak m}}$ can be computed by means of Koszul complexes $K^\bullet$ which are self-dual: $K^\bullet = \operatorname{Hom}_R(K^\bullet, R)[d]$. Finally, there is Tate duality: Let $G$ be a pro-$p$ duality group of dimension $s$. Then for discrete $G$-modules $A$ we have $H^i(G, \operatorname{Hom}(A,I)) \cong H^{s-i}(G,A)^*$ for a dualizing module $I$. Consider $\Lambda_R(G)=\varprojlim_U R[G/U]$ where $U$ runs through the open normal subgroups of $G$. It is well known that $\Lambda_R({\mathbb Z}_p^s)\cong R[[Y_1,\dots,Y_s]]$ and indeed $R=\Lambda_{\mathcal O}({\mathbb Z}_p^r)$. The maximal ideal of $\Lambda_R(G)$ is now generated by the regular sequence $(\pi, X_1, \dots, X_t, Y_1, \dots, Y_s)$ and no matter how we split up this regular sequence into two, they will remain regular. The Koszul complex then gives rise to a number of interesting spectral sequences and these should (at least morally) recover the spectral sequences $$\operatorname{Tor}_n^{{\mathbb Z}_p}(D_m(M^\vee), {\mathbb Q}_p/{\mathbb Z}_p) \Longrightarrow \operatorname{Ext}_{\Lambda_{{\mathbb Z}_p}(G)}^{n+m}(M, \Lambda_{{\mathbb Z}_p}(G))^\vee \label{eqn:jannsen-pont-of-iw-adjoints-is-tor-tate-pont}$$ and $$\varinjlim_k D_n(\operatorname{Tor}^{{\mathbb Z}_p}_m( {\mathbb Z}_p/p^k, M)^\vee \Longrightarrow \operatorname{Ext}_{\Lambda_{{\mathbb Z}_p}(G)}^{n+m}(M, \Lambda_{{\mathbb Z}_p}(G))^\vee), \label{eqn:jannsen-pont-of-iw-adjoints-is-colim-of-tate-of-pont-of-loc-coh}$$ which show up in Jannsen’s proof of [@MR1097615 2.1 and 2.2]. The functors $D_n$ stem from Tate’s spectral sequence and are a corner stone in the theory of duality groups. We will show in \[sec:iwasawa-adjoints\] that these spectral sequences (and many more) are consequences of the four duality principles laid out above. This also allows us to generalize Jannsen’s spectral sequences to more general coefficients. For example, generalisations of \[eqn:jannsen-pont-of-iw-adjoints-is-tor-tate-pont\] and \[eqn:jannsen-pont-of-iw-adjoints-is-colim-of-tate-of-pont-of-loc-coh\] are subject of \[prop:pont-of-iw-adjoints-is-tor-tate-pont\] and of \[prop:pont-of-iw-adjoints-is-colim-of-tate-of-pont-of-loc-coh\] respectively. While another spectral sequence for Iwasawa adjoints has already been generalized to more general coefficients (cf. \[thm:lim-sharifi-spec-seq-local\]), the generalizations of the aforementioned spectral sequences are missing in the literature. We can even generalize an explicit calculation of Iwasawa adjoints (cf. [@MR1097615 corollary 2.6], [@MR2392026 (5.4.14)]) in \[prop:dual-of-iw-adj-is-twist-of-local-coh-of-coeff\]. Furthermore, we generalize Venjakob’s result on local duality for Iwasawa algebras ([@MR1924402 theorem 5.6]) to more general coefficients (cf. \[thm:local-duality-for-iw-alg\]). As an application we determine the torsion submodule of local Iwasawa cohomology generalizing a result of Perrin-Riou in the case $R={\mathbb Z}_p$ in \[thm:torsion-in-local-iw-coh-for-poinc-grps\]. Conventions =========== A *ring* will always be unitary and associative, but not necessarily commutative. If not explicitly stated otherwise, “module” means left-module, “Noetherien” means left-Noetherien etc. We will furthermore use the language of derived categories. If ${\pmb{\mathrm{A}}}$ is an abelian category, we denote with ${{\pmb{\mathrm{D}}}}({\pmb{\mathrm{A}}})$ the derived category of unbounded complexes, with ${{\pmb{\mathrm{D}}}}^+({\pmb{\mathrm{A}}})$ the derived category of complexes bounded below, with ${{\pmb{\mathrm{D}}}}^-({\pmb{\mathrm{A}}})$ the derived category of complexes bounded above and with ${{\pmb{\mathrm{D}}}}^b({\pmb{\mathrm{A}}})$ the derived category of bounded complexes. As we simultaneously have to deal with left- and right-exact functors, both covariant and contravariant, recovering spectral sequences from isomorphisms in the derived category is a bit of a hassle regarding the indices. Suppose that ${\pmb{\mathrm{A}}}$ has enough injectives and projectives and that $M$ is a (suitably bounded) complex of objects of $A$. Then for a covariant functor $F\colon {\pmb{\mathrm{A}}}\ra{\pmb{\mathrm{A}}}$ we set ${{\pmb{\mathrm{R}}}}F=F(Q)$ and ${{\pmb{\mathrm{L}}}}F=F(P)$ with $Q$ a complex of injective objects, quasi-isomorphic to $A$ and $P$ a complex of projectives, quasi-isomorphic to $A$. If $F$ is contravariant, we set ${{\pmb{\mathrm{L}}}}F(A)=F(Q)$ and ${{\pmb{\mathrm{R}}}}F(A)=F(P)$. For indices, this implies the following: Assume that $A$ is concentrated in degree zero. Then for $F$ covariant, ${{\pmb{\mathrm{R}}}}F(A)$ has non-vanishing cohomology at most in non-negative degrees and ${{\pmb{\mathrm{L}}}}F(A)$ at most in non-positive degrees. For $F$ contravariant, it’s exactly the other way around. We set ${{\pmb{\mathrm{L}}}}^qF(A)=H^q({{\pmb{\mathrm{L}}}}F(A))$ and ${{\pmb{\mathrm{R}}}}^qF (A)=H^q({{\pmb{\mathrm{R}}}}F(A))$. Note that with these conventions - ${{\pmb{\mathrm{R}}}}^p (-)^G (A) = H^p(G,A)$ - ${{\pmb{\mathrm{L}}}}^q (-\otimes M)(A) = \operatorname{Tor}_{-q}(A,M)$ - ${{\pmb{\mathrm{R}}}}^p \operatorname{Hom}(-, M)(A) = \operatorname{Ext}^{p}(A,M)$ - ${{\pmb{\mathrm{L}}}}^q (\varinjlim_U (-^U)^*)(A) = \varinjlim_U H^{-q}(U,A)^*$ If $F\colon {\pmb{\mathrm{A}}}\ra{\pmb{\mathrm{A}}}$ is exact, then $F$ maps quasi-isomorphic complexes to quasi-isomorphic complexes. Its derivation ${{\pmb{\mathrm{R}}}}F$ (or ${{\pmb{\mathrm{L}}}}F$) is then given by simply applying $F$ and we will make no distinction between $F\colon {\pmb{\mathrm{A}}}\ra{\pmb{\mathrm{A}}}$ and ${{\pmb{\mathrm{R}}}}F\colon {{\pmb{\mathrm{D}}}}({\pmb{\mathrm{A}}})\ra {{\pmb{\mathrm{D}}}}({\pmb{\mathrm{A}}})$ in this case. For every integer $d\in{\mathbb Z}$ we have a *shift operator* $[d]$, so that for complexes $C$ and $n\in{\mathbb Z}$ the following holds: $$([d](C))^n = C^{n+d}.$$ We will at times write $C[d]$ instead of $[d](C)$. Note that although we occasionally cite , we deviate from Weibel’s conventions in this regard: Our $[d]$ is Weibel’s $[-d]$. We furthermore set $\operatorname{Hom}(C^\bullet, D^\bullet)$ to be the complex with entries $\operatorname{Hom}(C^\bullet, D^\bullet)^i = \bigoplus_{n\in{\mathbb Z}} \operatorname{Hom}(C^k, D^{k+1})$. Sign conventions won’t matter in this paper. Recall that an $R$-module is called *coherent* if it is finitely generated and all of its finitely generated submodules are finitely presented. The natural functor from coherent $R$-modules to $R$-modules then induces equivalences ${{\pmb{\mathrm{D}}}}^*({\pmb{\mathrm{Coh}}}(R))\cong {{\pmb{\mathrm{D}}}}^*_c({R\textrm{-}{\pmb{\mathrm{Mod}}}})$ for $*\in\{+,b\}$ where subscript “c” means complexes with coherent cohomology. If $R$ is Noetherian, then the notions of coherent, finitely generated and Noetherian modules coincide. A few facts on $R$-modules ========================== Noncommutative rings -------------------- Let $R$ be a ring. The intersection of all maximal left ideals coincides with the intersection of all maximal right ideals and is called the *Jacobson radical* of $R$ and is hence a two-sided ideal, denoted by $J(R)$. For every $r\in J(R)$ the element $1-r$ then has both a left and a right inverse and the following form of Nakayama’s lemma holds, cf. e.g. [@MR1125071 4.22]. Let $M\in{R\textrm{-}{\pmb{\mathrm{Mod}}}}$ be a finitely generated $R$-module. If $J(R)M=M,$ then $M=0$. Recall that a ring is called local if it has a unique maximal left-ideal. This unique left ideal is then also the ring’s unique maximal right-ideal and the group of two-sided units is the complement of this maximal ideal. The following is well known and gives rise to the notion of “finitely presented” (or “compact”) objects in arbitrary categories. \[lm:hom2-commutes-with-dir-lim-iff-m-fin-pres\] Let $R$ be a ring and $M$ an $R$-module. Then $\operatorname{Hom}_R(M, -)$ commutes with all direct limits if and only if $M$ is finitely presented. If $R$ is Noetherien, this isomorphism extends to higher Ext-groups. \[lm:ext-commutes-with-dir-lim-if-r-noeth-m-fg\] Let $R$ be a Noetherian ring, $M$ a finitely generated $R$-module, and $(N_i)_i$ a direct system of $R$-modules. Then $$\operatorname{Ext}^q_R(M, \varinjlim_i N_i) \cong \varinjlim_i \operatorname{Ext}^q_R(M, N_i).$$ As $R$ is Noetherian, there exists a resolution of $M$ of finitely generated projective $R$-Modules. As $\varinjlim$ commutes with homology, \[lm:hom2-commutes-with-dir-lim-iff-m-fin-pres\] yields the result. Recall the following subtleties: Let $R,S,T$ be rings, $N$ a $S$-$R$-bimodule and $P$ a $S$-$T$-bimodule. Then $\operatorname{Hom}_S(N,P)$ has the natural structure of an $R$-$T$-bimodule via $(rf)(n) = f(nr)$ and $(ft)(n)=f(n)t$. Furthermore let $M$ be a $R$-left-module. Then canonically $$\operatorname{Hom}_R(M, \operatorname{Hom}_S(N,P)) = \operatorname{Hom}_S(N\otimes_R M, P)$$ as $T$-modules. \[lm:hom-of-flat-inj-is-inj\] If $P$ is an $R$-$R$-bimodule that is flat as an $R$-right-module and $Q$ an injective $R$-module, then $\operatorname{Hom}_R(P,Q)$ is again an injective $R$-left-module. $\operatorname{Hom}_{R}(-,\operatorname{Hom}_R(P,Q)) = \operatorname{Hom}_R(-,Q) \circ (P\otimes_R -)$ is a composition of exact functors and hence exact. \[lm:tor-m-n-zero-then-ext-m-hom-n-q-zero\] Let $N$ be an $R$-$R$-bimodule and $M$ an $R$-modules. Then $$\operatorname{Tor}_q^R(N,M)=0 \text{ for } 1\leq q \leq n$$ if and only if $$\operatorname{Ext}^q_R(M,\operatorname{Hom}_R(N,Q))=0 \text{ for } 1 \leq q \leq n$$ and all injective $R$-left-modules $Q$. Let $$0 \ra K \ra P \ra N \ra 0$$ be an exact sequence of $R$-$R$-bimodules with $P$ flat from the right. First of all, applying $h_Q=\operatorname{Hom}_R(-,Q)$ yields the exact sequence $$0 \ra h_Q(N) \ra h_Q(P) \ra h_Q(K) \ra 0 \label[sequence]{seq:dual-of-resolution-of-n}$$ as $Q$ is injective. By \[lm:hom-of-flat-inj-is-inj\] $h_Q(P)$ is injective, so $\operatorname{Ext}_R^q(M,h_Q(P))=0$ for $q\geq 1$ and hence $\operatorname{Ext}^{q+1}_R(-,h_Q(N)) = \operatorname{Ext}^q_R(-,h_Q(K))$ for $q\geq 1$. Similarly $\operatorname{Tor}_{q+1}^R(N,-)=\operatorname{Tor}_q^R(K,-)$ for $q \geq 1$. Hence we can by dimension shifting reduce to the case $n=q=1$. Tensoring above exact sequence with $M$ from the right yields the exact sequence $$0 \ra \operatorname{Tor}_1^R(N,M) \ra K\otimes M \ra P \otimes M \ra N\otimes M\ra 0$$ and by injectivity of $Q$ the exact sequence $$0 \ra h_Q(N\otimes M) \ra h_Q(P\otimes M) \ra h_Q(K\otimes M) \ra h_Q(\operatorname{Tor}_1^R(N,M)) \ra 0.$$ We can also apply $\operatorname{Ext}^\bullet_R(M,-)$ to the exact \[seq:dual-of-resolution-of-n\]. By Tensor-Hom-adjointness and \[lm:hom-of-flat-inj-is-inj\], this implies that $$h_Q(\operatorname{Tor}_1^R(N,M)) = \operatorname{Ext}^1_R(M, h_Q(N)),$$ so if $\operatorname{Tor}_1^R(N,M)=0$, then clearly $\operatorname{Ext}^1_R(M, h_Q(N))=0$ for all injective $Q$. If on the other hand $\operatorname{Ext}^1_R(M, h_Q(N))=0$ for all injective $Q$, then $h_Q(\operatorname{Tor}_1^R(N,M))=0$ for all injective $Q$ and hence also $\operatorname{Tor}_1^R(N,M)=0$ as the category of $R$-modules has sufficiently many injectives. Let $R$ be a ring. A sequence $(r_1,\dots,r_d)$ of central elements in $R$ is called *regular*, if for each $i$ the residue class of $r_{i+1}$ in $R/(r_1,\dots,r_i)$ is not a zero-divisor. \[prop:tor-vanishes-regular-sequences\] Let $R$ be a ring and $(r_1,\dots,r_k,s_1,\dots,s_l)$ such a regular sequence that the sequence $(s_1,\dots,s_l)$ is itself regular. Let $I=(r_i)_i$ and $J=(s_i)_i$ be the ideals generated by the first and second part of the regular sequence. Then for all $q\geq 1$ $$\operatorname{Tor}^R_q(R/I, R/J) = 0$$ and $$\varprojlim_n \operatorname{Tor}^R_q(R/I^n, R/J^n)=0.$$ Let us first show that $\operatorname{Tor}^R_1(R/I, R/J) = 0$ and then reduce to this case by induction on $l$. Consider the exact sequence of $R$-modules $$0 \ra J \ra R \ra R/J \ra 0$$ and apply $\operatorname{Tor}^R_\bullet(R/I,-)$. As $R$ is flat, $$\operatorname{Tor}^R_1(R/I, R/J) = \ker \left(R/I\otimes_R J \ra R/I\right) = \frac{I\cap J}{IJ}.$$ We argue by induction on $l$ that this is zero: For $l=1$, $x\in I\cap J$ implies $x=\lambda s_1 = s_1\lambda \in I$, so $\lambda=0$ in $R/I$, so $\lambda\in I$ and hence $x\in IJ$. Denote with $J'$ the ideal generated by $s_1,\dots,s_{l-1}$. By induction $I\cap J'=IJ'$. Let $x\in I\cap J = I\cap (J'+s_lR)$, so $x= a+s_lb$ with $a\in J'$ and $b\in R$. Clearly $s_l b=0$ in $R/(I+J')$, so $b\in I+J'$ by regularity of the sequence and hence $$I\cap J = I\cap (J'+s_lR) = I \cap (J' + s_l(I+J')) = I \cap (J' + s_lI) = I\cap J' + I\cap s_l I$$ as $s_lI\subseteq I$. Now $I\cap J' + I\cap s_l I = IJ' +s_l I = I(J'+s_lR) = IJ$, and this was to be shown. We now argue by induction on $l$ that $\operatorname{Tor}^R_q(R/I,R/J)=0$ for all $q>0$. For $l=1$ we have the free resolution $$0 \ra R { {\mathop{\ra}\limits^{\vbox to -5\ex@{\kern-\tw@\ex@ \hbox{\hspace{-0.25em}\scriptsize$s_1$}\vss}}}} R \ra R/J \ra 0$$ hence $\operatorname{Tor}^R_q(R/I, R/J)=0$ for $q>1$ and for $q=1$ by what we saw above. Let $J'$ be again the ideal generated by $s_1,\dots,s_{l-1}$. By induction we can assume that all $\operatorname{Tor}_q^R(R/I,R/J')$ vanish. Consider the sequence $$0 \ra R/J' { {\mathop{\ra}\limits^{\vbox to -5\ex@{\kern-\tw@\ex@ \hbox{\hspace{-0.25em}\scriptsize$s_l$}\vss}}}} R/J' \ra R/J \ra 0,$$ which is exact as the subsequence $(s_1,\dots,s_l)$ is regular. Applying $\operatorname{Tor}^R_\bullet(R/I,-)$ shows that $\operatorname{Tor}^R_q(R/I,R/J)=0$ for $q>1$ by induction hypothesis — and by what we saw above also for $q=1$. It is easy to see that the sequence $(r_1^m, \dots, r_k^m,s_1^m,\dots,s_l^m)$ is again regular (cf. e.g.[@MR879273 theorem 16.1]). Denote the ideal generated by $(r_i^m)_i$ and $(s_i^m)_i$ by $I^{(m)}$ and $J^{(m)}$ respectively. Let $m'=\max\{k,l\}$. Clearly $I^{m'm}\subseteq I^{(m)}\subseteq I^m$ and the same is true for $J$, hence the natural map $$\operatorname{Tor}^R_q(R/I^{m'm}, R/J^{m'm}) \ra \operatorname{Tor}^R_q(R/I^m, R/J^m)$$ factors through $\operatorname{Tor}^R_q(R/I^{(m)}, R/J^{(m)})$, which is zero. The Koszul Complex ------------------ Recall the following (cf. e.g.[@MR1269324 section 4.5]). Denote by $$K_\bullet(x) = 0 \ra R { {\mathop{\ra}\limits^{\vbox to -5\ex@{\kern-\tw@\ex@ \hbox{\hspace{-0.25em}\scriptsize$x$}\vss}}}} R \ra 0$$ the chain complex (i.e., the degree decreases to the right) concentrated in degrees one and zero for a ring $R$ and a central element $x\in Z(R)$. For central elements $x_1,\dots,x_d$ the complex $$K_\bullet(x_1,\dots,x_d) = K_\bullet(x_1)\otimes\dots\otimes K_\bullet(x_d)$$ is called the *Koszul complex attached to $x_1,\dots,x_d$*. We will also consider the cochain complex $K^\bullet(x_1,\dots,x_d)$ with entries $K^p(x_1,\dots,x_d)=K_{-p}(x_1,\dots,x_d)$. While this definition is certainly elegant, a more down to earth description is given as follows: $K_{p}(x_1,\dots,x_d)$ is the free $R$-module generated by the symbols $e_{i_1}\wedge\dots\wedge e_{i_p}$ with $i_1<\dots<i_p$ with differential $$d(e_{i_1}\wedge\dots\wedge e_{i_p}) = \sum_{k=1}^p (-1)^{k+1} x_{i_k} e_{i_1}\wedge\dots\wedge\widehat{e_{i_k}}\wedge\dots\wedge e_{i_p}.$$ This description emphasizes the importance of using central elements $(x_i)_i$. The importance of the Koszul complex for our purposes stems from the following fact: If $x_1,\dots,x_d$ is a regular sequence, then $K_\bullet(x_1,\dots,x_d)$ is a free resolution of $R/(x_1,\dots,x_d)$, cf. e.g.[@MR1269324 corollary 4.5.5]. \[prop:koszul-poincare-duality\] The complex $K_\bullet=K_\bullet(x_1,\dots,x_d)$ is isomorphic to the complex $$0 \ra \operatorname{Hom}_R(K_0, R) \ra \dots \ra \operatorname{Hom}_R(K_{d}, R) \ra 0,$$ where $\operatorname{Hom}_R(K_0,R)$ is in degree $d$ and $\operatorname{Hom}_R(K_d,R)$ in degree zero. Analogously $$K^\bullet \cong \operatorname{Hom}_R(K^\bullet, R)[d].$$ We have to describe isomorphisms $K_{p} \cong \operatorname{Hom}_R(K_{d-p},R)$ such that all diagrams $$\begin{tikzcd} K_{p} \arrow{r}{d}\arrow{d}{\cong}& K_{p-1} \arrow{d}{\cong}\\ \operatorname{Hom}_R(K_{d-p},R)\arrow{r}{d^*}& \operatorname{Hom}_R(K_{d-p+1},R)\end{tikzcd}$$ commute. Consider the map $$e_{i_1}\wedge\dots\wedge e_{i_p} \mapsto \left( e_{j_1} \wedge \dots \wedge e_{j_{d-p}} \mapsto \mathrm{sgn}(\sigma) \right)$$ where $\mathrm{sgn}(\sigma)$ is the sign of the permutation $\sigma(1)=i_1, \dots, \sigma(p)=i_p, \sigma(p+1)=j_1,\dots, \sigma(d)=j_{d-p}$, i.e., in the exterior algebra $\bigwedge^dR^d$ we have $$e_{i_1}\wedge\dots\wedge e_{i_p} \wedge e_{j_1}\wedge \dots \wedge e_{j_{d-p}} = \mathrm{sgn}(\sigma) e_1\wedge\dots\wedge e_d.$$ (Note that $\mathrm{sgn}(\sigma)=0$ if $\sigma$ is not bijective.) It is then easy to verify that the diagram above does indeed commute: Identifying $R$ with $\bigwedge^dR^d$, all but at most one summand vanishes in the ensuing calculation and the difference in sign is precisely the difference in the permutations. \[prop:duality-rhom-tor-for-reg-seq-in-der-cat\] Let $R$ be a ring and $x_1,\dots,x_d$ a regular sequence of central elements in $R$. Then in the bounded derived category of $R$-modules $$[d]\circ \operatorname{{\pmb{\mathrm{R}}}Hom}_R(R/(x_1,\dots,x_n), -) \cong R/(x_1,\dots,x_n) \otimes^{{\pmb{\mathrm{L}}}}_R -.$$ Denote with $K_\bullet$ the Koszul (chain) complex $K_\bullet(x_1,\dots,x_d)$ (concentrated in degrees $d,d-1,\dots, 0$) and with $K^\bullet$ the Koszul (cochain) complex (concentrated in degrees $-d, -d+1, \dots, 0$). As $x_1,\dots,x_d$ form a regular sequence, $K^\bullet$ is a free resolution of $R/(x_1,\dots,x_n)$ and hence allows us to calculate the derived functors as follows. $$\begin{aligned} \operatorname{{\pmb{\mathrm{R}}}Hom}_R(R/(x_1,\dots,x_n), M)[d] &= \operatorname{Hom}_R(K^\bullet,M)[d]\\ &\cong \operatorname{Hom}_R(K^\bullet[-d],R) \otimes_R M\\ &\cong K^\bullet \otimes_R M\\ &= R/(x_1,\dots,x_d) \otimes^{{\pmb{\mathrm{L}}}}_R M, \end{aligned}$$ with the crucial isomorphisms being due to the fact that $K^\bullet$ is a complex of free modules and \[prop:koszul-poincare-duality\]. It is clear that these isomorphisms are functorial in $M$. \[prop:rhom-r-mod-regular-commutes-with-matlis\] Let $R$ be a commutative ring, $x_1,\dots,x_d$ a regular sequence in $R$ and $T=\operatorname{Hom}_R(-,Q)$ with $Q$ injective. Then $$\operatorname{{\pmb{\mathrm{R}}}Hom}_R(R/(x_1,\dots,x_d), -) \circ T = T \circ [d] \circ \operatorname{{\pmb{\mathrm{R}}}Hom}_R(R/(x_1,\dots,x_d), -)$$ on the derived category of $R$-modules. The functor $T$ is exact and $$\operatorname{Hom}_R(R/(x_1,\dots,x_d),-)\circ T = T \circ (R/(x_1,\dots,x_d)\otimes_R -)$$ by adjointness. Hence also $$\operatorname{{\pmb{\mathrm{R}}}Hom}_R(R/(x_1,\dots,x_d),-)\circ T = T \circ (R/(x_1,\dots,x_d)\otimes_R^{{{\pmb{\mathrm{L}}}}} -).$$ By \[prop:duality-rhom-tor-for-reg-seq-in-der-cat\], this is just $T \circ [d] \circ \operatorname{{\pmb{\mathrm{R}}}Hom}_R(R/(x_1,\dots,x_d), -).$ \[prop:duality-ext-tor-for-reg-seq-classical\] Let $R$ be a commutative ring and $x_1,\dots,x_d$ a regular sequence in $R$. Let further $M$ be an $R$-module. Then $$\operatorname{Ext}^{d-p}_R(R/(x_1,\dots,x_d), M)=\operatorname{Tor}^R_{p}(R/(x_1,\dots,x_d), M).$$ This is just \[prop:duality-rhom-tor-for-reg-seq-in-der-cat\], taking extra care of the indices: $$\begin{aligned} \operatorname{Ext}^{d-p}_R(R/(x_1,\dots,x_d), M) &= {\pmb{\mathrm{R}}}^{d-p}\operatorname{Hom}_R(R/(x_1,\dots,x_d), M)\\ &= H^{-p}\operatorname{{\pmb{\mathrm{R}}}Hom}_R(R/(x_1,\dots,x_d)[-d], M) \\&= H^{-p}(R/(x_1,\dots,x_d) \otimes^{{\pmb{\mathrm{L}}}}_R M) \\ &= \operatorname{Tor}^R_{p}(R/(x_1,\dots,x_d),M).\end{aligned}$$ Local Cohomology ---------------- Let $R$ be a ring and $\underline{J} = (J_n)_{n\in {\mathbb N}}$ a decreasing sequence of two-sided ideals. (The classical example is to take a two-sided ideal $J$ and set $\underline{J}=(J^n)_n$.) For an $R$-left-module $M$ set $$\Gamma_{\underline{J}}(M) = \{m\in M\mid J_n m = 0 \text{ for some $n$}\}.$$ It is clear that $\Gamma_{\underline{J}}$ is a left-exact functor with values in $R{\pmb{\mathrm{\text{-}Mod}}}$. Denote its right-derived functor in the derived category ${{\pmb{\mathrm{D}}}}^+(R{\pmb{\mathrm{\text{-}Mod}}})$ by ${{\pmb{\mathrm{R}}}}\Gamma_{\underline{J}}$. \[rem:different-expressions-of-loc-coh\] $\Gamma_{\underline{J}} = \varinjlim_n \operatorname{Hom}_R(R/J_n,-)$, so ${{\pmb{\mathrm{R}}}}\Gamma_{\underline{J}} = \varinjlim_n \operatorname{{\pmb{\mathrm{R}}}Hom}_R(R/J_n,-)$ and ${{\pmb{\mathrm{R}}}}^q\Gamma_{\underline{J}} = \varinjlim_n \operatorname{Ext}_R^q(R/J_n,-)$. \[prop:right-adj-preserves-inj-obj-if-left-adj-exact\] Let ${\pmb{\mathrm{A}}},{\pmb{\mathrm{B}}}$ be abelian categories, with additive functors $L\colon {\pmb{\mathrm{A}}}\ra{\pmb{\mathrm{B}}}$ left adjoint to $R\colon {\pmb{\mathrm{B}}}\ra{\pmb{\mathrm{A}}}$. If $L$ is exact, $R$ preserves injective objects. This is well known, cf. e.g.[@MR1269324 proposition 2.3.10]. \[prop:loc-coh-indep-of-base\] Let $\varphi\colon R\ra S$ be a homomorphism between unitary rings. Let $\underline{J}$ be decreasing sequence of two-sided ideals in $R$ and denote with $\underline{J}S$ the induced sequence of two-sided ideals in $S$. If $\varphi(R)$ lies in the centre of $S$, then $\Gamma_{\underline{J}} \circ \mathrm{res}_\varphi=\mathrm{res}_\varphi\circ \Gamma_{\underline{J}S}$. If furthermore injective $S$-modules are also injective as $R$-modules, e.g., if $S$ is a flat $R$-module via \[prop:right-adj-preserves-inj-obj-if-left-adj-exact\], then ${{\pmb{\mathrm{R}}}}\Gamma_{\underline{J}} \circ \mathrm{res}_\varphi=\mathrm{res}_\varphi\circ{{\pmb{\mathrm{R}}}}\Gamma_{\underline{J}S}$. Local cohomology is thus independent of the base ring for flat extensions and we will omit $\mathrm{res}_\varphi$ and the distinction between $\underline{J}S$ and $\underline{J}$ in the future. Note especially that if $R$ is complete, then $R[[G]]=R[G]^\wedge$ is a flat $R$-module. \[prop:loc-coh-is-max-fin-submod\] Let $R$ be a Noetherian local ring with maximal ideal $\mathfrak m$ and finite residue field. Let $M$ be a finitely generated $R$-module. Then $\Gamma_{\underline{\mathfrak m}}(M)$ is the maximal finite submodule of $M$. Denote with $T$ the maximal finite submodule of $M$ (which exists as $M$ is Noetherian). By Nakayama there exists a $k\in{\mathbb N}$ with $\mathfrak m^kT=0$ and hence $T\subseteq\Gamma_{\underline{\mathfrak m}}(M)$. Conversely $R/\mathfrak m^k$ is a finite ring for each $k$, hence $Rm$ is a finite module for each $m\in\Gamma_{\underline{\mathfrak m}}(M)$ and is thus contained in $T$. \[prop:local-coh-commutes-with-direct-limits\] If $R$ is a Noetherian ring and $\underline{J}$ a decreasing sequence of ideals, then ${{\pmb{\mathrm{R}}}}\Gamma_{\underline{J}}$ and ${{\pmb{\mathrm{R}}}}^q\Gamma_{\underline{J}}$ commute with direct limits. This is just \[lm:ext-commutes-with-dir-lim-if-r-noeth-m-fg\], as for Noetherian rings, direct limits of injective modules are again injective. For $\underline{I}$ and $\underline{J}$ decreasing sequences of two-sided ideals of a ring $R$ set $(\underline{I} + \underline{J})_n=I_n+J_n$. If $I$ and $J$ are two-sided ideals of a ring $R$, then generally $\underline{I}+\underline{J}\neq \underline{I+J}$. But as these two families are cofinal, $\Gamma_{\underline{I}+\underline{J}} = \Gamma_{\underline{I+J}}$. Clearly $\Gamma_{\underline{I}+\underline{J}}=\Gamma_{\underline{I}}\circ\Gamma_{\underline{J}}$, but regrettably $${{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}+\underline{J}} = {{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}} {{\pmb{\mathrm{R}}}}\Gamma_{\underline{J}}$$ is in general false if the families $\underline{I}$ and $\underline{J}$ are not sufficiently independent from one another: For $R={\mathbb Z}$, $\underline{I}=\underline{J}=(n_i{\mathbb Z})_i$ any descending sequence of non-trivial ideals and $M={\mathbb Q}/{\mathbb Z}$, the five-term-sequence in cohomology would start as follows: $$0 \ra \varinjlim_{i,j} \operatorname{Ext}^1_{{\mathbb Z}}({\mathbb Z}/n_i, \operatorname{Hom}({\mathbb Z}/n_j, {\mathbb Q}/{\mathbb Z})) \ra \varinjlim_i \operatorname{Ext}^1_{{\mathbb Z}}({\mathbb Z}/n_i, {\mathbb Q}/{\mathbb Z}) \ra \dots$$ But clearly $\operatorname{Ext}^1_{{\mathbb Z}}({\mathbb Z}/n_i, {\mathbb Q}/{\mathbb Z})=0$ and $$\operatorname{Ext}^1_{{\mathbb Z}}({\mathbb Z}/n_i, \operatorname{Hom}({\mathbb Z}/n_j, {\mathbb Q}/{\mathbb Z})) = \operatorname{Ext}^1_{{\mathbb Z}}({\mathbb Z}/n_i, {\mathbb Z}/n_j) = {\mathbb Z}/(n_i,n_j),$$ hence $$\varinjlim_{i,j} \operatorname{Ext}^1_{{\mathbb Z}}({\mathbb Z}/n_i, \operatorname{Hom}({\mathbb Z}/n_j, {\mathbb Q}/{\mathbb Z})) = \varinjlim_i {\mathbb Z}/n_i,$$ so the sequence above cannot possibly be exact. This argument of course generalizes: Were ${{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}+\underline{J}} = {{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}} {{\pmb{\mathrm{R}}}}\Gamma_{\underline{J}}$, then [@MR0077480 chapter XV, theorem 5.12] implied that $$\varinjlim_i \operatorname{Ext}^p_R(R/I_i, \varinjlim_j \operatorname{Hom}_R(R/J_j, Q)) = 0$$ for all $p>0$ and $Q$ injective, i.e., if the isomorphism in the derived category holds, then because ${{\pmb{\mathrm{R}}}}\Gamma_{\underline{J}}$ mapped injective objects to $\Gamma_{\underline{I}}$-acyclics. Using \[lm:tor-m-n-zero-then-ext-m-hom-n-q-zero\], a sufficient criterion for that to happen is that the transition maps eventually factor through $\operatorname{Ext}^p_R(R/\widetilde{I}_i, \operatorname{Hom}_R(R/\widetilde{J}_j, Q))$ for some $\widetilde{I}_i, \widetilde{J}_j$ with $\operatorname{Tor}_p^R(R/\widetilde{I}_i, R/\widetilde{J}_j)=0$ for all $p>0$ and this criterion appears to be close to optimal. The following proposition is a simple application of this principle. \[prop:spectral-sequence-for-local-cohomology-and-sum-of-ideals\] Let $R$ be a commutative ring and $(r_1,\dots,r_k,s_1,\dots,s_l)$ such a regular sequence, that $(s_1,\dots,s_l)$ is itself again regular. Then for the ideals $I=(r_i)_i$ and $J=(s_i)_i$ we have $${{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}+\underline{J}}={{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}}{{\pmb{\mathrm{R}}}}\Gamma_{\underline{J}} = {{\pmb{\mathrm{R}}}}\Gamma_{\underline{J}}{{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}} .$$ Denote the ideal generated by $(r_i^m)_i$ by $I^{(m)}$ and similarly for $J$. Then the transition maps in the system $$\varinjlim_i \operatorname{Ext}^p_R(R/I^i, \varinjlim_j \operatorname{Hom}_R(R/J^j, Q)) = \varinjlim_{i,j}\operatorname{Ext}^p_R(R/I^i, \operatorname{Hom}_R(R/J^j, Q))$$ eventually factor through $\operatorname{Ext}^p_R(R/I^{(n)}, \operatorname{Hom}_R(R/J^{(n)}, Q))$. But as we saw before, the sequences $(r_1^m,\dots,r_k^m,s_1^m,\dots,s_l^m)$ and $(s_1^m,\dots,s_l^m)$ are again regular, so this vanishes by \[lm:tor-m-n-zero-then-ext-m-hom-n-q-zero,prop:tor-vanishes-regular-sequences\] for $p\geq 1$. As $\operatorname{Tor}^R_\bullet(-,-)$ is symmetrical for commutative rings, the same argument also applies for ${{\pmb{\mathrm{R}}}}\Gamma_{\underline{J}}{{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}}.$ (Avoiding) Matlis Duality {#sec:matlis-duality} ========================= First recall Pontryagin duality. The functor $\Pi=\operatorname{Hom}_{\mathrm{cts}}(-,{\mathbb R}/{\mathbb Z})$ induces a contravariant auto-equivalence on the category of locally compact Hausdorff abelian groups and interchanges compact with discrete groups. The isomorphism $A\ra \Pi(\Pi(A))$ is given by $a \mapsto (\varphi\mapsto \varphi(a))$. If $G$ is pro-$p$, then $\Pi(G)=\operatorname{Hom}_{\mathrm{cts}}(G,{\mathbb Q}_p/{\mathbb Z}_p)$. If $D$ is a discrete torsion group or a topologically finitely generated profinite group, then $\Pi(D)=\operatorname{Hom}_{{\mathbb Z}}(D,{\mathbb Q}/{\mathbb Z})$. We will write $-^\vee$ for $\Pi$ if it is notationally more convenient. Matlis duality is commonly stated as follows: \[thm:matlis-duality\] Let $R$ be a complete Noetherian commutative local ring with maximal ideal $\mathfrak m$ and $\mathcal E$ a fixed injective hull of the $R$-module $R/\mathfrak m$. Then $\operatorname{Hom}_R(-,\mathcal E)$ induces an equivalence between the finitely generated modules and the Artinian modules with inverse $\operatorname{Hom}_R(-,\mathcal E)$. If $R$ is a discrete valuation ring, then $Q(R)/R$ is an injective hull of its residue field. Matlis duality – using an *abstract* dualizing module instead of a topological one – behaves very nicely in relation to local cohomology. In applications however the Matlis module $\mathcal E$ is cumbersome and in general not particularly easy to construct. Consider the rings $R={\mathbb Z}_p$, $S_1={\mathbb Z}_p[\pi]$ and $S_2={\mathbb Z}_p[[T]]$ with $\pi$ a uniformizer of ${\mathbb Q}_p(\sqrt{p})$. Clearly the homomorphisms $R\ra S_i$ are local and flat and their respective residue fields agree. But while $\mathcal E_R={\mathbb Q}_p/{\mathbb Z}_p, \mathcal E_{S_1}\cong {\mathbb Q}_p/{\mathbb Z}_p^{\oplus 2}$ as an abelian group. Furthermore, $\mathcal E_{S_2}\cong \bigoplus_{{\mathbb N}} {\mathbb Q}_p/{\mathbb Z}_p$ as an abelian group by \[prop:matlis-module-is-pont-of-ring\]. The best we can hope for in general is the following. Let $R\ra S$ be a flat and local homomorphism between Noetherian local rings with respective maximal ideals $\mathfrak m$ and $\mathfrak M$. Assume that $R/\mathfrak m^n \cong S/\mathfrak M^n$ for all $n$. Then an injective hull of $S/\mathfrak M$ as an $S$-module is also an injective hull of $R/\mathfrak m$ as an $R$-module. Starting with pro-$p$ local rings, Matlis modules are however intimately connected with Pontryagin duality. \[prop:qp-mod-zp-is-somewhat-dualizing-for-r-mod-m\] Let $R$ be a pro-$p$ local ring with maximal ideal $\mathfrak m$. Then there exists an isomorphism of $R$-modules $R/\mathfrak m\cong\operatorname{Hom}_{{\mathbb Z}_p}(R/\mathfrak m,{\mathbb Q}_p/{\mathbb Z}_p)$. As $R/\mathfrak m$ is finite and hence a commutative field, both objects are isomorphic as abelian groups. As vector spaces of the same finite dimension over $R/\mathfrak m$ they are hence isomorphic as $R/\mathfrak m$-modules and thus as $R$-modules. \[prop:pont-is-dual-wrt-r-vee\] Let $R$ be a pro-$p$ local ring with maximal ideal $\mathfrak m$ and $M$ a finitely presented or a discrete $R$-module. Then $\Pi(M)=\operatorname{Hom}_R(M,\Pi(R))$. Let first $M=\varinjlim_i M_i$ be an arbitrary direct limit of finitely presented $R$-modules. Then by \[lm:hom2-commutes-with-dir-lim-iff-m-fin-pres\] $$\begin{aligned} \operatorname{Hom}_R(\varinjlim_i M_i, \Pi(R)) &= \varprojlim_i \operatorname{Hom}_R(M_i,\varinjlim_k \Pi(R/\mathfrak M^k)) \\ &= \varprojlim_i \varinjlim_k \operatorname{Hom}_R(M_i,\operatorname{Hom}_{{\mathbb Z}_p}(R/\mathfrak M^k, {\mathbb Q}_p/{\mathbb Z}_p))\\ &= \varprojlim_i \varinjlim_k \operatorname{Hom}_{{\mathbb Z}_p}(M_i/\mathfrak M^k, {\mathbb Q}_p/{\mathbb Z}_p) \\ &= \varprojlim_i \Pi(M_i). \end{aligned}$$ If $M$ itself is finitely presented, this shows the proposition. If $M$ is discrete, we can take the $M_i$ to be discrete and finitely presented (i.e., finite). The projective limit of their duals exists in the category of compact $R$-modules and it follows that $\varprojlim_i \Pi(M_i)=\Pi(M)$. \[prop:matlis-module-is-pont-of-ring\] Let $R$ be a Noetherian pro-$p$ local ring with maximal ideal $\mathfrak m$. Then $\Pi(R)=\operatorname{Hom}_{\mathrm{cts}}(R,{\mathbb Q}_p/{\mathbb Z}_p)$ is an injective hull of $R/\mathfrak m$ as an $R$-module. $\Pi(R)$ is injective as an abstract $R$-module: By Baer’s criterion it suffices to show that $\operatorname{Hom}_R(R,\Pi(R))\ra\operatorname{Hom}_R(I,\Pi(R))$ is surjective for every (left-)ideal $I$ of $R$. By \[prop:pont-is-dual-wrt-r-vee\], this is equivalent to the surjectivity of $\Pi(R)\ra \Pi(I)$, which is clear. In lieu of \[prop:qp-mod-zp-is-somewhat-dualizing-for-r-mod-m\] it hence suffices to show that $\operatorname{Hom}_{{\mathbb Z}_p}(R/\mathfrak m,{\mathbb Q}_p/{\mathbb Z}_p)\subseteq \operatorname{Hom}_{\mathrm{cts}}(R,{\mathbb Q}_p/{\mathbb Z}_p)$ is an essential extension, so take $H$ to be an $R$-submodule of $\operatorname{Hom}_{\mathrm{cts}}(R,{\mathbb Q}_p/{\mathbb Z}_p)$ and $0\neq f\in H$. Then by continuity, $f$ descends to $$f\colon R/\mathfrak m^{k+1}\ra{\mathbb Q}_p/{\mathbb Z}_p$$ with $k$ minimal. It follows that there exists an element $r\in\mathfrak m^k$ with $f(r)\neq 0$. $rf\colon R\ra{\mathbb Q}_p/{\mathbb Z}_p$ is consequentially also not zero, lies in $H$ but now descends to $$rf\colon R/\mathfrak m\ra {\mathbb Q}_p/{\mathbb Z}_p,$$i.e., $H\cap \operatorname{Hom}_{{\mathbb Z}_p}(R/\mathfrak m,{\mathbb Q}_p/{\mathbb Z}_p)\neq 0$. \[prop:pont-is-matlis-for-fg-and-art-mods\] Let $R$ be a commutative pro-$p$ Noetherian commutative local ring. Then if $M$ is finitely generated or Artinian, Matlis and Pontryagin duality agree. Immediate from \[prop:pont-is-dual-wrt-r-vee,prop:matlis-module-is-pont-of-ring\]. \[prop:loc-coh-is-lim-of-pont-of-shift-of-invariants-of-pont\] Let $R$ satisfy Matlis duality via $T=\operatorname{Hom}_R(-,\mathcal E)$. Let $\underline{I}$ be a decreasing family of ideals generated by regular sequences of length $d$. Then $${{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}} = \varinjlim_n T \circ [d] \circ \operatorname{{\pmb{\mathrm{R}}}Hom}(R/ I_n, -) \circ T$$ on ${{\pmb{\mathrm{D}}}}^+_c({R\textrm{-}{\pmb{\mathrm{Mod}}}}).$ By \[prop:rhom-r-mod-regular-commutes-with-matlis\] it follows that ${{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}} = \varinjlim_n \operatorname{{\pmb{\mathrm{R}}}Hom}(R/I_n, -) \circ T \circ T = \varinjlim_n T \circ [d]\circ\operatorname{{\pmb{\mathrm{R}}}Hom}(R/I_n, -) \circ T$. Tate Duality and Local Cohomology ================================= Working in the derived category makes a number of subtleties more explicit than working only with cohomology groups. Assume that $R$ is a complete local commutative Noetherian ring with finite residue field of characteristic $p$ and $G$ an analytic pro-$p$ group. Then every $\Lambda=R[[G]]$-module has a natural topology via the filtration of augmentation ideals of $\Lambda$. It is furthermore obvious to consider the following two categories: - $\mathcal C(\Lambda)$, the category of compact $\Lambda$-modules (with continuous $G$-action), - $\mathcal D(\Lambda)$, the category of discrete $\Lambda$-modules (with continuous $G$-action). Pontryagin duality then induces equivalences between $\mathcal C(\Lambda)$ and $\mathcal D(\Lambda^\circ)$, where $-^\circ$ denotes the opposite ring. It is furthermore well-known that both categories are abelian, that $\mathcal C(\Lambda)$ has exact projective limits and enough projectives and analogously that $\mathcal D(\Lambda)$ has exact direct limits and enough injectives. It is important to note that the notion of continuous $\Lambda$-homomorphisms and abstract ones often coincides: If $M$ is finitely generated with the quotient topology and $N$ is either compact or discrete, every $\Lambda$-homomorphism $M\ra N$ is continuous, cf. [@MR2954997 lemma 3.1.4]. In what follows we want to compare Tate cohomology, i.e. ${{\pmb{\mathrm{L}}}}D$ as defined below, with other cohomology theories such as local cohomology. Now Tate cohomology is defined on the category of *discrete* $G$-modules and we hence have a contravariant functor ${{\pmb{\mathrm{L}}}}D\colon {{\pmb{\mathrm{D}}}}^+(\mathcal D(\Lambda)) \ra {{\pmb{\mathrm{D}}}}^-({\Lambda^\circ\textrm{-}{\pmb{\mathrm{Mod}}}})$. Local cohomology on the other hand is defined on ${{\pmb{\mathrm{D}}}}^+({\Lambda\textrm{-}{\pmb{\mathrm{Mod}}}})$ or any subcategory that contains sufficiently many acyclic (e.g. injective) modules. This is not necessarily satisfied for ${{\pmb{\mathrm{D}}}}^+(\mathcal C(\Lambda))$. A statement such as $${{\pmb{\mathrm{L}}}}D \circ \Pi = [d] \circ {{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}}$$ without further context hence does not make much sense: The implication would be that this would be an isomorphism of functors defined on ${{\pmb{\mathrm{D}}}}^b(\mathcal C(\Lambda))$, but ${{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}}$ doesn’t exist on ${{\pmb{\mathrm{D}}}}^b(\mathcal C(\Lambda))$. Let $G$ be a profinite group and $A$ a discrete $G$-module. Denote with $D$ the functor $$D\colon A \mapsto \varinjlim_U (A^U)^*$$ where $N^*=\operatorname{Hom}_{{\mathbb Z}}(N,{\mathbb Q}/{\mathbb Z})$, the limit runs over the open normal subgroups of $G$ with the dual of the corestriction being the transition maps (cf. [@MR2392026 II.5 and III.4]). $D$ is right exact and contravariant and $D(A)$ has a continuous action of $G$ *from the right*. Denote its left derivation in the derived category of discrete $G$-modules by $${{\pmb{\mathrm{L}}}}D\colon {{\pmb{\mathrm{D}}}}^+(\mathcal D(\widehat{{\mathbb Z}}[[G]])) \ra {{\pmb{\mathrm{D}}}}^-({\widehat{{\mathbb Z}}[[G]]^\circ\textrm{-}{\pmb{\mathrm{Mod}}}})$$ (where $-^\circ$ denotes the opposite ring), so $D_i(A)= {{\pmb{\mathrm{L}}}}^{-i} D(A) = \varinjlim_U H^i(U,A)^*$. If $G$ is a profinite group, $R$ a profinite ring and $A$ a discrete $R[[G]]$-module, then $D(A)$ is again an $R[[G]]$-module, so $${{\pmb{\mathrm{L}}}}D\colon {{\pmb{\mathrm{D}}}}^+(\mathcal D(R[[G]]))\ra {{\pmb{\mathrm{D}}}}^-({R[[G]]^\circ\textrm{-}{\pmb{\mathrm{Mod}}}}),$$ where $\mathcal D(R[[G]])$ denotes the category of discrete $R[[G]]$-modules. Furthermore, we can of course also look at the functor $${{\pmb{\mathrm{L}}}}D\colon {{\pmb{\mathrm{D}}}}^+({R[[G]]\textrm{-}{\pmb{\mathrm{Mod}}}})\ra {{\pmb{\mathrm{D}}}}^-({R[[G]]^\circ\textrm{-}{\pmb{\mathrm{Mod}}}}).$$ Naturally, these functors don’t necessarily coincide. \[prop:inj-objects-in-discrete-modules-indep-of-coeff\] Let $R$ be such a profinite ring, that the structure morphism $\widehat{{\mathbb Z}}\ra R$ gives it the structure of a finitely presented flat $\widehat{{\mathbb Z}}$-module. Let $G$ be a profinite group such that $R[[G]]$ is a Noetherian local ring with finite residue field. (This is the case if $G$ is a $p$-adic analytic group and $R$ is the valuation ring of a finite extension over ${\mathbb Z}_p$.) Then an injective discrete $R[[G]]$-module is an injective discrete $G$-module. By \[prop:right-adj-preserves-inj-obj-if-left-adj-exact\] it suffices to show that $?\colon\mathcal D(R[[G]])\ra \mathcal D(\widehat{{\mathbb Z}}[[G]])$ has an exact left adjoint. It is clear that $$M \mapsto R[[G]] \otimes_{\widehat{{\mathbb Z}}[[G]]} M$$ is an algebraic exact left adjoint, so it remains to show that $R \otimes_{\widehat{{\mathbb Z}}} M = R[[G]]\otimes_{\widehat{{\mathbb Z}}[[G]]} M$ is a discrete $R[[G]]$-module. Now $M$ is the direct limit of finite modules, hence so is $ R \otimes_{\widehat{{\mathbb Z}}} M$. But for a finite $R[[G]]$-module $N$ this is clear as then $\mathfrak m^k M_i=0$ for some $k$ with $\mathfrak m$ the maximal ideal of $R[[G]]$ by Nakayama. \[prop:tate-d-independent-of-coefficients-if-fg-flat-etc\] The following diagram commutes if $R$ is a finitely presented flat $\widehat{{\mathbb Z}}$-module with $R[[G]]$ Noetherian and local with finite residue field: $$\begin{tikzcd} {{\pmb{\mathrm{D}}}}^+(\mathcal D(R[[G]])) \arrow{d}{{{\pmb{\mathrm{L}}}}D} \arrow{r}{?} & {{\pmb{\mathrm{D}}}}^+(\mathcal D(\widehat{{\mathbb Z}}[[G]])) \arrow{d}{{{\pmb{\mathrm{L}}}}D} \\ {{\pmb{\mathrm{D}}}}^-({R[[G]]^\circ\textrm{-}{\pmb{\mathrm{Mod}}}}) \arrow{r}{?} & {{\pmb{\mathrm{D}}}}^-({\widehat{{\mathbb Z}}[[G]]^\circ\textrm{-}{\pmb{\mathrm{Mod}}}}) \end{tikzcd}$$ Clearly the forgetful functors and $D$ all commute on the level of categories of modules. The result then follows from \[prop:inj-objects-in-discrete-modules-indep-of-coeff\]. \[prop:fg-proj-is-compact-proj\] 1. Let $M$ be an Artinian $\Lambda$-module. Then $\Lambda\times M\ra M$ is continuous if we give $M$ the discrete topology. 2. Let $N$ be a Noetherian $\Lambda$-module. Then $N$ is compact if we give it the topology induced by $\Lambda$. 3. The functor $${\pmb{\mathrm{f}}}.{\pmb{\mathrm{g}}}.\textrm{-}{\Lambda\textrm{-}{\pmb{\mathrm{Mod}}}}\ra\mathcal C(\Lambda)$$ maps projective objects to projectives. \[prop:local-coh-of-torus-is-tate-coh-of-matlis\] For the $d$-dimensional torus $G\cong{\mathbb Z}_p^d$ and its Iwasawa algebra $\Lambda(G)=\varprojlim_i \mathcal O[G/G^{p^i}]$ over a discrete complete valuation ring $\mathcal O$ with residue characteristic $p$ and uniformizer $\pi$ and with augmentation ideals $I_i = \ker \Lambda(G) \ra \mathcal O[G/G^{p^i}]$ the following holds in ${{\pmb{\mathrm{D}}}}^b_c({\Lambda\textrm{-}{\pmb{\mathrm{Mod}}}})$: $${{\pmb{\mathrm{L}}}}D \circ T \cong [d] \circ {{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}},$$ Especially the following diagram commutes: $$\begin{tikzcd} {{\pmb{\mathrm{D}}}}^b_c({\Lambda\textrm{-}{\pmb{\mathrm{Mod}}}}) \arrow[dash]{r}{\cong} \arrow[hook]{d} & {{\pmb{\mathrm{D}}}}^b({\pmb{\mathrm{f}}}.{\pmb{\mathrm{g}}}.\textrm{-}{\Lambda\textrm{-}{\pmb{\mathrm{Mod}}}}) \arrow{r} & {{\pmb{\mathrm{D}}}}^b(\mathcal C(\Lambda)) \arrow{r}{T} & {{\pmb{\mathrm{D}}}}^b(\mathcal D(\Lambda^\circ)) \arrow{d}{{{\pmb{\mathrm{L}}}}D} \\ {{\pmb{\mathrm{D}}}}^b({\Lambda\textrm{-}{\pmb{\mathrm{Mod}}}}) \arrow{rrr}{[d]\circ{{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak m}}}&&& {{\pmb{\mathrm{D}}}}^b({\Lambda\textrm{-}{\pmb{\mathrm{Mod}}}}) \end{tikzcd}$$ $\Lambda(G)$ is a regular local ring with maximal ideal $(\pi, \gamma_1-1,\dots, \gamma_d-1)$ for any set of topological generators $(\gamma_i)_i$ of $G$. One immediately verifies that the sequences $\gamma_1^{p^i}-1,\dots, \gamma_d^{p^i}-1$ are again regular and generate the ideals $I_i$. By \[prop:loc-coh-is-lim-of-pont-of-shift-of-invariants-of-pont\], $$[d] \circ {{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}}= \varinjlim_n T\circ \operatorname{{\pmb{\mathrm{R}}}Hom}_\Lambda(\Lambda/I_n,-) \circ T.$$ Take a bounded complex $M$ of finitely generated $R$-modules that is quasi-isomorphic to a bounded complex $P$ of finitely generated projective modules. The resulting complex $T(P)$ is then not only a bounded complex of injective discrete modules by Pontryagin duality and \[prop:fg-proj-is-compact-proj\], but also a bounded complex of injective abstract $\Lambda$-modules by \[lm:hom-of-flat-inj-is-inj\]. In all relevant derived categories $T(M)\cong T(P)$ holds. As $\operatorname{Hom}_\Lambda(\Lambda/I_n,-)=(-)^{G^{p^n}}$ by construction, we can compute $[d] \circ {{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}} (M)$ as follows (keeping \[prop:pont-is-matlis-for-fg-and-art-mods\] in mind): $$\begin{aligned} [d] \circ {{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}} (M) &= [d] \circ {{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}} (P) \\ &= \varinjlim_n T\circ \operatorname{{\pmb{\mathrm{R}}}Hom}_\Lambda(\Lambda/I_n,-) \circ T(P)\\ & = \varinjlim_n T(\operatorname{Hom}_\Lambda(\Lambda/I_n,T(P))) \\ & = \varinjlim_n T(T(P)^{G^{p^n}}) \\ & = \varinjlim_n \Pi(T(P)^{G^{p^n}}) \\ & = D(T(P)) = {{\pmb{\mathrm{L}}}}D \circ T(M). \end{aligned}$$ \[prop:rhom-r-mod-m-k-restricts-to-der-coh-lambda-mod\] Let $R$ be a commutative Noetherian ring with unit and $R\ra S$ a flat ring extension with $R$ contained in the centre of $S$ and $S$ again (left-)Noetherian. Then $$\operatorname{{\pmb{\mathrm{R}}}Hom}_R\colon {{\pmb{\mathrm{D}}}}^-({R\textrm{-}{\pmb{\mathrm{Mod}}}})^{\mathrm{opp}}\times{{\pmb{\mathrm{D}}}}^+({R\textrm{-}{\pmb{\mathrm{Mod}}}}) \ra {{\pmb{\mathrm{D}}}}^+({R\textrm{-}{\pmb{\mathrm{Mod}}}})$$ extends to $$\operatorname{{\pmb{\mathrm{R}}}Hom}_R\colon {{\pmb{\mathrm{D}}}}^-({R\textrm{-}{\pmb{\mathrm{Mod}}}})^{\mathrm{opp}}\times{{\pmb{\mathrm{D}}}}^+({S\textrm{-}{\pmb{\mathrm{Mod}}}}) \ra {{\pmb{\mathrm{D}}}}^+({S\textrm{-}{\pmb{\mathrm{Mod}}}}),$$ which in turn restricts to $$\operatorname{{\pmb{\mathrm{R}}}Hom}_R\colon {{\pmb{\mathrm{D}}}}^b_c({R\textrm{-}{\pmb{\mathrm{Mod}}}})^{\mathrm{opp}}\times{{\pmb{\mathrm{D}}}}^b_c({S\textrm{-}{\pmb{\mathrm{Mod}}}}) \ra {{\pmb{\mathrm{D}}}}^b_c({S\textrm{-}{\pmb{\mathrm{Mod}}}}),$$ First note that if $M$ is an $R$-left-module and $N$ an $S$-left-module, then $\operatorname{Hom}_R(M,N)$ carries the structure of an $S$-left-module via $(s f)(m) = sf(m)$. Then $\operatorname{Hom}_R(R,S)\cong S$ as $S$-left-modules and the following diagram commutes:$$\begin{tikzcd} {S\textrm{-}{\pmb{\mathrm{Mod}}}} \arrow{rr}{\operatorname{Hom}_R(M,-)}\arrow{d}{?} && {S\textrm{-}{\pmb{\mathrm{Mod}}}}\arrow{d}{?} \\ {R\textrm{-}{\pmb{\mathrm{Mod}}}}\arrow{rr}{\operatorname{Hom}_R(M, -)} && {R\textrm{-}{\pmb{\mathrm{Mod}}}}\end{tikzcd}$$ As $S$ is a flat $R$-module, $?$ preserves injectives by \[prop:right-adj-preserves-inj-obj-if-left-adj-exact\] and we can compute $\operatorname{{\pmb{\mathrm{R}}}Hom}_R(M,-)$ in either category. If $M$ is a finitely generated $R$-module and $N$ a finitely generated $S$-module, then $\operatorname{Hom}_R(M,N)$ is again a finitely generated $S$-module, as $S$ is left-Noetherian. If $M$ is a bounded complex of finitely generated $R$-modules, then it is quasi-isomorphic to a bounded complex of finitely generated projective $R$-modules. The result then follows at once. Note however that $\operatorname{{\pmb{\mathrm{R}}}Hom}_R$ does *not* extend to a functor $$\operatorname{{\pmb{\mathrm{R}}}Hom}_R\colon {{\pmb{\mathrm{D}}}}^-({S\textrm{-}{\pmb{\mathrm{Mod}}}})^{\mathrm{opp}}\times{{\pmb{\mathrm{D}}}}^+({R\textrm{-}{\pmb{\mathrm{Mod}}}}) \ra {{\pmb{\mathrm{D}}}}^+({S\textrm{-}{\pmb{\mathrm{Mod}}}}).$$ Even in those cases where we can give $\operatorname{Hom}_R(M,A)$ the structure of an $S$-module (e.g. when $S$ has a Hopf structure with antipode $s\mapsto\overline{s}$ via $(sf)(m)=f(\overline{s}m)$), projective $S$-modules in general are not projective. This is specially true for $R[[G]]$, which is a flat, but generally not a projective $R$-module. An essential ingredient in the proof of this section’s main theorem is Grothendieck local duality. It is commonly stated as follows: \[thm:local-duality\] Let $R$ be a commutative regular local ring of dimension $d$ with maximal ideal $\mathfrak m$, and $\mathcal E$ a fixed injective hull of the $R$-module $R/\mathfrak m$. Denote with $R[d]$ the complex concentrated in degree $-d$ with entry $R$. Then $${{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak m}} \cong T \circ \operatorname{{\pmb{\mathrm{R}}}Hom}_R(-, R[d]) = [-d] \circ T \circ \operatorname{{\pmb{\mathrm{R}}}Hom}_R(-,R)$$ on ${{\pmb{\mathrm{D}}}}^b_c({R\textrm{-}{\pmb{\mathrm{Mod}}}})$. The regularity assumption on $R$ can be weakened if one is willing to deal with a dualizing complex that is not concentrated in just one degree (loc.cit.). Relaxing commutativity however is more subtle and will be the focus of \[sec:local-dual-iwas\]. \[prop:jannsen-spec-seq-in-der-cat\] Let $\mathcal O$ be a pro-$p$ discrete valuation ring, $R=\mathcal O[[X_1,\dots,X_t]]$ with maximal ideal $\mathfrak m$, $G={\mathbb Z}_p^s$ and $\Lambda = R[[G]]$. Then $$T \circ \operatorname{{\pmb{\mathrm{R}}}Hom}_\Lambda(-,\Lambda) \cong [t+1] \circ {{\pmb{\mathrm{R}}}}\Gamma_{\mathfrak m} \circ {{\pmb{\mathrm{L}}}}D \circ T$$ on ${{\pmb{\mathrm{D}}}}^b_c({\Lambda\textrm{-}{\pmb{\mathrm{Mod}}}})$. The right hand side can furthermore be expressed as $${{\pmb{\mathrm{R}}}}\Gamma_{\mathfrak m} \circ {{\pmb{\mathrm{L}}}}D \circ T \cong \varinjlim_k\, {{\pmb{\mathrm{L}}}}D \circ T \circ \operatorname{{\pmb{\mathrm{R}}}Hom}_R(R/\mathfrak m^k, -).$$ $\Lambda$ is again a regular local ring, now of dimension $t+s+1$. Denote its maximal ideal by $\mathfrak M$. By \[thm:local-duality\] $${{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak M}} \cong T\circ \operatorname{{\pmb{\mathrm{R}}}Hom}_\Lambda(-,\Lambda[s+t+1]) = [-s-t-1] \circ T \circ \operatorname{{\pmb{\mathrm{R}}}Hom}_\Lambda(-,\Lambda) .$$ Now $\mathfrak M= \mathfrak m + (\gamma_1-1,\dots,\gamma_s-1)$ and if $x_1,\dots,x_{t+1}$ is a regular sequence in $R$, then $x_1,\dots,x_{t+1}, \gamma_1-1,\dots,\gamma_s-1$ is a regular sequence in $\Lambda$. Furthermore, the sequence $\gamma_1-1,\dots,\gamma_s-1$ is of course itself regular in $\Lambda$. Let it generate the ideal $I$. Then we can apply \[prop:spectral-sequence-for-local-cohomology-and-sum-of-ideals\], i.e., $${{\pmb{\mathrm{R}}}}\Gamma_{\mathfrak M} \cong {{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak m}} \circ {{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}}.$$ By \[prop:local-coh-of-torus-is-tate-coh-of-matlis\], we have ${{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}} = [-s] \circ {{\pmb{\mathrm{L}}}}D \circ T$, which shows the first isomorphism. Consider furthermore the functor $\varinjlim_k {{\pmb{\mathrm{L}}}}D \circ T \circ \operatorname{{\pmb{\mathrm{R}}}Hom}_R(R/\mathfrak m^k, -)$. By \[prop:rhom-r-mod-m-k-restricts-to-der-coh-lambda-mod\] we can compute it on ${{\pmb{\mathrm{D}}}}^b_c({\Lambda\textrm{-}{\pmb{\mathrm{Mod}}}})$ as $$\begin{aligned} \varinjlim_k {{\pmb{\mathrm{L}}}}D \circ T \circ \operatorname{{\pmb{\mathrm{R}}}Hom}_R(R/\mathfrak m^k, -) & \cong \varinjlim_k [s] \circ{{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}} \circ \operatorname{{\pmb{\mathrm{R}}}Hom}_R(R/\mathfrak m^k, -)\\ &\cong [s] \circ {{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}} \circ \varinjlim_k \operatorname{{\pmb{\mathrm{R}}}Hom}_R(R/\mathfrak m^k, -) \\ & = [s] \circ {{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}} \circ {{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak m}} \\ &\cong [s] \circ {{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak m}} \circ {{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}}\\ &\cong [s] \circ {{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak m}} \circ [-s] \circ {{\pmb{\mathrm{L}}}}D \circ T = {{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak m}}\circ {{\pmb{\mathrm{L}}}}D \circ T, \end{aligned}$$ as by \[prop:local-coh-commutes-with-direct-limits\], local cohomology commutes with direct limits, ${{\pmb{\mathrm{R}}}}\Gamma_{\mathfrak m}$ and ${{\pmb{\mathrm{R}}}}\Gamma_{\underline{I}}$ commute by \[prop:spectral-sequence-for-local-cohomology-and-sum-of-ideals\] and two choice applications of \[prop:local-coh-of-torus-is-tate-coh-of-matlis\]. If we express \[prop:jannsen-spec-seq-in-der-cat\] in terms of a spectral sequence, it looks like this: $$\varinjlim_k{{\pmb{\mathrm{L}}}}^pD(T(\operatorname{Ext}^q_R(R/\mathfrak m^k, M))) \Longrightarrow T(\operatorname{Ext}^{t+1-p-q}_\Lambda(M,\Lambda)).$$ Writing $\operatorname{E}^\bullet_\Lambda$ for $\operatorname{Ext}^\bullet_\Lambda(-,\Lambda)$, flipping the sign of $p$ and shifting $q\mapsto t+1-q$ then yields $$\varinjlim_k D_p( \operatorname{Ext}^{t+1-q}_R(R/\mathfrak m^k, M)^\vee) \Longrightarrow \operatorname{E}^{p+q}_\Lambda(M)^\vee$$ and the following exact five term sequence: $$\begin{tikzcd}\operatorname{E}^2_\Lambda(M)^\vee \arrow{r} & \varinjlim_k D_2(\operatorname{Ext}_R^{t+1}(R/\mathfrak m^k, M)^\vee) \arrow{r} & \varinjlim_k D(\operatorname{Ext}_R^{t}(R/\mathfrak m^k, M)^\vee) \arrow[out=355, in=175, looseness=1.5, overlay]{lld} \\ \operatorname{E}^1_\Lambda(M)^\vee \arrow{r} & \varinjlim_k D_1(\operatorname{Ext}_R^{t+1}(R/\mathfrak m^k, M)^\vee) \arrow{r} & 0\end{tikzcd}$$ Iwasawa Adjoints {#sec:iwasawa-adjoints} ================ In this section let $R$ be a pro-$p$ commutative local ring with maximal ideal $\mathfrak m$ and residue field of characteristic $p$. Let $G$ be a compact $p$-adic Lie group and $\Lambda=\Lambda(G)=\varprojlim_U R[[G/U]]$, where $U$ ranges over the open normal subgroups of $G$. As is customary, we set again $\operatorname{E}^\bullet_\Lambda(M) = \operatorname{Ext}^\bullet_\Lambda(M,\Lambda)$. Even though $R$ is commutative, we will differentiate between $R$- and $R^\circ$-modules, as we will regard $R$ as a subring of $\Lambda$. Note that if $M$ is a left $\Lambda$-module, it also has an operation of $\Lambda$ from the right given by $mg=g^{-1}m$. This of course does not give $M$ the structure of a $\Lambda$-bimodule, as the actions are not compatible. We can however still give $\operatorname{Hom}_\Lambda(M,\Lambda)$ the structure of a left $\Lambda$-module by $(g.\varphi)(m)= \varphi(m)g^{-1}$. \[prop:e0-as-proj-lim-over-coinv\] $\operatorname{E}^0_\Lambda(M) = \varprojlim_U \operatorname{Hom}_R(M_U, R)$ for finitely generated $\Lambda$-modules $M$, where the transition map for a pair of open normal subgroups $U\leq V$ are given by the dual of the trace map $M_V\ra M_U, m\mapsto\sum_{g\in V/U} gm$. Note first that as $\operatorname{Hom}_\Lambda(M,-)$ commutes with projective limits, $$\operatorname{Hom}_\Lambda(M,\Lambda) = \varprojlim_U \operatorname{Hom}_\Lambda(M, R[G/U]) = \varprojlim_U \operatorname{Hom}_{R[G/U]}(M_U, R[G/U]).$$ For $U$ an open normal subgroups of $G$, consider the trace map $$\begin{gathered} \operatorname{Hom}_R(M_U, R)\ra\operatorname{Hom}_{R[G/U]}(M_U, R[G/U])\\ \varphi\mapsto \left(m\mapsto\sum_{g\in G/U} \varphi(g^{-1}m)\cdot g\right),\end{gathered}$$ which is clearly an isomorphism of $R$-modules and induces the required isomorphism to the projective system mentioned in the proposition. \[prop:pont-of-iw-adjoints-is-colim-pont-rhom-inv\] On ${{\pmb{\mathrm{D}}}}^b_c({\Lambda\textrm{-}{\pmb{\mathrm{Mod}}}})$ we have $$\Pi \circ \operatorname{{\pmb{\mathrm{R}}}Hom}_\Lambda(-, \Lambda) \cong \varinjlim_U \Pi \circ \operatorname{{\pmb{\mathrm{R}}}Hom}_R(-,R) \circ {{\pmb{\mathrm{L}}}}(-)_{U}.$$ Immediate by \[prop:e0-as-proj-lim-over-coinv\], as $(-)_U$ clearly maps finitely generated free $\Lambda$-modules to finitely generated free $R$-modules. The spectral sequence attached to \[prop:pont-of-iw-adjoints-is-colim-pont-rhom-inv\] looks like this: $$\varinjlim_U \operatorname{Ext}_R^p(H_{q}(U, M), R)^\vee \Longrightarrow \operatorname{E}^{p+q}_\Lambda(M)^\vee$$ Its five term exact sequence is given by $$\begin{tikzcd} \operatorname{E}^2_\Lambda(M)^\vee \arrow{r} & \varinjlim_U \operatorname{Ext}^2_R(M_U, R)^\vee \arrow{r} & \varinjlim_U \operatorname{Hom}_R(H_1(U,M), R)^\vee \arrow[looseness=1.5, overlay, out=355, in=175]{dll} \\ \operatorname{E}^1_\Lambda(M)^\vee \arrow{r} & \varinjlim_U \operatorname{Ext}^1_R(M_U, R)^\vee \arrow{r} & 0 \end{tikzcd}$$ \[prop:iw-dual-as-colim-of-coinv-otimes-r-vee\] $\operatorname{Hom}_\Lambda(M,\Lambda)^\vee \cong \varinjlim_U R^\vee \otimes_R M_U$ for finitely generated $\Lambda$-modules $M$. $$\begin{aligned} \operatorname{Hom}_\Lambda(M,\Lambda)^\vee & \cong \varinjlim_U \Pi(\operatorname{Hom}_R(M_U, R)) \\ & \cong\varinjlim_U \Pi\operatorname{Hom}_{R^\circ}(\Pi(R), \Pi(M_U))\\ &\cong \Pi \circ \Pi( \varinjlim_U M_U\otimes_{R^\circ} \Pi(R))\\ &\cong \varinjlim M_U \otimes_{R^\circ} R^\vee. \end{aligned}$$ \[prop:pont-of-iw-adjoints-is-tor-tate-pont\] $\Pi \circ \operatorname{{\pmb{\mathrm{R}}}Hom}_\Lambda(-,\Lambda) \cong \left(R^\vee \otimes^{{\pmb{\mathrm{L}}}}_R -\right) \circ {{\pmb{\mathrm{L}}}}D \circ \Pi$ on ${{\pmb{\mathrm{D}}}}^b_c({\Lambda\textrm{-}{\pmb{\mathrm{Mod}}}})$. Using \[prop:e0-as-proj-lim-over-coinv,prop:iw-dual-as-colim-of-coinv-otimes-r-vee\], usual Pontryagin duality (where we occasionally write $-^\vee$ for the dual), and the fact that tensor products commute with direct limits, we get: $$\begin{aligned} \operatorname{Hom}_\Lambda(M,\Lambda)^\vee & \cong \varinjlim_U R^\vee \otimes_R M_U\\ &\cong R^\vee \otimes_R \varinjlim \Pi(\Pi(M)^U)\\ &= R^\vee \otimes_R D(\Pi(M)). \end{aligned}$$ It hence remains to show that $(D\circ \Pi)$ maps projective objects to $R^\vee \otimes_R -$-acyclics and it actually suffices to check this for the module $\Lambda$. But $D(\Pi(\Lambda))=\varinjlim_U R[G/U]$ is clearly $R^\vee \otimes_R -$-acyclic. The spectral sequence attached to \[prop:pont-of-iw-adjoints-is-tor-tate-pont\] looks like this: $$\operatorname{Tor}_{p}^R(R^\vee,D_{q}(M^\vee)) \Longrightarrow \operatorname{E}^{p+q}_\Lambda(M)^\vee,$$ which yields the following five term exact sequence: $$\begin{tikzcd} \operatorname{E}^2_\Lambda(M)^\vee \arrow{r} & \operatorname{Tor}_2^R(R^\vee, D(M^\vee)) \arrow{r} & R^\vee \otimes_R D_1(M^\vee) \arrow[looseness=1.5, overlay, out=355, in=175]{lld} \\ \operatorname{E}^1_\Lambda(M)^\vee \arrow{r} & \operatorname{Tor}_1^R(R^\vee, D(M^\vee)) \arrow{r} & 0 \end{tikzcd}$$ This also proves that $\operatorname{E}^p_\Lambda(M)=0$ if $p>\dim G + \dim R$. If $\dim R=1$, the spectral sequence degenerates and we can compute $\operatorname{E}^p_\Lambda(\operatorname{tor}_R M)$ and $\operatorname{E}^p_\Lambda(M/\operatorname{tor}_R M)$ akin to [@MR2392026 (5.4.13)]. \[prop:r-vee-is-colim-over-residues\] $R^\vee \cong \varinjlim_k R/\mathfrak m^k$ if $R$ is regular. $R$ satisfies local duality by assumption, hence by \[prop:pont-is-matlis-for-fg-and-art-mods,prop:duality-ext-tor-for-reg-seq-classical\] $R^\vee = T(R) \cong {{\pmb{\mathrm{R}}}}^d\Gamma_{\underline{\mathfrak m}}(R) = \varinjlim_k \operatorname{Ext}^d_R(R/\mathfrak m^{(k)}, R) \cong \varinjlim_k R/\mathfrak m^{(k)},$ where we again denote with $\mathfrak m^{(k)}$ the ideal generated by $k$th powers of generators of $\mathfrak m$. \[prop:iw-dual-as-colim-of-coniv-of-residues\] $\operatorname{Hom}_\Lambda(M,\Lambda)^\vee \cong \varinjlim_{U,k} (M/\mathfrak m^k)_U$ for finitely generated $\Lambda$-modules $M$ and regular $R$. $$\begin{aligned} \operatorname{Hom}_\Lambda(M,\Lambda)^\vee &\cong R^\vee \otimes_R \varinjlim_U M_U \\ &\cong \varinjlim_U \varinjlim_k R/\mathfrak m^k \otimes_R M_U \\ &\cong \varinjlim_U \varinjlim_k (M/\mathfrak m^k M)_U\end{aligned}$$ using \[prop:iw-dual-as-colim-of-coinv-otimes-r-vee,prop:r-vee-is-colim-over-residues\]. \[prop:pont-of-iw-adjoints-is-colim-of-tate-of-pont-of-loc-coh\] If $R$ is regular, $ \Pi \circ \operatorname{{\pmb{\mathrm{R}}}Hom}_\Lambda(-, \Lambda) \cong \varinjlim_k {{\pmb{\mathrm{L}}}}D \circ \Pi \circ \left(R/\mathfrak m^k \otimes^{{\pmb{\mathrm{L}}}}_R -\right) \cong \varinjlim_k {{\pmb{\mathrm{L}}}}D \circ \Pi \circ [d] \circ \operatorname{{\pmb{\mathrm{R}}}Hom}_R(R/\mathfrak m^k, -)$ on ${{\pmb{\mathrm{D}}}}^b_c({\Lambda\textrm{-}{\pmb{\mathrm{Mod}}}}).$ By \[prop:iw-dual-as-colim-of-coniv-of-residues\] $$\begin{aligned} \operatorname{Hom}_\Lambda(M,\Lambda)^\vee &\cong \varinjlim_U \varinjlim_k (M/\mathfrak m^k M)_U \\ &\cong \varinjlim_{U,k} \Pi(\Pi(M/\mathfrak m^k M)^U)\\ &= \varinjlim_k D(\Pi(R/\mathfrak m^k \otimes_R M)). \end{aligned}$$ By \[prop:duality-rhom-tor-for-reg-seq-in-der-cat\] it suffices to show that $(\Lambda/\mathfrak m^k)^\vee$ is $D$-acyclic. But $${{\pmb{\mathrm{L}}}}^{-i}D((\Lambda/\mathfrak m^k)^\vee) = \varinjlim_U H^i(U, R/\mathfrak m^k[[G]]^\vee)^* = \varinjlim_U H_i(U, R/\mathfrak m^k[[G]])=0$$for $i>0$ by Shapiro’s Lemma. The other isomorphism now follows from \[prop:duality-rhom-tor-for-reg-seq-in-der-cat\]. The spectral sequences attached to \[prop:pont-of-iw-adjoints-is-colim-of-tate-of-pont-of-loc-coh\] look like this: $$\varinjlim_k D_{p}(\operatorname{Tor}_q^R(R/\mathfrak m^k, M)^\vee) \Longrightarrow \operatorname{E}^{p+q}_\Lambda(M)^\vee$$ and $$\varinjlim_k D_{p}(\operatorname{Ext}_R^{d-q}(R/\mathfrak m^k, M)^\vee) \Longrightarrow \operatorname{E}^{p+q}_\Lambda(M)^\vee$$ with exact five term sequences $$\begin{tikzcd} \operatorname{E}^2_\Lambda(M)^\vee \arrow{r} & \varinjlim_k D_2( (M/\mathfrak m^kM )^\vee)\arrow{r} & \varinjlim_k D(\operatorname{Tor}_1^R(R/\mathfrak m^k, M)^\vee) \arrow[looseness=1.5, overlay, out=355, in=175]{lld} \\ \operatorname{E}^1_\Lambda(M)^\vee \arrow{r} & \varinjlim_k D_1((M/\mathfrak m^kM)^\vee) \arrow{r} & 0 \end{tikzcd}$$ and $$\begin{tikzcd} \operatorname{E}^2_\Lambda(M)^\vee \arrow{r} & \varinjlim_k D_2(\operatorname{Ext}^d_R(R/\mathfrak m^k, M)^\vee)\arrow{r} & \varinjlim_k D(\operatorname{Ext}^{d-1}_R(R/\mathfrak m^k, M)^\vee) \arrow[looseness=1.5, overlay, out=355, in=175]{lld} \\ \operatorname{E}^1_\Lambda(M)^\vee \arrow{r} & \varinjlim_k D_1(\operatorname{Ext}^d_R(R/\mathfrak m^k , M)^\vee) \arrow{r} & 0 \end{tikzcd}$$ respectively. \[prop:fg-lambda-mods-are-complete-and-sep\] Let $M$ be a finitely generated $\Lambda$-module. Then $M\cong\varprojlim_k M/\mathfrak M^kM$ algebraically and topologically. \[prop:pont-of-rhom-r-mod-m-is-disc-p-tor\] $\Pi \circ \operatorname{{\pmb{\mathrm{R}}}Hom}_R(R/\mathfrak m^k, -)$ maps bounded complexes of $\Lambda$-modules with coherent cohomology to bounded complexes whose cohomology modules are discrete $p$-torsion $G$-modules. If $M$ is a complex in $\varinjlim_U {{\pmb{\mathrm{D}}}}^b_c({R[G/U]\textrm{-}{\pmb{\mathrm{Mod}}}})$, then all cohomology groups of $\operatorname{{\pmb{\mathrm{R}}}Hom}_R(R/\mathfrak m^k, M)^\vee $ are furthermore finite. The groups $\operatorname{Ext}^q_R(R/\mathfrak m^k, M)$ for $M$ finitely generated over $\Lambda$ are clearly $p$-torsion and finitely generated as $\Lambda$-modules, hence compact by \[prop:fg-lambda-mods-are-complete-and-sep\], and consequentially topologically profinite and pro-$p$. Their Pontryagin duals are thus discrete $p$-torsion $G$-modules. If $M$ is finitely generated over some $R[G/U]$, it is also finitely generated over $R$ and $\operatorname{Ext}^q_R(R/\mathfrak m^k, M)$ finitely generated over $R/\mathfrak m^k$, hence finite. \[prop:pont-of-iw-adjoint-with-duality-grp-has-deg-spec-seq\] Assume that $R$ is regular. Let $G$ be a duality group (cf. [@MR2392026 (3.4.6)]) of dimension $s$ at $p$. Then $$\begin{aligned} \operatorname{E}^m_\Lambda(M)^\vee &\cong \varinjlim_k {{\pmb{\mathrm{L}}}}^{-s} D \operatorname{Ext}^{d-(m-s)}_R(R/\mathfrak m^k, M)^\vee \\ & \cong \varinjlim_k {{\pmb{\mathrm{L}}}}^{-s} D \operatorname{Tor}_{m-s}^R(R/\mathfrak m^k, M)^\vee.\end{aligned}$$ for finitely generated $R[G/U]$-modules $M$. Especially $\Pi\circ \operatorname{E}^s_\Lambda$ is then right-exact. As $G$ is a duality group of dimension $s$ at $p$, the complex ${{\pmb{\mathrm{L}}}}D(M')$ has trivial cohomology outside of degree $-s$ if $M'$ is a finite discrete $p$-torsion $G$-module. Together with \[prop:pont-of-rhom-r-mod-m-is-disc-p-tor\] this implies that the spectral sequence attached to \[prop:pont-of-iw-adjoints-is-colim-of-tate-of-pont-of-loc-coh\] degenerates and gives $$\operatorname{E}^m_\Lambda(M)^\vee \cong \varinjlim_k {{\pmb{\mathrm{L}}}}^{-s} D \operatorname{Ext}^{d-(m-s)}_R(R/\mathfrak m^k, M)^\vee.$$ The other isomorphism follows with exactly the same argument. Note furthermore that as $\dim R=d$, $\operatorname{Ext}^{d-(m-s)}_R(R/\mathfrak m^k, M)=0$ if $m-s<0$, hence $\operatorname{E}^m_\Lambda(M)^\vee=0$ if $m<s$. \[prop:dual-of-iw-adj-is-twist-of-local-coh-of-coeff\] Assume that $R$ is regular and that $G$ is a Poincaré group at $p$ of dimension $s$ with dualizing character $\chi\colon G\ra {\mathbb Z}_p^\times$ (cf.[@MR2392026 (3.7.1)]), which gives rise to the “twisting functor” ${\pmb{\mathrm{\chi}}}\colon M\mapsto M(\chi)$. Assume that $R$ is a commutative complete Noetherian ring of global dimension $d$ with maximal ideal $\mathfrak m$. Then $$T \circ \operatorname{{\pmb{\mathrm{R}}}Hom}_\Lambda(-,\Lambda) = {\pmb{\mathrm{\chi}}} \circ [d+s] \circ {{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak m}}$$ on $\varinjlim_U {{\pmb{\mathrm{D}}}}^b_c({R[G/U]\textrm{-}{\pmb{\mathrm{Mod}}}}).$ Let $I={\mathbb Q}_p/{\mathbb Z}_p(\chi)={\pmb{\mathrm{\chi}}}({\mathbb Q}_p/{\mathbb Z}_p)$ be the dualizing module of $G$. For any $p$-torsion $G$-module $A$ we have by [@MR2392026 (3.7)] that ${{\pmb{\mathrm{L}}}}^{-s}D(A) = \varinjlim_U H^s(U,A)^* \cong \varinjlim_U \operatorname{Hom}_{{\mathbb Z}_p}(A,I)^U = \operatorname{Hom}_{{\mathbb Z}_p}(A,I)$, as $I$ is also a dualizing module for every open subgroup of $G$. Note that $$H^0 ( {\pmb{\mathrm{\chi}}} \circ [d] \circ {{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak m}}) = {\pmb{\mathrm{\chi}}}\circ {{\pmb{\mathrm{R}}}}^{d}\Gamma_{\underline{\mathfrak m}}$$ and that ${\pmb{\mathrm{\chi}}}\circ{{\pmb{\mathrm{R}}}}^{d}\Gamma_{\underline{\mathfrak m}}$ is hence right-exact. Note furthermore that $$\begin{aligned} H^0( [-s]\circ T \circ \operatorname{{\pmb{\mathrm{R}}}Hom}_\Lambda(-,\Lambda)) &= \operatorname{E}^s(-)^\vee \\ &\cong \varinjlim_k {{\pmb{\mathrm{L}}}}^{-s}D \operatorname{Ext}^d_R(R/\mathfrak m^k, -)^\vee \\ &\cong \varinjlim_k \operatorname{Hom}_{{\mathbb Z}_p} (\operatorname{Ext}^d_R(R/\mathfrak m^k, -)^\vee, I)\\ &\cong {\pmb{\mathrm{\chi}}}\circ{{\pmb{\mathrm{R}}}}^d\Gamma_{\underline{\mathfrak m}} \end{aligned}$$ using \[prop:pont-of-iw-adjoint-with-duality-grp-has-deg-spec-seq\] and Pontryagin duality. By [@MR0222093 I.7.4] the left derivation of ${{\pmb{\mathrm{R}}}}^d\Gamma_{\underline{\mathfrak m}}$ is $[d]\circ{{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak m}}$: The complex ${{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak m}}(R)$ is concentrated in degree $d$ and hence every module is a quotient of a module with this property, as local cohomology commutes with arbitrary direct limits. Note that even though this theorem suspiciously looks like local duality, the local cohomology on the right hand side is with respect to the maximal ideal of the coefficient ring, not the whole Iwasawa algebra. The local duality result is subject of the next section. \[prop:explicit-iw-adj\] In the setup of \[prop:dual-of-iw-adj-is-twist-of-local-coh-of-coeff\] assume that $M$ is a finitely generated $R[G/U]$-module. Then the following hold: 1. If $M$ is free over $R$, then $$\operatorname{E}_\Lambda^q(M)^\vee \cong \begin{cases}M\otimes_R R^\vee(\chi) & \text{if } q=s\\ 0 &\text{else}\end{cases}$$ 2. If $M$ is $R$-torsion, then $\operatorname{E}_\Lambda^q(M)=0$ for all $q\leq s$. 3. If $M$ is finite, then $$\operatorname{E}_\Lambda^q(M)^\vee \cong \begin{cases}M(\chi) & \text{if } q=d+s\\ 0 &\text{else}\end{cases}$$ By \[prop:dual-of-iw-adj-is-twist-of-local-coh-of-coeff\], we have $$\operatorname{E}_\Lambda^q(M)^\vee \cong {{\pmb{\mathrm{R}}}}^{d+s-q}\Gamma_{\underline{\mathfrak m}}(M)(\chi)$$ in any case. In the first case, this is just $M\otimes_R R^\vee$ for $q=s$ and zero else. In the second case, local duality yields ${{\pmb{\mathrm{R}}}}^d\Gamma_{\underline{\mathfrak m}}(M)\cong \operatorname{Hom}_R(M,R)^\vee=0$. In the third case, we note that $M$ has an injective resolution by modules that are the direct limit of finite modules. together with \[prop:loc-coh-is-max-fin-submod\] then imply the result. Local Duality for Iwasawa Algebras {#sec:local-dual-iwas} ================================== This section gives a streamlined proof of a local duality result for Iwasawa algebras as first published in and generalizes the result to more general coefficient rings. Let $G$ be a pro-$p$ Poincaré group of dimension $s$ with dualizing character $\chi\colon G\ra{\mathbb Z}_p^\times$ and $R$ a commutative Noetherian pro-$p$ regular local ring with maximal ideal $\mathfrak m$ of global dimension $d$. Set $\Lambda=R[[G]]$. \[prop:g-poincare-then-top-loc-coh-is-pont-dual-for-ring\] ${{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak M}}(\Lambda)\cong\Lambda^\vee[-d-s]$ and $\operatorname{Ext}^i_\Lambda(\Lambda/\mathfrak M^l,\Lambda)\cong {{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak m}}^{d+s-i}(\Lambda/\mathfrak M^l)(\chi).$ By \[prop:pont-of-iw-adjoint-with-duality-grp-has-deg-spec-seq\], $${{\pmb{\mathrm{R}}}}^{i}\Gamma_{\underline{\mathfrak M}}(\Lambda) = \varinjlim_l \operatorname{E}^i(\Lambda/\mathfrak M^l) \cong \varinjlim_{l} \left( \varinjlim_k {{\pmb{\mathrm{L}}}}^{-s}D (\operatorname{Ext}^{d-(i-s)}_R(R/\mathfrak m^k, \Lambda/\mathfrak M^l)^\vee)\right)^\vee.$$ As in the proof of \[prop:dual-of-iw-adj-is-twist-of-local-coh-of-coeff\], $$\begin{aligned} {{\pmb{\mathrm{L}}}}^{-s}D (\operatorname{Ext}^{d-(i-s)}_R(R/\mathfrak m^k, \Lambda/\mathfrak M^l)^\vee) &\cong \operatorname{Hom}_{{\mathbb Z}_p}(\operatorname{Ext}^{d-(i-s)}_R(R/\mathfrak m^k, \Lambda/\mathfrak M^l)^\vee, {\mathbb Q}_p/{\mathbb Z}_p(\chi))\\ &\cong \operatorname{Ext}^{d-(i-s)}_R(R/\mathfrak m^k, \Lambda/\mathfrak M^l)(\chi)\end{aligned}$$ and in the direct limit over $k$ this becomes ${{\pmb{\mathrm{R}}}}\Gamma^{d-(i-s)}_{\underline{\mathfrak m}}(\Lambda/\mathfrak M^l)(\chi)$. $\Gamma_{\underline{\mathfrak m}}$ restricted to the subcategory of finite $\Lambda$- (or $R$-)modules is the identity. As $\Gamma_{\underline{\mathfrak m}}$ commutes with arbitrary direct limits, this is also true for the category of discrete $\Lambda$-modules. As the latter category contains sufficiently many injectives, ${{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak m}}(N)=N$ if $N$ is a complex of discrete $\Lambda$-modules. Now $\Lambda/\mathfrak M^l$ is such a finite module, hence $${{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak M}}(\Lambda) = [-d-s]\circ \varinjlim_l ({{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak m}}(\Lambda/\mathfrak M^l)(\chi))^\vee = \Lambda(\chi)^\vee[-r].$$The proposition now follows at once if we observe that $\Lambda\cong\Lambda(\chi)$ as a $\Lambda$-module via $g\mapsto \chi(g)g$. \[thm:local-duality-for-iw-alg\] $${{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak M}} \cong [-r] \circ \Pi \circ \operatorname{{\pmb{\mathrm{R}}}Hom}_{\Lambda}(-,\Lambda)$$ on ${{\pmb{\mathrm{D}}}}^b_c(\Lambda)$. Because of \[prop:g-poincare-then-top-loc-coh-is-pont-dual-for-ring\], this follows verbatim as in [@MR0224620 theorem 6.3]: The functors ${{\pmb{\mathrm{R}}}}^r\Gamma_{\underline{\mathfrak M}}$ and $\operatorname{Hom}_\Lambda(-,\Lambda)^\vee$ are related by a pairing of $\operatorname{Ext}$-groups, are both covariant and right-exact and agree on $\Lambda$, hence also agree on finitely generated modules. As the complex ${{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak M}}(\Lambda)$ is concentrated in degree $r$, the same argument as in \[prop:dual-of-iw-adj-is-twist-of-local-coh-of-coeff\] shows that the left derivation of ${{\pmb{\mathrm{R}}}}^r\Gamma_{\underline{\mathfrak M}}$ is just $[r]\circ {{\pmb{\mathrm{R}}}}\Gamma_{\underline{\mathfrak M}}$ and the result follows. Torsion in Iwasawa Cohomology ============================= There are notions of both local and global Iwasawa cohomology. Our result about their torsion below holds in both cases and we will first deal with the local case. In both subsections, $R$ is a commutative Noetherian pro-$p$ local ring of residue characteristic $p$. Torsion in Local Iwasawa Cohomology ----------------------------------- Let $K$ be a finite extension of ${\mathbb Q}_p$ and $K_\infty|K$ a Galois extension with an analytic pro-$p$ Galois group $G$ without elements of finite order. Let $T$ be a finitely generated $\Lambda=R[[G]]$-module and set set $A=T\otimes_R R^\vee$. Due to Lim and Sharifi we have the following spectral sequence (stemming from an isomorphism of complexes in the derived category). Write $H^i_{\mathrm{Iw}}(K_\infty, T) = \varprojlim_{K'} H^{i}(G_{K'}, T)$ where the limit is taken with respect to the corestriction maps over all finite field extensions $K'|K$ contained in $K_\infty.$ \[thm:lim-sharifi-spec-seq-local\] There is a convergent spectral sequence $$\operatorname{E}^i_\Lambda(H^j(G_K, A)^\vee) \Longrightarrow H^{i+j}_{\mathrm{Iw}}(K,T).$$ This is [@MR3084561 theorem 4.2.2, remark 4.2.3]. \[thm:torsion-in-local-iw-coh-for-poinc-grps\] If $G$ is a pro-$p$ Poincaré group of dimension $s\geq 2$ with dualizing character $\chi\colon G\ra{\mathbb Z}_p^\times$ and if $R$ is regular, then $$\operatorname{tor}_\Lambda H^1_{\mathrm{Iw}}(K_\infty, T) = 0.$$ If $s=1$, then $$\operatorname{tor}_\Lambda H^1_{\mathrm{Iw}}(K_\infty, T) \cong \operatorname{Hom}_R( (T^*)_G, R)(\chi^{-1}),$$ where $T^* = \operatorname{Hom}_R(T,R)$. The exact five-term sequence attached to \[thm:lim-sharifi-spec-seq-local\] starts like this: $$\begin{tikzcd} 0 \arrow{r} & \operatorname{E}^1_\Lambda(H^0(G_K,A)^\vee) \arrow{r} & H^1_{\mathrm{Iw}}(K_\infty, T) \arrow{r} & \operatorname{E}^0_\Lambda(H^1(G_K,A)^\vee) \arrow{d}\\ & & & \operatorname{E}^2_\Lambda(H^0(G_K,A)^\vee) \end{tikzcd}$$ Note that $\operatorname{E}^2_\Lambda(M)$ is pseudo-null and hence $\Lambda$-torsion for every finitely generated module $M$, which follows from the spectral sequence attached to the isomorphism $\operatorname{{\pmb{\mathrm{R}}}Hom}_\Lambda(-,\Lambda)\circ \operatorname{{\pmb{\mathrm{R}}}Hom}_\Lambda(-,\Lambda) \cong \operatorname{id}$ on ${{\pmb{\mathrm{D}}}}^b_c({\Lambda\textrm{-}{\pmb{\mathrm{Mod}}}})$. Furthermore $\operatorname{E}^0_\Lambda(M)$ is always $\Lambda$-torsion free, as $\Lambda$ is integral. It follows that $\operatorname{tor}_\Lambda H^1_{\mathrm{Iw}}(K_\infty, T) \subseteq \operatorname{E}^1_\Lambda(H^0(G_K,A)^\vee)$. As the latter is $\Lambda$-torsion, $$\operatorname{tor}_\Lambda H^1_{\mathrm{Iw}}(K_\infty, T) = \operatorname{E}^1_\Lambda(H^0(G_K,A)^\vee).$$ The result now follows immediately from \[prop:explicit-iw-adj\] Torsion in Global Iwasawa Cohomology ------------------------------------ Let $K$ be a finite extension of ${\mathbb Q}$ and $S$ a finite set of places of $K$. Let $K_S$ be the maximal extension of $K$ which is unramified outside $S$ and $K_\infty|K$ a Galois extension contained in $K_S$. Suppose that $G=G(K_\infty|K)$ is an analytic pro-$p$ group without elements of finite order. Let $T$ be a finitely generated $\Lambda=R[[G]]$-module and set $A=T\otimes_R R^\vee$. Due to Lim and Sharifi we have the following spectral sequence (stemming from an isomorphism of complexes in the derived category). Write $H^i_{\mathrm{Iw}}(K_\infty, T) = \varprojlim_{K'} H^{i}(G(K_S|K'), T)$ where the limit is taken with respect to the corestriction maps over all finite field extensions $K'|K$ contained in $K_\infty.$ \[thm:lim-sharifi-spec-seq-global\] There is a convergent spectral sequence $$\operatorname{E}^i_\Lambda(H^j(G_S, A)^\vee) \Longrightarrow H^{i+j}_{\mathrm{Iw}}(K_\infty, T).$$ This follows from [@MR3084561 theorem 4.5.1]. From this we derive the following. \[thm:torsion-in-global-iw-coh-for-poinc-grps\] If $G$ is a pro-$p$ Poincaré group of dimension $s\geq 2$ with dualizing character $\chi\colon G\ra{\mathbb Z}_p^\times$ and if $R$ is regular, then $$\operatorname{tor}_\Lambda H^1_{\mathrm{Iw}}(K_\infty, T) = 0.$$ If $s=1$, then $$\operatorname{tor}_\Lambda H^1_{\mathrm{Iw}}(K_\infty, T) \cong \operatorname{Hom}_R( (T^*)_G, R)(\chi^{-1}),$$ where $T^* = \operatorname{Hom}_R(T,R)$. Replace “$G_K$” with “$G_S$” in the proof of \[thm:torsion-in-local-iw-coh-for-poinc-grps\].

--- abstract: 'Let $\K=\Q(\sqrt{d_1},\ldots, \sqrt{d_k})$ be a polyquadratic number field and $N$ be a squarefree positive integer with at least $k$ distinct factors. The Galois group, $\Gal(\K/\Q)$ is an elementary abelian two group generated by $\sigma_i$ such that $g_i(\sqrt{d_i})=-\sqrt{d_i}$. Let $\zeta:\Gal(\K/\Q) \rightarrow \Aut(X_0(N))$ be the cocycle that sends $\sigma_i$ to $w_{m_i}$ where $w_{m_i}$ are the Atkin-Lehner involutions of $\X$. In this paper, we study the $\Q_p$-rational points of the twisted modular curve $\Xc$ and give an algorithm to produce such curves which has $\Q_p$-rational points for all primes $p$. Then we investigate violations of the Hasse Principle for these curves and give an asymptotic for the number of such violations. Finally, we study reasons of such violations.' address: | Department of Mathematics\ University of Texas-Austin\ Texas, USA author: - Ekin Ozman title: 'On Polyquadratic Twists of $X_0(N)$' --- Introduction ============ Given $(m_1, \ldots, m_k)$ pairwise relatively prime, squarefree, positive integers and $(d_1,\ldots, d_k)$ relatively prime squarefree integers we construct a twisted modular curve $\Xc$ as follows: Let $N=\Pi_{i=1}^k m_i$ and $\K=\Q(\sqrt{d_1},\ldots, \sqrt{d_k})$. The Galois group of $\K/\Q$ is elementary abelian $2$ group generated by $\sigma_i$ for $1 \leq i \leq k$. The automorphism group of the modular curve $\X$ is generated by the Atkin-Lehner involutions $w_{p_i}$ for each $p_i|N$. Let $\zeta:\Gal(\K/\Q) \rightarrow \Aut(X_0(N))$ be the cocycle that sends $\sigma_i$ to $w_{m_i}$. The curve $\Xc$ is the twist of $\X$ by $\zeta$. In particular rational points of $\Xc$ are $\K$-rational point of $\X$ fixed by $\sigma_i \circ w_{m_i}$ for each $i$ in between $1$ and $k$. Like $\X$, the twisted curve $\Xc$ is also a moduli space. Rational points of $\Xc$ parametrize $\Q$-curves. Recall that a $\Q$-curve is an elliptic curve $E$ defined over a number field $\K$ which is isogenous to all of its Galois conjugates. It is a result of Elkies that every $\Q$-curve is geometrically isogenous to a $\Q$-curve defined over a polyquadratic field i.e. a field that is generated by quadratic fields. Therefore understanding rational points of $\Xc$ gives information about $\Q$-curves as well. Then one can naturally ask the following question, a particular case of which was first stated in [@Ell] and answered in [@Ozman] : \[Q1\] Given pairwise relatively prime, squarefree, positive integers $(m_1, \ldots, m_k)$, relatively prime squarefree integers $(d_1,\ldots, d_k)$ and a prime number $p$, what can be said about the points in $\Xc(\Q_p)$ where $N=\Pi_{i=1}^k m_i$, $\K=\Q(\sqrt{d_1},\ldots, \sqrt{d_k})$ and $\zeta$ is as described above? Understanding local points is the first step towards understanding global points of any curve. If it happens to be the case that $\Xc(\Q_p)$ is empty for some $p$, then $\Xc(\Q)$ is also empty and there is no such $\Q$-curve. However, having $\Q_p$-points for every prime $p$ does not guarentee the existence of $\Q$-points unless the genus of the curve is zero. For instance, in the case of quadratic twists there are many examples which have local points, but no global points. It is even possible to give an exact asymptotic formula for the number of such curves [@Ozman]. This raises the following question: Is there an asymptotic for the number of twists $\Xc$ which violate the Hasse principle? We address these two questions in the first five sections. In the last section, we discuss further directions and reasons of violations of the Hasse principle. More precisely, in Sections 2 and 3 we give conditions on the existence of local points. These conditions depend on splitting behavior of the prime $p$ in the given polyquadratic field $\K$. In some cases we are able to give necessary and sufficient conditions, in other cases we have only sufficient ones. However, this still allows us to give an algorithm which produces infinitely many $\Xc$ with local points everywhere, for any given $N$, as summarized in Section 4. In Section 5, we use this algorithm combined with the methods of [@Ozman] and [@Clark], and give an asymptotic formula for the number of biquadratic twists which has local points everywhere but no global points. In fact this can be generalized to higher degree twists as well. Section 6 is about further directions in this problem. For the twists with computationally feasible equations, we try to explain the lack of global points using Mordell Weil sieve. This is equivalent to Brauer Manin obstruction by the work of Scharaschkin [@sch]. One can not apply Mordell Weil sieve if $\Pic^1(\Xc)(\Q)$ is empty. Understanding the Picard group is usually hard for a generic curve. Note that we even do not have equations for an arbitrary member of the twisted family. Given $N$, we give sufficient conditions for $\Pic^1(\Xc)(\Q)$ to be nonempty in the case of quadratic and some biquadratic cases. We also find families of cases where $\Pic^1(\Xc)(\Q_p)=\emptyset$ when $p$ and $N$ satisfy certain arithmetic conditions. During the course of typing these results, the PhD thesis of Jim Stankewicz has been brought to author’s attention. Many of the results in Sections 2 and 3 can be concluded from this thesis. However, we still included them here since it may be hard to derive these results from [@Jim] for a reader who is not familiar with the subject and also the work was independent. The case of good reduction ========================== $p$ is unramified in $K$ ------------------------ : In this case the extension is unramified therefore we can use the theory of Galois descent. Since $p$ is a good prime, $\Xc$ has a smooth model over $\Z_p$. Therefore, by Hensel’s Lemma, $\Xc(\Q_p)$ is non-empty if and only if $\Xc(\F_p)$is non-empty. Let $\mathcal{P}$ be a prime of $\K$ lying over $p$. According to our notation given in the introduction section, each Galois map $\sigma_i$ is twisted by some $w_{m_i}$. Let $S$ be the set of indices $i$ such that $p$ is inert in $\Q(\sqrt{d_i})$. Then decomposition group of $\mathcal{P}$ is $\{1, \prod\limits_{i \in S} \sigma_i \}$. Note that since $\K/\Q$ is an abelian Galois extension, it doesn’t matter which $\mathcal{P}$ we choose. The map $\prod\limits_{i \in S} \sigma_i $ induce frobenius map on the level of residue fields. Note that the coycle $\zeta$ twists the action of $\prod\limits_{i \in S} \sigma_i $ by $\prod\limits_{i \in S} w_{m_i}$. Therefore $\Xc(\F_p)$ consists of $\F_{p^2}$-rational points of $\X$ that are fixed by $w_M \circ \frob$ where $M=\prod\limits_{i \in S}m_i$. Note that, if $S$ is empty then $p$ splits completely in $\K/\Q$ and $\K \hookrightarrow \K_{\mathcal{P}} \cong \Q_p$. Therefore, $\Xc(\Q_p)=\X(\Q_p)\neq \emptyset$. The other extreme case is when $p$ is inert in each $\Q(\sqrt{d_i})$, i.e. $S=\{1,2,\ldots,k\}$. We will show that in this case there are points in $\Xc(\F_p)$. Our strategy is to prove that there is a supersingular point fixed by $w_N \circ \frob$, or equivalently, by $w_N \circ w_p =w_{Np}$. This will be derived from well-known results in quaternion arithmetic, see [@Vig] page 152. Using more advanced tools of quaternion arithmetic we can give sufficient conditions for the existence of a $\Q_p$-point on $\Xc$ when $0<|S|<k$. \[unramifiedgood\] Let $p$ be a prime which doesn’t divide $N$ and unramified in $K$. Let $M=\prod\limits_{i \in S} m_i$ then if $\left ( \frac{-pM}{p_i} \right )=1$ for all $p_j |N$ and $j \notin S$ then $\Xc(\Q_p)$ is nonempty. Remark: If $S=\emptyset$ or $S=\{1,\ldots,k\}$ then Proposition \[unramifiedgood\] implies that $\Xc(\Q_p)$ is non empty. In other words, if $p$ splits in each $\Q(\sqrt{d_i})$ or inert in each $\Q(\sqrt{d_i})$ then $\Xc(\Q_p)$ is non-empty. The idea will be similar to the idea of Theorem 3.17 in [@Ozman]. We will show that there is a supersingular point in $\Xc(\F_p)$. For the convenience of the reader, we produce the related parts of the proof here again. Let $\Sigma_N$ be the set of tuples $(E,C)$ such that $E$ is a supersingular elliptic curve over characteristic $p$ and $C$ is cyclic group of order $N$. We start by studying the action of the involution $w_N \circ \sigma$ on $\Sigma_N$. Let $B:=\End(E) \otimes_{\Z} \Q$, then $B$ is the unique quaternion algebra over $\Q$ which is ramified only at $p$ and $\infty$, $\End(E)$ is a maximal order in $B$, and $\End(E,C)$ is an Eichler order of level $N$. The frobenius map acts on the set of singular points of $X_0(Np)_{/\F_p}$ as $w_p$ (see Chapter V, Section 1 of [@Deligne] or Proposition 3.8 in [@Ribet2]). The modular curve $X_0(Np)$ has bad reduction over $\F_p$. A regular model of $X_0(N)_{/ \F_p}$ is given by Deligne-Rapaport and consists of two copies of $X_0(N)_{/ \F_p}$ glued along supersingular points. Let $\psi$ be a map from $X_0(N)_{/ \F_p}$ to $X_0(Np)_{/ \F_p}$, an isomorphism onto one of the two components. The map $\psi$ takes the supersingular locus of $X_0(N)_{/ \F_p}$ to the supersingular locus of $X_0(Np)_{/ \F_p}$. The set $\Sigma_N$ is the supersingular locus of $X_0(N)_{/ \F_p}$. The Atkin-Lehner operators $w_{Mp}$ acts on $X_0(Np)$ for every $M|N$, in particular $w_{Mp}$ acts on $\psi(\Sigma_N)$. When we say the action of $w_{Mp}$ on $\Sigma_N$, it is meant the action of $w_{Mp}$ on $\psi(\Sigma_N)$. The involution $w_{Mp}$ has a fixed point on $\Sigma_N$ if and only if $\Z[\sqrt{-Mp}]$ embeds in $\End(E,C)$ for some $(E,C) \in \Sigma_N$. Since $\End(E,C)$ is an Eichler order of level $N$ in $\Q_{p,\infty}$ by Eichler’s optimal embedding theorem(see [@Vig]) $\Z[\sqrt{-Mp}] \hookrightarrow \End(E,C)$ if and only if we have the condition given in the proposition statement. This implies that there is a supersingular point in $\Xc(\F_p)$ and by Hensel’s lemma there is a point in $\Xc(\Q_p)$. By Proposition \[prop:unramifiedgood\] we have a sufficient condition for $\Xc(\Q_p) \neq \emptyset$ and if we have the conditions given in this proposition then reduction of any point in $\Xc(\Q_p)$ has to be a supersingular point. If the condition of the Proposition \[prop:unramifiedgood\] fails, there may still be ordinary points in $\Xc(\F_p)$ which are points in $\X(\F_{p^2})$ fixed by $\frob \circ w_M$. Now we will assume that the condition of the Proposition \[prop:unramifiedgood\] fails i.e. $\left ( \frac{-pM}{p_j} \right )=-1$ for some $p_j|N$ and $j \notin S$. In this case if there is $x$ in $\Xc(\F_p)$ then $x$ has to be an ordinary point. The following proposition gives the conditions for the existence of a $\F_p$-rational, $w_M$ fixed ordinary point of $\X$ which is another sufficient condition for $\Xc(\F_p)\neq \emptyset$. Note that if both Propositions \[prop:unramifiedgood\] and \[ordinarycm\] fail then $\Xc(\F_p)$ can still be non-empty but such a point has to be an ordinary point defined over $\F_{p^2}-\F_p$ and fixed by $w_M \circ \frob$. \[lemmacmlifting\] Let $p$ be an odd prime and $M,p_j$ be as above. There is a $w_M$-fixed $\Q_p$-rational point on $X_0(N)$ if and only if there is a $w_M$-fixed $\F_p$-rational point on $X_0(N)_{/\F_p}$. Say $(E,C^{N/M} \oplus C^M)$ is a $w_M$-fixed point of $X_0(N)(\F_p)$, where $C^{N/M}, C^M$ denotes cyclic subgroups of order $N/M,M$. In particular $C^{N/M}$ can be written as $\oplus C^{p_j}$ where $j \notin S$. Then $\ker(\lambda)=C^M$ and $\lambda(C^{p_j})=C^{p_j}$ for some endomorphism $\lambda$ of $E$. In particular, $p_j$ divides separable degree of $\lambda-[x]$ where $[x]$ denotes multiplication by $x$ map for some $1\leq x \leq p_j-1$. We have a short exact sequence as follows: $$0 \rightarrow C^M \rightarrow E \stackrel{\lambda}{\rightarrow} E \rightarrow 0$$ By Deuring’s Lifting Theorem([@Deuring]), this short exact sequence can be lifted to a short exact sequence as below where everything is defined over a number field $B$ and there is a prime $\nu$ of $B$ with residue degree $1$ and the reduction of $\tilde{E}, \tilde{\lambda}$ mod $\nu$ gives us the sequence above. $$0 \rightarrow \tilde{C^M} \rightarrow \tilde{E} \stackrel{\tilde{\lambda}}{\rightarrow} \tilde{E} \rightarrow 0.$$ The lifted curve $\tilde{E}$ and the lifted map $\tilde{\lambda}$ give rise to a $w_M$-fixed point on $\X(B)$ if and only if there exists cyclic groups of order $p_j$ in $\ker(\tilde{\lambda}-[x])$ for all $j$, where $[x]$ is multiplication by $x$ map for some integer $x$ in $\{1,\ldots,p_j-1\}$. By assumption, $p_j$ divides the degree of $\lambda-[x]$ for some $x$ in $\{1,\ldots,p_j-1\}$. Since $(p,M)=1$, degree of $\lambda=M$ is the same as degree of $\tilde{\lambda}$. Therefore $\tilde{E}$ gives rise to a $w_M$ fixed point in $\X(B)$ since both inertia and residual degrees of $p$ in $B$ are one, $B \hookrightarrow \Q_p$. Conversely, since the fixed locus of $w_M$ is proper, a $w_M$-fixed point on $X_0(N)(\Q_p)$ reduces to a $w_M$-fixed point on $X_0(N)(\F_p)$. Remark: For $p=2$, the argument goes as in [@Ozman] Section 4. \[ordinarycm\] Let $p$ be an odd prime. There is a $w_M$-fixed point in $X_0(N)(\Q_p)$ whose reduction to $\F_p$ is ordinary if and only if $\left( \frac{-M}{p} \right) = 1$ and the number field $\Q(H(-M))$ has a prime lying over $p$ with residue degree one where $H(-M)$ denotes the Hilbert class polynomial of reduced discriminant $M$. If $p=2$, since the unique ordinary elliptic curve over $\F_2$ has endomorphism ring $\Z[\frac{1+\sqrt{-7}}{2}]$, $M=7$. Note that since $\Xc$ has good reduction at $p$, the big primes(compared to the genus of $\Xc$) are not problematic, i.e. $\Xc(\F_p)\neq \emptyset$ for all such primes. Hence we have the following result for $p$ unramified in $K$ and not dividing $N$. \[prop:unramifiedgood\] Let $p$ be a prime not dividing $N$ and inert in the quadratic fields $\Q(\sqrt{d_i})$ for $i \in S$. Let $M$ be the product of $m_i$ for $i \in S$. Then $\Xc(\Q_p)$ is non-empty if one of the following holds: - $p > 4g^2$ where $g$ is the genus of $\X$. - $S$ is empty or $S=\{1,\ldots, k\}$. - $\left(\frac{-pM}{p_j}\right )=1$ for $p_j|N$ and $j \notin S$. - $\left(\frac{-pM}{p_j}\right )=-1$ for some $p_j|N$ and $j \notin S$: Hilbert class polynomial of reduced discriminant $M$ has a root mod $p$ if $p$ is odd and $M=7$ if $p=2$. $p$ is ramified in $\K$ ----------------------- : Let $p$ be ramified in only one of the fields $\Q(\sqrt{d_i})$ and splits in the others. Let $\nu$ be a prime of $\K$ lying over $p$. In this case we don’t have the theory of Galois descent since the extension is ramified. The $\Q_p$-rational points of $\Xc$ are points in $\X(K_\nu)$ that are fixed by $\sigma_{d_i} \circ w_{m_i}$. This case is quite similar to the case of quadratic twists, see [@Ozman]. By Lemma \[lemmacmlifting\], we have the necessary and sufficient conditions to lift a $w_{m_i}$ fixed point of $\X(\F_p)$ to a $w_{m_i}$-fixed point of $\X(\Q_p)$, which gives a point in $\Xc(\Q_p)$. The following proposition gives us the converse. The proof is very similar to the proof of Proposition 4.4 in [@Ozman] so we don’t reproduce the proof here. \[prop:diag\] Let $x$ be a point of $X_0(N)(\K_{\nu})$ such that $w_{m_i}(x^{\sigma_{d_i}})=x$ then $x$ reduces to a $w_{m_i}$-fixed point on the special fiber of $\mathcal{X}_0(N)_{/R}$ where $R$ is the integer ring of $\K_{\nu}$. Combining Lemma \[lemmacmlifting\] and Proposition \[prop:diag\] we can conclude the following: \[prop:inertram\] Let $p$ be an odd prime that doesn’t divide $N$ and ramified only in one of the quadratic fields, namely $\Q(\sqrt{d_i})$ and splits in the others. Then $\Xc(\Q_p)$ is non-empty if and only if there is a prime $\nu$ in $\Q(H(-M))$ lying over $p$ with inertia degree one. If $p=2$, by Lemma 4.9 [@Ozman], any $w_M$-fixed point of $\X(F_2)$ can be lifted to an elliptic curve $\tilde{E}$ over a number field $B$ such that $\tilde{E}$ has complex multiplication by the maximal order of $\Q(\sqrt{-M})$. Hence $2$ is unramified in $B$ and $ B \hookrightarrow \Q_2$, there is a $w_M$-fixed point in $\X(Q_2)$. Bad Primes ========== In this section we will deal with the primes diving $N$. We’ll assume that these primes are unramified in $\K$. Fix a prime divisor $p=p_{i_0}$ of $N$. As in the case of good primes, if $p$ splits totally in $\K$ then $\K$ embeds in $\Q_p$, hence $\Xc(\Q_p)=\X(\Q_p)\neq \emptyset$. The results in other cases can be summarized as below: \[thm:mainbad\] Let $N=p_1\ldots p_k$ and $p=p_{i_0}$ be an odd prime dividing $N$. Let $S$ be set of indices$i$ such that $p$ is inert in every $\Q(\sqrt{d_i})$ and splits in the rest. 1. If $i_0$ is in $S$ (i.e. $p$ is inert in the quadratic field twisting the Galois action by $w_p$) then: If $p$ is odd, $\Xc(\Q_p) \neq \emptyset$ if and only if $N=p\prod q_i$ or $N=2p\prod q_i$ where $p \equiv 3 \mod 4, q_i \equiv 1 \mod 4$ and $S$ consists of only $i_0$. If $p=2$, $\Xc(\Q_p) \neq \emptyset$ if $N=2\prod q_i$ where $q_i \equiv 1 \mod 4$ and $S$ contains all $i$. (i.e. $2$ is inert in all the quadratic number fields $\Q(\sqrt{d_i})$.) or $S$ contains only $i_0$ ($2$ is inert in only one of the quadratic fields.) 2. If $i_0$ is not in $S$ (i.e. $p$ spits in the quadratic field twisting the Galois action by $w_p$) then: If $p$ is odd and $S$ contains all $i$ such that $p_i$ is an odd prime dividing $N$ and $i \neq i_0$ i.e. $N=2pM$ or $N=pM$ where $M$ is the product of $q_i$ for $i \in S$ and $q_i \equiv 1 \mod 4$ then $\Xc(\Q_p) \neq \emptyset$. If $p=2$ and $S$ contains all $i$ such that $p_i$ is an odd prime dividing $N$ i.e. $N=pM$ where $M$ is the product of $q_i$ for $i \in S$ and $q_i \equiv 1 \mod 4$ then $\Xc(\Q_p) \neq \emptyset$. In this case we are restricting ourselves to primes unramified in each $\Q(\sqrt{d_i})$. Therefore, we have the theory of Galois descent, we can use Hensel’s Lemma. Note that since we are in the case of bad reduction, the first thing we need is a regular model for $\Xc_{/ \F_p}$ which is the twist of the regular model of $\X_{/\F_p}$. Deligne-Rapaport gives a regular model of $\X_{/\F_p}$. This model consists of two copies of $X_0(N/p)_{/\F_p}$ glued along supersingular points and becomes regular after blowing up $(\frac{|\Aut(x)|}{2}-1)$-many times at each intersection point. For instance if every intersection point has automorphism group $\{\pm 1\}$ then the model is already regular. Using this model and Hensel’s Lemma we can conclude that there is a $\Q_p$-point on $\Xc$ if and only if there is a smooth point in $\Xc(\F_p)$. Let $S$ be the set of indices $i$ such that $p=p_{i_0}$ is inert in $\Q(\sqrt{d_i})$. We will study the cases $i_0 \in S$ and $i_0 \notin S$ separately. $i_0 \in S$ ----------- : Then the decomposition group of $p$ in $\K$ is $\{1, \prod\limits_{i \in S} \sigma_{d_i}\}$ and an $\F_p$-rational of $\Xc$ is a point in $\X(\F_{p^2})$ fixed by $ \prod\limits_{i \in S} w_{p_i} \circ \frob=w_M \circ \frob$ where $M=\prod \limits_{i \in S} p_i$. Note that $p|M$. Since $w_p$ interchanges each branch, $X_0(N/p)$ and frobenius acts on each branch, a $w_M \circ \frob$-fixed point has to be an intersection point i.e. a supersingular point. Moreover frobenius acts as $w_p$ on the set of supersingular points(as mentioned in the section of good reduction), hence $\Xc(\F_p)$ is nonempty if and only is there is a smooth supersingular point in $X_0(N/p)(\F_{p^2})$ fixed by $w_p \circ w_M= w_{M/p}$. \[prop:degree\] Using the notation above, there is a $w_{M/p}$ fixed smooth supersingular point in $X_0(N/p)(\F_{p^2})$ if and only if there is a Eichler order $O$ of level $N/p$ in the quaternion algebra ramified at $p$ and infinity such that both $\Z[\sqrt{-M/p}]$ and $\Z[i]$ embed simultaneously into $O$. Let $x$ be a supersingular $w_{M/p}$-fixed point on $X_0(N)(\F_{p^2})$. This is equivalent to say that $\Z[\sqrt{-M/p}]$ embeds in $\End(x)$, which is an Eichler order of level $N/p$ in the quaternion algebra ramified at $p$. We will show that in order to be smooth $\Z[i]$ must embed in $\End(x)$ as well. Recall that at each singular(hence supersingular) point $x$ we have $(|\Aut(x)|/2-1)$-many exceptional lines. The automorphism group of an elliptic curve over a field of characteristic $q$ is $\mu_2,\mu_4$ or $\mu_6$ if $q$ is not $2$ or $3$ where $\mu_s$ denotes the group of primitive $s$-th roots of unity. If $q=2$ or $3$ and $E$ is the unique supersingular elliptic curve in characteristic $q$ then $\Aut(E)$ is $C_3 \rtimes \{\pm1,\pm i,\pm j,\pm k\}$ or $C_3 \rtimes C_4 $ respectively, where $C_m$ denotes the cyclic group of order $m$. Therefore if $|\Aut(x)|=4n$ for $n>1$, there is an element of order $4$ in $\Aut(x)$ and the number of blow-ups is $2n-1$ which is odd. Since we have odd number of exceptional lines, there is one line $L_{/ \F_p}$ that is fixed by the Galois action. On this line the there are $p+1$ rational points, two of which are singular. Therefore if $\Z[i]$ embeds in $\End(x)$ then $x$ is a smooth point. Conversely any smooth point is of this form. If $|\Aut(x)|$ is $2$, then the model of $X_0(N)/\Z_p$ is already regular hence no need to blow-up, $x$ is singular. If $|\Aut(x)|=6$, then we replace this point by 2 exceptional lines over $\F_p$ and $\frob \circ w_N$ interchanges these lines. Each of these exceptional lines cuts one of the branches and also the other exceptional line once. Denote the intersection point of these lines by $x'$. Then $x'$ induces an $\F_p$-rational point of $\Xc$. However, it is a singular point. Hence $x$ is a smooth point on $\Xc(\F_p)$ if and only if both $\Z[\sqrt{-M/p}]$ and $\Z[i]$ embed in $\End(x)$. There is an Eichler order $O$ of level $N/p$ in the quaternion algebra $\Q_{p,\infty}$ such that $\Z[i]$ embeds in $O$ if and only if $N=p\prod q_i$ or $N=2p\prod q_i$ where $p \equiv 3 \mod 4$ and $q_i \equiv 1 \mod 4$. Another order $\Z[\sqrt{-M/p}]$ also embeds into $O$ whenever $O$ has an element $a$ of norm $-M/p$. If we consider the $\Z$-module generated by $1,i,a,ia$ we get relations between discriminant of $O$ and the $\Z$-module. Using this idea and properties of quaternion arithmetic we can deduce the following result, a nice proof of which is also given in [@Jim]. \[prop:jim\]*([@Jim] Corollary 4.2.3)* Let $B$ be a definite quaternion algebra ramified at $D'$ and let $O$ be an Eichler order of $B$ of squarefree level $N'$ such that $\Z[i] \hookrightarrow O$. If $m | D'N'$ and $m \neq 1$ then $\Z[\sqrt{-m}] \hookrightarrow O$ if and only if $m=D'N'$ or $2|D'N'$ and $m=D'N'/2$. Combining Proposition \[prop:jim\], Proposition \[prop:degree\] and the observation following Proposition \[prop:degree\] we obtain the first part of Theorem \[thm:mainbad\]. $i_0 \notin S$ -------------- Then the decomposition group of $p=p_{i_0}$ in $\K$ is $\{1, \prod\limits_{i \in S} \sigma_{d_i}\}$ and an $\F_p$-rational of $\Xc$ is a point in $\X(\F_{p^2})$ fixed by $ \prod\limits_{i \in S} w_{p_i} \circ \frob=w_M \circ \frob$ where $M=\prod \limits_{i \in S} p_i$. Note that $p \nmid M$. Note that since $w_{p}$ is the only involution that interchanges the branches, $\frob \circ w_M$ doesn’t interchange the branches. Hence an $\frob \circ w_M$-fixed point need not be a supersingular point. This is the main difference with the case $i_0 \in S$. However, as the proposition below shows, we can still find a smooth $\frob \circ w_M$-fixed point among the supersingular points in some cases. \[onesplitsuper\] Using the notation above, $\Xc(\Q_p)$ is nonempty if we are in one of the following cases: - $p \equiv 3 \mod 4$, $N=p\prod q_i$ or $N=2p\prod q_i$ with $q_i \equiv 1 \mod 4$ and $M=\prod q_i$ - $p=2$ and $N=2\prod q_i$ with $q_i \equiv 1 \mod 4$ and $M=\prod q_i$ As noted above there exists a smooth supersingular point $x$ in $\Xc(\F_{p})$ if and only if $\Z[i]$ embeds in $\End(x)$ and $x$ is fixed by $\frob \circ w_M=w_{Mp}$ since frobenius acts on $x$ as $w_{p}$. This is equivalent to embed both $\Z[i]$ and $\Z[\sqrt{-Mp}]$ into $\End(x)$ which happens if and only if we have the conditions above as cited in Proposition \[prop:jim\]. Remark: Note that Proposition \[onesplitsuper\] is not an if and only if statement. If the conditions of the theorem fails, this only tells that there aren’t any smooth supersingular point in $\Xc(\F_{p})$. But there may still be ordinary points(which are necessarily smooth) which can be lifted to $\Xc(\Q_{p}$. For instance, by Proposition \[ordinarycm\], we have the necessary and sufficient conditions to lift a $w_{p}$ fixed point of $\X(\F_p)$ to a $w_{p}$-fixed point of $\X(\Q_p)$, which gives a point in $\Xc(Q_p)$. Remark: This corrects conditions given in Theorem 3.7 in [@Ozman]. Algorithm ========= In this section we give an algorithm (using the results of previous sections) which produces $\Xc$ with $\Q_p$ points for every prime $p$. More precisely the input of the algorithm is: Pairwise relatively prime squarefree positive integers $(m_1,\ldots,m_k)$ and its output is polyquadratic fields $\K=\Q(\sqrt{d_1},\ldots,\sqrt{d_k})$ such that $\Xc(\Q_p)\neq \emptyset$ for all primes $p$ where $N=\prod\limits_{i=1}^k m_i$ and $\zeta:G_{\Q} \rightarrow \Aut(\X)$ such that $\sigma \mapsto \sigma \circ w_{m_i}$ if $\sigma(\sqrt{d_i})=-\sqrt{d_i}$ and $1$ otherwise. Note that $\sigma_i$ is the Galois map that sends $\sqrt{d_i}$ to $-\sqrt{d_i}$. Given $N=\prod\limits_{i=1}^k m_i$ as above proceed as below: 1. Choose $d_i$ such that there is no prime simultaneously ramified in $\Q(\sqrt{d_i})$ and $\Q(\sqrt{d_j})$ for $i \neq j$. 2. For all $p|N$ choose $d_i$ such that $p$ splits in each $\Q(\sqrt{d_i})$. Since there are finitely many such $p$, it is possible to choose $d_i$ accordingly. 3. For all $p \nmid N$ and $p < 4g^2$ where $g$ is the genus of $X_0(N)$, choose $(d_1,\ldots,d_k)$ such that $p$ is inert in $\Q(\sqrt{d_i})$ for all $i$ in $[1, r]$ or $p$ splits completely in $\K$. 4. For all $p \nmid N$, $p > 4g^2$ and $p$ ramified in one(and only one by first step) of the $\Q(\sqrt{d_i})$ the Hilbert class polynomial of reduced discriminant $-m_i$ has a root mod $p$. Density Results =============== In this section we give an asymptotic for the number of bi-quadratic twists $\Xc$ that has local points everywhere but no global points. To make things more concrete we will make the following assumptions on $N$ but similar asymptotic can be find for other $N$ as well. - $m_1,m_2,d_1$ be primes congruent to $1$ mod $4$ and $m_1 \equiv m2 \mod 8$ such that $\left( \frac{d_1}{m_i} \right ) =1$ and $\left( \frac{-2m_1}{m_2} \right)=1$ - Hilbert class polynomial of discriminant $-4m_1$ has a root modulo $d_1$ - Every prime $p<4g^2$ splits in $\Q(\sqrt{d_1})$ where $g$ is the genus of $\X$. The curve $\Xc$ is the twist of $\X$ via the cocyle $\zeta$ which sends $\sigma$ to $w_{m_1}$ if $\sigma(\sqrt{d_1})=-\sqrt{d_1}$, to $w_{m_2}$ if $\sigma(\sqrt{d_2})=-\sqrt{d_2}$ and trivial otherwise. Given a positive integer $X$, let $A'$ be the set of positive squarefree integers $d_2\leq X$ such that $\Xc(\Q_p)$ is non-empty for all $p$ when there is no prime $p$ simultaneously ramified in $\Q(\sqrt{d_1},\sqrt{d_2})$ and $\Q(\sqrt{-N})$. \[prop:densityodd\] Keeping the notation as above, we have that $$|A'|=M_{S_N} \frac{X}{\log^{1-\alpha}X}+ O\left (\frac{X}{\log^{2-\alpha}X} \right )$$ where $\alpha$ is the density of the set of primes $p$ such that Hilbert class polynomial of reduced discriminant $-m_2$ has a root mod $p$ and $\left( \frac{d_1}{p} \right)=1$. In order to prove this proposition we need the following result of Serre: \[thm:serreden\] Let $0<\alpha <1$ be Frobenius density of a set of primes $S$ and $N_S(X)$ is the number of squarefree integers in $[1 \ldots X]$ all of whose prime factors lie in $S$. Then $N_S(X) = c_S\frac{X}{\log^{1-\alpha}X}+ O\left (\frac{X}{\log^{2-\alpha}X} \right )$ for some positive constant $c_S$. *Proof of proposition:* Let $S_N$ be the set of primes $p$ such that Hilbert class polynomial of reduced discriminant $-m_2$ has a root mod $p$ and $\left( \frac{d_1}{p}\right)=1$. Since $S_N$ is a Chebotarev set of primes, it has a well defined density $\alpha$. Let $A=\{d \in \Z | d \leq X,\textrm{squarefree},(d,N)=1, d=\Pi_i p_i, p_i \in S_N\}$ then the density of $A$ is given by the above theorem of Serre. Let $p=2$. Since $2$ is ramified in $\Q(\sqrt{-N})$, $2$ must be unramified in $\Q(\sqrt{d_1},\sqrt{d_2})$. Therefore we should consider $d_2$ in $A$ such that $d_2 \equiv 1$ mod $4$. Since $2 \nmid N$, $2$ is an unramified good prime. By Proposition \[prop:inertram\] if $2$ splits in both $Q(\sqrt{d_i})$ or inert in both, then $\Xc(N)(\Q_2)$ is nonempty. In the case that $2$ is inert in one of the quadratic fields and split in the other one, the conditions $m_i \equiv 1$ mod $4$, $m_1 \equiv m2 \mod 8$ and $\left( \frac{-2m_1}{m_2} \right)=1$ guarantees the existence of a $\Q_2$-point. Now we will show that for each $d_2$ in $A$ such that $d_2 \equiv 1$ mod $4$, $\Xc(\Q_p)$ is nonempty. We begin with $p \nmid N$. If $p$ is inert in both $\Q(\sqrt{d_1})$ and $\Q(\sqrt{d_2})$ or splits in both then $\Xc(\Q_p)$ is nonempty by Proposition \[prop:inertram\]. Let $p$ be inert in $\Q(\sqrt{d_1})$ and splits in $\Q(\sqrt{d_2})$. Then by assumption $p$ has to be greater than $4g^2$. Hence using Weil bounds and Hensel’s Lemma, $\Xc(\Q_p)$ is nonempty. Say $p$ is inert in $\Q(\sqrt{d_2})$ and splits in $\Q(\sqrt{d_1})$. If $p>4g^2$, there are local points, so say $p<4g^2$. If $\left( \frac{-pm_2}{p_i} \right)=1$ for all $p_i | m_1$, there are local points by Part c of Proposition \[prop:inertram\]. On the other hand if $\left( \frac{-pm_2}{p_i} \right)=-1$ for some $p_i |m_1$, then we are done with last part of the same proposition since $p \in S_N$ and $p$ is an odd prime. Let $p$ be ramified in $\Q(\sqrt{d_1})$. In particular $p=d_1$. Since $p \in S_N$, $\left( \frac{d_1}{p}\right)=1=\left( \frac{p}{d_1}\right)$, hence $p$ splits in $\Q(\sqrt{d_2})$. Then with the given conditions on $m_1,m_2$ and $d_1$ in the beginning, $\Xc(\Q_p)$ is nonempty by Proposition \[prop:unramifiedgood\]. Similarly, let $p$ be ramified in $\Q(\sqrt{d_2})$. Then $p|d_2$, $p\in S_N$, hence $\left( \frac{d_1}{p}\right)=1$, $p$ splits in $\Q(\sqrt{d_1})$ hence $\Xc(\Q_p)$ is nonempty by the same proposition. For $p=m_1$ or $p=m_2$ we show that $p$ splits in $\Q(\sqrt{d_1}, \sqrt{d_2})$. By given conditions, $p$ splits in $\Q(\sqrt{d_1})$. The genus field of $\Q(\sqrt{-N})$ is $\Q(\sqrt{-1},\sqrt{m_1},\sqrt{m_2})$ and ring class field of $\Z[\sqrt{-N}]$ is $\Q(\sqrt{-N},j(\sqrt{-N}))$ and $j(\sqrt{-N})$ is real([@Cox]). Therefore $\Q(\sqrt{m_1},\sqrt{m_2})$ lies in $\Q(j(\sqrt{-N}))$. Let $q$ be a prime divisor of of $d$. Note that $q$ has to be odd. Since $q$ is in $S_N$, there is a prime $\mathcal{Q}$ of $\Q(j(\sqrt{-N}))$ lying over $q$ with inertia degree $1$. Therefore, $\left ( \frac{p}{q} \right )=\left ( \frac{q}{p} \right )=1$, hence $p$ splits in $\Q(\sqrt{d_2})$. Therefore under the given conditions on $m_i$ and $d_1$, for any $d_2$ in $A$ and $d \equiv 1$ mod $4$, $\Xc(N)(\Q_p) \neq \emptyset$ for all $p$. This gives us the claimed asymptotic. Using this result, one can write down a curve $X_0(N)$ and compute an explicit asymptotic for the set of bi-quadratic twists of $\Xc$ violating the Hasse principle using Faltings’ finiteness results as done by Clark in the proof of Theorem 2 in [@Clark]. The following is essentially same as Theorem 2 in [@Clark]. \[fintwist\] Let $D,N$ be a squarefree integers and $m$ be a divisor of greater than $427$. Let $K=\Q(\sqrt{D})$ and $L$ be a quadratic extension of $K$. Consider the cocycle $\zeta':\sigma \mapsto w_m$ where $\sigma$ is the generator of the Galois group of $L/K$. Let $X^D/K$ denote the twist of $\X/K$ via $\zeta'$. Hence $X^D(K)=\{P \in \X(L)| \sigma(P)=w_m(P) \}$. Then there are only finitely many $D$ such that $X^D(K)$ is nonempty. Consider the following maps: $$\alpha_D: X^D(N)(K) \hookrightarrow \X(L)$$ and $$\beta_D: \X(L) \rightarrow \X/w_m (L)$$ Let $C_D$ be the image of compositions of $\beta_D$ and $\alpha_D$. Then $C_D$ is in $\X/w_m (K)$. Moreover, $\X/w_m (K)=\bigcup_{D} C_D \cup w_m(\X(K))$. Since $m>427$, no $w_m$-fixed point is defined over a quadratic number field (which is equivalent to say that the class number of the order $\Z[\sqrt{-m}]$ is bigger than $2$). Therefore, $S_D \cap S_{D'}=\emptyset$ for $D\neq D'$ since any element $P$ in the intersection $S_D \cap S_{D'}$ gives rise to a $K$ rational $w_m$-fixed point. Moreover, the curve $X_0(N)/w_m$ has genus bigger than one since $m$ is big enough. Therefore there are only finitely many $D$ such that $S_D$ is non-empty. This implies that there are only finitely many $D$ such that $X^D(K)$ is non-empty. \[thm:violation\] Assuming the conditions on $m_i$ and $d_1$ given in the beginning of the section and $N>427$, the number of the bi-quadratic twists $\Xc$ which violate the Hasse Principle when there is no prime simultaneously ramified in $\Q(\sqrt{d_1}, \sqrt{d_2})$ and $\Q(\sqrt{-N})$, is asymptotically $M_{S_N}\frac{X}{\log^{1-\alpha}X}$. Let $X^*(N)$ be the quotient of $X_0(N)$ by $<w_{m_1}, w_{m_2}>$. Gonzalez and Lario prove in [@GonzalezLario] that the genus of $X^*(N)$ is greater than $1$ when $N>159$ if $N$ is product of 2 primes $m_1$ and $m_2$. Therefore $X^*(N)$ has finitely many $K$-rational points for any number field $K$ when $N>159$. Let $\Xc$ be the bi-quadratic twist as defined above. In particular any element $P$ of $\Xc(\Q)$ is fixed by $\sigma_1\sigma_2w_N$. Consider $\X$ as a curve over $K=\Q(\sqrt{d_1d_2})$. Then any $P \in \Xc$ induces a $K$-rational point on the quadratic twist of $\X$ by $\zeta':\sigma_1\sigma_2 \mapsto w_N$. Let $D$ be $d_1d_2$ and $X^D(N)$ denote the quadratic twist of $\X_{/ K}$ by $\zeta'$. By Proposition \[fintwist\], there are only finitely many $D$ such that $X^D(K)$ is non-empty. This implies that there are only finitely many twists $\Xc$ with $\Q$-rational points since for each $D$, there are only finitely many relatively prime many tuples $(d_1,d_2)$ such that $d_1d_2=D$. Hence we have the claimed asymptotic for the number of bi-quadratic twists which has local points everywhere but no global points. Remark: It is possible to obtain generalization of Theorem \[thm:violation\] using polyquadratic twists. In order to do this, one first needs to generalize Proposition \[prop:densityodd\]. This is possible but we preferred to include only the biquadratic twist case for simplicity. Similarly, it is possible to generalize Theorem \[fintwist\] by replacing $K$ with the compositum field, $\langle \Q(\sqrt{d_id_j})\rangle_ {1\leq i<j\leq k}$ and $L:=\Q(\sqrt{d_1}, \ldots, \sqrt{d_k})$. In order to make the same proof work, we need to make $m$ big enough so that the class number of $\Z[\sqrt{-m}]$ is bigger than the degree of $K$. This is possible since we know that given any $d$ there exists finitely many $\Z[\sqrt{-m}]$ with class number $d$ by the work of Deuring, Mordell and Heilbronn in 1934. Examples and Other Directions ============================= In this section we study the existence of global points mostly through examples. We deal with twists which has local points everywhere but no global points. At the end, we give an example of a twist which doesn’t have any ‘’non-exceptional’ rational points. Let $C$ be a smooth, projective, geometrically integral curve over $\Q$ of genus greater than or equal to $2$ with a rational degree one divisor $D$. Then we can embed $C$ into its Jacobian $J$ via the map $P\mapsto [P]-D$. Let $S$ be a finite set of primes which $C$ has good reduction at and assume that we know the generators of Mordell Weil group, $J(\Q)$. Then for every $p$ in $S$ we can compute the finite abelian group $J(\F_p)$ and the set $C(\F_p)$. Let $inj_p$ denote the injection from $C(\F_p)$ to $J(\F_p)$ and $red_p$ be the reduction map from $J(\Q)$ to $J(\F_p)$. Then we obtain the following diagram: $$\begin{array}{ccc} C(\Q) & \stackrel{P \mapsto [P]-D}{\longrightarrow} & J(\Q) \\ \downarrow & & \downarrow_{red=\prod_{p \in S}red_p} \\ \prod_{p \in S} C(\F_p) & \stackrel{inj=\prod_{p \in S}inj_p}{\longrightarrow} & \prod_{p \in S} J(\F_p)\\ \end{array}$$ If there is a $P$ in $C(\Q)$ then $red_p([P]-D)$ is in $inj_p(C(\F_p))$ for any $p$ in $S$. In particular if images of $red$ and $inj$ do not intersect then $C(\Q)=\emptyset$. \[thm:sch\] *([@sch], [@sch2])* In the case of curves and under the assumption that the Tate-Shafarevich group of the jacobian of the curve is finite and the curve has a rational degree one divisor Mordell-Weil Sieve is equivalent to the Brauer-Manin obstruction. If we suspect that $C(\Q)$ is empty then we can try to show the non-existence of global points using Mordell-Weil Sieve which is equivalent to Brauer-Manin obstruction by Theorem \[thm:sch\]. This explicit method can be very hard to apply in practice since one needs to know equation of $C$ and generators of the Mordell Weil group of jacobian of $C$. For our family of twists, finding equations can be very tricky since we are dealing with modular curves which tend to have equations with big coefficients. Finding generators $J(\Q)$, even when the coefficients of the equation of the curve is small can be very hard as shown in [@Flynn] and [@BruinStoll]. Moreover, in order to start applying this method, we need to make sure that $\Pic^1(C)(\Q)\neq \emptyset$ which again is not easy in general even for a single curve given with an explicit equation. Moreover, we need to apply this to a family of curves most of which don’t have explicit equation(at least with small coefficients). Therefore, we start with the following study about $\Pic^1(C)(\Q)$. Then we will give some explicit examples as well. Since $C$ has local points everywhere, finding a rational degree one divisor class in enough by the following proposition: \[brauer\]([@CoMa], Proposition 2.4, Corollary 2.5) If $C$ has a $\Q_p$ point then every $\Q_p$ rational divisor class contains a $\Q_p$ rational divisor. If every $\Q_p$ rational divisor class contains a $\Q_p$ rational divisor for all primes $p$ then every $\Q$-rational divisor class contains a $\Q$-rational divisor. The involutions $w_m$ of $\X$ also acts on the divisor classes. We will denote by $w_m^0$ the action on $\Pic^0$ and by $w_m^1$ the action on $\Pic^1$. By definition the group of rational degree one divisor classes on $\Xc$, $$\Pic^1(\Xc)(\Q)=\{D \in \Pic^1(\Xc)(\bar{\Q}) | D^{\sigma_i}=w_{m_i}^1(D) \; \forall i\}.$$ Let $S$ be the class of $\infty-0$ in $\Pic^0(X_0(N))(\Q)$. Note that $w_N$ interchanges $0$ and $\infty$. \[lem:triv\] Let $\Xc$ be a quadratic twist of $\X$ by $w_N$. There is a $D$ in $\Pic^1(\Xc)(\Q)$ if and only if there is a $P$ in $\Pic^0(X_0(N))(\K)$ such that $P^{\sigma}=w_N^0P+S$. Say there is $P \in \Pic^0(X_0(N))(\K)$ such that $P^{\sigma}=w_N^0P+S$. Let $D=P+(0)$, hence $D \in \Pic^1(X_0(N))(\K)$. Now $D^{\sigma}=P^{\sigma}+(0)^{\sigma}=P^{\sigma}+(0)$ since $(0) \in \Pic^1(X_0(N))(\Q)$ and $w_N^1D=w_N^0P+(\infty)$ hence $D^{\sigma}-w_N^1D=P^{\sigma}+(0)-w_N^0P-(\infty)=0$. Conversely say there is $D \in \Pic^1(X_0(N))(\K)$ such that $D^{\sigma}-w_N^1D=0$. Let $P=D-(0)$. Then $P^{\sigma}=D^{\sigma}-(0)$ and $w_N^0P=w_N^1D-(\infty)$, $P^{\sigma}-w_N^0P=D^{\sigma}-w_N^1D+(\infty)-(0)=(\infty-0)$. Remark: We can extend the argument above to higher degree extensions. For instance consider the following biquadratic extension of $X_0(m_1m_2)$ by $K=\Q(\sqrt{d_1}, \sqrt{d_2})$. Let $C_1,C_2,C_3,C_4$ be the four cusps of $X_0(m_1m_2)$ such that $w_{m_1}(C_1)=C_2, w_{m_2}(C_1)=C_3$. Let $C$ denote the twisted curve, $S_1=(C_2-C_1)$ and $S_2=(C_3-C_1)$. Then $\Pic(C)(\Q)\neq \emptyset$ if and only if there exists a $P \in \Pic^0(K)$ such that $\sigma_1 P=w_{m_1}P+S_1$ and $\sigma_2 P=w_{m_2}P+S_2$. The differences of cusps are chosen so that each $w_{m_i}$ acts as $-1$ on $S_i$. For instance, let $m_1=13, m_2=2$. The cuspidal group of $X_0(26)$ is cyclic of order $21$ and the Atkin Lehner involutions act as multiplication by $8$ and $13$ on the cuspidal group. Hence $S_1=15S$ and $S_2=7S$ where $S$ is the generator of the cuspidal group. Then $P=11S$ satisfies the relations, hence any biquadratic twist of $X_0(26)$ has a degree one rational divisor class. Manin has showed that the difference of cusps of $X_0(N)$ have finite order. Therefore we get the following corollary: \[cor:div\] Let $\Xc$ be quadratic twist with local points everywhere. If the order of the divisor $S=(\infty-0)$ is odd in the cuspidal subgroup of $J_0(N)$ then there is a rational degree one divisor on $\Xd$. By Lemma \[lem:triv\] there is a rational degree one divisor class on $\Xc$ if and only if there is a $P$ in $\Pic^0(X_0(N))(\K)$ such that $P^{\sigma}=w_N^0P+S$. Since $w_N$ acts as $-1$ on $J_0(N)$ and the order of $S$ is odd, there is such $P$. Then by Proposition \[brauer\] this is equivalent to have a degree one rational divisor on $\Xc$. Since the cuspidal subgroup of $J_0(26)$ is cyclic of order $21$, by Corollary \[cor:div\] every quadratic twist $C$ has a rational degree one divisor if $C$ has local points everywhere. Note that since the equation of $X_0(26)$ is an irreducible sextic, it is not possible to see the existence of a degee one divisor class via elementary methods. When $N$ is prime, Ogg showed in [@OggC] that the order of $S$ is given by numerator of $ \frac{N-1}{12}$. Hence we can conclude that whenever $N$ is prime and not congruent to $13$ modulo $24$, $\Pic(C)(\Q)$ is nonempty under the assumption that $C(\Q_p)$ is nonempty. Hence this shows that the work is in the case when the order of $S$ is even. In this case we have the following result. \[thm:divinert\] Let $N>3$ be a prime that is inert in the quadratic number field $\K$. Let $X$ be quadratic twist of $X_0(N)$ by $K$ and $w_N$. Then $\Pic^1(X)(\Q_N)$ is empty if and only if $N \equiv 1 \mod 24$ or $N \equiv 17 \mod 24$. Say there is a $D$ in $\Pic^1(X)(\Q_N)$. Let $J_N$ be the special fiber of the Neron model of Jacobian of $X$. By Lemma \[lem:triv\] there is a $P$ in $J_N(\F_{N^2})$ such that the difference of $\frob(P)$ and $w_N^0(P)$ is equal to the class of the divisor $(0)-(\infty)$. By [@Deligne], $J_N$ has the form $J_N^0 \times C$ where $C$ is a cyclic group generated by the class of the divisor $(0)-(\infty)$. Moreover, the group of connected components of $J_N$, which will be denoted by $\phi_N$, is isomorphic to $C$. Let $I_P$ be the connected component in which $P$ is living and $c_P$ be the element of $C$ that corresponds to $I_P$. Then the difference of $\frob(P)$ and $w_N^0(P)$ translates into $\frob(c_P)-w_N(c_P)$. Since $w_N$ interchanges $(0)$ and $(\infty)$, it acts as $-1$ on $C$. Hence the relation translates into $\frob (c_P)+(c_P)$. The map $\frob$ induces the trivial action on $C$ since the generator of $C$ is rational, hence we end up with $c_P+c_P$. However, on the other side of the equality we had $S=(0)-(\infty)$. This gives a contradiction since the order of $S$(which is given by numerator of $\frac{N-1}{12}$, is even there cannot be an element $c_P$ in $C$ such that $c_P+c_P=S$. Conversely if $N$ is not $1$ mod $24$, then the order of $S$ is odd hence, there is a $c$ in the cuspidal group such that $2c=S$. **Remark:** In the course of the proof of Theorem \[thm:divinert\] it was crucial that $C$ was isomorphic to the component group of the special fiber of Neron model of $J_0(N)$. This is true for a special class of composite $N$ values as well, but not for every square-free value of $N$. Using Table 2 in [@Mazur2], we see that $\phi_p$ is isomorphic to $C$ when $N=p\prod q_i$ such that there is at least one $q_i$ that is not congruent to $1$ mod $4$ and one $q_j$ such that $q_j$ not congruent to $1$ mod $3$. If we are in this case, the order of $(0)-(\infty)$ is given by $Q\frac{p-1}{12}$, where $Q=\prod (q_i+1)$. Hence, the statement of Theorem \[thm:divinert\] can be generalized to this case as well and same proof applies. Let $\K$ be a polyquadratic field and say $p$ is inert in $\K$, $N=p\prod q_i$ such that $p>3$ and $q_i \equiv 3 \mod 4, q_j \equiv 2 \mod 3 $ for some $i,j$. Then $\Pic^1(X)(\Q_p) = \emptyset$, where $X$ denotes the polyquadratic twist of $X_0(N)$. Using Table 2 in [@Mazur2] and the given congruence relation on $N$, we see that $\phi_p/C$ is trivial. Then like in the proof of Theorem \[thm:divinert\], $\Pic^1(C)(\Q_p)$ is non-empty if and only if the order of $(0)-(\infty)$ is odd. However, in this case, the order of $(0)-(\infty)$ is given by $\prod(q_i+1)\frac{p-1}{12}$ which is always an even number under the assumption of the decomposition of $N$ given in the theorem statement. In the other cases of Mazur’s Table 2 in [@Mazur2], $\phi_p$ is $C \oplus D$ for some abelian group $D$. Therefore we cannot apply the same method in general, without knowing how $w_N$ and $\frob$ acts on $D$. Once we know $\Pic^1(C)(\Q)\neq \emptyset$, and $C(\Q_p)\neq \emptyset$ for all $p$, we can try different things to study $C(\Q)$. One of these is ‘descent theory’ (for details see for instance [@Stoll1]). For genus 2 curves over rationals, 2-decent has been implemented pretty efficiently in MAGMA. Using the command `TwoCoverDescent` we were able to see that in some examples $Sel^{\pi}(C)=0$ where $\pi:D \rightarrow C$ and $D$ is a two cover. When this method fails, one can try Mordell Weil Sieve or Chabuty methods. For the examples that we have encountered, if the descent failed, the rank of the jacobian was too big to apply Chabuty methods. Therefore, Mordell Weil sieve was our only choice. Note that this method requires explicit generators of $J(\Q)$ which is usually hard to compute. The following table summarizes the computations done for quadratic twists of $X_0(26)$. Note that $J_0(26)$ decomposes over $\Q$ as $E_1 \times E_2$ where $E_i$ are elliptic curves of conductor $26$. The twist of $J_0(26)$ by the cocycle $\zeta$ is isogenous to the product of twists $E_1^{\zeta} \times E_2^{\zeta}$. And since $w_{26}$ acts as multiplication by $-1$, $E_i^{\zeta}$ is the usual quadratic twist of $E_i$. Therefore the rank of $J_0^{\zeta}(\Q)$ is the sums of the ranks of $E_i^{\zeta}$. This lets us compute the exact rank of $J_0^{\zeta}(26)$. Once we know the exact rank of the jacobian then we look for generators and in the lucky cases which we were able to find the generators, we apply Mordell Weil Sieve(if the two cover descent doesn’t work). $N$ $d$ Rank of $J$ Descent Works? Set of primes for MW Sieve ------ ------- ------------- ---------------- ---------------------------- $26$ $-29$ $1$ Yes $26$ $-23$ $2$ No $17,31$ $26$ $23$ $1$ Yes $26$ $29$ $2$ No $5,11,23$ $26$ $-79$ $2$ Yes We now change the direction and give an example of a twist $S$ which has rational points. When the genus is bigger than one we know by Faltings’ theorem that $C(\Q)$ is finite however there is no known algorithm that will give us all the members of this finite set. However in some cases we can find the full list. For instance, for a genus 2 curve, we can use theory of heights to come up with the generators of the Mordell Weil group of jacobian of C(which is only feasible when the height bounds are small enough) and when the rank is $1$ we can apply Chabuty methods combined with Mordell Weil sieve. This has been implemented in MAGMA. For more details see [@Stoll1] and references there. Using these we get the following example: The quadratic twist of $X_0(26)$ by $\Q(\sqrt{-1})$ and $w_{26}$ has only two rational points and they parametrize the elliptic curve with j-invariant 1728, hence only rational points in the twisted curve are CM points. [HD]{} N. Bruin, M. Stoll,*The Mordell-Weil sieve: proving non-existence of rational points on curves*, LMS J. Comput. Math. 13 (2010), 272–306. P.L. Clark, *An “anti-Hasse principle” for prime twists*, Int. J. Number Theory 4 (2008), no. 4, 627–637. Coray, D. ; Manoil, C. *On large Picard groups and the Hasse principle for curves and $K3$ surfaces*, Acta Arith. 76 1996), no. 2, 165–189. D. Cox, *Primes of the Form $x^2+ny^2$*, John Wiley & Sons, Inc., 1989. P. Deligne and M. Rapoport, *Les schémas de modules de courbes elliptiques*, (French) Modular functions of one variable, II (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), pp. 143–316. Lecture Notes in Math., Vol. 349, Springer, Berlin, 1973. M. Deuring, *Die Typen der Multiplikatorenringe elliptischer Funktionenkörper*. (German) Abh. Math. Sem. Hansischen Univ. 14, (1941). 197–272. J.S. Ellenberg, *Q-curves and Galois representations,* in Modular Curves and Abelian Varieties, J. Cremona, J.C.Lario, J.Quer and K.Ribet, eds. 2004 E.V. Flynn, *The Hasse principle and the Brauer-Manin obstruction for curves*, Manuscripta Math. 115 (2004), no. 4, 437–466. J. González, J.C. Lario, *Rational and elliptic parametrizations of Q-curves*, J. Number Theory 72 (1998), no. 1, 13–31. B. Mazur, *Modular curves and the Eisenstein ideal*, Inst. Hautes Études Sci. Publ. Math. No. 47 (1977), 33–186 (1978). A.P. Ogg, *Rational Points on certain elliptic modular curves*, Proc. Sympos. Pure Math., vol. 24, AMS, 1973, 221-231 E. Ozman, *Points on Quadratic Twists of $X_0(N)$*, Acta Arithmetica, to appear. K.A. Ribet, *On modular representations of ${\rm \Gal}(\overline {Q}/Q)$ arising from modular forms*, Invent. Math. 100 (1990), no. 2, 431–476. V. Scharaschkin, *Local Global Problems and the Brauer-Manin Obstruction*, PhD Thesis, University of Michigan, 1999 V. Scharaschkin, *The Brauer Manin Obstruction for Curves*, J. Stankewicz, *Twists of Shimura Curves*, PhD Thesis, University of Georgia, 2012 J.P. Serre, *Divisibilité de certaines fonctions arithmétiques*, L’Enseign. Math. 22, 1976, 227-260. A.N. Skorobogatov, *Torsors and Rational Points*, CTM 144, Cambridge Univ. Press, 2001 M. Stoll, *Rational points on curves*, J. Théor. Nombres Bordeaux 23 (2011), no. 1, 257–277 M. Vignéras, *Arithmétique des Algèbres de Quaternions*, Springer-Verlag, New-York, 1980.

--- abstract: 'We report a measurement of the branching fractions of $\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell}$ decays based on 417 fb$^{-1}$ of data collected at the $\Upsilon(4S)$ resonance with the detector at the 2 $e^+e^-$ storage rings. Events are selected by fully reconstructing one of the $B$ mesons in a hadronic decay mode. A fit to the invariant mass differences $m(D^{(*)}\pi)-m(D^{(*)})$ is performed to extract the signal yields of the different $D^{**}$ states. We observe the $\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell}$ decay modes corresponding to the four $D^{**}$ states predicted by Heavy Quark Symmetry with a significance greater than six standard deviations including systematic uncertainties.' title: 'Measurement of the Branching Fractions of [$\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell}$ ]{} Decays in Events Tagged by a Fully Reconstructed [$B$]{} Meson' --- authors\_jun2008.tex Semileptonic $B$ decays to orbitally-excited P-wave charm mesons ($D^{**}$) are of interest for several reasons. Improved knowledge of the branching fractions for these decays is important to reduce the systematic uncertainty in the measurements of the Cabibbo-Kobayashi-Maskawa [@CKM] matrix elements $|V_{cb}|$ and $|V_{ub}|$. For example, one of the leading sources of systematic uncertainty on $|V_{cb}|$ measurements from $\Bbar \to D^* \ell^- \bar{\nu}_{\ell}$ decays [@ell] is the limited knowledge of the background due to $\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell}$ [@BaBarHQET]. The $D^{**}$ mesons contain one charm quark and one light quark with relative angular momentum $L=1$. According to Heavy Quark Symmetry (HQS) [@IW], they form one doublet of states with angular momentum $j \equiv s_q + L= 3/2$ $\left[D_1(2420), D_2^*(2460)\right]$ and another doublet with $j=1/2$ $\left[D^*_0(2400), D_1'(2430)\right]$, where $s_q$ is the light quark spin. Parity and angular momentum conservation constrain the decays allowed for each state. The $D_1$ and $D_2^*$ states decay through a D-wave to $D^*\pi$ and $D^{(*)}\pi$, respectively, and have small decay widths, while the $D_0^*$ and $D_1'$ states decay through an S-wave to $D\pi$ and $D^*\pi$ and are very broad. $\Bbar \rightarrow D^{**} \ell^- \bar{\nu}_{\ell}$ decays constitute a significant fraction of $B$ semileptonic decays [@pdg] and may help to explain the discrepancy between the inclusive $\Bbar \to X\ell^- \bar{\nu}_{\ell}$ rate and the sum of the measured exclusive decay rates [@pdg; @babar-2; @babar-3]. The measured decay properties for $\Bbar \rightarrow D^{**} \ell^- \bar{\nu}_{\ell}$ can be compared with the predictions of the Heavy Quark Effective Theory (HQET) [@LLSW]. QCD sum rules [@uraltsev] imply the strong dominance of $B$ decays to the narrow $D^{**}$ states over those to the wide ones, while some experimental data show the opposite trend [@belle; @delphi2005]. In this letter, we present the observation of $B$ semileptonic decays into the four excited $D$ mesons predicted by HQS and measure the ${\cal B}(\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell})$ branching fractions. The analysis is based on data collected with the  detector [@detector] at the 2 asymmetric-energy $e^+e^-$ storage rings at SLAC. The data consist of a total of 417 fb$^{-1}$ recorded at the $\Upsilon(4S)$ resonance, corresponding to approximately 460 million  pairs. An additional 40 fb$^{-1}$, taken at a center-of-mass (CM) energy 40 MeV below the $\Upsilon(4S)$ resonance, is used to study background from $e^+e^- \to f\bar{f}~(f=u,d,s,c,\tau)$ continuum events. A detailed GEANT4-based Monte Carlo (MC) simulation [@Geant] of  and continuum events is used to study the detector response, its acceptance, and to validate the analysis techniques. The simulation describes $\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell}$ decays using the ISGW2 model [@ISGW], and non-resonant $\Bbar \to D^{(*)} \pi \ell^- \bar{\nu}_{\ell}$ decays using the model of Goity and Roberts [@Goity]. We select semileptonic $\Bbar \to D^{**}\ell^-\bar{\nu}_{\ell}$ decays with $\ell=e, \mu$ in events containing a fully reconstructed $B$ meson ($B_\mathrm{tag}$), which allows us to constrain the kinematics, reduce the combinatorial background, and determine the charge and flavor of the signal $B$ meson. $D^{**}$ mesons are reconstructed in the $D^{(*)}\pi^{\pm}$ decay modes and the different $D^{**}$ states are identified by a fit to the invariant mass differences $m(D^{(*)}\pi) - m(D^{(*)})$. We first reconstruct the semileptonic $B$ decay, selecting a lepton with momentum $p^*_{\ell}$ in the CM frame larger than 0.6 GeV/$c$. We search for pairs of oppositely-charged tracks that form a vertex and remove those with an invariant mass consistent with a photon conversion or a $\pi^0$ Dalitz decay. Candidate $D^0$ mesons that have the correct charge correlation with the lepton are reconstructed in the $K^-\pi^+$, $K^- \pi^+ \pi^0$, $K^- \pi^+ \pi^+ \pi^-$, $K^0_S \pi^+ \pi^-$, $K^0_S \pi^+ \pi^- \pi^0$, $K^0_S \pi^0$, $K^+ K^-$, $\pi^+ \pi^-$, and $K^0_S K^0_S$ channels, and $D^+$ mesons in the $K^- \pi^+ \pi^+$, $K^- \pi^+ \pi^+ \pi^0$, $K^0_S \pi^+$, $K^0_S \pi^+ \pi^0$, $K^+ K^- \pi^+$, $K^0_S K^+$, and $K^0_S \pi^+ \pi^+ \pi^-$ channels. In events with multiple $D\ell^-$ combinations, the candidate with the best $D$-$\ell$ vertex fit is selected. Candidate $D^*$ mesons are reconstructed by combining a $D$ candidate with a pion or a photon in the $D^{*+} \rightarrow D^0 \pi^+ $, $D^{*+} \rightarrow D^+ \pi^0$, $D^{*0} \rightarrow D^0 \pi^0$, and $D^{*0} \rightarrow D^0 \gamma$ channels. In events with multiple $D^{*}\ell^-$ combinations, we choose the candidate with the smallest $\chi^2$ based on the deviations from the nominal values of the $D$ invariant mass and the invariant mass difference between the $D^*$ and the $D$, using the resolution measured in each mode. We reconstruct $B_\mathrm{tag}$ decays [@BrecoVub] in charmed hadronic modes $\Bbar \rightarrow D Y$, where $Y$ represents a collection of hadrons, composed of $n_1\pi^{\pm}+n_2 K^{\pm}+n_3 K^0_S+n_4\pi^0$, where $n_1+n_2 =1,3,5$, $n_3 \leq 2$, and $n_4 \leq 2$. Using $D^0(D^+)$ and $D^{*0}(D^{*+})$ as seeds for $B^-(\Bzb)$ decays, we reconstruct about 1000 different decay chains. The kinematic consistency of a $B_\mathrm{tag}$ candidate with a $B$ meson decay is evaluated using two variables: the beam-energy substituted mass $m_{ES} \equiv \sqrt{s/4-|p^*_B|^2}$, and the energy difference $\Delta E \equiv E^*_B -\sqrt{s}/2$. Here $\sqrt{s}$ is the total CM energy, and $p^*_B$ and $E^*_B$ denote the momentum and energy of the $B_\mathrm{tag}$ candidate in the CM frame. For correctly identified $B_\mathrm{tag}$ decays, the $m_{ES}$ distribution peaks at the $B$ meson mass, while $\Delta E$ is consistent with zero. We select $B_\mathrm{tag}$ candidates in the signal region defined as 5.27 GeV/$c^2$ $< m_{ES} <$ 5.29 GeV/$c^2$, excluding those with daughter particles in common with the charm meson or the lepton from the semileptonic $B$ decay. In the case of multiple $B_\mathrm{tag}$ candidates in an event, we select the one with the smallest $|\Delta E|$ value. The $B_\mathrm{tag}$ and the $D^{(*)}\ell$ candidates are required to have the correct charge-flavor correlation. We account for mixing effects in the $\Bzb$ sample as described in Ref. [@BBmixing]. Cross-feed effects, $i.e.$, $B^-_\mathrm{tag} (\Bzb_\mathrm{tag})$ candidates erroneously reconstructed as a neutral (charged) $B$, are subtracted using estimates from the simulation. We reconstruct $B^- \to D^{(*)+}\pi^- \ell^- \bar{\nu}_{\ell}$ and $\Bzb \to D^{(*)0}\pi^+ \ell^- \bar{\nu}_{\ell}$ decays starting from the corresponding $B_\mathrm{tag}+D^{(*)} \ell^-$ combinations. We select events with only one additional reconstructed charged track, correctly matched to the $D^{(*)}$ flavor, that has not been used for the reconstruction of the $B_\mathrm{tag}$, the signal $D^{(*)}$, or the lepton. $D(D^{*})$ candidates are selected within 2$\sigma$ (1.5-2.5$\sigma$, depending on the $D^*$ decay mode) of the $D$ mass ($D^{*}-D$ mass difference), where the resolution $\sigma$ is typically around 8 (1-7) MeV$/c^{2}$. For the $\Bzb \rightarrow D^{(*)0}\pi^+ \ell^- \bar{\nu}_{\ell}$ decay, we additionally require the invariant mass difference $m(D^0\pi^+)-m(D^0)$ to be greater than 0.18 GeV/$c^2$ to veto $\Bzb \rightarrow D^{*+} \ell^- \bar{\nu}_{\ell}$ events. Semileptonic $\Bbar \rightarrow D^{**}\ell^- \bar{\nu}_{\ell}$ decays are identified by the missing mass squared in the event, $m^2_\mathrm{miss} = \left[p(\Upsilon(4S)) -p(B_\mathrm{tag}) - p(D^{(*)}\pi) - p(\ell)\right]^2$, defined in terms of the particle four-momenta. For correctly reconstructed signal events, the only missing particle is the neutrino, and $m^2_\mathrm{miss}$ peaks at zero. Other $B$ semileptonic decays, where one particle is not reconstructed (feed-down) or is erroneously added to the charm candidate (feed-up), exhibit higher or lower values in $m^2_\mathrm{miss}$ [@babar-3]. In feed-down cases where both a $D$ and a $D^*$ candidate have been reconstructed, we keep only the latter candidate. Mode Selection Criteria ------------------------------------------------ ------------------------------------------------ $B^- \to D^{*+}\pi^- \ell^- \bar{\nu}_{\ell}$ $-0.25 < m^2_\mathrm{miss} < 0.25$ GeV$^2/c^4$ $B^- \to D^{+}\pi^- \ell^- \bar{\nu}_{\ell}$ $-0.25 < m^2_\mathrm{miss} < 0.8$ GeV$^2/c^4$ $\Bzb \to D^{*0}\pi^+ \ell^- \bar{\nu}_{\ell}$ $-0.2 < m^2_\mathrm{miss} < 0.35$ GeV$^2/c^4$ $\Bzb \to D^{0}\pi^+ \ell^- \bar{\nu}_{\ell}$ $-0.15 < m^2_\mathrm{miss} < 0.85$ GeV$^2/c^4$ : $m^2_\mathrm{miss}$ selection criteria. \[tab:MMcuts\] The $m^2_\mathrm{miss}$ selection criteria are listed in Table \[tab:MMcuts\]. The $m^2_\mathrm{miss}$ region between 0.2 and 1 GeV$^{2}/c^{4}$ for $\Bbar \rightarrow D \pi\ell^- \bar{\nu}_{\ell}$ events is dominated by feed-down from $\Bbar \to D^{**} (\to D^* \pi) \ell^- \bar{\nu}_{\ell}$ semileptonic decays where the soft pion from the $D^*$ decay is not reconstructed. In order to retain these events we apply an asymmetric cut on $m^2_\mathrm{miss}$ for these modes. The signal yields for the $\Bbar \to D^{**}\ell^- \bar{\nu}_{\ell}$ decays are extracted through a simultaneous unbinned maximum likelihood fit to the four $m(D^{(*)}\pi) - m(D^{(*)})$ distributions. With the current statistics, validation studies on MC samples show that our sensitivity to non-resonant $\Bbar \to D^{(*)}\pi \ell^- \bar{\nu}_{\ell}$ decays is limited. Including hypotheses for these components results in a fitted contribution that is consistent with zero. Thus we assume that these non-resonant contributions are negligible. The probability that $\Bbar \to D^{**} (\to D^* \pi) \ell^- \bar{\nu}_{\ell}$ decays are reconstructed as $\Bbar \to D^{**} (\to D \pi) \ell^- \bar{\nu}_{\ell}$ is determined with the MC simulation to be 26%(59%) for the $B^-$($\Bzb$) sample and held fixed in the fit. {width="50.00000%"} \[fig:Fit\] The Probability Density Functions (PDFs) for the $D^{**}$ signal components are determined using MC $\Bbar \to D^{**}\ell^- \bar{\nu}_{\ell}$ signal events. A convolution of a Breit-Wigner function with a Gaussian, whose resolution is determined from the simulation, is used to model the $D^{**}$ resonances. The $D^{**}$ masses and widths are fixed to measured values [@pdg]. We rely on the MC prediction for the shape of the combinatorial and continuum background. A non-parametric KEYS function [@keys] is used to model this component for the $D^* \pi \ell^- \bar{\nu}_{\ell}$ sample, while for the $D \pi \ell^- \bar{\nu}_{\ell}$ sample we use the convolution of an exponential with a Gaussian to model the tail from virtual $D^*$ mesons. The combinatorial and continuum background yields are estimated from data. We fit the hadronic $B_\mathrm{tag}$ $m_{ES}$ distributions for $\Bbar \to D^{**}\ell^- \bar{\nu}_{\ell}$ events as described in [@babar-3], and we obtain the number of background events from the integral of the background function in the $m_{ES}$ signal region. Table \[tab:results\] summarizes the results from two fits: one in which we fit the charged and neutral $B$ samples separately, and one in which we impose the isospin constraints ${\cal B} (B^- \to D^{**}\ell^- \bar{\nu}_{\ell})/{\cal B} (\Bzb \to D^{**}\ell^- \bar{\nu}_{\ell}) = \tau_{B^-}/\tau_{\Bzb}$. The latter fit yields a significance greater than 6 standard deviations for all four $D^{**}$ states including systematic uncertainties. The results of this fit are shown in Fig. 1\[fig:Fit\]. The $D_2^*$ contributes to both the $D\pi$ and the $D^*\pi$ samples. In the nominal fit we fix the ratio ${\cal B} (D^*_2 \to D\pi)/{\cal B} (D^*_2 \to D^*\pi)$ to $2.2$ [@pdg]. When we allow this ratio to float we obtain $1.9 \pm 0.6$. To reduce systematic uncertainties we measure the ratios of the ${\cal B} (\Bbar \rightarrow D^{**} \ell^- \bar{\nu}_{\ell})$ branching fractions to the inclusive $\Bzb$ and $B^-$ semileptonic branching fractions. A sample of $\Bbar \to X \ell^- \bar{\nu}_{\ell}$ events is selected by identifying a charged lepton with $p^*_{\ell}>0.6$ GeV/$c$ and the correct charge correlation with the $B_\mathrm{tag}$ candidate. In the case of multiple $B_\mathrm{tag}$ candidates in an event, we select the one reconstructed in the decay channel with the highest purity, defined as the fraction of signal events in the $m_{ES}$ signal region. Background components that peak in the $m_{ES}$ signal region include cascade $B$ meson decays ($i.e.$, the lepton does not come directly from the $B$) and hadronic decays, and are subtracted using the corresponding MC predictions. Decay Mode Yield $\epsilon_\mathrm{sig} (\times 10^{-4})$ ${\cal B}$ ($\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell}$ ) $\times$ ${\cal B} (D^{**} \to D^{(*)} \pi^{\pm})$ % $S_\mathrm{tot} (S_\mathrm{stat})$ ${\cal B}$ % $S_\mathrm{tot} (S_\mathrm{stat})$ --------------------------------------------- -------------- ------------------------------------------ --------------------------------------------------------------------------------------------------------------- ------------------------------------ -------------------------- ------------------------------------ $B^- \to D^{0}_1 \ell^- \bar{\nu}_{\ell}$ $165 \pm 18$ 1.24 $0.29 \pm 0.03 \pm 0.03$ 9.9 (12.7) $0.29 \pm 0.03 \pm 0.03$ 10.7 (15.2) $B^- \to D^{*0}_2 \ell^- \bar{\nu}_{\ell}$ $97 \pm 16$ 1.44 $0.15 \pm 0.02 \pm 0.01$ 6.3 (7.3) $0.12 \pm 0.02 \pm 0.01$ 6.0 (7.4) $B^- \to D^{'0}_1 \ell^- \bar{\nu}_{\ell}$ $142 \pm 21$ 1.13 $0.27 \pm 0.04 \pm 0.05$ 5.4 (8.0) $0.30 \pm 0.03 \pm 0.04$ 6.4 (10.0) $B^- \to D^{*0}_0 \ell^- \bar{\nu}_{\ell}$ $137 \pm 26$ 1.15 $0.26 \pm 0.05 \pm 0.04$ 4.5 (5.8) $0.32 \pm 0.04 \pm 0.04$ 6.1 (8.3) $\Bzb \to D^{+}_1 \ell^- \bar{\nu}_{\ell}$ $88 \pm 14$ 0.70 $0.27 \pm 0.04 \pm 0.03$ 7.0 (8.4) $\Bzb \to D^{*+}_2 \ell^- \bar{\nu}_{\ell}$ $29 \pm 13$ 0.91 $0.07 \pm 0.03 \pm 0.01$ ($< 0.11$ @90% CL) 2.3 (2.5) $\Bzb \to D^{'+}_1 \ell^- \bar{\nu}_{\ell}$ $86 \pm 18$ 0.60 $0.31 \pm 0.07 \pm 0.05$ 4.6 (5.8) $\Bzb \to D^{*+}_0 \ell^- \bar{\nu}_{\ell}$ $142 \pm 26$ 0.70 $0.44 \pm 0.08 \pm 0.06$ 4.7 (6.0) \[tab:results\] The total yield for the inclusive $\Bbar \to X \ell^- \bar{\nu}_{\ell}$ decays is obtained from a maximum likelihood fit to the $m_{ES}$ distribution of the $B_\mathrm{tag}$ candidates, as described in  [@babar-3]. The fit yields 198,897 $\pm$ 1,578 events for the $B^- \to X \ell^- \bar{\nu}_{\ell}$ sample and 120,168 $\pm$ 1,036 events for the $\Bzb \to X \ell^- \bar{\nu}_{\ell}$ sample. The ratios ${\cal B}(\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell})/{\cal B}(\Bbar \to X \ell^- \bar{\nu}_{\ell})= (N_\mathrm{sig}/\epsilon_\mathrm{sig})\cdot (\epsilon_\mathrm{sl}/N_\mathrm{sl})$ are obtained by correcting the signal yields for the reconstruction efficiencies (estimated from  MC events). Here, $N_\mathrm{sig}$ is the number of $\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell}$ signal events, reported in Table \[tab:results\] together with the corresponding reconstruction efficiencies $\epsilon_\mathrm{sig}$, $N_\mathrm{sl}$ is the $\Bbar \to X \ell^- \bar{\nu}_{\ell}$ signal yield, and $\epsilon_\mathrm{sl}$ is the corresponding reconstruction efficiency including the $B_\mathrm{tag}$ reconstruction, equal to 0.39% and 0.25% for the $B^- \to X \ell^- \bar{\nu}_{\ell}$ and $\Bzb \to X \ell^- \bar{\nu}_{\ell}$ decays, respectively. The absolute branching fractions ${\cal B} (\Bbar \rightarrow D^{**} \ell^- \bar{\nu}_{\ell})$ are then determined using the semileptonic branching fraction ${\cal B}(\Bbar \to X \ell^- \bar{\nu}_{\ell})= ( 10.78 \pm 0.18)\%$ and the ratio of the $\Bzb$ and the $B^-$ lifetimes $\tau_{B^-}/\tau_{\Bzb} = 1.071 \pm 0.009$ [@pdg]. Numerous sources of systematic uncertainties have been investigated. The largest uncertainty is due to the determination of the $\Bbar \to D^{**}\ell^- \bar{\nu}_{\ell}$ signal yields (resulting in 5.5-17.0% relative systematic uncertainty depending on the $D^{**}$ state). This uncertainty is estimated using ensembles of fits to the data in which the input parameters are varied within the known uncertainties in the PDF parameterization (0.2-8.7%), the shape and yield of the combinatorial and continuum background (0.2-10.4%), the modeling of the broad $D^{**}$ states (4.5-13.8%), and the $D^{*}$ feed-down rate (0.5-4.0%). We check that the combinatorial and continuum background shape is well reproduced by the simulation by verifying that the MC samples of right-sign and wrong-sign $D^{(*)}\pi$ combinations have similar shapes, and that the wrong-sign distribution in the data agrees well with that in the simulation. We observe an excess of events in the low invariant mass difference region in the four samples that is not accounted for by the background PDF. We study $\Bbar \to D^{(*)}n\pi \ell^-\bar{\nu}_{\ell}$ ($n>1$) decays, not included in our standard MC simulation, as a possible source of this excess. We use different MC models for these decays, and find that they do not account for all the observed excess. We evaluate a corresponding systematic uncertainty (0.1-3.2%), included in the yield uncertainty above. The uncertainties due to the detector simulation are determined by varying, within bounds given by data control samples, the charged track reconstruction efficiency (1.3-2.0%), the photon reconstruction efficiency (0.2-4.8%), the lepton identification efficiency (1.2-1.6%), and the reconstruction efficiency for low momentum charged (1.2%) and neutral pions (1.3%). We use an HQET model [@LLSW] to test the model dependence of the $\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell}$ simulation (0.8-2.5%). We include the uncertainty on the branching fractions of the reconstructed $D$ and $D^{*}$ modes (3.0-4.5%), and on the absolute branching fraction ${\cal B} (\Bbar \to X \ell^- \bar{\nu}_{\ell})$ used for the normalization (1.9%). We also include a systematic uncertainty due to differences in the efficiency of the $B_\mathrm{tag}$ selection in the exclusive selection of $\Bbar \to D^{**} \ell^- \bar{\nu}_{\ell}$ decays and the inclusive $\Bbar \to X \ell^- \bar{\nu}_{\ell}$ reconstruction (4.0-5.6%). In conclusion, we report the simultaneous observation of $\Bbar \to D^{**}\ell^-\bar{\nu}_{\ell}$ decays into the four $D^{**}$ states predicted by HQS. The measured branching fractions are reported in Table \[tab:results\]. We find results consistent with Ref. [@babar-3] for the sum of the different $D^{**}$ branching fractions. The rate for the $D^{**}$ narrow states is in good agreement with recent measurements [@D0]; the one for the broad states is in agreement with DELPHI [@delphi2005] but does not agree with the $D'_1$ limit of Belle [@belle]. The rate for the broad states is found to be large. If these broad states are indeed due to $\Bbar \to D'_1 \ell^- \bar{\nu}_{\ell}$ and $\Bbar \to D^*_0 \ell^- \bar{\nu}_{\ell}$ decays, this is in conflict with the expectations from QCD sum rules. acknow\_PRL.tex [99]{} M. Kobayashi and T. Maskawa, Prog. Theor. Phys. [**49**]{}, 652 (1973). The charge conjugate state is always implied unless stated otherwise. B. Aubert [*et al.*]{} ( Collab.), Phys. Rev. D[**77**]{}, 032002 (2008). N. Isgur and M. B. Wise, Phys. Rev. Lett. [**66**]{}, 1130 (1991). W. -M. Yao [*et al.*]{} (Particle Data Group), J. Phys. G [**33**]{}, 1 (2006). B. Aubert [*et al.*]{} ( Collab.), Phys. Rev. D[**76**]{}, 051101 (2007). B. Aubert [*et al.*]{} ( Collab.), Phys. Rev. Lett. [**100**]{}, 151802 (2008). A. K. Leibovich, Z. Ligeti, I. W. Stewart and M. B. Wise, Phys. Rev. D[**57**]{}, 308 (1998). N. Uraltsev, Phys.  Lett.  B. [**501**]{}, 86 (2001). D. Liventsev [*et al.*]{} (Belle Collab.), Phys. Rev. D[**77**]{}, 091503 (2008). J. Abdallah [*et al.*]{} (DELPHI Collab.), Eur.  Phys.  J. [**C45**]{} 35 (2006). B. Aubert [*et al.*]{} ( Collab.), Nucl. Instrum. Methods A[**479**]{}, 1 (2002). S. Agostinelli [*et al.*]{}, Nucl. Instrum. Methods A[**506**]{}, 250 (2003). D. Scora and N. Isgur, Phys. Rev. D[**52**]{}, 2783 (1995). See also N. Isgur [*et al.*]{}, Phys. Rev. D[**39**]{}, 799 (1989). J. L. Goity and W. Roberts, Phys. Rev. D[**51**]{}, 3459 (1995). B. Aubert [*et al.*]{} ( Collab.), Phys. Rev. Lett. [**92**]{}, 071802 (2004). B. Aubert [*et al.*]{} ( Collab.), Phys. Rev. D[**69**]{}, 111104 (2004). K. Cranmer, Comput. Phys. Commun. [**136**]{}, 198 (2001). V. Abazov [*et al.*]{} (DØ Collab.), Phys. Rev. Lett. [**95**]{}, 171803 (2005).

--- abstract: | A new symplectic N-body integrator is introduced, one designed to calculate the global $360^\circ$ evolution of a self-gravitating planetary ring that is in orbit about an oblate planet. This freely-available code is called [epi\_int]{}, and it is distinct from other such codes in its use of streamlines to calculate the effects of ring self-gravity. The great advantage of this approach is that the perturbing forces arise from smooth wires of ring matter rather than discreet particles, so there is very little gravitational scattering and so only a modest number of particles are needed to simulate, say, the scalloped edge of a resonantly confined ring or the propagation of spiral density waves. The code is applied to the outer edge of Saturn’s B ring, and a comparison of Cassini measurements of the ring’s forced response to simulations of Mimas’ resonant perturbations reveals that the B ring’s surface density at its outer edge is $\sigma_0=195\pm 60$ gm/cm$^2$ which, if the same everywhere across the ring would mean that the B ring’s mass is about $90\%$ of Mimas’ mass. Cassini observations show that the B ring-edge has several free normal modes, which are long-lived disturbances of the ring-edge that are not driven by any known satellite resonances. Although the mechanism that excites or sustains these normal modes is unknown, we can plant such a disturbance at a simulated ring’s edge, and find that these modes persist without any damping for more than $\sim10^5$ orbits or $\sim100$ yrs despite the simulated ring’s viscosity $\nu_s=100$ cm$^2$/sec. These simulations also indicate that impulsive disturbances at a ring can excite long-lived normal modes, which suggests that an impact in the recent past by perhaps a cloud of cometary debris might have excited these disturbances which are quite common to many of Saturn’s sharp-edged rings. author: - 'Joseph M. Hahn' - 'Joseph N. Spitale' - | Submitted for publication in the\ [*Astrophysical Journal*]{} on December 28, 2012\ Revised April 26, 2013\ Accepted June 1, 2013   bibliography: - 'biblio.bib' title: | An N-body Integrator for Gravitating Planetary Rings,\ and the Outer Edge of Saturn’s B Ring --- Introduction {#intro_section} ============ A planetary ring is often coupled dynamically to a satellite via orbital resonances. The ring’s response to resonant perturbations varies with the forcing, and if the ring is for instance composed of low optical depth dust, then the ring’s response will vary with the satellite’s mass and its proximity. But in an optically thick planetary ring, such as Saturn’s main A and B rings or its many dense narrow ringlets, the ring is also interacting with itself via self gravity, so its response is also sensitive to the ring’s mass surface density $\sigma_0$ [@S84; @MB05; @HSP09]. So by measuring a dense ring’s response to satellite perturbations, and comparing that measurement to a model for the ring-satellite system, one can then infer the ring’s physical properties, such as its surface density $\sigma_0$, and perhaps other quantities too [@MB05; @TBN07; @HSP09]. Recently [@HSP09] developed a semi-analytic model of the outer edge of Saturn’s B ring, which is confined by an $m=2$ inner Lindblad resonance with the satellite Mimas. The resonance index $m$ also describes the ring’s anticipated equilibrium shape, with the ring-edge’s deviations from circular motion expected to have an azimuthal wavenumber of $m=2$. So the B ring’s expected shape is a planet-centered ellipse, which has $m=2$ alternating inward and outward excursions. The model of [@HSP09] also calculates the ring’s equilibrium $m=2$ response excited by Mimas, but that comparison between theory and observation was done during the early days of the Cassini mission when that spacecraft’s measurement of the ring-edge’s semimajor axis $a_{\mbox{\scriptsize edge}}$ was still rather uncertain. It turns out that the ring’s inferred surface density is very sensitive to how far the B ring’s outer edge extends beyond the resonance, which was quite uncertain then due to the uncertainty in $a_{\mbox{\scriptsize edge}}$, so the uncertainty in the ring’s inferred $\sigma_0$ was also relatively large. Now however $a_{\mbox{\scriptsize edge}}$ is known with much greater precision, so a re-examination of this system is warranted. Cassini’s monitoring of the B ring also reveals that the ring’s outer edge exhibits several normal modes, which are unforced disturbances that are not associated with any known satellite resonances. Figure \[Bring\_fig\] illustrates this phenomenon with a mosaic of images that Cassini acquired of the B ring’s edge on 28 January 2008. [@SP10] have also fit a kinematic model to four years worth of Cassini images of the B ring; that model is composed of four normal modes having azimuthal wavenumbers $m=1,2,2,3$ that steadily rotate over time at distinct rates. In the best-fitting kinematic model there are two $m=2$ modes, one that is forced by and corotating with Mimas, as well as a free $m=2$ mode that rotates slightly faster. The amplitudes and orientations of all the modes as they appear in the 28 January 2008 data is also shown in Fig. \[Bring\_fit\_fig\]. Note that although the B ring’s outer edge, as seen in Fig. \[Bring\_fig\], might actually resemble a simple $m=2$ shape on 28 January 2008, at other times the ring-edge’s shape is much more complicated than a simple $m=2$ configuration, yet at other times the ring-edge is relatively smooth and nearly circular; see for example Fig. 1 of [@SP10]. This behavior is due to the superposition of the normal modes that are rotating relative to each other, which causes the B ring’s edge to evolve over time. Since this system is not in simple equilibrium, a time-dependent model of the ring that does not assume equilibrium is appropriate here. So the following develops a new N-body method that is designed specifically to track the time evolution of a self-gravitating planetary ring, and that model is then applied to the latest Cassini results. Section \[method\_section\] describes in detail the N-body model that can simulate all $360^\circ$ of a narrow annulus in a self-gravitating planetary ring using a very modest number of particles. Section \[B ring\] then shows results from several simulations of the outer edge of Saturn’s B ring, and demonstrates how a ring’s observed epicyclic amplitudes and pattern speeds can be compared to N-body simulations to determine the ring’s physical properties. Results are then summarized in Section \[summary\]. Numerical method {#method_section} ================ The following briefly summarizes the theory of the symplectic integrator that [@DLL98] use in their [SYMBA]{} code and [@C99] use in the [MERCURY]{} integrator to calculate the motion of objects in nearly Keplerian orbits about a point-mass star. That numerical method is adapted here so that one can study the evolution of a self-gravitating planetary ring that is in orbit about an oblate planet. symplectic integrators {#symplectic} ---------------------- The Hamiltonian for a system of N bodies in orbit about a central planet is $$\begin{aligned} H &=& \sum_{i=0}^{N}\frac{p_i^2}{2m_i} + \sum_{i=0}^{N}\sum_{j>i}^{N} V_{ij},\end{aligned}$$ where body $i$ has mass $m_i$ and momentum $\mathbf{p}_i = m_i\mathbf{v}_i$ where $\mathbf{v}_i=\mathbf{\dot{r}}_i$ is its velocity and $V_{ij}$ is the potential such that $\mathbf{f}_{ij}=-\nabla_{\mathbf{r}_i}V_{ij}$ is the force on $i$ due to body $j$ where $\nabla_{\mathbf{r}_i}$ is the gradient with respect to coordinate $\mathbf{r}_i$, and the index $i=0$ is reserved for the central planet whose mass is $m_0$. Next choose a coordinate system such that all velocities are measured with respect to the system’s barycenter, so $\mathbf{p}_0 = -\sum_{j=1}^N\mathbf{p}_j$, and the Hamiltonian becomes $$\begin{aligned} H &=& \sum_{i=1}^{N}\left(\frac{p_i^2}{2m_i} + V_{i0}\right) + \sum_{i=1}^{N}\sum_{j>i}^NV_{ij} + \frac{1}{2m_0}\left(\sum_{i=1}^N\mathbf{p}_i\right)^2 \equiv H_A + H_B + H_C\end{aligned}$$ since $V_{ij} = V_{ji}$. This Hamiltonian has three parts, \[H\] $$\begin{aligned} H_A &=& \sum_{i=1}^N\left(\frac{p_i^2}{2m_i} + V_{i0}\right)\\ H_B &=& \sum_{i=1}^N\sum_{j>i}^NV_{ij}\\ H_C &=& \frac{1}{2m_0}\left(\sum_{i=1}^N\mathbf{p}_i\right)^2, \end{aligned}$$ and the following will employ spatial coordinates such that all $\mathbf{r}_i$ are measured relative to the central planet. This combination of planetocentric coordinates and barycentric velocities is referred to as ‘democratic-heliocentric’ coordinates in [@DLL98] and ‘mixed-center’ coordinates in [@C99]. In the above, $H_A$ is the sum of two-body Hamiltonians, $H_B$ represents the particles’ mutual interactions, and $H_C$ accounts for the additional forces that arise in this particular coordinate system that are due to the central planet’s motion about the barycenter. Hamilton’s equations for the evolution of the coordinates $\mathbf{r}_i$ and momenta $\mathbf{p}_i$ for particle $i\ge1$ are $\mathbf{\dot{r}}_i = \nabla_{\mathbf{p}_i}H$ and $\mathbf{\dot{p}}_i = -\nabla_{\mathbf{r}_i}H$. So a particle that is subject only to Hamiltonian $H_B$ during short time interval $\delta t$ would experience the velocity kick $$\begin{aligned} \label{dv} \mathbf{\delta v}_i &=& \frac{{\mathbf{\dot{p}}}_i \delta t}{m_i} = -\nabla_{\mathbf{r}_i}H_B\frac{\delta t}{m_i} = \frac{\delta t}{m_i}\sum_{j=1}^N\mathbf{f}_{ij},\end{aligned}$$ which of course is $i$’s response to the forces exerted by all the other small particles in the system. And since $H_C$ is a function of momenta only, a particle subject to $H_C$ during time $\delta t$ will see its spatial coordinate kicked by $$\begin{aligned} \label{dx} \mathbf{\delta r}_i &=& \frac{\delta t}{m_0}\sum_{j=1}^{N} \mathbf{p}_j\end{aligned}$$ due to the planet’s motion about the barycenter. Now let $\xi_i(t)$ represent any of particle $i$’s coordinates $x_i$ or momenta $p_i$; that quantity evolves at the rate [@G80] $$\begin{aligned} \label{eom} \frac{d\xi_i}{dt} &=& [\xi_i, H] = [\xi_i, H_A + H_B + H_C] =(A + B + C) \xi_i\end{aligned}$$ where the brackets are a Poisson bracket, and the operator $A$ is defined such that $A\xi_i = [\xi_i, H_A]$, with operators $B$ and $C$ defined similarly. The solution to Eqn. (\[eom\]) for $\xi_i$ evaluated at the later time $t + \Delta t$ is formally $$\begin{aligned} \label{eom_soln} \xi_i(t + \Delta t) &=& e^{(A + B + C)\Delta t}\xi_i(t)\end{aligned}$$ [@G80], but this exact expression is in general not analytic and not in a useful form. However [@DLL98] and [@C99] show that the above is approximately $$\begin{aligned} \label{eom_soln_approx} \xi_i(t + \Delta t) &\simeq& e^{B\Delta t/2}e^{C\Delta t/2}e^{A\Delta t} e^{C\Delta t/2}e^{B\Delta t/2}\xi_i(t),\end{aligned}$$ which indicates that five actions that are to occur as the system of orbiting bodies are advanced one timestep $\Delta t$ by the integrator. First ([*i.*]{}) the operator $e^{B\Delta t/2}$ acts on $\xi_i(t)$, which increments ([*i.e.*]{} kicks) particle $i$’s velocity $\mathbf{v}_i$ by Eqn. (\[dv\]) due to the system’s interparticle forces with $\delta t=\Delta t/2$. Then ([*ii.*]{}) the $e^{C\Delta t/2}$ operator acts on the result of substep ([*i.*]{}) and kicks the particle’s spatial coordinates $\mathbf{r}_i$ according to Eqn. (\[dx\]) due to the central planet’s motion about the barycenter. Then in substep ([*iii.*]{}) the $e^{A\Delta t}$ operation advances the particle along its unperturbed epicyclic orbit about the central planet during a full timestep $\Delta t$, with this substep is referred to below as the orbital ‘drift’ step. Step ([*iv.*]{}) is another coordinate kick $\delta\mathbf{r}_i$ and the last step ([*v.*]{}) is the final velocity kick. In a traditional symplectic N-body integrator the planet’s oblateness is treated as a perturbation whose effect would be accounted for during steps ([*i.*]{}) and ([*v.*]{}) which provide an extra kick to a particle’s velocity every timestep. Those kicks cause a particle in a circular orbit to have a tangential speed that is faster than the Keplerian speed by the fractional amount that is of order $\sim J_2(R/r)^2\sim3\times10^{-3}$ where $J_2\simeq0.016$ is Saturn’s second zonal harmonic and $r/R\sim2$ is a B ring particle’s orbit radius $r$ in units of Saturn’s radius $R$. The particle’s circular speed is super-Keplerian, and if its coordinates and velocities were to be converted to Keplerian orbit elements, its Keplerian eccentricity would also be of order $e\sim3\times10^{-3}$. This putative eccentricity should be compared to the observed eccentricity of Saturn’s B ring, which is the focus of this study and is of order $e\sim10^{-4}$, about 30 times smaller than the particle’s Keplerian eccentricity. The main point is, that one does not want to use Keplerian orbit elements when describing a particle’s nearly circular motions about an oblate planet because the Keplerian eccentricity is dominated by planetary oblateness whose effects obscures the ring’s much smaller forced motions. To sidestep this problem, the following algorithm uses the [*epicyclic*]{} orbit elements of [@BL94] which provide a more accurate representation of an unperturbed particle’s orbit about an oblate planet. Note that this use of epicyclic orbit elements effectively takes the effects of oblateness out of the integrator’s velocity kick steps ([*i.*]{}) and ([*v.*]{}) and places oblateness effects in the integrator’s drift step ([*iii.*]{}), which is preferable because the forces in the B ring that are due to oblateness are about $\sim10^4$ times larger than any satellite perturbation. The following details how these epicyclic orbit elements are calculated and are used to evolve the particle along its unperturbed orbit during the drift substep. epicyclic drift {#drift} --------------- This 2D model will track a particle’s motions in the ring plane, so the particle’s position and velocity relative to the central planet can be described by four epicyclic orbit elements: semimajor axis $a$, eccentricity $e$, longitude of periapse $\tilde{\omega}$, and mean anomaly $M$. For a particle in a low eccentricity orbit about an oblate planet, the relationship between the particle’s epicyclic orbit elements and its cylindrical coordinates $r, \theta$ and velocities $v_r, v_\theta$ are \[rv\] $$\begin{aligned} \label{r} r &=& a\left[1 - e\cos M + \left(\frac{\eta_0}{\kappa_0}\right)^2(2 - \cos^2M)e^2 \right] \\ \theta &=& \tilde{\omega} + M + \frac{\Omega_0}{\kappa_0}\left\{2e\sin M + \left[\frac{3}{2} + \left(\frac{\eta_0}{\kappa_0}\right)^2\right]e^2\sin M\cos M\right\}\\ v_r &=& a\kappa_0\left[e\sin M + 2\left(\frac{\eta_0}{\kappa_0}\right)^2e^2\sin M\cos M\right]\\ \label{v_t} v_\theta &=& a\Omega_0\left\{1 + e\cos M - 2\left(\frac{\eta_0}{\kappa_0}\right)^2e^2 + \left[1 + \left(\frac{\eta_0}{\kappa_0}\right)^2\right]e^2\cos^2 M\right\}, \end{aligned}$$ which are adapted from Eqns. (47-55) of [@BL94]. These equations are accurate to order ${\cal O}(e^2)$ and require $e\ll1$. Here $\Omega_0(a)$ is the angular velocity of a particle in a circular orbit while $\kappa_0(a)$ is its epicyclic frequency and the frequency $\eta_0(a)$ is defined below, all of which are functions of the particle’s semimajor axis $a$. Also keep in mind that when the following refers to the particle’s orbit elements, it is the [*epicyclic*]{} orbit elements that are intended[^1], which are distinct from the [*osculating*]{} orbit elements that describe pure Keplerian motion around a spherical planet. But these distinctions disappear in the limit that the planet becomes spherical and the orbit frequencies $\Omega_0, \kappa_0$, and $\eta_0$ all converge on the mean motion $\sqrt{Gm_0/a^3}$, where $G$ is the gravitational constant and $m_0$ is the central planet’s mass; in that case, Eqns. (\[rv\]) recover a Keplerian orbit to order ${\cal O}(e^2)$. The three orbit frequencies $\Omega_0$, $\kappa_0$, and $\eta_0$ appearing in Eqns. (\[rv\]) are obtained from spatial derivatives of the oblate planet’s gravitational potential $\Phi$, which is $$\begin{aligned} \Phi(r) &=& -\frac{Gm_0}{r} + \frac{Gm_0}{r}\sum_{k=1}^{\infty}J_{2k}P_{2k}(0) \left(\frac{R_p}{r}\right)^{2k}\end{aligned}$$ where $R_p$ is the planet’s effective radius, $J_{2k}$ is one of the oblate planet’s zonal harmonics, and $P_{2k}(0)$ is a Legendre polynomial with zero argument. For reasons that will be evident shortly, these calculations will only preserve the $J_{2}$ term in the above sum, so $$\begin{aligned} \Phi(r) &=& -\frac{Gm_0}{r}\left[ 1 + \frac{1}{2}J_2\left(\frac{R_p}{r}\right)^2\right]\end{aligned}$$ and the orbital frequencies are \[orbit\_frequencies\] $$\begin{aligned} \label{Omega^2} \Omega_0^2(a) &=& \left.\frac{1}{r}\frac{\partial\Phi}{\partial r}\right|_{r=a} = \frac{Gm_0}{a^3}\left[1 + \frac{3}{2}J_2\left(\frac{R_p}{a}\right)^2 \right]\\ \label{kappa^2} \kappa_0^2(a) &=& \left.\frac{3}{r}\frac{\partial\Phi}{\partial r}\right|_{r=a} + \left.\frac{\partial^2\Phi}{\partial r^2}\right|_{r=a} = \frac{Gm_0}{a^3}\left[1 - \frac{3}{2}J_2\left(\frac{R_p}{a}\right)^2 \right]\\ \eta_0^2(a) &=& \left.\frac{2}{r}\frac{\partial\Phi}{\partial r}\right|_{r=a} - \left.\frac{r}{6}\frac{\partial^3\Phi}{\partial r^3}\right|_{r=a} = \frac{Gm_0}{a^3}\left[1 - 2J_2\left(\frac{R_p}{a}\right)^2 \right]\\ \beta_0^2(a) &=& - \left.\frac{r^4}{24}\frac{\partial^4\Phi}{\partial r^4}\right|_{r=a} = \frac{Gm_0}{a^3}\left[1 + \frac{15}{2}J_2\left(\frac{R_p}{a}\right)^2 \right] \end{aligned}$$ where the additional frequency $\beta_0(a)$ is needed below. During the particle’s unperturbed epicyclic drift phase its angular orbit elements $M$ and $\tilde{\omega}$ advance during timestep $\Delta t$ by amount \[dM\] $$\begin{aligned} \Delta M &=& \kappa\Delta t\\ \Delta \tilde{\omega} &=& (\Omega - \kappa)\Delta t \end{aligned}$$ where the frequencies $\Omega$ and $\kappa$ in Eqns. (\[dM\]) differ slightly from Eqns. (\[orbit\_frequencies\]) due to additional corrections that are of order ${\cal O}(e^2)$: \[Omega\_kappa\] $$\begin{aligned} \Omega(a,e) &=& \Omega_0\left\{1 + 3\left[\frac{1}{2} - \left(\frac{\eta_0}{\kappa_0}\right)^2\right]e^2 \right\}\\ \kappa(a,e) &=& \kappa_0\left(1 + \left\{\frac{15}{4}\left[\left(\frac{\Omega_0}{\kappa_0}\right)^2 - \left(\frac{\eta_0}{\kappa_0}\right)^4\right] - \frac{3}{2}\left(\frac{\beta_0}{\kappa_0}\right)^2\right\}e^2 \right) \end{aligned}$$ [@BL94]. [@BL94] also show that the above equations have three integrals of the motion: the particle’s specific energy $E$, its specific angular momentum $h$, and its epicyclic energy $I_3$. Those integrals are \[integrals\] $$\begin{aligned} E &=& \frac{1}{2}(v_r^2 + v_\theta^2) + \Phi(r) = \frac{1}{2}(a\Omega_0)^2 + \Phi(a) + \frac{1}{2}(a\kappa_0)^2 e^2 + {\cal O}(e^4)\\ h &=& r v_\theta = a^2\Omega_0 + {\cal O}(e^4)\\ \label{I3} \mbox{and}\quad I_3 &=& \frac{1}{2}[v_r^2 + \kappa_0^2(r-a)^2] - \eta_0^2(r-a)^3/a = \frac{1}{2}(a\kappa_0e)^2 + {\cal O}(e^4). \end{aligned}$$ Advancing the particle along its epicyclic orbit require converting its cylindrical coordinates and velocities into epicyclic orbit elements. To obtain the particle’s semimajor axis, solve the angular momentum integral $h(a)=a^2\Omega_0$, which is quadratic in $a$ so $$\begin{aligned} \label{a} a &=&g\left(1 + \sqrt{1 - \frac{3J_2}{2g^2}}\right) R_p\end{aligned}$$ where $g=(rv_\theta)^2/2Gm_0R_p$. Note though that if the $J_4$ and higher oblateness terms had been preserved in the planet’s potential, then the angular momentum polynomial would be of degree 4 and higher in $a$, for which there is no known analytic solution. That equation could still be solved numerically, but that step would have to be performed for all particles at every timestep, which would slow the N-body algorithm so much as to make it useless. So only the $J_2$ term is preserved here, which nonetheless accounts for the effects of planetary oblateness in a way that is sufficiently realistic. To calculate the particle’s remaining orbit elements, use Eqn. (\[I3\]) to obtain the $I_3$ integral which then provides its eccentricity via $$\begin{aligned} \label{e} e &=& \frac{\sqrt{2I_3}}{a\kappa_0}.\end{aligned}$$ Then set $x=e\cos M$ and $y=e\cos M$ and solve Eqns. (\[r\]) and (\[v\_t\]) for $x$ and $y$: \[xy\] $$\begin{aligned} x &=& \left(\frac{\eta_0}{\kappa_0}\right)^2 \left[2(1+e^2) - \frac{v_\theta}{a\Omega_0} - \frac{r}{a}\right] + 1 - \frac{r}{a}\\ \mbox{and}\quad y &=& \frac{v_r/a\kappa_0}{1 + 2(\eta_0/\kappa_0)^2x}, \end{aligned}$$ which then provides the mean anomaly via $\tan M=y/x$. To summarize, the epicyclic drift step uses Eqns. (\[integrals\]–\[xy\]) to convert each particle’s cylindrical coordinates into epicyclic orbit elements. The particles’ orbit frequencies $\Omega(a,e)$ and $\kappa(a,e)$ are obtained via Eqns. (\[orbit\_frequencies\]) and (\[Omega\_kappa\]), and Eqns. (\[dM\]) are then used to advance each particle’s orbit elements $M$ and $\tilde{\omega}$ during timestep $\Delta t$, with Eqns. (\[rv\]) used to convert the particles’ orbit elements back into cylindrical coordinates. velocity kicks due to the ring’s internal forces {#kicks} ------------------------------------------------ The N-body code developed here is designed to follow the dynamical evolution of all $360^\circ$ of a narrow annulus within a planetary ring, and it is intended to deliver accurate results quickly using a desktop PC. The most time consuming part of this algorithm is the calculation of the accelerations that the gravitating ring exerts on all of its particles, so the principal goal here is to design an algorithm that will calculate these accelerations with sufficient accuracy while using the fewest possible number of simulated particles. ### streamlines The dominant internal force in a dense planetary ring is its self gravity, and the representation of the ring’s full $360^\circ$ extent via a modest number of [*streamlines*]{} provides a practical way to calculate rapidly the acceleration that the entire ring exerts on any one particle. A streamline is the closed path through the ring that is traced by those particles that share a common initial semimajor axis $a$. The simulated portion of the planetary ring will be comprised of $N_r$ discreet streamlines that are spaced evenly in semimajor axis $a$, with each streamline comprised of $N_\theta$ particles on each streamline, so a model ring consists of $N_rN_\theta$ particles. Simulations typically employ $N_r\sim100$ streamlines with $N_\theta\sim50$ particles along each streamline, so a typical ring simulation uses about five thousand particles. Note though that the assignment of particles to a given streamline is merely labeling; particles are still free to wander over time in response to the ring’s internal forces: gravity, pressure, and viscosity. But as the following will show, the simulated ring stays coherent and highly organized throughout the run, in the sense that particles on the same streamline do not pass each other longitudinally, nor do adjacent streamlines cross. Because the simulated ring stays so highly organized, there is no radial or transverse mixing of the ring particles, and the particles will preserve over time membership in their streamline[^2]. ### ring self gravity {#ring_gravity} The concept of gravitating streamlines is widely used in analytic studies of ring dynamics [@GT79; @BGT83aj; @BGT86; @LR95; @HSP09], and the concept is easily implemented in an N-body code. Because the simulated portion of the ring is narrow, its streamlines are all close in the radial sense. Consequently the gravitational pull that one streamline exerts on a particle is dominated by the nearest part of the streamline, with that acceleration being quite insensitive to the fact that the more distant and unimportant parts of the perturbing streamline are curved. So the perturbing streamline can be regarded as a straight and infinitely long wire of matter whose linear density is $\lambda\simeq m_pN_\theta /2\pi a$ to lowest order in the streamline’s small eccentricity $e$, where $m_p$ is the mass of a single particle. The gravitational acceleration that a wire of matter exerts on the particle is $$\begin{aligned} \label{Ag} A_g &=& \frac{2G\lambda}{\Delta}\end{aligned}$$ where $\Delta$ is the separation between the particle and the streamline. However the particles in that streamline only provide $N_\theta$ discreet samplings of a streamline that is after all slightly curved over larger spatial scales. So to find the distance to nearest part of the perturbing streamline, the code identifies at every timestep the three perturbing particles that are nearest in longitude to the perturbed particle. A second-degree Lagrange polynomial is then used to fit a smooth continuous curve through those three particles [@KS], and this polynomial provides a convenient method for extrapolating the perturbing streamline’s distance $\Delta$ from the perturbed particle. This procedure is also illustrated in Fig. \[streamline\_fig\], which shows that the radial and tangential components of that acceleration are \[Ag\_r\_theta\] $$\begin{aligned} A_{g,r} &\simeq& A_g\\ \mbox{and}\qquad A_{g,\theta} &\simeq& -A_g v_r'/v_\theta' \end{aligned}$$ to lowest order in the perturbing streamline’s eccentricity $e'$, where $v_r'$ and $v_\theta'$ are the radial and tangential velocity components of that streamline. Equation (\[Ag\_r\_theta\]) is then summed to obtain the gravitational acceleration that all other streamlines exerts on the particle. To obtain the gravity that is exerted by the streamline that the particle inhabits, treat the particle as if it resides in a gap in that streamline that extends midway to the adjacent particles. The nearby portions of that streamline can be regarded as two straight and semi-infinite lines of matter pointed at the particle whose net gravitational acceleration is $$\begin{aligned} \label{Ag_streamline} A_g &=& 2G\lambda\left(\frac{1}{\Delta_+} - \frac{1}{\Delta_-}\right)\end{aligned}$$ where $\Delta_+$ and $\Delta_-$ are the particle’s distance from its neighbors in the leading (+) and trailing (-) directions. The radial and tangential components of that streamline’s gravity are \[Ag\_r\_theta\_streamline\] $$\begin{aligned} A_{g,r} &\simeq& A_g v_r/v_\theta\\ \mbox{and}\qquad A_{g,\theta} &\simeq& A_g \end{aligned}$$ where $v_r, v_\theta$ are the perturbed particle’s velocity components. A major benefit of using Eqn. (\[Ag\]) to calculate the ring’s gravitational acceleration is that there is no artificial gravitational stirring. This is in contrast to a traditional N-body model that would use discreet point masses to represent what is really a continuous ribbon of densely-packed ring matter. Those gravitating point masses then tug on each other in amounts that very rapidly in magnitude and direction as they drift past each other in longitude, and those rapidly varying tugs will quickly excite the simulated particles’ dispersion velocity. As a result, the particles’ unphysical random motions tend to wash out the ring’s large-scale coherent forced motions, which is usually the quantity that is of interest. So, although Eqn. (\[Ag\]) is only approximate because it does not account for the streamline’s curvature that occurs far away from a perturbed ring particle, Eqn. (\[Ag\]) is still much more realistic and accurate than the force law that would be employed in a traditional global N-body simulation of a planetary ring, which out of computational necessity would treat a continuous stream of ring matter as discreet clumps of overly massive gravitating particles. ### ring pressure {#pressure} A planetary ring is very flat and its vertical structure will be unresolved in this model, so a 1D pressure $p$ is employed here. That pressure $p$ is the rate-per-length that a streamline segment communicates linear momentum to the adjacent streamline orbiting just exterior to it, with that momentum exchange being due to collisions occurring among particles on adjacent streamlines. So for a small streamline segment of length $\delta\ell$ that resides somewhere in the ring’s interior, the net force on that segment due to ring pressure is $\delta f = [p(r-\Delta) - p(r)]\delta\ell$ since $p(r-\Delta)$ is the pressure or force-per-length exerted by the streamline that lies just interior and a distance $\Delta$ away from segment $\delta\ell$, and $p(r)$ is the force-per-length that segment $\delta\ell$ exerts on the exterior streamline. And since force $\delta f = A_p \delta m$ where $\delta m=\lambda\delta\ell$ is the segment’s mass, the acceleration on a particle due to ring pressure is $$\begin{aligned} \label{Ap} A_p &=& \frac{\delta f}{\delta m} = \frac{p(r-\Delta) - p(r)}{\lambda} \simeq-\frac{\Delta}{\lambda}\frac{\partial p}{\partial r} =-\frac{1}{\sigma}\frac{\partial p}{\partial r}\end{aligned}$$ since the ring’s surface density $\sigma=\lambda/\Delta$. Formulating the acceleration in terms of pressure differences across adjacent streamlines is handy because the model can then easily account for the large pressure drop that occurs at a planetary ring’s edge, which can be quite abrupt when the ring’s edge is sharp. For a particle orbiting at the ring’s innermost streamline, the acceleration there is simply $A_p=-p(r)/\lambda$ since there is no ring matter orbiting interior to it so $p(r-\Delta)=0$ there; likewise the acceleration of a particle in the ring’s outermost streamline is $A_p = p(r-\Delta)/\lambda$. Pressure is exerted perpendicular to the streamline and hence it is predominantly a radial force, so by the geometry of Fig. \[streamline\_fig\] the radial component of the acceleration due to pressure is $A_{p,r}\simeq A_p$ while the tangential component $A_{p,\theta}\simeq-A_pv_r/v_\theta$ is smaller by a factor of $e$, where $v_r$ and $v_\theta$ are the perturbed particle’s radial and tangential velocities. This accounts for the pressure on the particle due to adjacent streamlines. The acceleration on the particle due to pressure gradients in the particle’s streamline is simply $A_p=-(\partial p/\partial\theta)/(r\sigma)$. This acceleration points in the direction of the particle’s motion, so the radial and tangential components of that acceleration are $A_{p,r}\simeq A_p v_r/v_\theta$ and $A_{p,\theta}\simeq A_p$. Acceleration due to pressure requires selecting an equation of state (EOS) that relates the pressure $p$ to the ring’s other properties, and this study will treat the ring as a dilute gas of colliding particles for which the 1D pressure is $p=c^2\sigma$ where $c$ is the particles dispersion velocity. However alternate EOS exist for planetary rings, and that possibility is discussed in Section \[EOS\]. A simple finite difference scheme is used to calculate the pressure gradient in Eqn. (\[Ap\]) in the vicinity of particle $i$ in streamline $j$ that lies at at longitude $\theta_{i,j}$. Lagrange polynomials are again used to evaluate the adjacent streamlines’ planetocentric distances $r_{i, j-1}$ and $r_{i, j+1}$ along the particle’s longitude $\theta_{i,j}$, so the pressure gradient at particle $i$ in streamline $j$ is $$\begin{aligned} \label{dp_dr} \left.\frac{\partial p}{\partial r}\right|_{i,j} &\simeq& \frac{p_{i,j+1} - p_{i, j-1}}{r_{i, j+1} - r_{i, j-1}}\end{aligned}$$ where the pressures in the adjacent streamlines $p_{i, j+1}$ and $p_{i, j-1}$ are also determined by interpolating those quantities to the perturbed particle’s longitude $\theta_{i,j}$. The surface density $\sigma_{i,j}$ in the vicinity of particle $i$ in streamline $j$ is determined by centering a box about that particle whose radial extent spans half the distance to the neighboring streamlines, so $$\begin{aligned} \label{sigma} \sigma_{i,j} &=& \frac{2\lambda_j}{r_{i,j+1} - r_{i,j-1}}.\end{aligned}$$ If however streamline $j$ lies at the ring’s inner edge where $j=0$ then the surface density there is $\sigma_{i,0}=\lambda_0/(r_{i,1} - r_{i,0})$ while the surface density at the outermost $j=N_r -1$ streamline is $\sigma_{i, N_r - 1}=\lambda_{N_r - 1}/(r_{i, N_r - 1} - r_{i, N_r - 2})$. ### ring viscosity {#viscosity} Viscosity has two types, shear viscosity and bulk viscosity. Shear viscosity is the friction that results as particles on adjacent streamlines collide as they flow past each other. The friction due to this shearing motion causes adjacent streamlines to torque each other, so shear viscosity communicates a radial flux of angular momentum through the ring. A particle on a streamline experiences a net torque and hence a tangential acceleration when there is a radial gradient in that angular momentum flux. And if there are additional spatial gradients in the ring’s velocities that cause ring particles to converge towards or diverge away from each other, then these relative motions will cause ring particles to bump each other as they flow past, which transmits momentum through the ring via the pressure forces discussed above. However the ring particles’ viscous bulk friction tends to retard those relative motions, and that friction results in an additional flux of linear momentum through the ring. Any radial gradients in that linear momentum flux then results in a radial acceleration on a ring particle. The 1D radial flux of the $z$ component of angular momentum due to the ring’s shear viscosity is derived in Appendix \[shear\_appendix\]: $$\begin{aligned} \label{F_shear} F &=& -\nu_s\sigma r^2\frac{\partial\dot{\theta}}{\partial r}\end{aligned}$$ (see Eqn. \[F\_app\]) where $\nu_s$ is the ring’s kinematic shear viscosity and $\dot{\theta}=v_\theta/r$ is the angular velocity. The quantity $F$ is the rate-per-length that one streamline segment communicates angular momentum to the adjacent streamline orbiting just exterior, so the net torque on a streamline segment of length $\delta\ell$ is $\delta\tau=[F(r-\Delta) - F(r)]\delta\ell$ but $\delta\tau=rA_{\nu,\theta}\delta m$ where $\delta m = \lambda\delta\ell$ so the tangential acceleration due to the ring’s shear viscosity is $$\begin{aligned} \label{A_vs} A_{\nu,\theta} &=& \frac{F(r-\Delta) - F(r)}{\lambda r} =-\frac{1}{\sigma r}\frac{\partial F}{\partial r}.\end{aligned}$$ Again this differencing approach is useful because it easily accounts for the large viscous torque that occurs at a ring’s sharp edge since $A_{\nu,\theta}=- F(r)/\lambda r$ at the ring’s inner edge and $A_{\nu,\theta}=F(r - \Delta)/\lambda r$ at the ring’s outer edge. Appendix \[bulk\_appendix\] shows that the radial flux of linear momentum due to the ring’s shear and bulk viscosity is $$\begin{aligned} \label{G} G &=& -\left(\frac{4}{3}\nu_s + \nu_b\right)\sigma\frac{\partial v_r}{\partial r} - \left(\nu_b - \frac{2}{3}\nu_s\right)\frac{\sigma v_r}{r}\end{aligned}$$ (Eqn. \[G\_appendix2\]) where $\nu_b$ is the ring’s bulk viscosity. This quantity is analogous to a 1D pressure so the corresponding acceleration is $$\begin{aligned} \label{A_vb} A_{\nu,r} &=& \frac{G(r-\Delta) - G(r)}{\lambda} =-\frac{1}{\sigma}\frac{\partial G}{\partial r}\end{aligned}$$ in the ring’s interior and $A_{\nu,r}=-G(r)/\lambda$ or $A_{\nu,r}=G(r-\Delta)/\lambda$ along the ring’s inner or outer edges. To evaluate the partial derivatives that appear in the flux equations (\[F\_shear\]) and (\[G\]), Lagrange polynomials are again used to determine the angular and radial velocities $\dot{\theta}$ and $v_r$ in the adjacent streamlines, interpolated at the perturbed particle’s longitude, with finite differences used to calculate the radial gradients in those quantities. ### satellite gravity {#sat_gravity} All ring particles are also subject to each satellite’s gravitational acceleration, $A_s=Gm_s/\Delta^2$, where $m_s$ is the satellite’s mass and $\Delta$ is the particle-satellite separation. Satellites also feel the gravity exerted by all the ring particles, as well as the satellites’ mutual gravitational attractions. And once all of the accelerations of each ring particle and satellite are tallied, each body is then subject to the corresponding velocity kicks of steps ([*i.*]{}) and ([*v.*]{}) that are described just below Eqn. (\[eom\_soln\_approx\]). tests of the code {#tests} ----------------- The N-body integrator developed here is called [epi\_int]{}, which is shorthand for [*epicyclic integrator*]{}, and the following briefly describes the suite of simulations whose known outcomes are used to test all of the code’s key parts. [**Forced motion at a Lindblad resonance:**]{} numerous massless particles are placed in circular orbits at Mimas’ $m=2$ inner Lindblad resonance. In this test, Mimas’ initially zero mass is slowly grown to its current mass over an exponential timescale $\tau_s=1.6\times10^4$ ring orbits, which excites adiabatically the ring particle’s forced eccentricities to levels that are in excellent agreement with the solution to the linearized equations of motion, Eqn. (42) of [@GT82]. Similar results are also obtained for the particle’s response to Janus’ $m=7$ inner Lindblad resonance, which is responsible for confining the outer edge of Saturn’s A ring. These simulations test the implementation of the integrator’s kick-step-drift scheme as well as the satellite’s forcing of the ring. [**Precession due to oblateness:**]{} this simple test confirms that the orbits of massless particles in low eccentricity orbits precess at the expected rate, $\dot{\tilde{\omega}}(a)=\Omega - \kappa = \frac{3}{2}J_2(R_p/a)^2\Omega(a)$, due to planetary oblateness $J_2$. [**Ringlet eccentricity gradient and libration:**]{} when a narrow eccentric ringlet is in orbit about an oblate planet, dynamical equilibrium requires the ringlet to have a certain eccentricity gradient so that differential precession due to self-gravity cancels that due to oblateness. And when the ringlet is composed of only two streamlines then this scenario is analytic, with the ringlet’s equilibrium eccentricity gradient given by Eqn. (28b) of [@BGT83]. So to test [epi\_int]{}’s treatment of ring self-gravity, we perform a suite of simulations of narrow eccentric ringlets that have surface densities $40<\sigma<1000$ gm/cm$^2$ with initial eccentricity gradients given by Eqn. (28b), and integrate over time to show that these pairs of streamlines do indeed precess in sync with no relative precession, as expected, over runtimes that exceed of the timescale for massless streamlines to precess differentially. And when we repeat these experiments with the ringlets displaced slightly from their equilibrium eccentricity gradients, we find that the simulated streamlines librate at the frequency given by Eqn. (30) of [@BGT83], as expected. [**Density waves in a pressure-supported disk:**]{} this test examines the model’s treatment of disk pressure, and uses a satellite to launch a two-armed spiral density wave at its $m=2$ ILR in a non-gravitating pressure supported disk. The resulting pressure wave has a wavelength and amplitude that agrees with Eqn. (46) of [@W86], as expected. [**Viscous spreading of a narrow ring:**]{} in this test [epi\_int]{} follows the radial evolution of an initially narrow ring as it spreads radially due to its viscosity, and the simulated ring’s surface density $\sigma(r,t)$ is in excellent agreement with the expected solution, Eqn. (2.13) of [@P81]. Simulations of the Outer Edge of Saturn’s B Ring {#B ring} ================================================ The semimajor axis of the outer edge of Saturn’s B ring is $a_{\mbox{\scriptsize edge}}=117568\pm4$ km, and that edge lies $\Delta a_2=12\pm4$ km exterior to Mimas’ $m=2$ inner Lindblad resonance (ILR) (@SP10, hereafter SP10). Evidently Mimas’ $m=2$ ILR is responsible for confining the B ring and preventing it from viscously diffusing outwards and into the Cassini Division. Mimas’ $m=2$ ILR excites a forced disturbance at the ring-edge whose radius–longitude relationship $r(\theta)$ is expected to have the form $r(\theta, t) = a_{\mbox{\scriptsize edge}} - R_m\cos m(\theta - \tilde{\omega}_m)$ where $R_m$ is the epicyclic amplitude of the mode whose azimuthal wavenumber is $m$ and whose orientation at time $t$ is given by the angle $\tilde{\omega}_m(t)$. This forced disturbance is expected to corotate with Mimas’ longitude, and such a pattern would have a pattern speed $\dot{\tilde{\omega}}_m=d\tilde{\omega}_m/dt$ that satisfies $\dot{\tilde{\omega}}_m=\Omega_s$ where $\Omega_s$ is satellite Mimas’ angular velocity. SP10 have analyzed the many images of the B ring’s edge that have been collected by the Cassini spacecraft, and they show that this ring-edge does indeed have a forced $m=2$ shape that corotates with Mimas as expected. But they also show that the B ring’s edge has an additional [*free*]{} $m=2$ pattern that rotates slightly faster than the forced pattern. SP10 also detect two additional modes, a slowly rotating $m=1$ pattern as well as a rapidly rotating $m=3$ pattern. These findings are confirmed by stellar occulation observations of the B ring’s outer edge that also detect additional lower-amplitude $m=4$ and $m=5$ modes [@NFH12]. The following will use the N-body model to investigate the higher amplitude $m=1,2$, and 3 modes seen at the B ring’s edge. But keep in mind that only the $m=2$ forced pattern has a known driver, namely, Mimas’ $m=2$ ILR, while the nature of the perturbation that launched the other three free modes in the B ring is quite unknown. So to study the B ring’s behavior when those free modes are present, an admittedly ad hoc method is used. Specifically, the simulated ring particles’ initial conditions are constructed in a way that plants a free $m=1,2$, or 3 pattern at the simulated ring’s edge at time $t=0$. The N-body integrator then advances the system over time, which then reveals how those free patterns evolve over time. And to elucidate those findings most simply, the following subsections first consider the B ring’s $m=1$, 2, and 3 patterns in isolation. All simulations use a timestep $\Delta t=0.2/2\pi=0.0318$ orbit periods, so there are 31.4 timesteps per orbit of the simulated B ring, and nearly all simulations use oblateness $J_2=0.01629071$, which is the same value we used in previous work [@HSP09]. And lastly, these simulations also zero the viscous acceleration that is exerted at the simulated B ring’s innermost and outermost streamlines, to prevent them from drifting radially due to the ring’s viscous torque. This is in fact appropriate for the simulation’s innermost streamline, since in reality the viscous torque from the unmodeled part of the B ring should deliver to the inner streamline a constant angular momentum flux $F$ that it then communicates to the adjacent streamline, so the viscous acceleration $A_{\nu,\theta}\propto \partial F/\partial r$ at the simulation’s inner edge really should be zero. But zeroing the viscous acceleration of outer streamline might seem like a slight-of-hand since it should be $A_{\nu,\theta}=F/\lambda r$ according Section \[viscosity\]. But setting $A_{\nu,\theta}=0$ is done because, if not, then the outermost streamline will slowly but steadily drifts radially outwards past Mimas’ $m=2$ ILR, which also causes that streamline’s forced eccentricity to slowly and steadily grow as the streamline migrates. This happens because the model does not settle into a balance where the ring’s positive viscous torque on its outermost streamline is opposed by a negative torque exerted by the satellite’s gravity. We also note that the semi-analytic model of this resonant ring-edge, which is described in [@HSP09], also had the same difficulty in finding a torque balance. So to sidestep this difficulty, this model zeros the viscous acceleration at the outermost streamline, which keeps its semimajor axis static as if it were in the expected torque balance. This then allows us to compare simulations to the B ring’s forced $m=2$ pattern to that measured by the Cassini spacecraft. The validity of this approximation is also assessed below in Section \[force\]. the forced and free patterns {#m=2} ---------------------------- SP10 detect a forced $m=2$ pattern at the B ring’s outer edge that has an epicyclic amplitude $R_2=34.6\pm0.4$ km, and that forced pattern corotates with the satellite Mimas. They also detect a free pattern whose epicyclic amplitude is $2.7$ km larger, so the forced and free patterns are nearly equal in amplitude, and the free pattern rotates slightly faster than the forced pattern by $\Delta\dot{\tilde{\omega}}_2=0.0896\pm0.0007$ degrees/day (SP10). The radius-longitude relationship for a ring-edge that experiences these two modes can be written $$\begin{aligned} \label{dr_2} r(\theta, t) &=& a - R_2\cos m(\theta - \theta_s) - \tilde{R}_2\cos m(\theta - \tilde{\omega}_2)\end{aligned}$$ where $R_2$ is the epicyclic amplitude of the forced pattern that corotates with Mimas whose longitude is $\theta_s(t)$ at time $t$, and $\tilde{R}_2$ is the epicyclic amplitude of the free pattern with $\tilde{\omega}_2(t)$ being the free pattern’s longitude. The N-body integrator [epi\_int]{} is used to simulate the forced and free $m=2$ patterns that are seen at the outer edge of the B ring, for simulated rings having a variety of initial surface densities $\sigma_0$. These simulations use $N_r=130$ streamlines that are distributed uniformly in the radial direction with spacings $\Delta a=5.13$ km, so the radial width of the simulated portion of the B ring is $w=(N_r - 1)\Delta a=662$ km. Each streamline is populated with $N_\theta=50$ particles that are initially distributed uniformly in longitude $\theta$ and in circular coplanar orbits. These simulations use a total of $N_rN_\theta=6500$ particles, which is more than sufficient to resolve the $m=2$ patterns seen here. These systems are evolved for $t=41.5$ years, which corresponds to $3.2\times10^4$ orbits, and is sufficient time to see the simulation’s slightly faster free $m=2$ pattern lap the forced $m=2$ pattern several times. The execution time for these high resolution, publication-quality simulations is 1.5 days on a desktop PC, but sufficiently useful preliminary results from lower-resolution simulations can be obtained in just a few hours. The B ring’s viscosity is unknown, so these simulations will employ a value for the kinematic shear viscosity $\nu_s$ and bulk viscosity $\nu_b$ that are typical of Saturn’s A ring, $\nu_s=\nu_b=100$ cm$^2$/sec [@TBN07]. The simulated particles’ dispersion velocity $c$ is also chosen so that the ring’s gravitational stability parameter $Q=c\kappa/\pi G\sigma_0=2$, since Saturn’s main rings likely have $1\lesssim Q\lesssim2$ [@S95]. Mimas’ mass is $m_s=6.5994\times10^{-8}$ Saturn masses, and its semimajor axis $a_s$ is chosen so that its $m=2$ inner Lindblad resonance lies $\Delta a_{\mbox{\scriptsize res}}=12.2$ km interior to the simulated B ring’s outer edge. This model only accounts for the $J_2=0.01629071$ part of Saturn’s oblateness, so the constraint on the resonance location puts the simulated Mimas at $a_s=185577.0$ km, which is 38 km exterior to its real position. Starting the ring particles in circular orbits provides an easy way to plant equal-amplitude free and forced $m=2$ patterns in the ring. This creates a free $m=2$ pattern that at time $t=0$ nulls perfectly the forced $m=2$ pattern due to Mimas. However the free pattern rotates slightly faster than the forced pattern, so the ring’s epicyclic amplitude varies between near zero and $\sim2R_2$ as the rotating patterns interfere constructively or destructively over time. This behavior is illustrated in Fig. \[m=2\_fig\] which shows results from a simulation of a B ring whose undisturbed surface density is $\sigma_0=280$ gm/cm$^2$. The wire diagrams show the ring’s streamlines via radius versus longitude plots, with dots indicating individual particles, and the adjacent grayscale map shows the ring’s surface density at that instant. Figure \[m=2\_fig\] shows snapshots of the system at five distinct times that span one cycle of the ring’s circulation: at time $t=26.4$ yr when the ring’s outermost streamline is nearly circular due to the forced and free patterns being out of phase by nearly $180^\circ/m=90^\circ$ and interfering destructively, to time $t=28.2$ yr when the forced and free patterns are in phase and interfere constructively, to nearly circular again at time $t=30.0$ yr. The circulation cycle seen in Fig. \[m=2\_fig\] repeats for the duration of the integration, which spans about 10 cycles. The gray lines in Fig. \[m=2\_fig\] show the semimajor axes $a$ of all particles on each streamline; note that all particles on a given streamline preserve a common semimajor axes, and this is also true of their eccentricities $e$. In the simulations shown here, the two orbit elements $a$ and $e$ do not vary with the particle’s longitude $\theta$. This however is distinct from the particles’ angular orbit elements $M$ and $\tilde{\omega}$, which do vary linearly with longitude $\theta$ along each streamline. Recall that the [epi\_int]{} code does not in any way force or require particles to inhabit a given streamline. The streamline concept is only used when calculating the forces that all of the ring’s streamlines exert on each particle, which the symplectic integrator then uses to advance these particles forwards in time. Although a particle’s $e$ and $a$ are in principle free to drift away from that of the other streamline-members, that does not happen in the simulations shown here; evidently the particles’ $a$ and $e$ evolve slowly in the orbit-averaged sense, with that time-averaged evolution being independent of longitude $\theta$. This accounts for why all particles on the same streamline have the same evolution in $a$ and $e$. This time-averaged evolution is also a standard assumption that is routinely invoked in analytic models of planetary rings (see [*cf*]{}. @GT79 [@BGT86; @HSP09]), and the simulations shown here confirm the validity of that assumption. A suite of seven B ring simulations is performed for rings whose undisturbed surface densities range over $120\le\sigma_0\le360$ gm/cm$^2$. Results are summarized in Fig. \[R2\_fig\] which shows the forced epicyclic amplitude $R_2$ (solid curve) and the free epicyclic amplitude $\tilde{R}_2$ (dashed curve) from each simulation. These amplitudes are obtained by fitting Eqn. (\[dr\_2\]) to the simulated B ring’s outermost streamline assuming that the free pattern there rotates at a constant rate, $\tilde{\omega}_2(t) = \tilde{\omega}_0 + \dot{\tilde{\omega}}_2t$ where $\tilde{\omega}_0$ is the free pattern’s angular offset at time $t=0$ and $\dot{\tilde{\omega}}_2$ is the free mode’s pattern speed. Equation (\[dr\_2\]) provides an excellent representation of the ring-edge’s behavior over time, and that equation has four parameters $R_2, \tilde{R}_2, \tilde{\omega}_0$, and $\dot{\tilde{\omega}}_2$ that are determined by least squares fitting. The observed epicyclic amplitude of the B ring’s forced $m=2$ component is $R_2=34.6\pm0.4$ km (SP10), and the gray bar in Fig. \[R2\_fig\] indicates that the outer edge of the B ring has a surface density of about $\sigma_0=195$ gm/cm$^2$. And if we naively assume that the ring’s surface density is everywhere the same, then its total mass of Saturn’s B ring is about $90\%$ of Mimas’ mass. Figure \[R2\_fig\] also shows that the amplitude of the forced pattern $R_2$ gets larger for rings that have a smaller surface density $\sigma_0$, due to the ring’s lower inertia, with the forced response varying roughly as $R_2\propto\sigma_0^{-0.67}$. This also makes lighter rings more difficult to simulate, because their larger epicyclic amplitudes also causes the ring’s streamlines to get more bunched up at periapse. For instance in the $\sigma_0=280$ gm/cm$^2$ simulation of Fig. \[m=2\_fig\], the ring’s edge at longitudes $\theta=\theta_s$ and $\theta=\theta_s\pm\pi$ are overdense by a factor of 3 at time $t=28.2$ yr, which is when the force and free patterns add constructively. Streamline bunching in lighter rings is even more extreme, which is also more problematic, because streamlines that are too compressed can at times cross in these overdense sites, and the simulated ring’s subsequent evolution becomes unreliable. To avoid the streamline crossing that occurs in simulations of lower surface density, the model also grows the mass of Mimas exponentially over the timescale $\tau_s$ that takes values of $0.41\le \tau_s\le6.2$ years, with faster satellite growth ($\tau_s=0.41$ yrs or 320 B ring orbits) occurring in simulations of a heavy B ring having $\sigma_0\ge280$ gm/cm$^2$ and slower growth ($\tau_s=6.2$ yrs or 4800 B ring orbits) for the lighter $\sigma_0\le240$ gm/cm$^2$ ring simulations. The satellite growth timescale $\tau_s$ controls the amplitude of the free pattern $\tilde{R}_2$, with the ring having a smaller free epicyclic amplitude $\tilde{R}_2$ when $\tau_s$ is larger; see the dashed curve in Fig. \[R2\_fig\]. Indeed, when the satellite grows over a timescale $\tau_s\gg6.2$ yrs ([*i.e.*]{} $\tau_s\gg4800$ orbits), the ring responds adiabatically to forcing by the slowly growing Mimas, and shows only a forced $m=2$ pattern that corotates with Mimas, with the free $m=2$ pattern having a negligible amplitude. Consequently, only the $\sigma_0=280, 320$, and $360$ gm/cm$^2$ simulations in Fig. \[R2\_fig\] are faithful in their attempt to reproduce a B ring whose free epicyclic amplitude $\tilde{R}_2$ is slightly larger than the forced amplitude $R_2$. However the lower-surface density simulations have free patterns whose amplitudes are smaller than the forced patterns, and these simulated rings have outer edges whose longitude of periapse librate about Mimas’ longitude, rather than circulate. Also of interest here is the so-called radial depth of the $m=2$ disturbance, $\Delta a_{e/10}$, which is defined as the semimajor axis separation between the ring’s outer edge and the streamline whose mean eccentricity is one-tenth that of the edge’s eccentricity. For these $m=2$ simulations the radial depth is $\Delta a_{e/10}=154$km, so the radial width of the simulated part of the ring is $w=4.3\Delta a_{e/10}$. ### sensitivity to resonance location and other factors {#edge_sensitivity} The surface density $\sigma_0$ that is inferred from the amplitude of the ring’s forced motion $R_2$ is very sensitive to the uncertainty in the ring’s semimajor axis, which is $\delta a_{\mbox{\scriptsize edge}}$. For example, when the B ring is simulated again but with its outer edge instead extending further out by $\delta a_{\mbox{\scriptsize edge}}=4$ km, those simulations show that the ring’s forced amplitude $R_2$ is larger by about 6 km, which requires increasing $\sigma_0$ by $\delta\sigma_0=60$ gm/cm$^2$ so that the simulated $R_2$ is in agreement with the observed value. Similarly, when the simulated ring’s edge is moved inwards by $\delta a_{\mbox{\scriptsize edge}}=4$ km, the forced amplitude $R_2$ is smaller and the ring’s surface density $\sigma_0$ must be decreased by $\delta\sigma_0$ to compensate. So the surface density of the B ring-edge is $\sigma_0=195\pm60$ gm/cm$^2$, and this value represents the mean surface density of outer $\Delta a_{e/10}\simeq150$km that is most strongly disturbed by Mimas’ $m=2$ resonance. These results are also in excellent agreement with the semi-analytic model of [@HSP09], which calculated only the ring’s forced motion. However these results are very insensitive to the model’s other main unknown, the ring’s viscosity $\nu$. For instance, when we re-run the $\sigma_0=200$ gm/cm$^2$ simulation with the ring’s shear and bulk viscosities increases as well as decreased by a factor of 10, we obtain the same forced response $R_2$. So these findings are insensitive to range of ring viscosities considered here, $10<\nu<1000$ cm$^2$/sec. ### free $m=2$ pattern {#free m=2} The dotted curve in Fig. \[m2\_fig\] shows the simulations’ free $m=2$ pattern speeds $\dot{\tilde{\omega}}_2$, which is also sensitive to the ring’s undisturbed surface density $\sigma_0$. The purpose of this subsection is to illustrate how a free normal mode can also be used to determine the ring’s surface density. Although these result will not be as definitive as the value of $\sigma_0$ that was inferred from the ring’s forced pattern, due to a greater sensitivity to the observational uncertainties, the following illustrates an alternate technique that in principle can be used to infer the surface density of other rings, such as the many narrow ringlets orbiting Saturn that also exhibit free normal modes. But first note the models’ large discrepancy with the observed free $m=2$ pattern speed reported in SP10, which is the upper horizontal bar in Fig. \[m2\_fig\]. This discrepancy is [*not*]{} due to the $\delta a_{\mbox{\scriptsize edge}}=\pm4$km uncertainty in the ring-edge’s semimajor axis, which makes the simulated ring particles’ mean angular velocity uncertain by the fraction $\delta\Omega/\Omega=1.5\delta a_{\mbox{\scriptsize edge}}/a_{\mbox{\scriptsize edge}} \simeq 0.005\%$. We find empirically that the simulations’ pattern speeds are also uncertain by this fraction, so $\delta\dot{\tilde{\omega}}_2\simeq0.02$ deg/day, which is the vertical extent of the gray band around the simulated data in Fig. \[m2\_fig\]. Rather, this discrepancy is indirectly due to the absence of the $J_4$ and higher terms from the N-body simulations. To demonstrate this, repeat the $\sigma_0=200$ gm/cm$^2$ simulation with $J_2$ boosted slightly by factor $f^\star=1.0395013$ so that the second zonal harmonic is $J_2^\star=f^\star J_2=0.016934294$. This increases the simulated B ring-edge’s angular velocity slightly to $\Omega_{\mbox{\scriptsize edge}}=758.8824$ deg/day, which is in fact the ring’s true angular velocity at $a=a_{\mbox{\scriptsize edge}}$ when the higher order $J_4$ and $J_6$ terms are also accounted for[^3]. And since Saturn’s gravitational force there is $a_{\mbox{\scriptsize edge}}\Omega_{\mbox{\scriptsize edge}}^2$, this means that Saturn’s gravity on the simulated particles at $r=a_{\mbox{\scriptsize edge}}$ is in fact the true value. Note that boosting $J_2$ to the slightly larger value $J_2^\star$ also requires bringing the simulated Mimas inwards and just interior to its true semimajor axis by 2km. Which speeds up both the forced and free pattern speeds, and is why this simulation’s free $m=2$ pattern speed $\dot{\tilde{\omega}}_2$, which is the cross in Fig. \[m2\_fig\], is in better agreement with the observed pattern speed. So the discrepancy between all the other simulated and observed pattern speeds $\dot{\tilde{\omega}}_2$ is due to those models’ not accounting for the additional gravity that is due to the $J_4$ and higher terms in Saturn’s oblate figure. Compensating for the absence of those oblateness effects requires altering the simulated satellite’s orbits slightly, which in turn alters the forced and free pattern speeds slightly. But the following will show that these two patterns’ [*relative*]{} speeds are quite insensitive to the particular value of $J_2$ and the absence of the $J_4$ and higher terms. The best way to compare simulated to observed free $m=2$ patterns is to consider the free $m=2$ pattern speed relative to the forced pattern speed, which is the satellite’s mean angular velocity $\Omega_{\mbox{\scriptsize sat}}$. That frequency difference is $\Delta\dot{\tilde{\omega}}_2=\dot{\tilde{\omega}}_2-\Omega_{\mbox{\scriptsize sat}}$, and is plotted versus ring surface density $\sigma_0$ in Fig. \[m2\_rel\_fig\]. Black dots are from the simulation and the light gray band indicates the $\delta\dot{\tilde{\omega}}_2\simeq0.02$ deg/day spread that results from the $\delta a_{\mbox{\scriptsize edge}}=\pm4$ km uncertainty in the ring-edge’s semimajor axis. The relatively large uncertainty in $a_{\mbox{\scriptsize edge}}$ means that one can only conclude from Fig. \[m2\_rel\_fig\] that $\sigma_0\lesssim210$ gm/cm$^2$. If however the uncertainty in $a_{\mbox{\scriptsize edge}}$ were instead $\delta a_{\mbox{\scriptsize edge}}=\pm1$ km, then the uncertainty in $\Delta\dot{\tilde{\omega}}_2$ would be 4 times smaller (darker gray band), which would have allowed us to determine the ring surface density with a much smaller uncertainty of only $\pm20$ gm/cm$^2$. The lesson here is that if one wishes to use models of free patterns to infer $\sigma_0$ in, say, narrow ringlets, then one will likely need to know the ring-edge’s semimajor axis with a precision of $\delta a_{\mbox{\scriptsize edge}}\simeq\pm1$ km. The cross in Fig. \[m2\_rel\_fig\] indicates that the the free $m=2$ pattern speed relative to the forced is unchanged when Saturn’s oblateness is boosted to $J_2^\star$. And to demonstrate that this kind of plot is rather insensitive to oblateness effects, the white dot in Fig. \[m2\_rel\_fig\] shows that these relative pattern speeds change only very slightly even when $J_2$ is set to zero. Note though that there will be instances where there is no forced mode with which to compare pattern speeds. In that case it will be convenient to convert the free pattern speed $\dot{\tilde{\omega}}_m=\Omega_{ps}$ into a radius by solving the Lindblad resonance criterion $$\begin{aligned} \label{resonance_eqn} \kappa(r) &= \epsilon m[\Omega(r) - \Omega_{ps}]\end{aligned}$$ for the resonance radius $r=a_m$, where $\kappa(r)$ is the ring particles’ epicyclic frequency (Eqn. \[kappa\^2\]), and $\epsilon=+1 (-1)$ at an inner (outer) Lindblad resonance. So for the simulated B ring’s free $m=2$ mode, Eqn. (\[resonance\_eqn\]) is solved for the radius $r=\tilde{a}_2$ of the $\epsilon=+1$ inner Lindblad resonance that is associated with this mode. That quantity is to be compared to a nearby reference distance, which in this case would be the semimajor axis of the B ring’s outer edge $a_{\mbox{\scriptsize edge}}$. Results are shown in Fig. \[da2\_fig\], which shows the simulations’ distance from the B ring’s outer edge to the free $m=2$ pattern’s ILR , $\Delta a_2 = a_{\mbox{\scriptsize edge}} - \tilde{a}_2$, plotted versus ring surface density $\sigma_0$. Heavier rings have a faster pattern speeds (Fig. \[m2\_fig\]–\[m2\_rel\_fig\]), and so the pattern’s resonance resides at a higher orbital frequency $\Omega(r)$ and thus must lie further inwards of the ring’s outer edge in order to satisfy the resonance condition, Eqn. (\[resonance\_eqn\]). Figure \[da2\_fig\] has the same information content as Fig. \[m2\_rel\_fig\], which is why it also tells us that the B ring’s outer edge has $\sigma_0\lesssim210$ gm/cm$^2$. However a plot like Fig. \[da2\_fig\] will also provide the best way to interpret the B ring’s free $m=3$ mode, which is examined below in subsection \[m=3\]. Lastly, note that the free $m=2$ patterns seen in these simulations persist for $3\times10^4$ orbits or 40 years without any sign of damping, despite the ring’s viscosity $\nu=100$ cm$^2$/sec. This is illustrated in Fig. \[R\_epi\_fig\], which plots the ring-edge’s epicyclic amplitude over time for the nominal $\sigma_0=200$ gm/cm$^2$ simulation. Indeed we have also rerun this simulation using a viscosity that is ten times larger and still saw no damping. These experiments reveal a possibly surprising result, that a free pattern can persist at a ring-edge for a considerable length of time, likely hundreds of years or longer, and Section \[force\] will show that this longevity is due to the viscous forces being several orders or magnitude weaker than the ring’s other interval forces. So one possible interpretation of the free modes seen at the B ring and at other ring edges is that they are relics from past disturbances in Saturn’s ring that may have happened hundreds or more years ago. This possibility is discussed further in Section \[impulse\]. the free pattern {#m=3} ---------------- The B ring’s free $m=3$ mode has an epicyclic amplitude of $\tilde{R}_3=11.8\pm0.2$ km, a pattern speed $\dot{\tilde{\omega}}_3=507.700\pm0.001$ deg/day, and the inner Lindblad resonance associated with this pattern speed lies $\Delta a_3=24\pm4$ km interior to the B ring’s outer edge (SP10). To excite a free $m=3$ pattern at the ring-edge, place a fictitious satellite in an orbit that has an $m=3$ inner Lindblad resonance $\Delta a_3=24$ km interior to the ring’s outer edge. Noting that the satellite Janus happens to have an $m=3$ resonance in the vicinity, about 2000 km inwards of the B ring’s edge, these simulations use a Janus-mass satellite to perturb the simulated ring for about $1650$ orbits (about 2 years), which excites an $m=3$ pattern at the ring’s outer edge. The satellite is then removed from the system, which converts the pattern into a free normal mode, and [epi\_int]{} is then used to evolve the now unperturbed ring for another $1.8\times10^4$ orbits (about 23 years). Figure \[m3\_fig\] plots the ring-edge’s epicyclic amplitude, where it is shown that the free mode persists at the B ring’s outer edge, undamped over time, despite the simulated ring’s viscosity of $\nu=100$ cm$^2$/sec. A suite of such B ring simulations is performed, with ring surface densities $120\le\sigma_0\le360$ gm/cm$^2$ and all other parameters identical to the nominal model of Section \[m=2\] except where noted in Fig. \[da3\_fig\] caption. The pattern speed $\Omega_{ps}=\dot{\tilde{\omega}}_3$ of the $m=3$ normal mode is then extracted from each simulation, with those speeds again being slightly faster in the heavier rings. Those pattern speeds are then inserted into Eqn. (\[resonance\_eqn\]) which is solved for the radius of the inner Lindblad resonance $\tilde{a}_3$, each of which lies a distance $\Delta a_3 = a_{\mbox{\scriptsize edge}} - \tilde{a}_3$ inwards of the ring’s outer edge, and those distances are plotted in Fig. \[da3\_fig\] versus ring surface density $\sigma_0$. The simulated distances $\Delta a_3$ are compared to the observed edge-resonance distance reported in SP10, which indicates a ring surface density $160\le\sigma_0\le310$ gm/cm$^2$. This finding is consistent with the the results from the $m=2$ patterns, but this constraint on $\sigma_0$ is again rather loose due to the $\delta a_{\mbox{\scriptsize edge}}=\pm4$ km uncertainty in the ring-edge’s semimajor axis. But our purpose here is to show how one might use models of free normal modes to infer the surface density of other rings and narrow ringlets, which again will likely require knowing the ring-edge’s semimajor axis to $\pm1$ km or better. Also note that the radial depth of this $m=3$ disturbance is $\Delta a_{e/10}=50$ km, about three times smaller than the radial depth of the $m=2$ disturbance. the free pattern {#m=1} ---------------- The B ring’s free $m=1$ mode has an epicyclic amplitude of $\tilde{R}_1=20.9\pm0.4$ km and a pattern speed $\dot{\tilde{\omega}}_1=5.098\pm0.003$ deg/day that is slightly faster than the local precession rate, and the inner Lindblad resonance that is associated with this pattern speed lies $\Delta a_1=253\pm4$ km interior to the B ring’s outer edge (SP10). Several simulations of the B ring’s $m=1$ pattern are evolved for model rings having surface densities of $120\le\sigma_0\le360$ gm/cm$^2$. To excite the $m=1$ pattern at the simulated ring’s edge, again arrange a fictitious satellite’s orbit so that its $m=1$ ILR lies $\Delta a_1=253$ km interior to the B ring’s edge, which is the site where the resonance condition (Eqn. \[resonance\_eqn\]) is satisfied when the satellite’s mean angular velocity matches the ring particles’ precession rate, $\Omega_s = \dot{\tilde{\omega}}=\Omega_{ps}$. The simulated ring is perturbed by a satellite whose mass is about 20% that of Mimas, for $1.6\times10^4$ orbits or 21 years, which excites a forced $m=1$ pattern at the ring’s edge that corotates with the satellite. The satellite is then removed, which converts the forced $m=1$ pattern into a free pattern, and the ring is evolved for another $6.4\times10^4$ orbits or 83 years. For each simulation the free pattern speed is measured, and Eqn. (\[resonance\_eqn\]) is then used to convert the free pattern speed into a resonance radius $\tilde{a}_1$, which is displayed in Fig. \[da1\_fig\] that shows that resonance’s distance from the ring’s outer edge, $\Delta a_1=a_{\mbox{\scriptsize edge}} - \tilde{a}_1$. As the figure shows, the free $m=1$ pattern rotates slightly faster in the heavier ring and thus the associated $m=1$ ILR must lie further inwards in order to satisfy the resonance condition $\Omega_{ps}=\dot{\tilde{\omega}}=\frac{3}{2}J_2(R_p/a)^2\Omega$. Again there is no damping of the free $m=1$ pattern, which stays localized at the ring’s outer edge over the simulation’s 83 yr timespan, despite the simulated ring’s viscosity $\nu=100$ cm$^2$/sec. The radial depth of this $m=1$ disturbance is much greater than the others, $\Delta a_{e/10}=614$ km, which is about four times larger than the $m=2$ disturbance. Comparing Fig. \[da1\_fig\] to Figs. \[da2\_fig\] and \[da3\_fig\] also shows that the LR associated with the $m=1$ disturbance lies about 10 times further from the ring-edge than the $m=1$ and $m=2$ resonances. Which is why the $m=1$ simulation uses streamlines whose width $\Delta a$ is $\sim10\times$ larger, since a wider portion of the B ring-edge must be simulated in order to capture this disturbances’ deeper reach into the B ring. Note also that the $\pm4$ km uncertainty in this resonance’s position relative to the B ring edge, which is entirely due to the uncertainty in the B ring-edge’s semimajor axis, is in this case relatively small. Which is why the ring’s free $m=1$ mode can also be used to probe its surface density with some precision (unlike the free $m=2$ and $m=3$ modes), and is consistent with a B ring surface density of $\sigma_0\simeq200$ gm/cm$^2$, convergence tests {#convergence} ----------------- A number of simulations have also been performed, which repeat the ring simulations using various particle numbers $N_r$ and $N_\theta$ and various widths $w$ of the simulated ring. We find that the results reported here do not change significantly when the simulated ring is populated densely with enough particles, and when the radial width of the simulated B ring is sufficiently wide. Those convergence tests reveal that the number of particles along each streamline must satisfy $N_\theta\ge 20m$, that the radial width of each streamline should satisfy $\Delta a\le 0.04\Delta a_{e/10}$, and that the total width of the simulated ring should satisfy $w>4\Delta a_{e/10}$. All of the simulations reported here satisfy these requirements. Discussion ========== This section re-examines the model’s treatment of viscous effects at the ring’s edge, and also describes related topics that will be considered in followup work. the ring’s internal forces {#force} -------------------------- Figure \[forces\_fig\] plots the accelerations that the ring’s internal forces—gravity, pressure, and viscosity—exert on each ring particle. These accelerations are from the nominal $\sigma_0=200$ gm/cm$^2$ simulation that is described in Figs. \[R2\_fig\]–\[R\_epi\_fig\], and these accelerations are plotted versus each particle’s distance from the ring’s edge, so those forces obviously get larger closer to the ring’s disturbed outer edge. But the main point of Fig. \[forces\_fig\] is that the ring’s self gravity is the dominant internal force in the ring, exceeding the pressure force by a factor of $\sim100$ at the ring’s outer edge and by a larger factor elsewhere. Those pressure forces are also about $\sim10\times$ larger than the ring’s viscous forces. But recall that those simulations had zeroed the viscous acceleration that the ring exerts on its outermost streamline (Section \[B ring\]), when that acceleration should instead be $A_{\nu,\theta}=F/\lambda r$ as indicated by the large blue dot at the right edge of Fig. \[forces\_fig\]. Note though that the neglected viscous acceleration of the ring’s edge is still about $\sim1000\times$ smaller than that due to ring gravity and $\sim10\times$ smaller than that due to ring pressure. So this justifies neglecting, at least for the short-term $t\sim100$ yr simulations considered here, the much smaller viscous forces at the ring’s outer edge. Nonetheless this study’s neglect of the small viscous force at the ring’s outer edge implies that this model does not yet account for the B ring’s radial confinement by Mimas’ $m=2$ ILR. So there appears to be some missing physics that will be necessary if one is interested in the ring’s resonant confinement or the ring’s long-term evolution over $t\gg100$ yr timescales. The suspected missing physics is described below. ### unmodeled effects: the viscous heating of a resonantly confined ring-edge {#viscous heating} The model’s inability to confine the B ring’s outer edge at Mimas’ $m=2$ ILR may be a consequence of the ring’s kinematic viscosity $\nu$ being treated here as a constant parameter everywhere in the simulated ring. Although treating $\nu$ as a constant is a simple and plausible way to model the effects of the ring’s viscous friction, it might not be adequate or accurate if one wishes to simulate the resonant confinement of a planetary ring. This is because the ring’s viscosity transports both energy and angular momentum radially outwards through the ring. So if the ring’s outer edge is to be confined by a satellite’s $m^{\mbox{\scriptsize th}}$ Lindblad resonance, the satellite must absorb the ring’s outward angular momentum flux, which it can do by exerting a negative gravitational torque at the ring’s edge. But [@BGT82] show via a simple Jacobi-integral argument that resonant interactions only allow the satellite to absorb but a fraction of the energy that viscosity delivers to the ring-edge. Consequently the ring’s viscous friction still delivers some orbital energy to the ring-edge where it accumulates and heats up the ring particles’ random velocities $c$. And if collisions among particles are the main source of the ring’s viscosity, then $\nu_s\simeq c^2\tau/2\Omega(1+\tau^2)$ where $\tau\propto\sigma$ is the ring’s optical depth [@GT82]. In this case viscous heating would increase $c$ as well as $\nu_s$ at the ring’s edge. The enhanced dissipation there should also increase the angular lag $\phi$ between the ring-edge’s forced pattern and the satellite’s longitude (see Eqn. 83b of @HSP09). Which will also be important because the gravitational torque that the satellite exerts on the ring-edge varies as $\sin\phi$ [@HSP09], and that torque needs to be boosted if the satellite is to confine the spreading ring. To model this phenomenon properly, the [epi\_int]{} code also needs to employ an energy equation, one that accounts for how viscous heating tends to increase the ring particles’ dispersion velocity $c$ and viscosity $\nu_s$ nearer the ring’s edge. The increased dissipation and the resulting orbital lag will allow the satellite to exert a greater torque on the ring which, we suspect, will enable the satellite to resonantly confine the simulated ring’s outer edge. The derivation of this energy equation and its implementation in [epi\_int]{} are ongoing, and those results will be reported on in a followup study. an alternate equation of state {#EOS} ------------------------------ The EOS adopted here is appropriate for a dilute gas of colliding ring particles whose mutual separations greatly exceed their sizes. This should be regarded as a limiting case since ring particles can of course be packed close to each other in the ring. But [@BGT85] consider the other extreme limiting case, with close-packed particles that reside shoulder to shoulder in the ring. In that case the ring is expected to behave as an incompressible fluid whose volume density $\rho=\sigma/2h$ stays constant. So when some perturbation causes ring streamlines to bunch up and increases the ring’s surface density $\sigma$, the ring’s vertical scale height $h$ also increases as ring particles are forced to accumulate along the vertical direction. This in turn increases the ring’s pressure due to the larger gravitational force along the vertical direction. [@BGT85] show that infinitesimal density waves in an incompressible disk are unstable and grow in amplitude over time. This phenomenon is related to the viscous overstability, and [@LR95] show that it can distort a narrow eccentric ringlet’s streamlines in a way that accounts for its $m=1$ shapes. [@BGT85] also suggest that unstable density waves can be trapped between a Lindblad resonance and the B ring’s outer edge, which might explain the normal modes seen there, and [@SP10] use this concept to estimate the ring’s surface density there. But keep in mind that this instability only occurs when the ring particles are densely packed to the point of being incompressible, which requires the ring to be very thin and dynamically cold. We have shown here that the amplitude of the B ring’s forced motions indicates that the ring-edge has a surface density $\sigma\simeq200$ gm/cm$^2$. So if this ring is incompressible and composed of icy spheres having a mean volume density of $\rho=\sigma/2h\simeq0.5$ gm/cm$^3$, this then requires a B ring thickness of only $h\sim2$ meters, which is rather thin compared to other estimates [@CBC10]. Similarly the ring particles’ dispersion velocity $c$ must be small compared to that expected for a dilute particle gas, so $c\ll (h\Omega\sim0.3$ mm/sec), which again is cold compared to all other estimates for Saturn’s rings [@CBC10]. The upshot is that an incompressible EOS requires the ring to be very thin and dynamically cold, likely much colder and thinner than is generally thought. Consequently we are optimistic that the compressible EOS used here, $p=\sigma c^2$, is the appropriate choice for simulations of the outer edge of Saturn’s B ring. Nonetheless in a followup investigation we do intend to encode the incompressible EOS into [epi\_int]{}, to see if the BGT instability can account for the higher $m\ge2$ free modes that are seen at the outer edge of the B ring and in many other narrow ringlets. impulse origin for normal modes {#impulse} ------------------------------- The simulations of Section \[B ring\] used a fictitious temporary satellite to excite the free modes that occur at many Saturnian ring edges. These simulations used an admittedly ad hoc method—the sudden appearance and disappearance of a satellite—to excite these modes. Nonetheless these models demonstrate that transient and impulsive events can excite normal modes at ring edges, and those simulations show that normal modes can persist at the ring’s edge for hundreds of years after the disturbance has occurred. Which suggests that an impulsive event in the recent past, perhaps an impact into Saturn’s rings, might be responsible for exciting the normal modes that are seen at the outer edge of the B ring, as well as the normal modes that are also seen along the edges of several narrower ringlets [@FMR10; @HNB10; @FNC11; @NFH12] The possibility that normal modes are due to an impact is motivated by the discovery of vertical corrugations in Saturn’s C and D rings [@HBS07; @HBE11] and in Jupiter’s main dust ring [@SHB11]. These vertical structures are spirals that span a large swath of each ring, and they are observed to wind up over time due to the central planet’s oblateness. Evolving the vertical corrugations backwards in time also unwinds their spiral pattern until some moment when the affected region is a single tilted plane. Unwinding the Jovian corrugation shows that that disturbance occurred very close to the date when the tidally disrupted comet Shoemaker-Levy 9 impacted Jupiter in 1994, which suggests an impact from a tidally disrupted comet as the origin of these ring-tilts [@SHB11]. However a single sub-km comet fragment cannot tilt a large $\sim2\times10^5$ km-wide planetary ring. But a disrupted comet can produce an extended cloud of dust, and if that disrupted dust cloud returns to the planet with enough mass and momentum, then it might tilt a ring that at a later date would be observed as a spiral corrugation. However the tidal disruption of comet about a low-density planet like Saturn is more problematic, because tidal disruption only occurs when the comet’s orbit is truly close to parabolic and not too hyperbolic, and with periapse just above the planet’s atmosphere [@ST92; @RBL98]. But it is easy to envision an alternate scenario that might be more likely, with a small km-sized comet originally in a heliocentric orbit coming close enough to Saturn to instead strike the main A or B rings. This scenario is more probable because the cross-section available to orbits impacting the main rings is significantly larger than those resulting in tidal disruption. The impacting comet’s considerably greater momentum will nonetheless carry the impactor through the dense A or B rings, but the collision itself is likely energetic enough to shatter the comet. And if that collision is sufficiently dissipative, then the resulting cometary debris will then stay bound to Saturn, and in an orbit that will return that debris back into the ring system on its next orbit. Small differences among the orbits of individual debris particles’ means that, when the debris encounters the rings again, that impacting debris will be spread across a much larger footprint on the ring, which presumably will allow any dense rings or ringlets to absorb the debris’ mass and momentum in a way that effectively gives the ring particles there a sudden velocity kick $\mathbf{\Delta v}$ in proportion to the comet debris density $\rho$ and velocity $\mathbf{v}_r$ relative to the ring matter. But if comet Shoemake-Levy 9’s (SL9) impact with Jupiter is any guide, then impact by a cloud of comet debris could last as long week of time, which might tend to smear this effect out due to the ring’s orbital motion. But that effect would be offset if the debris train’s dust cloud is also rather clumpy, like the SL9 debris train was. Indeed, it is possible that this scenario might also account for the spiral corrugations of Saturn’s C and D ring. It is also conceivable that an inclined cloud of impacting comet debris might also excite the vertical analog of normal modes—long-lived vertical oscillations of a ring’s edge. This admittedly speculative scenario will be pursued in a followup study, to determine whether debris from an impact-disrupted comet can excite the normal modes seen at ring edges, and to determine the mass of the progenitor comet that would be needed to account for these modes’ observed amplitudes. Summary of results {#summary} ================== We have developed a new N-body integrator that calculates the global evolution of a self-gravitating planetary ring as it orbits an oblate planet. The code is called [epi\_int]{}, and it uses the same kick-drift-step algorithm as is used in other symplectic integrators such as [SYMBA]{} and [MERCURY]{}. However the velocity kicks that are due to ring gravity are computed via an alternate method that assumes that all particles inhabit a discreet number of streamlines in the ring. The use of streamlines to calculate ring self gravity has been used in analytic studies of rings [@GT79; @BGT83aj; @BGT86], and the streamline concept is easily implemented in an N-body code. A streamline is the closed path through the ring that is traced by particles having a common semimajor axis. All streamlines are radially close to each other, so the gravitation acceleration due to a streamline is simply that due to a long wire, $A=2G\lambda/\Delta$ where $\lambda$ is the streamline’s linear density and $\Delta$ is the particle’s distance from the streamline. Which is very useful since particles are responding to the pull of smooth wires rather than discreet clumps of ring matter so there is no gravitational scattering. Which means that only a modest number of particles are needed, typically a few thousand, to simulate all $360^\circ$ of a scalloped ring like the outer edges of Saturn’s A and B ring. Only a few thousand particles are also needed to simulate linear as well as nonlinear spiral density waves, and execution times are just a few hours on a desktop PC. Another distinction occurs during the particles’ unperturbed drift step when particles follow the epicyclic orbit of [@LR95] about an oblate planet, rather than the usual Keplerian orbit about a spherical planet. This effectively moves the perturbation due to the planet’s oblate figure out of the integrator’s kick step and into the drift step. The code also employs hydrodynamic pressure and viscosity to account for the transport of linear and angular momentum through the ring that arises from collisions among ring particles. Another convenience of the streamline formulation is that it easily accounts for the large pressure drop that occurs at a ring’s sharp edge, as well as the large viscous torque that the ring exerts there. The model also accounts for the mutual gravitational perturbations that the ring and the satellites exert on each other. The [epi\_int]{} code is written in IDL, and the source code is available for download at [http://gemelli.spacescience.org/‘hahnjm/software.html](http://gemelli.spacescience.org/~hahnjm/software.html). This integrator is used to simulate the forced response that the satellite Mimas excites at its $m=2$ inner Lindblad resonance (ILR) that lies near the outer edge of Saturn’s B ring. That resonance lies $\Delta a_2=12 \pm4$ km inwards of the ring’s edge, and simulations show that the ring’s forced epicyclic amplitude varies with the ring’s surface density $\sigma_0$ as $R_2\propto\sigma_0^{0.67}$. Good agreement with Cassini measurements of $R_2$ occurs when the simulated ring has a surface density of $\sigma_0=195\pm 60$ gm/cm$^2$ (see Fig. \[R2\_fig\]), where the uncertainty in $\sigma_0$ is dominated by the $\delta a_{\mbox{\scriptsize edge}}=4$ km uncertainty that [@SP10] find in the ring-edge’s semimajor axis. This $\sigma_0$ is the mean surface density over that part of the B ring that is disturbed by this resonance, whose influence in the ring extends to a radial distance of $\Delta a_{e/10}\sim150$ km from the B ring’s outer edge. And if we naively assume that this surface density is the same everywhere across Saturn’s B ring, then its total mass is about $90\%$ of Mimas’ mass. Cassini observations reveal that the outer edge of Saturn’s B ring also has several free normal modes that are not excited by any known satellite resonances. Although the mechanism that excites these free modes is uncertain, we are nonetheless able to excite free modes in a simulated ring via various ad-hoc methods. For instance, a fictitious satellite’s $m^{\mbox{\scriptsize th}}$ Lindblad resonance is used to excite a forced pattern at the ring edge. Removing that satellite then converts the forced patten into a free normal mode that persists in these simulations for up to $\sim100$ years or $\sim10^5$ orbits without any damping, despite the simulated ring having a kinematic viscosity of $\nu=100$ cm$^2$/sec; see Fig. \[m3\_fig\] for one example. Alternatively, starting the ring particles in circular orbits while subject to Mimas’ $m=2$ gravitational perturbation excites both a forced and a free $m=2$ pattern that initially null each other precisely at the start of the simulation. But the forced patten corotates with Mimas’ longitude while the free pattern rotates slightly faster in a heavier ring, which suggests that a free mode’s pattern speed can also be used to infer a ring’s surface density $\sigma_0$. However the free pattern speed is also influenced by the $J_4$ and higher terms in the oblate planet’s gravity field, which are absent from this model which only accounts for the $J_2$ component. So the simulated pattern speed cannot be compared directly to the observed pattern speed; see Fig. \[m2\_fig\]. To avoid this difficulty, the resonance condition (Eqn. \[resonance\_eqn\]) is used to calculate the radius of the Lindblad resonance that is associated with the free normal mode. Plotting the distances of the simulated and observed resonances from the B ring’s edge (Figs. \[da2\_fig\], \[da3\_fig\], and \[da1\_fig\]) then provides a convenient way to compare simulations to observations of free modes in a way that is insensitive to the planet’s oblateness. Simulations of the B ring’s free $m=2$ and $m=3$ patterns are consistent with Cassini measurements of the B ring’s normal modes when the simulated ring-edge again has a surface density of $\sigma_0\sim200$ gm/cm$^2$, which is a nice consistency check. But these particular measurements do not provide tight constraint on the ring’s $\sigma_0$, due to the fact that the $m=2$ and $m=3$ Lindblad resonances only lie $\Delta a_m\sim25$ km from the outer edge of a ring whose semimajor axis $a$ is uncertain by $\delta a_{\mbox{\scriptsize edge}}=4$ km. However the B ring’s free $m=1$ normal mode does lie much deeper in the ring’s interior, $\Delta a_1=253\pm4$, so the uncertainly in its location is fractionally much smaller, and this normal mode does confirm the $\sigma_0\simeq200$ gm/cm$^2$ value that was inferred from simulations of the B ring’s forced response $R_2$. One of the goals of this study is to determine whether simulations of free modes can be used to determine the surface density and mass of a narrow ringlet. Such ringlets show a broad spectrum of free normal models over $0\le m\le 5$ [@FMR10; @HNB10; @FNC11; @NFH12], and the answer appears to be yes since free pattern speeds do increase with $\sigma_0$. However Section \[free m=2\] shows that the semimajor axes of the ringlet’s edges likely need to be known to a precision of $\delta a_{\mbox{\scriptsize edge}}\sim1$ km in order for a free mode to provide a useful measurement of the ringlet’s $\sigma_0$. The origin of these free modes, which are quite common along the edges of Saturn’s broad rings and its many narrow ringlets, is uncertain. [@BGT85] show that, if a planetary ring’s particles are packed shoulder to shoulder such that the ring behaves like an incompressible fluid, then that ring is unstable to the growth of density waves, a phenomenon also termed viscous overstability, and they suggest that the B ring’s normal modes might be due to unstable waves that are trapped between a Lindblad resonance and the ring’s edge. To study this further, we will in a followup study adapt [epi\_int]{} to employ an incompressible equation of state, to see if the viscous overstability can in fact account for the free normal modes seen along the Saturnian ring edges. Although the current version of [epi\_int]{} does not account for the origin of these free modes, one can still plant a free mode along the edge of a simulated ring by temporarily perturbing a ring at a fictitious satellite’s Lindblad resonance, and then removing that satellite, which creates an unforced mode that persists undamped at the ring-edge for more than $\sim10^5$ orbits or $\sim100$ yrs despite the simulated ring having a kinematic viscosity of $\nu=100$ cm$^2$/sec. Because this forcing is suddenly turned on and off, this suggests that any sudden or impulsive disturbance of the ring can excite normal modes, with those disturbances possibly persisting for hundreds or maybe thousands of years. And in Section \[impulse\] we suggest that the Saturnian normal modes might be excited by an impact with a collisionally disrupted cloud of comet dust. This is a slight variation of the scenario that [@HBS07] and [@SHB11] propose for the origin of corrugated planetary rings, and in a followup investigation we intend to determine whether such impacts can also account for the normal modes seen in Saturn’s rings. And lastly, we find that [epi\_int]{}’s treatment of ring viscosity has difficulty accounting for the radial confinement of the B ring’s outer edge by Mimas’ $m=2$ inner Lindblad resonance. This model employs a kinematic shear viscosity $\nu_s$ that is everywhere a constant, which causes the simulation’s outermost streamline to slowly but steadily drift radially outwards. Which in turn causes the ring’s forced epicyclic amplitude $R_2$ to slowly grow over time, and makes difficult any comparison to Cassini’s measurement of $R_2$. To sidestep this difficulty, the model zeros the torque that the simulated ring exerts on its outermost streamline, which does allow the ring to settle into a static configuration that can be compared to Cassini observations and yields a measurement of the ring’s surface density $\sigma_0$. This approximate treatment is also examined in in Section \[force\], which shows that the viscous acceleration of the ring-edge, had it been included in the simulation, is still orders of magnitude smaller than that due to ring self gravity. So this study of the dynamics of the B ring’s forced and free modes is not adversely impacted by this approximate treatment. But this does mean that the B ring’s radial confinement is still an unsolved problem, and Section \[viscous heating\] suggests that this might be a consequence of treating $\nu_s$ as a constant. [@BGT82] show that viscosity’s outward transport of energy should also heat the ring’s outer edge and increase the ring particles’ dispersion velocity $c$ there. And if collisions among ring particles are the dominant source of ring viscosity, then $\nu_s\propto c^2$ and viscous dissipation would be enhanced at the ring edge, which in turn would increase the angular lag between the ring’s forced response and the Mimas’ longitude. That then would increase the gravitational torque that that satellite exerts on the ring-edge. So in a followup study we will modify [epi\_int]{} to address this problem in a fully self-consistent way, to see if enhanced dissipation at the ring-edge also increases Mimas’ gravitational torque there sufficiently to prevent the B ring’s outer edge from flowing viscously beyond that satellite’s $m=2$ inner Lindblad resonance. [**Acknowledgments**]{} J. Hahn’s contribution to this work was supported by grant NNX09AU24G issued by NASA’s Science Mission Directorate via its Outer Planets Research Program. The authors thank Denise Edgington of the University of Texas’ Center for Space Research (CSR) for composing Fig. \[streamline\_fig\], and J. Hahn thanks Byron Tapley for graciously providing office space and the use of the facilities at CSR. The authors are also grateful for the helpful suggestions provided by an anonymous reviewer. Appendix \[shear\_appendix\] {#shear_appendix} ============================ The following calculates the flux of angular momentum that is communicated via a disk’s viscosity. The disk is flat and thin and has a vertical halfwidth $h$ and constant volume density $\rho$ that is related to its surface density $\sigma$ via $\rho=\sigma/2h$. The disk is assumed viscous, and its gravity is ignored here since this Appendix is only interested in the angular momentum flux that is transported solely by viscosity. The density of angular momentum in the disk is $\bm{\ell} = \mathbf{r}\times\rho\mathbf{v}$, and the vertical component along the $z=x_3$ axis is $\ell_3 = x_1\rho v_2 - x_2\rho v_1$ in Cartesian coordinates $x=x_1$ and $y=x_2$ where $\rho$ and $v_i$ are functions of position and time, so the time rate of change of $\ell_3$ is $$\begin{aligned} \label{dl3/dt} \frac{\partial\ell_3}{\partial t} &= x_1\displaystyle\frac{\partial}{\partial t}(\rho v_2) - x_2\displaystyle\frac{\partial}{\partial t}(\rho v_1).\end{aligned}$$ The time derivatives in the above are Euler’s equation, $$\begin{aligned} \label{EEqn} \frac{\partial}{\partial t}(\rho v_i) &=-\displaystyle\sum_{k=1}^3 \frac{\partial\Pi_{ik}}{\partial x_k}\end{aligned}$$ where the $\Pi_{ik}$ are the elements of the momentum flux density tensor $$\begin{aligned} \label{Pi} \Pi_{ik} &= \rho v_iv_k + \delta_{ik}p - \sigma'_{ik}\end{aligned}$$ where $p$ is the pressure and $\sigma'_{ik}$ are the elements of the viscous stress tensor [@LL87]. Inserting Eqn. (\[Pi\]) into (\[dl3/dt\]) yields $$\begin{aligned} \label{dl3/dt_v2} \frac{\partial\ell_3}{\partial t} &= -x_1\nabla\cdot\bm{\Pi}_2 + x_2\nabla\cdot\bm{\Pi}_1\end{aligned}$$ where the vector $$\begin{aligned} \label{Pi_vector} \bm{\Pi}_i=\displaystyle\sum_{k=1}^3\Pi_{ik}\bm{\hat{x}}_k\end{aligned}$$ is the flux density of the $i$ component of linear momentum and $\bm{\hat{x}}_k$ is the unit vector along the $x_k$ axis. Equation (\[dl3/dt\_v2\]) can be rewritten $$\begin{aligned} \label{dl3/dt_v3} \frac{\partial\ell_3}{\partial t} &= -\nabla\cdot(x_1\bm{\Pi}_2 - x_2\bm{\Pi}_1) +\bm{\Pi}_2\cdot\nabla x_1 - \bm{\Pi}_1\cdot\nabla x_2\end{aligned}$$ but note that $\bm{\Pi}_1\cdot\nabla x_2 - \bm{\Pi}_2\cdot\nabla x_1=\Pi_{21}-\Pi_{12}= \sigma'_{12} - \sigma'_{21}=0$ since the viscous stress tensor is symmetric (Eqn. \[sigma’\]), so $$\begin{aligned} \label{dl3/dt_v4} \frac{\partial\ell_3}{\partial t} &= -\nabla\cdot \bm{F}_3\end{aligned}$$ where $$\begin{aligned} \label{F3} \bm{F}_3 &= x_1\bm{\Pi}_2 - x_2\bm{\Pi}_1.\end{aligned}$$ Integrating Eqn. (\[dl3/dt\_v4\]) over some volume $V$ that is bounded by area $A$ yields $$\begin{aligned} \label{F3_flux} \frac{\partial}{\partial t}\int_V \ell_3 dV &= -\int_V\nabla\cdot \bm{F}_3 dV = -\int_A \bm{F}_3\cdot \bm{dA}\end{aligned}$$ by the divergence theorem, so Eqn. (\[F3\_flux\]) indicates that $\bm{F}_3$ is the flux of the $x_3$ component of angular momentum out of volume $V$ that is being transported by advection, pressure, and viscous effects. This Appendix is interested in the part of $\bm{F}_3$ that is due to viscous effects, which will be identified as $\bm{F}'_3$ and is obtained by replacing $\Pi_{ik}$ in Eqn. (\[Pi\]) with $-\sigma'_{ik}$ so $$\begin{aligned} \label{F3'} \bm{F}'_3 &= (x_2\sigma'_{11} - x_1\sigma'_{21})\bm{\hat{x}}_1 + (x_2\sigma'_{12} - x_1\sigma'_{22})\bm{\hat{x}}_2.\end{aligned}$$ This is the 2D flux of the $x_3$ component of angular momentum that is transported by the disk’s viscosity whose horizontal components in Cartesian coordinates are $\bm{F}'_3=F'_1\bm{\hat{x}}_1 + F'_2\bm{\hat{x}}_2$ where $F'_1=x_2\sigma'_{11} - x_1\sigma'_{21}$ and $F'_2=x_2\sigma'_{12} - x_1\sigma'_{22}$. However this Appendix desires the radial component of $\bm{F}'_3$ are some site $r,\theta$ in the disk, which is $F'_r=F'_1\cos\theta + F'_2\sin\theta$. The elements of the viscous stress tensor are [@LL87] $$\begin{aligned} \label{sigma'} \sigma'_{ik} &= \eta\left(\displaystyle\frac{\partial v_i}{\partial x_k} + \frac{\partial v_k}{\partial x_i}\right) + (\zeta - \frac{2}{3}\eta)\delta_{ik}\nabla\cdot\bm{v}\end{aligned}$$ where $\eta$ is the shear viscosity, $\zeta$ is the bulk viscosity, and $\delta_{ik}$ is the Kronecker delta. Inserting this into $F'_r$ and replacing $x_1=r\cos\theta$ and $x_2=r\sin\theta$ then yields $$\begin{aligned} \label{F'_r} F'_r &=-\eta r\left(\displaystyle\frac{\partial v_1}{\partial x_2} + \frac{\partial v_2}{\partial x_1}\right)\cos2\theta + \eta r\left(\displaystyle\frac{\partial v_1}{\partial x_1} - \frac{\partial v_2}{\partial x_2}\right)\sin2\theta.\end{aligned}$$ The horizontal velocities are $v_1=v_r\cos\theta - v_\theta\sin\theta $ and $v_2 = v_r\sin\theta + v_\theta\cos\theta $ when written in terms of their radial component $v_r$ and tangential component $v_\theta=r\dot{\theta}$. The derivatives in Eqn. (\[F’\_r\]) are $$\begin{aligned} \label{dv/dx} \begin{split} \frac{\partial v_1}{\partial x_1} &= \left(\cos\theta\frac{\partial}{\partial r} -\frac{\sin\theta}{r}\frac{\partial}{\partial \theta}\right)v_1\\ &= \cos^2\theta\frac{\partial v_r}{\partial r} - \sin\theta\cos\theta r\frac{\partial\dot{\theta}}{\partial r} + \frac{\sin^2\theta}{r}v_r - \frac{\sin\theta\cos\theta}{r}\frac{\partial v_r}{\partial\theta} + \frac{\sin^2\theta}{r}\frac{\partial v_\theta}{\partial\theta}\\ \frac{\partial v_2}{\partial x_2} &= \left(\sin\theta\frac{\partial}{\partial r} +\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}\right)v_2\\ &= \sin^2\theta\frac{\partial v_r}{\partial r} + \sin\theta\cos\theta r\frac{\partial\dot{\theta}}{\partial r} + \frac{\cos^2\theta}{r}v_r + \frac{\sin\theta\cos\theta}{r}\frac{\partial v_r}{\partial\theta} + \frac{\cos^2\theta}{r}\frac{\partial v_\theta}{\partial\theta}\\ \frac{\partial v_1}{\partial x_2} &= \left(\sin\theta\frac{\partial}{\partial r} +\frac{\cos\theta}{r}\frac{\partial}{\partial \theta}\right)v_1\\ \frac{\partial v_2}{\partial x_1} &= \left(\cos\theta\frac{\partial}{\partial r} -\frac{\sin\theta}{r}\frac{\partial}{\partial \theta}\right)v_2 \end{split}\end{aligned}$$ when written in terms of cylindrical coordinates, and the combinations of derivatives in Eqn. (\[F’\_r\]) are \[dv/dx +- dv/dx\] $$\begin{aligned} \label{dv1/dx2 + dv2/dx1} \frac{\partial v_1}{\partial x_2} + \frac{\partial v_2}{\partial x_1} &=& \left(\frac{\partial v_r}{\partial r} -\frac{1}{r}\frac{\partial v_\theta}{\partial \theta} - \frac{v_r}{r}\right)\sin2\theta + \left(\frac{\partial v_\theta}{\partial r} +\frac{1}{r}\frac{\partial v_r}{\partial \theta} - \frac{v_\theta}{r}\right)\cos2\theta\\ \label{dv1/dx1 - dv2/dx2} \frac{\partial v_1}{\partial x_1} - \frac{\partial v_2}{\partial x_2} &=& \left(\frac{\partial v_r}{\partial r} -\frac{1}{r}\frac{\partial v_\theta}{\partial \theta} - \frac{v_r}{r}\right)\cos2\theta - \left(\frac{\partial v_\theta}{\partial r} +\frac{1}{r}\frac{\partial v_r}{\partial \theta} - \frac{v_\theta}{r}\right)\sin2\theta, \end{aligned}$$ Inserting these into Eqn. (\[F’\_r\]) then yields a result that is thankfully much more compact, $$\begin{aligned} \label{F'_r2} F'_r &=-\eta\left( \displaystyle r^2\frac{\partial\dot{\theta}}{\partial r} + \frac{\partial v_r}{\partial \theta}\right) \simeq -\eta r^2\displaystyle \frac{\partial\dot{\theta}}{\partial r},\end{aligned}$$ noting that the second term in Eqn. (\[F’\_r2\]) may be neglected since the azimuthal gradient is much smaller than the radial gradient for the disks considered here. This is the radial component of the disk’s 2D viscous angular momentum flux density, so the 1D viscous angular momentum flux density is Eqn. (\[F’\_r2\]) integrated through the disk’s vertical cross section: $$\begin{aligned} \label{F_app} F &= \int_{-h}^{h} F'_rdx_3= -\nu_s\sigma r^2\displaystyle \frac{\partial\dot{\theta}}{\partial r}\end{aligned}$$ where $\nu_s=\eta/\rho$ is the disk’s kinematic shear viscosity. Appendix \[bulk\_appendix\] {#bulk_appendix} =========================== The flux density of $x_1$-type momentum is $\bm{\Pi}_1$ (see Eqn. \[Pi\_vector\]) while the flux density of $x_2$-type momentum is $\bm{\Pi}_2$, so the flux density of radial momentum is $\bm{G} = \cos\theta\bm{\Pi}_1 + \sin\theta\bm{\Pi}_2$ and the radial component of this momentum flux density is $$\begin{aligned} \label{G_appendix} G_r &= \bm{G\cdot\hat{r}} = (\cos\theta\Pi_{11} + \sin\theta\Pi_{21}) \bm{\hat{x_1}\cdot\hat{r}} + (\cos\theta\Pi_{12} + \sin\theta\Pi_{22})\bm{\hat{x_2}\cdot\hat{r}}\\ &= \cos^2\theta\Pi_{11} + \sin\theta\cos\theta(\Pi_{12} + \Pi_{21}) + \sin^2\theta\Pi_{22}\end{aligned}$$ where $\bm{\hat{r}}$ is the unit vector in the radial direction. The part of that momentum flux that is transported solely by viscous effects will be called $G_r'$ and is again obtained by replacing the $\Pi_{ik}$ in the above with $-\sigma'_{ik}$: $$\begin{aligned} \label{G'_r} G_r'&=& -\cos^2\theta\sigma_{11} - \sin\theta\cos\theta(\sigma'_{12} + \sigma'_{21}) - \sin^2\theta\sigma'_{22}\\ &=& - 2\eta\left[ \displaystyle\cos^2\theta\frac{\partial v_1}{\partial x_1} + \sin^2\theta\frac{\partial v_2}{\partial x_2} + \sin\theta\cos\theta \left( \frac{\partial v_1}{\partial x_2} + \frac{\partial v_2}{\partial x_1}\right)\right] - (\zeta - \frac{2}{3}\eta)\bm{\nabla\cdot v}.\end{aligned}$$ Equations (\[dv/dx\]) provide the combination $$\begin{aligned} \begin{split} \cos^2\theta\frac{\partial v_1}{\partial x_1} + \sin^2\theta\frac{\partial v_2}{\partial x_2} &= \left(\frac{3}{4} + \frac{1}{4}\cos4\theta\right)\frac{\partial v_r}{\partial r} -\frac{1}{4}\sin4\theta r\frac{\partial \dot{\theta}}{\partial r} +\frac{1}{2r}\sin^22\theta v_r -\frac{1}{4r}\sin4\theta \frac{\partial v_r}{\partial \theta}\\ &+\frac{1}{2r}\sin^22\theta \frac{\partial v_\theta}{\partial \theta}, \end{split}\end{aligned}$$ and inserting this plus Eqn. (\[dv1/dx2 + dv2/dx1\]) into Eqn. (\[G’\_r\]) then yields $$\begin{aligned} \label{G'_r2} G_r' &= -\displaystyle\left(\frac{4}{3}\eta +\zeta\right)\frac{\partial v_r}{\partial r} -\left(\zeta-\frac{2}{3}\eta\right) \left(\frac{v_r}{r} + \frac{1}{r}\frac{\partial v_\theta}{\partial \theta}\right)\end{aligned}$$ but the $\partial v_\theta/\partial \theta$ term is again neglected in the streamline approximation. This is the 2D radial momentum flux due to viscous transport, so the vertically integrated linear momentum flux due to viscosity is $$\begin{aligned} \label{G_appendix2} G &=& \int_{-h}^{h} G'_rdx_3= -\displaystyle\left(\frac{4}{3}\nu_s +\nu_b\right)\sigma\frac{\partial v_r}{\partial r} -\left(\nu_b-\frac{2}{3}\nu_s\right)\frac{\sigma v_r}{r}.\end{aligned}$$ [^1]: Actually what we identify here as the semimajor axis $a$ is called $r_0$ in [@BL94], which differs slightly from what they identify as the epicyclic semimajor axis $a_e$ where $a_e=r_0(1 + e^2)$. [^2]: But if the simulated ring is instead initialized with all particles on a given streamline having distinct (rather than common) values for $a$ and $e$, then the resulting streamlines can appear ragged in longitude $\theta$. And if that initial ring is sufficiently ragged or non-smooth, then that raggedness can grow over time as the particles $a$’s and $e$’s evolve independently. The main point is that the streamline model employed here succeeds when all streamlines are sufficiently smooth, and that is accomplished by initializing all particles in a given streamline with commmon $a,e$. [^3]: This mean angular velocity is obtained using the physical constants given in the 25 August 2011 Cassini SPICE kernel file: $Gm_0=37940585.47323534$ km$^3$/sec$^2$, $J_2=0.016290787119$, $J_4=-0.000934741301$, and $J_6=0.000089240275$.

--- abstract: 'We investigate the motion of a thin rigid body in Stokes flow and the corresponding slender body approximation used to model sedimenting fibers. In particular, we derive a rigorous error bound comparing the rigid slender body approximation to the classical PDE for rigid motion in the case of a closed loop with constant radius. Our main tool is the slender body PDE framework established by the authors and D. Spirn in [@closed_loop; @free_ends], which we adapt to the rigid setting.' author: - | Yoichiro Mori, Laurel Ohm [^1]\ *School of Mathematics, University of Minnesota, Minneapolis, MN 55455* bibliography: - 'rigid\_bib.bib' title: 'An error bound for the slender body approximation of a thin, rigid fiber sedimenting in Stokes flow' --- Introduction ============ Determining the motion of a three-dimensional rigid body sedimenting in a Stokesian fluid is an important problem in both theoretical and computational fluid mechanics. This motion is described by a classical PDE [@corona2017integral; @galdi1999steady; @weinberger1972variational], which we write below in the case of a thin rigid body. We use ${\mathcal{E}}({\bm{u}}) = \frac{1}{2}(\nabla{\bm{u}}+(\nabla{\bm{u}})^{\rm T})$ to denote the symmetric gradient, and $\bm{\sigma}=\bm{\sigma}({\bm{u}},p) = 2{\mathcal{E}}({\bm{u}})-p{\bf I}$ to denote the stress tensor. Let $\Sigma_\epsilon$ denote a closed loop slender body of radius $\epsilon>0$ (to be made precise in Section \[geometry\]) and let $\Omega_\epsilon ={\mathbb{R}}^3 \setminus \overline{\Sigma_\epsilon}$ and $\Gamma_\epsilon={\partial}\Sigma_\epsilon$ (see Figure \[fig:coord\_sys\]). The full PDE description of a slender body undergoing a rigid motion in Stokes flow may be written as follows: $$\label{rigid} \begin{aligned} -\Delta {\bm{u}}^{\rm r} +\nabla p^{\rm r} &=0 \hspace{2.6cm} \text{ in } \Omega_\epsilon \\ {{\rm{div}\,}}{\bm{u}}^{\rm r} &= 0 \hspace{2.6cm} \text{ in } \Omega_\epsilon \\ {\bm{u}}^{\rm r}({\bm{x}}) &= {\bm{v}}^{\rm r} + \bm{\omega}^{\rm r}\times {\bm{x}}, \qquad {\bm{x}}\in \Gamma_\epsilon \\ {\bm{u}}^{\rm r}({\bm{x}}) &\to 0 \hspace{2.6cm} \text{as }{\left\lvert {\bm{x}}\right\rvert}\to \infty \end{aligned}$$ and $$\begin{aligned} \int_{\Gamma_\epsilon} \bm{\sigma}^{\rm r}\bm{n} \; dS &= \bm{F}, \quad \int_{\Gamma_\epsilon} {\bm{x}}\times (\bm{\sigma}^{\rm r}\bm{n}) \; dS = \bm{T} .\end{aligned}$$ Here the total force $\bm{F}\in {\mathbb{R}}^3$ and torque $\bm{T}\in {\mathbb{R}}^3$ are given and we aim to solve for the linear velocity ${\bm{v}}^{\rm r}\in {\mathbb{R}}^3$ and angular velocity $\bm{\omega}^{\rm r}\in{\mathbb{R}}^3$ of the body. Note that the boundary value problem is in fact valid for rigid bodies of arbitrary shape, but for the purposes of this paper we specifically consider here a slender closed loop. Using the variational framework of [@galdi1999steady; @gonzalez2004dynamics; @weinberger1972variational], it can be shown that is a well-posed PDE.\ On the computational side, there has been much recent interest in numerical simulations of rigid particle sedimentation [@guazzelli2006sedimentation; @guazzelli2011fluctuations], and various tools have been developed to facilitate these simulations [@corona2017integral; @jung2006periodic; @mitchell2015sedimentation].\ ![The geometry of the rigid fiber may be parameterized with respect to the orthogonal frame ${\bm{e}}_t(s)$, ${\bm{e}}_{n_1}(s)$, ${\bm{e}}_{n_2}(s)$ defined in Section \[geometry\].[]{data-label="fig:coord_sys"}](SB_geometry_rigid "fig:")\ For a thin rigid body, a commonly-used tool for simplifying simulations is slender body theory, which exploits the thin geometry of the body by approximating the filament as a one-dimensional force density distributed along the fiber centerline. Slender body theory is a popular method for modeling sedimentation of thin fibers, both rigid [@butler2002dynamic; @park2010cloud; @saintillan2005smooth; @shin2009structure] and semi-flexible [@li2013sedimentation; @manikantan2014instability]. Here we will specifically consider the slender body theory established by Keller and Rubinow [@keller1976slender] and further developed in [@gotz2000interactions; @johnson1980improved; @tornberg2004simulating].\ Let ${\bm{X}}: {\mathbb{T}}\equiv {\mathbb{R}}/ {\mathbb{Z}}\to {\mathbb{R}}^3$ denote the coordinates of the slender body centerline, parameterized by arclength $s$ and defined more precisely in Section \[geometry\]. Given a line force density $\bm{f}^{\rm s}(s)$, $s\in {\mathbb{T}}$, the slender body approximation yields a direct expression approximating the velocity of the fiber, given by [@shelley2000stokesian]: $$\label{SBT_expr} \begin{aligned} {\bm{u}}^{\rm s}_{\rm C}(s) &= \bm{\Lambda}[\bm{f}^{\rm s}](s) + \bm{K}[\bm{f}^{\rm s}](s),\\ \bm{\Lambda}[\bm{f}](s) &:= \frac{1}{8\pi}\big[({\bf I}- 3{\bm{e}}_t{\bm{e}}_t^{\rm T})-2({\bf I}+{\bm{e}}_t{\bm{e}}_t^{\rm T}) \log(\pi\epsilon/4) \big]{\bm f}(s) \\ \bm{K}[\bm{f}](s) &:= \frac{1}{8\pi}\int_{{\mathbb{T}}} \left[ \left(\frac{{\bf I}}{|\bm{R}_0|}+ \frac{\bm{R}_0\bm{R}_0^{\rm T}}{|\bm{R}_0|^3}\right){\bm f}(s') - \frac{{\bf I}+{\bm{e}}_t(s){\bm{e}}_t(s)^{\rm T} }{|\sin (\pi(s-s'))/\pi|} {\bm f}(s)\right] \, ds'. \end{aligned}$$ Here ${\bm{e}}_t(s)$ is the unit tangent vector to ${\bm{X}}(s)$ and $\bm{R}_0(s,s') = {\bm{X}}(s) - {\bm{X}}(s')$. The slender body approximation generally allows for bending and flexing of the filament along its centerline and requires specifying the one-dimensional force density over the length of the fiber centerline. If the fiber is constrained to be fully rigid, only the total force $\bm{F}$ and torque $\bm{T}$ must be specified, where $$\label{SBT_cond1} \int_{{\mathbb{T}}} \bm{f}^{\rm s}(s) \, ds = \bm{F}, \quad \int_{{\mathbb{T}}} {\bm{X}}(s)\times \bm{f}^{\rm s}(s) \, ds = \bm{T}.$$ Additionally, we constrain the motion of the fiber centerline to be rigid, i.e. $$\label{SBT_cond2} {\bm{u}}^{\rm s}_{\rm C}(s) = {\bm{v}}^{\rm s}+\bm{\omega}^{\rm s}\times {\bm{X}}(s)$$ for constant vectors ${\bm{v}}^{\rm s}$, $\bm{\omega}^{\rm s}$. These constraints give rise to a system of integral equations which must be solved to obtain the line force density along the slender body (see [@gustavsson2009gravity; @tornberg2006numerical]).\ The aim of this paper is to establish a rigorous error bound between the slender body approximation for rigid motion - and the classical PDE describing the sedimentation of a rigid fiber immersed in Stokes flow. We show the following theorem. \[rigid\_theorem\] Let $\Sigma_\epsilon$ be a slender body as defined in Section \[geometry\]. Suppose the total force $\bm{F}\in {\mathbb{R}}^3$ and torque $\bm{T}\in {\mathbb{R}}^3$ are given, and assume that rigid slender body approximation - is satisfied by some $\bm{f}^{\rm s}\in C^1({\mathbb{T}})$. Then the difference ${\bm{v}}^{\rm r}-{\bm{v}}^{\rm s}$, $\bm{\omega}^{\rm r}-\bm{\omega}^{\rm s}$ between the linear and angular velocities of true rigid motion and the slender body approximation - satisfies $$\label{rigid_err} {\left\lvert {\bm{v}}^{\rm r}-{\bm{v}}^{\rm s} \right\rvert} + {\left\lvert \bm{\omega}^{\rm r}-\bm{\omega}^{\rm s} \right\rvert} \le C\sqrt{\epsilon}{\left\lvert \log\epsilon \right\rvert}^{3/2} \big({\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert \bm{f}^{\rm s} \right\rVert}_{C^1({\mathbb{T}})} + {\left\lvert \bm{F} \right\rvert}+{\left\lvert \bm{T} \right\rvert} \big)$$ for $C$ depending on $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$. The constants $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$ have to do only with the shape of the fiber centerline and are defined in Section \[geometry\]. Notice that we must assume that we can find $\bm{f}^{\rm s}\in C^1({\mathbb{T}})$ satisfying -, and that this $\bm{f}^{\rm s}$ then appears in the final error bound. In this sense, our bound should be considered as an [*a posteriori*]{} error estimate, similar to the type of estimates commonly used in finite element analysis [@ainsworth2011posteriori]. To obtain an [*a priori*]{} bound, we would need to able to say that such an $\bm{f}^{\rm s}$ is then bounded by the given $\bm{F}$ and $\bm{T}$, but the proof of such a bound is hindered by the likely ill-posedness of the rigid slender body approximation itself. Before discussing this issue in greater detail, we note how it arises in the error analysis.\ In order to compare the classical PDE with the rigid slender body approximation, we introduce an intermediary PDE which we will call the [*slender body PDE for rigid motion*]{}. The idea follows from the notion of slender body PDE proposed by the authors and D. Spirn in [@closed_loop] and [@free_ends] as a framework for analyzing the error introduced by the Keller-Rubinow slender body approximation for closed-loop and open-ended fibers, respectively. To construct the rigid slender body PDE, we impose that the velocity of the slender body is uniform over each cross section $s$ of the fiber. In particular, we approximate ${\bm{x}}\in \Gamma_\epsilon$ as its $L^2$ projection onto the fiber centerline ${\bm{X}}(s)$, thereby ignoring slight differences in torque across the slender body. Note that the slender body geometry is defined in Section \[geometry\] such that this projection onto the fiber centerline is unique; i.e. the notion of “fiber cross section" is well-defined. We define the slender body PDE for rigid motion as follows: $$\label{SB_PDE} \begin{aligned} -\Delta {\bm{u}}^{\rm p} +\nabla p^{\rm p} &=0 \hspace{3.27cm} \text{in } \Omega_\epsilon \\ {{\rm{div}\,}}{\bm{u}}^{\rm p} &=0 \hspace{3.27cm} \text{in } \Omega_\epsilon \\ {\bm{u}}^{\rm p}({\bm{x}}) &= {\bm{v}}^{\rm p} + \bm{\omega}^{\rm p}\times {\bm{X}}(s) \qquad \text{on } \Gamma_\epsilon \\ {\bm{u}}^{\rm p}({\bm{x}}) &\to 0 \hspace{3.23cm} \text{as }{\left\lvert {\bm{x}}\right\rvert}\to \infty \end{aligned}$$ and $$\begin{aligned} \int_{\Gamma_\epsilon} \bm{\sigma}^{\rm p}\bm{n} \; {\mathcal{J}}_\epsilon(s,\theta) \, d\theta \, ds &= \bm{F}, \qquad \int_{{\mathbb{T}}} {\bm{X}}(s)\times \bigg(\int_0^{2\pi} \bm{\sigma}^{\rm p}\bm{n} \, {\mathcal{J}}_\epsilon(s,\theta) d\theta \bigg) ds = \bm{T}.\end{aligned}$$ Here we have written $dS = {\mathcal{J}}_\epsilon(s,\theta)\, d\theta\, ds$, where ${\mathcal{J}}_\epsilon$ is the Jacobian factor on the slender body surface, which we parameterize as a tube about ${\bm{X}}(s)$ using surface angle $\theta$ (see Section \[geometry\] and expression ). We show that for a closed filament, the rigid slender body PDE is in fact close to the classical PDE for rigid motion [@galdi1999steady; @weinberger1972variational] – in particular, the variation in torque over any cross section of the slender body is higher order in $\epsilon$.\ In the case of a flexible filament with a prescribed force density per unit length along the centerline, the slender body PDE of [@closed_loop; @free_ends] is well-posed, and the difference between the slender body approximation and the PDE solution can be estimated in terms of the slender body radius and the given line force density. We would like to use the existing error analysis in [@closed_loop] to bound the difference between the rigid slender body approximation and the rigid slender body PDE solution. The rigid case is complicated by the fact that the existing error bound relies on knowledge of line force density along the filament, and therefore on the solvability of the integral equation for this force density arising from -.\ This raises the issue of the lack of a general solution theory for this particular type of integral equation. Specifically, a detailed spectral analysis by Götz [@gotz2000interactions] in the case of a straight slender body centerline shows that the slender body operator $(\bm{\Lambda}+\bm{K})$ in is not necessarily invertible for all values of $\epsilon$ and all centerline velocities ${\bm{u}}^{\rm s}_{\rm C}$. A similar result for a perfectly circular, planar centerline was shown by Shelley-Ueda in [@shelley2000stokesian]. For fibers with more general centerline curvature, a spectral analysis of the slender body integral operator is complicated and the invertibility properties remain unclear.\ In rigid slender body theory, this same integral operator arises in the integral equations for the line force density $\bm{f}^{\rm s}$. Therefore, to carry out an error analysis for the rigid slender body problem we must assume from the start that we are considering a slender body approximation that gives rise to $\bm{f}^{\rm s}\in C^1({\mathbb{T}})$ satisfying -. This $\bm{f}^{\rm s}$ must be included in the final error bound along with the total force $\bm{F}$ and torque $\bm{T}$, giving rise to a type of [*a posteriori*]{} error estimate. Furthermore, the $\sqrt{\epsilon}$ bound is the best we can do using our current techniques, but it is possible that a more complete solution theory for the slender body approximation could improve this bound.\ In practice, various regularizations of the slender body integral operator are used to combat this ill-posedness, although rigorous justification for these regularizations is still needed. Given that the slender body PDE framework of [@closed_loop; @free_ends] [*is*]{} well-posed, we should be able to use this framework to come up with the best regularization for the ill-posed slender body approximation. However, this is truly a deeper issue that we plan to explore in future work. Outline of the proof of Theorem \[rigid\_theorem\] -------------------------------------------------- The strategy for proving Theorem \[rigid\_theorem\] is to show that, given $\bm{F}$ and $\bm{T}$, the solution to the rigid slender body PDE is close to both the classical rigid PDE solution and the rigid slender body approximation - .\ First, we must show that the rigid slender body PDE is well-posed. Using Definition \[rigid\_weak\_p\] of a weak solution to the rigid slender body PDE , we show the following. \[SB\_PDE\_well\] Let $\Sigma_\epsilon$ be a slender body as defined in Section \[geometry\]. Given $\bm{F}$ and $\bm{T}\in{\mathbb{R}}^3$, there exists a unique weak solution $({\bm{u}}^{\rm p},p^{\rm p})\in {\mathcal{R}}_\epsilon^{{{\rm{div}\,}}} \times L^2(\Omega_\epsilon)$ to the slender body PDE for rigid motion satisfying the estimate $$\label{totalPest} {\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} + {\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \le C{\left\lvert \log\epsilon \right\rvert}^{1/2} ( {\left\lvert \bm{F} \right\rvert}+{\left\lvert \bm{T} \right\rvert} )$$ for $C$ depending on $c_\Gamma$ and $\kappa_{\max}$. Theorem \[SB\_PDE\_well\] can be established using many of the same tools from the well-posedness theory in [@closed_loop]. In addition, we will make use of the following bound along the slender body centerline ${\bm{X}}(s)$: \[v\_omega\_bd\] Let ${\bm{X}}$ be as in Section \[geometry\] and consider constant vectors ${\bm{v}}$, $\bm{\omega} \in {\mathbb{R}}^3$. Then $${\left\lvert {\bm{v}}\right\rvert}+{\left\lvert \bm{\omega} \right\rvert} \le C{\left\lVert {\bm{v}}+ \bm{\omega}\times {\bm{X}}\right\rVert}_{L^2({\mathbb{T}})}$$ for $C$ depending only on $c_\Gamma$ and $\kappa_{\max}$. We will first prove Lemma \[v\_omega\_bd\] in Section \[2ndLemma\]; then Theorem \[SB\_PDE\_well\] quickly follows using some of the key inequalities collected in Section \[ineq\].\ With the variational framework for , comparing to is relatively straightforward. Using Lemma \[v\_omega\_bd\], we show that the difference between the true rigid motion and the slender body PDE description satisfies the following lemma. \[true\_vs\_SB\] Let ${\bm{X}}$ be as in Section \[geometry\]. Given $\bm{F}$ and $\bm{T}\in {\mathbb{R}}^3$, let $({\bm{v}}^{\rm r},\bm{\omega}^{\rm r})$ be the corresponding boundary values satisfying and let $({\bm{v}}^{\rm p},\bm{\omega}^{\rm p})$ be the boundary values satisfying . Then $$\label{true_vs_SB_bd} {\left\lvert \bm{\omega}^{\rm r}-\bm{\omega}^{\rm p} \right\rvert}+ {\left\lvert {\bm{v}}^{\rm r}-{\bm{v}}^{\rm p} \right\rvert} \le \epsilon{\left\lvert \log\epsilon \right\rvert}C( {\left\lvert \bm{T} \right\rvert}+ {\left\lvert \bm{F} \right\rvert})$$ where $C$ depends only on $c_\Gamma$ and $\kappa_{\max}$. The main difficulties in proving Theorem \[rigid\_theorem\] arise in comparing to - . As discussed, the solvability issue for the rigid slender body approximation is addressed by assuming that we are only considering a rigid slender body approximation - that gives rise to a force density $\bm{f}^{\rm s}\in C^1({\mathbb{T}})$. An additional difficulty arises in that in order to use the error analysis framework of [@closed_loop], the line force density along the slender body must be the same for both the slender body approximation and the slender body PDE. Therefore we need to define yet another intermediary PDE.\ Given $\bm{f}^{\rm s}\in C^1({\mathbb{T}})$ satisfying - for given $\bm{F}$ and $\bm{T}\in {\mathbb{R}}^3$, we define ${\bm{u}}^{\rm p,s}$ as the solution to the PDE: $$\label{SB_PDE_ps} \begin{aligned} -\Delta {\bm{u}}^{\rm p,s} +\nabla p^{\rm p,s} &=0 \hspace{2.75cm} \text{in } \Omega_\epsilon \\ {{\rm{div}\,}}{\bm{u}}^{\rm p,s} &=0 \hspace{2.75cm} \text{in } \Omega_\epsilon \\ \int_0^{2\pi} (\bm{\sigma}^{\rm p,s}\bm{n}) \; {\mathcal{J}}_\epsilon(s,\theta) \, d\theta &= \bm{f}^{\rm s}(s) \hspace{2.05cm} \text{on }\Gamma_\epsilon \\ {\bm{u}}^{\rm p,s}\big|_{\Gamma_\epsilon} &= {\rm Tr}({\bm{u}}^{\rm p,s})(s), \qquad \text{ unknown but independent of } \theta \\ {\bm{u}}^{\rm p,s}&\to 0 \hspace{2.65cm} \text{as } {\left\lvert {\bm{x}}\right\rvert}\to\infty. \end{aligned}$$ Here ${\rm Tr}({\bm{u}}^{\rm p,s})(s)$ denotes the trace of ${\bm{u}}^{\rm p,s}$ on $\Gamma_\epsilon$, and $\theta$ refers to the parameterization of $\Gamma_\epsilon$ as a tube about ${\bm{X}}(s)$ (see Section \[geometry\]). By [@closed_loop], we know that a (weak) solution $({\bm{u}}^{\rm p,s},p^{\rm p,s})$ exists and is unique. Now, ${\rm Tr}({\bm{u}}^{\rm p,s})(s)$ may not be precisely a rigid motion, but we can show that it is close. In particular, by Theorem 1.3 in [@closed_loop], we may bound the difference between ${\rm Tr}({\bm{u}}^{\rm p,s})(s)$ and ${\bm{u}}^{\rm s}_{\rm C}(s) = {\bm{v}}^{\rm s}+\bm{\omega}^{\rm s}\times {\bm{X}}(s)$ by $$\label{centerline_err} {\left\lVert {\rm Tr}( {\bm{u}}^{\rm p,s}) - ({\bm{v}}^{\rm s}+\bm{\omega}^{\rm s} \times {\bm{X}}) \right\rVert}_{L^2({\mathbb{T}})} \le C\epsilon {\left\lvert \log\epsilon \right\rvert}^{3/2}{\left\lVert \bm{f}^{\rm s} \right\rVert}_{C^1({\mathbb{T}})}$$ for $C$ depending only on $c_\Gamma$ and $\kappa_{\max}$.\ A further technical issue arises in comparing to . In order to obtain a useful estimate of the difference between $({\bm{u}}^{\rm p,s}-{\bm{u}}^{\rm p},p^{\rm p,s}-p^{\rm p})$ in terms of only $\bm{F}$, $\bm{T}$, and $\bm{f}^{\rm s}(s)$, we will need a careful characterization of the $\epsilon$-dependence in a higher regularity estimate for solutions to (see Lemma \[high\_reg\]). Note that for a (sufficiently smooth) sedimenting rigid body, once well-posedness of the PDE has been established, higher regularity of the solution follows by standard arguments for a Stokes Dirichlet boundary value problem. In our case, the novelty is determining how the higher regularity bound scales with $\epsilon$. Our proof (see Appendix \[reg\_lem\]) makes use of the local coordinate system valid near the slender body. We obtain commutator estimates for the tangential derivatives along the slender body surface and use an integration by parts argument, along with the form of the Stokes equations in local coordinates, to show that the bound for an additional derivative of the rigid slender body PDE solution scales like $1/\epsilon$, up to logarithmic corrections.\ Now, using the variational framework for the slender body PDE along with this higher regularity lemma, we can show the following estimate. \[ups\_up\_err\] Let ${\bm{u}}^{\rm p,s}$ satisfy and let $({\bm{v}}^{\rm p},\bm{\omega}^{\rm p})$ denote the rigid slender body PDE boundary values satisfying . Then $${\left\lVert {\rm Tr}( {\bm{u}}^{\rm p,s}) - ( {\bm{v}}^{\rm p} + \bm{\omega}^{\rm p}\times {\bm{X}}) \right\rVert}_{L^2({\mathbb{T}})} \le C\sqrt{\epsilon}{\left\lvert \log\epsilon \right\rvert}^{3/2} \big({\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert \bm{f}^{\rm s} \right\rVert}_{C^1({\mathbb{T}})} + {\left\lvert \bm{F} \right\rvert}+{\left\lvert \bm{T} \right\rvert} \big)$$ for $C$ depending on $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$. Combining estimate with Lemma \[ups\_up\_err\] and using Lemma \[v\_omega\_bd\] with ${\bm{v}}^{\rm p}-{\bm{v}}^s$ and $\bm{\omega}^{\rm p}-\bm{\omega}^{\rm s}$ in place of ${\bm{v}}$ and $\bm{\omega}$, we obtain the following bound for the difference between the slender body approximation - and the slender body PDE : $$\label{PDE_vs_SBT} {\left\lvert {\bm{v}}^{\rm p}-{\bm{v}}^{\rm s} \right\rvert} + {\left\lvert \bm{\omega}^{\rm p}-\bm{\omega}^{\rm s} \right\rvert} \le C\sqrt{\epsilon}{\left\lvert \log\epsilon \right\rvert}^{3/2} \big({\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert \bm{f}^{\rm s} \right\rVert}_{C^1({\mathbb{T}})} + {\left\lvert \bm{F} \right\rvert}+{\left\lvert \bm{T} \right\rvert} \big).$$ Finally, combining the estimate with Lemma \[true\_vs\_SB\] yields Theorem \[rigid\_err\]. The remainder of this paper is thus devoted to showing Lemmas \[v\_omega\_bd\] - \[ups\_up\_err\]. We will begin by introducing the variational framework for and noting some key inequalities in Section \[variational0\]. In Section \[2ndLemma\], we show Lemma \[v\_omega\_bd\] and use it to derive estimates for $({\bm{u}}^{\rm p},p^{\rm p},{\bm{v}}^{\rm p},\bm{\omega}^{\rm p})$ satisfying . These estimates can then be used to show Theorem \[SB\_PDE\_well\]. In Section \[3rdLemma\], we use the variational framework for the rigid slender body PDE to prove Lemma \[true\_vs\_SB\]. Finally, in Section \[1stLemma\], we prove Lemma \[ups\_up\_err\] to complete the proof of Theorem \[rigid\_err\]. Geometry and variational framework {#variational0} ================================== We begin in Section \[geometry\] with a precise definition of the slender body geometry. In Section \[variational\], we introduce the variational form of the slender body PDE for rigid motion , which, along with the variational form of , will provide the framework for obtaining Theorem \[rigid\_theorem\]. Finally, in Section \[ineq\], we make note of some key inequalities that will be used throughout the remainder of this paper. Slender body geometry {#geometry} --------------------- As in [@closed_loop], we let ${\bm{X}}: {\mathbb{T}}\equiv {\mathbb{R}}/ {\mathbb{Z}}\to {\mathbb{R}}^3$ denote the coordinates of a closed, non-self-intersecting $C^3$ curve in ${\mathbb{R}}^3$, parameterized by arclength $s$. We require that $$\label{cgamma} \inf_{s\neq s'}\frac{{\left\lvert {\bm{X}}(s)-{\bm{X}}(s') \right\rvert}}{{\left\lvert s-s' \right\rvert}} \ge c_\Gamma$$ for some constant $c_\Gamma>0$.\ Along ${\bm{X}}(s)$ we consider the orthonormal frame $({\bm{e}}_t(s),{\bm{e}}_{n_1}(s),{\bm{e}}_{n_2}(s))$ defined in [@closed_loop]. Here ${\bm{e}}_t(s) = \frac{d{\bm{X}}}{ds}$ is the unit tangent vector to ${\bm{X}}(s)$ and $({\bm{e}}_{n_1}(s),{\bm{e}}_{n_2}(s))$ span the plane normal to ${\bm{e}}_t(s)$. The frame satisfies the ODEs $$\label{C2_frame} \frac{d}{ds}{\bm{e}}_t = \kappa_1{\bm{e}}_{n_1}+\kappa_2{\bm{e}}_{n_2}, \quad \frac{d}{ds}{\bm{e}}_{n_1} = -\kappa_1{\bm{e}}_t+\kappa_3{\bm{e}}_{n_2}, \quad \frac{d}{ds}{\bm{e}}_{n_2} = -\kappa_2{\bm{e}}_t-\kappa_3{\bm{e}}_{n_1}$$ where $\kappa_1^2(s) +\kappa_2^2(s) = \kappa^2(s)$, the fiber curvature, and $\kappa_3$ is a constant satisfying ${\left\lvert \kappa_3 \right\rvert}\le \pi$. We require the orthonormal frame to be $C^2$ and denote $$\kappa_{\max} := \max_{s\in{\mathbb{T}}} {\left\lvert \kappa(s) \right\rvert}, \quad \xi_{\max} = \max_{s\in{\mathbb{T}}} {\left\lvert \frac{{\partial}^3{\bm{X}}}{{\partial}s^3} \right\rvert} .$$ Note that ${\left\lvert {\partial}\kappa_1 /{\partial}s \right\rvert} +{\left\lvert {\partial}\kappa_2/{\partial}s \right\rvert} \le \xi_{\max}+2(\kappa_{\max}+\pi)$.\ We define $${\bm{e}}_\rho(s,\theta) := \cos\theta{\bm{e}}_{n_1}(s)+\sin\theta{\bm{e}}_{n_2}(s)$$ and, for some $r_{\max}=r_{\max}(c_\Gamma,\kappa_{\max})\le \frac{1}{2\kappa_{\max}}$, we can uniquely parameterize points ${\bm{x}}$ within a neighborhood ${\rm dist}({\bm{x}},{\bm{X}})< r_{\max}$ of ${\bm{X}}(s)$ as $${\bm{x}}= {\bm{X}}(s) + \rho{\bm{e}}_\rho(s,\theta), \quad 0\le \rho <r_{\max}.$$ For $\epsilon<r_{\max}/4$, we may then define a slender body of uniform radius $\epsilon$ as $$\label{SB_def} \Sigma_\epsilon:= \big\{{\bm{x}}\in {\mathbb{R}}^3 \; : \; {\bm{x}}= {\bm{X}}(s) + \rho{\bm{e}}_\rho(s,\theta), \; \rho <\epsilon \big\}.$$ We parameterize the slender body surface $\Gamma_\epsilon = {\partial}\Sigma_\epsilon$ as $$\Gamma_\epsilon = {\bm{X}}(s) + \epsilon{\bm{e}}_\rho(s,\theta).$$ In addition, we may parameterize the Jacobian factor ${\mathcal{J}}_\epsilon(s,\theta)$ on the slender body surface as $$\label{jac_fac} {\mathcal{J}}_\epsilon(s,\theta) = \epsilon\big(1-\epsilon(\kappa_1(s)\cos\theta+\kappa_2(s)\sin\theta) \big).$$ Variational form of {#variational} -------------------- Letting $\Omega_\epsilon={\mathbb{R}}^3\backslash\overline{\Sigma_\epsilon}$ for $\Sigma_\epsilon$ as in Section \[geometry\], we recall the following function spaces, used in [@closed_loop] to study a slender body PDE of the form . We use $D^{1,2}(\Omega_\epsilon)$ to denote the homogeneous Sobolev space $$\label{D12} D^{1,2}(\Omega_\epsilon) = \big\{ {\bm{u}}\in L^6(\Omega_\epsilon) \; : \; \nabla {\bm{u}}\in L^2(\Omega_\epsilon) \big\},$$ which, due to the Sobolev inequality in $\Omega_\epsilon \subset {\mathbb{R}}^3$ (see Lemma \[sobolev\]), is a Hilbert space with norm ${\left\lVert \nabla {\bm{u}}\right\rVert}_{L^2(\Omega_\epsilon)}$. We define $D^{1,2}_0(\Omega_\epsilon)$ as the closure of $C_0^\infty(\Omega_\epsilon)$ (smooth, compactly supported test functions) in $D^{1,2}(\Omega_\epsilon)$.\ We also recall the space ${\mathcal{A}}_\epsilon$, the subspace of $D^{1,2}(\Omega_\epsilon)$ with $\theta$-independent boundary values: $$\label{Aepsilon} {\mathcal{A}}_\epsilon = \big\{ {\bm{u}}\in D^{1,2}(\Omega_\epsilon) \; : \; {\bm{u}}\big|_{\Gamma_\epsilon} = {\bm{u}}(s) \big\}.$$ Here the boundary value ${\bm{u}}\big|_{\Gamma_\epsilon} = {\bm{u}}(s)$ is not directly specified but is required to be independent of the surface angle $\theta$. We define ${\mathcal{A}}_\epsilon^{{\rm{div}\,}}$ to be the divergence-free subspace of ${\mathcal{A}}_\epsilon$.\ We also recall the variational form of , examined in detail in [@closed_loop]. A weak solution ${\bm{u}}^{\rm p,s}\in {\mathcal{A}}_\epsilon^{{{\rm{div}\,}}}$ to satisfies $$\label{weak_no_p} \int_{\Omega_\epsilon} 2 {\mathcal{E}}({\bm{u}}^{\rm p,s}): {\mathcal{E}}({\bm{v}}) \, d{\bm{x}}= \int_{\mathbb{T}}{\bm{v}}(s)\cdot\bm{f}^{\rm s}(s) \, ds$$ for any ${\bm{v}}\in{\mathcal{A}}_\epsilon^{{{\rm{div}\,}}}$. In addition, for ${\bm{u}}^{\rm p,s}$ satisfying , there exists a unique pressure $p^{\rm p,s}\in L^2(\Omega_\epsilon)$ satisfying $$\label{weak_with_p} \int_{\Omega_\epsilon} \big( 2 {\mathcal{E}}({\bm{u}}^{\rm p,s}): {\mathcal{E}}({\bm{v}}) - p^{\rm p,s}\, {{\rm{div}\,}}{\bm{v}}\big)\, d{\bm{x}}= \int_{\mathbb{T}}{\bm{v}}(s)\cdot\bm{f}^{\rm s}(s) \, ds$$ for any ${\bm{v}}\in {\mathcal{A}}_\epsilon$. To study , we define the following subspace of ${\mathcal{A}}_\epsilon$, where we further restrict the boundary value to be a rigid motion: $$\label{Repsilon} {\mathcal{R}}_\epsilon = \big\{ {\bm{u}}\in D^{1,2}(\Omega_\epsilon) \; : \; {\bm{u}}\big|_{\Gamma_\epsilon} = {\bm{v}}+ \bm{\omega}\times{\bm{X}}(s) \text{ for }{\bm{v}},\, \bm{\omega}\in {\mathbb{R}}^3 \big\}.$$ Again, ${\bm{v}}$ and $\bm{\omega}$ are not directly specified but are required to be constant vectors in ${\mathbb{R}}^3$. We let ${\mathcal{R}}_\epsilon^{{\rm{div}\,}}$ denote the divergence-free subspace of ${\mathcal{R}}_\epsilon$.\ We then define a weak solution to the rigid motion slender body PDE as follows. \[rigid\_weak\] A weak solution ${\bm{u}}^{\rm p}\in {\mathcal{R}}_\epsilon^{{\rm{div}\,}}$ to satisfies $$\int_{\Omega_\epsilon} 2\, {\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}(\bm{\varphi}) \, d{\bm{x}}= {\bm{v}}_\varphi \cdot\bm{F} + \bm{\omega}_\varphi \cdot\bm{T}$$ for any $\bm{\varphi}\in {\mathcal{R}}_\epsilon^{{\rm{div}\,}}$, where we denote $\bm{\varphi}\big|_{\Gamma_\epsilon} = {\bm{v}}_\varphi + \bm{\omega}_\varphi \times {\bm{X}}(s)$. Given the existence and uniqueness of ${\bm{u}}^{\rm p}$ satisfying Definition \[rigid\_weak\], using an essentially identical proof to that in Section 2.2 of [@closed_loop], we can establish an equivalent notion of weak solution that includes a corresponding weak pressure $p^{\rm p}\in L^2(\Omega_\epsilon)$ and removes the divergence-free restriction on test functions $\bm{\varphi}$. \[rigid\_weak\_p\] Given ${\bm{u}}^{\rm p}\in {\mathcal{R}}_\epsilon^{{\rm{div}\,}}$ satisfying Definition \[rigid\_weak\], there exists a unique $p^{\rm p}\in L^2(\Omega_\epsilon)$ satisfying $$\int_{\Omega_\epsilon} \big(2\, {\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}(\bm{\varphi}) - p\, {{\rm{div}\,}}\bm{\varphi} \big) \, d{\bm{x}}= {\bm{v}}_\varphi \cdot\bm{F} + \bm{\omega}_\varphi\cdot \bm{T}$$ for any $\bm{\varphi}\in {\mathcal{R}}_\epsilon$. Here we again denote $\bm{\varphi}\big|_{\Gamma_\epsilon} = {\bm{v}}_\varphi + \bm{\omega}_\varphi \times {\bm{X}}(s)$. Important inequalities {#ineq} ---------------------- In addition to the definitions of Section \[variational\], we collect the statements of various inequalities that are used throughout the paper, keeping track of the $\epsilon$-dependence in any constants that arise. The proofs of these inequalities are mostly contained in [@closed_loop], with the exception of Lemma \[trace2\], which appears in Appendix \[appendix\].\ First, we note the following ${\mathcal{A}}_\epsilon$ trace inequality, which holds for functions ${\bm{u}}\in {\mathcal{A}}_\epsilon$ due to $\theta$-independence on $\Gamma_\epsilon$. As a slight abuse of notation, the trace operator ${\rm Tr}$, when applied to ${\mathcal{A}}_\epsilon$ functions, will be considered as both a function on $\Gamma_\epsilon$ and on ${\mathbb{T}}$. *($L^2({\mathbb{T}})$ trace inequality)*\[trace1\] Let $\Omega_{\epsilon}={\mathbb{R}}^3 \backslash \overline{\Sigma_{\epsilon}}$ be as in Section \[geometry\]. Then any ${\bm{u}}\in {\mathcal{A}}_\epsilon$ satisfies $$\|{\rm Tr}({\bm{u}})\|_{L^2({\mathbb{T}})} \le C |\log\epsilon|^{1/2} \| \nabla {\bm{u}}\|_{L^2(\Omega_{\epsilon})},$$ where the constant $C$ depends on $\kappa_{\max}$ and $c_{\Gamma}$ but is independent of $\epsilon$. The proof of this lemma appears in Appendix A.2.1 of [@closed_loop].\ On the other hand, for general $D^{1,2}(\Omega)$ functions, the following trace inequality holds over the surface $\Gamma_\epsilon$: *($L^2(\Gamma_\epsilon)$ trace inequality)*\[trace2\] Let $\Omega_{\epsilon}={\mathbb{R}}^3 \backslash \overline{\Sigma_{\epsilon}}$ be as in Section \[geometry\]. Then any ${\bm{u}}\in D^{1,2}(\Omega_\epsilon)$ satisfies $$\|{\rm Tr}({\bm{u}})\|_{L^2(\Gamma_\epsilon)} \le C \sqrt{\epsilon}|\log\epsilon|^{1/2} \| \nabla {\bm{u}}\|_{L^2(\Omega_{\epsilon})},$$ where the constant $C$ depends on $\kappa_{\max}$ and $c_{\Gamma}$ but is independent of $\epsilon$. The proof of Lemma \[trace2\] appears in Appendix \[appendix\].\ We will also need the following Korn inequality. *(Korn inequality)*\[korn\] Let $\Omega_{\epsilon}={\mathbb{R}}^3 \backslash \overline{\Sigma_{\epsilon}}$ be as in Section \[geometry\]. Then any ${\bm{u}}\in D^{1,2}(\Omega_{\epsilon})$ satisfies $$\|\nabla {\bm{u}}\|_{L^2(\Omega_{\epsilon})} \le C\|{\mathcal{E}}({\bm{u}})\|_{L^2(\Omega_{\epsilon})},$$ where the constant $C$ depends only on $\kappa_{\max}$ and $c_{\Gamma}$. The proof of $\epsilon$-independence in the Korn constant is given in Appendix A.2.2 - A.2.3 in [@closed_loop].\ Finally, we make use of the following pressure estimate. \[pressure\] For $({\bm{u}},p)$ satisfying the Stokes equations in $\Omega_\epsilon$, we have $${\left\lVert p \right\rVert}_{L^2(\Omega_\epsilon)} \le C {\left\lVert {\mathcal{E}}({\bm{u}}) \right\rVert}_{L^2(\Omega_\epsilon)}$$ for $C$ independent of $\epsilon$. The proof of this lemma exactly follows the proof of estimate (2.17) in [@closed_loop]. Proof of Lemma \[v\_omega\_bd\] and a corollary {#2ndLemma} =============================================== Here we prove Lemma \[v\_omega\_bd\] and make note of a corollary which allows us to obtain a useful bound for functions in $\rm{R}_\epsilon$. This corollary, along with the Korn inequality (Lemma \[korn\]) and pressure estimate (Lemma \[pressure\]), then allows us to prove Theorem \[SB\_PDE\_well\]. Note that Lemma \[v\_omega\_bd\] is obviously true when ${\bm{v}}=\bm{\omega}=0$; thus we can assume that at least one of ${\bm{v}},\bm{\omega}$ is nonzero. Suppose that Lemma \[v\_omega\_bd\] does not hold. Then we may choose a sequence of triples $({\bm{v}}_k,\bm{\omega}_k,{\bm{X}}_k(s))$ such that the following properties hold for each $k=1,2,3,\dots$. First, ${\bm{v}}_k,\bm{\omega}_k\in{\mathbb{R}}^3$ satisfy ${\left\lvert {\bm{v}}_k \right\rvert}^2+{\left\lvert \bm{\omega}_k \right\rvert}^2=1$, and ${\bm{X}}_k(s)$ is a closed curve satisfying the geometric constraints of Section \[geometry\] – in particular, ${\left\lvert {\bm{X}}_k'' \right\rvert}\le \kappa_{\max}$. For simplicity, we also take $\int_{{\mathbb{T}}}{\bm{X}}_k(s) \, ds=0$. In addition, $$1 = {\left\lvert {\bm{v}}_k \right\rvert}^2+{\left\lvert \bm{\omega}_k \right\rvert}^2 > k^2\int_{{\mathbb{T}}}{\left\lvert {\bm{v}}_k+\bm{\omega}_k\times {\bm{X}}_k(s) \right\rvert}^2 \, ds.$$ Then $$\int_{{\mathbb{T}}}{\left\lvert {\bm{v}}_k+\bm{\omega}_k\times {\bm{X}}_k(s) \right\rvert}^2 \, ds < \frac{1}{k^2} \to 0$$ as $k\to\infty$. Since ${\bm{v}}_k,\bm{\omega}_k$ are just vectors in ${\mathbb{R}}^3$, some limit ${\bm{v}}_\infty,\bm{\omega}_\infty$ exists. Furthermore, since each ${\bm{X}}_k$ is controlled in $C^2$ by $\kappa_{\max}$, we have that (passing to a subsequence) ${\bm{X}}_k\to {\bm{X}}_\infty$ in $C^1$ for some closed, unit length curve ${\bm{X}}_\infty(s)$. Thus $$\int_{{\mathbb{T}}}{\left\lvert {\bm{v}}_\infty +\bm{\omega}_\infty \times {\bm{X}}_\infty(s) \right\rvert}^2 \, ds =0,$$ and therefore $\bm{\omega}_\infty\times {\bm{X}}_\infty(s)\equiv -{\bm{v}}_\infty$. But $\bm{\omega}_\infty$ and ${\bm{v}}_\infty$ are both constant vectors with ${\left\lvert {\bm{v}}_\infty \right\rvert}^2+{\left\lvert \bm{\omega}_\infty \right\rvert}^2=1$, while ${\bm{X}}_\infty(s)$ necessarily has nonzero curvature. Thus $\bm{\omega}_\infty\times {\bm{X}}_\infty(s)$ cannot identically equal the constant vector $-{\bm{v}}_\infty$. Furthermore, because ${\bm{X}}_k$ was allowed to vary among curves satisfying the constraints of Section \[geometry\], the constant $C$ arising in Lemma \[v\_omega\_bd\] depends only on $c_\Gamma$ and $\kappa_{\max}$. Given $\bm{F}$ and $\bm{T}$, as an immediate corollary to Lemma \[v\_omega\_bd\] we obtain the following useful bound for any function $\bm{\varphi}\in{\mathcal{R}}_\epsilon$. \[omegaP\] Consider $\bm{\varphi}\in {\mathcal{R}}_\epsilon$ with boundary value denoted by ${\bm{v}}_\varphi+\bm{\omega}_\varphi\times {\bm{X}}(s)$. Then $$\label{curved_est} |{\bm{v}}_\varphi|+|\bm{\omega}_\varphi| \le C |\log\epsilon|^{1/2} {\left\lVert {\mathcal{E}}(\bm{\varphi}) \right\rVert}_{L^2(\Omega_\epsilon)}$$ where $C$ depends only on $c_\Gamma$ and $\kappa_{\max}$. Using Lemma \[v\_omega\_bd\] along with the slender body trace estimate (Lemma \[trace1\]) and Korn inequality (Lemma \[korn\]), we immediately obtain $$\begin{aligned} {\left\lvert {\bm{v}}_\varphi \right\rvert}+ {\left\lvert \bm{\omega}_\varphi \right\rvert} &\le {\left\lVert {\bm{v}}_\varphi+\bm{\omega}_\varphi\times{\bm{X}}\right\rVert}_{L^2({\mathbb{T}})} \le C{\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert \nabla \bm{\varphi} \right\rVert}_{L^2(\Omega_\epsilon)} \\ &\le C{\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert {\mathcal{E}}(\bm{\varphi}) \right\rVert}_{L^2(\Omega_\epsilon)}\end{aligned}$$ for $C$ depending only on $c_\Gamma$ and $\kappa_{\max}$. Using Corollary \[omegaP\] and the variational formulation of , we may now prove Theorem \[SB\_PDE\_well\]. We first show the existence of a weak solution ${\bm{u}}^{\rm p}\in {\mathcal{R}}_\epsilon^{{{\rm{div}\,}}}$ satisfying Definition \[rigid\_weak\]. Note that the bilinear form appearing on the left hand side of Definition \[rigid\_weak\] is bounded on ${\mathcal{R}}_\epsilon^{{{\rm{div}\,}}}$, as $${\left\lvert \int_{\Omega_\epsilon} 2\, {\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}(\bm{\varphi}) \, d{\bm{x}}\right\rvert} \le 2{\left\lVert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rVert}_{L^2(\Omega_\epsilon)}{\left\lVert {\mathcal{E}}(\bm{\varphi}) \right\rVert}_{L^2(\Omega_\epsilon)} \le 2{\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)}{\left\lVert \nabla\bm{\varphi} \right\rVert}_{L^2(\Omega_\epsilon)}.$$ Coercivity of the bilinear form also follows by the Korn inequality (Lemma \[korn\]). Furthermore, using Corollary \[omegaP\], the linear functional on the right hand side of Definition \[rigid\_weak\] is bounded for $\bm{\varphi}\in {\mathcal{R}}_\epsilon^{{{\rm{div}\,}}}$, as $${\left\lvert {\bm{v}}_\varphi \cdot\bm{F} + \bm{\omega}_\varphi\cdot \bm{T} \right\rvert} \le C{\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert {\mathcal{E}}(\bm{\varphi}) \right\rVert}_{L^2(\Omega_\epsilon)}({\left\lvert \bm{F} \right\rvert} +{\left\lvert \bm{T} \right\rvert}) \le C{\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert \nabla\bm{\varphi} \right\rVert}_{L^2(\Omega_\epsilon)}({\left\lvert \bm{F} \right\rvert} +{\left\lvert \bm{T} \right\rvert}).$$ Then, by the Lax-Milgram theorem, there exists a unique weak solution ${\bm{u}}^{\rm p}\in {\mathcal{R}}_\epsilon^{{{\rm{div}\,}}}$ to .\ In addition, using the variational form of along with Corollary \[omegaP\], we have that ${\bm{u}}^{\rm p}$ satisfies $$\begin{aligned} \int_{\Omega_\epsilon} {\left\lvert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rvert}^2 \, d{\bm{x}}&= \int_{\Gamma_\epsilon} ({\bm{v}}^{\rm p}+\bm{\omega}^{\rm p}\times {\bm{X}}(s))\cdot(\bm{\sigma}^{\rm p}\bm{n}) \, dS \\ &= {\bm{v}}^{\rm p}\cdot\int_{\Gamma_\epsilon} \bm{\sigma}^{\rm p}\bm{n} \, dS + \bm{\omega}^{\rm p}\cdot \int_{{\mathbb{T}}} {\bm{X}}(s)\times \bigg(\int_0^{2\pi} (\bm{\sigma}^{\rm p}\bm{n}) \, {\mathcal{J}}_\epsilon(s,\theta) d \theta \bigg) ds \\ &\le {\left\lvert {\bm{v}}^{\rm p} \right\rvert}{\left\lvert \bm{F} \right\rvert} + {\left\lvert \bm{\omega}^{\rm p} \right\rvert}{\left\lvert \bm{T} \right\rvert} \le C{\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rVert}_{L^2(\Omega_\epsilon)}({\left\lvert \bm{F} \right\rvert} +{\left\lvert \bm{T} \right\rvert}) \\ &\le \frac{1}{2}{\left\lVert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rVert}_{L^2(\Omega_\epsilon)}^2 + C{\left\lvert \log\epsilon \right\rvert}({\left\lvert \bm{F} \right\rvert}^2 +{\left\lvert \bm{T} \right\rvert}^2),\end{aligned}$$ where we have used Young’s inequality in the last line. We thus obtain the estimate $$\label{Eup_est} {\left\lVert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rVert}_{L^2(\Omega_\epsilon)} \le C{\left\lvert \log\epsilon \right\rvert}^{1/2} ( {\left\lvert \bm{F} \right\rvert}+{\left\lvert \bm{T} \right\rvert} ).$$ As noted after Definition \[rigid\_weak\], the existence of a unique corresponding weak pressure $p^{\rm p}\in L^2(\Omega_\epsilon)$ satisfying Definition \[rigid\_weak\_p\] as well as Lemma \[pressure\] follows by an essentially identical proof to that appearing in Section 2.2 of [@closed_loop].\ Combining with Lemmas \[korn\] and \[pressure\] then yields the bound . Classical versus slender body PDE description of rigid motion {#3rdLemma} ============================================================= Using the variational framework of Section \[variational0\] along with Lemma \[v\_omega\_bd\], we prove Lemma \[true\_vs\_SB\] comparing the classical PDE and slender body PDE descriptions of rigid slender body motion. The difference $\overline{\bm{u}}= {\bm{u}}^{\rm r}- {\bm{u}}^{\rm p}$, $\overline p= p^{\rm r}- p^{\rm p}$, $\overline{\bm{\sigma}}=\bm{\sigma}^{\rm r}- \bm{\sigma}^{\rm p}$, $\overline{\bm{\omega}}= \bm{\omega}^{\rm r}- \bm{\omega}^{\rm p}$, $\overline{\bm{v}}= {\bm{v}}^{\rm r}- {\bm{v}}^{\rm p}$ satisfies the PDE $$\label{diff_PDE} \begin{aligned} -\Delta \overline{\bm{u}}+\nabla \overline p &=0, \quad {{\rm{div}\,}}\overline{\bm{u}}=0 \hspace{2.3cm} \text{in } \Omega_\epsilon \\ \overline{\bm{u}}({\bm{x}}) &= \overline{\bm{v}}+ \overline{\bm{\omega}}\times {\bm{x}}+ \epsilon \bm{\omega}^{\rm p} \times {\bm{e}}_\rho, \qquad {\bm{x}}\in \Gamma_\epsilon \\ \overline{\bm{u}}({\bm{x}}) &\to 0 \hspace{4.3cm} \text{as }{\left\lvert {\bm{x}}\right\rvert}\to \infty \\ \int_{\Gamma_\epsilon} \overline{\bm{\sigma}}\bm{n} \, dS &=0, \quad \int_{\Gamma_\epsilon} {\bm{x}}\times (\overline{\bm{\sigma}}\bm{n}) \, dS = - \epsilon \int_{\Gamma_\epsilon} {\bm{e}}_\rho \times (\bm{\sigma}^{\rm p}\bm{n}) \; dS. \end{aligned}$$ Then, (formally) multiplying by $\overline{\bm{u}}$ and integrating by parts, we have that $\overline{\bm{u}}$ satisfies $$\label{diff_bound0} \begin{aligned} \int_{\Omega_\epsilon} 2|{\mathcal{E}}(\overline{\bm{u}})|^2 \, d{\bm{x}}&= \int_{\Gamma_\epsilon} \big( \overline{\bm{v}}+ \overline{\bm{\omega}}\times {\bm{x}}+ \epsilon\bm{\omega}^{\rm p}\times{\bm{e}}_\rho \big) \cdot(\overline{\bm{\sigma}}\bm{n}) \, dS \\ &= \overline{\bm{v}}\cdot\int_{\Gamma_\epsilon}\overline{\bm{\sigma}}\bm{n} \, dS + \overline{\bm{\omega}}\cdot\int_{\Gamma_\epsilon} {\bm{x}}\times(\overline{\bm{\sigma}}\bm{n}) \, dS + \epsilon\bm{\omega}^{\rm p}\cdot\int_{\Gamma_\epsilon} {\bm{e}}_\rho\times(\overline{\bm{\sigma}}\bm{n}) \, dS \\ &= -\epsilon \overline{\bm{\omega}}\cdot\int_{\Gamma_\epsilon} {\bm{e}}_\rho \times (\bm{\sigma}^{\rm p}\bm{n}) \, dS + \epsilon \bm{\omega}^{\rm p}\cdot\int_{\Gamma_\epsilon} {\bm{e}}_\rho \times(\overline{\bm{\sigma}}\bm{n}) \, dS. \end{aligned}$$ To estimate the right hand side of , we first need to define a smooth cutoff function $\phi(\rho)$ satisfying $$\label{cutoff_def} \phi(\rho) = \begin{cases} 1, & \rho < 2 \\ 0, & \rho >4 \end{cases}$$ with smooth decay satisfying $$\label{phi_decay} {\left\lvert \frac{d\phi}{d\rho} \right\rvert} \le c_\phi.$$ Then for ${\bm{x}}={\bm{X}}(s)+\rho{\bm{e}}_\rho(\theta,s)$ in a neighborhood of $\Gamma_\epsilon$, we define $\phi_\epsilon(\rho):= \phi(\rho/\epsilon)$.\ We estimate the second term on the right hand side first, noting that the estimation of the first term will be essentially identical. Using index notation (the subscript $\cdot_{,j}$ signifies $\frac{{\partial}\cdot}{{\partial}x_j}$; sum over repeated indices) along with the divergence theorem, we may write $$\label{cross_est0} \begin{aligned} \int_{\Gamma_\epsilon} \big({\bm{e}}_\rho \times(\overline{\bm{\sigma}}\bm{n}) \big)_i \, dS &= \int_{\Gamma_\epsilon} \varepsilon_{ijk} (e_\rho)_j \overline\sigma_{k\ell} n_\ell \, dS = \int_{\Omega_\epsilon} (\phi \, \varepsilon_{ijk}(e_\rho)_j \overline\sigma_{k\ell})_{,\ell} \, d{\bm{x}}\\ &= \int_{\Omega_\epsilon} \varepsilon_{ijk}\big( \phi_{,\ell}(e_\rho)_j \overline\sigma_{k\ell} + \phi \,(e_\rho)_{j,\ell} \overline\sigma_{k\ell} \big)\, d{\bm{x}}. \end{aligned}$$ Here $\varepsilon_{ijk}$ is the alternating symbol $$\varepsilon_{ijk} = \begin{cases} 1, & \text{ for even permutations of }i,j,k \\ -1, & \text{ for odd permutations of }i,j,k \\ 0, & \text{ if } i=j,j=k, \text{ or }k=i, \end{cases}$$ and we have used that $\overline{\bm{\sigma}}$ is divergence-free. Although $\overline{\bm{\sigma}}$ only belongs to $L^2(\Omega_\epsilon)$ [*a priori*]{}, we may justify equation because ${{\rm{div}\,}}\overline{\bm{\sigma}}=0\in L^2(\Omega_\epsilon)$, and therefore $\overline{\bm{\sigma}}$ belongs to the space $H_{{{\rm{div}\,}}}(\Omega_\epsilon):= \{\bm{\varphi}\in L^2(\Omega_\epsilon) \, : \, {{\rm{div}\,}}\bm{\varphi}\in L^2(\Omega_\epsilon) \}$. Then we may use the divergence theorem for $H_{{{\rm{div}\,}}}$ functions given in Lemma IV.3.3 of [@boyer2012mathematical] to obtain . Now, due to the cutoff $\phi_\epsilon$, the integrand on the right hand side of is supported only within the region $${\mathcal{O}}_\epsilon := \big\{{\bm{X}}(s)+\rho {\bm{e}}_\rho(s,\theta) \; : \; s\in{\mathbb{T}}, \; \epsilon \le \rho\le4\epsilon, \; 0\le \theta<2\pi \big\}$$ with ${\left\lvert {\mathcal{O}}_\epsilon \right\rvert}= C\epsilon^2$ for some $C$ depending only on $c_\Gamma$ and $\kappa_{\max}$.\ Within ${\mathcal{O}}_\epsilon$, defining ${\widehat{\kappa}}(s,\theta):= \kappa_1(s)\cos\theta+\kappa_2(s)\sin\theta$, we have $$\label{grad_rho} \begin{aligned} {\left\lvert \nabla {\bm{e}}_\rho(s,\theta) \right\rvert} &= {\left\lvert \frac{1}{\rho}\frac{{\partial}{\bm{e}}_\rho}{{\partial}\theta}{\bm{e}}_\theta^{\rm T}+ \frac{1}{1-\rho{\widehat{\kappa}}}\bigg(\frac{{\partial}{\bm{e}}_\rho}{{\partial}s} - \kappa_3\frac{{\partial}{\bm{e}}_\rho}{{\partial}\theta} \bigg){\bm{e}}_t^{\rm T} \right\rvert} \\ &= {\left\lvert \frac{1}{\rho}{\bm{e}}_\theta{\bm{e}}_\theta^{\rm T} - \frac{1}{1-\rho{\widehat{\kappa}}}(\kappa_1\cos\theta+\kappa_2\sin\theta) {\bm{e}}_t{\bm{e}}_t^{\rm T} \right\rvert} \le \frac{1}{\epsilon}+ 4\kappa_{\max}, \end{aligned}$$ where the final $\kappa_{\max}$ bound is shown in Appendix \[reg\_lem\].\ Using , , and Cauchy-Schwarz, we may estimate as $$\label{cross_bound} \begin{aligned} {\left\lvert \int_{\Gamma_\epsilon} {\bm{e}}_\rho \times(\overline{\bm{\sigma}}\bm{n}) \, dS \right\rvert} &\le \int_{{\mathcal{O}}_\epsilon} \big({\left\lvert \phi_\epsilon\nabla{\bm{e}}_\rho \right\rvert}+{\left\lvert \nabla\phi_\epsilon \right\rvert}){\left\lvert \overline{\bm{\sigma}} \right\rvert} \, d{\bm{x}}\\ &\le {\left\lvert {\mathcal{O}}_\epsilon \right\rvert}^{1/2} \bigg(\frac{1}{\epsilon} + 4\kappa_{\max} + \frac{c_\phi}{\epsilon} \bigg)\bigg(\int_{\Omega_\epsilon} {\left\lvert \overline{\bm{\sigma}} \right\rvert}^2 \, d{\bm{x}}\bigg)^{1/2} \\ & \le C \bigg(\int_{\Omega_\epsilon} {\left\lvert \overline{\bm{\sigma}} \right\rvert}^2 \, d{\bm{x}}\bigg)^{1/2} \end{aligned}$$ where $C$ depends only on the shape of ${\bm{X}}$ – in particular, $c_\Gamma$ and $\kappa_{\max}$. Finally, using Lemma \[pressure\], we obtain $$\label{cross_bound2} {\left\lvert \int_{\Gamma_\epsilon} {\bm{e}}_\rho \times(\overline{\bm{\sigma}}\bm{n}) \, dS \right\rvert} \le C \bigg(\int_{\Omega_\epsilon} \big({\left\lvert {\mathcal{E}}(\overline{\bm{u}}) \right\rvert}^2 + \overline p^2 \big) \, d{\bm{x}}\bigg)^{1/2} \le C \bigg(\int_{\Omega_\epsilon} {\left\lvert {\mathcal{E}}(\overline{\bm{u}}) \right\rvert}^2 \, d{\bm{x}}\bigg)^{1/2}.$$ Following exactly the same procedure, we can also show $$\label{cross_bound3} {\left\lvert \int_{\Gamma_\epsilon} {\bm{e}}_\rho \times(\bm{\sigma}^{\rm p}\bm{n}) \, dS \right\rvert} \le C \bigg(\int_{\Omega_\epsilon} {\left\lvert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rvert}^2 \, d{\bm{x}}\bigg)^{1/2}.$$ Furthermore, in the same way as in Lemma \[omegaP\], it can be shown that $$\label{omegav_bound} {\left\lvert \overline{\bm{\omega}} \right\rvert}+ {\left\lvert \overline{\bm{v}}\right\rvert} \le C{\left\lVert \overline{\bm{v}}+\overline{\bm{\omega}}\times{\bm{x}}+ \epsilon\bm{\omega}^{\rm p}\times {\bm{e}}_\rho \right\rVert}_{L^2(\Gamma_\epsilon)} \le C\sqrt{\epsilon}{\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert {\mathcal{E}}(\overline{\bm{u}}) \right\rVert}_{L^2(\Omega_\epsilon)},$$ where we have used the $L^2(\Gamma_\epsilon)$ trace estimate (Lemma \[trace2\]).\ Then, using and in along with Lemma \[omegaP\] and , we have $$\label{diff_bound1} \begin{aligned} \int_{\Omega_\epsilon} 2|{\mathcal{E}}(\overline{\bm{u}})|^2 \, d{\bm{x}}&\le \epsilon C{\left\lvert \overline{\bm{\omega}} \right\rvert}\bigg(\int_{\Omega_\epsilon}{\left\lvert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rvert}^2 \, d{\bm{x}}\bigg)^{1/2} + \epsilon C{\left\lvert \bm{\omega}^{\rm p} \right\rvert} \bigg(\int_{\Omega_\epsilon}{\left\lvert {\mathcal{E}}(\overline{\bm{u}}) \right\rvert}^2 \, d{\bm{x}}\bigg)^{1/2} \\ &\le \epsilon{\left\lvert \log\epsilon \right\rvert}^{1/2}C\bigg(\int_{\Omega_\epsilon}{\left\lvert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rvert}^2 \, d{\bm{x}}\bigg)^{1/2}\bigg(\int_{\Omega_\epsilon}{\left\lvert {\mathcal{E}}(\overline{\bm{u}}) \right\rvert}^2 \, d{\bm{x}}\bigg)^{1/2} \\ &\le \epsilon^2{\left\lvert \log\epsilon \right\rvert}C\int_{\Omega_\epsilon}{\left\lvert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rvert}^2 \, d{\bm{x}}+ \int_{\Omega_\epsilon}{\left\lvert {\mathcal{E}}(\overline{\bm{u}}) \right\rvert}^2 \, d{\bm{x}}, \end{aligned}$$ where we have used Young’s inequality in the last line. Then, using , we obtain $${\left\lVert {\mathcal{E}}(\overline{\bm{u}}) \right\rVert}_{L^2(\Omega_\epsilon)} \le \epsilon{\left\lvert \log\epsilon \right\rvert}C( {\left\lvert \bm{T} \right\rvert}+ {\left\lvert \bm{F} \right\rvert}).$$ Finally, using again, we obtain Lemma \[true\_vs\_SB\]. Proof of Lemma \[ups\_up\_err\] {#1stLemma} ================================ Finally, we prove Lemma \[ups\_up\_err\] comparing the rigid slender body PDE to the intermediary slender body PDE .\ We begin by defining $$\label{fp_def} \bm{f}^{\rm p}(s):=\int_0^{2\pi} (\bm{\sigma}^{\rm p}\bm{n}) \, {\mathcal{J}}_\epsilon(s,\theta) d\theta$$ for $\bm{\sigma}^{\rm p}$ as in , and establish the following: \[fp\_FT\] Suppose the slender body $\Sigma_\epsilon$ is as in Section \[geometry\] – in particular, ${\bm{X}}\in C^3({\mathbb{T}})$. Let the total force $\bm{F}$ and torque $\bm{T}$ be given, and let $\bm{f}^{\rm p}$ be as defined in . Then $${\left\lVert \bm{f}^{\rm p} \right\rVert}_{L^2({\mathbb{T}})} \le C{\left\lvert \log\epsilon \right\rvert}^{3/2}({\left\lvert \bm{F} \right\rvert}+{\left\lvert \bm{T} \right\rvert})$$ for $C$ depending only on $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$. The proof of this lemma relies on a higher regularity estimate for $\bm{\sigma}^{\rm p}$. Note that once Theorem \[SB\_PDE\_well\] has been established, we immediately obtain that ${\bm{u}}^{\rm p}\big|_{\Gamma_\epsilon} = {\bm{v}}^{\rm p}+\bm{\omega}^{\rm p}\times {\bm{X}}(s)$ is in $C^3(\Omega_\epsilon)$, since ${\bm{v}}^{\rm p}$ and $\bm{\omega}^{\rm p}$ are just constants in ${\mathbb{R}}^3$ and the fiber centerline ${\bm{X}}$ is in $C^3({\mathbb{T}})$. Given this $C^3$ Dirichlet data, $\bm{\sigma}^{\rm p}\in H^1(\Omega_\epsilon)$ follows by standard higher regularity arguments for the exterior Stokes Dirichlet boundary value problem (see the proof of Lemma V.4.3 in [@galdi2011introduction] or Theorem IV.5.8 in [@boyer2012mathematical]). Note that since ${\bm{X}}\in C^3({\mathbb{T}})$, $\bm{\sigma}^{\rm p}$ should in fact be even more regular, but the method we use to show Lemma \[high\_reg\] only allows us to quantify the $\epsilon$-dependence in the estimate for ${\left\lVert \nabla\bm{\sigma}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)}$. In particular, we can show the following bound on $\nabla \bm{\sigma}^{\rm p}$. \[high\_reg\] Given $\Omega_\epsilon$ as in Section \[geometry\], the solution $\bm{\sigma}^{\rm p}$ to belongs to $H^1(\Omega_\epsilon)$ and satisfies $$\label{high_reg_eqn} {\left\lVert \nabla\bm{\sigma}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \le {\left\lVert \nabla^2{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} + {\left\lVert \nabla p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \le \frac{C}{\epsilon}{\left\lvert \log\epsilon \right\rvert}^{1/2}\big({\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} + {\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \big),$$ where $C$ depends on $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$. The proof of the $\epsilon$-dependence in Lemma \[high\_reg\] is given in Appendix \[reg\_lem\].\ Using Lemma \[high\_reg\] and Corollary \[omegaP\], we have the higher regularity estimate $$\label{high_reg2} \begin{aligned} {\left\lVert \nabla\bm{\sigma}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} &\le \frac{C}{\epsilon}{\left\lvert \log\epsilon \right\rvert}^{1/2}\bigg({\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} + {\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \bigg) \le \frac{C}{\epsilon}{\left\lvert \log\epsilon \right\rvert}({\left\lvert \bm{F} \right\rvert}+{\left\lvert \bm{T} \right\rvert}). \end{aligned}$$ Now, using that ${\mathcal{J}}_\epsilon(s,\theta)>0$ for each $(s,\theta)\in \Gamma_\epsilon$ and the surface measure ${\left\lvert \Gamma_\epsilon \right\rvert} = \int_{\mathbb{T}}\int_0^{2\pi} {\mathcal{J}}_\epsilon(s,\theta) d\theta ds = \epsilon$, we have $$\begin{aligned} {\left\lVert \bm{f}^{\rm p} \right\rVert}_{L^2({\mathbb{T}})}^2 &= \int_{\mathbb{T}}{\left\lvert \int_0^{2\pi} \bm{\sigma}^{\rm p}\bm{n} \, {\mathcal{J}}_\epsilon(s,\theta) d\theta \right\rvert}^2 ds \le {\left\lvert \Gamma_\epsilon \right\rvert} \int_{\mathbb{T}}\int_0^{2\pi} {\left\lvert {\rm Tr}(\bm{\sigma}^{\rm p}) \right\rvert}^2 \, {\mathcal{J}}_\epsilon(s,\theta) d\theta \, ds \\ &\le C\epsilon^2{\left\lvert \log\epsilon \right\rvert}{\left\lVert \nabla\bm{\sigma}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)}^2 \le C{\left\lvert \log\epsilon \right\rvert}^3({\left\lvert \bm{F} \right\rvert}+{\left\lvert \bm{T} \right\rvert})^2.\end{aligned}$$ Here we have applied both the $L^2(\Gamma_\epsilon)$ trace inequality (Lemma \[trace2\]) and the higher regularity estimate in the last line. With Lemma \[fp\_FT\], we are now equipped to show Lemma \[ups\_up\_err\]. The proof relies on estimates for the PDE satisfied by the difference between solutions to and . Letting ${\widetilde{{\bm{u}}}} = {\bm{u}}^{\rm p,s}-{\bm{u}}^{\rm p}$, ${\widetilde{p}} = p^{\rm p,s}- p^{\rm p} $, ${\widetilde{{\bm{v}}}} = {\bm{v}}^{\rm s}-{\bm{v}}^{\rm p}$, ${\widetilde{\bm{\omega}}} = \bm{\omega}^{\rm s}-\bm{\omega}^{\rm p}$, ${\widetilde{\bm{\sigma}}}=\bm{\sigma}^{\rm p,s} - \bm{\sigma}^{\rm p}$, we consider the following boundary value problem: $$\label{wtbu_eqn} \begin{aligned} -\Delta {\widetilde{{\bm{u}}}} +\nabla {\widetilde{p}} &=0, \quad {{\rm{div}\,}}{\widetilde{{\bm{u}}}} =0 \hspace{2.25cm} \text{in } \Omega_\epsilon \\ {\widetilde{{\bm{u}}}}({\bm{x}}) &= {\widetilde{{\bm{v}}}} + {\widetilde{\bm{\omega}}}\times {\bm{X}}(s) + \bm{R}(s), \qquad {\bm{x}}\in \Gamma_\epsilon \\ {\widetilde{{\bm{u}}}}({\bm{x}}) &\to 0 \hspace{4.3cm} \text{as }{\left\lvert {\bm{x}}\right\rvert}\to \infty \\ \int_{\Gamma_\epsilon} {\widetilde{\bm{\sigma}}}\bm{n} \; dS &= 0, \quad \int_{{\mathbb{T}}} {\bm{X}}(s)\times \bigg(\int_0^{2\pi} {\widetilde{\bm{\sigma}}}\bm{n} \, {\mathcal{J}}_\epsilon(s,\theta) d\theta \bigg) ds = 0, \end{aligned}$$ where $\bm{R}(s):={\rm Tr}({\bm{u}}^{\rm p,s})(s) - \big( {\bm{v}}^{\rm s} + \bm{\omega}^{\rm s}\times {\bm{X}}(s)\big)$ satisfies $$\label{Rest} {\left\lVert \bm{R} \right\rVert}_{L^2({\mathbb{T}})} \le C\epsilon{\left\lvert \log\epsilon \right\rvert}^{3/2}{\left\lVert \bm{f}^{\rm s} \right\rVert}_{C^1({\mathbb{T}})},$$ by . We consider the variational form of : formally, multiplying by by ${\widetilde{{\bm{u}}}}$ and integrating by parts, we have $$\begin{aligned} \int_{\Omega_\epsilon} 2|{\mathcal{E}}({\widetilde{{\bm{u}}}})|^2 \, d{\bm{x}}&= \int_{\Gamma_\epsilon} \big({\widetilde{{\bm{v}}}} + {\widetilde{\bm{\omega}}}\times {\bm{X}}(s) +\bm{R}(s) \big) \cdot({\widetilde{\bm{\sigma}}}\bm{n}) \; dS \\ &= {\widetilde{{\bm{v}}}} \cdot \int_{\Gamma_\epsilon}({\widetilde{\bm{\sigma}}}\bm{n}) \; dS + {\widetilde{\bm{\omega}}}\cdot\int_{{\mathbb{T}}} {\bm{X}}(s)\times \bigg(\int_0^{2\pi} {\widetilde{\bm{\sigma}}}\bm{n}\, {\mathcal{J}}_\epsilon(s,\theta) d\theta \bigg) ds \\ &\qquad + \int_{\Gamma_\epsilon} \big({\widetilde{{\bm{v}}}} + {\widetilde{\bm{\omega}}}\times {\bm{X}}(s) +\bm{R}(s) \big) \cdot({\widetilde{\bm{\sigma}}}\bm{n}) \; dS \\ &=\int_{\mathbb{T}}\bm{R}(s) \cdot \big( \bm{f}^{\rm s} -\bm{f}^{\rm p} \big) \; ds \le {\left\lVert \bm{R} \right\rVert}_{L^2({\mathbb{T}})} \big( {\left\lVert \bm{f}^{\rm s} \right\rVert}_{L^2({\mathbb{T}})} + {\left\lVert \bm{f}^{\rm p} \right\rVert}_{L^2({\mathbb{T}})} \big) \\ &\le C\epsilon{\left\lvert \log\epsilon \right\rvert}^{3/2}{\left\lVert \bm{f}^{\rm s} \right\rVert}_{C^1({\mathbb{T}})}\big({\left\lVert \bm{f}^{\rm s} \right\rVert}_{L^2({\mathbb{T}})} + {\left\lvert \log\epsilon \right\rvert}^{3/2}({\left\lvert \bm{F} \right\rvert}+{\left\lvert \bm{T} \right\rvert}) \big). \end{aligned}$$ Here we have used and Lemma \[fp\_FT\] in the final line. It would seem to make sense to try to bound the difference $\bm{f}^{\rm s} -\bm{f}^{\rm p}$ appearing in the second-to-last line by ${\left\lVert {\mathcal{E}}({\widetilde{{\bm{u}}}}) \right\rVert}_{L^2(\Omega_\epsilon)}$, or try to use an extension $\overline{\bm{R}}({\bm{x}})\in D^{1,2}(\Omega_\epsilon)$ with $\overline{\bm{R}}\big|_{\Gamma_\epsilon} = \bm{R}(s)$ and instead take ${\widetilde{{\bm{u}}}} - \overline{\bm{R}}$ as a test function in the above variational estimate to get rid of the boundary term. In either case, we run into difficulties in that we only have an $L^2({\mathbb{T}})$ estimate for $\bm{R}(s)$, when at least an $H^{1/2}({\mathbb{T}})$ estimate would be needed. However, as noted in Lemma \[high\_reg\], bounding the gradient of a function on $\Omega_\epsilon$ incurs an additional factor of $1/\epsilon$. By scaling, an $H^{1/2}({\mathbb{T}})$ estimate for $\bm{R}(s)$ would likely yield the same $\sqrt{\epsilon}$ factor appearing in Lemma \[ups\_up\_err\]. Now, using the $L^2({\mathbb{T}})$ trace inequality (Lemma \[trace1\]), the Korn inequality (Lemma \[korn\]), and Young’s inequality, along the with above ${\left\lVert {\mathcal{E}}({\widetilde{{\bm{u}}}}) \right\rVert}_{L^2(\Omega_\epsilon)}$ estimate, we have $$\begin{aligned} {\left\lVert {\rm Tr}({\bm{u}}^{\rm p,s}) - ({\bm{v}}^{\rm p}+\bm{\omega}^{\rm p}\times{\bm{X}}) \right\rVert}_{L^2({\mathbb{T}})} &\le C{\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert \nabla{\widetilde{{\bm{u}}}} \right\rVert}_{L^2(\Omega_\epsilon)} \le C{\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert {\mathcal{E}}({\widetilde{{\bm{u}}}}) \right\rVert}_{L^2(\Omega_\epsilon)} \\ &\le C\sqrt{\epsilon}{\left\lvert \log\epsilon \right\rvert}^{3/2} \big({\left\lvert \log\epsilon \right\rvert}^{1/2}{\left\lVert \bm{f}^{\rm s} \right\rVert}_{C^1({\mathbb{T}})} + {\left\lvert \bm{F} \right\rvert}+{\left\lvert \bm{T} \right\rvert} \big),\end{aligned}$$ yielding Lemma \[ups\_up\_err\]. Appendix ======== Here we provide proofs for the $L^2(\Gamma_\epsilon)$ trace inequality (Lemma \[trace2\]) and the higher regularity estimate (Lemma \[high\_reg\]).\ We first recall the following lemma, which will be used throughout the appendix. *(Sobolev inequality)*\[sobolev\] Let $\Omega_{\epsilon}={\mathbb{R}}^3\backslash\overline{\Sigma_{\epsilon}}$ be as in Section \[geometry\]. For any ${\bm{u}}\in D^{1,2}(\Omega_{\epsilon})$, we have $$\label{sobolev_const} \| {\bm{u}}\|_{L^6(\Omega_{\epsilon})} \le C\|\nabla{\bm{u}}\|_{L^2(\Omega_{\epsilon})}$$ where $C$ depends only on $c_\Gamma$ and $\kappa_{\max}$. The proof of $\epsilon$-independence of $C$ appears in Appendix A.2.4 of [@closed_loop]. Proof of Lemma \[trace2\] ------------------------- The proof of the $L^2(\Gamma_\epsilon)$ trace inequality follows the same outline as the proof of Lemma \[trace1\], contained in Appendix A.2.1 of [@closed_loop]. In particular, using the $\epsilon$-independent $C^2$-diffeomorphisms $\psi_j$ (defined in Appendix A.2.1, [@closed_loop]) which map segments of the curved slender body $\Sigma_\epsilon$ to a straight cylinder, it suffices to show the $\sqrt{\epsilon}{\left\lvert \log\epsilon \right\rvert}$ dependence of the trace constant for a straight cylinder.\ Accordingly, let ${\mathcal{D}}_\rho\subset{\mathbb{R}}^2$ denote the open disk of radius $\rho$ in ${\mathbb{R}}^2$, centered at the origin, and, for some $a<\infty$, define the cylindrical surface $\Gamma_{\epsilon,a}={\partial}{\mathcal{D}}_\epsilon\times [-a,a]$ and the cylindrical shell ${\mathcal{C}}_{\epsilon,a}= ({\mathcal{D}}_1\backslash\overline{{\mathcal{D}}_\epsilon}) \times [-a,a]$. Consider the function space $$D^{1,2}_\Gamma({\mathcal{C}}_{\epsilon,a})= \big\{ {\bm{u}}\in D^{1,2}({\mathcal{C}}_{\epsilon,a}) \; : \; {\bm{u}}\big|_{{\partial}{\mathcal{C}}_{\epsilon,a}\backslash\Gamma_{\epsilon,a}} = 0 \big\}.$$ As in the proof of Lemma \[trace1\], it suffices to show the $\sqrt{\epsilon}{\left\lvert \log\epsilon \right\rvert}$ dependence of the $L^2(\Gamma_{\epsilon,a})$ trace constant for functions belonging to $D^{1,2}_\Gamma({\mathcal{C}}_{\epsilon,a})$.\ By estimate (A.4) in [@closed_loop], any ${\bm{u}}\in C^1({\mathcal{C}}_{\epsilon,a}) \cap C^0(\overline{{\mathcal{C}}_{\epsilon,a}})\cap D^{1,2}_\Gamma({\mathcal{C}}_{\epsilon,a})$ satisfies $$\begin{aligned} {\left\lvert {\rm Tr}({\bm{u}}) \right\rvert}^2 \le {\left\lvert \log\epsilon \right\rvert} \int_\epsilon^1 {\left\lvert \frac{{\partial}{\bm{u}}}{{\partial}\rho} \right\rvert}^2 \rho \, d\rho.\end{aligned}$$ Then, noting that the surface element on $\Gamma_{\epsilon,a}$ is simply $\epsilon$, we have $$\begin{aligned} {\left\lVert {\rm Tr}({\bm{u}}) \right\rVert}_{L^2(\Gamma_{\epsilon,a})}^2 &= \int_{-a}^a \int_0^{2\pi} {\left\lvert {\rm Tr}({\bm{u}}) \right\rvert}^2 \epsilon \, d\theta\, ds \\ &\le \epsilon{\left\lvert \log\epsilon \right\rvert} \int_{-a}^a \int_0^{2\pi}\int_\epsilon^1 {\left\lvert \frac{{\partial}{\bm{u}}}{{\partial}\rho} \right\rvert}^2 \rho \, d\rho\, d\theta\, ds \le \epsilon{\left\lvert \log\epsilon \right\rvert}{\left\lVert \nabla{\bm{u}}\right\rVert}_{L^2({\mathcal{C}}_{\epsilon,a})}^2.\end{aligned}$$ The same result for ${\bm{u}}\in D^{1,2}_\Gamma({\mathcal{C}}_{\epsilon,a})$ follows by density. Proof of Lemma \[high\_reg\] {#reg_lem} ---------------------------- To determine the $\epsilon$-dependence of the constant in , it suffices to work locally near the slender body surface and show that Lemma \[high\_reg\] holds within an $\epsilon$-independent region about the slender body centerline. We define the region $$\label{mcO} {\mathcal{O}} = \big\{{\bm{x}}\in \Omega_{\epsilon} \; : \; {\bm{x}}= {\bm{X}}(s) + \rho {\bm{e}}_{\rho}(s,\theta), \quad \epsilon < \rho<r_{\max} \big\},$$ where $r_{\max}$ is as in Section \[geometry\]. Within ${\mathcal{O}}$, we can use the orthonormal frame . We will use the notation ${\partial}_s,{\partial}_\theta,{\partial}_\rho$ to denote derivatives ${\partial}/{\partial}s$, ${\partial}/{\partial}\theta$, ${\partial}/{\partial}\rho$ with respect to the variables $s,\theta,\rho$, defined with respect to the orthonormal frame. We verify the $\epsilon$-dependence in the bound for $\nabla^2{\bm{u}}^{\rm p}$ and $\nabla p^{\rm p}$ in two parts: we first show an $L^2$ bound for derivatives $\nabla({\partial}_s{\bm{u}}^{\rm p})$, $\nabla({\partial}_\theta{\bm{u}}^{\rm p})$, ${\partial}_s p^{\rm p}$, and ${\partial}_\theta p^{\rm p}$ in directions tangent to the slender body surface $\Gamma_\epsilon$, and then use these bounds to estimate the derivatives $\nabla({\partial}_\rho{\bm{u}}^{\rm p})$, ${\partial}_\rho p^{\rm p}$ normal to $\Gamma_\epsilon$.\ We begin by estimating the tangential derivatives $\nabla({\partial}_s{\bm{u}}^{\rm p})$ and $\nabla({\partial}_\theta{\bm{u}}^{\rm p})$. Since the derivatives ${\partial}_s$ and ${\partial}_\theta$ with respect to the orthonormal frame do not commute with the “straight” differential operators $\nabla$ and ${{\rm{div}\,}}$, we will need to make use of the following commutator bounds. \[comm\_ests\] For any function ${\bm{u}}\in D^{1,2}_0({\mathcal{O}})$ and for each of the differential operators $D={{\rm{div}\,}},\, \nabla,\, {\mathcal{E}}(\cdot)$, the following commutator estimates hold: $$\begin{aligned} {\left\lVert [D,{\partial}_\theta]{\bm{u}}\right\rVert}_{L^2({\mathcal{O}})} &\le C{\left\lVert \nabla{\bm{u}}\right\rVert}_{L^2({\mathcal{O}})}, \quad {\left\lVert [D,{\partial}_s]{\bm{u}}\right\rVert}_{L^2({\mathcal{O}})} \le C{\left\lVert \nabla{\bm{u}}\right\rVert}_{L^2({\mathcal{O}})}, \end{aligned}$$ where the constant $C$ depends only on $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$. We begin by denoting $$\begin{aligned} {\bm{e}}_\theta(s,\theta) &= -\sin\theta{\bm{e}}_{n_1}(s) + \cos\theta {\bm{e}}_{n_2}(s),\\ u_\rho &= {\bm{u}}\cdot{\bm{e}}_\rho, \; u_\theta={\bm{u}}\cdot{\bm{e}}_\theta, \; u_s = {\bm{u}}\cdot{\bm{e}}_t. \end{aligned}$$ Then, with respect to the orthonormal frame , the divergence and gradient are given by $$\begin{aligned} {{\rm{div}\,}}{\bm{u}}&= \frac{1}{1-\rho{\widehat{\kappa}}} \bigg( \frac{1}{\rho} \frac{{\partial}(\rho(1-\rho{\widehat{\kappa}})u_\rho)}{{\partial}\rho} + \frac{1}{\rho}\frac{{\partial}((1-\rho{\widehat{\kappa}})u_\theta)}{{\partial}\theta} + \frac{{\partial}u_s}{{\partial}s} \bigg) \\ \nabla {\bm{u}}&= {\bm{e}}_\rho(s,\theta)\frac{{\partial}{\bm{u}}}{{\partial}\rho}^{\rm T} + {\bm{e}}_\theta(s,\theta)\frac{1}{\rho}\frac{{\partial}{\bm{u}}}{{\partial}\theta}^{\rm T} + {\bm{e}}_t(s)\frac{1}{1-\rho{\widehat{\kappa}}} \bigg(\frac{{\partial}{\bm{u}}}{{\partial}s} -\kappa_3 \frac{{\partial}{\bm{u}}}{{\partial}\theta} \bigg)^{\rm T},\end{aligned}$$ where $$\label{kappahat} {\widehat{\kappa}}(s,\theta) = \kappa_1(s)\cos\theta + \kappa_2(s)\sin\theta.$$ Direct computation of the commutators yields $$\begin{aligned} [{{\rm{div}\,}},{\partial}_\theta]{\bm{u}}&= \frac{({\partial}_\theta{\widehat{\kappa}})}{1-\rho{\widehat{\kappa}}}\bigg( \rho\,{{\rm{div}\,}}{\bm{u}}- \frac{1}{\rho}\frac{{\partial}}{{\partial}\rho} \big(\rho^2 u_\rho \big) - \frac{{\partial}u_\theta}{{\partial}\theta} \bigg) - \frac{({\partial}_\theta^2{\widehat{\kappa}})}{1-\rho{\widehat{\kappa}}}u_\theta \\ [{{\rm{div}\,}},{\partial}_s]{\bm{u}}&= \frac{({\partial}_s{\widehat{\kappa}})}{1-\rho{\widehat{\kappa}}}\bigg( \rho\,{{\rm{div}\,}}{\bm{u}}- \frac{1}{\rho}\frac{{\partial}}{{\partial}\rho} \big(\rho^2 u_\rho \big) - \frac{{\partial}u_\theta}{{\partial}\theta} \bigg) - \frac{({\partial}_\theta{\partial}_s{\widehat{\kappa}})}{1-\rho{\widehat{\kappa}}}u_\theta \\ [\nabla,{\partial}_\theta]{\bm{u}}&= {\bm{e}}_\theta\frac{{\partial}{\bm{u}}}{{\partial}\rho}^{\rm T} - {\bm{e}}_\rho\frac{1}{\rho}\frac{{\partial}{\bm{u}}}{{\partial}\theta}^{\rm T} + {\bm{e}}_t\frac{\rho({\partial}_\theta{\widehat{\kappa}})}{(1-\rho{\widehat{\kappa}})^2} \bigg(\frac{{\partial}{\bm{u}}}{{\partial}s} -\kappa_3 \frac{{\partial}{\bm{u}}}{{\partial}\theta} \bigg)^{\rm T} \\ [\nabla,{\partial}_s]{\bm{u}}&= ({\partial}_s{\bm{e}}_\rho)\frac{{\partial}{\bm{u}}}{{\partial}\rho}^{\rm T} + ({\partial}_s{\bm{e}}_\theta)\frac{1}{\rho}\frac{{\partial}{\bm{u}}}{{\partial}\theta}^{\rm T} + \bigg({\bm{e}}_t \frac{\rho({\partial}_s{\widehat{\kappa}})}{1-\rho{\widehat{\kappa}}}+({\partial}_s{\bm{e}}_t) \bigg)\frac{1}{1-\rho{\widehat{\kappa}}} \bigg(\frac{{\partial}{\bm{u}}}{{\partial}s} -\kappa_3 \frac{{\partial}{\bm{u}}}{{\partial}\theta} \bigg)^{\rm T} \end{aligned}$$ Using and the orthonormal frame ODEs , we have $$\begin{aligned} {\left\lvert {\partial}_\theta {\widehat{\kappa}} \right\rvert} &= {\left\lvert -\kappa_1\sin\theta+ \kappa_2\cos\theta \right\rvert}\le \kappa_{\max}, \quad {\left\lvert {\partial}_s {\widehat{\kappa}} \right\rvert} = {\left\lvert \kappa_1'\cos\theta+\kappa_2'\sin\theta \right\rvert}\le \xi_{\max} + 2(\kappa_{\max}+\pi), \\ {\left\lvert {\partial}_\theta^2{\widehat{\kappa}} \right\rvert} &= {\left\lvert -{\widehat{\kappa}} \right\rvert}\le \kappa_{\max}, \quad {\left\lvert {\partial}_\theta{\partial}_s {\widehat{\kappa}} \right\rvert} = {\left\lvert -\kappa_1'\sin\theta+ \kappa_2'\cos\theta \right\rvert}\le \xi_{\max}+ 2(\kappa_{\max}+\pi), \\ {\left\lvert {\partial}_s {\bm{e}}_\rho \right\rvert} &= {\left\lvert -{\widehat{\kappa}}{\bm{e}}_t + \kappa_3{\bm{e}}_\theta \right\rvert}\le \kappa_{\max} +\pi, \quad {\left\lvert {\partial}_s{\bm{e}}_\theta \right\rvert} = {\left\lvert -({\partial}_\theta {\widehat{\kappa}}){\bm{e}}_t -\kappa_3{\bm{e}}_t \right\rvert}\le \kappa_{\max} +\pi, \\ {\left\lvert \frac{1}{1-\rho{\widehat{\kappa}}} \right\rvert} &\le \frac{1}{1- r_{\max}\kappa_{\max}(\cos\theta+\sin\theta) } \le \frac{1}{1- \frac{1}{2\kappa_{\max}}\kappa_{\max}\sqrt{2} } \le 4 .\end{aligned}$$ Finally, noting that, by Lemma \[sobolev\], $${\left\lVert u_\theta \right\rVert}_{L^2({\mathcal{O}})} \le {\left\lvert {\mathcal{O}} \right\rvert}^{1/3} {\left\lVert {\bm{u}}\right\rVert}_{L^6({\mathcal{O}})} \le C{\left\lVert \nabla{\bm{u}}\right\rVert}_{L^2({\mathcal{O}})},$$ the desired $L^2(\Omega)$ bounds follow for each of $D={{\rm{div}\,}},\nabla$. The estimate for the symmetric gradient ${\mathcal{E}}({\bm{u}})$ then follows from the gradient commutator bound. Now, to derive an estimate for $\nabla({\partial}_s{\bm{u}}^{\rm p})$, we will make use of Definition \[rigid\_weak\_p\] with a particular test function $\bm{\varphi}$, which we will construct here. First, we want our test function to be supported only within ${\mathcal{O}}$. We define a smooth cutoff function $$\label{Ocutoff} \psi(\rho) = \begin{cases} 1, & \rho < r_{\max}/4 \\ 0, & \rho > r_{\max}/2, \end{cases} \quad {\left\lvert \frac{{\partial}\psi}{{\partial}\rho} \right\rvert} \le C,$$ where $C$ depends only on $r_{\max}$. Note that $\psi(\rho)$ commutes with both ${\partial}_\theta$ and ${\partial}_s$.\ We would like to use ${\partial}_s^2(\psi{\bm{u}}^{\rm p})$ as a test function in Definition \[rigid\_weak\_p\], but it will be more convenient to work with a function which vanishes on $\Gamma_\epsilon$. We therefore construct a correction ${\bm{g}}\in C^2(\Omega_\epsilon)$ supported only in ${\mathcal{O}}$ and satisfying $$\label{g_correction} {\bm{g}}\big|_{\Gamma_\epsilon} = ({\partial}_s {\bm{u}}^{\rm p})\big|_{\Gamma_\epsilon} = \bm{\omega}^{\rm p}\times{\bm{e}}_t(s), \quad {\left\lVert \nabla {\bm{g}}\right\rVert}_{L^2({\mathcal{O}})} \le C{\left\lvert \bm{\omega} \right\rvert},$$ where $C$ depends on $c_\Gamma$ and $\kappa_{\max}$. To build ${\bm{g}}$, we follow a similar construction used in Section 4.1 of [@closed_loop]. We define $${\bm{g}}_0(\rho,\theta,s) = \begin{cases} \bm{\omega}^{\rm p}\times {\bm{e}}_t(s) & \text{if } \rho<4\epsilon \\ 0 & \text{otherwise} \end{cases}$$ and take $${\bm{g}}(\rho,\theta,s):= \phi_\epsilon(\rho){\bm{g}}_0(\rho,\theta,s),$$ where $\phi_\epsilon(\rho)$ is the smooth cutoff defined in -. Note that ${\bm{g}}\in C^2$ and is supported within the region $${\mathcal{O}}_\epsilon := \big\{ {\bm{X}}(s) + \rho {\bm{e}}_\rho(s,\theta) \; : \; s\in {\mathbb{T}}, \, \epsilon \le \rho \le 4\epsilon, \, 0\le \theta <2\pi \big\},$$ where ${\left\lvert {\mathcal{O}}_\epsilon \right\rvert} \le C\epsilon^2$. Then, using and , we have $$\begin{aligned} {\left\lVert \nabla {\bm{g}}\right\rVert}_{L^2({\mathcal{O}})} &\le \sqrt{{\left\lvert {\mathcal{O}}_\epsilon \right\rvert}}{\left\lVert \nabla {\bm{g}}\right\rVert}_{C({\mathcal{O}}_\epsilon)} \\ & \le \sqrt{{\left\lvert {\mathcal{O}}_\epsilon \right\rvert}}\bigg( {\left\lVert \frac{{\partial}\phi_\epsilon}{{\partial}\rho} \right\rVert}_{C({\mathcal{O}}_\epsilon)}{\left\lVert {\bm{g}}_0 \right\rVert}_{C({\mathcal{O}}_\epsilon)}+ {\left\lVert \frac{1}{1-\rho{\widehat{\kappa}}}\frac{{\partial}{\bm{g}}_0}{{\partial}s} \right\rVert}_{C({\mathcal{O}}_\epsilon)} \bigg) \le C{\left\lvert \bm{\omega}^{\rm p} \right\rvert}.\end{aligned}$$ Now, we could just use ${\partial}_s({\partial}_s(\psi{\bm{u}}^{\rm p}) -{\bm{g}})$ as a test function in Definition \[rigid\_weak\_p\], but it will actually be useful to include a second correction term in the following way. We consider ${\bm{z}}\in D^{1,2}_0({\mathcal{O}})$ satisfying $$\label{zee_def} \begin{aligned} {{\rm{div}\,}}{\bm{z}}&= {{\rm{div}\,}}(\psi{\partial}_s{\bm{u}}^{\rm p} -{\bm{g}}) \quad \text{in } {\mathcal{O}} \\ {\left\lVert \nabla {\bm{z}}\right\rVert}_{L^2({\mathcal{O}})} &\le C{\left\lVert {{\rm{div}\,}}(\psi{\partial}_s{\bm{u}}^{\rm p} -{\bm{g}}) \right\rVert}_{L^2({\mathcal{O}})} \end{aligned}$$ for $C$ depending only on $c_\Gamma$ and $\kappa_{\max}$. We know that such a ${\bm{z}}$ exists due to [@galdi2011introduction], Section III.3, and the constant $C$ is independence of $\epsilon$ due to Appendix A.2.5 of [@closed_loop]. Furthermore, since ${{\rm{div}\,}}{\bm{u}}^{\rm p}=0$, by Proposition \[comm\_ests\] we have $$\begin{aligned} {\left\lVert {{\rm{div}\,}}(\psi{\partial}_s{\bm{u}}^{\rm p}-{\bm{g}}) \right\rVert}_{L^2({\mathcal{O}})} &\le {\left\lVert {{\rm{div}\,}}({\partial}_\theta{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})}+ C{\left\lVert {\partial}_\theta{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}+{\left\lVert \nabla {\bm{g}}\right\rVert}_{L^2({\mathcal{O}})} \\ &\le {\left\lVert [{{\rm{div}\,}},{\partial}_\theta]{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} + C{\left\lVert {\partial}_\theta{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} + C{\left\lvert \bm{\omega}^{\rm p} \right\rvert} \\ &\le C{\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} + C{\left\lvert \bm{\omega}^{\rm p} \right\rvert}.\end{aligned}$$ Here we have also used that ${\left\lVert {\partial}_\theta{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} \le {\left\lVert \rho \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} \le r_{\max}{\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}$. In particular, ${\bm{z}}$ satisfying also satisfies $$\label{zestimate} {\left\lVert \nabla{\bm{z}}\right\rVert}_{L^2({\mathcal{O}})} \le C{\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} + C{\left\lvert \bm{\omega}^{\rm p} \right\rvert}.$$ Using extension by zero to consider ${\bm{z}}$ as a function over all $\Omega_\epsilon$, we can now construct our desired test function for use in Definition \[rigid\_weak\_p\]. In particular, we will use the function ${\partial}_s({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}})$ in place of $\bm{\varphi}$ in Definition \[rigid\_weak\_p\]. Note that by definition of ${\bm{z}}$, this function may only belong to $L^2(\Omega_\epsilon)$. In this case, we can make sense of the following integration-by-parts argument using finite differences rather than full derivatives (see [@boyer2012mathematical], Section III.2.7 for construction of finite difference operators along a curved boundary). Thus we really only need ${\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}}\in D^{1,2}(\Omega_\epsilon)$ to make sense of the following result. Note that in integrating by parts, we will also need to make use of the fact that, for $i=s,\theta$, $$\label{jacfac_i} {\partial}_i (d{\bm{x}}) = -\frac{\rho{\partial}_i{\widehat{\kappa}}}{1-\rho{\widehat{\kappa}}} d{\bm{x}}:= {\mathcal{J}}_i \, d{\bm{x}}, \quad {\left\lvert {\mathcal{J}}_i \right\rvert} \le C; \quad i=s,\theta ,$$ where $C$ depends on $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$.\ Then, using ${\partial}_s({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}})$ in Definition \[rigid\_weak\_p\], we have $$\begin{aligned} 0 &= \int_{{\mathcal{O}}} \bigg(2{\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}\big({\partial}_s({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}}) \big) - p^{\rm p} \,{{\rm{div}\,}}({\partial}_s({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}})) \bigg) \, d{\bm{x}}\\ &= \int_{{\mathcal{O}}} 2{\mathcal{E}}({\bm{u}}^{\rm p}): {\partial}_s{\mathcal{E}}({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}}) \, d{\bm{x}}+ \int_{{\mathcal{O}}} 2{\mathcal{E}}({\bm{u}}^{\rm p}): [{\mathcal{E}}(\cdot),{\partial}_s]({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}}) \, d{\bm{x}}\\ &\qquad - \int_{{\mathcal{O}}} p^{\rm p} \,{\partial}_s({{\rm{div}\,}}({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}})) \, d{\bm{x}}- \int_{{\mathcal{O}}} p^{\rm p} \,[{{\rm{div}\,}},{\partial}_s]({\partial}_s(\psi{\bm{u}}^{\rm p})- {\bm{g}}-{\bm{z}}) \, d{\bm{x}}\\ &= -\int_{{\mathcal{O}}} 2{\partial}_s{\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}}) \, d{\bm{x}}-\int_{{\mathcal{O}}} 2{\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}}) \, {\mathcal{J}}_s \, d{\bm{x}}\\ &\qquad + \int_{{\mathcal{O}}} 2{\mathcal{E}}({\bm{u}}^{\rm p}): [{\mathcal{E}}(\cdot),{\partial}_s]({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}}) \, d{\bm{x}}- \int_{{\mathcal{O}}} p^{\rm p} \,[{{\rm{div}\,}},{\partial}_s]({\partial}_s(\psi{\bm{u}}^{\rm p})- {\bm{g}}-{\bm{z}}) \, d{\bm{x}}\\ &= -\int_{{\mathcal{O}}} 2{\mathcal{E}}({\partial}_s{\bm{u}}^{\rm p}):{\mathcal{E}}({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}}) \, d{\bm{x}}- \int_{{\mathcal{O}}} 2{\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}}) \, {\mathcal{J}}_s \, d{\bm{x}}\\ &\qquad +\int_{{\mathcal{O}}} 2[{\mathcal{E}}(\cdot),{\partial}_s]{\bm{u}}^{\rm p}: {\mathcal{E}}({\partial}_s(\psi{\bm{u}}^{\rm p})- {\bm{g}}-{\bm{z}}) \, d{\bm{x}}+ \int_{{\mathcal{O}}} 2{\mathcal{E}}({\bm{u}}^{\rm p}): [{\mathcal{E}}(\cdot),{\partial}_s]({\partial}_s(\psi{\bm{u}}^{\rm p})-{\bm{g}}-{\bm{z}}) \, d{\bm{x}}\\ &\qquad - \int_{{\mathcal{O}}} p^{\rm p} \,[{{\rm{div}\,}},{\partial}_s]({\partial}_s(\psi{\bm{u}}^{\rm p})- {\bm{g}}-{\bm{z}}) \, d{\bm{x}}.\end{aligned}$$ Note that the first integral in the third line vanishes due to the definition of ${\bm{z}}$. In this way we we can avoid having to deal with a ${\partial}_s p^{\rm p}$ term in the resulting estimate.\ Then, using Proposition \[comm\_ests\], estimates and , and Lemma \[korn\], we have $$\begin{aligned} {\left\lVert {\mathcal{E}}({\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})}^2 &\le C{\left\lVert {\mathcal{E}}({\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} \big( {\left\lVert {\partial}_s {\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} + {\left\lVert {\mathcal{E}}({\bm{z}}) \right\rVert}_{L^2({\mathcal{O}})} + {\left\lVert {\mathcal{E}}({\bm{g}}) \right\rVert}_{L^2({\mathcal{O}})} \big) \\ &\quad + C{\left\lVert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} \big({\left\lVert {\mathcal{E}}(\psi{\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} +{\left\lVert {\mathcal{E}}({\bm{g}}) \right\rVert}_{L^2({\mathcal{O}})} +{\left\lVert {\mathcal{E}}({\bm{z}}) \right\rVert}_{L^2({\mathcal{O}})} \big) \\ &\quad + 2{\left\lVert [{\mathcal{E}}(\cdot),{\partial}_s]{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}\big( {\left\lVert {\mathcal{E}}(\psi{\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} + {\left\lVert {\mathcal{E}}({\bm{z}}) \right\rVert}_{L^2({\mathcal{O}})} + {\left\lVert {\mathcal{E}}({\bm{g}}) \right\rVert}_{L^2({\mathcal{O}})} \big) \\ &\quad +2{\left\lVert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} \big( {\left\lVert [{\mathcal{E}}(\cdot),{\partial}_s](\psi{\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} + {\left\lVert [{\mathcal{E}}(\cdot),{\partial}_s]({\bm{z}}) \right\rVert}_{L^2({\mathcal{O}})} + {\left\lVert [{\mathcal{E}}(\cdot),{\partial}_s]({\bm{g}}) \right\rVert}_{L^2({\mathcal{O}})} \big) \\ &\quad + {\left\lVert p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} \big( {\left\lVert [{{\rm{div}\,}},{\partial}_s](\psi{\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} + {\left\lVert [{{\rm{div}\,}},{\partial}_s]({\bm{z}}) \right\rVert}_{L^2({\mathcal{O}})} +{\left\lVert [{{\rm{div}\,}},{\partial}_s]({\bm{g}}) \right\rVert}_{L^2({\mathcal{O}})} \big) \\ &\le C({\left\lVert {\mathcal{E}}({\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} + {\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} +{\left\lvert \bm{\omega} \right\rvert})\big({\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}+{\left\lVert p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} +{\left\lvert \bm{\omega}^{\rm p} \right\rvert} \big) \\ &\le \delta{\left\lVert {\mathcal{E}}({\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})}^2 + C(\delta) \big({\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}^2+{\left\lVert p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}^2 + {\left\lvert \bm{\omega}^{\rm p} \right\rvert}^2\big)\end{aligned}$$ for any $0<\delta\in{\mathbb{R}}$, by Young’s inequality. Taking $\delta=\frac{1}{2}$ and using Lemma \[korn\], we obtain $$\label{ps_est} \begin{aligned} {\left\lVert \nabla({\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} &\le {\left\lVert {\mathcal{E}}({\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} \le C \big({\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}+{\left\lVert p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} + {\left\lvert \bm{\omega}^{\rm p} \right\rvert} \big) \\ & \le C{\left\lvert \log\epsilon \right\rvert}^{1/2} \big({\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)}+{\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \big), \end{aligned}$$ where we have used Corollary \[omegaP\] to bound ${\left\lvert \bm{\omega}^{\rm p} \right\rvert}$. Here $C$ depends only on $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$.\ We may estimate ${\partial}_\theta{\bm{u}}^{\rm p}$ in a similar way. In fact, the construction of the analogous test function is simpler since $({\partial}_\theta{\bm{u}}^{\rm p})\big|_{\Gamma_\epsilon} = {\partial}_\theta({\bm{v}}+\bm{\omega}\times {\bm{X}}(s))=0$ and thus we do not need to correct for a nonzero boundary value. Following the same steps used to estimate ${\partial}_s{\bm{u}}^{\rm p}$, we obtain $$\label{ptheta_est} {\left\lVert \nabla({\partial}_\theta{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} \le C \big({\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)}+{\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \big),$$ where $C$ depends only on $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$.\ In addition to the estimates and , we need bounds for the tangential derivatives ${\partial}_s p^{\rm p}$ and ${\partial}_\theta p^{\rm p}$ of the pressure. We begin by estimating ${\partial}_s p^{\rm p}$; the bound for ${\partial}_\theta p^{\rm p}$ is similar. Since we already know that ${\partial}_s p^{\rm p}\in L^2(\Omega_\epsilon)$, we may consider ${\widetilde{{\bm{z}}}} \in D^{1,2}_0({\mathcal{O}})$ satisfying $$\label{wtzee_def} \begin{aligned} {{\rm{div}\,}}{\widetilde{{\bm{z}}}} &= \psi {\partial}_s p^{\rm p} \quad \text{in }{\mathcal{O}}, \\ {\left\lVert \nabla{\widetilde{{\bm{z}}}} \right\rVert}_{L^2({\mathcal{O}})} &\le C{\left\lVert \psi {\partial}_s p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}, \end{aligned}$$ where $\psi$ is as in . Again, we know that such a ${\widetilde{{\bm{z}}}}$ exists due to [@galdi2011introduction], Section III.3 and [@closed_loop], Appendix A.2.5.\ Using ${\partial}_s{\widetilde{{\bm{z}}}}$ as a test function in Definition \[rigid\_weak\_p\] (again, we can make sense of the following computation using finite differences, and thus only require ${\widetilde{{\bm{z}}}}\in D^{1,2}({\mathcal{O}})$), we have $$\begin{aligned} 0&= \int_{{\mathcal{O}}} \bigg(2{\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}({\partial}_s {\widetilde{{\bm{z}}}}) - p^{\rm p}\, {{\rm{div}\,}}({\partial}_s{\widetilde{{\bm{z}}}})\bigg) \, d{\bm{x}}= \int_{{\mathcal{O}}}2{\mathcal{E}}({\bm{u}}^{\rm p}): {\partial}_s{\mathcal{E}}({\widetilde{{\bm{z}}}}) \, d{\bm{x}}\\ &\quad + \int_{{\mathcal{O}}}2{\mathcal{E}}({\bm{u}}^{\rm p}): [{\mathcal{E}}(\cdot),{\partial}_s]{\widetilde{{\bm{z}}}} \, d{\bm{x}}- \int_{{\mathcal{O}}} p^{\rm p} \,{\partial}_s{{\rm{div}\,}}{\widetilde{{\bm{z}}}} \, d{\bm{x}}- \int_{{\mathcal{O}}} p^{\rm p} \, [{{\rm{div}\,}},{\partial}_s]{\widetilde{{\bm{z}}}} \, d{\bm{x}}\\ &= -\int_{{\mathcal{O}}}2{\partial}_s{\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}({\widetilde{{\bm{z}}}}) \, d{\bm{x}}- \int_{{\mathcal{O}}}2{\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}({\widetilde{{\bm{z}}}}) \, {\mathcal{J}}_s \, d{\bm{x}}+ \int_{{\mathcal{O}}}2{\mathcal{E}}({\bm{u}}^{\rm p}): [{\mathcal{E}}(\cdot),{\partial}_s]{\widetilde{{\bm{z}}}} \, d{\bm{x}}\\ &\quad - \int_{{\mathcal{O}}} p^{\rm p} \, [{{\rm{div}\,}},{\partial}_s]{\widetilde{{\bm{z}}}} \, d{\bm{x}}+ \int_{{\mathcal{O}}}({\partial}_s p){{\rm{div}\,}}{\widetilde{{\bm{z}}}} \, d{\bm{x}}+ \int_{{\mathcal{O}}} p^{\rm p} \,{{\rm{div}\,}}{\widetilde{{\bm{z}}}} \, {\mathcal{J}}_s \, d{\bm{x}}\\ &= \int_{{\mathcal{O}}}\psi({\partial}_s p)^2 \, d{\bm{x}}-\int_{{\mathcal{O}}}2{\partial}_s{\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}({\widetilde{{\bm{z}}}}) \, d{\bm{x}}- \int_{{\mathcal{O}}}2{\mathcal{E}}({\bm{u}}^{\rm p}): {\mathcal{E}}({\widetilde{{\bm{z}}}}) \, {\mathcal{J}}_s \, d{\bm{x}}\\ &\quad + \int_{{\mathcal{O}}}2{\mathcal{E}}({\bm{u}}^{\rm p}): [{\mathcal{E}}(\cdot),{\partial}_s]{\widetilde{{\bm{z}}}} \, d{\bm{x}}- \int_{{\mathcal{O}}} p^{\rm p} \, [{{\rm{div}\,}},{\partial}_s]{\widetilde{{\bm{z}}}} \, d{\bm{x}}+ \int_{{\mathcal{O}}} p^{\rm p} \,{{\rm{div}\,}}{\widetilde{{\bm{z}}}} \, {\mathcal{J}}_s \, d{\bm{x}},\end{aligned}$$ where ${\mathcal{J}}_s \, d{\bm{x}}$ is as in and we have used . Then, using that $\psi^2\le \psi$, we have $$\begin{aligned} {\left\lVert \psi{\partial}_s p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}^2 &\le 2{\left\lVert {\mathcal{E}}({\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} {\left\lVert {\mathcal{E}}({\widetilde{{\bm{z}}}}) \right\rVert}_{L^2({\mathcal{O}})} + 2{\left\lVert [{\mathcal{E}}(\cdot),{\partial}_s]{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} {\left\lVert {\mathcal{E}}({\widetilde{{\bm{z}}}}) \right\rVert}_{L^2({\mathcal{O}})} \\ &\quad + C{\left\lVert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} {\left\lVert {\mathcal{E}}({\widetilde{{\bm{z}}}}) \right\rVert}_{L^2({\mathcal{O}})} + 2{\left\lVert {\mathcal{E}}({\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})}{\left\lVert [{\mathcal{E}}(\cdot),{\partial}_s]{\widetilde{{\bm{z}}}} \right\rVert}_{L^2({\mathcal{O}})} \\ &\quad +{\left\lVert p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} {\left\lVert [{{\rm{div}\,}},{\partial}_s]{\widetilde{{\bm{z}}}} \right\rVert}_{L^2({\mathcal{O}})} + C{\left\lVert p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} {\left\lVert {{\rm{div}\,}}{\widetilde{{\bm{z}}}} \right\rVert}_{L^2({\mathcal{O}})}\\ &\le C\big({\left\lVert \nabla({\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})} + {\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} +{\left\lVert p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} \big) {\left\lVert \psi{\partial}_s p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} \\ &\le \delta {\left\lVert \psi{\partial}_s p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}^2 + C(\delta)\big({\left\lVert \nabla({\partial}_s{\bm{u}}^{\rm p}) \right\rVert}_{L^2({\mathcal{O}})}^2 + {\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}^2 +{\left\lVert p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})}^2 \big)\end{aligned}$$ for $0<\delta\in {\mathbb{R}}$. Here we have used , , Proposition \[comm\_ests\], and Young’s inequality. Taking $\delta=\frac{1}{2}$ and using , we obtain $${\left\lVert \psi{\partial}_s p^{\rm p} \right\rVert}_{L^2({\mathcal{O}})} \le C{\left\lvert \log\epsilon \right\rvert}^{1/2}\big( {\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} +{\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \big).$$ Then, using , within the region $${\mathcal{O}}' = \bigg\{{\bm{x}}\in \Omega_{\epsilon} \; : \; {\bm{x}}= {\bm{X}}(s) + \rho {\bm{e}}_{\rho}(s,\theta), \quad \epsilon < \rho<\frac{r_{\max}}{4} \bigg\},$$ we have $$\label{psp_est} {\left\lVert {\partial}_s p^{\rm p} \right\rVert}_{L^2({\mathcal{O}}')} \le C{\left\lvert \log\epsilon \right\rvert}^{1/2}\big( {\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} +{\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \big)$$ for $C$ depending only on $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$.\ We can similarly use to show $$\label{pthetap_est} {\left\lVert {\partial}_\theta p^{\rm p} \right\rVert}_{L^2({\mathcal{O}}')} \le C\big( {\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} +{\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \big).$$ Now we can use the tangential bounds , , , and to obtain an estimate for derivatives $\nabla({\partial}_\rho{\bm{u}}^{\rm p})$ normal to $\Gamma_\epsilon$. For this, we will use the full Stokes equations , written with respect to the orthonormal frame ${\bm{e}}_t$, ${\bm{e}}_\rho$, ${\bm{e}}_\theta$ in ${\mathcal{O}}$ as $$\begin{aligned} - \Delta {\bm{u}}^{\rm p} +\nabla p^{\rm p} &= -\Delta {\bm{u}}^{\rm p} + \frac{{\partial}p^{\rm p}}{{\partial}\rho}{\bm{e}}_{\rho} + \frac{1}{\rho}\frac{{\partial}p^{\rm p}}{{\partial}\theta}{\bm{e}}_{\theta} + \frac{1}{1-\rho{\widehat{\kappa}}}\bigg(\frac{{\partial}p^{\rm p}}{{\partial}s}-\kappa_3\frac{{\partial}p^{\rm p}}{{\partial}\theta}\bigg){\bm{e}}_t =0 \\ {{\rm{div}\,}}{\bm{u}}^{\rm p} &= \frac{1}{1-\rho{\widehat{\kappa}}}\bigg(\frac{1}{\rho}\frac{{\partial}(\rho (1-\rho{\widehat{\kappa}}) u_{\rho})}{{\partial}\rho}+\frac{1}{\rho}\frac{{\partial}((1-\rho{\widehat{\kappa}}) u_{\theta})}{{\partial}\theta} + \frac{{\partial}u_s}{{\partial}s} \bigg) = 0.\end{aligned}$$ Here ${\widehat{\kappa}}$ is as in and we recall the notation $u_\rho = {\bm{u}}^{\rm p}\cdot{\bm{e}}_\rho$, $u_\theta={\bm{u}}^{\rm p}\cdot{\bm{e}}_\theta$, $u_s={\bm{u}}^{\rm p}\cdot{\bm{e}}_t$.\ From the divergence-free condition on ${\bm{u}}^{\rm p}$, after multiplying through by $\rho (1-\rho{\widehat{\kappa}})$ and differentiating once with respect to $\rho$, we obtain $$\begin{aligned} {\left\lVert \frac{{\partial}^2 u_{\rho}}{{\partial}^2 \rho} \right\rVert}_{L^2(\mathcal{O})} &\le C\bigg({\left\lVert \frac{1}{\rho} \nabla {\bm{u}}^{\rm p} \right\rVert}_{L^2(\mathcal{O})} + \bigg\|\frac{1}{\rho}\bigg\|_{L^{\infty}(\mathcal{O})}|\mathcal{O}|^{1/3}\big\|{\bm{u}}^{\rm p} \big\|_{L^6(\mathcal{O})} \\ &\hspace{2cm}+ \bigg\|\frac{1}{\rho}\frac{{\partial}}{{\partial}\rho}\bigg(\frac{{\partial}u_{\theta}}{{\partial}\theta}\bigg)\bigg\|_{L^2(\mathcal{O})} +\bigg\|\frac{{\partial}}{{\partial}\rho}\bigg(\frac{{\partial}u_s}{{\partial}s}\bigg)\bigg\|_{L^2(\mathcal{O})} \bigg)\\ &\le \frac{C}{\epsilon}{\left\lvert \log\epsilon \right\rvert}^{1/2}\big( \|\nabla {\bm{u}}^{\rm p}\|_{L^2(\Omega_\epsilon)} +{\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \big),\end{aligned}$$ where we have used and along with the Sobolev inequality on $\Omega_\epsilon$.\ Furthermore, using the ${\bm{e}}_\rho$ component of $-\Delta {\bm{u}}^{\rm p} +\nabla p=0$, we have $$\begin{aligned} \frac{{\partial}p^{\rm p}}{{\partial}\rho} &= (\Delta{\bm{u}}^{\rm p}) \cdot{\bm{e}}_{\rho} \\ &= \frac{1}{\rho (1-\rho{\widehat{\kappa}})}\frac{{\partial}}{{\partial}\rho}\left(\rho (1-\rho{\widehat{\kappa}})\frac{{\partial}{\bm{u}}^{\rm p}}{{\partial}\rho}\right)\cdot{\bm{e}}_{\rho} +\frac{1}{\rho^2(1-\rho{\widehat{\kappa}})}\frac{{\partial}}{{\partial}\theta}\bigg((1-\rho{\widehat{\kappa}})\frac{{\partial}{\bm{u}}^{\rm p}}{{\partial}\theta}\bigg)\cdot{\bm{e}}_{\rho} \\ &\hspace{4cm} +\frac{1}{1-\rho{\widehat{\kappa}}} \frac{{\partial}}{{\partial}s}\bigg( \frac{1}{1-\rho{\widehat{\kappa}}}\bigg[ \frac{{\partial}{\bm{u}}^{\rm p}}{{\partial}s}- \kappa_3\frac{{\partial}{\bm{u}}^{\rm p}}{{\partial}\theta} \bigg] \bigg)\cdot{\bm{e}}_{\rho} \\ &= \frac{1}{\rho (1-\rho{\widehat{\kappa}})}\frac{{\partial}}{{\partial}\rho}\left(\rho (1-\rho{\widehat{\kappa}})\frac{{\partial}u_{\rho}}{{\partial}\rho}\right) +\frac{1}{\rho^2(1-\rho{\widehat{\kappa}})}\frac{{\partial}}{{\partial}\theta}\bigg((1-\rho{\widehat{\kappa}})\frac{{\partial}{\bm{u}}^{\rm p}}{{\partial}\theta}\bigg)\cdot{\bm{e}}_{\rho} \\ &\hspace{4cm} +\frac{1}{1-\rho{\widehat{\kappa}}} \frac{{\partial}}{{\partial}s}\bigg( \frac{1}{1-\rho{\widehat{\kappa}}}\bigg[ \frac{{\partial}{\bm{u}}^{\rm p}}{{\partial}s}- \kappa_3\frac{{\partial}{\bm{u}}^{\rm p}}{{\partial}\theta} \bigg] \bigg)\cdot{\bm{e}}_{\rho}, \end{aligned}$$ since ${\bm{e}}_{\rho}(s,\theta)$ does not vary with $\rho$. Therefore, using , , , and , along with the the bound on $\frac{{\partial}^2 u_{\rho}}{{\partial}\rho^2}$, we have $$\|\nabla p^{\rm p}\|_{L^2(\mathcal{O}')} \le \frac{C}{\epsilon}{\left\lvert \log\epsilon \right\rvert}^{1/2}\big( \|\nabla {\bm{u}}^{\rm p}\|_{L^2(\Omega_\epsilon)} +{\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \big).$$ Finally, to estimate $\frac{{\partial}^2 u_j}{{\partial}\rho^2}$, $j=\theta,s$, we again use that $$\begin{aligned} \nabla p^{\rm p}\cdot{\bm{e}}_j &= (\Delta {\bm{u}}^{\rm p})\cdot {\bm{e}}_j(s,\theta) \\ &= \frac{1}{\rho (1-\rho{\widehat{\kappa}})}\frac{{\partial}}{{\partial}\rho}\bigg(\rho (1-\rho{\widehat{\kappa}})\frac{{\partial}u_j}{{\partial}\rho}\bigg) +\frac{1}{\rho^2(1-\rho{\widehat{\kappa}})}\frac{{\partial}}{{\partial}\theta}\bigg((1-\rho{\widehat{\kappa}})\frac{{\partial}{\bm{u}}^{\rm p}}{{\partial}\theta}\bigg)\cdot{\bm{e}}_j \\ &\hspace{4cm}+\frac{1}{1-\rho{\widehat{\kappa}}} \frac{{\partial}}{{\partial}s}\bigg( \frac{1}{1-\rho{\widehat{\kappa}}}\bigg[ \frac{{\partial}{\bm{u}}^{\rm p}}{{\partial}s}- \kappa_3\frac{{\partial}{\bm{u}}^{\rm p}}{{\partial}\theta} \bigg] \bigg)\cdot{\bm{e}}_j , \quad j=\theta,s,\end{aligned}$$ since each of ${\bm{e}}_t(s)$, ${\bm{e}}_\rho(s,\theta)$ and ${\bm{e}}_\theta(s,\theta)$ are independent of $\rho$. Then we have $$\begin{aligned} \bigg\|\frac{{\partial}^2 u_j}{{\partial}\rho^2}\bigg\|_{L^2(\mathcal{O}')} &\le C \bigg( \bigg\|\frac{1}{\rho}\bigg\|_{L^{\infty}(\mathcal{O}')} \| \nabla {\bm{u}}^{\rm p}\|_{L^2(\mathcal{O}')} + \bigg\|\frac{{\partial}^2{\bm{u}}^{\rm p}}{{\partial}s^2}\bigg\|_{L^2(\mathcal{O}')} \\ &\hspace{2cm} + \bigg\|\frac{{\partial}^2{\bm{u}}^{\rm p}}{{\partial}s{\partial}\theta}\bigg\|_{L^2(\mathcal{O}')} +\bigg\|\frac{{\partial}^2{\bm{u}}^{\rm p}}{{\partial}\theta^2}\bigg\|_{L^2(\mathcal{O}')}+\|\nabla p^{\rm p}\|_{L^2(\mathcal{O}')} \bigg) \\ &\le \frac{C}{\epsilon}{\left\lvert \log\epsilon \right\rvert}^{1/2} \big({\left\lVert \nabla{\bm{u}}^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} + {\left\lVert p^{\rm p} \right\rVert}_{L^2(\Omega_\epsilon)} \big), \quad j=\theta, s, \end{aligned}$$ where $C$ depends only on $c_\Gamma$, $\kappa_{\max}$, and $\xi_{\max}$. Altogether, we obtain Lemma \[high\_reg\]. $\square$ We note that the factor of $\frac{1}{\epsilon}$ in Lemma \[high\_reg\] is necessary. As a heuristic, we consider an infinite straight cylinder of radius $\epsilon$ and take ${\bm{u}}= (\frac{1}{\rho}-\frac{1}{\epsilon}) {\bm{e}}_{\theta}$, where ${\bm{e}}_{\theta}$ is now the (constant) angular vector in straight cylindrical coordinates, and $p\equiv$ constant. Ignoring decay conditions toward infinity along the cylinder, $({\bm{u}},p)$ solves the Stokes equations with ${\bm{u}}=0$ on the cylinder surface. Then $$\begin{aligned} |\nabla^2{\bm{u}}| =\bigg| \frac{{\partial}^2}{{\partial}\rho^2} \frac{1}{\rho}\bigg| = \bigg|\frac{2}{\rho^3}\bigg| = \frac{2}{\rho}\big| \nabla {\bm{u}}\big|,\end{aligned}$$ and within the region $\epsilon < \rho \le 2 \epsilon$, we have $|\nabla^2{\bm{u}}| \ge \frac{1}{\epsilon}|\nabla {\bm{u}}|$. [^1]: This research was supported in part by NSF grant DMS-1620316 and DMS-1516978, awarded to Y.M., and a Torske Klubben Fellowship, awarded to L.O. L.O. also thanks Dallas Albritton for helpful discussion.

--- abstract: 'If a long chain is held in a pot elevated a distance $h_1$ above the floor, and the end of the chain is then dragged over the rim of the pot and released, the chain flows under gravity down into a pile on the floor. Not only does the chain flow out of the pot, it also leaps above the pot in a “chain-fountain”. I predict and observe that the steady state shape of the fountain is an inverted catenary, and discuss how to apply boundary conditions to this solution. In the case of a level pot, the fountain shape is completely vertical. In this case I predict and observe both how fast the fountain grows to its steady state height, and how it grows $\propto t^2$ if there is no floor. The fountain is driven by an unexpected push force from the pot that acts on the link of chain about to come into motion. I confirm this by designing two new chains, one consisting of hollow cylinders threaded on a string and one consisting of heavy beads separated by long flexible threads. The former is predicted to produce a pot-push and hence a fountain, while the latter will not. I confirm these predictions experimentally. Finally I directly observe the anomalous push in a horizontal chain-pick up experiment.' author: - John S Biggins title: Growth and Shape of a Chain Fountain --- The mechanics of chains is one of the oldest fields in physics. Galileo observed that hanging chains approximate parabolas, particularly when the curvature is small[@galilei1974two], while the true shape was proved to be a catenary by Huygens Leibniz and John Bernoulli[@lockwood1971book]. A chain hanging in a catenary is a structure supporting its weight with pure tension. In 1675 Hooke discovered that a thin arch supporting its own weight with pure compression must follow the inverted shape of a hanging chain[@hookedescription], that is, an inverted catenary. Ever since architects from Wren to Gaudi have incorporated inverted catenary arches into their buildings and even used hanging strings to build inverted architectural prototypes. We might expect such a venerable and technologically important field to have few remaining surprises, but chain mechanics has recently produced several. A chain falling onto a table accelerates faster than $g$, leading inexorably to the conclusion that the table must pull down on the falling chain[@grewal2011chain; @hamm2010weight]. If a pile of chain rests on a surface, and the end is then pulled in the plane of the surface to deploy the chain, an unexpected noisy chain arch has been observed to form perpendicular to the surface of the chain immediately beyond of the pile[@Santagelochainarch], that is, in the portion of chain that has just come into motion. There is also recent work on the rich dynamics of whips and free ends[@shapeofawhip; @HannaSantangelofreeend; @tomaszewski2006motion]. The most recent surprise comes via Mould’s videos of a chain fountain[@mouldwebsite], shown in fig.  \[photoanddiagram\]a, in which a chain not only flows from an elevated pot to the floor under gravity but leaps above the pot. These videos have surprised and delighted almost 3.5 million viewers. In this letter I demonstrate that chain in such a fountain traces Hook’s inverted catenary, but as a structure of pure tension stabilized by the motion of the chain. ![a) Steve Mould demonstrating a chain fountain. Photo courtesy of J. Sanderson. b) Diagram of a chain fountain. A chain with mass per unit length $\lambda$ flows at speed $v$ along a curved trajectory from a pot tilted to an angle $\theta_p$ and elevated to an height $h_1$, to the floor. The fountain has height $h_2$ and width $w$. At each point $x$ the chain has a height $y(x)$ a tension $T(x)$ and makes an angle $\theta(x)$ with the vertical.[]{data-label="photoanddiagram"}](photoanddiagram.pdf){width="30.00000%"} In a chain fountain, the leaping of the chain above the pot requires that when a link of the chain is brought into motion, it must not only be pulled into motion by the moving chain but also pushed into motion by the pot[@BigginsWarnerChain]. This anomalous push is expected to arise whenever a pile of chain is deployed and, as such, has a wide field of potential applications. However, the analysis in [@BigginsWarnerChain] infers the existence of the anomalous push from a simplified model of a zero-width steady-state fountain, leading to questions about whether the anomalous push is an artifact of these assumptions. In this letter I consider fountains of finite width and the dynamics of fountain growth. The extended theory does not mitigate the need for an anomalous force, and explains the observed fountain behavior well. I also confirm the anomalous force hypothesis experimentally, both by direct observation in a horizontal pickup geometry, and by comparing the fountains made by radically different sorts of chain. A non-vertical chain fountain is sketched in fig. \[photoanddiagram\]b. We expect that, after the fountain reaches the floor, it will tend to a steady shape.To find this equilibrium curve, consider an element of chain with horizontal extent $\mathrm{d}x$, which has length $\mathrm{d}s=\mathrm{d}x/\sin{(\theta)}$ and mass $\lambda \mathrm{d}s$. Tangentially there is no acceleration so the tension gradient balances gravity, $$T'(x)=\frac{\lambda g}{\sin{\theta}} \cos{\left(\theta\right)} .$$ Since $\cot{\left(\theta\right)}=y'(x)$ this can be integrated to give $$T(x)=\lambda g y +\lambda(v^2-c g),$$ where we have written the constant of integration as $\lambda(v^2-c g)$ and $c$ is a constant. Perpendicularly, there is the inward force $T(x)/r(x)$ (where $r(x)$ is the radius of curvature), a Laplace-pressure like term that arrises whenever one has tension in a curved surface. This force and gravity supply the centripetal acceleration: $$\frac{T(x)}{ r(x)}-\lambda g \sin{\left(\theta\right)} =\lambda \frac{v^2}{r(x)}.$$ Recalling that in Cartesians $1/r=y''(x)/(1+y'(x)^2)^{3/2}$ and $\sin{\left(\theta\right)}=1/\sqrt{1+y'(x)^2}$, this simplifies to $$(T(x)-\lambda v^2)y''(x)=g \lambda (1+y'(x)^2).\label{cateqn1}$$ Substituting in our result for $T(x)$ and solving for $y(x)$ reveals that a chain moving along its own length under gravity in an unchanging shape must trace a catenary.[@Tripos_1854; @airy1858mechanical; @perkins1989theoretical]. Curiously, this result was first published as a question in the 1854 Cambridge University maths examination[@Tripos_1854]. In the case of the fountain, this catenary must be an inverted one, viz. $$y(x)=-a \cosh{\left(\left(x-b\right)/a\right)}+c$$ where $a$, $b$ are new constants of integration. The simplest inverted catenary is $y(x)=-\cosh{(x)}$. The above solution is simply this curve translated and with a “zoom” by a factor of $a$ equal to the radius of curvature of the catenary at its apex. All steady-state chain fountains should therefore produce shapes that, after zooming and translating, collapse onto $-\cosh(x)$. To test this, a 50m long brass ball-chain was put in a 1L beaker, elevated to 1.72m above the ground and tilted by an angle $\theta_p$. The end of the chain was then pulled over the rim and released initiating a chain fountain. The experiment was repeated with different tilt angles, resulting in different fountain shapes, which were photographed towards the end of each run to ensure the fountain was in its steady state. Runs with significant tangles were disregarded. Two examples are shown in fig.  \[chaincats\]a, one thin and one wide. The fountains undulate locally but macroscopically trace a catenary. In fig.  \[chaincats\]b many chain fountains with different widths are rescaled onto a single inverted catenary, demonstrating that the chain fountain is well described by Hook’s inverted catenary. To determine the parameters, $a$, $b$ and $c$, $v$ and $w$ (the width of the fountain, see fig. \[photoanddiagram\]b) we require five boundary conditions. The first two, $y(0)=0$, and $y(w)=-h_1$, fix the coordinate origin and the fountain drop. To find the remaining three we must examine the pickup and putdown processes. In a time $\mathrm{d}t$ a length of chain $v\mathrm{d}t$ is picked up, acquiring a momentum $\lambda v^2 \mathrm{d}t$. If the links are accelerated solely by the tension, then the third boundary condition is $T(0)=\lambda v^2$. However, inspecting eqn. (\[cateqn1\]), we see that the right-hand side is always finite so setting $T(0)=\lambda v^2$ require $y''(0)\to \infty$, corresponding to the chain reversing direction immediately above the pot. For a fountain we must have $T(0)<\lambda v^2$. However, the total force must still be $\lambda v^2$, so we must introduce an anomalous reaction force from the pot pushing the links into motion $f_p=\alpha \lambda v^2$, reducing the tension to $T(0)=(1-\alpha)\lambda v^2$. We understand this anomalous pot push by modeling the chain as freely jointed rigid rods being picked up vertically from a horizontal surface, fig.  \[widefig\]ai. The next rod is pulled upward into motion by a tension force $T$ at its end, sketched in fig.  \[widefig\]aii, causing the center of mass of the rod to lift and the rod to rotate. In isolation this would cause the far end of the rod to move down. In reality the surface prevents this by pushing up with a reaction force $f_p$. We can estimate the size of this force, and hence $\alpha$, by considering the forces on the rod at the first moment of pickup when the rod is still horizontal. The initial linear and angular accelerations of the rod are $m \dot{v}=T+f_p$ and $I \dot{\omega}=(f_p-T)l/2$, where $v$ is the velocity of the rod’s center of mass, and $I$ and $\omega$ are the rod’s moment of inertia and angular velocity, both around the center of mass. The initial acceleration of the left hand tip of the rod must be zero, requiring $\dot{v}+\dot{\omega}l/2=0$. Writing $f_p=\alpha \lambda v^2$ and $T=(1-\alpha)\lambda v^2$ we can rearrange this to estimate $\alpha$ as $$\alpha=\half\left(1-I/\left(\quarter m l^2\right)\right).\label{alphaeq}$$ This calculation is not intended as a precise calculation of the value of $\alpha$, but rather to illustrate how a surface can push on a departing chain, and hence motivate taking a non-zero value for $\alpha$ at all. This simple calculation demonstrates that the pot can push, justifies the functional form $f_p=\alpha \lambda v^2$ and reveals that $\alpha$ depends on the details of the chain, here via the ratio $I/(m l^2)$. A better estimate would require us to include angle fluctuations in both the pile of rods and the departing chain, and to consider the complete pickup-process rather than just the first moment. We can use the above form to crudely estimate $\alpha$ for the ball-chain used in this paper. It takes 6 balls for the chain to turn $180\,^{\circ}$ so, in a freely jointed rod-model, the rod-like unit must be three balls connected by two rods. Treating the balls as point masses and the rods as light, this gives $I=2(m/3)(l/2)^2=m l^2/6$, and hence $\alpha=1/6=0.166..$ which, as expected, is close to but somewhat larger than the best-fit experimental value of $\alpha=0.12$. If the chain departs at an angle, sketched in fig.  \[widefig\]aiii, the velocity and tension point along the chain’s tangent. Momentum conservation requires any force from the surface to point in the same tangential direction and hence have an in-surface component which can only arise as a frictional force. Since friction always acts to hinder or prevent motion, it will always oppose the in-plane component of the tension, shown in fig. \[widefig\]aiv, the total surface force will only point in the tangential direction if pickup is perpendicular to the surface, giving a fourth boundary condition $\theta(0)=\theta_p$. At the floor a force $\lambda v^2$ is required to halt the chain. If all this force comes from the floor then the tension at the end is $T(w)=0$. However, careful observations of free-falling chains dropping onto hard surfaces reveal that they accelerate faster chains that fall freely [@grewal2011chain; @hamm2010weight]. This demands $T(w)>0$ so that the force on the falling chain is greater than its weight, so we set $T(w)=\beta \lambda v^2$. If the chain hits the floor at an angle, the friction and reaction from the floor are both opposite to the chain’s velocity and can add to point along the chain’s tangent, as sketched in fig.  \[widefig\]v-vi. There is thus is no constraint on $\theta(w)$. The five boundary conditions $$\begin{aligned} y(0)&=0\hspace{2em} \theta(0)=\theta_p\hspace{2em}T(0)=(1-\alpha)\lambda v^2\notag \\ y(w)&=-h_1\hspace{ 2 em} T(w)=\beta \lambda v^2,\end{aligned}$$ determine the five unknowns in the catenary solution as $$\begin{aligned} a&=\frac{\alpha h_1 \sin (\theta_p)}{1-\alpha -\beta }\hspace{0.5em}c=\frac{\alpha h_1}{1-\alpha -\beta }\hspace{0.5em}v=\sqrt{\frac{g h_1}{1-\alpha -\beta }}\\ b&=a \sinh ^{-1}(\cot (\theta_p))\hspace{1.5em}w=a\cosh ^{-1}\left((c+h_1)/a\right)+b.\notag\end{aligned}$$ We see the chain’s trajectory is independent of $g$ though the chain’s speed is not. We plot the fountain width, $w$, as a function of $\theta_p$ in fig.  \[widefig\]b. We see that for small tilt angles the width of the fountain is zero, but that the gradient $\mathrm{d}w/\mathrm{d}\theta_p$ is divergent, albeit weakly as $\sinh^{-1}(\cot(\theta_p))\sim\log(\theta_p)$, and that for large tilt angles the fountain width starts to decrease. A unit length of chain releases gravitational energy $g h_1 \lambda$ and receives kinetic energy $\half g h_1 \lambda/(1-\alpha-\beta)$. In the traditional regime ($\alpha=\beta=0$) half the gravitational energy is thus lost in the chain pickup process, as classically expected when picking up a chain at constant velocity[@BigginsWarnerChain]. The additional anomalous forces reduce this energy loss. The requirement that the kinetic energy not be larger than the released gravitational potential energy imposes the bound $\alpha+\beta<\half$. The fountain height above the pot is simply $h_2=y(b)$: $$h_2=\frac{\alpha h_1 (1-\sin (\theta_p))}{1-\alpha -\beta }.$$ Both the height and the width of the fountain vanish if $\alpha=0$ (but not if $\beta=0$) establishing that the push from the pot during the chain pickup is the driver of a chain fountain. If the pot is level ($\theta_p=0$) the height is maximal and $w=0$ so the fountain is vertical. We compare the predicted and experimental shapes of chain fountains in fig.  \[widefig\]c. Fixing $h_1$ and $\theta_p$ to their experimental values, a good match is achieved for large-angle fountains, and the height of all fountains, by taking $\alpha=0.12$ and $\beta=0.11$, consistent with [@grewal2011chain] and [@BigginsWarnerChain]. However, although the small angle fountains are catenaries, the predictions are too thin. This is true for all physical values of $\alpha$ and $\beta$ so we conclude that $\theta(0)$ is actually slightly larger than $\theta_p$, and thus that the argument for perpendicular pickup must be lacking. This may be because it neglects the pot’s finite width, which for small $\theta_p$ is comparable to the pot-level width of the catenary, leading to substantial variation in the pickup angle as the pickup point move around the pot. It may also be because the rods do not lie flat on a flat pot surface, but at at somewhat random angles on a rough bed of other rods, so the rods can experience non-frictive reaction forces that are not perpendicular to the pot base, or because the rods come into motion prior to being picked up. Divergence of $\mathrm{d}w/\mathrm{d}\theta_p$ for $\theta_p\to0$ makes the system sensitive to any deviation of $\theta(0)$ from $\theta_p$ at small angles, where the discrepancy between actual and predicted width is seen. We now consider the growth of the fountain. We focus on the case of a level pot where the fountain is vertical. We sketch such a fountain in fig.  \[vertfountain\]a, labeling the speeds of the rising and falling legs $v_1$ and $v_2$. As before, the tension above the pot is $T_P=(1-\alpha )\lambda v_1^2$ and that above the floor is $T_F=\beta\lambda v_2^2$. Fig. \[vertfountain\]b shows two diagrams, separated by a time $\mathrm{d}t$, of a section of chain, length $2l$, traversing the apex. Inextensibility requires $2 \mathrm{d}l+ v_2\mathrm{d}t=v_1\mathrm{d}t$. However, $\mathrm{d}l/\mathrm{d}t=\dot{h}_2$, so this constraint gives $$\dot{h}_2=\half\left(v_1 - v_2\right).\label{h2d}$$ Further, a length $v_1\mathrm{d}t-\mathrm{d}l=\half(v_1+v_2)\mathrm{d}t$ is converted from moving upward at $v_1$ to downward at $v_2$, requiring a momentum change of $\mathrm{d}p=\half\lambda \left(v_1+v_2\right)^2\mathrm{d}t$. This momentum is provided by the tension $T_T$ which acts on both sides of the apex, requiring $2 T_T=\half \lambda \left(v_1+v_2\right)^2.$ Momentum balance in the two legs then requires $$\begin{aligned} - g h_2+\quarter \left(v_1+v_2\right)^2-(1-\alpha) v_1^2&=h_2 \dot{v}_1\label{eqn3}\\ - g (h_1+h_2)\hspace{-0.1 em}+\quarter \left(v_1+v_2\right)^2-\beta v_2^2&=-(h_1+h_2)\dot{v}_2.\label{eqn4}\end{aligned}$$ The dynamic equations (\[h2d\]-\[eqn4\]) are not analytically integrable, though in the steady state they reproduce the earlier analysis. Dimensional analysis reveals the fountain must grow to a height proportional to $h_1$ in a time proportional to $\sqrt{h_1/g}\sim 0.5\mathrm{s}$ . We confirm this via numerical solutions, shown in fig.  \[examplenumintegrate\], obtained using the initial conditions $h_2=v_1=v_2=0$ at $t=0$. Experimental data for fountain growth was obtained, again using a $1.8$m drop, by recording the height of the apex above the level of beads in the pot every 0.17s. The pot was tilted to a small angle $\theta_p\sim 1.5^{\circ}$ to ensure the chain exited on one side. The fountain was initiated with the chain touching the ground to match the theory. The total length of the chain was 50m, and the fountain lasted 12s. We compare this data with the theoretical prediction in fig.  \[expgrowthgraphs\]a, using the same chain parameters ($\alpha=0.12$, $\beta=0.11$) as in the catenary analysis. We see that the experimental line is quite noisy, but the theoretical line, with no new fitting parameters, matches well. If there is no floor the end of the chain falls ever lower. The relevant equations are still eqns (\[h2d\]-\[eqn4\]) but are augmented by $\dot{h}_1=v_2$, and modified by setting $\beta=0$ since, as there is no floor, it cannot provide any force. These new equations admit a continuously growing analytic solution with $v_1\propto v_2\propto gt$ and $h_1\propto h_2\propto g t^2$, where the constants of proportionality only depend on $\alpha$. In particular, $$\begin{aligned} h_2=\frac{\left( 4 \alpha +3 \sqrt{4-3 \alpha }-6 \right) g t^2}{42-32 \alpha }.\end{aligned}$$ We test this in fig. \[expgrowthgraphs\]b using an 8m drop, provided by a 3-story stairwell in the Canvendish Laboratory, and an 8m length of the same ball chain. We again achieve a good fit using $\alpha=0.12$. The driver of the chain fountain is the surprising extra push from the pot that helps launch links of the chain into motion. We test the suggested origin of this force by considering two very different chains. The first consists of small (diameter 4mm) spherical tungsten beads separated by 2.5cm lengths of thread. This does not produce a push since the beads are picked up individually and their round shape means any rotation induced does not result in them pushing down, so this chain should not produce a fountain. The second chain consists of short hollow cylinders (in reality pieces of uncooked macaroni pasta, 2cm long and 4mm diameter) strung end to end on fine thread. It resembles the chain of freely-jointed rods used to derive eqn. (\[alphaeq\]), and hence should produce a reaction and a fountain. The chains, shown in fig.  \[chaintests\]a, had similar mass densities, were 10m long and were dropped from a pot 1.8m above the ground. The fountains produced are shown in figs. \[chaintests\]b-c. The result that the bead chain does not produce a fountain while the freely-joined-rod chain does is unambiguous. To directly observe the push, 8m of ball-chain was closely packed in serpentine rows on a table, and the end was then released over the end of the table, causing the chain to deploy horizontally perpendicular to the rows. As first reported online[@Geminardtablefall], and seen in supplementary video 1, during the experiment the rows of chain moves backwards, implying that they push forwards on the departing chain. Observation of the chain fountain’s catenary shape and its growth and saturation reveal that, although it fluctuates considerably, it does so around the steady state discussed in this paper. However, the angle the chain fountain leaves a slightly tilted pot remains unquantified and the noisy nature of the fountain is poorly understood. The energetic origin of the noise is clear — it comes the gravitational potential energy that is “dissipated” during the pick-up process — however, its amplitude and wavelength are unquantified. These questions probably relate to the finite pot width and the chaotic ordering of the chain pile, suggesting further work varying pot width and using ordered piles. The relationship between $\alpha$ and $\beta$ also merits further work. They are very similar for the ball chain, leading one to wonder whether they are always equal. This work also complicates the idea of a perfectly flexible string. Eqn. (\[alphaeq\]) shows that $\alpha$ for a chain of rods depends only on $I/(ma^2)$. This will remain finite and chain-dependent as the link length tends to zero, so different perfectly flexible strings will produce different fountains heights. The rope fountain, with finite bending stiffness rather than links, is also an open problem. This paper confirms that when a chain is picked up, part of its momentum comes from the surface it is picked up from. In any industrial or technological setting where a chain is being deployed, accurate predictions about how much force is required will have to include this force. Further work on how to maximize this force may find application. Finally, picking up a chain has traditionally been thought to belong to a wide class of problems in which half the work done is dissipated. Charging a capacitor at constant voltage is a better known electrical example. The surprising upwards push during chain pickup increases the fraction of energy that is retained to $1/(2(1-\alpha))$. Perhaps it is worth revisiting other traditional problems in this class to see whether similar effects can be harnessed to reduce energy dissipation. 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--- abstract: 'The $SL(2)$-type of any smooth, irreducible and unitarizable representation of $GL_n$ over a $p$-adic field was defined by Venkatesh. We provide a natural way to extend the definition to all smooth and irreducible representations. For unitarizable representations we show that the $SL(2)$-type of a representation is preserved under base change with respect to any finite extension. The Klyachko model of a smooth, irreducible and unitarizable representation $\pi$ of $GL_n$ depends only on the $SL(2)$-type of $\pi.$ As a consequence we observe that the Klyachko model of $\pi$ and of its base-change are of the same type.' author: - Omer Offen and Eitan Sayag date: - - - title: 'The $SL(2)$-type and base change' --- introduction ============ Let $F$ be a finite extension of ${\mathbb{Q}}_{p}$. In [@MR2133760], Venkatesh assigned a partition of $n,$ the *$SL(2)$-type* of $\pi,$ to any smooth, irreducible and unitarizable representation $\pi$ of $GL_n(F).$ For a representation of Arthur type the $SL(2)$-type encodes the combinatorial data in the Arthur parameter. In general, the $SL(2)$-type is defined in terms of Tadic’s classification of the unitary dual. The reciprocity map for $GL_n(F)$ is a bijection from the set of isomorphism classes of smooth irreducible representations of $GL_n(F)$ to the set of isomorphism classes of $n$-dimensional Weil-Deligne representations (cf. [@MR1876802] and [@MR1738446]). Applying the reciprocity map we observe that there is a natural way to extend the definition of the $SL(2)$-type to all smooth and irreducible representations of $GL_n(F)$ (see Theorem \[thm: SL(2)-type\] and Remark \[rmk: SL(2)-type\]). The reciprocity map also allows the definition of base change with respect to any finite extension $E$ of $F.$ It is a map ${\operatorname{bc}}_{E/F}$ from isomorphism classes of smooth irreducible representation of $GL_n(F)$ to isomorphism classes of smooth irreducible representation of $GL_n(E)$ that is the ‘mirror image’ of restriction with respect to $E/F$ of Weil-Deligne representations. The content of Theorem \[thm: main\], our main result, is that for any smooth, irreducible and unitarizable representation $\pi$ of $GL_n(F)$ the representations $\pi$ and ${\operatorname{bc}}(\pi)$ have the same $SL(2)$-type. In [@MR2332593], [@os], [@OS3] we studied the Klyachko models of smooth irreducible representations of $GL_{n}(F),$ that is, distinction of a representation with respect to certain subgroups that are a semi direct product of a unipotent and a symplectic group. Our results are also described in terms of Tadic’s classification and depend, in fact, only on the $SL(2)$-type of a representation. For example, a smooth, irreducible and unitarizable representation $\pi$ of $GL_{2n}(F)$ is $Sp_{2n}(F)$-distinguished, i.e. it satisfies ${\operatorname{Hom}}_{Sp_{2n}(F)}(\pi,{\mathbb{C}}) \ne 0,$ if and only if the $SL(2)$-type of $\pi$ consists entirely of even parts (and in this case ${\operatorname{Hom}}_{Sp_{2n}(F)}(\pi,{\mathbb{C}})$ is one dimensional [@MR1078382 Theorem 2.4.2]). For unitarizable representations, our results on Klyatchko models are reinterpreted here in terms of the $SL(2)$-type. As a consequence we show that Klyachko models are preserved under base-change with respect to any finite extension. In particular, we have \[thm: symplectic main\] Let $E/F$ be a finite extension of $p$-adic fields. A smooth, irreducible and unitarizable representation $\pi$ of $GL_{2n}(F)$ is $Sp_{2n}(F)$-distinguished if and only if ${\operatorname{bc}}_{E/F}(\pi)$ is $Sp_{2n}(E)$-distinguished. The rest of this note is organized as follows. After setting some general notation in Section \[sec: notation\], in Section \[sec: bc\] we recall the definition of the reciprocity map. In Section \[sec: SL(2)-type\] we recall the definition of Venkatesh for the $SL(2)$-type of a unitarizable representation and extend it to all smooth irreducible representations. We recall (and reformulate in terms of the $SL(2)$-type) our results on symplectic (and more generally on Klyachko) models in Section \[sec: Klyachko\]. Our main observation Theorem \[thm: main\] and its application to Klyachko models Corollary \[cor: main\] are stated in Section \[sec: statements\] and proved in Section \[sec: proofs\]. The main theorem says that base change respects $SL(2)$-types and its corollary says that base change respects Klyachko types. Theorem \[thm: symplectic main\] is a special case where the Klyachko type is purely symplectic. Notation {#sec: notation} ======== Let $F$ be a finite extension of ${\mathbb{Q}}_p$ for some prime number $p$ and let ${\left|{\,\cdot}\right|}_F:F^\times \to {\mathbb{C}}^\times$ denote the standard absolute value normalized so that the inverse of uniformizers are mapped to the size of the residual field. Denote by $W_F$ the Weil group of $F$ and by $I_F$ the inertia subgroup of $W_F.$ We normalize the reciprocity map $T_F:W_F \to F^\times,$ given by local class field theory, so that geometric Frobenius elements are mapped to uniformizers. The map $T_F$ defines an isomorphism from the abelianization $W_F^{ab}$ of $W_F$ to $F^\times$ (this is the inverse of the Artin map). Let ${\left|{\,\cdot}\right|}_{W_F}={\left|{\,\cdot}\right|}_{F}\circ T_F$ denote the associated absolute value on $W_F.$ Denote by ${{\bf 1}}_\Omega$ the characteristic function of a set $\Omega.$ Let ${\operatorname{MS}}_{{\operatorname{fin}}}(\Omega)$ be the set of finite multisets of elements in $\Omega,$ that is, the set of functions $f:\Omega \to {\mathbb{Z}}_{\ge 0}$ of finite support. When convenient we will also denote $f$ by $\{\omega_1,\dots,\omega_1,\omega_2,\dots,\omega_2,\dots\}$ where $\omega\in \Omega$ is repeated $f(\omega)$ times. Let ${\mathcal{P}}={\operatorname{MS}}_{{\operatorname{fin}}}({\mathbb{Z}}_{>0})$ be the set of partitions of positive integers and let $${\mathcal{P}}(n)=\{f \in {\mathcal{P}}: \sum_{k=1}^\infty k\,f(k)=n\}$$ denote the subset of partitions of $n.$ For $n,\,m \in {\mathbb{Z}}_{>0}$ let $(n)_m=m\,{{\bf 1}}_n=\{n,\dots,n\}$ be the partition of $nm$ with ‘$m$ parts of size $n$’. Let ${\operatorname{odd}}:{\mathcal{P}}\to {\mathbb{Z}}_{\ge 0}$ be defined by $${\operatorname{odd}}(f)=\sum_{k=0}^\infty f(2k+1),$$ i.e. ${\operatorname{odd}}(f)$ is the number of odd parts of the partition $f.$ Reciprocity and base-change for $GL_n(F)$ {#sec: bc} ========================================= Weil-Deligne representations ---------------------------- An $n$-dimensional *Weil-Deligne* representation is a pair $((\rho,V),N)$ where $(\rho,V)$ is an $n$-dimensional representation of $W_{F}$ that decomposes as a direct sum of irreducible representations and $N:V \to V$ is a linear operator such that $${\left|{w}\right|}_{W_F}\,N \circ \rho(w)=\rho(w)\circ N,\ w \in W_F.$$ The map $((\rho,V),N) \mapsto ([\rho],f),$ where $[\rho]$ denotes the isomorphism class of the $n$-dimensional representation $(\rho,V)$ of $W_F$ and $f \in {\mathcal{P}}(n)$ is the partition of $n$ associated to the Jordan decomposition of $N,$ defines an injective map on isomorphism classes of Weil-Deligne representations. Denote its image by ${\mathcal{G}}_{F}(n).$ In this way we identify the set ${\mathcal{G}}_F(n)$ with the set of isomorphism classes of $n$-dimensional Weil-Deligne representations. Let $P_{F,n}: {\mathcal{G}}_F(n) \to {\mathcal{P}}(n)$ be the projection to the second coordinate. Let ${\mathcal{G}}_F=\cup_{n=1}^\infty {\mathcal{G}}_F(n)$ be the set of isomorphism classes of all finite dimensional Weil-Deligne representations and let $P_F:{\mathcal{G}}_F \to {\mathcal{P}}$ be the map such that ${P_F}_{|{\mathcal{G}}_F(n)}=P_{F,n}.$ The local Langlands correspondence ---------------------------------- Let ${\mathcal{A}}_F(n)$ be the set of isomorphism classes of smooth and irreducible representations of $GL_n(F)$ and set ${\mathcal{A}}_F=\cup_{n=1}^\infty {\mathcal{A}}_F(n).$ For every $\pi \in {\mathcal{A}}_F$ we denote by $\omega_\pi$ the central character of (any representation in the isomorphism class of) $\pi.$ Fix a non trivial additive character $\psi$ of $F.$ Due to Harris-Taylor [@MR1876802] and independently to Henniart [@MR1738446] there exists a unique sequence of bijections $${\operatorname{rec}}_{F,n}: {\mathcal{A}}_{F}(n) \to {\mathcal{G}}_{F}(n)$$ for all $n\ge 1$ satisfying the following properties: $$\begin{aligned} \label{eq: LCFT}& &{\operatorname{rec}}_{F}(\chi)=\chi \circ T_{F};\\& & \label{eq: L-id}L(\pi_{1} \times \pi_{2},s)=L({\operatorname{rec}}_{F}(\pi_{1})\otimes {\operatorname{rec}}_{F}(\pi_{2}),s);\\& & \label{eq: epsilon-id}\epsilon(\pi_{1} \times \pi_{2},s,\psi)=\epsilon({\operatorname{rec}}_{F}(\pi_{1})\otimes {\operatorname{rec}}_{F}(\pi_{2}),s,\psi);\\& & \label{eq: central char}\det \circ {\operatorname{rec}}_{F}(\pi)=rec_{F}(\omega_\pi);\\& &\label{eq: dual-id} {\operatorname{rec}}_{F}(\pi^\vee)={\operatorname{rec}}_{F}(\pi)^\vee.\end{aligned}$$ Here $\chi \in {\mathcal{A}}_F(1),\,\pi,\,\pi_1,\,\pi_2 \in {\mathcal{A}}_F,$ $\pi^\vee$ is the contragredient of $\pi,$ ${\operatorname{rec}}_{F}(\pi)^\vee$ is the dual of ${\operatorname{rec}}_{F}(\pi)$ and ${\operatorname{rec}}_F:{\mathcal{A}}_F \to {\mathcal{G}}_F$ is such that ${{\operatorname{rec}}_F}_{|{\mathcal{A}}_F(n)}={\operatorname{rec}}_{F,n}.$ Expressing ${\operatorname{rec}}_F$ in terms of ${\operatorname{rec}}_F^\circ$ ------------------------------------------------------------------------------ Let ${\mathcal{A}}_F^\circ(n)\subseteq {\mathcal{A}}_F(n)$ be the subset of isomorphism classes of supercuspidal representations and let ${\mathcal{G}}_F^\circ(n)\subseteq {\mathcal{G}}_F(n)$ be the subset of isomorphism classes $([\rho],f)$ such that $\rho$ is irreducible and $f={{\bf 1}}_n=\{n\}.$ The set ${\mathcal{G}}_F^\circ(n)$ is identified with the set of isomorphism classes of irreducible and $n$-dimensional representations of $W_F.$ It follows from the work of Harris-Taylor and independently of Henniart that there exists a unique sequence of bijections $${{\operatorname{rec}}_{F,n}}_{|{\mathcal{A}}_F^\circ(n)}={\operatorname{rec}}_{F,n}^\circ:{\mathcal{A}}_F(n)\to {\mathcal{G}}_F^\circ(n)$$ satisfying , , , and . The work of Zelevinsky [@MR584084] allows the extention of ${\operatorname{rec}}_F^\circ$ to the map ${\operatorname{rec}}_F$ on ${\mathcal{A}}_F.$ This is also explained in [@MR902829] and we now recall the construction of ${\operatorname{rec}}_F$ in terms of ${\operatorname{rec}}^\circ_F.$ For $s \in {\mathbb{C}}$ and every isomorphism class $\varpi=[\pi] \in {\mathcal{A}}_F$ (resp. $\varrho=([\rho],f) \in {\mathcal{G}}_F$) let $\varpi[s]=[\pi \otimes {\left|{\det}\right|}_F^s]$ (resp. $\varrho[s]=([\rho\otimes {\left|{\,\cdot}\right|}_{W_F}^s],f)$). A *segment* in ${\mathcal{A}}_F^\circ$ (resp. ${\mathcal{G}}_F^\circ$) is a set of the form $$\Delta[\sigma,r]=\{\sigma[\frac{1-r}2],\sigma[\frac{3-r}2],\dots,\sigma[\frac{r-1}2]\}$$ (resp. $$\Delta[\rho,r]=\{\rho[\frac{1-r}2],\rho[\frac{3-r}2],\dots,\rho[\frac{r-1}2]\})$$ for some $\sigma \in {\mathcal{A}}_F^\circ$ (resp. $\rho \in {\mathcal{G}}_F^\circ$) and $r\in {\mathbb{Z}}_{>0}.$ Let ${\mathcal{S}}$ (resp. ${\mathcal{S}}'$) denote the set of all segments in ${\mathcal{A}}_F^\circ$ (resp. ${\mathcal{G}}_F^\circ$) and let ${\mathcal{O}}={\operatorname{MS}}_{{\operatorname{fin}}}({\mathcal{S}})$ (resp. ${\mathcal{O}}'={\operatorname{MS}}_{{\operatorname{fin}}}({\mathcal{S}}')$). The bijection ${\operatorname{rec}}^\circ_F:{\mathcal{A}}_F^\circ \to {\mathcal{G}}_F^\circ$ defines a bijection ${\operatorname{rec}}_F^\circ:{\mathcal{S}}\to {\mathcal{S}}'$ given by ${\operatorname{rec}}_F^\circ(\Delta[\sigma,r])=\Delta[{\operatorname{rec}}_F^\circ(\sigma),r]$ and a bijection ${\operatorname{rec}}_F^\circ:{\mathcal{O}}\to {\mathcal{O}}'$ given by ${\operatorname{rec}}_F^\circ(a)({\operatorname{rec}}_F^\circ(\Delta))=a(\Delta),\,\Delta\in {\mathcal{S}}.$ In [@MR584084 Section 6.5] Zelevinsky defines a bijection $a \mapsto {\langle{a}\rangle}$ from ${\mathcal{O}}$ to ${\mathcal{A}}_F.$ The Zelevinsky involution is defined in [@MR584084 Section 9.12] as an involution on the Grothendick group associated with ${\mathcal{A}}_F.$ It is proved by Aubert [@MR1285969], [@MR1390967b] and independently by Procter [@MR1625483] that the Zelevinsky involution restricts to a bijection from ${\mathcal{A}}_F$ to itself that we denote by $\pi \mapsto \pi^t.$ In [@MR584084 Section 10.2] Zelevinsky defines a bijection $\tau:{\mathcal{O}}' \to {\mathcal{G}}_F$ as follows. For a segment $\Delta[\rho,r]\in {\mathcal{S}}'$ where $\rho \in {\mathcal{G}}_F^\circ(t)$ let $$\tau(\Delta[\rho,r])=(\oplus_{i=1}^r \,\rho, (r)_t)$$ and for $a' \in {\mathcal{O}}'$ set $$\tau(a')=\oplus_{\Delta' \in {\mathcal{O}}'} \tau(\Delta')$$ where for $([\rho_1],f_1),\dots,([\rho_m],f_m) \in {\mathcal{G}}_F$ the direct sum is given by $$([\rho_1],f_1)\oplus\cdots \oplus ([\rho_m],f_m)=([ \rho_1 \oplus\cdots \oplus \rho_m],f_1+\cdots+f_m).$$ The reciprocity map ${\operatorname{rec}}_F$ is given by $${\operatorname{rec}}_F({\langle{a}\rangle}^t)=\tau({\operatorname{rec}}_F^\circ(a)),\ a \in {\mathcal{O}}.$$ The $SL(2)$-type of a representation {#sec: SL(2)-type} ==================================== Denote by ${\mathcal{A}}^u_F(n)$ the subset of ${\mathcal{A}}_F(n)$ consisting of all isomorphism classes of unitarizable representations and let ${\mathcal{A}}_F^u=\cup_{n=1}^\infty {\mathcal{A}}_F(n).$ For $[\pi_1],\dots,[\pi_m] \in {\mathcal{A}}_F$ we denote by $\pi_1 \times \cdots \times \pi_m$ the representation parabolically induced from $\pi_1 \otimes \cdots \otimes \pi_m$ and by $[\pi_1] \times \cdots \times [\pi_m]$ its isomorphism class. For $\sigma \in {\mathcal{A}}_F^\circ$ and integers $n,\,r >0$ let $$\delta[\sigma,n]={\langle{\Delta[\sigma,n]}\rangle}^t,$$ $$a(\sigma,n,r)=\{\Delta[\sigma[\frac{1-r}{2}],n],\Delta[\sigma[\frac{3-r}{2}],n], \cdots,\Delta[\sigma,n](\frac{r-1}{2})\}\in {\mathcal{O}}$$ and $$U(\delta[\sigma,n],r)={\langle{a(\sigma,n,r)}\rangle}.$$ Tadic’s classification of the unitary dual of $GL_n(F)$ [@MR870688] implies that if $\sigma \in {\mathcal{A}}_F^\circ \cap {\mathcal{A}}_F^u$ then $U(\delta[\sigma,n],r) \in {\mathcal{A}}_F^u$ and that for any $\pi\in {\mathcal{A}}_F^u$ there exist $\sigma_1,\dots,\sigma_m \in {\mathcal{A}}_F^\circ$ and integers $n_1,\dots,n_m,\,r_1,\dots,r_m >0$ such that $$\label{eq: general unitary} \pi=U(\delta[\sigma_1,n_1],r_1)\times \cdots \times U(\delta[\sigma_m,n_m],r_m).$$ It further follows from [@MR1359141 Lemma 3.3] that $$\label{eq: transpose of U} U(\delta[\sigma,n],r)^t=U(\delta[\sigma,r],n).$$ The $SL(2)$ of a representation $\pi\in {\mathcal{A}}_F^u$ of the form is defined in [@MR2133760 Definition 1] to be the partition $$\label{eq: explicit SL(2)-type} \{(r_1)_{n_1},\dots,(r_m)_{n_m}\}.$$ \[thm: SL(2)-type\] The $SL(2)$-type of a representation $\pi\in {\mathcal{A}}_F^u$ equals $P_F({\operatorname{rec}}_F(\pi^t)).$ \[rmk: SL(2)-type\] Theorem \[thm: SL(2)-type\] allows us to define the $SL(2)$-type of any $\pi \in {\mathcal{A}}_F$ by the formula $P_F({\operatorname{rec}}_F(\pi^t)).$ Note further that given a reciprocity map (local Langlands conjecture), this provides a recipe to define the $SL(2)$-type of an irreducible representation for any reductive group! Based on Tadic’s classification of the unitary dual of $GL_n(F),$ the proof of Theorem \[thm: SL(2)-type\] is merely a matter of following the definitions. For convenience, we provide the proof. The key is in the following simple observations. \[lemma: rec of unitary\] Let $\pi\in {\mathcal{A}}_F^u$ be of the form . Then $$\label{eq: rec of unitary} {\operatorname{rec}}_F(\pi)=\oplus_{i=1}^m \oplus_{j=1}^{r_i} \tau(\Delta[\sigma_i[\frac{r_i+1}2-j],n_i])$$ and $$\label{eq: transpose of unitary} \pi^t=U(\delta[\sigma_1,r_1],n_1)\times \cdots \times U(\delta[\sigma_m,r_m],n_m)\in {\mathcal{A}}_F^u.$$ Let $a_i=a(\sigma_i,r_i,n_i).$ It follows from that $$\pi={\langle{a_1}\rangle}^t \times \cdots \times {\langle{a_m}\rangle}^t= ({\langle{a_1}\rangle} \times \cdots \times {\langle{a_m}\rangle})^t$$ and since $t$ is an involution on ${\mathcal{A}}_F$ that ${\langle{a_1}\rangle} \times \cdots \times {\langle{a_m}\rangle} \in {\mathcal{A}}_F.$ Thus, it follows from [@MR584084 Proposition 8.4] that ${\langle{a_1}\rangle} \times \cdots \times {\langle{a_m}\rangle}={\langle{a_1+\cdots+a_m}\rangle}.$ In other words $\pi={\langle{a_1+\cdots+a_m}\rangle}^t$ and therefore by definition $${\operatorname{rec}}_F(\pi)=\tau({\operatorname{rec}}_F^\circ(a_1+\cdots+a_m))=\oplus_{i=1}^m \tau({\operatorname{rec}}_F^\circ(a_i)).$$ The identity now follows from the definition of $\tau({\operatorname{rec}}_F^\circ(a_i)).$ Note that implies that $$\pi^t=U(\delta[\sigma_1,r_1],n_1)\times \cdots \times U(\delta[\sigma_m,r_m],n_m)$$ and the classification of Tadic therefore implies that $\pi^t \in {\mathcal{A}}_F^u.$ Thus we get . Applying to $\pi^t$ and comparing with Theorem \[thm: SL(2)-type\] follows from the definitions. From now on for every $\pi \in {\mathcal{A}}_F$ we denote by $$\label{eq: def of vv} {\mathcal{V}}(\pi)=P_F({\operatorname{rec}}_F(\pi^t))$$ the $SL(2)$-type of $\pi.$ Klyachko models {#sec: Klyachko} =============== For positive integers $r$ and $k$ denote by $U_r$ the subgroup of upper triangular unipotent matrices in $GL_r(F)$ and by $Sp_{2k}(F)$ the symplectic group in $GL_{2k}(F).$ Fix a decomposition $n=r+2k.$ Let $$H_{r,2k} =\{ \left( \begin{array}{cc} u & X \\ 0 & h \\ \end{array} \right): u \in U_r,\, X \in M_{r \times 2k}(F),\,h \in Sp_{2k}(F)\}.$$ Let $\psi$ be a non trivial character of $F.$ For $u=(u_{i,j}) \in U_r$ let $$\psi_r(u)=\psi(u_{1,2}+\cdots+u_{r-1, r})$$ and let $\psi_{r,2k}$ be the character of $H_{r,2k}$ defined by $$\psi_{r,2k} \left( \begin{array}{cc} u & X \\ 0 & h \\ \end{array} \right)=\psi_r(u).$$ We refer to the space $${\mathcal{M}}_{r,2k}=\mbox{Ind}_{H_{r,2k}}^{GL_{n}(F)}(\psi_{r,2k})$$ as a *Klyachko model* for $GL_n(F).$ Here ${\operatorname{Ind}}$ denotes the functor of non-compact smooth induction. In [@OS3 Corollary 1] we showed that for any $\pi \in {\mathcal{A}}_F^u(n)$ there exists a unique decomposition $$n=r(\pi)+2k(\pi)$$ such that $${\operatorname{Hom}}_{GL_n(F)}(\pi,{\mathcal{M}}_{r(\pi),2k(\pi)})\ne 0$$ and that in fact $\dim_{\mathbb{C}}({\operatorname{Hom}}_{GL_n(F)}(\pi,{\mathcal{M}}_{r(\pi),2k(\pi)}))=1.$ For $\pi \in {\mathcal{A}}_F^u,$ the *Klyachko type* of $\pi$ is the ordered pair $(r(\pi),2k(\pi)).$ In fact, for ${\mathcal{A}}_F^u$ [@os Theorem 8] provides a receipt in order to read the Klyachko type off Tadic’s classification. Based on , our results can be reinterpreted by the formula $$\label{eq: kly in terms of SL(2)-type} r(\pi)={\operatorname{odd}}({\mathcal{V}}(\pi)),\ \pi \in {\mathcal{A}}_F^u.$$ Base change-The main results {#sec: statements} ============================ Let $E$ be a finite extension of $F.$ Denote by ${\operatorname{res}}_{E/F,n}:{\mathcal{G}}_F(n)\to {\mathcal{G}}_E(n)$ the map defined by ${\operatorname{res}}_{E/F,n}(([\rho],f))=([\rho_{|W_E}],f).$ For $n\ge 1$ the *base change* ${\operatorname{bc}}_{E/F}(\pi)\in {\mathcal{A}}_E(n)$ of $\pi \in {\mathcal{A}}_F(n)$ is defined by $${\operatorname{rec}}_E({\operatorname{bc}}_{E/F}(\pi))={\operatorname{res}}_{E/F}({\operatorname{rec}}_F(\pi)).$$ \[thm: main\] Let $E/F$ be a finite extension of $p$-adic fields and let $\pi$ be a smooth, irreducible and unitarizable representation of $GL_n(F).$ Then ${\operatorname{bc}}_{E/F}(\pi)$ is a smooth, irreducible and unitarizable representation of $GL_n(E)$ and $${\mathcal{V}}(\pi)={\mathcal{V}}({\operatorname{bc}}_{E/F}(\pi)),$$ i.e. $\pi$ and ${\operatorname{bc}}_{E/F}(\pi)$ have the same $SL(2)$-type. As a consequence we have the following. \[cor: main\] Under the assumptions of Theorem \[thm: main\] we have $$r(\pi)=r({\operatorname{bc}}_{E/F}(\pi)),$$ i.e. $\pi$ and ${\operatorname{bc}}_{E/F}(\pi)$ have the same Klyachko type. Corollary \[cor: main\] is straightforward from Theorem \[thm: main\] and . Proof of the main result {#sec: proofs} ======================== \[lemma: bc for cuspidal\] Let $E/F$ be a finite extension. For $\sigma \in {\mathcal{A}}_F^\circ\cap {\mathcal{A}}_F^u$ there exist $\sigma_1,\dots,\sigma_m \in {\mathcal{A}}_E^\circ\cap {\mathcal{A}}_E^u$ such that $${\operatorname{bc}}_{E/F}(\sigma)=\sigma_1\times \cdots \times \sigma_m.$$ Recall that a representation in $ {\mathcal{A}}_F^\circ$ is unitarizable if and only if its central character is unitary. Let $\rho$ be the irreducible representation of $W_F$ such that ${\operatorname{rec}}_F(\sigma)=([\rho],{{\bf 1}}_n).$ It follows from that $\rho$ has a unitary central character and therefore it has a unitary structure. Thus, the restriction $\rho_{|W_E}$ to $W_E$ also has a unitary structure and therefore each of its irreducible componencts has a unitary central character. The lemma follows by applying to ${\operatorname{res}}_{E/F}({\operatorname{rec}}_F(\sigma))$. \[prop: bc of Zel inv\] Let $E/F$ be a finite extension and let $\pi\in {\mathcal{A}}_F^u$ then ${\operatorname{bc}}(\pi)\in {\mathcal{A}}_E^u$ and $$\label{eq: bc and transpose commute} {\operatorname{bc}}_{E/F}(\pi^t)={\operatorname{bc}}_{E/F}(\pi)^t.$$ Let $\pi \in {\mathcal{A}}_F^u$ be of the form . By Lemma \[lemma: bc for cuspidal\] there exist $\sigma_{i,k}\in {\mathcal{A}}_E^\circ,\,i=1,\dots,m,\,k=1,\dots,t_i$ such that $${\operatorname{bc}}_{E/F}(\sigma_i)=\sigma_{i,1}\times \cdots \times \sigma_{i,t_i}.$$ Let $\rho_i={\operatorname{rec}}_F^\circ(\sigma_i)$ and $\rho_{i,k}={\operatorname{rec}}_E^\circ(\sigma_{i,k}).$ Thus, $${\operatorname{res}}_{E/F}(\rho_i)=\oplus_{k=1}^{t_i}\rho_{i,k}.$$ It follows from that $$\label{eq: res of rec} {\operatorname{res}}_{E/F}({\operatorname{rec}}_F(\pi))=\oplus_{i=1}^m \oplus_{j=1}^{r_i}\oplus_{k=1}^{t_i} \tau(\Delta[\sigma_{i,k}[\frac{r_i+1}2-j],n_i]).$$ On the other hand, let $$\Pi=\times_{i=1}^m \times_{k=1}^{t_i}U(\delta[\sigma_{i,k},n_i],r_i)$$ Since $\pi\in {\mathcal{A}}_F^u,$ the classification of Tadic implies that $\Pi \in {\mathcal{A}}_E^u$ and by applied to $E$ instead of $F$ we have $$\label{eq: rec_E id} {\operatorname{rec}}_E(\Pi)=\oplus_{i=1}^m \oplus_{j=1}^{r_i}\oplus_{k=1}^{t_i} \tau(\Delta[\sigma_{i,k}[\frac{r_i+1}2-j],n_i]).$$ Comparing with we obtain that $\Pi={\operatorname{bc}}_{E/F}(\pi)$ and in particular that ${\operatorname{bc}}_{E/F}(\pi) \in{\mathcal{A}}_E^u.$ Applying this to $\pi^t$ expressed by gives $${\operatorname{bc}}_{E/F}(\pi^t)=\times_{i=1}^m \times_{k=1}^{t_i}U(\delta[\sigma_{i,k},r_i],n_i).$$ Applying now to ${\operatorname{bc}}_{E/F}(\pi)^t$ we obtain the identity . It is straightforward from the definitions that $$\label{eq: trivial id} P_F({\operatorname{rec}}_F(\pi))=P_E({\operatorname{rec}}_E({\operatorname{bc}}_{E/F}(\pi)),\ \pi \in {\mathcal{A}}_F.$$ For $\pi \in {\mathcal{A}}_F^u,$ applying to $\pi^t$ and then we get that $$P_F({\operatorname{rec}}_F(\pi^t))=P_E({\operatorname{rec}}_E({\operatorname{bc}}_{E/F}(\pi)^t).$$ The identity ${\mathcal{V}}(\pi)={\mathcal{V}}({\operatorname{bc}}_{E/F}(\pi))$ is now immediate from . This completes the proof of Theorem \[thm: main\]. [OS07b]{} Anne-Marie Aubert. Dualité dans le groupe de [G]{}rothendieck de la catégorie des représentations lisses de longueur finie d’un groupe réductif [$p$]{}-adique. , 347(6):2179–2189, 1995. Anne-Marie Aubert. Erratum: “[D]{}uality in the [G]{}rothendieck group of the category of finite-length smooth representations of a [$p$]{}-adic reductive group” \[[T]{}rans. [A]{}mer. [M]{}ath. [S]{}oc. [**347**]{} (1995), no. 6, 2179–2189; [@MR1285969]\]. , 348(11):4687–4690, 1996. Guy Henniart. Le point sur la conjecture de [L]{}anglands pour [${\rm GL}(N)$]{} sur un corps local. In [*Séminaire de théorie des nombres, Paris 1983–84*]{}, volume 59 of [*Progr. Math.*]{}, pages 115–131. Birkhäuser Boston, Boston, MA, 1985. Guy Henniart. Une preuve simple des conjectures de [L]{}anglands pour [${\rm GL}(n)$]{} sur un corps [$p$]{}-adique. , 139(2):439–455, 2000. Michael J. Heumos and Stephen Rallis. Symplectic-[W]{}hittaker models for [${\rm Gl}\sb n$]{}. , 146(2):247–279, 1990. Michael Harris and Richard Taylor. , volume 151 of [*Annals of Mathematics Studies*]{}. Princeton University Press, Princeton, NJ, 2001. With an appendix by Vladimir G. Berkovich. Omer Offen and Eitan Sayag. Global mixed periods and local klyachko models for the general linear group . , 2007:article ID rnm136, 25 pages, doi:10.1093/imrn/rnm136, 2007. Omer Offen and Eitan Sayag. On unitary representations of [${\rm GL}\sb {2n}$]{} distinguished by the symplectic group. , 125(2):344–355, 2007. Omer Offen and Eitan Sayag. Uniqueness and disjointness of klyachko models . , 254:2846–2865, 2008. Marko Tadi[ć]{}. Classification of unitary representations in irreducible representations of general linear group (non-[A]{}rchimedean case). , 19(3):335–382, 1986. M. Tadi[ć]{}. On characters of irreducible unitary representations of general linear groups. , 65:341–363, 1995. Akshay Venkatesh. The [B]{}urger-[S]{}arnak method and operations on the unitary dual of [${\rm GL}(n)$]{}. , 9:268–286 (electronic), 2005. A. V. Zelevinsky. Induced representations of reductive [${\frak{p}}$]{}-adic groups. [II]{}. [O]{}n irreducible representations of [${\rm GL}(n)$]{}. , 13(2):165–210, 1980.

--- abstract: | A gambler moves on the vertices $1, \ldots, n$ of a graph using the probability distribution $p_{1}, \ldots, p_{n}$. A cop pursues the gambler on the graph, only being able to move between adjacent vertices. What is the expected number of moves that the gambler can make until the cop catches them? Komarov and Winkler proved an upper bound of approximately $1.97n$ for the expected capture time on any connected $n$-vertex graph when the cop does not know the gambler’s distribution. We improve this upper bound to approximately $1.95n$ by modifying the cop’s pursuit algorithm. author: - | Jesse Geneson\ geneson@gmail.com title: 'An anti-incursion algorithm for unknown probabilistic adversaries on connected graphs' --- Introduction ============ Games with cops and robbers on graphs, which can be applied for designing anti-incursion programs, have been studied for several decades [@T1; @T2; @T3; @T4; @NW; @Q]. We investigate a version of the game where the adversary moves among the vertices $1, \ldots, n$ following a probability distribution $p_{1}, \ldots, p_{n}$. Before the game starts, the cop picks and occupies a vertex from $G$. In each round of the game, the cop selects and moves to an adjacent vertex or stays at the same vertex. The gambler chooses to occupy a vertex randomly based on a time-independent distribution, not restricted to only adjacent vertices. Whenever both players occupy the same vertex at the same time, the cop wins. The gambler is called a known gambler if the cop knows their probability distribution. Otherwise the gambler is called unknown. Gambler-pursuit games model anti-incursion programs navigating a linked list of ports, trying to minimize interception time for enemy packets. Komarov and Winkler proved that the expected capture time on any connected $n$-vertex graph is exactly $n$ for a known gambler [@KW], assuming that both players use optimal strategies. For an unknown gambler, Komarov and Winkler proved an upper bound of approximately $1.97n$ [@KW]. Komarov and Winkler conjectured that the general upper bound for the unknown gambler on a connected $n$-vertex graph can be improved from about $1.97n$ to $3n/2$, and that the star is the worst case for this bound. In Sections \[unkn\] and \[unkn1\], we improve the upper bound for the unknown gambler’s expected capture time to approximately $1.95n$ by using a different strategy for the cop. Unknown Gambler Pursuit Algorithm {#unkn} ================================= Let $G$ be a connected $n$-vertex graph. As in [@KW], let $T$ be a spanning subtree of $G$. We describe the cop’s pursuit algorithm, and then we prove an upper bound of approximately $1.95n$ on the expected capture time. Suppose that the cop performs a depth first search of $T$, except the cop stays at some leaves for two turns instead of one. Specifically, the cop uniformly at random selects a subset $U$ of ${\lceil0.72912 n\rceil}$ vertices and stays at the vertices in $U$ for an extra turn if the vertices are leaves. If there is a vertex $v$ in $U$ that is not a leaf, then the depth first search would already go twice through $v$, so the cop does not need to stay an extra turn at $v$. After the proof, we explain the reason for using the number $0.72912$. The cop flips a coin to decide whether to perform the depth first search forward or backward. Thus the total number of turns in a single depth first search (including the extra turns for the leaves in $U$) is at most $1+2(n-1)+{\lceil0.72912 n\rceil} \leq 2.72912 n$. The search is repeated until capture. Since the cop flips a coin to decide whether to search forward or backward, the expected number of turns in the successful depth first search is at most $1.36456n$. Analysis {#unkn1} ======== Let the vertices of the graph be named $1, \ldots, n$. Suppose that the unknown gambler chooses vertex $i$ with probability $p_{i}$. We split the proof into two cases to show that the probability of evasion in a single depth first search is at most $0.17745$. If there are two vertices $i$ and $j$ that the cop visits at least twice each such that $p_{i}+p_{j} \geq 0.732$, then the probability of evasion in a single depth first search is less than $0.162$. If the cop visits $i$ and $j$ both at least twice, then the probability of evasion is at most $(1-p_{i})^{2}(1-p_{j})^{2} \leq (1-p_{i})^{2}(0.268+p_{i})^{2}$, which has a maximum value of approximately $0.16157$ on the interval $[0,1]$ at $p_{i} = 0.366$. Next we show that the probability of evasion is at most $e^{-1.72912} < 0.17745$ when there are no vertices $i$ and $j$ that the cop visits at least twice each such that $p_{i}+p_{j} \geq 0.732$. \[didj\] Suppose that there are no vertices $x$ and $y$ that the cop visits at least twice each such that $p_{x}+p_{y} \geq 0.732$. Then the probability of evasion for a single depth first search is at most $(1-\frac{1}{n})^{1.72912 n}$. Let $i, j$ be any two vertices of $G$ and suppose that $p_{i}+p_{j} = a$ and let $t_{1}, \ldots, t_{n-2}$ denote the vertices of $G$ not equal to $i$ or $j$. Given the condition that there are no vertices $x$ and $y$ that the cop visits at least twice each such that $p_{x}+p_{y} \geq 0.732$, then the probability of evasion for a single depth first search can be bounded by performing the following reduction to obtain shorter searches called $S$ and $S'$. First we define $S'$. If the cop visits a vertex $v$ not in $U$ more than once in the original depth first search, skip the cop’s visits to $v$ after the first visit to $v$ in $S'$; if the cop visits a vertex $v$ in $U$ more than twice in the original depth first search, skip the cop’s visits to $v$ after the second visit to $v$ in $S'$. Note that with $S'$, vertices in $U$ are visited exactly twice, and vertices not in $U$ are visited exactly once. The number of turns in $S'$ is thus $n+{\lceil0.72912n\rceil}={\lceil1.72912n\rceil}$. To obtain $S$ from $S'$, skip all visits to vertices $i$ and $j$. Note that the reduction can only increase the probably of evasion, so the probability of evasion for the original depth first search is at most the probability of evasion for $S'$, which is at most the probability of evasion for $S$. Note also that the searches $S$ or $S'$ could be impossible for the cop to perform, since consecutive vertices in $S$ or $S'$ might not be adjacent. The searches $S$ and $S'$ are only used in this proof to obtain an upper bound on the probability of evasion for the original depth first search. For $c, d \in \left\{1,2\right\}$, define $f_{c,d}(p_{t_{1}}, \ldots, p_{t_{n-2}})$ to be the probability that the gambler evades the cop in search $S$ and that the cop makes $c$ visits to vertex $i$ and $d$ visits to vertex $j$ in search $S'$, conditioned on the fact that there are no vertices $x$ and $y$ that the cop visits at least twice each such that $p_{x}+p_{y} \geq 0.732$ in the original depth first search. Then the probability of evasion in search $S'$ is $p = p(p_{i},p_{j},p_{t_{1}},\ldots,p_{t_{n-2}})$ of the form $(1-p_{i})(1-a+p_{i}) f_{1,1}(p_{t_{1}}, \ldots, p_{t_{n-2}})+(1-p_{i})^{2}(1-a+p_{i}) f_{2,1}(p_{t_{1}}, \ldots, p_{t_{n-2}})+(1-p_{i})(1-a+p_{i})^{2} f_{1,2}(p_{t_{1}}, \ldots, p_{t_{n-2}})+(1-p_{i})^{2}(1-a+p_{i})^{2} f_{2,2}(p_{t_{1}}, \ldots, p_{t_{n-2}})$. Note that $f_{1,2} = f_{2,1}$ by symmetry and $p(\frac{1}{n},\ldots,\frac{1}{n}) = (1-\frac{1}{n})^{{\lceil1.72912 n\rceil}}$. Then $\frac{d p}{d p_{i}} = (a-2p_{i})f_{1,1}(p_{t_{1}}, \ldots, p_{t_{n-2}})+(a-2)(2p_{i}-a)f_{1,2}(p_{t_{1}}, \ldots, p_{t_{n-2}})+2(1-p_{i})(1-a+p_{i})(a-2p_{i})f_{2,2}(p_{t_{1}}, \ldots, p_{t_{n-2}})$, which is equal to zero at $p_{i} = \frac{a}{2}$. Moreover $\frac{d^{2} p}{d p_{i}^{2}} = -2 f_{1,1}(p_{t_{1}}, \ldots, p_{t_{n-2}})+2(a-2)f_{1,2}(p_{t_{1}}, \ldots, p_{t_{n-2}})+(12(p_{i}-\frac{a}{2})^{2}-(a-2)^{2})f_{2,2}(p_{t_{1}}, \ldots, p_{t_{n-2}})$, which is at most $-2 f_{1,1}(p_{t_{1}}, \ldots, p_{t_{n-2}})+2(a-2)f_{1,2}(p_{t_{1}}, \ldots, p_{t_{n-2}})+(2a^{2}+4a-4)f_{2,2}(p_{t_{1}}, \ldots, p_{t_{n-2}})$. If $a \geq 0.732$, then $f_{2,2}(p_{t_{1}}, \ldots, p_{t_{n-2}}) = 0$ since there are no vertices $x$ and $y$ that the cop visits at least twice each such that $p_{x}+p_{y} \geq 0.732$. Thus $\frac{d^{2} p}{d p_{i}^{2}} = -2 f_{1,1}(p_{t_{1}}, \ldots, p_{t_{n-2}})+2(a-2)f_{1,2}(p_{t_{1}}, \ldots, p_{t_{n-2}}) \leq 0$ for all $p_{i} \in [0,a]$, so $p$ is maximized when $p_{i} = \frac{a}{2}$. If $a \leq 0.732$, then $2a^{2}+4a-4 < 0$. Thus $\frac{d^{2} p}{d p_{i}^{2}} \leq 0$ for all $p_{i} \in [0,a]$, so $p$ is maximized when $p_{i} = \frac{a}{2}$. This implies that $p$ is maximized only when $p_i=p_j$ for any two vertices $i$ and $j$ of $G$. Thus if there are no vertices $x$ and $y$ that the cop visits at least twice each such that $p_{x}+p_{y} \geq 0.732$, then $p$ is maximized when $p_{i} = \frac{1}{n}$ for all $i$, so the probability of evasion is at most $(1-\frac{1}{n})^{{\lceil1.72912 n\rceil}} \leq (1-\frac{1}{n})^{1.72912 n}$. By the last lemma, the probability of evasion for a single depth first search is at most $(1-\frac{1}{n})^{1.72912 n} \leq e^{-1.72912} < 0.17745$. Thus the expected number of depth first searches until the cop catches the robber is at most $\frac{1}{1-0.17745} < 1.21574$. Let $X$ denote the random variable for the total number of turns in all of the cop’s depth first searches, not including the successful depth first search. Let $Y$ denote the random variable for the number of turns in the successful search. By linearity of expectation, the expected capture time is equal to $E(X)+E(Y)$. We proved above that $E(X) \leq (2.72912 n)(1.21574-1) < 0.58879n$ and $E(Y) \leq 1.36456n$. Thus the upper bound on the expected capture time is less than $1.95335 n$. Comments ======== The reason why we chose $0.72912$ for the constant in the pursuit algorithm was because the function $\frac{x}{1-e^{1-x}}-\frac{x}{2}$ has a minimum value of $1.95328$ on the interval $(1,\infty)$ at $x = 2.72912$. Despite this fact, it seems likely that our upper bound is not tight. The best current bounds on the maximum possible expected capture time for any connected $n$-vertex graph are between approximately $1.082n$ and $1.953n$. The lower bound follows from Komarov’s proof for the cycle $C_{n}$ [@KT]. However, there are a few families of graphs for which there are already tighter bounds. Komarov and Winkler proved an upper bound of approximately $1.082n$ for the expected capture time on the cycle $C_{n}$ [@KW] to match the lower bound from [@KT]. It is also easy to show that the expected capture time for the path $P_{n}$ is between $1.082n$ and $1.313n$ using the method in [@KW]. Acknowledgments =============== The author thanks Shen-Fu Tsai for helpful questions and suggestions on the exposition of this paper. [7]{} A. Berarducci and B. Intrigila, On the cop number of a graph, Adv. Appl. Math. 14 (1993), 389-403. A. Bonato, P. Golovach, G. Hahn, and J. Kratochvil, The capture time of a graph, Discrete Math. 309 (2009), 5588-5595. G. Hahn, F. Laviolette, N. Sauer, and R.E. Woodrow, On cop-win graphs, Discrete Math. 258 (2002), 27-41. G. Hahn and G. MacGillivray, A note on k-cop, l robber games on graphs, Discrete Math. 306 (2006), 2492-2497. N. Komarov and P. Winkler, Cop vs. Gambler, Discrete Math. 339 (2016), 1677-1681. N. Komarov, Capture time in variants of cops and robbers games, Ph.D. thesis, Dartmouth College, 2013. R. Nowakowski and P. Winkler, Vertex to vertex pursuit in a graph, Discrete Math. 43 (1983), 235-239. A. Quilliot, Homomorphismes, points fixes, rétractations et jeux de poursuite dans les graphes, les ensembles ordonnés et les espaces métriques, Ph.D. thesis, Université de Paris VI., 1983.

--- bibliography: - 'BrainrefRMTGFP.bib' --- =1 [**Organization and hierarchy of the human functional brain network lead to a chain-like core** ]{} Rossana Mastrandrea$^{*1}$, Andrea Gabrielli$^{1,2}$, Fabrizio Piras$^{3,4}$, Gianfranco Spalletta$^{4,5}$, Guido Caldarelli$^{1,2}$ Tommaso Gili$^{3,4}$\ **[1]{} IMT School for Advanced Studies, Lucca, piazza S. Ponziano 6, 55100 Lucca, Italy\ **[2]{} Istituto dei Sistemi Complessi (ISC) - CNR, UoS Sapienza, Dipartimento di Fisica, Universitá Sapienza; P.le Aldo Moro 5, 00185 - Rome, Italy\ **[3]{} Enrico Fermi Center, Piazza del Viminale 1, 00184 Rome, Italy\ **[4]{} IRCCS Fondazione Santa Lucia, Via Ardeatina 305, 00179 Rome, Italy\ **[5]{} Menninger Department of Psychiatry and Behavioral Sciences, Baylor College of Medicine, Houston, Tx, USA\ $*$ E-mail: rossana.mastrandrea@imtlucca.it\ ********** Abstract {#abstract .unnumbered} ======== The brain is a paradigmatic example of a complex system as its functionality emerges as a global property of local mesoscopic and microscopic interactions. Complex network theory allows to elicit the functional architecture of the brain in terms of links (correlations) between nodes (grey matter regions) and to extract information out of the noise. Here we present the analysis of functional magnetic resonance imaging data from forty healthy humans during the resting condition for the investigation of the basal scaffold of the functional brain network organization. We show how brain regions tend to coordinate by forming a highly hierarchical chain-like structure of homogeneously clustered anatomical areas. A maximum spanning tree approach revealed the centrality of the occipital cortex and the peculiar aggregation of cerebellar regions to form a closed core. We also report the hierarchy of network segregation and the level of clusters integration as a function of the connectivity strength between brain regions. Introduction {#introduction .unnumbered} ============ The intrinsic functional architecture of the brain and its changes due to cognitive engagement, ageing and diseases are nodal topics in neuroscience, attracting considerable attention from many disciplines of scientific investigation. Complex network theory [@caldarelli2007scale; @barabasi2009scale; @newman2010networks] provides tools representing the state-of-the-art of multivariate analysis of local cortical and subcortical mesoscopic interactions. The new amount of data available from a variety of different sources showed clearly that the complex network description of both the structural and the functional organization of the brain demonstrates a repertoire of unexpected properties of brain connectomics [@bullmore2009complex]. Accordingly network theory allows a description of brain architecture in terms of patterns of communication between brain regions, treated as evolving networks and associate this evolution to behavioral outcomes [@bassett2011dynamic]. Brain networks are characterized by a balance of dense relationships among areas highly engaged in processing roles, as well as sparser relationships between regions belonging to systems with different processing roles. This segregation facilitates communication among brain areas that may be distributed anatomically but are needed for sets of processing operations [@sporns2013network]. Along with that the integrated functional organization of the brain involves each network component executing a discrete cognitive function mostly autonomously or with the support of other components where the computational load in one is not heavily influenced by processing in the others [@bertolero2015modular]. In this quest for a constantly improving quantitative description of the brain and of the cardinal features of its functioning complex networks plays a crucial role [@rubinov2010complex; @deco2011emerging; @van2011rich; @sporns2012simple]. Specifically, it proved to be able to elicit both the scaffold of the mutual interactions among different areas in healthy brains [@bullmore2009complex; @bassett2016small] and the local failure of the global functioning in diseased brains [@Bassett2009; @rosazza2011resting; @aerts2016brain]. Network representation describes the brain as a graph with a set of nodes - a variable number of brain areas (from $10\sp{2}$ to $10\sp{4}$) - connected with links representing functional, structural or effective interactions. The use of complex network theory passed progressively from an initial assessment of basic topological properties [@salvador2005neurophysiological; @van2008small; @telesford2010reproducibility] to a more sophisticated description of global features of the brain, as small-worldness [@achard2006resilient], rich-club organization [@van2011rich] and topology [@petri2014homological; @tiz2016]. However, a clear understanding of the functional advantage of a network organization in the brain, the characterization of its substrate and a description of the network structure as a function of the level of brain regions interaction are still missing. In this paper, we investigate resting state functional networks, where links represent the strength of correlation between time series of spontaneous neural activity as measured by blood oxygen-level-dependent (BOLD) functional MRI (fMRI) [@eguiluz2005]. Specifically we interpret a functional connection between two nodes as the magnitude of the synchrony of their low-frequency oscillations, which is associated with the modulation of activity in one node by another one [@wang2012electrophysiological; @honey2012slow] largely constrained by anatomical connectivity [@hagmann2008mapping; @honey2009predicting]. A percolation analysis of the functional network [@gallos2012small; @tiz2016] is used to highlight the progressive engagement of brain regions in the whole network as a function of the connectivity strength. Subsequently, by means of the maximum spanning forest (MSF) and the maximum spanning tree (MST) representations [@caldarelli2007scale; @caldarelli2012network] we obtain the basal scaffold of the brain network, that shows to be characterized by a linear backbone in which few nodes (cerebellar and occipital regions) play a central role, even progressively increasing the spatial resolution or reducing nodes size. Results {#results .unnumbered} ======= Percolation Analysis {#percolation-analysis .unnumbered} -------------------- The representative human functional brain network is often analyzed introducing specific thresholds to map the fully-connected correlation matrix in a sparse binary matrix [@bullmore2009complex]. Here, to avoid any arbitrary assumption we perform a percolation analysis [@gallos2012small; @tiz2016] on the whole network. We rank correlations in increasing order, one at a time we remove from the network the link corresponding to the observed value and explore the global organization of the remaining network. Specifically, in fig.\[perc\] (a) we show the number of connected components updated each time a new link is removed. The emergence and the number of *plateaux* shed light on the hierarchical structure of the network, their length unveils the intrinsic stability of certain network configurations after the removal of links. For the human functional brain network a remarkable hierarchy in the disaggregation process emerges from the comparison with a proper null model. The same percolation analysis is performed on the ensemble of 100 randomizations (Methods) of the observed correlation matrix showing a faster disaggregation in disconnected components with plateaux absent or of small length. We compute the distribution of plateaux length looking at the increment of correlation values when the network passes from $n$ to $n+1$ components and show it in fig.\[perc\] (b). In the same figure we also report the average plateaux length computed on the ensemble of randomizations. A great variability characterizes the percolation curve of the real network with significant deviations of the plateaux length from the random case. Chain-like modules {#chain-like-modules .unnumbered} ================== The percolation analysis highlights the existence of a not trivial functional organization of the human brain network. Here, we investigate it considering a filtered network where for each area all weighted links but the strongest are discarded. This approach gives rise to 36 disconnected components forming a Maximum Spanning Forest (Methods) with a new information on link directionality. It simply indicates that each source points toward its maximally correlated brain area and it is not necessarily reciprocated. Figure \[MSF\] shows the abundance of modules with size 2 and reciprocated links: those are meanly mirror-areas of the right and left brain hemispheres or adjacent/very close ROIs belonging to the same hemisphere. Nodes are coulored according to the anatomical regions reported in figure \[MSF\] (a) (detailed names of ROIs in table 1 in SI). A noteworthy result concerns groups of size greater than 3 exhibiting a chain-like structure, sometimes very long as for the Cerebellum. This implies that most of the nodes in the MSF have in-degree[^1] equal to one, few equal to 2 and very few greater than 2. Furthermore, nodes tend to connect with nodes belonging to the same anatomical region. The only exception is represented by the Temporal Lobe: ROIs in this region are linked with all the other anatomical areas except for the Cerebellum and the Deep Grey Matter ones. We build the MSF of the 100 randomizations of the real network obtaining a number of components varying in the range $[12,24]$. All randomized correlation matrices exhibit a star-like organization of components in the MSF, while the number of modules of size 2 is dramatically reduced. Moreover, colors of linked nodes are completely random. In figure S1 we show the MSF performed on one of such randomizations. We compute the distribution of node in-degree of the observed MSF and of the ensemble of its randomizations (fig. 2(c)). In the real case the in-degree is small with almost the $70\%$ of values equal to 1. The ensemble of randomizations shows more variability and a left-skewed distribution of in-degree, with almost the $50\%$ of values equal to 0, the $30\%$ equal to 1 and a maximum of 15. The outcome confirms that the MSF of all the randomized correlation matrices shows similar features: star-like organization of modules with few in-degree hubs representing the strongest connected partner for several brain areas. This result highlights the fundamental hierarchical organization of brain functional areas. ### From forest to tree {#from-forest-to-tree .unnumbered} As next step, we link together all groups of nodes in fig.\[MSF\](b) such that (i) we add only one link between two modules previously disconnected and with the greatest weight; (ii) we end up with a unique connected component (Methods). The resulting network is a Maximum Spanning Tree as all nodes are connected through a maximal weighted path without forming loops. The MST in fig.\[Tree\](a) preserves the chain-like organization of nodes and mainly reproduces the anatomical division in regions. Some star-like aggregations emerge revealing the centrality of certain nodes as ROI 99, belonging to the Cerebellum and 82, part of the Temporal Lobe. Remarkable is the separation of the Deep Grey Matter, which represents the ancient part of the human brain, connecting to the periphery of the tree with the Temporal Lobe, the Insula and Cingulate areas. Nor chain-like organization neither relevant agglomeration of nodes are observable in the tree obtained from the MSF of the randomized correlation matrices (one examples in fig. S2). Also in this case the distribution of node degree characterizes the two kinds of network organization: star-like (null model) and chain-like (real brain network). The comparison between the real network and the ensemble of its randomization (fig. 3(b)) confirms the star-like organization of the random MST, with half of nodes having degree equal to 1 and maximum degree equal to 17. Hierarchical integration of brain areas {#hierarchical-integration-of-brain-areas .unnumbered} ---------------------------------------- The maximum spanning tree approach doesn’t provide any evidence about the internal connectivity of components observed in the MSF, as loops are not allowed. Here we showed a number of snapshots of the network associated with specific percolation phases. By selecting appropriate thresholds and discarding correlation values below them, we inspected the network formation process according to the strength of edges. The rationale behind this approach comes from the possibility to explore the level of segregation of brain areas and how they functionally and hierarchically integrate. Thresholds are chosen looking at figure \[perc\] (b) and considering only correlation values associated to significant deviations of plateaux length of the real network from the random case. Panels from (a) to (n) in fig.\[MSTsnap\] show the human functional brain network related to the aforementioned thresholds. As the threshold value decreases (from (a) to (n)), the number of links increases showing how groups of nodes appear and their between and within connections intensify. The sequence of snapshots reveal the hierarchical functional organization of the brain network. Nodes do not connect forming several large and disconnected components, but: (i) nodes belonging to the Occipital Lobe start to connect each other and the anatomical region is almost completely connected after few steps; (ii) some nodes belonging to the same region form small chains; (iii) several nodes directly connect to the Occipital Lobe. Therefore, the most central area is represented by the Occipital Lobe, while ROIs belonging to the Deep Grey Matter lie in the periphery connecting to the network at the last moment in the percolation phases. The Cerebellum forms a quite long chain before joining the Occipital Lobe through the ROI 99 already emerged in the MST for its centrality. [![[**Snapshots of the human functional brain network.**]{} Each snapshot is realized thresholding the correlation matrix. Thresholds are chosen looking at the most significant deviation of plateaux length of the real percolation curve from its ensemble of randomization (fig.\[perc\](b). ) Colors represent anatomical regions according to the grouping of AAL parcellation in fig.\[MSF\](a): $\textcolor{Blue}{\mathlarger{\mathlarger{\mathlarger{\bullet}}}}$ Frontal Lobe; $\textcolor{Orange}{\mathlarger{\mathlarger{\mathlarger{\bullet}}}}$ Insula; $\textcolor{Cyan}{\mathlarger{\mathlarger{\mathlarger{\bullet}}}}$ Cingulate; $\textcolor{LimeGreen}{\mathlarger{\mathlarger{\mathlarger{\bullet}}}}$ Temporal Lobe; $\textcolor{Red}{\mathlarger{\mathlarger{\mathlarger{\bullet}}}}$ Occipital Lobe; $\textcolor{Yellow}{\mathlarger{\mathlarger{\mathlarger{\bullet}}}}$ Parietal Lobe; $\textcolor{Gray}{\mathlarger{\mathlarger{\mathlarger{\bullet}}}}$ Deep Grey Matter; $\textcolor{VioletRed}{\mathlarger{\mathlarger{\mathlarger{\bullet}}}}$ Cerebellum. \[MSTsnap\]](Snap1v2.pdf "fig:"){width="28.00000%"}]{} [![[**Snapshots of the human functional brain network.**]{} Each snapshot is realized thresholding the correlation matrix. Thresholds are chosen looking at the most significant deviation of plateaux length of the real percolation curve from its ensemble of randomization (fig.\[perc\](b). ) Colors represent anatomical regions according to the grouping of AAL parcellation in fig.\[MSF\](a): $\textcolor{Blue}{\mathlarger{\mathlarger{\mathlarger{\bullet}}}}$ Frontal Lobe; $\textcolor{Orange}{\mathlarger{\mathlarger{\mathlarger{\bullet}}}}$ Insula; $\textcolor{Cyan}{\mathlarger{\mathlarger{\mathlarger{\bullet}}}}$ Cingulate; $\textcolor{LimeGreen}{\mathlarger{\mathlarger{\mathlarger{\bullet}}}}$ Temporal Lobe; $\textcolor{Red}{\mathlarger{\mathlarger{\mathlarger{\bullet}}}}$ Occipital Lobe; $\textcolor{Yellow}{\mathlarger{\mathlarger{\mathlarger{\bullet}}}}$ Parietal Lobe; $\textcolor{Gray}{\mathlarger{\mathlarger{\mathlarger{\bullet}}}}$ Deep Grey Matter; $\textcolor{VioletRed}{\mathlarger{\mathlarger{\mathlarger{\bullet}}}}$ Cerebellum. \[MSTsnap\]](Snap2.pdf "fig:"){width="30.00000%"}]{} Inter-subject variability {#inter-subject-variability .unnumbered} ========================= We explore the inter-subject variability with several approaches. We compute the standard deviation of pairwise correlation values across the 40 individuals and report it together with the average correlation matrix used for the analysis. Figure \[std\] (a) shows that there are few negative correlation values, however small in absolute term. The standard deviation ranges in the interval $[0.039, 0.55]$, with most of the highest values concerning correlation averages close to zero (Fig. \[std\] (b)). Those are few links which show positive and negative values across the 40 individuals and average around zero. They mainly belong to the Cerebellum (see SI) and the inter-subjects variability in this case could be due to the specific anatomical location with consequences on the resulting fMRI. However, they do not affect our results as in our analysis only the greatest correlation values play a key-role (percolation, MSF, MST), while the smallest threshold considered in the hierarchical integration of brain areas is 0.33. In figure \[std\] (c) we also report the coefficient of variation for the correlation matrix thresholded using the False Discovery Rate (FDR). Most of the values concentrate around zero, while the greatest ones correspond to correlation values close to zero. As next step, we compute the pairwise cosine similarity between the correlation matrices of subjects. We find high similarity between subjects (fig.S15 (a)), with few exceptions (mainly, subjects 35 and 38). However, their presence does not affect the average correlation matrix as explicitly shown by applying the Jackknife resampling technique. One at a time, we remove the correlation matrix corresponding to a subject from the sample and average the remaining 39 correlation matrices. Then, we compute the cosine similarity between the 40 resulting correlation matrices (fig.S15 (b)). The similarity values belong to $[0.9996, 0.9999]$, revealing that the emerged diversity for some subjects is negligible when we compute the average correlation matrix. Discussion {#discussion .unnumbered} ========== In this paper we have reported the results of a network theory-based investigation of a large human fMRI dataset, aimed at assessing the basal architecture of the resting state functional connectivity network. We have computed the Maximal Spanning Forest (MSF) and the Maximal Spanning Tree (MST) for our population-wise brain. The former showed a peculiar composition of the basic modules composing the network, by revealing their structure as linear chains. The MSF reveals both intra- and inter-hemispherical modules, and the presence of small modules alongside with larger sub-networks. The MST, on the other hand, enables the classification of connectors and provincial areas. We also used a modified percolation analysis that retains the information on all the connected components of a given network. Such variation of the percolation analysis does not require the application of a threshold to obtain binary connectivity networks. By means of this technique, applicable to experimental correlation matrices, we showed hierarchically the progressive integration of brain regions in the whole organization of the connectivity network, which does not appear in a null model defined by constraining the spectrum of the observed correlation matrix. The percolation analysis here used represents a generalization of the classical one [@gallos2012small] and is characterised by some improvements [@tiz2016] among which it must be recognized the ability to detect newly disconnected modules from secondary components. As a complex system, the human brain is intrinsically organized into modules, creating a high degree of flexibility and adaptability of the system without fundamentally altering underlying structure [@bassett2011understanding]. Remarkably, our percolation analysis unearthed the existence of a hierarchical organization of areas, clusters of areas and modules within the human functional brain network. The process of network aggregation from isolated components appeared slow - compared to the null-model - with the emergence of significant stable configurations identified by several plateaux in the percolation curve. This outcomeda suggests a progressive engagement of brain regions according to the local amount of connectivity, namely a network organization in hierarchical modules whose participation to the entire structure is progressively included as the links strength decreases. The sequence of stable network configurations that, from the single areas, leads to the fully connected network, can be achieved by choosing values associated with specific plateaux of the percolation curve. Although this is technically topological modularity it is remarkable that it reflects functional and anatomical features. For instance bilateral correspondence of regions belonging to the same module was obtained without the introduction of symmetry constraints imposed in the analyses, supporting the conclusion that a hallmark of large-scale resting-state networks is their inter-hemispheric symmetry [@smith2009correspondence]. Interestingly, descending the hierarchical staircase of the percolation we found that occipital regions as the cuneus, the lingual gyrus, the occipital superior and middle gyrus and the calcarine sulcus tended to form a densely interconnected module. Along with that, frontal, cerebellar and temporal regions resulted involved in different modules characterised by linear chains, which joined the occipital cluster, in a progressive integration process, through a nodal bridge represented by the precuneus. In particular, temporal and occipital lobes resulted connected thanks to the bridging action of the fusiform and the inferior occipital gyrus as expected by considering the local cytoarchitecture [@caspers2013cytoarchitectonical]. Thalamus, putamen, pallidus, hippocampus, amygdala and caudate (here refereed to as deep grey matter) was found to form binary bilateral modules that joined the whole structure at the end of the integration process. If on one hand the remarkable connectivity within the occipital lobe can be ascribed to an efficient coupling via the interplay of tangential intracortical and callosal connections [@gencc2015functional], on the other hand such connectedness could be ascribed to fixational eye movements [@martinez2004role]. In support to this observation, it must be noticed that in Parkinson’s disease patients, which present with eye movement disturbances that accompany the cardinal motor symptoms, the worse the oculomotor performance is, the more the regional functional connectivity is decreased [@gorges2016association]. A second striking result has been obtained from MSF and MST analysis of the whole brain network. For the first time we report a non-trivial and ordered structure for the basal scaffold of the functional network of the brain. Keeping track only of the strongest connection for each region, the network appeared organised in components exhibiting a linear chain-like structure. When the same procedure has been applied to a proper randomisation of the real human brain functional network, the outcome strongly differed: hubs and star-like structure emerged. This result was tested for robustness by improving the number of regions included in the grey matter parcellation. We calculated correlation matrices from two further templates composed of 276 and 531 regions of interest, respectively, each obtained as a sub-parcellation of the AAL template. The chain-like organization has been proved to persist as the topological outcome of the MSF and the MST analyses of the human brain functional network, against the randomised counterpart that showed a star-like organization (figures S3-S14 in the Supplementary Information). Firstly, modules composition within the MSF deserves a comment. We found linear clusters of regions grouped according neuroanatomical criteria, except the temporal and insular regions. While it is expected to find close portions of the cortex strongly coupled, it is not obvious that the same amount of coupling encompasses regional bilaterality embedded in the modularity of the whole brain network. The exception represented by the temporal and insular regions can be ascribed to different reasons. The insular lobes showed to be more connected with frontal regions, through the Rolandic opercula, than with each other (each one respecting its own laterality). Concurrently, the fronto-insular cortex, wherein generation and experience of emotion are found [@gu2013anterior], is implicated in the elaborate circuitry associated with awareness [@craig2009you] and uniquely (together with the anterior cingulate cortex and the dorsolateral prefrontal cortex) characterized by the presence of Von Economo neurons [@fajardo2008economo; @allman2011economo]. On the other hand the fragmentation of the temporal lobe can follow the mosaic structure of its functional specialization. In fact it includes auditory sensory and association cortices, part of the posterior language cortex, visual and higher order association cortices, primary and association olfactory cortices and enthorinal cortex [@frackowiak2004human]. The grouping of other heterogeneous clusters is due both to spatial closeness (precentral and postcentral gyrus) and to homofunctional specialization (hippocampus and parahippocampal gyrus). The MST representation of the functional brain network scaffold was found characterized by a long linear structure, whose skeleton follows the anatomical spatial distribution of lobes (from the frontal to the occipital lobe, passing through the temporal and parietal lobes), small chains linked alongside and a star-like structure (composed of cerebellar regions) at the end of one side. The main connectors along the main chain included: the cuneus/precuneus , the superior temporal gyrus, the superior occipital gyrus, the inferior parietal lobule, the medial orbitofrontal cortex/gyrus rectus and the VI lobule of the cerebellum. Notably, precuneus, inferior parietal cortex and medial orbitofrontal cortex constitute the Default Mode Network [@greicius2003functional; @raichle2007default], which is a crucial module of the whole brain network both for its functioning and as the first target of metabolic alterations both in neurological and psychiatric diseases [@buckner2008brain; @broyd2009default]. The large connectivity with the Default Mode Network showed by the superior temporal gyrus, which confers to this region a central role in the MST, has been observed in fMRI data [@zhang2012functional] but never found in FDG-PET data. Since one of the differences between the two techniques is the level of noise produced during the recordings (relatively silent during FDG-PET vs intensively noisy during MRI), the absence of overlap of the patterns of connectivity reported in that region might be explained by the auditory engagement included in the MRI experiments [@passow2015default]. The same argument works as far as the central role of the superior occipital cortex, being the visual activity engaged by the fixation during the resting-state fMRI experiment. A final comment must be reserved to the cerebellar module. The understanding of human cerebellar function has undergone a paradigm shift. No longer considered purely devoted to motor control, a wide role for the cerebellum in cognitive and affective functions is supported by anatomical, clinical and functional neuroimaging data [@picerni2013cerebellum; @laricchiuta2014cerebellum]. Recent evidence from functional connectivity studies in humans indicates that the cerebellum participates in functional networks with sensorimotor areas engaged in motor control and with association cortices that are involved in cognitive processes [@habas2009distinct; @krienen2009segregated; @o2010distinct]. The evidence of the cerebellar engagement in higher cognitive functions as well as in sensorimotor processes might explain its central role in the basal scaffold of the whole functional brain network. Specifically, it has been reported that the lobule VI of the cerebellum, here found as a main connector of the network, has been found to be involved in different cognitive tasks, from mental rotation to working memory processing [@stoodley2012functional]. In addition, it has been showed that during the first stages of consciousness loss, induced by mild sedation, the pattern of thalamic functional disconnections included regions belonging to the cerebellum [@gili2013thalamus], supporting the evidence of a strong cerebello-thalamic interaction showed by the MST analysis. Lastly it must be noticed that both in MSF and MST analysis, as weights of our graphs, the square of the correlation coefficient (r2) was used. Positive and negative correlations, characterized by the same absolute value, refer to the same amount of information in terms of connectivity strength. Since in this study we considered undirected weighted graphs, the phase-shift between time series, which gives rise to the sign of the correlation coefficient, was not significantly informative. Indeed, @goelman2014maximizing showed that both positive and negative correlations are related to synchronized neural activity and that the sign of correlations can result from neural-mediated hemodynamic mechanisms, which lead to temporal and spatial heterogeneity. Moreover, r2 was chosen since it has a strict relationship with the coefficient of determination R2. Specifically, after a simple linear regression, R2 equals the square of the correlation coefficient between the observed and modelled data values [@draper1998applied]. In terms of a correlation analysis, r2 estimates how well a time series can predict another one and can be considered as a measure of the magnitude of the relationship between time series. In conclusion we have reported that the whole architecture of the brain activity at rest is characterised by chain-like structures organized in a hierarchical modular arrangement. This kind of organization is thought to be stable under large-scale reconnection of substructure [@robinson2009dynamical], and efficient in terms of wiring costs [@bullmore2012economy]. In addition the chain-like feature of the basal modules facilitate the wiring and reconnection processes, not asking for a specific region as a hub, but functioning the whole module as a multiple access point for the functional plasticity. Methods {#methods .unnumbered} ------- Forty healthy subjects (mean age=38, SD=10; mean educational attainment=15, SD=3; male N=19) participated in this study. All subjects were carefully screened for a current or past diagnosis of any DSM-5 Axis I or II disorder using the SCID-5 Research Version edition (SCID-5-RV: [@first2015structured]) and the SCID-5 Personality Disorders (SCID-5-PD:[@first2016structured]). Exclusion criteria included: (1) a history of psychoactive substance dependence or abuse during the last one year period evaluated by the structured interview SCID-5-RV, (2) a history of neurologic illness or brain injury, (3) major medical illnesses, that is, diabetes not stabilized, obstructive pulmonary disease or asthma, hematological/oncological disorders, B12 or folate deficiency as evidenced by blood concentrations below the lower normal limit, pernicious anemia, clinically significant and unstable active gastrointestinal, renal, hepatic, endocrine or cardiovascular system disease, newly treated hypothyroidism, (4) the presence of any brain abnormality and microvascular lesion apparent on conventional FLAIR-scans. The presence, severity and location of vascular lesions was computed using a semi-automated method[@iorio2013white], (5) IQ below the normal range according to TIB (Test Intelligenza Breve, Italian analog of the National Adult Reading Test – NART –) (6) global cognitive deterioration according to a Mini-Mental State Examination (MMSE) [@folstein1975mini] score lower than 26, (7) dementia diagnosis according with DSM-5 criteria [@american2013diagnostic] and (8) non-Italian language native speaker. All participants were right-handed, with normal or corrected-to-normal vision. They gave written informed consent to participate after the procedures had been fully explained. The study was approved and carried out in accordance with the guidelines of the IRCCS Santa Lucia Foundation Ethics Committee. Data acquisition and preprocessing {#data-acquisition-and-preprocessing .unnumbered} ---------------------------------- FMRI data were collected using gradient-echo echo-planar imaging at 3T (Philips Achieva) using a (T2\*)-weighted imaging sequence sensitive to blood oxygen level-dependent (BOLD) (TR = 3 s, TE = 30 ms, matrix = 80 x 80, FOV=224x224, slice thickness = 3 mm, flip angle = 90°, 50 slices, 240 vol). A thirty-two channel receive-only head coil was used. A high-resolution T1-weighted whole-brain structural scan was also acquired (1x1x1 mm voxels). Subjects were instructed to lay in the scanner at rest with eyes open. For the purposes of accounting for physiological variance in the time-series data, cardiac and respiratory cycles were recorded using the scanner’s built-in photoplethysmograph and a pneumatic chest belt, respectively. The human brain was segmented into 116 macro-regions from the AAL template [@tzourio2002automated]. Resting state fMRI signals were averaged across each region to generate 116 time-series, which in turn were pairwise correlated calculating the Pearson’s coefficient and organized in a symmetric matrix. Two further templates were used to test the robustness of results at increasing spatial resolution by subdividing AAL regions as described in @fornito2010network. The number of regions included in the two templates was 276 and 531 respectively. Several sources of physiological variance were removed from each individual subject’s time-series fMRI data. For each subject, physiological noise correction consisted of removal of time-locked cardiac and respiratory artifacts (two cardiac harmonics and two respiratory harmonics plus four interaction terms) using linear regression [@glover2000image] and of low-frequency respiratory and heart rate effects [@birn2006separating; @shmueli2007low; @chang2009effects]. FMRI data were then preprocessed as follows: correction for head motion and slice-timing and removal of non-brain voxels (performed using FSL: FMRIB’s Software Library, www.fmrib.ox.ac.uk/fsl). Head motion estimation parameters were used to derive the frame-wise displacement (FD): time points with high FD (FD $>$ 0.2 mm) were replaced through a least-squares spectral decomposition as described in @power2014methods. Data were then demeaned, detrended and band-pass filtered in the frequency range 0.01-0.1 Hz, using custom software written in Matlab (The Math Works). For group analysis, a two-step registration process was performed. fMRI data were transformed first from functional space to individual subjects’ structural space using FLIRT (FMRIB’s Linear Registration Tool) and then non-linearly to a standard space (Montreal Neurological Institute MNI152 standard map) using Advanced Normalization Tools (ANTs; Penn Image Computing & Science Lab, http://www.picsl.upenn.edu/ANTS/). Finally data were spatially smoothed (5x5x5 mm full-width half-maximum Gaussian kernel). In order to create an average adjacency matrix at the population level, subject-wise matrices were Fisher-transformed, averaged across subjects and back-transformed. Percolation Analysis and Thresholding {#percolation-analysis-and-thresholding .unnumbered} ------------------------------------- We used a variation of the percolation analysis proposed by @gallos2012small, recently developed by @tiz2016. We ranked all experimentally determined correlation coefficients in increasing order and one at a time we removed from the network the link corresponding to the observed value in order to explore the global organization of the remaining network. by considering the connected components. This enabled us to calculate connected components by systematically varying the threshold on the network. Among the possible thresholds we looked at those values associated with the plateaux showed in the curve number of connected components vs thresholds obtained by the percolation process. Specifically, we considered all correlation values corresponding to points in the distribution of the real plateaux length overcoming the mean plus four standard deviations of the distribution of plateaux length of the ensemble of randomizations. This procedure led to the definition of 23 values, for which stable configurations of brain networks were found. Maximum Spanning Forest and Maximum Spanning Tree {#maximum-spanning-forest-and-maximum-spanning-tree .unnumbered} ------------------------------------------------- Starting from the initial correlation matrix we built a new matrix keeping only the maximum values along each row and sending all the rest to zero. We construct a directed network whose links connect each source-ROI with its maximally correlated target-ROI. This approach is equivalent to create a network from scratch in three steps: (i) to rank in decreasing order all the pairwise correlation values; (ii) to look at the list from the top and adding a new link if at least one of the two involved nodes has degree zero, otherwise discarding such correlation; (iii) to update the node degrees. The procedure ends when all nodes have at least degree equal to one. In both cases, the resulting network is made up by several components of nodes connected through their strongest links and without forming loops. In other terms, we have a Maximum Spanning Forest (MSF) of the initial correlation network. Furthermore, the first procedure introduced a directionality, simply revealing for each brain area which is its maximally correlated counterpart. The correlation values discarded during the construction of the MSF and ranked in increasing order were then used to build a Maximum Spanning Tree (MST). Starting from the top of the list, a link between the two involved nodes was drawn if they did not belong to the same group, then the two sets were merged into one; otherwise the correlation value was discarded. The procedure ended when all the nodes belonged to the same connected component. A statistical benchmark for human brains {#a-statistical-benchmark-for-human-brains .unnumbered} ---------------------------------------- In order to define to what extent the stepwise structure highlighted by the percolation analysis, the MSF and MSF were significant, we compared the results with a proper statistical benchmark. Specifically, in order to understand whether the results were a proper representation of the real intrinsic organization of the brain functioning at rest we needed to define an appropriate null model. A matrix has to satisfy some requirements in order to be a correlation matrix: it must be symmetric, with diagonal elements equal to 1, with off-diagonal elements in the range \[-1,1\] and it has to be positive semidefinite. Not all standard randomization techniques are suitable in this case. Here, we kept fix the spectrum distribution of the observed correlation matrix applying a series of givens rotations. This choice avoids to introduce further procedures to adjust reshuffled matrices in order to be positive semidefinite. We generated 100 randomizations of the observed correlation matrix. All comparisons between the ensemble of randomizations and the real network were performed on the pointwise squared matrix in order to neglect signs but to keep the ranking of correlation values. Relevant quantities were computed on each randomization, than averaged over the ensemble. [^1]: Out-degree is equal to 1 for construction.

--- abstract: 'We study the production of three gauge bosons at the next generation of linear $e^+e^-$ colliders operating in the $\gamma\gamma$ mode. The processes $\gamma\gamma \rightarrow W^+W^-V$ ($V=Z^0$, or $\gamma$) can provide direct information about the quartic gauge-boson couplings. We analyze the total cross section as well as several dynamical distributions of the final state particles including the effect of kinematical cuts. We find out that a linear $e^+e^-$ machine operating in the $\gamma\gamma$ mode will produce 5–10 times more three-gauge-boson states compared to the standard $e^+e^-$ mode at high energies.' address: | $^a$ Instituto de Física, Universidade de São Paulo,\ C.P. 20516, 01498-970 São Paulo, Brazil\ $^b$ Instituto de Física Teórica, Universidade Estadual Paulista,\ Rua Pamplona 145, 01405-900 São Paulo, Brazil. author: - | F. T. Brandt $^a$, O. J. P. Éboli $^a$, E. M. Gregores $^b$,\ M. B. Magro $^a$, P. G. Mercadante $^a$, and S. F. Novaes $^b$ title: 'Triple Vector Boson Processes in $\gamma\gamma$ Colliders' --- Introduction {#sec:int} ============ The multiple vector-boson production will be a crucial test of the gauge structure of the Standard Model since the triple and quartic vector-boson couplings involved in this kind of reaction are strictly constrained by the $SU(2)_L \otimes U(1)_Y$ gauge invariance. Any small deviation from the Standard Model predictions for these couplings spoils the intimate cancellations of the high energy behaviour between the various diagrams, giving rise to an anomalous growth of the cross section with energy. It is important to measure the vector-boson selfcouplings and look for deviations from the Standard Model, which would provide indications for a new physics. The production of several vector bosons is the ideal place to search directly for any anomalous behaviour of the triple and quartic couplings. The reaction $e^+ e^- \rightarrow W^+ W^-$ will be accessible at LEP200 and important information about the $WW\gamma$ and $WWZ$ vertices will be available in the near future [@ano:ee]. Nevertheless, due to its limited center of mass energy available, we will have to wait for colliders with higher center of mass energy in order to produce a final state with three or more gauge bosons and to test the quartic gauge-boson coupling. The measurement of the three-vector-boson production cross section can provide a non-trivial test of the Standard Model that is complementary to the analyses of the production of vector-boson pairs. Previously, the cross sections for triple gauge boson production in the framework of the Standard Model were presented for $e^+e^-$ colliders [@bar:plb; @bar:num; @gunion] and hadronic colliders [@bar:plb; @golden]. An interesting option that is deserving a lot of attention nowadays is the possibility of transforming a linear $e^+e^-$ collider in a $\gamma\gamma$ collider. By using the old idea of Compton laser backscattering [@las0], it is possible to obtain very energetic photons from an electron or positron beam. The scattering of a laser with few GeV against a electron beam is able to give rise to a scattered photon beam carrying almost all the parent electron energy with similar luminosity of the electron beam [@laser]. This mechanism can be employed in the next generation of $e^+e^-$ linear colliders [@pal; @bur] (NLC) which will reach a center of mass energy of 500–2000 GeV with a luminosity of $\sim 10^{33}$ cm$^{-2}$ s$^{-1}$. Such machines operating in $\gamma\gamma$ mode will be able to study multiple vector boson production with high statistic. In this work, we examine the production of three vector bosons in $\gamma\gamma$ collisions through the reactions $$\eqnum{I} \label{z} \gamma + \gamma \rightarrow W^+ + W^- + Z^0 \; ,$$ $$\eqnum{II} \label{g} \gamma + \gamma \rightarrow W^+ + W^- + \gamma \; .$$ These processes involve only interactions of between the gauge bosons making more evident any deviation from predictions of the Standard Model gauge structure. Besides that, there is no tree-level contribution involving the Higgs boson which eludes all the uncertainties coming from the scalar sector, like the Higgs boson mass. Nevertheless, the production of multiple longitudinal gauge bosons can shed light on the symmetry breaking mechanism even when there is no contribution coming from the standard Higgs boson. For instance, in models where the electroweak-symmetry breaking sector is strongly interacting there is an enhancement of this production [@golden; @strong]. We analyze the total cross section of the processes above, as well as the dynamical distributions of the final state vector bosons. We concentrate on final states where the $W$ and $Z^0$ decay into identifiable final states. We conclude that for a center of mass energy $\sqrt{s} \gtrsim 500$ GeV and an annual integrated luminosity of 10 fb$^{-1}$, there will be a promising number of fully reconstructible events. Moreover, we find out that a linear $e^+e^-$ machine operating in the $\gamma\gamma$ mode will produce 5–10 times more three-gauge-boson states compared to the standard $e^+e^-$ mode at high energies. The outline is as follows. In Sec. \[sec:res\], we introduce the laser backscattering spectrum, and present the details of the calculational method. Section \[cs:dis\] contains our results for the total cross section and the kinematical distributions of the final state gauge bosons for center of mass energies $\sqrt{s} = 0.5$ and $1$ TeV. This paper is supplemented by an appendix which gives the invariant amplitudes for the above processes. Calculational Method {#sec:res} ==================== The cross section for the triple-vector-boson production via $\gamma\gamma$ fusion can be obtained by folding the elementary cross section for the subprocesses $\gamma\gamma \rightarrow WWV$ ($V= Z^0,~ \gamma$) with the photon luminosity ($dL_{\gamma\gamma}/dz$), $$d\sigma (e^+e^-\rightarrow \gamma\gamma \rightarrow WWV)(s) = \int_{z_{\text{min}}}^{z_{\text{max}}} dz ~ \frac{dL_{\gamma\gamma}}{dz} ~ d \hat\sigma (\gamma\gamma \rightarrow WWV) (\hat s=z^2 s) \; ,$$ where $\sqrt{s}$ ($\sqrt{\hat{s}}$) is the $e^+e^-$ ($\gamma\gamma$) center of mass energy and $z^2= \tau \equiv \hat{s}/s$. Assuming that the whole electron beam is converted into photons via the laser backscattering mechanism, the relation connecting the photon structure function $F_{\gamma/e} (x,\xi)$ to the photon luminosity is $$\frac{d L_{\gamma\gamma}}{dz} = 2 ~ \sqrt{\tau} ~ \int_{\tau/x_{\text{max}}}^{x_{\text{max}}} \frac{dx}{x} F_{\gamma/e} (x,\xi)F_{\gamma/e} (\tau/x,\xi) \; . \label{lum}$$ For unpolarized beams the photon-distribution function [@laser] is given by $$F_{\gamma/e} (x,\xi) \equiv \frac{1}{\sigma_c} \frac{d\sigma_c}{dx} = \frac{1}{D(\xi)} \left[ 1 - x + \frac{1}{1-x} - \frac{4x}{\xi (1-x)} + \frac{4 x^2}{\xi^2 (1-x)^2} \right] \; , \label{f:l}$$ with $$D(\xi) = \left(1 - \frac{4}{\xi} - \frac{8}{\xi^2} \right) \ln (1 + \xi) + \frac{1}{2} + \frac{8}{\xi} - \frac{1}{2(1 + \xi)^2} \; ,$$ where $\sigma_c$ is the Compton cross section, $\xi \simeq 4 E\omega_0/m_e^2$, $m_e$ and $E$ are the electron mass and energy respectively, and $\omega_0$ is the laser-photon energy. The fraction $x$ represents the ratio between the scattered photon and initial electron energy for the backscattered photons traveling along the initial electron direction. The maximum value of $x$ is $$x_{\text{max}} = \frac{\omega_{\text{max}}}{E} = \frac{\xi}{1+\xi} \; ,$$ with $\omega_{\text{max}}$ being the maximum scattered photon energy. The fraction of photons with energy close to the maximum value grows with $\sqrt{s}$ and $\omega_0$. Nevertheless, the bound $\xi < 2(1 + \sqrt{2})$ should be respected in order to avoid the reduction in the efficiency of the $e\to\gamma$ conversion due to the creation of $e^+e^-$ pairs in collisions of the laser with backscattered photons. We assumed that $\omega_0$ has the maximum value compatible with the above constraint, [*e.g.*]{} for $\sqrt{s} = 500 $ GeV, $\omega_0 = 1.26$ eV and $x_{\text{max}} \simeq 0.83$. With this choice, more than half of the scattered photons are emitted inside a small angle ($\theta < 5 \times 10^{-6}$ rad) and carry a large amount of the electron energy. Due to this hard photon spectrum, the luminosity Eq.(\[lum\]) is almost constant for $z < x_{\text{max}}$. The analytical calculation of the cross section for the process $\gamma\gamma \rightarrow W^+W^- \gamma$ ($\gamma\gamma \rightarrow W^+W^-Z^0$) requires the evaluation of twelve Feynman diagrams in the unitary gauge, which is a tedious and lengthy calculation despite of being straightforward. For the sake of completeness, we exhibit in the Appendix the expression of the amplitudes of these processes. In order to perform these calculations in a efficient and reliable way [@red], we used an improved version of the numerical technique presented in Ref. [@bar:num; @zep:num]. The integrations were also performed numerically using a Monte Carlo routine [@lepage] and we tested the Lorentz and $U(1)_{\text em}$ gauge invariances of our results for the amplitudes. Cross Sections and Gauge-boson Distributions {#cs:dis} ============================================ We have evaluated the total cross section for the processes $\gamma\gamma \rightarrow W^+W^-V$ imposing kinematical cuts on the final state particles. Our first cut required that the produced gauge bosons are in the central region of the detector, [*i.e.*]{} we imposed that the angle of vector boson with the beam pipe is larger than $30^\circ$, which corresponds to a cut in the pseudo-rapidity of $|\eta| < 1.32$. We further required the isolation of the final particles by demanding that all vector bosons make an angle larger than $25^\circ$ between themselves. Moreover, for the process \[g\], we imposed a cut on the photon transverse momentum, $p_T^\gamma >10$ GeV, to guarantee that the results are free of infrared divergences and to mimic the performance of a typical electromagnetic calorimeter. In Tables \[total:z\] and \[total:g\] we exhibit the results for the total cross section of the processes \[z\] and \[g\], with and without the above cuts. As we can see from these tables, the two-gauge-boson cross section ($\gamma + \gamma \rightarrow W^+ + W^-$), which is the main reaction in a $\gamma\gamma$ collider [@gin], is from 2 to 4 orders of magnitude above those for three gauge bosons depending upon $\sqrt{s}$. Nevertheless, we still find promising event rates for final states $W^+W^-V$ for an $e^+e^-$ collider with an annual integrated luminosity of 10 fb$^{-1}$. Moreover, the triple-gauge-boson production in $e^+e^-$ and $\gamma\gamma$ colliders are comparable at $\sqrt{s}=500$ GeV, while the event rate in $\gamma\gamma$ collider is a factor of 5–10 larger than the one in a $e^+e^-$ machine at $\sqrt{s}=1$ TeV. The observed growth of the total cross section for the production of three gauge bosons is due to gauge-boson exchange in the $t$ and $u$ channels. Since we are interested in final states where all the gauge bosons are identified, the event rate is determined not only by the total cross section, but also by the reconstruction efficiency that depends on the particular decay channels of the vector bosons. In principle, charged lepton and light quark jet pairs can be easily identified. However, in the semileptonic decay of heavy quark the presence of unmeasurable neutrinos spoils the invariant mass measurement, and we adopt, as in Ref. [@bar:num], that the efficiency for reconstruction of a $W^\pm$ ($Z^0$) is 0.61 (0.65). In general, final-state photons can be identified with high efficiency as an electromagnetic shower with a neutral initiator. Combining the reconstruction efficiencies for individual particles, we obtain that the process \[z\] (\[g\]) has a detection efficiency of 0.24 (0.37). Once the reconstruction efficiency is substantial, the crucial factor for event rates is the production cross section. Assuming the above cuts and efficiencies we expect, for a 500 (1000) GeV collider with an annual integrated luminosity of $10$ fb$^{-1}$, a total yield of 25 (198) $\gamma + \gamma \rightarrow W^+ + W^- + Z^0$ fully reconstructed events per year and 428 (714) $\gamma + \gamma \rightarrow W^+ + W^- + \gamma$ reconstructed events per year with $P^\gamma_T > 10$ GeV. In order to reach a better understanding of these reactions, we present in Fig. \[fig:1\]–\[fig:6\] various distributions of the final state gauge bosons. In Fig. \[fig:1\] we show the distribution in $\cos\theta$, where $\theta$ is the polar angle of the particles ($W^\pm$, and $V= \gamma, Z^0$) with the beam pipe. The results are presented with and without the angular cuts described above. The $W^+$ and $W^-$ curves coincide due to the charge conjugation invariance. We should notice that these processes are particularly sensitive to central region requirement since, analogously to what happens in the reaction $\gamma\gamma \rightarrow W^+ W^-$, the $W$’s go preferentially along the beam pipe direction. This fact can also be seen from the rapidity distribution of the final state particles (Fig. \[fig:2\]). Therefore, the requirement that the gauge bosons are produced in the central region of the detector implies in a loss of $1/2$ to $5/6$ of the total number of events. Increasing the center of mass energy, the $W$’s tend to populate the high rapidity region while the $V= \gamma, Z^0$ distribution maintains its shape. Consequently, the cut in the $W$ angle with beam pipe discards most of the high energy events. In order to estimate the importance of the isolation cut on the final particles, we present in Fig. \[fig:3\] the distributions in the angle between the vector bosons. Charge conjugation invariance of the processes implies that the distribution for $W^+Z^0$ and $W^-Z^0$ are the same. In both processes \[z\] and \[g\], the $W$’s tend to be back-to-back, while the $WV$ ($V=Z^0$ or $\gamma$) is relatively flat, demonstrating that the isolation cut is not very restrictive. The distribution for different energies of the collider are quite similar, apart from a constant factor due to the growth of the total cross section. The invariant mass distributions of the $W^+W^-$ and $W^\pm Z^0~(\gamma)$ pairs are presented in Fig. \[fig:4\]. Once again the $W^+Z^0~(\gamma)$ and $W^-Z^0~(\gamma)$ curves coincide. From this Figure we can learn that the average invariant mass of the pairs $W^+W^-$ is higher then the one for $WZ^0~(\gamma)$ pairs. As the center of mass energy of the collider is increased the distributions grows due to the growth of the total cross section. Moreover, the invariant mass distribution for $WZ^0~(\gamma)$ and $W^+W^-$ pairs are considerably different: the former is rather narrow and peaked at small invariant masses while the later one is broader and peaked at high invariant masses. Figure \[fig:5\] shows the laboratory energy distributions the of the $W^\pm$ and $Z^0~(\gamma)$ gauge bosons. In the process $\gamma\gamma \rightarrow W^+W^-Z^0$, the $E_Z$ and $E_{W^\pm}$ distributions are rather similar, with the average energy of the $W^\pm$ being larger than the average $Z^0$ energy. As the center of mass energy of the collider is increased the distributions grow and become rather isolated, while the peaks broaden systematically. In the process $\gamma\gamma \rightarrow W^+W^- \gamma$, the distributions in $E_\gamma$ and $E_{W^\pm}$ are very different due to the infrared divergences: the $E_\gamma$ is strongly peaked towards small energies while $E_{W^\pm}$ is rather broad and peaked at high energies. With the increase of the collider energy the difference between the distribution become clearer. We exhibit in Fig. \[fig:6\] the transverse-momentum distribution for the $W^\pm$ and $Z^0~(\gamma)$ vector bosons. There are no distinctive difference between the distribution for $W^\pm$ and $Z^0$ in process \[z\], apart from the fact that the $Z^0$’s exhibit a smaller average $p_T$ than the $W$’s. In the case of process \[g\], the distributions for $\gamma$ and $W^\pm$ are very different since the first is peaked at very small $p_T$ due to the infrared divergences. [*Note added. *]{} After completing this work, we came across an estimate of the total elementary cross section for the processes studied here done by M. Baillargeon and F. Boudjema [@boudjema]. This work was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP). {#section .unnumbered} We collect in this appendix the expressions for the amplitudes of the processes $\gamma\gamma \rightarrow W^+W^-V$, with $V=Z^0$ or $\gamma$. The Feynman diagrams contributing to these processes are given in Fig.\[fig:7\]. The momenta and polarizations of the initial photons where denoted by ($k_1$, $k_2$) and ($\epsilon_\mu(k_1)$, $\epsilon_\nu(k_2)$), while the momenta and polarizations of the final state $W^+$, $W^-$ and $V$ are given by ($p_+$, $p_-$, $k_3$) and ($\epsilon_\alpha(p_+)$, $\epsilon_\beta(p_-)$, $\epsilon_\gamma(k_3)$) respectively. For a given choice of the initial and final polarizations the amplitude of these processes can be written as $$M={G_{v} \epsilon_{\mu}(k_1)\epsilon_{\nu}(k_2)\epsilon_{\alpha} (p_+) \epsilon_{\beta}(p_-)\epsilon_{\gamma}(k_3)M_T^{\mu\nu\alpha\beta\gamma}}\;,$$ with $$M_T^{\mu\nu\alpha\beta\gamma}=\sum_{\imath=1}^{7} {M_{\imath}^{\mu\nu\alpha\beta\gamma}} \; ,$$ where the ${M_{\imath}^{\mu\nu\alpha\beta\gamma}}$ is the contribution of the set of diagrams $\imath$ to the processes. The factor $G_v$ depends upon the process, assuming the value $e^3$ for the production of $W^+W^-\gamma$ and the value $e^3 \cot^2{\theta_W}$, with $\theta_W$ being the Weinberg angle, for the final state $W^+W^-Z^0$. In order to write a compact expression for the amplitude, it is convenient to define the triple-gauge-boson coupling coefficient as $$\Gamma_3^{\alpha\beta\gamma}(P_1,P_2)=\left[(2P_1+P_2)^{\beta}g^{\alpha\gamma} -(2P_2+P_1)^{\alpha}g^{\beta\gamma}+(P_2-P_1)^{\gamma} g^{\beta\alpha}\right]\;,$$ the quartic-gauge-boson coupling $$\Gamma_4^{\mu\nu\alpha\beta}=g^{\mu\alpha}g^{\nu\beta}+ g^{\mu\beta} g^{\nu\alpha}-2g^{\mu\nu}g^{\alpha\beta}\; ,$$ and the propagator tensor $$D^{\mu\nu}(k)= \frac{(g^{\mu\nu} - k^\mu k^\nu / m^2)}{k^2-m^2} \;.$$ Using the above definitions, the contributions of the different set of diagrams can be written as $$\begin{aligned} %\label {fy} M_1^{\mu\nu\alpha\beta\gamma}&=&\Gamma_3^{\alpha\gamma\xi}(p_+,k_3) D_{\xi\sigma}(p_++k_3)\Gamma_3^{\mu\sigma\rho}(k_1,-(p_++k_3))\nonumber \\ & & D_{\rho\lambda}(p_--k_2)\Gamma_3^{\beta\nu\lambda}(-p_-,k_2) + \; [ k_{1\leftrightarrow 2}\; ; \;\mu\leftrightarrow\nu ]\end{aligned}$$ $$\begin{aligned} %\label {fy} M_2^{\mu\nu\alpha\beta\gamma}&=& \Gamma_3^{\alpha\beta\xi}(k_3,p_-) D_{\xi\sigma}(p_-+k_3)\Gamma_3^{\sigma\nu\rho}(-p_--k_3,k_2) \nonumber \\ & & D_{\rho\lambda}(k_1-p_+)\Gamma_3^{\mu\alpha\lambda}(-p_+,k_2) +\; [ k_{1\leftrightarrow 2}\; ; \;\mu\leftrightarrow\nu ]\end{aligned}$$ $$\begin{aligned} %\label {fy} M_3^{\mu\nu\alpha\beta\gamma}&=&\Gamma_3^{\mu\alpha\xi}(k_1,-p_+) D_{\xi\sigma}(k_1-p_+)\Gamma_3^{\gamma\sigma\rho} (-k_3,(k_1-p_+)) \nonumber \\ & & D_{\rho\lambda}(p_--k_2)\Gamma_3^{\nu\beta\lambda}(-k_2,p_-) +\; [ k_{1\leftrightarrow 2}\; ; \;\mu\leftrightarrow\nu ]\end{aligned}$$ $$\begin{aligned} %\label {fy} M_4^{\mu\nu\alpha\beta\gamma}=\Gamma_3^{\beta\nu\xi}(-p_-,k_2) D_{\xi\lambda}(k_2-p_-) \Gamma_4^{\lambda\alpha\mu\gamma} +\; [ k_{1\leftrightarrow 2}\; ; \;\mu\leftrightarrow\nu ]\end{aligned}$$ $$\begin{aligned} %\label {fy} M_5^{\mu\nu\alpha\beta\gamma}=\Gamma_3^{\mu\alpha\xi}(k_1,-p_+) D_{\xi\lambda}(k_1-p_+) \Gamma_4^{\lambda\beta\nu\gamma} +\; [ k_{1\leftrightarrow 2}\; ; \;\mu\leftrightarrow\nu ]\end{aligned}$$ $$\begin{aligned} %\label {fy} M_6^{\mu\nu\alpha\beta\gamma}= \Gamma_3^{\alpha\gamma\xi}(p_+,k_3) D_{\xi\lambda}(p_++k_3)\Gamma_4^{\lambda\beta\nu\mu}\end{aligned}$$ $$\begin{aligned} %\label {fy} M_7^{\mu\nu\alpha\beta\gamma}= \Gamma_3^{\gamma\beta\xi}(k_3,p_-) D_{\xi\lambda}(-p_--k_3)\Gamma_4^{\lambda\alpha\nu\mu}\end{aligned}$$ where $[ k_{1\leftrightarrow 2}\; ; \;\mu\leftrightarrow\nu ]$ indicates the crossed contributions of the initial photons. K. Hagiwara, R. D. Peccei, D. Zeppenfeld, and K. Hikasa, Nucl. Phys. [**B282**]{}, 253 (1987); Proceedings of the ECFA Workshop on LEP200, CERN Report 87-08, ECFA Report 87/108, ed.A. Böhm and W. Hoogland, Aachen 1987. V. Barger and T. Han, Phys. Lett. [**212B**]{}, 117 (1988). V. Barger, T. Han, and R. J. N. Phillips, Phys.Rev. D [**39**]{}, 146 (1989). A. Tofighi-Niaki and J. F. Gunion, Phys.Rev. D [**39**]{}, 720 (1989). M.  Golden and S. Sharpe, Nucl. Phys. [**B261**]{}, 217 (1985). F. R. Arutyunian, and V. A. Tumanian, Phys. Lett. [**4**]{}, 176 (1963); R. H. Milburn, Phys. Rev. Lett. [**10**]{}, 75 (1963); see also C. Akerlof, University of Michigan Report No. UMHE 81-59 (1981), unpublished. I. F. Ginzburg, G. L. Kotkin, V. G. Serbo, and V. I.Telnov, Nucl. Instrum.  Methods [**205**]{}, 47 (1983); [**219**]{}, 5 (1984); V. I. Telnov, Nucl. Instrum. Methods [**A294**]{}, 72 (1990). R. B. Palmer, Annu.  Rev. Nucl. Part. Sci. [**40**]{}, 529 (1990). D. L. Burke, “Linear Colliders: When? – How?“, to appear in the Proceedings of the XXVI International Conference on High Energy Physics”, Dallas (1992). M. Chanowitz and M.K. Gaillard, Phys. Lett., 85 (1984). We have also checked our results using the analytical amplitudes obtained using REDUCE and FORM. It turned out that the numerical evaluation of cross sections and distributions by the numerical technique is at least ten times faster than the analytical results obtained with REDUCE and FORM. K. Hagiwara, and D. Zeppenfeld, Nucl. Phys.  [**B274**]{}, 1 (1986). G. P. Lepage, J. Comp. Phys. [**27**]{}, 192 (1978). I. F.  Ginzburg, G. L. Kotkin, S. L. Panfil, and V. G. Serbo, Nucl. Phys. [**B228**]{}, 285 (1983). M. Baillargeon and F. Boudjema in Proceedings of the “Beyond the Standard Model III", Ottawa, Ontario, June 1992, edited by S. Godfrey (World Scientific). ------------------ -------------- ----------- $\sqrt{s}$ (GeV) without cuts with cuts 500 20.4 10.2 1000 289 81.9 ------------------ -------------- ----------- : Total cross section in fb for the process $\gamma \gamma \rightarrow W^+ W^-Z^0$.[]{data-label="total:z"} ------------------ -------------- ----------- -------------- ----------- $\sqrt{s}$ (GeV) without cuts with cuts without cuts with cuts 500 296 115 167 69 1000 1162 192 748 138 ------------------ -------------- ----------- -------------- ----------- : Total cross section in fb for the process $\gamma \gamma \rightarrow W^+W^-\gamma$[]{data-label="total:g"}

--- abstract: 'We study the waves at the interface between two thin horizontal layers of immiscible fluids subject to high-frequency horizontal vibrations. Previously, the variational principle for energy functional, which can be adopted for treatment of quasi-stationary states of free interface in fluid dynamical systems subject to vibrations, revealed existence of standing periodic waves and solitons in this system. However, this approach does not provide regular means for dealing with evolutionary problems: neither stability problems nor ones associated with propagating waves. In this work, we rigorously derive the evolution equations for long waves in the system, which turn out to be identical to the ‘plus’ (or ‘good’) Boussinesq equation. With these equations one can find all time-independent-profile solitary waves (standing solitons are a specific case of these propagating waves), which exist below the linear instability threshold; the standing and slow solitons are always unstable while fast solitons are stable. Depending on initial perturbations, unstable solitons either grow in an explosive manner, which means layer rupture in a finite time, or falls apart into stable solitons. The results are derived within the long-wave approximation as the linear stability analysis for the flat-interface state \[D.V. Lyubimov, A.A. Cherepanov, Fluid Dynamics [**21**]{}, 849–854 (1987)\] reveals the instabilities of thin layers to be long-wavelength.' author: - 'D. S. Goldobin' - 'A. V. Pimenova' - 'K. V. Kovalevskaya' - 'D. V. Lyubimov' - 'T. P. Lyubimova' title: 'Running interfacial waves in two-layer fluid system subject to longitudinal vibrations' --- Introduction {#sec_intro} ============ In [@Wolf-1961; @Wolf-1970] Wolf reported experimental observations of the occurrence of steady wave patterns on the interface between immiscible fluids subject to horizontal vibrations. The build-up of the theoretical basis for these experimental findings was initiated with the linear instability analysis of the flat state of the interface [@Lyubimov-Cherepanov-1987; @Khenner-Lyubimov-Shotz-1998; @Khenner-etal-1999] (see Fig.\[fig1\] for the sketch of the system considered in these works). Specifically, it was found that in thin layers the instability is a long-wavelength one [@Lyubimov-Cherepanov-1987]. In [@Khenner-Lyubimov-Shotz-1998; @Khenner-etal-1999], the linear stability was determined for the case of arbitrary frequency of vibrations. In spite of the substantial advance in theoretical studies, the problem proved to require subtle approaches; a comprehensive straightforward weakly-nonlinear analysis of the system subject to high-frequency vibrations still remains lacking in the literature (as well as the long-wavelength one). The approach employed in [@Lyubimov-Cherepanov-1987] can be (and was) used for analysis of time-independent quasi-steady patterns (including non-linear ones) only, but not the evolution of these patterns over time. This “restricted” analysis of the system revealed that quasi-steady patterns can occur both via sub- and supercritical pitchfork bifurcations, depending on the system parameters. Later on, specifically for thin layers, which will be the focus of our work, the excitation of patterns was shown to be always subcritical [@Zamaraev-Lyubimov-Cherepanov-1989] (paper [@Zamaraev-Lyubimov-Cherepanov-1989] is published only in Russian, although the result can be derived from [@Lyubimov-Cherepanov-1987] as well). Within the approach of [@Lyubimov-Cherepanov-1987; @Zamaraev-Lyubimov-Cherepanov-1989] neither time-dependent patterns nor the stability of time-independent patterns can be analyzed. Specifically for the case of subcritical excitation, time-independent patterns may belong to the stability boundary between the attraction basins of the flat-interface state and the finite-amplitude pattern state in the phase space. [^1] In this work we accomplish the task of derivation of the governing equations for dynamics of patterns on the interface of two-layer fluid system within the approximation of inviscid fluids. In Wolf’s experiments [@Wolf-1961; @Wolf-1970], the viscous boundary layer in the most viscous liquid was an order of magnitude thinner than the liquid layer, meaning the approximation of inviscid liquid is relevant. The layer is assumed to be thin enough for the evolving patterns to be long-wavelength [@Lyubimov-Cherepanov-1987]. With the governing equations we analyze the dynamics of the system below the linear instability threshold, where the system turns out to be identical to the ‘plus’ Boussinesq equation. The system admits soliton solutions, these solutions are parameterized with single parameter, soliton speed. The maximal speed of solitons equals the minimal group velocity of linear waves in the system; the soliton waves move always slower than the packages of linear waves. Stability analysis reveals that the standing and slow solitons are unstable while fast solitons are stable. The system, as the ‘plus’ Boussinesq equation, is known to be fully integrable. Recently, the problem of stability of a liquid film on a horizontal substrate subject to tangential vibrations was addressed in the literature [@Shklyaev-Alabuzhev-Khenner-2009]. The stability analysis for space-periodic patterns and solitary waves for the latter system was reported in [@Benilov-Chugunova-2010]. The similarity of this problem with the problem we consider and expected similarity of results are illusive. Firstly, for the problem of [@Shklyaev-Alabuzhev-Khenner-2009] the liquid film is involved into oscillating motion only due to viscosity, an inviscid liquid will be motionless over the tangentially vibrating substrate, while in the system we consider the inviscid fluid layers will oscillate due to motion of the lateral boundaries of the container and fluid incompressibility [@Lyubimov-Cherepanov-1987; @Khenner-Lyubimov-Shotz-1998; @Khenner-etal-1999]. Secondly, the single-film case corresponds to the case of zero density of the upper layer in a two-layer system; in the system we consider this is a very specific case. These dissimilarities have their reflection in the resulting mathematical models; the governing equations for long-wavelength patterns derived in [@Shklyaev-Alabuzhev-Khenner-2009] are of the 1st order with respect to time and the 4th order with respect to the space coordinate and describes purely dissipative patterns in the viscous fluid, while the equation we will report is of 2nd order in time, 4th order in space and describes non-dissipative dynamics. The paper is organized as follows. In Sec.\[sec\_statement\] we provide a physical description and mathematical model for the system under consideration. In Sec.\[sec\_deriv\] the governing equations for long-wavelength patterns are derived and discussed. In Sec.\[sec\_solitons\] soliton solutions are presented and their stability properties are analyzed. Conclusions are drawn in Sec.\[sec\_concl\]. Problem statement and governing equations {#sec_statement} ========================================= We consider a system of two horizontal layers of immiscible inviscid fluid, confined between two impermeable horizontal boundaries (see Fig.\[fig1\]). The system is subject to high-frequency longitudinal vibrations of linear polarization; the velocity of vibrational motion of the system is $be^{i\omega t}+c.c.$ (here “$c.c.$” stands for complex conjugate). For simplicity, we consider the case of equal thickness, say $h$, of two layers, which is not expected to change the qualitative picture of the system behavior [^2] but makes calculations simpler. The density of upper liquid $\rho_1$ is smaller than the density of the lower one $\rho_2$. We choose the horizontal coordinate $x$ along the direction of vibrations, the $z$-axis is vertical with origin at the unperturbed interface between layers. In this system, at the limit of infinitely extensive layers, the state with flat interface $z=\zeta(x,y)=0$ is always possible. In real layers of finite extent, the oscillating lateral boundaries enforce liquid waves perturbing the interface; however, at a distance from these boundaries the interface will be nearly flat as well. For inviscid fluids, this state (the ground state) features spatially homogeneous pulsating velocity fields $\vec{v}_{j0}$ in both layers; $$\begin{array}{c} \displaystyle \vec{v}_{j0}=a_j(t)\vec{e}_x,\qquad a_j(t)=A_je^{i\omega t}+c.c.,\\[10pt] \displaystyle A_1=\frac{\rho_2 b}{\rho_1+\rho_2},\qquad A_2=\frac{\rho_1 b}{\rho_1+\rho_2}, \end{array} \label{eq01}$$ where $j=1,2$ and $\vec{e}_x$ is the unit vector of the $x$-axis. All equations and parameters in this subsection are dimensional. The time instant $t=0$ is chosen so that $b$ and $A_j$ are real. The result (\[eq01\]) follows from the condition of zero pressure jump across the uninflected interface and the condition of the total fluid flux through the vertical cross-section being equal $\int_{-h}^{+h}v^{(x)}dz=2h(be^{i\omega t}+c.c.)$ (which is due to the system motion with velocity $be^{i\omega t}+c.c.$). Considering the flow of inviscid fluid, it is convenient to introduce the potential $\phi_j$ of the velocity field; $$\vec{v}_j=-\nabla\phi_j\,. \label{eq02}$$ The mass conservation law for incompressible fluid, $\nabla\cdot\vec{v}_j=0$, yields the Laplace equation for the potential, $\Delta\phi_j=0$. The kinematic conditions on the top and bottom boundaries $$\phi_{1z}(z=h)=\phi_{2z}(z=-h)=0 \label{eq03}$$ and on the interface $z=\zeta(x,y)$ $$\begin{aligned} \dot{\zeta}&=&-\phi_{1z}+\nabla\phi_1\cdot\nabla\zeta\,, \label{eq04} \\[5pt] \dot{\zeta}&=&-\phi_{2z}+\nabla\phi_2\cdot\nabla\zeta \label{eq05}\end{aligned}$$ are also to be taken into account. (In what follows, the upper dot stands for the time-derivative and letter in subscript denotes partial derivative with respect to the corresponding coordinate.) Equations (\[eq04\]) and (\[eq05\]) can be derived from the condition that the points of zero value of the distance function $F=z-\zeta(x,y)$, which correspond to the position of the interface, move with fluid, i.e., the Lagrangian derivative (material derivative) $dF/dt=\partial F/\partial t+\vec{v}\cdot\nabla{F}$ is zero on the interface: $-\dot\zeta+v^{(z)}-\vec{v}\cdot\nabla\zeta=0$, and this holds for both fluids. After substitution of the potential flow, the Euler equation takes the following form: $$\nabla\left(-\dot{\phi}_j+\frac{1}{2}\left(\nabla\phi_j\right)^2\right) =\nabla\left(-\frac{1}{\rho_j}p_j-gz\right),$$ where $g$ is the gravity. The latter equation provides the expression for the pressure field in the volume of two fluids for a given flow field; $$p_j=p_{j0}+\rho_j\left(\dot{\phi}_j -\frac{1}{2}\left(\nabla\phi_j\right)^2-gz\right). \label{eq06}$$ Now the stress on the interface needs to be included to make the equation system self-contained, by providing the required boundary conditions for $\phi_j$ on the interface between the two fluids. The pressure jump across the interface is due to the surface tension; $$z\!=\!\zeta(x,y): \quad p_1-p_2 =-\alpha\nabla\cdot\vec{n}\quad \mbox{with }\vec{n}=\frac{\nabla F}{\left|\nabla F\right|}, \label{eq07}$$ where $\alpha$ is the surface tension coefficient and $\vec{n}$ is the unit vector normal to the interface. The system we consider does not possess any internal instability mechanisms in the absence of vibrations (unlike, e.g., [@Thiele-Vega-Knobloch-2006; @Nepomnyashchy-Simanovskii-2013]). Vibrations discriminate one of horizontal directions and there are no reasons to expect that close to the threshold of vibration-induced instabilities the excited patterns will experience spatial modulation along the $y$-direction, which is perpendicular to the vibration polarization direction. Furthermore, the linear stability analysis revealed the marginal vibration-induced instability of the flat-interface state to be long-wavelength [@Lyubimov-Cherepanov-1987; @Zamaraev-Lyubimov-Cherepanov-1989]. Hence, we restrict our consideration to the case of $(x,z)$-geometry and the long-wavelength approximation, $\left|{\partial_x\vec{v}}\right|\ll\left|{\partial_z\vec{v}}\right|$. Governing equations for large-scale patterns {#sec_deriv} ============================================ Derivation of equations ----------------------- In this section we derive the governing equation for long-wavelength ([*or*]{} large-scale) patterns. We employ the standard multiscale method with small parameters $(T^{-1}/\omega)$ and $(h/l)$, where $T$ is the characteristic time scale of the evolution of interface patterns (to be specified below), and $l$ is the reference horizontal length of patterns, $\partial_x\sim l^{-1}$. The hierarchy of small parameters and the orders of magnitude of fields will be determined in the course of derivation. Within the long-wavelength approximation, the solutions to the Laplace equation for $\phi_j(x,t)$ satisfying boundary conditions (\[eq03\]) in the most general form read $$\begin{aligned} \phi_1=-a_1(t)x+\Phi_1(x,t)-\frac{1}{2}(h-z)^2\Phi_{1xx}(x,t) \nonumber\\[5pt] {}+\frac{1}{4!}(h-z)^4\Phi_{1xxxx}(x,t)-\dots\,, \label{eqa01}\end{aligned}$$ $$\begin{aligned} \phi_2=-a_2(t)x+\Phi_2(x,t)-\frac{1}{2}(h+z)^2\Phi_{2xx}(x,t) \nonumber\\[5pt] {}+\frac{1}{4!}(h+z)^4\Phi_{2xxxx}(x,t)-\dots\,. \label{eqa02}\end{aligned}$$ Here the ground state (the flat-interface state) is represented by the terms $-a_j(t)x$; $\Phi_j(x,t)$ describe perturbation flow, they are as yet arbitrary functions of $x$ and $t$. After substitution of $p_j$ from expression (\[eq06\]) and $\phi_j$ from expressions (\[eqa01\])–(\[eqa02\]), the condition of stress balance on the interface (\[eq07\]) reads $$\begin{aligned} \displaystyle \hspace{-40pt} p_{1\infty}-p_{2\infty} +\rho_1\Bigg[-\dot{a}_1x+\dot{\Phi}_1-\frac{(h-\zeta)^2}{2}\dot{\Phi}_{1xx} -\frac{1}{2}\left(-a_1+\Phi_{1x}-\frac{(h-\zeta)^2}{2}\Phi_{1xxx}\right)^2\quad \nonumber\\[10pt] \displaystyle \qquad {}-\frac{((h-\zeta)\Phi_{1xx})^2}{2}+\dots\Bigg] \nonumber\\[10pt] \displaystyle {}-\rho_2\Bigg[-\dot{a}_2x+\dot{\Phi}_2-\frac{(h+\zeta)^2}{2}\dot{\Phi}_{2xx} -\frac{1}{2}\left(-a_2+\Phi_{2x}-\frac{(h+\zeta)^2}{2}\Phi_{2xxx}\right)^2\quad \nonumber\\[10pt] \displaystyle {}-\frac{((h+\zeta)\Phi_{2xx})^2}{2}+\dots\Bigg] \nonumber\\[10pt] \displaystyle {}+(\rho_2-\rho_1)g\zeta=\alpha\frac{\zeta_{xx}}{(1+\zeta_x^2)^{3/2}}\,. \nonumber\end{aligned}$$ Here “…” stand for terms $\mathcal{O}_1(\dot\Phi_jh^4/l^4)+\mathcal{O}_2(\Phi_j^2h^4/l^6)+\mathcal{O}_3(a_j\Phi_jh^4/l^5)$; here and in what follows, $\mathcal{O}_j(Z)$ stand for unspecified contributions of the same order of smallness as their argument $Z$, and index $j$ is used to distinguish several nonidentical contributions to one and the same equation. We specify the order of the neglected terms so as to facilitate tracking the correctness of the derivations. The difference of constants $p_{1\infty}-p_{2\infty}$ is to be determined from the condition that in the area of vanishing perturbations of the pulsation flow, i.e. $\Phi_j(x,t)=const$, the interface remains flat, i.e. $\zeta(x,t)=0$. This condition yields $p_{1\infty}-p_{2\infty}-(\rho_1a_1^2(t)-\rho_2a_2^2(t))/2=0$. One can choose the following units of measurements for length: $L=\sqrt{\alpha/[(\rho_2-\rho_1)g]}$, for time: $T=L/b$, and for the fluid densities: $\rho_\ast$—which mean replacement $$\begin{array}{c} (x,z)\to(Lx,Lz),\quad t\to Tt,\quad \zeta\to L\zeta,\\[5pt] \Phi_j\to(L^2/T)\Phi_j,\quad \rho_i\to\rho_\ast\rho_i \end{array} \label{rescaling1}$$ in equations—and rewrite the last equation in the dimensionless form $$\begin{array}{l} \displaystyle B\Bigg[\frac{\rho_1a_1^2-\rho_2a_2^2}{2}+\rho_1\dot{\Phi}_1-\rho_2\dot{\Phi}_2 -\frac{\rho_1(h-\zeta)^2}{2}\dot{\Phi}_{1xx}+\frac{\rho_2(h+\zeta)^2}{2}\dot{\Phi}_{2xx} -\frac{\rho_1}{2}\left(a_1-\Phi_{1x}+\frac{1}{2}(h-\zeta)^2\Phi_{1xxx}\right)^2 \\[15pt] \displaystyle\quad {}+\frac{\rho_2}{2}\left(a_2-\Phi_{2x}+\frac{1}{2}(h+\zeta)^2\Phi_{2xxx}\right)^2 -\frac{\rho_1}{2}\left((h-\zeta)\Phi_{1xx}\right)^2 +\frac{\rho_2}{2}\left((h+\zeta)\Phi_{2xx}\right)^2 +\dots\Bigg]+\zeta =\frac{\zeta_{xx}}{(1+\zeta_x^2)^{3/2}}\,. \end{array} \label{eqa03}$$ Here the dimensionless vibration parameter $$B\equiv\frac{\rho_\ast b^2}{\sqrt{\alpha(\rho_2-\rho_1)g}}=B_0+B_1 \label{eqa04}$$ ($\rho_j$ is dimensional here), where $B_0 $ is the critical value of the vibration parameter above which the flat-interface state becomes linearly unstable, $B_1$ is a small deviation of the vibration parameter from the critical value. Further, kinematic conditions (\[eq04\]) and (\[eq05\]) turn into $$\dot\zeta=\left(-(h-\zeta)\Phi_{1x}+\frac{1}{3!}h^3\Phi_{1xxx} -a_1\zeta+\dots\right)_x , \label{eqa05}$$ $$\dot\zeta=\left((h+\zeta)\Phi_{2x}-\frac{1}{3!}h^3\Phi_{2xxx} -a_2\zeta+\dots\right)_x . \label{eqa06}$$ Here “…” stand for $\mathcal{O}_1(\Phi_jh^2\zeta/l^3)+\mathcal{O}_2(\Phi_jh^4/l^5)$. Equations (\[eqa03\]), (\[eqa05\]) and (\[eqa06\]) form a self-contained equation system. It is convenient to distinguish two main time-modes in various fields: the average over vibration period part and the pulsation part; $$\begin{array}{l} \zeta=\eta(\tau,x)+\xi(\tau,x)e^{i\omega t}+c.c.+\dots\,, \\[10pt] \Phi_j=\varphi_j(\tau,x)+\psi_j(\tau,x)e^{i\omega t}+c.c.+\dots\,, \end{array}$$ where $\tau$ is a “slow” time related to the average over vibration period evolution and “…” stand for higher powers of $e^{i\omega t}$. In order to develop an expansion in small parameter $\omega^{-1}$, we have to adopt a certain hierarchy of smallness of parameters, fields, etc. We adopt small deviation from the instability threshold $B_1\sim\omega^{-1}$. Then $\eta\sim\omega^{-1}$ and $\partial_x\sim\omega^{-1/2}$ (cf. [@Lyubimov-Cherepanov-1987; @Zamaraev-Lyubimov-Cherepanov-1989]). It is as well established (e.g., [@Lyubimov-Cherepanov-1987; @Zamaraev-Lyubimov-Cherepanov-1989]) that for finite wavelength perturbations (finite $k\ne0$) $B_0(k)=B_0(0)+Ck^2+\mathcal{O}(k^4)$. Generally, the expansion of the exponential growth rate of perturbations in the series for $B_1$ near the instability threshold possesses a non-zero linear part, and $B_0(k)-B_0(0)\sim k^2$; therefore, $\partial_\tau\sim\mathcal{O}_1(B_1)+\mathcal{O}_2(k^2)\sim\omega^{-1}$. It is more convenient to determine the order of magnitude of $\xi$, $\varphi_j$ and $\psi_j$ in the course of development of the expansion. Collecting terms with $e^{i\omega t}$ in equations (\[eqa05\]) and (\[eqa06\]), one finds $$\begin{aligned} i\omega\xi+\xi_\tau=\Big(-(h-\eta)\psi_{1x}+\frac{1}{3!}h^3\psi_{1xxx}\qquad \nonumber\\[5pt] {}+\xi\varphi_{1x}-A_1\eta+\dots\Big)_x\,, \label{eqa07}\end{aligned}$$ $$\begin{aligned} i\omega\xi+\xi_\tau=\Big((h+\eta)\psi_{2x}-\frac{1}{3!}h^3\psi_{2xxx}\qquad \nonumber\\[5pt] {}+\xi\varphi_{2x}-A_2\eta+\dots\Big)_x\,, \label{eqa08}\end{aligned}$$ where “…” stand for $\mathcal{O}_1((\xi\varphi+\eta\psi)h^2/l^4)+\mathcal{O}_2(\psi\,h^4/l^6)$. Terms constant with respect to $t$ sum up to $$\eta_\tau=\Big(-(h-\eta)\varphi_{1x}+\xi\psi_{1x}^\ast+c.c.-A_1\xi^\ast+c.c.+\dots\Big)_x\,, \label{eqa09}$$ $$\eta_\tau=\Big((h+\eta)\varphi_{2x}+\xi\psi_{2x}^\ast+c.c.-A_2\xi^\ast+c.c.+\dots\Big)_x\,, \label{eqa10}$$ where the superscript “$*$” stands for complex conjugate and “…” stand for $\mathcal{O}_1((\eta\varphi+\xi\psi)h^2/l^4)+\mathcal{O}_2(\varphi h^4/l^6)$. The difference of equations (\[eqa07\]) and (\[eqa08\]) yields $\psi_j\sim\omega^{-1/2}$, and the difference of (\[eqa09\]) and (\[eqa10\]) yields $\varphi_j\sim\omega^{-1}$. For dealing with non-linear terms in what follows, it is convenient to extract the first correction to $\psi_j$ explicitly, i.e. write $\psi_j=\psi_j^{(0)}+\psi_j^{(1)}+\dots$, where $\psi_j^{(1)}\sim\omega^{-1}\psi _j^{(0)}\sim\omega^{-3/2}$. Equation (\[eqa07\]) (or (\[eqa08\])) yields in the leading order ($\sim\omega^{-3/2}$) $$\xi=\frac{i}{\omega}(h\psi_{1x}+A_1\eta)_x\sim\omega^{-\frac{5}{2}}\,. \label{eqa11}$$ Considering the difference of (\[eqa08\]) and (\[eqa07\]), one has to keep in mind that we are interested in localized patterns for which $\Phi_{jx}(x=\pm\infty)=0$, $\zeta(x=\pm\infty)=0$. Hence, this difference can be integrated with respect to $x$, taking the form $$\begin{aligned} h(\psi_1+\psi_2)_x-\eta(\psi_1-\psi_2)_x -\frac{1}{6}h^3(\psi_1+\psi_2)_{xxx}\qquad \nonumber\\[5pt] {}-\xi(\varphi_1-\varphi_2)_x+(A_1-A_2)\eta+\dots=0\,, \nonumber\end{aligned}$$ which yields in the first two orders of smallness $$h(\psi_1^{(0)}+\psi_2^{(0)})_x=-(A_1-A_2)\eta=-\frac{\rho_2-\rho_1}{\rho_2+\rho_1}\eta\,, \label{eqa12}$$ $$h(\psi_1^{(1)}+\psi_2^{(1)})_x=(\psi_1^{(0)}-\psi_2^{(0)})_x\eta +\frac{1}{6}h^3(\psi_1^{(0)}+\psi_2^{(0)})_{xxx}\,. \label{eqa13}$$ The difference and the sum of equations (\[eqa09\]) and (\[eqa10\]) yield in the leading order, respectively, $$\varphi_1=-\varphi_2\equiv\varphi\,, \label{eqa14}$$ $$\eta_\tau=-h\varphi_{xx}\,. \label{eqa15}$$ Let us now consider equation (\[eqa03\]). We will collect groups of terms with respect to power of $e^{i\omega t}$ and the order of smallness in $\omega^{-1}$.\ $\underline{\sim\omega^{+\frac{1}{2}}e^{i\omega t}}$: $$i\omega B_0(\rho_1\psi_1^{(0)}-\rho_2\psi_2^{(0)})=0\,.$$ We introduce $$\psi^{(0)}\equiv\rho_j\psi_j^{(0)}\,. \label{eqa16}$$ The last equation and equation (\[eqa12\]) yield $$\psi_x^{(0)}=-\frac{1}{h}\frac{\rho_1\rho_2(\rho_2-\rho_1 )}{(\rho_2+\rho_1)^2}\eta\,. \label{eqa17}$$ $\underline{\sim\omega^0e^{i\omega t}}$: $$\mbox{No contributions.}$$ $\underline{\sim\omega^{-\frac{1}{2}}e^{i\omega t}}$: $$\begin{array}{l} \displaystyle i\omega B_1\underbrace{(\rho_1\psi_1^{(0)}-\rho_2\psi_2^{(0)})}_{\quad=0} +i\omega B_0(\rho_1\psi_1^{(1)}-\rho_2\psi_2^{(1)}) \\[20pt] \displaystyle\qquad {}+B_0\underbrace{(\rho_1\psi_1^{(0)}-\rho_2\psi_2^{(0)})_\tau}_{\quad=0} \\[20pt] \displaystyle\qquad\qquad {}+i\omega B_0\frac{h^2}{2}\underbrace{(\rho_2\psi_{2xx}^{(0)}-\rho_1\psi_{1xx}^{(0)})}_{\quad=0}=0\,. \end{array}$$ (We marked the combinations which are known to be zero from the leading order of expansion.) Similarly to (\[eqa16\]), we introduce $$\psi^{(1)}\equiv\rho_j\psi_j^{(1)}\,. \label{eqa18}$$ The last equation and equation (\[eqa13\]) yield $$\begin{aligned} &&\hspace{-20pt}\displaystyle \psi_x^{(1)}=\frac{1}{h}\frac{\rho_2-\rho_1}{\rho_2+\rho_1}\psi_x^{(0)}\eta +\frac{h^2}{6}\psi_{xxx}^{(0)} \nonumber\\[10pt] &&\displaystyle =-\frac{\rho_1\rho_2(\rho_2-\rho_1)^2}{h^2(\rho_2+\rho_1)^3}\eta -\frac{h\rho_1\rho_2(\rho_2-\rho_1)}{6(\rho_2+\rho_1)^2}\eta_{xx}\,. \label{eqa19}\end{aligned}$$ $\underline{\sim\omega^{-1}(e^{i\omega t})^0}$: $$B_0[-\rho_2(A_2\psi_{2x}^{(0)\ast}+c.c.)+\rho_1(A_1\psi_{1x}^{(0)\ast}+c.c.)]+\eta=0\,.$$ Substituting (\[eqa16\]) and (\[eqa17\]) into the last equation gives $$\left[-\frac{2B_0\rho_1\rho_2(\rho_2-\rho_1)^2}{h(\rho_2+\rho_1)^3}+1\right]\eta=0\,.$$ Thus we obtain the solvability condition, which poses a restriction on $B_0$; this restriction determines the linear instability threshold $$B_0=\frac{(\rho_2+\rho_1)^3h}{2\rho_1\rho_2(\rho_2-\rho_1)^2}\,. \label{eqa20}$$ $\underline{\sim\omega^{-2}(e^{i\omega t})^0}$: (using (\[eqa14\]) for $\varphi_j$) $$\begin{array}{l} \displaystyle B_1\underbrace{[-\rho_2(A_2\psi_{2x}^{(0)\ast}+c.c.) +\rho_1(A_1\psi_{1x}^{(0)\ast}+c.c.)]}_{\qquad=-\eta/B_0}\\[5pt] \displaystyle \qquad {}+B_0\bigg[(\rho_2+\rho_1)\varphi_\tau+\rho_2|\psi_{2x}^{(0)}|^2 \\[15pt] \displaystyle {}-\rho_2\Big(A_2\psi_{2x}^{(1)\ast}+c.c.-A_2\frac{h^2}{2}\psi_{2xxx}^{(0)\ast}+c.c.\Big) -\rho_1|\psi_{1x}^{(0)}|^2 \\[15pt] \displaystyle\quad {}+\rho_1\Big(A_1\psi_{1x}^{(1)\ast}+c.c. -A_1\frac{h^2}{2}\psi_{1xxx}^{(0)\ast}+c.c.\Big)\bigg]=\eta_{xx}\,. \end{array}$$ Substituting $\psi_j^{(n)}$ from (\[eqa16\])–(\[eqa19\]), one can rewrite the latter equation as $$\begin{array}{r} \displaystyle -\frac{B_1}{B_0}\eta+B_0\Bigg[(\rho_2\!+\!\rho_1)\varphi_\tau -\frac{\rho_2\!-\!\rho_1}{\rho_2\rho_1}\left(\!\frac{\rho_1\rho_2(\rho_2\!-\!\rho_1)\eta}{h(\rho_2\!+\!\rho_1)^2}\!\right)^2 \\[15pt] \displaystyle {}-\frac{2\rho_1\rho_2(\rho_2-\rho_1)^3}{h^2(\rho_2+\rho_1)^4}\eta^2 -\frac{h\rho_1\rho_2(\rho_2-\rho_1)^2}{3(\rho_2+\rho_1)^3}\eta_{xx} \qquad\\[15pt] \displaystyle {}+h^2\frac{\rho_1\rho_2(\rho_2-\rho_1)^2}{h(\rho_2+\rho_1)^3}\eta_{xx}\Bigg]=\eta_{xx}\,. \end{array}$$ Together with equation (\[eqa15\]) the latter equation form the final [*system of governing equations for long-wavelength perturbations*]{} of the flat-interface state: $$\left\{ \begin{array}{rcl} \displaystyle B_0(\tilde\rho_2\!+\!\tilde\rho_1)\tilde\varphi_{\tilde\tau}& \displaystyle \!\!=\!\!& \displaystyle \left[1\!-\!\frac{\tilde{h}^2}{3}\right]\tilde\eta_{\tilde x\tilde x} +\frac{3}{2\tilde h}\frac{\tilde\rho_2\!-\!\tilde\rho_1}{\tilde\rho_2\!+\!\tilde\rho_1}\tilde\eta^2 +\frac{B_1}{B_0}\tilde\eta\,,\\[15pt] \displaystyle \tilde\eta_{\tilde\tau}& \displaystyle \!\!=\!\!& \displaystyle -\tilde h\tilde\varphi_{\tilde x\tilde x}\,. \end{array} \right. \label{eq08}$$ Here we explicitly mark the dimensionless variables and parameters with the tilde sign to distinguish them from original dimensional variables and parameters. Above in this paragraph, the tilde sign was omitted to make calculations possibly less laborious. For convenience we explicitly specify how to read rescaling (\[rescaling1\]) with the tilde-notation: $x=L\tilde x$, $t=(L/b)\tilde t$, $\rho_i=\rho_\ast\tilde\rho_i$, etc. The expression for $B_0$ (\[eqa20\]) in the original dimensional terms reads $$B_0=\frac{\rho_\ast(\rho_2+\rho_1)^3h}{2\rho_1\rho_2(\rho_2-\rho_1)^2} \sqrt{\frac{(\rho_2-\rho_1)g}{\alpha}}\,. \label{eq10}$$ We remark that equation system (\[eq08\]) is valid for $B_1$ small compared to $B_0$, otherwise one cannot stay within the long-wavelength approximation. On rare occasions it is possible to use long-wavelength for finite deviations from the linear instability threshold and derive certain information on the system dynamics (e.g., in [@Goldobin-Lyubimov-2007] for Soret-driven convection from localized sources of heat or solute in a thin porous layer, an unavoidable appearance of patterns similar to hydraulic jumps [@Watson-1964] was predicted within the long-wavelength approximation though for a finite deviation from the linear instability threshold). On the long-wavelength character of the linear instability ---------------------------------------------------------- In the text above, we relied on the fact that instability is long-wavelength for thin enough layers. Now we have appropriate quantifiers to specify quantitatively, what“thin enough” means. According to [@Lyubimov-Cherepanov-1987], we require that $\tilde h<\sqrt{3}$. Remarkably, we can see a footprint of this fact from equation system (\[eq08\]) with multiplier $[1-\tilde{h}^2/3]$ ahead of $\tilde\eta_{\tilde x\tilde x}$. Indeed, the exponential growth rate $\tilde\lambda$ of linear normal perturbations $(\tilde\eta,\tilde\varphi)\propto\exp(\tilde\lambda\tilde t+i\tilde k\tilde x)$ of the trivial state obeys $$\tilde\lambda^2=\frac{\tilde h\,\tilde{k}^2}{B_0(\tilde\rho_2+\tilde\rho_1)} \left(-\Bigg[1-\frac{\tilde{h}^2}{3}\Bigg]\tilde{k}^2+\frac{B_1}{B_0}\right). \label{eq11}$$ Below the linear instability threshold of infinitely long wavelength perturbations, i.e. for $B_1<0$, there are no growing perturbations for $\tilde{h}<\sqrt{3}$, while the perturbations with large enough $\tilde k$ grow for $\tilde{h}>\sqrt{3}$. Of course, this analysis of equation system (\[eq08\]) only highlights the long-wavelength character of the linear instability, since it deals with the limit of small $\tilde{k}$ and does not provide information on the linear stability for finite $\tilde{k}$. A comprehensive proof of the long-wavelength character of the instability for $\tilde{h}<\sqrt{3}$ comes from [@Lyubimov-Cherepanov-1987]. In the following we will consider system behavior below the linear instability threshold, i.e. for negative $B_1$. It is convenient to make further rescaling of coordinates and variables: $$\begin{array}{l} \displaystyle \tilde x\to x\sqrt{\frac{B_0}{(-B_1)}\Bigg[1-\frac{\tilde{h}^2}{3}\Bigg]}\,, \\[20pt] \displaystyle \tilde t\to t\sqrt{\frac{\tilde\rho_2-\tilde\rho_1}{\tilde{h}}\frac{B_0^3}{B_1^2} \Bigg[1-\frac{\tilde{h}^2}{3}\Bigg]}\,,\\[20pt] \displaystyle \tilde\eta\to\eta\,\tilde{h}\frac{\tilde\rho_2+\tilde\rho_1}{\tilde\rho_2-\tilde\rho_1}\frac{(-B_1)}{B_0}\,, \\[20pt] \displaystyle \tilde\varphi\to\varphi\sqrt{\frac{(\tilde\rho_2+\tilde\rho_1)^2}{(\tilde\rho_2-\tilde\rho_1)^3} \frac{B_1^2}{\tilde{h}B_0^3}\Bigg[1-\frac{\tilde{h}^2}{3}\Bigg]}\,. \end{array} \label{eq12}$$ We note that this implies the following rescaling of [*initial dimensional*]{} coordinates and variables: $$\begin{array}{l} \displaystyle x\to x\,L\sqrt{\frac{B_0}{(-B_1)}\Bigg[1-\frac{h^2}{3L^2}\Bigg]}\,,\\[20pt] \displaystyle t\to t\sqrt{\frac{\rho_2-\rho_1}{\rho_\ast}\frac{L^3B_0^3}{h\,b^2B_1^2} \Bigg[1-\frac{h^2}{3L^2}\Bigg]}\,,\\[20pt] \displaystyle \eta\to\eta\,h\frac{\rho_2+\rho_1}{\rho_2-\rho_1}\frac{(-B_1)}{B_0}\,,\\[20pt] \displaystyle \varphi\to\varphi\sqrt{\frac{\rho_\ast(\rho_2+\rho_1)^2}{(\rho_2-\rho_1)^3} \frac{L^3B_1^2}{h\,b^2B_0^3}\Bigg[1-\frac{h^2}{3L^2}\Bigg]}\,. \end{array} \label{eq13}$$ After this rescaling, equation system (\[eq08\]) takes the zero-parametric form; $$\begin{aligned} \dot\varphi&=&\eta_{xx}+\frac{3}{2}\eta^2-\eta\,, \label{eq14} \\ \dot\eta&=&-\varphi_{xx}\,. \label{eq15}\end{aligned}$$ The derivation of the latter equation system itself is one of the main results we report with this paper, as it allows consideration of the evolution of quasi-steady patterns in the two-layer fluid system under the action of the vibration field. The ‘plus’ Boussinesq equation and the original Boussinesq equation for gravity waves in shallow water ------------------------------------------------------------------------------------------------------ The equation system (\[eq14\])–(\[eq15\]) can be rewritten in the form of a ‘plus’ Boussinesq equation (plus BE); $$\ddot{\eta}-\eta_{xx}+\left(\frac{3}{2}\eta^2+\eta_{xx}\right)_{xx}=0\,. \label{eq_plBE}$$ Meanwhile, the original Boussinesq equation B (BE B) for gravity waves in a shallow water layer [@Boussinesq-1872] or in a two-layer system without vibrations [@Choi-Camassa-1999] reads $$\ddot{\eta}-\eta_{xx}-\left(\frac{3}{2}\eta^2+\eta_{xx}\right)_{xx}=0\,. \label{eq_BEB}$$ Both systems are fully integrable and multi-soliton solutions are known for them from the literature (e.g., [@Manoranjan-etal-1984; @Manoranjan-etal-1988; @Bogdanov-Zakharov-2002]). However, their dynamics is essentially different; the original BE B suffers from the short-wave instability, while the plus BE is free from this instability. Solitons in the plus BE can be unstable, decaying into pairs of stable solitons or experiencing explosive formation of sharp peaks in finite time [@Bogdanov-Zakharov-2002; @Bona-Sachs-1988; @Liu-1993]. In the sections below we will provide overview of the soliton dynamics for equation (\[eq\_plBE\]) in relation to the fluid dynamical system we deal with. Prior to doing so, in this subsection, we would like to focus more on discussion of different kinds of the generalized Boussinesq equation and their relationships with dynamics of systems of fluid layers. Small-amplitude gravity waves in shallow water are governed by the set A Boussinesq equations (equation system (25) in [@Boussinesq-1872]) which read in our terms after proper rescaling as $$\left\{ \begin{array}{rcl} \displaystyle\dot{\eta}+\varphi_{xx}&\!\!=\!\!& \displaystyle-(\eta\,\varphi_x)_x+\frac{1}{6}\varphi_{xxxx}\,, \\[10pt] \displaystyle\dot{\varphi}+\eta&\!\!=\!\!& \displaystyle-\frac{1}{2}(\varphi_x)^2+\frac{1}{2}\dot\varphi_{xx}\,, \end{array} \right.$$ where the terms in the right hand side of equations are small, i.e., both nonlinearity and dispersion are small. To the leading corrections pertaining to nonlinearity and dispersion, the latter equation system can be recast as $$\ddot{\eta}-\eta_{xx}=\left((\varphi_x)^2+\frac{1}{2}\eta^2+\eta_{xx}\right)_{xx}\,,$$ where small terms are collected in the r.h.s. part of the equation. For waves propagating in one direction $\partial_x\approx\pm\partial_t$ and, to the leading corrections, one can make substitution $(\varphi_x)^2\approx(\dot\varphi)^2\approx\eta$, which yields equation (\[eq\_BEB\]). Thus, the Boussinesq equation for the classical problem of waves in shallow water is not only inaccurate far from the edge of the spectrum of soliton speed (near $c=1$) but is also inappropriate for consideration of collisions of counterpropagating waves (as $|\varphi_x|\ne|\dot\varphi|$ for them). In contrast, the equations we derived for our physical system are accurate close to the vibration-induced instability threshold for the entire range of soliton speeds and all kinds of soliton interactions as long as the profile remains smooth. It is also noteworthy, that character of the original Boussinesq equation B is inherent to the dynamics of inviscid fluid layers in force fields and does not change without special external fields, the action of which cannot be formally represented by any correction to the gravity. The case of a vibration field turns out to be one such and yields the dynamics governed by the plus BE. Long waves below the linear instability threshold {#sec_solitons} ================================================= In this section we consider waves in the dynamic system (\[eq14\])–(\[eq15\]). In equations (\[eq13\]), one can see how the rescaling of each coordinate and variable depends on $(-B_1)=B_0-B$. From these dependencies it can be seen, that for patterns in the dynamic system (\[eq14\])–(\[eq15\]) the corresponding patterns in real time–space will obey the following scaling behavior near the linear instability threshold: spatial extent $x_\ast\propto1/\sqrt{B_0-B}$, reference time $t_\ast\propto1/(B_0-B)$, reference profile deviation $\eta_\ast\propto(B_0-B)$. Linear waves: dispersion equation, group velocity ------------------------------------------------- Let us first describe propagation of small perturbation, linear waves, in the dynamic system (\[eq14\])–(\[eq15\]). For normal perturbations $(\eta,\varphi)\propto\exp(-i\Omega t+ikx)$ the oscillation frequency reads $$\Omega(k)=k\sqrt{1+k^2}\,. \label{eq16}$$ The corresponding phase velocity is $$v_\mathrm{ph}=\Omega/k=\sqrt{1+k^2}\,, \label{eq17}$$ and the group velocity, which describes propagation of envelopes of wave packages, is $$v_\mathrm{gr}=\frac{\mathrm{d}\Omega}{\mathrm{d}k}=\frac{1+2k^2}{\sqrt{1+k^2}}\,. \label{eq18}$$ One can see that the minimal group velocity is 1 and the group velocity $v_\mathrm{gr}$ monotonously increases as wavelength decreases (see Fig.\[fig2\]). Solitons -------- The dynamic system (\[eq14\])–(\[eq15\]) admits time-independent-profile solutions, solitons $\eta(x,t)=\eta(x-ct)$, where $c$ is the soliton velocity. With identical equality $\partial_t\eta(x-ct)=-c\partial_x\eta(x-ct)$, for localized patterns, which vanish at $x\to\pm\infty$, equation (\[eq15\]) can be once integrated and yields $\varphi^\prime=c\eta$ (here the prime denotes the differentiation with respect to argument). Eq.(\[eq14\]) takes the form $$0=\eta^{\prime\prime}+\frac32\eta^2-(1-c^2)\eta\,. \label{eq19}$$ The latter equation admits the soliton solution $$\eta_0(x,t)=\frac{1-c^2}{\displaystyle\cosh^2\frac{\sqrt{1-c^2}(x\pm ct)}{2}}\,, \label{eq20}$$ the propagation direction ($+c$ or $-c$) is determined by the flow, $\varphi^\prime=\pm c\eta$. The family of soliton solutions turns out to be one-parametric, parameterized by the speed $c$ only. The speed $c$ varies within the range $[0,1]$; standing soliton ($c=0$) is the sharpest and the tallest one and for the fastest solitons, $c\to1$, the spatial extent tends to infinity, while the height tends to $0$. Considering in the same way a non-rescaled equation system (\[eq08\]), one can see, that for a given physical system with vibration parameter $B$ as a control parameter, the shape of a soliton solution is controlled by combination $$[(-B_1)/B_0-\tilde{c}^2\tilde{h}^{-1}B_0(\tilde\rho_2+\tilde\rho_1)]\,. \label{eq21}$$ This means that one and the same interface inflection soliton can exist for different values of $B$, though, since the shape-controlling parameter (\[eq21\]) should be the same, the non-rescaled soliton run speed $\tilde{c}$ grows as the departure from the threshold $(-B_1)$ increases. Since $v_\mathrm{gr}\ge1$ (see Fig.\[fig2\]) and $c^2\le1$, solitons of arbitrary height travel slower than any small perturbations of the flat-interface state. The maximal speed of solitons, $c_{\max}=1$, coincides with the minimal group velocity of linear waves. This yields notable information on the system dynamics. Fast solitons with $c$ tending to $1$ from below are extended and have a small height (see equation (\[eq20\])), while envelopes of long linear waves propagate with velocity $v_\mathrm{gr}$ tending to $1$ from above. This means that envelopes of small-height soliton packages travel faster than solitons in these packages. The issue of the generality of situations where the ranges of the possible soliton velocities and the group velocities of linear waves do not overlap but only touch each other is interestingly addressed in [@Akylas-1993; @Longuet-Higgins-1993] from the view point of emission of wave packages by the soliton (or the impossibility of such an emission). Stability of solitons --------------------- The stability properties of solitons in the ‘plus’ Boussinesq equation were addressed in literature [@Manoranjan-etal-1988; @Bona-Sachs-1988; @Liu-1993]; in [@Bona-Sachs-1988] the solitons with $1/2\le c\le1$ were proved to be stable and in [@Liu-1993] the solitons with $c<1/2$ were proved to be unstable. One can add more subtle details to this information: the spectrum of Lyapunov exponents (exponential growth rate) and the dependence of the scenario of nonlinear growth of perturbations on the initial perturbation. The problem of linear stability of the soliton $\eta_0(x-ct)$ to perturbations $\big(e^{\lambda t}\eta_1(x_1),e^{\lambda t}\varphi_1(x_1)\big)$ in the copropagating reference frame $x_1=x-ct$ reads $$\begin{aligned} \lambda\varphi_1+c\varphi_1^{\prime}&=& \eta_1^{\prime\prime}+3\eta_0(x_1)\,\eta_1-\eta_1\,, \label{eq22} \\ \lambda\eta_1+c\eta_1^{\prime}&=&-\varphi_1^{\prime\prime} \label{eq23}\end{aligned}$$ with boundary conditions $$\eta_1(\pm\infty)=\varphi_1(\pm\infty)=0\,. \label{eq24}$$ The eigenvalue problem (\[eq22\])–(\[eq24\]) was solved numerically with employment of the shooting method. The spectra of eigenvalues $\lambda$ for different $c$ are plotted in Fig.\[fig3\] and the first two eigenmodes of perturbations of the standing soliton ($c=0$) are plotted in Fig.\[fig4\](b). In Fig.\[fig4\](a), one can see the exponential growth rate ${\mathrm{Re}}(\lambda)$ of perturbations; the standing and slow solitons with $c<0.5$ are unstable, while the fast solitons with $c\ge 0.5$ are stable. The scenarios of evolution of unstable solitons were observed numerically by means of direct numerical simulation of the dynamic system (\[eq14\])–(\[eq15\]) with the finite difference method in an $x$-domain of length $200$ with periodic boundary conditions and the space step size $h_x=0.05$. [^3] As in [@Manoranjan-etal-1984; @Manoranjan-etal-1988; @Bogdanov-Zakharov-2002], two possible scenarios were observed: (i) soliton explosion with formation of a finite amplitude relief or, possibly, layer rupture; (ii) falling-apart of the soliton into two stable solitons (Fig.\[fig5\]). Since the phase space of the system is infinite-dimensional, the problem of discrimination of the initial perturbations leading to explosion and those leading to falling-apart may be generally nontrivial. However, in Fig.\[fig3\] one can see that there is only one instability mode for $c<1/2$ and the nonlinear evolution of perturbations turns out to depend only on projection of the small initial perturbation on this unstable direction. If the instability mode is normalized in such a way that $\eta_1(x_1=0)>0$ (cf. Fig.\[fig4\](b)), the initial perturbations with a positive scalar product with the instability mode lead to explosion, while the perturbations with a negative scalar product lead to falling-apart into two stable solitons. Soliton gas ----------- In Fig. \[fig6\], a sample of the system dynamics from arbitrary initial conditions is presented in domain $x\in[0;250]$ with periodic boundary conditions. One can see that, beyond the locations of formation of singularities [@Goldobin-etal-EPL-2014-solitons], this dynamics can be well treated as the kinetics of a gas of stable solitons. For wave dynamics in soliton-bearing systems, statistical physics approaches which describe the dynamics of dense soliton gases can be developed [@El-Kamchatnov-2005]. Knowing that the system dynamics can be viewed as a kinetics of a soliton gas, we can readdress the question of relationships between the group velocity of linear waves and the speed of solitons. The fast solitons with $c\to1$ have height $\eta_\mathrm{max}=[1-c^2]\to0$ and width $\delta\propto1/\sqrt{1-c^2}\to\infty$, i.e. must obey the laws established for the linear waves with wavenumber $k\to0$. Meanwhile for the latter waves we know the group velocity $v_\mathrm{gr}=1+(3/2)k^2+\mathcal{O}(k^4)$ \[see Eq.(\[eq18\])\]. Thus, the envelopes of traveling solitons always travel faster than these solitons. For an envelope of a nearly monochromatic wave the possibility of such a behavior is obvious, while for a gas of quasi-particles additional explanations are needed. For waves of density of quasi-particles (correspond to waves of envelope) it is actually possible to travel faster than the fastest particles if these particles have a finite “collision diameter”. (For a better intuition on this, one can imagine the elastic collision of two hard spheres moving along a line. These spheres exchange their momentums from the distance equal to the sum of their radii. If they are identified only by the momentum, they effectively jump for the distance of their interaction and proceed to move with the initial momentums.) Indeed, colliding copropagating solitons exchange their momentums, which means they efficiently exchange their locations, not crossing one another, but approaching for a certain finite distance, the collision diameter. Thus the momentum efficiently jumps in the direction of soliton motion for this distance and, during one and the same time interval, the wave in a gas can cover a longer distance than the gas quasi-particles. As soon as one can speak of the soliton density waves in a soliton gas, the question of the criterion for this gas to be considered as a continuous medium or a vacuum arises. For instance, it is obvious that one cannot speak of density waves or envelope waves for a system state with a single soliton; this is a vacuum. Whereas, for a continuous medium the concept of the group velocity should work well. Let us consider a gas of solitons with characteristic width $\delta\gg1$, i.e., quasi-particle speed $c^2=1-\delta^{-2}$ \[see Eq.(\[eq20\])\]. The signal transfer speed due to collisions (as described in the paragraph above) is larger that the particle speed, according to $v_\mathrm{gr}\approx c/(1-n\delta)$, where $n$ is the soliton number density. Mathematically, the criterion for $n$ is of interest. For us to be able to consider the soliton gas as a continuous medium, with the group velocity featured by Eq.(\[eq18\]), $v_\mathrm{gr}$ should at least reach $1$. Hence, $c/(1-n_\mathrm{min}\delta)=1$, which can be rewritten as $1-\delta^{-2}/2=1-n_\mathrm{min}\delta$, and one finally finds $$n_\mathrm{min}\sim\delta^{-3}\,. \label{eq:g01}$$ Interestingly, the last equation means that the maximal characteristic inter-soliton distance $\delta_\ast=1/n_\mathrm{min}$ for the gas to be a continuous medium but not a vacuum scales with the soliton width $\delta$ as $$\delta_\ast\sim\delta^3\,. \label{eq:g02}$$ Conclusion {#sec_concl} ========== We have considered the dynamics of patterns on the internal surface of the horizontal two-layer system of inviscid fluids subject to tangential vibrations. For thin layers ($h<\sqrt{3\alpha/[(\rho_2-\rho_1)g]}$) the instability is known to be long-wavelength and subcritical [@Lyubimov-Cherepanov-1987; @Zamaraev-Lyubimov-Cherepanov-1989]. The governing equations for long-wavelength patterns below the linear instability threshold have been derived—equation system (\[eq14\])–(\[eq15\])—allowing for the first time theoretical analysis for time-dependent patterns in the system and for stability of time-independent (quasi-steady) patterns. We note that the stability analysis for the only time-independent localized patterns in the system, standing solitons, has revealed them to be unstable. The system dynamics is found to be governed by dynamic system (\[eq14\])–(\[eq15\]) which is equivalent to the ‘plus’ Boussinesq equation. For dynamic system (\[eq14\])–(\[eq15\]), one-parametric family of localized solutions of time-independent profile, solitons, exists (equation (\[eq20\])). These solitons are up-standing embossments of the interface (cf. black curve in Fig.\[fig4\](b)) and are parameterized by the soliton speed $c$ only, which varies from $c=0$ (the tallest and sharpest solitons) to $c=1$ (solitons with width tending to infinity and height tending to zero). The standing and slow solitons ($c<1/2$) are unstable [@Liu-1993], while the fast solitons ($c\ge 1/2$) are stable [@Bona-Sachs-1988]. The group velocity of linear waves in the system is $v_\mathrm{gr}\ge1$, meaning that all the solitons travel more slowly than any wave packages of small perturbations of the flat-interface state. Two scenarios of development of the instability of slow solitons are possible, depending on the initial perturbation: explosion (probably leading to further layer rupture) and splitting into a pair of fast stable solitons. No other localized waves have been detected with direct numerical simulation, meaning that this one-parameter family of solitons is the only localized waves in the system. The system dynamics can be fully represented as the kinetics of gas of solitons before an explosion (and after it). It is not possible to compare our results to the results presented by Wolf [@Wolf-1961; @Wolf-1970] in detail. Wolf presented the wave patterns of the interface for the inverted state (the heavy liquid above the light one) above the linear instability threshold of the flat-interface non-inverted state. Meanwhile, we consider waves on the interface for the non-inverted state below the threshold, and our non-trivial findings pertain specifically to this case but not to the case of the inverted stratification. We are thankful to Dr. Maxim V. Pavlov and Dr. Takayuki Tsuchida for their useful comments on the work and drawing our attention to the fact that system (\[eq14\])–(\[eq15\]) is identical to the ‘plus’ Boussinesq equation. We thank Prof.Jeremy Levesley for his help with manuscript preparation. The work has been supported by the Russian Science Foundation (grant no. 14-21-00090). To the memory of our teachers and friends A. A. Cherepanov, D. V. Lyubimov, and S. V. Shklyaev. [25]{} G. H. Wolf, [*The dynamic stabilization of the Rayleigh-Taylor instability and the corresponding dynamic equilibrium*]{}, Z. Phys. [**227**]{}, 291 (1969). G. H. Wolf, [*Dynamic Stabilization of the Interchange Instability of a Liquid-Gas Interface*]{}, Phys. Rev. Lett. [**24**]{}, 444 (1970). D. V. Lyubimov and A. A. Cherepanov, [*Development of a steady relief at the interface of fluids in a vibrational field*]{}, Fluid Dynamics [**21**]{}, 849 (1986). D. V. Lyubimov, M. V. Khenner, and M. M. Shotz, [*Stability of a Fluid Interface Under Tangential Vibrations*]{}, Fluid Dynamics [**33**]{}, 318 (1998). M. V. Khenner, D. V. Lyubimov, T. S. Belozerova, and B. Roux, [*Stability of Plane-Parallel Vibrational Flow in a Two-Layer System*]{}, European Journal of Mechanics B/fluids [**18**]{}, 1085 (1999). A. V. Zamaraev, D. V. Lyubimov, and A. A. Cherepanov, [*On equlibrium shapes of the interface between two fluids in vibrational field*]{}, in [*Hydrodynamics and Processes of Heat and Mass Transfer*]{} (Ural Branch of Acad. of Science of USSR, Sverdlovsk, 1989), pp. 23–26. \[In Russian. Since the translation of this paper into English in not available in the literature, it may be suitable to notice that the results of this work related to the problem we consider can be as well deduced from [@Lyubimov-Cherepanov-1987].\] S. V. Shklyaev, A. A. Alabuzhev, and M. Khenner, [*Influence of a longitudinal and tilted vibration on stability and dewetting of a liquid film*]{}, Phys. Rev. E [**79**]{}, 051603 (2009). E. S. Benilov and M. Chugunova, [*Waves in liquid films on vibrating substrates*]{}, Phys. Rev. E [**81**]{}, 036302 (2010). U. Thiele, J. M. Vega, and E. Knobloch, [*Long-wave Marangoni instability with vibration*]{}, J. Fluid Mech. [**546**]{}, 61 (2006). A. A. Nepomnyashchy and I. B. Simanovskii, [*The influence of vibration on marangoni waves in two-layer films*]{}, J. Fluid Mech. [**726**]{}, 476 (2013). D. S. Goldobin and D. V. Lyubimov, [*Soret-Driven Convection of Binary Mixture in a Horizontal Porous Layer in the Presence of a Heat or Concentration Source*]{}, JETP [**104**]{}, 830 (2007). E. J. Watson, [*The radial spread of a liquid jet over a horizontal plane*]{}, J. Fluid Mech. [**20**]{}, 481 (1964). J. Boussinesq, [*Th[é]{}orie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond*]{}, Journal de Math[é]{}matiques Pures et Appliqu[é]{}es [**17**]{}, 55 (1872). W. Choi and R. Camassa, [*Fully nonlinear internal waves in a two-fluid system*]{}, J. Fluid Mech. [**396**]{}, 1 (1999). V. S. Manoranjan, A. R. Mitchell, and J. L. Morris, [*Numerical solutions of the good Boussinesq equation*]{}, SIAM J. Sci. Statist. Comput. [**5**]{}(4), 946 (1984). V. S. Manoranjan, T. Ortega, and J. M. Sanz-Serna, [*Soliton and antisoliton interactions in the “good” Boussinesq equation*]{}, J. Math. Phys. [**29**]{}(9), 1964 (1988). L. V. Bogdanov and V. E. Zakharov, [*The Boussinesq equation revisited*]{}, Physica D [**165**]{}, 137 (2002). J. L. Bona and R. L. Sachs, [*Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation*]{}, Comm. Math. Phys. [**118**]{}, 15 (1988). Y. Liu, [*Instability of solitary waves for generalized Boussinesq equations*]{}, J. Dynam. Differential Equations [**5**]{}, 537 (1993). T. R. Akylas, [*Envelope solitons with stationary crests*]{}, Phys. Fluids [**5**]{}, 789 (1993). M. S. Longuet-Higgins, [*Capillary–gravity waves of solitary type and envelope solitons on deep water*]{}, J. Fluid Mech. [**252**]{}, 703 (1993). D. S. Goldobin, K. V. Kovalevskaya, and D. V. Lyubimov, [*Elastic and inelastic collisions of interfacial solitons and integrability of a two-layer fluid system subject to horizontal vibrations*]{}, EPL (Europhys. Lett.) [**108**]{}, 54001 (2014). G. El and A. Kamchatnov, [*Kinetic Equation for a Dense Soliton Gas*]{}, Phys. Rev. Lett. [**95**]{}, 204101 (2005). [^1]: In the idealistic dissipation-free case, when there are no attracting states, the basins of attraction are replaced with the basins of dynamics around corresponding states; i.e., of running perturbations around the flat-interface state and of evolving finite-amplitude patterns. [^2]: For problems considered in [@Lyubimov-Cherepanov-1987; @Khenner-Lyubimov-Shotz-1998; @Khenner-etal-1999; @Zamaraev-Lyubimov-Cherepanov-1989], moderate discrepancies between two thicknesses resulted only in quantitative corrections. [^3]: The fact, that unperturbed unstable analytical solution used as initial conditions could persist for up to $100$ time units, suggests the discretisation and numerical integration schemes introduce quite a small inaccuracy into the system simulation.

--- author: - | Michele Pepe, [^1], Uwe-Jens Wiese\ Bern University, Switzerland\ E-mail: , , - | Bernard B. Beard\ Christian Brothers University, Memphis, USA\ E-mail: title: 'An Efficient Cluster Algorithm for CP(N-1) Models ' --- Standard Formulation of $CP(N-1)$ Models ======================================== The manifold $CP(N-1) = SU(N)/U(N-1)$ is a $(2N-2)$-dimensional coset space relevant in the context of the spontaneous breakdown of an $SU(N)$ symmetry to a $U(N-1)$ subgroup. In particular, in more than two space-time dimensions ($d > 2$) the corresponding Goldstone bosons are described by $N \times N$ matrix-valued fields $P(x) \in CP(N-1)$ which obey $$P(x)^2 = P(x), \ P(x)^\dagger = P(x), \ \mbox{Tr} P(x) = 1.$$ For $d = 2$ the Hohenberg-Mermin-Wagner-Coleman theorem implies that the $SU(N)$ symmetry cannot break spontaneously. Correspondingly, similar to 4-dimensional non-Abelian gauge theories, the fields $P(x)$ develop a mass-gap nonperturbatively. Motivated by these observations, D’Adda, Di Vecchia, and Lüscher [@DAd78] introduced $CP(N-1)$ models as interesting toy models for QCD. The corresponding Euclidean action is given by $$\label{CPNaction} S[P] = \int d^2x \ \frac{1}{g^2} \mbox{Tr}[\partial_\mu P \partial_\mu P],$$ where $g^2$ is the dimensionless coupling constant. Note that this action is invariant under global $\Omega \in SU(N)$ transformations $$P(x)' = \Omega P(x) \Omega^\dagger,$$ and under charge conjugation $C$ which acts as $^CP(x) = P(x)^*$. D-Theory Formulation of $CP(N-1)$ Models ======================================== In this section we describe an alternative formulation of field theory in which the $2$-dimensional $CP(N-1)$ model emerges from the dimensional reduction of discrete variables — in this case $SU(N)$ quantum spins in $(2+1)$ space-time dimensions. The dimensional reduction of discrete variables is the key ingredient of D-theory, which provides an alternative nonperturbative regularization of field theory. In D-theory we start from a ferromagnetic system of $SU(N)$ quantum spins located at the sites $x$ of a $2$-dimensional periodic square lattice. The $SU(N)$ spins are represented by Hermitean operators $T_x^a = \frac{1}{2} \lambda_x^a$ (Gell-Mann matrices for the triplet representation of $SU(3)$) that generate the group $SU(N)$ and thus obey $$[T_x^a,T_y^b] = i \delta_{xy} f_{abc} T_x^c, \ \mbox{Tr}(T_x^a T_y^b) = \frac{1}{2} \delta_{xy} \delta_{ab}.$$ In principle, these generators can be taken in any irreducible representation of $SU(N)$. However, as we will see later, not all representations lead to spontaneous symmetry breaking from $SU(N)$ to $U(N-1)$ and thus to $CP(N-1)$ models. The Hamilton operator for an $SU(N)$ ferromagnet takes the form $$H = - J \sum_{x,i} T_x^a T_{x+\hat i}^a,$$ where $J>0$ is the exchange coupling. By construction, the Hamilton operator is invariant under the global $SU(N)$ symmetry, i.e. it commutes with the total spin given by $$T^a = \sum_x T_x^a.$$ The Hamiltonian $H$ describes the evolution of the quantum spin system in an extra dimension of finite extent $\beta$. In D-theory this extra dimension is not the Euclidean time of the target theory, which is part of the $2$-dimensional lattice. Instead, it is an additional compactified dimension which ultimately disappears via dimensional reduction. The quantum partition function $$Z = \mbox{Tr} \exp(- \beta H)$$ (with the trace extending over the Hilbert space) gives rise to periodic boundary conditions in the extra dimension. The ground state of the quantum spin system has a broken global $SU(N)$ symmetry. The choice of the $SU(N)$ representation determines the symmetry breaking pattern. We choose a totally symmetric $SU(N)$ representation corresponding to a Young tableau with a single row containing $n$ boxes. It is easy to construct the ground states of the $SU(N)$ ferromagnet, and one finds spontaneous symmetry breaking from $SU(N)$ to $U(N-1)$. Consequently, there are $(N^2 - 1) - (N-1)^2 = 2N - 2$ massless Goldstone bosons described by fields $P(x)$ in the coset space $SU(N)/U(N-1) = CP(N-1)$. In the leading order of chiral perturbation theory the Euclidean action for the Goldstone boson fields is given by $$\label{ferroaction} S[P] = \int_0^\beta dt \int d^2x \ \mbox{Tr} [\rho_s \partial_\mu P \partial_\mu P - \frac{2 n}{a^2} \int_0^1 d\tau \ P \partial_t P \partial_\tau P].$$ Here $\rho_s$ is the spin stiffness, which is analogous to the pion decay constant in QCD. The second term in eq.(\[ferroaction\]) is a Wess-Zumino-Witten term which involves an integral over an interpolation parameter $\tau$. For $\beta = \infty$ the system then has a spontaneously broken global symmetry and thus massless Goldstone bosons. However, as soon as $\beta$ becomes finite, due to the Hohenberg-Mermin-Wagner-Coleman theorem, the symmetry can no longer be broken, and, consequently, the Goldstone bosons pick up a small mass $m$ nonperturbatively. As a result, the corresponding correlation length $\xi = 1/m$ becomes finite and the $SU(N)$ symmetry is restored over that length scale. The question arises if $\xi$ is bigger or smaller than the extent $\beta$ of the extra dimension. When $\xi \gg \beta$ the Goldstone boson field is essentially constant along the extra dimension and the system undergoes dimensional reduction. Since the Wess-Zumino-Witten term vanishes for field constant in $t$, after dimensional reduction the action reduces to $$\label{targetaction} S[P] = \beta \rho_s \int d^2x \ \mbox{Tr}[\partial_\mu P \partial_\mu P],$$ which is just the action of the 2-d target $CP(N-1)$ model. The coupling constant of the 2-d model is determined by the extent of the extra dimension and is given by $$\frac{1}{g^2} = \beta \rho_s.$$ Due to asymptotic freedom of the 2-d $CP(N-1)$ model, for small $g^2$ the correlation length is exponentially large, i.e.$$\xi \propto \exp(4 \pi \beta \rho_s/N).$$ Here $N/4 \pi$ is the 1-loop coefficient of the perturbative $\beta$-function. Indeed, one sees that $\xi \gg \beta$ as long as $\beta$ itself is sufficiently large. In particular, somewhat counter-intuitively, dimensional reduction happens in the large $\beta$ limit because $\xi$ then grows exponentially. In D-theory one approaches the continuum limit not by varying a bare coupling constant but by increasing the extent $\beta$ of the extra dimension. This mechanism of dimensional reduction of discrete variables is generic and occurs in all asymptotically free D-theory models [@Bro99; @Bro04]. It should be noted that (just like in the standard approach) no fine-tuning is needed to approach the continuum limit. Path Integral Representation of $SU(N)$ Quantum Spin Systems ============================================================ Let us construct a path integral representation for the partition function $Z$ of the $SU(N)$ quantum spin ferromagnet introduced above. In an intermediate step we introduce a lattice in the Euclidean time direction, using a Trotter decomposition of the Hamiltonian. However, since we are dealing with discrete variables, the path integral is completely well-defined even in continuous Euclidean time. Also the cluster algorithm to be described in the following section can operate directly in the Euclidean time continuum [@Beard96]. Hence, the final results are completely independent of the Trotter decomposition. In $2$ spatial dimensions (with an even extent) we decompose the Hamilton operator into $4$ terms $$H = H_1 + H_2 + H_3 + H_4,$$ with $$H_{1,2} = \!\! \sum_{\stackrel{x = (x_1,x_2)}{x_i \rm{even}}} \!\! h_{x,i}, \ \ H_{3,4} = \!\! \sum_{\stackrel{x = (x_1,x_2)}{x_i \rm{odd}}} \!\! h_{x,i}.$$ The individual contributions $$h_{x,i} = - J \ T_x^a T_{x+\hat i}^a,$$ to a given $H_i$ commute with each other, but two different $H_i$ do not commute. Using the Trotter formula, the partition function then takes the form $$\begin{aligned} Z\!\!\!&=&\!\!\!\lim_{M \rightarrow \infty} \! \mbox{Tr} \left\{\exp(- \epsilon H_1) \exp(- \epsilon H_2) \exp(- \epsilon H_3) \exp(- \epsilon H_4) \right\}^M.\end{aligned}$$ We have introduced $M$ Euclidean time-slices with $\epsilon = \beta/M$ being the lattice spacing in the Euclidean time direction. Inserting complete sets of spin states $q \in \{u,d,s,...\}$ the partition function takes the form $$Z = \sum_{[q]} \exp(- S[q]).$$ The sum extends over configurations $[q]$ of spins $q(x,t)$ on a $(2+1)$-dimensional space-time lattice of points $(x,t)$. The Boltzmann factor is a product of space-time plaquette contributions with $$\begin{aligned} \label{Boltzmannf} &&\exp(- s[u,u,u,u]) = \exp(- s[d,d,d,d]) = 1, \nonumber \\ &&\exp(- s[u,d,u,d]) = \exp(- s[d,u,d,u]) = \frac{1}{2}[1 + \exp(- \epsilon J)], \nonumber \\ &&\exp(- s[u,d,d,u]) = \exp(- s[d,u,u,d]) = \frac{1}{2}[1 - \exp(- \epsilon J)].\end{aligned}$$ In these expressions the flavors $u$ and $d$ can be permuted to other values. All the other Boltzmann factors are zero, which implies several constraints on allowed configurations. Cluster Algorithm for $SU(N)$ Quantum Ferromagnets ================================================== Let us now discuss the cluster algorithm for the $SU(N)$ quantum ferromagnet. Just like the original $SU(2)$ loop-cluster algorithm [@Eve93; @Wie94], the $SU(N)$ cluster algorithm builds a closed loop connecting neighboring lattice points with the spin in the same quantum state, and then changes the state of all those spins to a different randomly chosen common value. To begin cluster growth, an initial lattice point $(x,t)$ is picked at random. The spin located at that point participates in two plaquette interactions, one before and one after $t$. One picks one interaction arbitrarily and considers the states of the other spins on that plaquette. One of the corners of this interaction plaquette will be the next point on the loop. For configurations $C_1 = [u,d,u,d]$ or $[d,u,d,u]$ the next point is the time-like neighbor of $(x,t)$ on the plaquette, while for configurations $C_2 = [u,d,d,u]$ or $[d,u,u,d]$ the next point is the diagonal neighbor. If the states are all the same, i.e. for $C_3 = [u,u,u,u]$ or $[d,d,d,d]$, with probability $$p = \frac{1}{2}[1 + \exp(-\epsilon J)]$$ the next point on the loop is again the time-like neighbor, and with probability $(1 - p)$ it is the diagonal neighbor. The next point on the loop belongs to another interaction plaquette on which the same process is repeated. In this way the loop grows until it finally closes. Critical slowing down in the continuum limit ============================================ In order to determine the efficiency of this algorithm one has to study its critical slowing down when one approaches the continuum limit. We have used a multi-cluster algorithm for an $SU(3)$ quantum ferromagnet which corresponds to a $CP(2)$ model. As an observable, we have chosen the uniform magnetization which gives the cleanest signal, $$M= \sum_{x,t} (\delta_{q(x,t),u}-\delta_{q(x,t),d}).$$ The autocorrelation time $\tau$ of the magnetization is determined from the exponential fall-off of the autocorrelation function. The simulations have been performed at fixed ratio $\xi/\/L\approx2.5$, for lattice sizes $L/a$ = 20, 40, 80, 160, 320, 640 and the corresponding correlation lengths $\xi/a$ = 8.87(1), 16.76(1), 32.26(3), 64.6(1), 123.4(2), 253(1). Remarkably, the autocorrelation time doesn’t change when one varies the size of the system and the correlation length and stays close to $\tau\approx1$ sweep. This is a strong indication for an almost perfect algorithm where the critical slowing down is completely eliminated. Conclusions =========== Due to a no-go theorem [@Caracciolo93], so far no efficient cluster algorithm has been developed for $CP(N-1)$ models in the usual Wilson formulation. In the D-theory formulation, one has been able to perform simulations using a multi-cluster algorithm for large correlation lengths and with a corresponding autocorrelation time of about one sweep. Remarkably, there is almost no variation of the autocorrelation time when one spans a factor of about 30 in the correlation length. The critical slowing down of the algorithm in the continuum limit is hence completely eliminated. These results can be compared to the ones obtained with the efficient multigrid algorithm [@has93]. Our method has the advantage to obtain autocorrelation times more than 20 times smaller for similar correlation lengths, the multi-cluster algorithm is in addition straightforward to implement.\ With the D-theory regularization, it has recently also been possible to simulate $CP(N-1)$ models at non-trivial $\theta$-vacuum angle [@bbb04] which is normally impossible due to a severe sign problem. [99]{} A. D’Adda, P. Di Vecchia, and M. Lüscher, *An $1/N$ expandable series of nonlinear sigma model with instantons*, Nucl.Phys. [**B146**]{} (1978) 63. R. Brower, S. Chandrasekharan, and U.-J. Wiese, *$QCD$ as a quantum link model*, Phys. Rev. [**D60**]{} (1999) 094502 \[[ hep-th/9704106]{}\]. R. Brower, S. Chandrasekharan, S. Riederer, and U.-J. Wiese, *D-Theory: Field quantization by dimensional reduction of discrete variables*, Nucl. Phys. [**B693**]{} (2004) 149 \[[hep-lat/0309182]{}\]. B. B. Beard, U.-J. Wiese, *Simulations of discrete quantum systems in continuous Euclidean time*, Phys. Rev. Lett. [**77**]{} (1996) 5130 \[[cond-mat/9602164]{}\]. H. G. Evertz, G. Lana, and M. Marcu, *Cluster algorithm for vertex models*, Phys. Rev. Lett. [**70**]{} (1993) 875-879 \[[cond-mat/9211006]{}\]. U.-J. Wiese and H.-P. Ying, *A determination of the low-energy parameters of the 2-d Heisenberg antiferromagnet*, Z. Phys. [**B93**]{} (1994) 147 \[[cond-mat/9212006]{}\]. S. Caracciolo, R. G. Edwards, A. Pelissetto, A. D. Sokal, *Wolff type embedding algorithm for general nonlinear sigma models*, Nucl. Phys. [**B403**]{} (1993) 475-541 \[[hep-lat/9205005]{}\]. M. Hasenbusch and S. Meyer, *Testing accelerated algorithms in the lattice $CP(3)$ model*, Phys. Rev. [**D45**]{} (1992) 4376-4380. B.B. Beard, M. Pepe, S. Riederer, U.-J. Wiese, *Study of $CP(N-1)$ theta-vacua by cluster-simulation of $SU(N)$ quantum spin ladders*, Phys. Rev. Lett. [**94**]{} (2005) 010603 \[[hep-lat/0406040]{}\]. [^1]: This work was supported in part by the Swiss Nationalfonds for Scientific Research.

--- abstract: 'Statistical divergence is widely applied in multimedia processing, basically due to regularity and explainable features displayed in data. However, in a broader range of data realm, these advantages may not out-stand, and therefore a more general approach is required. In data detection, statistical divergence can be used as an similarity measurement based on collective features. In this paper, we present a collective detection technique based on statistical divergence. The technique extracts distribution similarities among data collections, and then uses the statistical divergence to detect collective anomalies. Our technique continuously evaluates metrics as evolving features and calculates adaptive threshold to meet the best mathematical expectation. To illustrate details of the technique and explore its efficiency, we case-studied a real world problem of click farming detection against malicious online sellers. The evaluation shows that these techniques provided efficient classifiers. They were also sufficiently sensitive to a much smaller magnitude of data alteration, compared with real world malicious behaviours. Thus, it is applicable in the real world.' author: - | [Ruoyu Wang[$^{1,2}$]{}, Daniel Sun[$^{2,3}$]{}, Guoqiang Li[$^{1*}$]{} ]{}\ *$^{1}$School of Software, Shanghai Jiao Tong University, China\ $^{2}$School of Computer Science and Engineering, University of New South Wales, Australia\ $^{3}$Data61, CSIRO, Australia\ {ruoyu.wang,li.g}@sjtu.edu.cn, daniel.sun@data61.csiro.au* title: Statistical Detection of Collective Data Fraud --- Introduction ============ Statistical divergence is widely applied in multimedia processing. Prevalent applications include multimedia event detection [@amid2014unsupervised], content classification [@moreno2004kullback; @park2005classification] and qualification [@pheng2016kullback; @goldberger2003efficient]. It has been attracting more attention since the dawn of big data era, basically due to regularity and interpretable features displayed in the data. However, in a broader range of data realm, these advantages may not out-stand (e.g. in online sales data records). It requires a more general approach. Currently, there are more than 2.7ZB data in the digital universe [@bigDataStatistics] and the growing speed is doubling every two years. It has already been hard and will be much harder in the future to harness the exploding volume of data that has resulted in many problems in data management and engineering, threatening trustworthiness and reliability of data flows inside working systems. Data error rate in enterprises is approximately 1% to 5%, and for some, even above 30% [@saha2014data]. Those data anomalies may arise due to both internal and external reasons. On one hand, components inside systems may generate problematic source data. For example, in a sensor network, some sensors may generate erroneous data when it experiences power failure or other extreme conditions [@rassam2014adaptive]. Data packages will be lost if sensor nodes fail to connect to network or some sensor hubs break down [@herodotou2014scalable]. Also, human operators act as a heavily vulnerable part to bugs and mistakes. Malicious insiders even deliberately modify system configurations for fatal compromises [@schuster2015vc3]. A study shows that 65% of organizations state that human errors are the main cause of data problems [@humanError] . On the other hand, data manipulation [@dataManipulation] from outside hackers composes another potential threat of data quality and reliability. *Data Manipulation* here, according to a NSA definition, refers to that “hackers can infiltrate networks via any attack vector, find their way into databases and applications and change information contained in those systems, rather than stealing data and holding it for ransom”. If data is compromised, it will severely affect mining and learning algorithms and further change the final decision driven by the data. In 2013, hackers from Syria put up fake reports via Associated Press’ Twitter account and caused a 150-point drop in the Dow [@SyriaHacker]. It is hard to detect a single record that is alerted but still remain in correct value scopes, but if sufficient data records are altered to change a final decision, we can still detect malicious data manipulation behaviours. According to our observation, typical manipulations on numerical data will lead to a drift or distortion of its original distribution. For measurable reshaping, we can enclose data collections with similar distribution patterns and filter out those strangely shaped ones. To address problems caused by data manipulation, we proposed a novel technique which sorts out manipulated data collections from normal ones by adopting statistical divergence. In this paper, we focus on a concrete data manipulation problem: click farming in online shops, and try to apply our technique to pick out those dishonest sellers. Our technique maps data collections to points in distribution spaces and reduce the problem to classical point anomaly detection. Optimizations estimate ground truth, mapping each data collection into a single real number within a definite interval. Then a Gaussian classifier can be applied to detect outliers derived from manipulated data. To automatically calculate adaptive threshold for the classifier, we keep two evidence sets for both normal points and anomalies, taking advantage of the property provided by statistical divergence. In the dynamic environments, these evidence sets are modified after every data collection is checked, in which manner they act intuitively as slide windows and keep up to the evolving features in dynamic scenarios. Our contribution includes: 1) A brief review on data anomaly detection and a study on the problem of click farming; 2) Detailed description of both basic and optimized framework of our technique, resolving several technical difficulties such as automated adaptive threshold; 3) Real world and synthetic data experiments that test efficiency of our technique and a comparison with a previous work on the same topic. The rest of the paper is organised as follows: Section \[sec:related-work\] states related work on data anomaly detection and describes a real world problem. Section \[sec:preliminaries\] introduces statistical distance. Details of proposed technique are introduced in section \[sec:algorithm-details\]. Then section \[sec:evaluation\] presents evaluation results and further findings of the algorithm. Finally, the paper is concluded in section \[sec:conclusion\]. Related Work {#sec:related-work} ============ Data Anomaly Detection ---------------------- Statistical divergence was applied mainly as classifiers on multimedia content [@park2005classification], especially as kernels in SVMs [@moreno2004kullback]. As a similarity measurement, it can also be used in qualitative and quantitative analysis in image evaluation [@pheng2016kullback; @goldberger2003efficient]. [@amid2014unsupervised] adopted divergence to detect events in multimedia streams. Anomaly detection, also known as outlier detection, has been studied for a long time and discussed in diverse research domains, such as fraud detection, intrusion detection, system monitoring, fault detection and event detection in sensor networks. Anomaly detection algorithms deal with input data in the form of points (or records), sequences, graphs and spatial and geographical relationships. [@chandola2009anomaly] According to relationships within data records, outliers can be classified into *point anomalies*, *contextual (or conditional) anomalies* and *collective anomalies*. [@goldberger2000components] Currently, distance based [@cao2014scalable; @cao2017multi] and feature evolving algorithms [@masud2013classification; @li2015discovery; @shao2014prototype] catch most attention. Others adopted tree isolation [@zhang2017lshiforest], model based [@yin2016model] and statistical methods [@zhu2002statstream] in certain applications. To detect collective anomalies, [@caudell1993adaptive] adopts the *ART (Adoptive Resonance Theory)* neural networks to detect time-series anomalies. *Box Modeling* is proposed in  [@chan2005modeling]. *Longest Common Subsequence* was leveraged in [@budalakoti2006anomaly] as similarity metric for symbolic sequence. Markovian modeling techniques are also popular in this domain[@ye2000markov; @warrender1999detecting; @pavlov2003sequence]. [@yu2015glad] depicts groups in social media as combinations of different “roles” and compare groups according to the proportion of each role within each group. Wang et al. proposed a technique, *Multinomial Goodness-of-Fit* (MGoF), to analyze likelihood ratio of distributions via Kullback-Leibler divergence, and is fundamentally a hypothesis test on distributions [@wang2011statistical]. MGoF divides the observed data sequence into several windows. It quantifies data in each window into a histogram and check these estimated distributions against several hypothesis. If the target distribution rejects all provided hypothesis, it is considered an anomaly and preserved as a new candidate of null hypothesis. If the target distribution failed to reject some hypothesis, then it is considered a supporting evidence of the one that yields most similarity. Furthermore, if the number of supporting evidence is larger than a threshold $c_{th}$, it is classified as non-anomaly. MGoF is the best competitor out of the similar techniques, and we use it as our baseline against our approach. Real World Problem: Click Farming Detection {#sec:related-realworld} ------------------------------------------- Taobao possesses a market share of 50.6% to 56.2% in China by 2016 [@iresearch2016b2c]. Currently, there are more than 9.4 million sellers in Taobao, providing more than 1 billion different products. Under the super-pressure caused by massive competitors, a number of the sellers choose to use some cheating techniques to raise reputation and sale volumes, then improve rankings in search lists. The most popular approach to manipulate transaction and reputation data is *Click Farming*, where sellers use a large number of customer accounts to create fake transaction records and give high remarks on products. Professional click farmers are usually well organized groups or companies containing thousands of people. Some companies even develop professional applications that can be deployed on common PCs to improve productivity [@zhao2016on]. There are two types of click farming behaviours: Centralized and Equalized. Centralized click farming refers to the scenarios that transactions are randomly generated throughout the day. A significant feature of this approach is that the cheating transactions usually assemble together in a short period of time since most workers work at the same time. Equalized click farming refers to the circumstances that click farms are arranged by some well programmed applications or teams carefully managed and strictly commit transactions according to a timetable. Thus the transaction distribution may not vary too much with and without click farming. A research performed in China showed that 81.9% of investigated people had heard of the behaviour of click farming, 51.2% are aware of click farm and 18.9% of them had experience of click farming themselves [@yan2015report]. American researchers reported in 2015 that over 11000 sellers on Taobao were detected to have click farmed records and only 2.2% of 4000 investigated dishonest sellers had been penalized because of the cheating attempts [@netease2015research]. Current detection techniques for click farming mainly focus on user behaviours, such as browsing frequencies and periods, most common purchasing time, favourite products, remarks and whether they communicate with sellers [@simpleDetection]. Those techniques require the platform to keep lots of records and user features. However, the detection can be easily bypassed by trained workers and some well programmed applications. ![Example cumulative distribution function of original and click farmed daily transaction data[]{data-label="fig:example-ecdf"}](./ExampleCDF.pdf){width="\linewidth"} Although it is hard to classify users as honest or malicious, we can still find clues from the sellers’ aspect. For normal sellers, their customers are usually similar since choices of products are seldom changed. Therefore, the distribution of transactions in a fixed period of time, say one day, is relatively stable. No matter how much alike between honest users and robots or the employed workers, the fake transaction records will always cause a bias or distortion of the original transaction distribution. To better observe the problem, we downloaded a real world data set containing Taobao online sellers’ transaction records and emulated the circumstances if it had been click farmed (see section \[sec:exp-methodology\]). Fig. \[fig:example-ecdf\] shows the difference between normal and click farmed distributions of one day in the data set. Thus, if we can measure the similarity between different transaction distributions, there is still a chance for us to detect dishonest sellers. Preliminaries {#sec:preliminaries} ============= Statistical divergence, also called statistical distance, measures the similarity between two or more distributions. Mathematically, statistical divergence is a function which describes the “distance” of one probability distribution to the other on a statistical manifold. Let $\mathbb{S}$ be a space of probability distributions, then a divergence is a function from $\mathbb{S}$ to non-negative real numbers: $$D(\cdot || \cdot): \mathbb{S} \times \mathbb{S} \rightarrow \mathbb{R^+} \cup \{0\}$$ Divergence between two distributions $P$ and $Q$, written as $D(P||Q)$, satisfies: 1. $D(P||Q) \ge 0, \forall P, Q \in \mathbb{S}$ 2. $D(P||Q) = 0$, if and only if $P=Q$ For our purposes, we do not require the function $D$ to have the property: $D(P||Q) = D(Q||P)$. But we do need it to be true that if $Q$ is more similar with $P$ than $U$, then $D(Q||P) < D(U||P)$. There are ways to calculate divergence, several frequently used divergence metrics are as follows: Kullback-Leibler Divergence --------------------------- Let $P,Q$ be discrete probability distributions, $Q(x)=0$ implies $P(x)=0$ for $\forall x$, the *Kullback-Leibler Divergence* from $Q$ to $P$ is defined to be: $$KLD(P||Q) = \sum_{Q(x)\ne 0} P(x)log\Big(\frac{P(x)}{Q(x)}\Big)$$ For $P,Q$ being continuous distributions: $$KLD(P||Q) = \int_{q(x) \ne 0} p(x)log\frac{p(x)}{q(x)}dx$$ Jensen-Shannon Divergence ------------------------- Let $P,Q$ be discrete probability distributions, *Jensen-Shannon Divergence* between $P$ and $Q$ is defined to be: $$JSD(P||Q) = \frac{1}{2}KLD(P||M) + \frac{1}{2}KLD(Q||M)$$ where $\displaystyle M = \frac{1}{2}(P+Q)$. A more generalized form is defined to be: $$JSD(P_1, \dots, P_n) = H\Big(\sum_{i=1}^n\pi_i P_i\Big) - \sum_{i=1}^n\pi_iH(P_i)$$ where $H$ is Shannon Entropy, $\displaystyle M = \sum_{i=1}^{n}\pi_iP_i$ and $\displaystyle \sum_{i=1}^{n}\pi_i = 1$. Especially, if $\displaystyle \pi_i = \frac{1}{n}$, then: $$JSD(P_1, \dots, P_n) = \frac{1}{n}\sum_{i=1}^{n}KLD(P_i||M)$$ Jensen-Shannon divergence has some fine properties: 1. $JSD(P||Q) = JSD(Q||P), \forall P, Q\in \mathbb{S}$. 2. $0 \le JSD(P_1, \dots, P_n) \le log_k(n)$. If a $k$ based algorithm is adopted. 3. To calculate $JSD(P||Q)$, it need not necessarily to be true that $Q(x)=0$ implies $P(x)=0$. Bhattacharyya Distance ---------------------- Let $P,Q$ be discrete probability distributions over same domain $X$, *Bhattacharyya Distance* between $P$ and $Q$ is defined to be: $$BD(P||Q) = -ln\Big(\sum_{x\in X}\sqrt{P(x)Q(x)}\Big)$$ Hellinger Distance ------------------ Let $P,Q$ be discrete probability distributions, *Hellinger Distance* between $P$ and $Q$ is defined to be: $$HD(P||Q) = \frac{1}{\sqrt{2}}\sqrt{\sum_x\bigg(\sqrt{P(x)} - \sqrt{Q(x)}\bigg)^2}$$ Kolmogorov-Smirnov Statistic ---------------------------- Let $P,Q$ be discrete one-dimensional probability distributions, $CDF_P$ and $CDF_Q$ are their cumulative probability functions respectively, *Kolmogorov-Smirnov Statistic* between $P$ and $Q$ is defined to be: $$KSS(P||Q) = \sup_x | CDF_P(x) - CDF_Q(x) |$$ Statistical Detection {#sec:algorithm-details} ===================== Diverse data sets in the real world show certain structures caused by hidden patterns or relationships among records. For example, traffic volume in the highway and the business transaction records, they may show a relatively stable distribution in the daily scale. Manipulation on those data (e.g. Fig. \[fig:example-ecdf\]) results in a drift or distortion of the distribution, which can be captured to trigger an alarm. Statistical Divergence Detection with Reference(SDD-R) {#sec:alg-opt-reference} ------------------------------------------------------ From Section \[sec:preliminaries\] we know that statistical divergence only provides a distance between two or more distributions. In a set of data collections, we can only draw a complete graph where nodes denote data collections and edges refer to the symmetric divergence between two connected nodes. From the graph we can find some points that have apparently larger distances with most of other points and return them as anomalies. This may work if anomalous nodes do not compose a large proportion. However the procedure will be too complicated to work out with large amounts of data. If it is assured that data collections form only one cluster, some optimizations can be applied to reduce complexity. Alternatively we can provide a frame of reference that generates absolute coordinates rather than the relative ones. This optimization is feasible if data collections form one single cluster in distribution space. This is true in most reality scenarios given that distribution is adopted to depict a macro property which comes out as one universal conclusion. In other words , if multiple distributions are used to describe subgroups of entire sample space, then a conclusive one can be obtained by averaging all these sub-distributions. Therefore, we can use an estimate cluster center as reference and test distances between the reference and each other data collections(Algorithm \[alg:sdd-r\]), yielding absolute distances. Data Collections $\mathbb{D} = \{D_1, \dots, D_n\}$ Estimated anomalous probability $\alpha$ Anomalous Data Collections $P_i \gets$ the distribution of $D_i$ $P_R \gets \frac{1}{n}\sum_{i = 1}^n P_i$ $d_i \gets D(P_i||P_R)$ $\mathcal{N}(\mu, \sigma) \gets$ Gaussian distribution estimated by $d_i$ $\{D_i | \frac{d_i - \mu}{\sigma} > 3 \}$ ![Distribution of Jensen-Shannon divergence on Taobao data set(without click farming) used in the experiments.[]{data-label="fig:jsd-dist"}](./JSD-Dist.pdf){width="\linewidth"} Fig. \[fig:jsd-dist\] shows distribution of all divergences against the reference. It can be approximated as a Gaussian distribution though the true one may differ a little more from the standard Gaussian than the expected estimation error. That is due to the unknown randomness within real world data. Few assumptions can be applied in real world data sets, no mention that data volume is sometimes relatively low. This topic is out of the domain discussed in this paper and we here only introduce the technique instead of the specific distribution model. Certainly, if stronger assumptions can be included to provide a more precise model, this component in the framework can be replaced to give better results. For the simplicity of our proposal, we deem the distributions of divergences to be Gaussian. By this approach, time complexity can be reduced from quadratic to linear. Fig. \[fig:raw-overview\] in Section \[sec:exp-raw\] demonstrates the result of the above process. Red distribution refer to the distances calculated from normal data collections, blue and green ones are from click-farmed data collections. Clearly, distances of normal data collections assembles together around a small value while anomalous ones lay around a larger distance value. Optimization: Statistical Divergence Detection with Evidence(SDD-E) ------------------------------------------------------------------- It is possible to further optimize SDD-R if we can provide this algorithm with evidence(Algorithm \[alg:sdd-e\]). Evidence set with normal data collections $\mathbb{E}_N = \{N_1, \dots, N_n\}$ Evidence set with anomalous data collections $\mathbb{E}_A = \{A_1, \dots, A_m\}$ Estimated anomalous probability $\alpha$ New data collection $\mathbb{D} = \{D_1, \dots, D_l\}$ Anomalous data collections in $\mathbb{D}$ $P_{N_i} \gets$ distribution of $D_{N_i}$\[line:hist-1\] $P_{A_i} \gets$ distribution of $D_{A_i}$\[line:hist-2\] $P_R \gets \frac{1}{n}\sum_{i=1}^{n}P_{N_i}$ $d_{N_i} \gets D(P_{N_i}||P_R)$ $d_{A_i} \gets D(P_{A_i}||P_R)$ $\mathcal{N}_N(\mu_N, \sigma_N) \gets$ normal distribution estimated from $\{d_{N_1}, \dots, d_{N_n}\}$ $\mathcal{N}_A(\mu_A, \sigma_A) \gets$ normal distribution estimated from $\{d_{A_1}, \dots, d_{A_m}\}$ $T \gets$ proper threshold derived from $\mathcal{N}_N$, $\mathcal{N}_A$ and $\alpha$ $P_i \gets$ distribution of $D_i$ $d_i \gets D(P_i||P_R)$ $\{D_i|d_i > T\}$ Evidences enables the algorithm to not only refine estimation of real distribution but also build knowledge of anomalous collections, which is similar to the parameter estimation within a certain sample set. According to the property of statistical divergence, we can infer that the true distribution of divergences calculated from normal data collections are close to but not exactly a Gaussian distribution $\mathcal{N}(\mu, \sigma)$ since for each point, there are both definite upper and lower bounds instead of infinities. Therefore, $\mu$ should be slightly larger than zero($\mu = 0 \iff P_i = P_j, \forall P_i, P_j \in \mathbb{E}_N$, for real world data sets, this is highly unlikely). Time complexity for this algorithm is still linear but with a larger coefficient. For certain divergence, it is possible to compare similarity from one distribution against multiple others, such as Jensen-Shannon Divergence. Although it reduces time complexity, it sacrifices unaffordable accuracy because divergence among multiple distribution dilutes differences. Take JSD as an example, suppose $P(1) = P(2) = P(3) = \frac{1}{3}$ and $Q(1) = \frac{1}{6}, Q(2) = \frac{1}{3}, Q(3) = \frac{1}{2}$, then $JSD(P||Q) \approx 0.033$ and $JSD(P, P, P, Q) \approx 0.024$. This algorithm can be slightly modified to deal with concept drift(for example, trading trend changes over time for online shops as they are often in the process of expanding or dwindling) by turning the two evidence sets as sliding windows and adopting certain update strategies such as *Least Recently Used*(LRU). Time complexity for this optimization is $O(n\cdot(|\mathbb{E}_N|+|\mathbb{E}_A|)\cdot T_D)$, where $T_D$ denotes time complexity of divergence calculation. Threshold {#sec:alg-threshold} --------- One important factor in algorithm SDD-E is the value of threshold. A lower threshold rejects more instances, improving the sensitivity of anomalous data while increasing the number of false alarms. A higher threshold provide higher true negative rates yet neglecting more possible threats. A naive but prevalent approach is to set a fixed value as the threshold(As is shown in Algorithm \[alg:sdd-r\]). This approach is easy to implement and may give satisfying results in specific cases. However, a fixed threshold requires specific analysis in the certain scenario, manual observations and tuning of parameters, which involves lots of human labour. The rule of “$3 \sigma$” declares all instances outside $[\mu - 3\sigma, \mu + 3\sigma]$ to be anomalous. It can be used to automatically determine a threshold. But as a rigid metric, it is merely an estimation of a suitable boundary considering average situations, which is far from optimality when concrete data is provided. It would be either lower than the optimum if anomalous data lies far away from the normal cluster, or higher than the optimum if the anomalies sit close to the cluster centre. Fortunately, applying divergence as the distance measurement among data collections provides a fine property. With a reference distribution, divergences of normal data collections form a quasi-Gaussian distribution as we have seen in section \[sec:alg-opt-reference\]. The same applies to those anomalous ones. ![Threshold can be determined either by a probability density value, or a radius from the centre. In the scenario shown in the figure, the threshold can be determined by the position between two centres of the distribution, denoted as “T” here. This form of threshold can also be applied to other types of distributions even there is no intersection between these two.[]{data-label="fig:example-threshold"}](./ExampleThreshold.pdf){width="\linewidth"} Moreover, as is verified in experiments, the meta-distribution of anomalous data collections lies in the right-hand-side to the normal one on the real line. As shown in Fig \[fig:example-threshold\], the black curve ($PDF_{normal}(x)$ or in short $PDF_n(x)$) displays the probability density function (PDF) fitting those divergences calculated from normal data collections; the blue curve ($PDF_{anomalous}(x)$ or in short $PDF_a(x)$) displays the PDF derived from anomalous data collections. Threshold is chosen to minimize total errors(both false negative and false positive). Suppose: $$\begin{aligned} PDF_n(x) &\approx \mathcal{N}(\mu_n, \sigma_n)\\ PDF_a(x) &\approx \mathcal{N}(\mu_a, \sigma_a) \end{aligned}$$ Then the optimal threshold $T$ is calculated by E.q.(\[equ:equal-weight\]). The optimal threshold will minimize total errors and yield an optimal outcome. However, this is not accurate enough, since E.q.(\[equ:equal-weight\]) implicates an assumption that chances are the same for a new data collection to be either anomalous or not. If we can determine the probability for a new data collection to be anomalous in any segment of data sequence, the equation should be modified as E.q.(\[equ:linear-weight\]), where $\alpha$ is the anomaly probability. $$\begin{aligned} \label{equ:equal-weight} T &= \mathop{\arg\min}_{T} \int_{0}^{T}PDF_{a}(x)dx + \int_{T}^{\sup(D)}PDF_{n}(x)dx \nonumber\\ & \approx \mathop{\arg\min}_{T} \int_{-\infty}^{T} \frac{e^{-\frac{(x - \mu_a)^2}{2\sigma_a^2}}}{\sqrt{2\pi} \sigma_a}dx + \int_{T}^{+\infty} \frac{e^{-\frac{(x - \mu_n)^2}{2\sigma_n^2}}}{\sqrt{2\pi} \sigma_n}dx \nonumber\\ & = \begin{cases} \displaystyle \frac{1}{\sigma_a^2 - \sigma_n^2}\left[(\sigma_a^2\mu_n - \sigma_n^2\mu_a) \pm \sigma_a\sigma_n\sqrt{(\mu_a - \mu_n)^2 + 2(\sigma_a^2 - \sigma_n^2)ln\frac{\sigma_a}{\sigma_n}}\right], & \sigma_a \ne \sigma_n\\ \displaystyle \frac{\mu_n + \mu_a}{2}, & \sigma_a = \sigma_n \end{cases} \end{aligned}$$ Note: when $\sigma_a \ne \sigma_n$, keep the root s.t. $\displaystyle \frac{T - \mu_a}{\sigma_a^3}e^{-\frac{(T - \mu_a)^2}{2\sigma_a^2}} < \frac{T - \mu_n}{\sigma_n^3}e^{-\frac{(T - \mu_n)^2}{2\sigma_n^2}}$ $$\begin{aligned} \label{equ:linear-weight} T & = \mathop{\arg\min}_{T} \alpha\int_{0}^{T}PDF_{a}(x)dx + (1-\alpha)\int_{T}^{\sup(D)}PDF_{n}(x)dx\nonumber\\ & \approx \mathop{\arg\min}_{T} \alpha\int_{-\infty}^{T} \frac{e^{-\frac{(x - \mu_a)^2}{2\sigma_a^2}}}{\sqrt{2\pi} \sigma_a}dx + (1-\alpha)\int_{T}^{+\infty} \frac{e^{-\frac{(x - \mu_n)^2}{2\sigma_n^2}}}{\sqrt{2\pi} \sigma_n}dx\nonumber\\ & = \begin{cases} \displaystyle \frac{1}{\sigma_a^2 - \sigma_n^2}\left[(\sigma_a^2\mu_n - \sigma_n^2\mu_a) \pm \sigma_a\sigma_n\sqrt{(\mu_a - \mu_n)^2 + 2(\sigma_a^2 - \sigma_n^2)ln\frac{(1 - \alpha)\sigma_a}{\alpha\sigma_n}}\right], & \sigma_a \ne \sigma_n\\ \displaystyle \frac{\mu_n + \mu_a}{2} + \frac{k^2ln\frac{1 - \alpha}{\alpha}}{\mu_a - \mu_n}, & \sigma_a = \sigma_n = k \end{cases} \end{aligned}$$ Note: when $\sigma_a \ne \sigma_n$, keep the root s.t. $\displaystyle \frac{\alpha (T - \mu_a)}{\sigma_a^3}e^{-\frac{(T - \mu_a)^2}{2\sigma_a^2}} < \frac{(1 - \alpha) (T - \mu_n)}{\sigma_n^3}e^{-\frac{(T - \mu_n)^2}{2\sigma_n^2}}$ Moreover, with an estimated anomaly probability, SDD-R can be also optimized by ranking all data collections according to their divergence value and select first $n \cdot \alpha$ ones with highest values as anomalies. Evaluation {#sec:evaluation} ========== Our algorithm was implemented and interpreted in Python 3.5.2. All experiments were tested on Ubuntu 16.04. In the following experiments, we figured out properties of real world data and performance of our technique against anomalous data collections. We also made a comparison among variations of SDD algorithms and MGoF.[^1] Methodology {#sec:exp-methodology} ----------- We adopted two data sets: 1) Koubei sellers’ transaction records[^2]; 2) Synthetic random distribution data set. Koubei data set was provided by Alibaba Tian Chi big data competition where all records were collected from real world business scenarios. It contained information about seller features, user payments and browsing behaviour. We randomly chose one seller (ID: 1629) and extracted transaction history of this seller, records ranging from Nov. 11th 2015 to Oct. 31st 2016. Entire transaction set was then divided into 325 collections, each containing records in one day. Fig. \[fig:daily-transaction-volume\] and \[fig:sale-distribution-sample\] give an overview of it. ![Changing of the daily sales volume shows that environment of online sales had been changing all the time.[]{data-label="fig:daily-transaction-volume"}](./DailyTransactionVolume.pdf){width="0.75\linewidth"} ![We selected 3 days randomly and drew sales distribution by counting hourly volume. Although sales volume has changed from day to day, the shape of the distribution remain almost alike.[]{data-label="fig:sale-distribution-sample"}](./SaleDistributions.pdf){width="\linewidth"} Two types of click-farmed data was generated according to patterns described in section \[sec:related-realworld\]. To emulate centralized click farming, we randomly inserted some Gaussian-distributed transactions in the chosen collection. As for emulating the equalized click farmers, we simply doubled each record in the chosen collection to make the new distribution exactly the same as the original one, which is harder for the online platform to discover. Usually, the click-farmed transactions are several times more than the volume it originally has, if the seller hires a group of organized workers. In our experiments, we use $\nu$ to denote the magnitude coefficient of click farming. Hence $|D_{anomalous}| = (1 + \nu)|D_{normal}|$. In the following experiments without extra illustration, we adopted $\nu = 1$. One defect of this data set is that the detailed time stamp is aligned at each hour of the day due to desensitization. We constructed an enhanced data set by assigning every time stamp a random value for minutes and seconds. Therefore, the enhanced data set should be closer to the reality. The synthetic data set was divided into four sections. First two sections contained sample sets drawn from a uniform and a Gaussian distribution respectively. The third section used a mixture of one uniform distribution and two Gaussian distributions to simulate a random-shaped distribution. Moreover, we made the random-shaped distribution drift slightly to form the last section of test data. Corresponding anomalies were drown from distributions with deviated parameters respectively. We adopted histograms to depict distributions of any shape. Surely, the kernel density estimation approaches will give smooth and continuous estimations on any sampled data. But the computational cost will be too much to afford. Step size of histograms is chosen by: $$\label{equ:step-size} l = c \sigma k^{-0.2},$$ where $k$ is sample size, $c$ is a constant relative to the shape of distribution (e.g. for normal distribution, $c=1.05$) and $\sigma$ the standard deviation. For data sets with a large number of elements, a random sampling method, such as Monte-Carlo method, can be applied to speed up the estimation procedure. Divergence metric adopted in each SDD algorithms was Jensen-Shannon divergence if no specific notation is made. However, MGoF used only Kullback-Leibler divergence due to its special mechanism. We use a “+” to denote algorithms optimized by a given $\alpha$. Experiments on Koubei Data Set {#sec:exp-raw} ------------------------------ We first tested our algorithms on Koubei data set in order to see whether and why the algorithm works. Anomalies were random selected days replaced by corresponding click farmed version. To play the role of purchasing platform, we investigated two levels of transaction distribution. The first level is to simply draw a histogram aligned to time spans. The second level is to draw a histogram on the sub-volumes in each time span(i.e. a histogram on frequencies in the first level histogram, as shown in Fig. \[fig:histogram-example\]). ![Examples of 1st and 2nd Level Histogram[]{data-label="fig:histogram-example"}](./HistogramExample.pdf){width="\linewidth"} On the raw data set, we had no choice but to set one hour a basket. While on the enhanced data set, we adopted E.q.(\[equ:step-size\]) to determine step size automatically. To test SDD-E, we randomly selected 30 correct days and 10 click farmed days as normal and anomalous evidence respectively. Here $\alpha = 0.2$. The results are shown in Table \[tab:result-koubei-raw\] and \[tab:result-koubei-enhanced\]. -------------------- ------------ ------------ ----------- ------------ ------------ ------------ ----------- ----------- ------------ ------------ ----------- ------------ ------------ ------------ ----------- ----------- **Pre(%)** **Rec(%)** **F1(%)** **T(ms)** **Pre(%)** **Rec(%)** **F1(%)** **T(ms)** **Pre(%)** **Rec(%)** **F1(%)** **T(ms)** **Pre(%)** **Rec(%)** **F1(%)** **T(ms)** **SDD-R** 89.51 48.75 63.12 266.77 21.97 **99.38** 35.98 11.12 6.67 0.63 1.14 249.15 21.22 72.50 32.83 7.81 **SDD-R+** 91.25 91.25 **91.25** **265.96** **61.88** 61.88 61.88 10.08 9.38 9.38 9.38 **247.31** **44.38** 44.38 44.38 6.92 **SDD-E Static** **92.46** 68.75 78.86 292.50 36.55 98.13 53.26 5.75 6.67 0.63 1.14 271.45 36.07 86.25 50.86 5.64 **SDD-E Static+** 85.02 32.50 47.02 293.77 46.24 91.88 61.52 5.95 10.00 0.63 1.18 272.71 43.60 76.25 **55.48** 5.68 **SDD-E Dynamic** 49.11 **99.38** 65.73 699.97 23.01 **99.38** 37.37 245.65 10.36 **18.13** **13.18** 681.09 22.09 **93.13** 35.71 242.85 **SDD-E Dynamic+** 73.21 98.75 84.09 701.06 48.02 96.25 **64.07** 255.43 8.15 6.88 7.46 681.89 40.79 78.13 53.59 253.03 **MGoF** 14.08 21.88 17.13 292.14 13.01 4.38 6.55 **3.64** **12.50** 3.13 5.00 250.42 12.50 3.13 5.00 **3.71** -------------------- ------------ ------------ ----------- ------------ ------------ ------------ ----------- ----------- ------------ ------------ ----------- ------------ ------------ ------------ ----------- ----------- -------------------- ------------ ------------ ----------- ------------ ------------ ------------ ----------- ----------- ------------ ------------ ----------- ------------ ------------ ------------ ----------- ----------- **Pre(%)** **Rec(%)** **F1(%)** **T(ms)** **Pre(%)** **Rec(%)** **F1(%)** **T(ms)** **Pre(%)** **Rec(%)** **F1(%)** **T(ms)** **Pre(%)** **Rec(%)** **F1(%)** **T(ms)** **SDD-R** 81.54 41.88 55.33 238.61 17.49 92.50 29.42 13.86 6.67 0.63 1.14 239.44 21.15 71.88 32.69 12.08 **SDD-R+** **91.25** 91.25 **91.25** **236.94** 36.88 36.88 36.88 12.98 8.13 8.13 8.13 **236.47** 38.75 38.75 38.75 11.03 **SDD-E Static** 88.31 67.50 76.52 257.60 31.99 92.50 47.54 9.74 6.67 0.63 1.14 257.65 34.49 91.25 50.06 9.37 **SDD-E Static+** 69.67 13.13 22.09 259.30 **42.12** 73.75 **53.62** 9.59 10.00 0.63 1.18 258.45 **44.17** 82.50 **57.54** 9.32 **SDD-E Dynamic** 42.93 **99.38** 59.96 1106.25 20.12 **95.63** 33.25 277.70 10.57 **17.50** **13.18** 1108.93 20.30 **94.38** 33.42 271.74 **SDD-E Dynamic+** 69.18 98.13 81.15 1110.81 33.25 87.50 48.19 292.60 7.06 3.75 4.90 1118.00 37.21 80.63 50.92 283.50 **MGoF** 13.97 17.50 15.54 294.08 14.66 8.13 10.45 **8.01** **12.50** 3.13 5.00 250.10 18.42 8.75 11.86 **7.55** -------------------- ------------ ------------ ----------- ------------ ------------ ------------ ----------- ----------- ------------ ------------ ----------- ------------ ------------ ------------ ----------- ----------- ![1st level histogram of day 5, 113 and 287, each day in a column. Distributions after centralized and equalized click farming are in 1st and 3rd rows correspondingly. And the original distributions are shown in the 2nd row.[]{data-label="fig:raw-hist-1st"}](./Raw1stLevelHist.pdf){width="\linewidth"} ![This figure shows distribution of JSD values(on 2nd level histograms) of normal and two types of click farming data. Divergences were calculated according to a reference averaged among all correct distributions.[]{data-label="fig:raw-overview"}](./RawOverview2nd.pdf){width="\linewidth"} When classifying toward 1st level histograms, centralized click farming behaviours can be easily discovered. As displayed in the first two rows in Fig. \[fig:raw-hist-1st\], normal collections share a similar distribution while centralized click-farmed ones abruptly violated the original shape. However, as a clever click farmer, equalized click farming did not in the least distort the distribution. Most of them escaped the check under perfect disguises. But when it came to 2nd level histograms, the “clever disguise” did not work any longer. It can be clearly seen in Fig. \[fig:raw-overview\] that distribution of divergence of both click farming types shows an obvious deviation from the normal one. The result showed that our technique outperformed MGoF in every real world cases. SDD-E provided best performance, yet it consumed the most computing power. Comparison among SDD-R revealed improvement of reference as well as the importance of threshold under this technique. Although dynamic SDD-E consumes more computation power, it is clear that dynamic SDD-E is capable of tracing the gradual shift of environment. MGoF turned out to be the worst since it always mark several false positive when $c_{th}$ had not been met and much more false negatives when similar errors occurred too many. Parameter $\alpha$ improved total accuracy of dynamic SDD-E algorithm by 10-20% as was supposed. It also increased its F1 by more than 20%. $\alpha$ made a great difference in SDD-R as well, which illustrated that divergence sorted almost all collections in correct order according to the averaged reference. However, static SDD-E did not show the same improvement. Since environment drift took greater influence in the result. In comparison with $\alpha$, adaptive threshold given by evidence sets did not bring the most improvement. But this threshold can be applied together with other optimizations such as slide windows. Test against Anomaly Proportion and Magnitude {#sec:exp-synthetic} --------------------------------------------- In this experiment, we tested algorithm performance under various anomaly proportion and magnitude. $\alpha$ ranged from 0.1 to 0.9 when $\nu = 1$ and $\nu \in [0.1, 0.9]$ when $\alpha = 0.1$, other settings remains the same. ![Accuracy and F1 on Different Anomaly Probabilities[]{data-label="fig:anomaly-probability"}](./AccuracyOnAnomalyProbability.pdf){width="\linewidth"} ![Accuracy and F1 on Different Anomaly Probabilities[]{data-label="fig:anomaly-probability"}](./F1OnAnomalyProbability.pdf){width="\linewidth"} Fig. \[fig:anomaly-probability\] shows that our technique outperformed MGoF and was relatively stable when dealing with all proportions of 1st level centralized anomalies. SDD-E performed even better since it maintains knowledge of both normal and anomalous distributions and calculates the threshold according to the best expectation. However, it relies on the accuracy of distribution estimation. When it came to 2nd level distributions, histograms became much coarser since data available was highly limited and thus its performance suffered dramatically. MGoF tended to classify every distribution as anomaly, therefore benefited most by larger $\alpha$. It always classifies as anomalous the first $c_{th}$ distributions supporting every null hypothesis. Thus when $\alpha$ increased, the proportion of misclassified normal collections also became larger, while those anomalies were still considered anomalous. And given that the total number of normal collections drops down, the overall accuracy tended to increase as more instances are correctly classified as anomalous. However, the right half shows a different trend. One reason is that MGoF uses KLD other than JSD. In the Koubei dataset, the discrete estimation of distributions oscillated in a wide range, leading to that the prerequisite of KLD is often unsatisfied. Thus the calculation of KLD may not give a correct measurement. Furthermore, the 2nd histogram provided fewer probability entries than the 1st level did. Thus it shows a more significant deviation from our expectation. For the classifiers of MGoF, they compromised to a high error rate. Because more anomalies gathered together and the algorithm recognized them as clusters of normal data. ![Accuracy and F1 on Different Anomaly Magnitudes[]{data-label="fig:anomaly-magnitude"}](./PerformanceOnAnomalyMagnitude.pdf){width="\linewidth"} From Fig. \[fig:anomaly-magnitude\] we can conclude that our algorithms are still the best, given that they are most sensitive toward tiny anomalous variations. However, static SDD-E did not rise until $\nu > 1$, this is because it suffered from fluctuation on the trade environment at the mean time. MGoF is not sensitive toward minor anomalies either. For a relatively small magnitude of click farming, the classifiers of MGoF quickly degrade to be trivial. The rigid threshold could not automatically rise up and was thus far from to optimal. Results on Synthetic Data Set ----------------------------- In this experiment, we tested all seven algorithms on totally synthetic data sets. Results are shown in TABLE \[tab:performance-synthetic\]. It shows that our technique can be applied towards any kind of distributions. And these techniques worked better under irregular distributions since difference were clearer among these. Comparison between SDD-R and static SDD-E shows that adaptive thresholds provided more flexible classifiers. Results under random-shape drifting proves the efficiency of sliding windows toward drifting context. -------------------- ------------ ------------ ----------- ------------ ------------ ------------ ----------- ------------ ------------ ------------ ----------- ------------ ------------ ------------ ----------- ------------ **Pre(%)** **Rec(%)** **F1(%)** **T(ms)** **Pre(%)** **Rec(%)** **F1(%)** **T(ms)** **Pre(%)** **Rec(%)** **F1(%)** **T(ms)** **Pre(%)** **Rec(%)** **F1(%)** **T(ms)** **SDD-R** 10.00 **100.00** 18.18 349.42 24.78 69.20 36.49 401.82 10.00 **100.00** 18.18 351.65 9.97 **99.00** 18.11 353.65 **SDD-R+** 22.40 22.40 22.40 348.49 34.40 34.40 34.40 **398.02** 58.40 58.40 58.40 **350.87** 1.20 1.20 1.20 **346.67** **SDD-E Static** 43.71 82.60 57.17 372.74 33.34 66.60 44.44 410.72 69.24 93.20 79.45 372.77 12.34 97.20 21.91 369.63 **SDD-E Static+** **75.10** 60.20 **66.83** 371.72 **45.93** 45.80 **45.86** 412.95 **89.52** 85.40 **87.41** 368.97 12.60 94.20 22.23 370.12 **SDD-E Dynamic** 18.53 81.80 30.21 6808.58 20.28 **71.40** 31.58 8985.92 25.90 97.40 40.92 5881.37 13.44 97.20 23.61 5747.19 **SDD-E Dynamic+** 27.85 25.60 26.68 8019.23 26.19 56.40 35.77 9197.94 77.60 79.40 78.49 6408.12 **50.55** 84.00 **63.11** 6176.57 **MGoF** 10.94 4.40 6.28 **347.97** 10.08 51.60 16.87 571.93 6.40 13.60 8.70 440.26 2.75 11.60 4.45 509.58 -------------------- ------------ ------------ ----------- ------------ ------------ ------------ ----------- ------------ ------------ ------------ ----------- ------------ ------------ ------------ ----------- ------------ [2]{} {width="\linewidth"} \[fig:divergence-metric-f1\] {width=".8\linewidth"} \[fig:divergence-metric-roc\] The last experiment was carried out on random-shaped distribution data set, with $alpha=0.1$ and rest parameters the same. Under different divergence metrics mentioned in section \[sec:preliminaries\], F1 scores were calculated among all SDD algorithms and ROC curves were recorded on SDD-R and static SDD-E. Given that MGoF was defined specifically on Kullback-Leibler divergence, it cannot be tested in the same way. Results are shown in Fig. \[fig:divergence-metric-f1\] and Fig. \[fig:divergence-metric-roc\]. It is indicated that Jensen-Shannon divergence is suited to all techniques due to its symmetry. Kullback-Leibler divergence provides more evident differences when references were given. Bhattacharyya distance and Hellinger distance turned out almost as good as Jensen-Shannon divergence, but they consumed less time. Kolmogorov-Smirnov Statistic performed relatively poor since it considers only the largest gap between two distributions, which provides little information. Discussion ---------- MGoF’s learning procedure of anomalous probability hypothesis is inefficient. To maintain a comprehensive knowledge of anomalies, MGoF has to reserve a single hypothesis entry for every type of them. But in reality, it is always the case that we face the heterogeneity of outliers. In the Koubei data set, there can be tens of anomalous distributions caused solely by centralized click farming. It takes a long time to discover every possible type of anomaly. Besides, if there happens to be more than $c_{th}$ anomalous distributions of the same type, later discovered collections will no longer declared to be anomalous any more. However, in SDD-R and SDD-E, that is not a problem since it can map and gather all anomalies together and draw a universal boundary between them and all normal collections. These techniques are suitable to all typical divergence metrics and consume little computation power(except dynamic SDD-E). The only drawback is that they require comprehensive estimation of target distributions. Although other parameters need estimation as well, they are naturally addressable under big data circumstances. Conclusion {#sec:conclusion} ========== This paper proposes a series of collective anomaly detection techniques, which helps detect data manipulations in modern data pipelines and data centres. Different from existing algorithms designed for collective anomalies, our approach employs statistical distance as the similarity measurement. We explored several technical points involved in the design of the algorithm and performed a thorough experiment to test its efficiency. The comparison experiment also illustrated the advantages of our technique. It can be concluded that the our technique can efficiently discover anomalies within the data collections and the classifier is sensitive enough toward real world data manipulations. [10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[ l@\#1 =l@\#1 \#2]{}]{} E. Amid, A. Mesaros, K. J. Palom[ä]{}ki, J. Laaksonen, and M. Kurimo, “Unsupervised feature extraction for multimedia event detection and ranking using audio content,” in *2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)*.1em plus 0.5em minus 0.4emIEEE, 2014, pp. 5939–5943. P. J. Moreno, P. P. Ho, and N. Vasconcelos, “A kullback-leibler divergence based kernel for svm classification in multimedia applications,” in *Advances in neural information processing systems*, 2004, pp. 1385–1392. D.-C. Park, D.-H. Nguyen, S.-H. Beack, and S. Park, “Classification of audio signals using gradient-based fuzzy c-means algorithm with divergence measure,” in *Pacific-Rim Conference on Multimedia*.1em plus 0.5em minus 0.4emSpringer, 2005, pp. 698–708. H. S. Pheng, S. M. Shamsuddin, W. Y. Leng, and R. Alwee, “Kullback leibler divergence for image quantitative evaluation,” in *AIP Conference Proceedings*, vol. 1750, no. 1.1em plus 0.5em minus 0.4emAIP Publishing, 2016, p. 020003. J. Goldberger, S. Gordon, and H. Greenspan, “An efficient image similarity measure based on approximations of kl-divergence between two gaussian mixtures,” in *null*.1em plus 0.5em minus 0.4emIEEE, 2003, p. 487. \(2017) Big data statistics and facts for 2017. \[Online\]. Available: <https://www.waterfordtechnologies.com/big-data-interesting-facts/> B. Saha and D. Srivastava, “Data quality: The other face of big data,” in *Data Engineering (ICDE), 2014 IEEE 30th International Conference on*.1em plus 0.5em minus 0.4emIEEE, 2014, pp. 1294–1297. M. A. Rassam, M. A. Maarof, and A. Zainal, “Adaptive and online data anomaly detection for wireless sensor systems,” *Knowledge-Based Systems*, vol. 60, pp. 44–57, 2014. H. Herodotou, B. Ding, S. Balakrishnan, G. Outhred, and P. Fitter, “Scalable near real-time failure localization of data center networks,” in *Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining*.1em plus 0.5em minus 0.4em ACM, 2014, pp. 1689–1698. F. Schuster, M. Costa, C. Fournet, C. Gkantsidis, M. Peinado, G. Mainar-Ruiz, and M. Russinovich, “Vc3: Trustworthy data analytics in the cloud using sgx,” in *Security and Privacy (SP), 2015 IEEE Symposium on*.1em plus 0.5em minus 0.4emIEEE, 2015, pp. 38–54. TowerData. (2013) 4 steps to eliminating human error in big data. \[Online\]. Available: <http://www.towerdata.com/blog/bid/113787/4-Steps-to-Eliminating-Human-Error-in-Big-Data> Is data manipulation the next step in cyber crime. \[Online\]. Available: <https://www.cloudmask.com/blog/is-data-manipulation-the-next-step-in-cybercrime> (2016). \[Online\]. Available: <https://www.cnbc.com/2016/03/09/the-next-big-threat-in-hacking--data-sabotage.html> V. Chandola, A. Banerjee, and V. Kumar, “Anomaly detection: A survey,” *ACM computing surveys (CSUR)*, vol. 41, no. 3, p. 15, 2009. A. L. Goldberger *et al.*, “Components of a new research resource for complex physiologic signals, physiobank, physiotoolkit, and physionet, american heart association journals,” *Circulation*, vol. 101, no. 23, pp. 1–9, 2000. L. Cao, D. Yang, Q. Wang, Y. Yu, J. Wang, and E. A. Rundensteiner, “Scalable distance-based outlier detection over high-volume data streams,” in *Data Engineering (ICDE), 2014 IEEE 30th International Conference on*.1em plus 0.5em minus 0.4emIEEE, 2014, pp. 76–87. L. Cao, Y. Yan, C. Kuhlman, Q. Wang, E. A. Rundensteiner, and M. Eltabakh, “Multi-tactic distance-based outlier detection,” in *Data Engineering (ICDE), 2017 IEEE 33rd International Conference on*.1em plus 0.5em minus 0.4emIEEE, 2017, pp. 959–970. M. M. Masud, Q. Chen, L. Khan, C. C. Aggarwal, J. Gao, J. Han, A. Srivastava, and N. C. Oza, “Classification and adaptive novel class detection of feature-evolving data streams,” *IEEE Transactions on Knowledge and Data Engineering*, vol. 25, no. 7, pp. 1484–1497, 2013. Y. Li, Q. Li, J. Gao, L. Su, B. Zhao, W. Fan, and J. Han, “On the discovery of evolving truth,” in *Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining*.1em plus 0.5em minus 0.4emACM, 2015, pp. 675–684. J. Shao, Z. Ahmadi, and S. Kramer, “Prototype-based learning on concept-drifting data streams,” in *Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining*.1em plus 0.5em minus 0.4emACM, 2014, pp. 412–421. X. Zhang, W. Dou, Q. He, R. Zhou, C. Leckie, R. Kotagiri, and Z. Salcic, “Lshiforest: A generic framework for fast tree isolation based ensemble anomaly analysis,” in *Data Engineering (ICDE), 2017 IEEE 33rd International Conference on*.1em plus 0.5em minus 0.4emIEEE, 2017, pp. 983–994. J. Yin and J. Wang, “A model-based approach for text clustering with outlier detection,” in *Data Engineering (ICDE), 2016 IEEE 32nd International Conference on*.1em plus 0.5em minus 0.4emIEEE, 2016, pp. 625–636. Y. Zhu and D. Shasha, “Statstream: Statistical monitoring of thousands of data streams in real time,” in *Proceedings of the 28th international conference on Very Large Data Bases*.1em plus 0.5em minus 0.4em VLDB Endowment, 2002, pp. 358–369. T. Caudell and D. Newman, “An adaptive resonance architecture to define normality and detect novelties in time series and databases,” in *IEEE World Congress on Neural Networks, Portland, Oregon*, 1993, pp. 166–176. P. K. Chan and M. V. Mahoney, “Modeling multiple time series for anomaly detection,” in *Data Mining, Fifth IEEE International Conference on*.1em plus 0.5em minus 0.4emIEEE, 2005, pp. 8–pp. S. Budalakoti, A. N. Srivastava, R. Akella, and E. Turkov, “Anomaly detection in large sets of high-dimensional symbol sequences,” 2006. N. Ye *et al.*, “A markov chain model of temporal behavior for anomaly detection,” in *Proceedings of the 2000 IEEE Systems, Man, and Cybernetics Information Assurance and Security Workshop*, vol. 166.1em plus 0.5em minus 0.4emWest Point, NY, 2000, p. 169. C. Warrender, S. Forrest, and B. Pearlmutter, “Detecting intrusions using system calls: Alternative data models,” in *Security and Privacy, 1999. Proceedings of the 1999 IEEE Symposium on*.1em plus 0.5em minus 0.4emIEEE, 1999, pp. 133–145. D. Pavlov, “Sequence modeling with mixtures of conditional maximum entropy distributions,” in *Data Mining, 2003. ICDM 2003. Third IEEE International Conference on*.1em plus 0.5em minus 0.4emIEEE, 2003, pp. 251–258. R. Yu, X. He, and Y. Liu, “Glad: group anomaly detection in social media analysis,” *ACM Transactions on Knowledge Discovery from Data (TKDD)*, vol. 10, no. 2, p. 18, 2015. C. Wang, K. Viswanathan, L. Choudur, V. Talwar, W. Satterfield, and K. Schwan, “Statistical techniques for online anomaly detection in data centers,” in *Integrated Network Management (IM), 2011 IFIP/IEEE International Symposium on*.1em plus 0.5em minus 0.4emIEEE, 2011, pp. 385–392. Iresearch. (2016). \[Online\]. Available: <http://report.iresearch.cn/content/2016/11/265551.shtml> M. Zhao, “On the phenomenon of click farming in taobao,” *CHINA BUSINESS*, no. 12, pp. 31–33, 2016. Z. Yan, “Report of click farming phenomenon in taobao–protection for consumers’ right to know in online shopping,” *FaZhi Yu JingJi*, vol. 20, pp. 195–196, 2015. T. L. Mirror. (2016). \[Online\]. Available: <http://tech.163.com/15/0405/14/AMENOJ7D00094ODV.html> (2015). \[Online\]. Available: <http://www.ebrun.com/20150906/147724.shtml> [^1]: All resources and more detailed experiment results can be retrieved online: https://github.com/TramsWang/StatisticalAnomalyDetection [^2]: https://tianchi.aliyun.com/competition/information.htm?raceId=231591

--- abstract: 'Given a generalized $\overline M^{n+1}=I\times_{\phi}F^n$ Robertson-Walker spacetime we will classify strongly stable spacelike hypersurfaces with constant mean curvature whose warping function verifies a certain convexity condition. More precisely, we will show that given $x:M^n\rightarrow\overline M^{n+1}$ a closed spacelike hypersurfaces of $\overline M^{n+1}$ with constant mean curvature $H$ and the warping function $\phi$ satisfying $\phi''''\geq\max\{H\phi'',0\}$, then $M^{n}$ is either minimal or a spacelike slice $M_{t_0}=\{t_0\}\times F$, for some $t_0\in I$.' author: - 'A. Barros, A. Brasil and A. Caminha' title: Stability of Spacelike Hypersurfaces in Foliated Spacetimes --- Introduction ============ Spacelike hypersurfaces with constant mean curvature in Lorentz manifolds have been object of great interest in recent years, both from physical and mathematical points of view. In [@ABC:03], the authors studied the uniqueness of spacelike hypersurfaces with CMC in generalized Robertson-Walker (GRW) spacetimes, namely, Lorentz warped products with 1-dimensional negative definite base and Riemannian fiber. They proved that in a GRW spacetime obeying the timelike convergence condition (i.e, the Ricci curvature is non-negative on timelike directions), every compact spacelike hypersurface with CMC must be umbilical. Recently, Alías and Montiel obtained, in [@AM:01], a more general condition on the warping function $f$ that is sufficient in order to guarantee uniqueness. More precisely, they proved the following Let $f:I\rightarrow\mathbb R$ be a positive smooth function defined on an open interval, such that $ff^{''}-(f^{'})^{2}\leq 0$, that is, such that $-\log f$ is convex. Then, the only compact spacelike hypersurfaces immersed into a generalized Robertson-Walker spacetime $I\times_fF^{n}$ and having constant mean curvature are the slices $\{t\}\times F$, for a [(]{}necessarily compact[)]{} Riemannian manifold $F$. Stability questions concerning CMC, compact hypersurfaces in Riemannian space forms began with Barbosa and do Carmo in [@BdC:84], and Barbosa, Do Carmo and Eschenburg in [@BdCE:88]. In the former paper, they introduced the notion of stability and proved that spheres are the only stable critical points for the area functional, for volume-preserving variations. In the setting of spacelike hypersurfaces in Lorentz manifolds, Barbosa and Oliker proved in [@Barbosa:93] that CMC spacelike hypersurfaces are critical points of volume-preserving variations. Moreover, by computing the second variation formula they showed that CMC embedded spheres in the de Sitter space $S_1^{n+1}$ maximize the area functional for such variations. In this paper, we give a characterization of [*strongly stable*]{}, CMC spacelike hypersurfaces in GRW spacetimes, the essential tool for the proof being a formula for the Laplacian of a new support function. More precisely, it is our purpose to show the following Let $\overline M^{n+1}=I\times_{\phi}F^n$ be a generalized Robertson-Walker spacetime, and $x:M^n\rightarrow\overline M^{n+1}$ be a closed spacelike hypersurface of $\overline M^{n+1}$, having constant mean curvature $H$. If the warping function $\phi$ satisfies $\phi''\geq\max\{H\phi',0\}$ and $M^n$ is strongly stable, then $M^{n}$ is either minimal or a spacelike slice $M_{t_0}=\{t_0\}\times F$, for some $t_0\in I$. Stable spacelike hypersurfaces ============================== In what follows, $\overline M^{n+1}$ denotes an orientable, time-oriented Lorentz manifold with Lorentz metric $\overline g=\langle\,\,,\,\,\rangle$ and semi-Riemannian connection $\overline\nabla$. If $x:M^n\rightarrow\overline M^{n+1}$ is a spacelike hypersurface of $\overline M^{n+1}$, then $M^n$ is automatically orientable ([@O'Neill:83], p. 189), and one can choose a globally defined unit normal vector field $N$ on $M^n$ having the same time-orientation of $V$, that is, such that $$\langle V,N\rangle<0$$ on $M$. One says that such an $N$ [*points to the future*]{}. A [*variation*]{} of $x$ is a smooth map $$X:M^n\times(-\epsilon,\epsilon)\rightarrow\overline M^{n+1}$$ satisfying the following conditions: 1. For $t\in(-\epsilon,\epsilon)$, the map $X_t:M^n\rightarrow\overline M^{n+1}$ given by $X_t(p)=X(t,p)$ is a spacelike immersion such that $X_0=x$. 2. $X_t\big|_{\partial M}=x\big|_{\partial M}$, for all $t\in(-\epsilon,\epsilon)$. The [*variational field*]{} associated to the variation $X$ is the vector field $\frac{\partial X}{\partial t}$. Letting $f=-\langle\frac{\partial X}{\partial t},N\rangle$, we get $$\frac{\partial X}{\partial t}\Big|_M=fN+\left(\frac{\partial X}{\partial t}\right)^T,$$ where $T$ stands for tangential components. The [*balance of volume*]{} of the variation $X$ is the function $\mathcal V:(-\epsilon,\epsilon)\rightarrow\mathbb R$ given by $$\mathcal V(t)=\int_{M\times[0,t]}X^*(d\overline M),$$ where $d\overline M$ denotes the volume element of $\overline M$. The [*area functional*]{} $\mathcal A:(-\epsilon,\epsilon)\rightarrow\mathbb R$ associated to the variation $X$ is given by $$\mathcal A(t)=\int_MdM_t,$$ where $dM_t$ denotes the volume element of the metric induced in $M$ by $X_t$. Note that $dM_0=dM$ and $\mathcal A(0)=\mathcal A$, the volume of $M$. The following lemma is classical: \[lemma:first variation\] Let $\overline M^{n+1}$ be a time-oriented Lorentz manifold and $x:M^n\rightarrow\overline M^{n+1}$ a spacelike closed hypersurface having mean curvature $H$. If $X:M^n\times(-\epsilon,\epsilon)\rightarrow\overline M^{n+1}$ is a variation of $x$, then $$\frac{d\mathcal V}{dt}\Big|_{t=0}=\int_MfdM,\ \ \ \ \frac{d\mathcal A}{dt}\Big|_{t=0}=\int_MnHfdM.$$ Set $H_0=\frac{1}{\mathcal A}\int_MdM$ and $\mathcal J:(-\epsilon,\epsilon)\rightarrow\mathbb R$ given by $$\mathcal J(t)=\mathcal A(t)-nH_0\mathcal V(t).$$ $\mathcal J$ is called the [*Jacobi functional*]{} associated to the variation, and it is a well known result [@BdCE:88] that $x$ has constant mean curvature $H_0$ if and only if $\mathcal J'(0)=0$ for all variations $X$ of $x$. We wish to study here immersions $x:M^n\rightarrow\overline M^{n+1}$ that maximize $\mathcal J$ for all variations $X$. Since $x$ must be a critical point of $\mathcal J$, it thus follows from the above discussion that $x$ must have constant mean curvature. Therefore, in order to examine whether or not some critical immersion $x$ is actually a maximum for $\mathcal J$, one certainly needs to study the second variation $\mathcal J''(0)$. We start with the following Let $x:M^n\rightarrow\overline M^{n+1}$ be a closed spacelike hypersurface of the time-oriented Lorentz manifold $\overline M^{n+1}$, and $X:M^n\times(-\epsilon,\epsilon)\rightarrow\overline M^{n+1}$ be a variation of $x$. Then, $$\label{eq:fundamental relation} n\frac{\partial H}{\partial t}=\Delta f-\left\{\overline{Ric}(N,N)+|A|^2\right\}f-n\langle\left(\frac{\partial X}{\partial t}\right)^T,\nabla H\rangle.$$ Although the above proposition is known to be true, we believe there is a lack, in the literature, of a clear proof of it in this degree of generality, so we present a simple proof here. Let $p\in M$ and $\{e_k\}$ be a moving frame on a neighborhood $U\subset M$ of $p$, geodesic at $p$ and diagonalizing $A$ at $p$, with $Ae_k=\lambda_ke_k$ for $1\leq k\leq n$. Extend $N$ and the $e_k's$ to a neighborhood of $p$ in $\overline M$, so that $\langle N,e_k\rangle=0$ and $(\overline\nabla_Ne_k)(p)=0$. Then $$\begin{aligned} n\frac{\partial H}{\partial t}&=&-{\rm tr}\left(\frac{\partial A}{\partial t}\right)=-\sum_k\langle\frac{\partial A}{\partial t}e_k,e_k\rangle=-\sum_k\langle\left(\overline\nabla_{\frac{\partial X}{\partial t}}A\right)e_k,e_k\rangle\\ &=&-\sum_k\left\{\langle\overline\nabla_{\frac{\partial X}{\partial t}}Ae_k,e_k\rangle-\langle A\overline\nabla_{\frac{\partial X}{\partial t}}e_k,e_k\rangle\right\}\\ &=&\sum_k\langle\overline\nabla_{\frac{\partial X}{\partial t}}\overline\nabla_{e_k}N,e_k\rangle+\sum_k\langle A\overline\nabla_{e_k}\frac{\partial X}{\partial t},e_k\rangle,\end{aligned}$$ where in the last equality we used the fact that $[\frac{\partial X}{\partial t},e_k]=0$. Letting $$I=\sum_k\langle\overline\nabla_{\frac{\partial X}{\partial t}}\overline\nabla_{e_k}N,e_k\rangle\ \ \text{and}\ \ II=\sum_k\langle A\overline\nabla_{e_k}\frac{\partial X}{\partial t},e_k\rangle,$$ we have $$\begin{aligned} I&=&\sum_k\left\{\langle\overline\nabla_{\frac{\partial X}{\partial t}}\overline\nabla_{e_k}N-\overline\nabla_{e_k}\overline\nabla_{\frac{\partial X}{\partial t}}N+\overline\nabla_{[e_k,\frac{\partial X}{\partial t}]}N,e_k\rangle+\overline\nabla_{e_k}\overline\nabla_{\frac{\partial X}{\partial t}}N,e_k\rangle\right\}\\ &=&\sum_k\left\{\langle\overline R\left(e_k,\frac{\partial X}{\partial t}\right)N,e_k\rangle+\langle\overline\nabla_{e_k}\overline\nabla_{\frac{\partial X}{\partial t}}N,e_k\rangle\right\}\\ &=&-\overline{Ric}\left(\frac{\partial X}{\partial t},N\right)+\sum_k\langle\overline\nabla_{e_k}\overline\nabla_{\frac{\partial X}{\partial t}}N,e_k\rangle.\end{aligned}$$ Since the frame $\{e_k\}$ is geodesic at $p$, it follows that $$\langle\overline\nabla_{\frac{\partial X}{\partial t}}N,\overline\nabla_{e_k}e_k\rangle=\langle\overline\nabla_{\frac{\partial X}{\partial t}}N,N\rangle\langle\overline\nabla_{e_k}e_k,N\rangle=0$$ at $p$, and hence $$\begin{aligned} \langle\overline\nabla_{e_k}\overline\nabla_{\frac{\partial X}{\partial t}}N,e_k\rangle&=&e_k\langle\overline\nabla_{\frac{\partial X}{\partial t}}N,e_k\rangle=-e_k\langle N,\overline\nabla_{\frac{\partial X}{\partial t}}e_k\rangle=-e_k\langle N,\overline\nabla_{e_k}\frac{\partial X}{\partial t}\rangle\\ &=&-e_ke_k\langle N,\frac{\partial X}{\partial t}\rangle+e_k\langle\overline\nabla_{e_k}N,\frac{\partial X}{\partial t}\rangle\\ &=&e_ke_k(f)+e_k\langle\overline\nabla_{e_k}N,\left(\frac{\partial X}{\partial t}\right)^T\rangle\\ &=&e_ke_k(f)+\langle\overline\nabla_{e_k}\overline\nabla_{e_k}N,\left(\frac{\partial X}{\partial t}\right)^T\rangle-\langle Ae_k,\overline\nabla_{e_k}\left(\frac{\partial X}{\partial t}\right)^T\rangle.\end{aligned}$$ For $II$, we have $$\begin{aligned} II&=&\sum_k\langle Ae_k,\overline\nabla_{e_k}\frac{\partial X}{\partial t}\rangle=\sum_k\langle Ae_k,\overline\nabla_{e_k}(fN+\left(\frac{\partial X}{\partial t}\right)^T)\rangle\\ &=&\sum_k\langle Ae_k,f\overline\nabla_{e_k}N\rangle+\sum_k\langle Ae_k,\overline\nabla_{e_k}\left(\frac{\partial X}{\partial t}\right)^T\rangle\\ &=&-f|A|^2+\sum_k\langle Ae_k,\overline\nabla_{e_k}\left(\frac{\partial X}{\partial t}\right)^T\rangle\\\end{aligned}$$ Therefore, $$\label{eq:aux_I} n\frac{\partial H}{\partial t}=-\overline{Ric}\left(\frac{\partial X}{\partial t},N\right)+\Delta f-f|A|^2+\sum_k\langle\overline\nabla_{e_k}\overline\nabla_{e_k}N,\left(\frac{\partial X}{\partial t}\right)^T\rangle.$$ Now, letting $$\frac{\partial X}{\partial t}=\sum_l^n\alpha_le_l+fN$$ and $Ae_k=\sum_jh_{jk}e_j$, one successively gets $$\begin{aligned} \overline{Ric}\left(\frac{\partial X}{\partial t},N\right)&=&\sum_l\alpha_l\overline{Ric}(N,e_l)+f\overline{Ric}(N,N)\\ &=&\sum_{k,l}\alpha_l\langle\overline R(e_k,e_l)e_k,N\rangle+f\overline{Ric}(N,N)\\\end{aligned}$$ and, since $(\overline\nabla_Ne_k)(p)=0$, $$\begin{aligned} \langle\overline R(e_k,e_l)e_k,N\rangle_p&=&\langle\overline\nabla_{e_l}\overline\nabla_{e_k}e_k-\overline\nabla_{e_k}\overline\nabla_{e_l}e_k,N\rangle_p\\ &=&e_l\langle\overline\nabla_{e_k}e_k,N\rangle_p-\langle\overline\nabla_{e_k}e_k,\overline\nabla_{e_l}N\rangle_p-e_k\langle\overline\nabla_{e_l}e_k,N\rangle_p\\ &=&-e_l\langle e_k,\overline\nabla_{e_k}N\rangle_p+e_k\langle e_k,\overline\nabla_{e_l}N\rangle_p\\ &=&e_l(h_{kk})-e_k(h_{kl}),\end{aligned}$$ so that $$\label{eq:aux_II} \overline{Ric}\left(\frac{\partial X}{\partial t},N\right)_p=\sum_{k,l}\alpha_le_l(h_{kk})-\sum_{k,l}\alpha_le_k(h_{kl})+f\overline{Ric}(N,N)_p.$$ Also, $$\begin{aligned} \alpha_l\langle\overline\nabla_{e_k}\overline\nabla_{e_k}N,e_l\rangle&=&\alpha_l\langle\nabla_{e_k}\overline\nabla_{e_k}N,e_l\rangle=-\alpha_l\sum_j\langle\nabla_{e_k}h_{kj}e_j,e_l\rangle\\ &=&-\alpha_l\sum_j\left\{e_k(h_{kj})\delta_{lj}+h_{kj}\langle\nabla_{e_k}e_j,e_l\rangle\right\}\\ &=&-\alpha_le_k(h_{kl}),\end{aligned}$$ and hence $$\label{eq:aux_III} \sum_k\langle\overline\nabla_{e_k}\overline\nabla_{e_k}N,\left(\frac{\partial X}{\partial t}\right)^T\rangle=-\sum_{k,l}\alpha_le_k(h_{kl}).$$ Substituting (\[eq:aux\_II\]) and (\[eq:aux\_III\]) into (\[eq:aux\_I\]), we finally arrive at $$\begin{aligned} n\frac{\partial H}{\partial t}&=&-\sum_{k,l}\alpha_le_l(h_{kk})-f\overline{Ric}(N,N)_p+\Delta f-f|A|^2\\ &=&-\left(\frac{\partial X}{\partial t}\right)^T(nH)-f\overline{Ric}(N,N)_p+\Delta f-f|A|^2.\end{aligned}$$ Let $\overline M^{n+1}$ be a Lorentz manifold and $x:M^n\rightarrow\overline M^{n+1}$ be a closed spacelike hypersurface having constant mean curvature $H$. If $X:M^n\times(-\epsilon,\epsilon)\rightarrow\overline M^{n+1}$ is a variation of $x$, then $$\label{eq:second variation of J} \mathcal J''(0)(f)=\int_Mf\left\{\Delta f-\left(\overline{Ric}(N,N)+|A|^2\right)f\right\}dM.$$ In the notations of the above discussion, set $f=f(0)$ and note that $H(0)=H$. It follows from lemma \[lemma:first variation\] that $$\mathcal J'(t)=\int_Mn\left\{H(t)-H\right\}f(t)dM_t.$$ Therefore, differentiating with respect to $t$ once more $$\begin{aligned} \mathcal J''(0)&=&\int_MnH'(0)f(0)dM_0+\int_Mn\left\{H(0)-H\right\}\frac{d}{dt}f(t)dM_t\Big|_{t=0}\\ &=&\int_MnH'(0)fdM.\end{aligned}$$ Taking into account that $H$ is constant, relation (\[eq:fundamental relation\]) finally gives formula \[eq:second variation of J\] It follows from the previous result that $\mathcal J''(0)=\mathcal J''(0)(f)$ depends only on $f\in C^{\infty}(M)$, for which there exists a variation $X$ of $M^n$ such that $\left(\frac{\partial X}{\partial t}\right)^{\bot}=fN$. Therefore, the following definition makes sense: Let $\overline M^{n+1}$ be a Lorentz manifold and $x:M^n\rightarrow\overline M^{n+1}$ be a closed spacelike hypersurface having constant mean curvature $H$. We say that $x$ is strongly stable if, for every function $f\in C^{\infty}(M)$ for which there exists a variation $X$ of $M^n$ such that $\left(\frac{\partial X}{\partial t}\right)^{\bot}=fN$, one has $\mathcal J''(0)(f)\leq 0$. Conformal vector fields ======================= As in the previous section, let $\overline M^{n+1}$ be a Lorentz manifold. A vector field $V$ on $\overline M^{n+1}$ is said to be [*conformal*]{} if $$\mathcal L_V\langle\,\,,\,\,\rangle=2\psi\langle\,\,,\,\,\rangle$$ for some function $\psi\in C^{\infty}(\overline M)$, where $\mathcal L$ stands for the Lie derivative of the Lorentz metric of $\overline M$. The function $\psi$ is called the [*conformal factor*]{} of $V$. Since $\mathcal L_V(X)=[V,X]$ for all $X\in\mathcal X(\overline M)$, it follows from the tensorial character of $\mathcal L_V$ that $V\in\mathcal X(\overline M)$ is conformal if and only if $$\label{eq:1.1} \langle\overline\nabla_XV,Y \rangle+\langle X,\overline\nabla_YV\rangle=2\psi\langle X,Y\rangle,$$ for all $X,Y\in\mathcal X(\overline M)$. In particular, $V$ is a Killing vector field relatively to $\overline g$ if and only if $\psi\equiv 0$. Any Lorentz manifold $\overline M^{n+1}$, possessing a globally defined, timelike conformal vector field is said to be a [*conformally stationary spacetime*]{}. \[prop:Laplacian of conformal vector field\] Let $\overline M^{n+1}$ be a conformally stationary Lorentz manifold, with conformal vector field $V$ having conformal factor $\psi:\overline M^{n+1}\rightarrow\mathbb R$. Let also $x:M^n\rightarrow\overline M^{n+1}$ be a spacelike hypersurface of $\overline M^{n+1}$, and $N$ a future-pointing, unit normal vector field globally defined on $M^n$. If $f=\langle V,N\rangle$, then $$\label{eq:Laplacian formula_I} \Delta f=n\langle V,\nabla H\rangle+f\left\{\overline{Ric}(N,N)+|A|^2\right\}+n\left\{H\psi-N(\psi)\right\},$$ where $\overline{Ric}$ denotes the Ricci tensor of $\overline M$, $A$ is the second fundamental form of $x$ with respect to $N$, $H=-\frac{1}{n}{\rm tr}(A)$ is the mean curvature of $x$ and $\nabla H$ denotes the gradient of $H$ in the metric of $M$. Fix $p\in M$ and let $\{e_k\}$ be an orthonormal moving frame on $M$, geodesic at $p$. Extend the $e_k$ to a neighborhood of $p$ in $\overline M$, so that $(\overline\nabla_Ne_k)(p)=0$, and let $$V=\sum_l^n\alpha_le_l-fN.$$ Then $$\begin{aligned} f=\langle N,V\rangle\Rightarrow e_k(f)&=&\langle\overline\nabla_{e_k}N,V\rangle+\langle N,\overline\nabla_{e_k}V\rangle\\ &=&-\langle Ae_k,V\rangle+\langle N,\overline\nabla_{e_k}V\rangle,\end{aligned}$$ so that $$\begin{aligned} \label{eq:I} \Delta f&=&\sum_ke_k(e_k(f))=-\sum_ke_k\langle Ae_k,V\rangle+\sum_ke_k\langle N,\overline\nabla_{e_k}V\rangle\nonumber\\ &=&-\sum_k\langle\overline\nabla_{e_k}Ae_k,V\rangle-2\sum_k\langle Ae_k,\overline\nabla_{e_k}V\rangle+\sum_k\langle N,\overline\nabla_{e_k}\overline\nabla_{e_k}V\rangle.\end{aligned}$$ Now, differentiating $Ae_k=\sum_lh_{kl}e_l$ with respect to $e_k$, one gets at $p$ $$\begin{aligned} \label{eq:II} \sum_k\langle\overline\nabla_{e_k}Ae_k,V\rangle&=&\sum_{k,l}e_k(h_{kl})\langle e_l,V\rangle+\sum_{k,l}h_{kl}\langle\overline\nabla_{e_k}e_l,V\rangle\nonumber\\ &=&\sum_{k,l}\alpha_le_k(h_{kl})-\sum_{k,l}h_{kl}\langle\overline\nabla_{e_k}e_l,N\rangle\langle V,N\rangle\nonumber\\ &=&\sum_{k,l}\alpha_le_k(h_{kl})-\sum_{k,l}h_{kl}^2f\nonumber\\ &=&\sum_{k,l}\alpha_le_k(h_{kl})-f|A|^2.\end{aligned}$$ Asking further that $Ae_k=\lambda_ke_k$ at $p$ (which is always possible), we have at $p$ $$\label{eq:III} \sum_k\langle Ae_k,\overline\nabla_{e_k}V\rangle=\sum_k\lambda_k\langle e_k,\overline\nabla_{e_k}V\rangle=\sum_k\lambda_k\psi=-nH\psi.$$ In order to compute the last summand of (\[eq:I\]), note that the conformality of $V$ gives $$\langle\overline\nabla_NV,e_k\rangle+\langle N,\overline\nabla_{e_k}V\rangle=0$$ for all $k$. Hence, differentiating the above relation in the direction of $e_k$, we get $$\langle\overline\nabla_{e_k}\overline\nabla_NV,e_k\rangle+\langle\overline\nabla_NV,\overline\nabla_{e_k}e_k\rangle+\langle\overline\nabla_{e_k}N,\overline\nabla_{e_k}V\rangle+\langle N,\overline\nabla_{e_k}\overline\nabla_{e_k}V\rangle=0.$$ However, at $p$ one has $$\begin{aligned} \langle\overline\nabla_NV,\overline\nabla_{e_k}e_k\rangle&=&-\langle\overline\nabla_NV,\langle\overline\nabla_{e_k}e_k,N\rangle N\rangle=-\langle\overline\nabla_NV,\lambda_kN\rangle\\ &=&-\lambda_k\psi\langle N,N\rangle=\lambda_k\psi\end{aligned}$$ and $$\langle\overline\nabla_{e_k}N,\overline\nabla_{e_k}V\rangle=-\lambda_k\langle e_k,\overline\nabla_{e_k}V\rangle=-\lambda_k\psi,$$ so that $$\label{eq:IV} \langle\overline\nabla_{e_k}\overline\nabla_NV,e_k\rangle+\langle N,\overline\nabla_{e_k}\overline\nabla_{e_k}V\rangle=0$$ at $p$. On the other hand, since $$[N,e_k](p)=(\overline\nabla_Ne_k)(p)-(\overline\nabla_{e_k}N)(p)=\lambda_ke_k(p),$$ it follows from (\[eq:IV\]) that $$\begin{aligned} \langle\overline R(N,e_k)V,e_k\rangle_p&=&\langle\overline\nabla_{e_k}\overline\nabla_NV-\overline\nabla_N\overline\nabla_{e_k}V+\overline\nabla_{[N,e_k]}V,e_k\rangle_p\\ &=&-\langle N,\overline\nabla_{e_k}\overline\nabla_{e_k}V\rangle_p-N\langle\overline\nabla_{e_k}V,e_k\rangle_p+\langle\overline\nabla_{\lambda_ke_k}V,e_k\rangle_p\\ &=&-\langle N,\overline\nabla_{e_k}\overline\nabla_{e_k}V\rangle_p-N(\psi)+\lambda_k\psi,\end{aligned}$$ and hence $$\label{eq:V} \sum_k\langle N,\overline\nabla_{e_k}\overline\nabla_{e_k}V\rangle_p=-nN(\psi)-nH\psi-\overline{Ric}(N,V)_p$$ Finally, $$\begin{aligned} \overline{Ric}(N,V)&=&\sum_l\alpha_l\overline{Ric}(N,e_l)-f\overline{Ric}(N,N)\\ &=&\sum_{k,l}\alpha_l\langle\overline R(e_k,e_l)e_k,N\rangle-f\overline{Ric}(N,N),\end{aligned}$$ and $$\begin{aligned} \langle\overline R(e_k,e_l)e_k,N\rangle_p&=&\langle\overline\nabla_{e_l}\overline\nabla_{e_k}e_k-\overline\nabla_{e_k}\overline\nabla_{e_l}e_k,N\rangle_p\\ &=&e_l\langle\overline\nabla_{e_k}e_k,N\rangle_p-\langle\overline\nabla_{e_k}e_k,\overline\nabla_{e_l}N\rangle_p-e_k\langle\overline\nabla_{e_l}e_k,N\rangle_p\\ &&+\langle\overline\nabla_{e_l}e_k,\overline\nabla_{e_k}N\rangle_p\\ &=&-e_l\langle e_k,\overline\nabla_{e_k}N\rangle_p+e_k\langle e_k,\overline\nabla_{e_l}N\rangle_p\\ &=&e_l(h_{kk})-e_k(h_{kl}),\end{aligned}$$ so that $$\overline{Ric}(N,V)_p=\sum_{k,l}\alpha_le_l(h_{kk})-\sum_{k,l}\alpha_le_k(h_{kl})-f\overline{Ric}(N,N)_p,$$ and it follows from (\[eq:V\]) that $$\begin{aligned} \label{eq:VI} \sum_k\langle N,\overline\nabla_{e_k}\overline\nabla_{e_k}V\rangle_p&=&-nN(\psi)-nH\psi+V^T(nH)\nonumber\\ &&+\sum_{k,l}\alpha_le_k(h_{kl})+f\overline{Ric}(N,N).\end{aligned}$$ Substituting (\[eq:II\]), (\[eq:III\]) and (\[eq:VI\]) into (\[eq:I\]), one gets the desired formula (\[eq:Laplacian formula\_I\]). Applications ============ A particular class of conformally stationary spacetimes is that of [*generalized Robertson-Walker*]{} spacetimes [@ABC:03], namely, warped products $\overline M^{n+1}=I\times_{\phi}F^n$, where $I\subseteq\mathbb R$ is an interval with the metric $-dt^2$, $F^n$ is an $n$-dimensional Riemannian manifold and $\phi:I\rightarrow\mathbb R$ is positive and smooth. For such a space, let $\pi_I:\overline M^{n+1}\rightarrow I$ denote the canonical projection onto the $I-$factor. Then the vector field $$V=(\phi\circ\pi_I)\frac{\partial}{\partial t}$$ is conformal, timelike and closed (in the sense that its dual $1-$form is closed), with conformal factor $\psi=\phi'$, where the prime denotes differentiation with respect to $t$. Moreover, according to [@Montiel:99], for $t_0\in I$, orienting the (spacelike) leaf $M_{t_0}^n=\{t_0\}\times F^n$ by using the future-pointing unit normal vector field $N$, it follows that $M_{t_0}$ has constant mean curvature $$H=\frac{\phi'(t_0)}{\phi(t_0)}.$$ If $\overline M^{n+1}=I\times_{\phi}F^n$ is a generalized Robertson-Walker spacetime and $x:M^n\rightarrow\overline M^{n+1}$ is a complete spacelike hypersurface of $\overline M^{n+1}$, such that $\phi\circ\pi_I$ is limited on $M$, then $\pi_F\big|_M:M^n\rightarrow F^n$ is necessarily a covering map ([@ABC:03]). In particular, if $M^n$ is closed, then $F^n$ is automatically closed. One has the following corollary of proposition \[prop:Laplacian of conformal vector field\]: Let $\overline M^{n+1}=I\times_{\phi}F^n$ be a generalized Robertson-Walker spacetime, and $x:M^n\rightarrow\overline M^{n+1}$ a spacelike hypersurface of $\overline M^{n+1}$, having constant mean curvature $H$. Let also $N$ be a future-pointing unit normal vector field globally defined on $M^n$. If $V=(\phi\circ\pi_I)\frac{\partial}{\partial t}$ and $f=\langle V,N\rangle$, then $$\label{eq:Laplacian formula_II} \Delta f=\left\{\overline{Ric}(N,N)+|A|^2\right\}f+n\left\{H\phi'+\phi''\langle N,\frac{\partial}{\partial t}\rangle\right\}.$$ where $\overline{Ric}$ denotes the Ricci tensor of $\overline M$, $A$ is the second fundamental form of $x$ with respect to $N$, and $H=-\frac{1}{n}{\rm tr}(A)$ is the mean curvature of $x$. First of all, $f=\langle V,N\rangle=\phi\langle N,\frac{\partial}{\partial t}\rangle$, and it thus follows from (\[eq:Laplacian formula\_I\]) that $$\Delta f=\left\{\overline{Ric}(N,N)+|A|^2\right\}f+n\left\{H\phi'-N(\phi')\right\}.$$ However, $$\overline\nabla\phi'=-\langle\overline\nabla\phi',\frac{\partial}{\partial t}\rangle\frac{\partial}{\partial t}=-\phi''\frac{\partial}{\partial t},$$ so that $$N(\phi')=\langle N,\overline\nabla\phi'\rangle=-\phi''\langle N,\frac{\partial}{\partial t}\rangle$$ We can now state and prove our main result: Let $\overline M^{n+1}=I\times_{\phi}F^n$ be a generalized Robertson-Walker spacetime, and $x:M^n\rightarrow\overline M^{n+1}$ be a closed spacelike hypersurface of $\overline M^{n+1}$, having constant mean curvature $H$. If the warping function $\phi$ satisfies $\phi''\geq\max\{H\phi',0\}$ and $M^n$ is strongly stable, then $M^{n}$ is either minimal or a spacelike slice $M_{t_0}=\{t_0\}\times F$, for some $t_0\in I$. Since $M^n$ is strongly stable, we have $$0\geq\mathcal J''(0)(g)=\int_Mg\left\{\Delta g-\left(\overline{Ric}(N,N)+|A|^2\right)g\right\}dM$$ for all $g\in C^{\infty}(M)$ for which $gN$ is the normal component of the variational field of some variation of $M^n$. In particular, if $f=\langle V,N\rangle=\phi\langle N,\frac{\partial}{\partial t}\rangle$, where $V=(\phi\circ\pi_I)\frac{\partial}{\partial t}$, and $g=-f=-\langle V,N\rangle$, then $$\Delta g=\left\{\overline{Ric}(N,N)+|A|^2\right\}g-n\left\{H\phi'+\phi''\langle N,\frac{\partial}{\partial t}\rangle\right\}.$$ Therefore, $M^n$ stable implies $$0\geq\int_M\phi\langle N,\frac{\partial}{\partial t}\rangle\left\{H\phi'+\phi''\langle N,\frac{\partial}{\partial t}\rangle\right\}dM$$ Letting $\theta$ be the hyperbolic angle between $N$ and $\frac{\partial}{\partial t}$, it follows from the reversed Cauchy-Schwarz inequality that $\cosh\theta=-\langle N,\frac{\partial}{\partial t}\rangle$, with $\cosh\theta\equiv 1$ if and only if $N$ and $\frac{\partial}{\partial t}$ are collinear at every point, that is, if and only if $M^n$ is a spacelike leaf $M_{t_0}$ for some $t_0\in I$. Hence, $$0\geq\int_M\phi\cosh\theta\left\{-H\phi'+\phi''\cosh\theta\right\}dM.$$ Now, notice that $-H\phi'+\phi''\cosh\theta\geq-\phi''+\phi''\cosh\theta$, which gives $$\phi\cosh\theta(-H\phi'+\phi''\cosh\theta)\geq\phi\phi''\cosh\theta(\cosh\theta-1).$$ Therefore, $$0\geq\int_M\phi\cosh\theta(-H\phi'+\phi''\cosh\theta)dM\geq\int_M\phi\phi''\cosh\theta(\cosh\theta-1)\geq 0,$$ and hence $$\phi''(\cosh\theta-1)=0\ \ \text{and}\ \ \phi''=H\phi'$$ on $M$. If, for some $p\in M$, one has $\phi''(p)=0$, then $\phi'H=0$ at $p$. If $H\neq 0$, then $\phi'(p)=0$. But if this is the case, then proposition 7.35 of [@O'Neill:83] gives that $$\overline\nabla_V\frac{\partial}{\partial t}=\frac{\phi'}{\phi}V=0$$ at $p$ for any $V$, and $M$ is totally geodesic at $p$. In particular, $H=0$, a contradiction. Therefore, either $\phi''(p)=0$ for some $p\in M$, and $M$ is minimal, or $\phi''\neq 0$ on all of $M$, whence $\cosh\theta=1$ always, and $M$ is an umbilical leaf such that $\phi''=H\phi'$. Note that $\frac{\phi''}{\phi'}=H=\frac{\phi'}{\phi}$, i.e., $\phi''\phi-(\phi')^2=0$, which is a limit case of Alías and Montiel’s timelike convergent condition. [10]{} L. J. Alías, A. Brasil Jr. and A. G. Colares, [*Integral Formulae for Spacelike Hypersurfaces in Conformally Stationary Spacetimes and Applications*]{}, Proc. Edinburgh Math. Soc. 46, (2003) 465-488 . L. J. Alías S. Montiel, [*Uniqueness of Spacelike Hypersurfaces with Constant Mean Curvature in Generalized Robertson-Walker Spacetimes*]{}, Proceedings of the International Conference held to honour the 60th birthday of A.M.Naveira, World Scientific, (2001) 59-69. J. L. M. Barbosa A. G. Colares, [*Stability of Hypersurfaces with Constant $r-$Mean Curvature*]{}, Ann. Global Anal. Geom. 15, (1997) 277-297. J. L. M. Barbosa M. do Carmo, [*Stability of Hypersurfaces with Constant Mean Curvature*]{}, Math. Z. 185, (1984) 339-353. J. L. M. Barbosa, M. do Carmo J. Eschenburg, [*Stability of Hypersurfaces with Constant Mean Curvature*]{}, Math. Z. 197, (1988) 123-138. J. L. M. Barbosa V. Oliker, [*Spacelike Hypersurfaces with Constant Mean Curvature in Lorentz Spaces*]{}, Matem. Contemporânea 4, (1993) 27-44. S. Montiel, [*Uniqueness of Spacelike Hypersurfaces of Constant Mean Curvature in foliated Spacetimes*]{}, Math. Ann. 314, (1999) 529-553. B. O’Neill, [*Semi-Riemannian Geometry with Applications to Relativity*]{}, London, Academic Press (1983).

--- abstract: '[Multisymplectic systems, partial differential equations, fluid dynamics, conservation laws, potential vorticity]{} We construct multisymplectic formulations of fluid dynamics using the inverse of the Lagrangian path map. This inverse map – the “back-to-labels” map – gives the initial Lagrangian label of the fluid particle that currently occupies each Eulerian position. Explicitly enforcing the condition that the fluid particles carry their labels with the flow in Hamilton’s principle leads to our multisymplectic formulation. We use the multisymplectic one-form to obtain conservation laws for energy, momentum and an infinite set of conservation laws arising from the particle-relabelling symmetry and leading to Kelvin’s circulation theorem. We discuss how multisymplectic numerical integrators naturally arise in this approach.' author: - 'C. J. Cotter, D. D. Holm and P. E. Hydon' title: Multisymplectic formulation of fluid dynamics using the inverse map --- \[firstpage\] Introduction ============ A system of partial differential equations (PDEs) is said to be *multisymplectic* if it is of the form $$K^{\alpha}_{ij}(\MM{z})z^j_{,\alpha} = {\frac{\partial H}{\partial z^i}},$$ where each of the two-forms $$\kappa^\alpha = \frac{1}{2}K_{ij}^\alpha(\MM{z})\, {\mathrm{d}}z^i\wedge{\mathrm{d}}z^j$$ is closed. Here $\MM{z}$ is an ordered set of dependent variables, total differentiation with respect to each independent variable $q^\alpha$ is denoted by the subscript $\alpha$ after a comma, and the Einstein summation convention is used. The closed two-form $\kappa^\alpha$ is associated with the independent variable $q^\alpha$; it is analogous to the symplectic two-form for a Hamiltonian ordinary differential equation. Hence there is a symplectic structure associated with each independent variable. In the first of a series of papers, Bridges (1997) pioneered the development of multisymplectic systems, showing that the rich geometric structure that is endowed by the symplectic two-forms can be used to understand the interaction and stability of nonlinear waves. For many important PDEs, the multisymplectic formulation has revealed hidden features that are important in stability analysis. In order to preserve at least some of these features in numerical simulations, Bridges & Reich (2001) introduced multisymplectic integrators, which generalise the symplectic methods that have been widely used in numerical Hamiltonian dynamics. Hydon (2005) showed that multisymplectic systems of PDEs may be derived from Hamilton’s principle whenever the Lagrangian is affine in the first-order derivatives and contains no higher-order derivatives. This can usually be achieved by introducing auxiliary variables to eliminate the derivatives. The aim of this paper is to provide a unified approach to producing multisymplectic formulations of fluid dynamics, based on the inverse map. Our approach covers all fluid dynamical equations that are written in Euler-Poincaré form (Holm *et al.*, 1998), *i.e.* all equations which arise due to the advection of fluid material. First we use the inverse map to form a canonical Euler-Lagrange equation (following the Clebsch representation given in Holm & Kupershmidt, 1983). Then the Lagrangian is made affine in the space and time derivatives by using constraints that introduce additional variables. Following Hydon (2005), we obtain a one-form quasi-conservation law which, when it is pulled back to the space and time coordinates, gives conservation laws for momentum and energy. We also obtain a two-form conservation law that represents conservation of symplecticity; when this is pulled back to the spatial coordinates, it leads to a conservation law for vorticity. The multisymplectic version of Noether’s Theorem yields an infinite space of conservation laws from the particle-relabelling symmetry for fluid dynamics; these conservation laws imply Kelvin’s circulation theorem. The conserved momentum that is canonically conjugate to the back-to-labels map plays a key role in the derivation of the conservation laws. The corresponding velocity is the convective velocity, whose geometric properties are discussed in Holm *et al.* (1986). In this paper we show how the above constructions are made in general, illustrating this with examples. We also discuss how multisymplectic integrators can be constructed using these methods. Sections \[review\] and \[inverse map sec\] review the relations among multisymplectic structures, the Clebsch representation and the momentum map associated with particle relabelling. Section \[inverse map EPDiff\] shows how to construct a multisymplectic formulation of the Euler-Poincaré equation for the diffeomorphism group (EPDiff), and derives the corresponding conservation laws, including the infinite set of conservation laws that yield Kelvin’s circulation theorem. Section \[advected\] extends this formulation to the Euler-Poincaré equation with advected quantities. This is illustrated by the incompressible Euler equation, showing how the circulation theorem arises in the multisymplectic formulation. Section \[numerics\] sketches numerical issues in the multisymplectic framework. Finally, Section \[summary\] summarises and outlines directions for future research. Review of multisymplectic structures {#review} ==================================== This section reviews the formulation of multisymplectic systems and their conservation laws, following Hydon (2005). A system of partial differential equation (PDEs) is multisymplectic provided that it can be represented as a variational problem with a Lagrangian that is affine in the first derivatives of the dependent variables: $$\label{mslag} L = L_j^\alpha(\MM{z})z^j_{,\alpha} - H(\MM{z}).$$ The Euler-Lagrange equations are then $$\label{Eul-Lag-eqns}K^{\alpha}_{ij}(\MM{z})z^j_{,\alpha} = {\frac{\partial H}{\partial z^i}},$$ where the functions $$\label{Msymp-struct-matrix} K^{\alpha}_{ij}(\MM{z}) = {\frac{\partial L^\alpha_j}{\partial z^i}}-{\frac{\partial L^\alpha_i}{\partial z^j}}$$ are coefficients of the multisymplectic structure matrix. We define the (closed) symplectic two-forms $$\label{kappa} \kappa^\alpha = \frac{1}{2}K_{ij}^\alpha(\MM{z})\, {\mathrm{d}}z^i\wedge{\mathrm{d}}z^j,$$ and obtain the structural conservation law (Bridges, 1997). $$\label{kappa law} \kappa^\alpha_{,\alpha} = 0.$$ Hydon showed that the Poincaré Lemma leads to a one-form quasi-conservation law $$(L^{\alpha}_jdz^j)_{,\alpha} = {\mathrm{d}}(L^\alpha_jz^j_{,\alpha}-H(\MM{z}))= {\mathrm{d}}{L}, \label{ofcl}$$ whose exterior derivative is (\[kappa law\]). Every one-parameter Lie group of point symmetries of the multisymplectic system (\[Eul-Lag-eqns\]) is generated by a differential operator of the form $$\label{X} X = Q^i(\MM{q},\MM{z}){\frac{\partial }{\partial z^i}} + (Q^i(\MM{q},\MM{z}))_{,\alpha} {\frac{\partial }{\partial z^i_{,\alpha}}}.$$ Noether’s Theorem implies that if $X$ generates variational symmetries, that is, if $$\label{varsym} XL = B^\alpha_{,\alpha}$$ for some functions $B^\alpha$, then the interior product of $X$ with the one-form quasi-conservation law yields the conservation law $$\label{noethm} (L_j^{\alpha}Q^j-B^{\alpha})_{,\alpha}=0.$$ This is the multisymplectic form of Noether’s theorem. Every multisymplectic system is invariant under translations in the independent variables $\MM{q}$. For each of these symmetries, Noether’s theorem yields a conservation law $$(L_j^\alpha z^j_{,\beta}-L\delta^\alpha_\beta)_{,\alpha}=0.$$ Such conservation laws can equally well be obtained by pulling back the quasi-conservation law (\[ofcl\]) to the base space of independent variables. Commonly, the independent variables are spatial position $\MM{x}$ and time $t$. Pulling back (\[ofcl\]) to these base coordinates yields the energy conservation law from the ${\mathrm{d}}{t}$ component, and the momentum conservation law from the remaining components. We shall see the form of these conservation laws for fluid dynamics in later sections. The inverse map and Clebsch representation {#inverse map sec} ========================================== Lagrangian fluid dynamics and the inverse map --------------------------------------------- Lagrangian fluid dynamics provides evolution equations for particles moving with a fluid flow. This is typically done by writing down a flow map $\Phi$ from some reference configuration to the fluid domain $\Omega$ at each instance in time. As the fluid particles cannot cavitate, superimpose or jump, this map must be a diffeomorphism. For an $n$-dimensional fluid flow, the flow map $\Phi:\,\mathbb{R}^n\times\mathbb{R}\mapsto\mathbb{R}^n$ given by $\MM{x}=\Phi(\MM{l},t)$ specifies the spatial position at time $t$ of the fluid particle that has *label* $\MM{l}=\Phi(\MM{x},0)$. The *inverse map* $\Phi^{-1}$ gives the label of the particle that occupies position $\MM{x}$ at time $t$ as the function $\MM{l}=\Phi^{-1}(\MM{x},t)$. The Eulerian velocity field $\MM{u}(\MM{x},t)$ gives the velocity of the fluid particle that occupies position $\MM{x}$ at time $t$ as follows: $$\MM{\dot{x}}(\MM{l},t)=\MM{u}(\MM{x}(\MM{l},t),t).$$ Each label component $l_k(\MM{x},t)$ satisfies the advection law $$\label{label eqn} l_{k,t} + u_il_{k,i} = 0.$$ Here ${}_{,t}$ and ${}_{,i}$ denote differentiation with respect to $t$ and $x_i$ respectively. We use Cartesian coordinates and the Euclidean inner product[^1], so we shall not generally distinguish between ‘up’ and ‘down’ indices; summation from 1 to $n$ is implied whenever an index is repeated. Clebsch representation using the inverse map -------------------------------------------- A canonical variational principle for fluid dynamics may be formulated by following the standard Clebsch procedure using the inverse map (Seliger & Whitham (1968), Holm & Kupershmidt, 1983). The Clebsch procedure begins with a functional $\ell[\MM{u}]$ of the Eulerian fluid velocity $\MM{u}$, which is known as the *reduced Lagrangian* in the context of Euler-Poincaré reduction (Holm *et al.*, 1998). One then enforces stationarity of the action $S=\int\ell[\MM{u}]{\mathrm{d}}t$ under the constraint that equation (\[label eqn\]) is satisfied by using a vector of $n$ Lagrange multipliers, which is denoted as $\MM{\pi}$. These Lagrange multipliers are the conjugate momenta to $\MM{l}$ in the course of the Legendre transformation to the Hamiltonian formulation. One may choose $\ell[\MM{u}]$ to be solely the kinetic energy, which depends only on $\MM{u}$. More generally, $\ell$ will also depend on thermodynamic Eulerian variables such as density, whose evolution may also be accommodated by introducing constraints. These constraints are often called the “Lin constraints” (Serrin, 1959). This idea was also used in reformulating London’s variational principle for superfluids (Lin, 1963). \[clebsch\] The Clebsch variational principle using the inverse map is $$\delta \int_{t_0}^{t_1} \ell[\MM{u}] + \int_{\Omega} \MM{\pi}\cdot(\MM{l}_t+\MM{u}\cdot\nabla\MM{l}){\mathrm{d}}V(\MM{x}) {\mathrm{d}}{t}=0,$$ where $\MM{\pi}(\MM{x},t)$ are Lagrange multipliers which enforce the constraint that particle labels $\MM{l}(\MM{x},t)$ are advected by the flow. Taking the indicated variations leads to the following equations: $$\begin{aligned} \label{momentum map} \delta \MM{u}:&& {\frac{\delta \ell}{\delta \MM{u}}} + (\nabla\MM{l})^T\cdot\MM{\pi} = 0, \\ \nonumber \delta \MM{\pi}:&& \MM{l}_t+(\MM{u}\cdot\nabla)\MM{l} = 0, \\ \nonumber \delta \MM{l}:&& \MM{\pi}_t+\nabla\cdot(\MM{u}\MM{\pi}) = 0,\end{aligned}$$ where $$\begin{aligned} \label{notation} \left((\nabla\MM{l})^T\cdot\MM{\pi}\right)_i := \pi_kl_{k,i}, \qquad (\nabla\cdot(\MM{u}\MM{\pi}))_k := (u_j\pi_k)_{,j},\end{aligned}$$ and the variational derivative $\delta{\ell}/\delta{\MM{u}}$ is defined by $$\ell[\MM{u}+\epsilon\MM{u}'] = \ell[\MM{u}] + \epsilon\int_{\Omega} {\frac{\delta \ell}{\delta \MM{u}}}\cdot\MM{u}' \,{\mathrm{d}}V(\MM{x}) + \mathcal{O}(\epsilon^2) \,.$$ In the language of fluid mechanics, the expression (\[momentum map\]) for the spatial momentum $\MM{m}=\delta{\ell}/\delta\MM{u}$ in terms of canonically conjugate variables $(\MM{l}, \MM{\pi})$ is an example of a “Clebsch representation,” which expresses the solution of the EPDiff equations (see below) in terms of canonical variables that evolve by standard canonical Hamilton equations. This has been known in the case of fluid mechanics for more than 100 years. For modern discussions of the Clebsch representation for ideal fluids, see, for example, Holm & Kupershmidt (1983) and Marsden & Weinstein (1983). In the language of geometric mechanics, the Clebsch representation is a momentum map. Particle relabelling -------------------- As the physics of fluids should be independent of the labelling of particles, one may relabel the particles (by a diffeomorphism of the flow domain) without changing the dynamics. This is called the *particle relabelling symmetry*; Noether’s theorem applied to this symmetry leads to the Kelvin circulation theorem. See Holm *et al.* (1998) for a modern description. Clebsch momentum map -------------------- A [[******]{}momentum map]{} is a map $\mathbf{J}: T^{\ast}Q \rightarrow \mathfrak{g}^\ast$ from the cotangent bundle $T^*Q$ of the configuration manifold $Q$ to the dual $\mathfrak{g}^\ast$ of the Lie algebra $\mathfrak{g}$ of a Lie group $G$ that acts on $Q$. The momentum map is defined by the formula, $$\label{momentummapdef} \mathbf{J} (\nu_q) \cdot \xi = \left\langle \nu_q, \xi_Q (q) \right\rangle,$$ where $\nu_q \in T ^{\ast} _q Q $ and $\xi \in \mathfrak{g}$. In this formula $\xi _Q $ is the infinitesimal generator of the action of ${G}$ on $Q$ associated with the Lie algebra element $\xi$, and $\left\langle \nu_q, \xi_Q (q) \right\rangle$ is the natural pairing of an element of $T ^{\ast}_q Q $ with an element of $T _q Q $. The Clebsch relation (\[momentum map\]) defines a momentum map for the right action $\operatorname{Diff} (\Omega)$ of the diffeomorphisms of the domain $\Omega$ on the back-to-labels map $\MM{l}$. [^2] The spatial momentum in equation (\[momentum map\]) may be rewritten as a map $\MM{J}_\Omega:\,T^*\Omega\mapsto\mathfrak{X}^*(\Omega)$ from the cotangent bundle of $\Omega$ to the dual $\mathfrak{X}^*(\Omega)$ of the vector fields $\mathfrak{X}(\Omega)$ given by $$\begin{aligned} \label{rightmommap} \MM{J}_\Omega:\,\MM{m}\cdot {\mathrm{d}}\MM{x} = - \Big( (\nabla\MM{l})^T\cdot\MM{\pi} \Big)\cdot {\mathrm{d}}\MM{x} = - \,\MM{\pi} \cdot {\mathrm{d}}\MM{l} =: - \,\pi_k {\mathrm{d}}l_k \,.\end{aligned}$$ That is, $\MM{J}_\Omega$ maps the space of labels and their conjugate momenta $(\MM{l},\MM{\pi})\in T^*\Omega$ to the space of one-form densities $\MM{m}\in\mathfrak{X}^*(\Omega)$ on $\Omega$. The map (\[rightmommap\]) may be associated with the [*right action*]{} $\MM{l}\cdot\eta$ of smooth invertible maps (diffeomorphisms) $\eta$ of the back-to-labels maps $\MM{l}$ by composition of functions, as follows, $$\label{rightDiff} \operatorname{Diff}(\Omega):\ \MM{l}\cdot\eta=\MM{l}\circ\eta \,.$$ The infinitesimal generator of this right action is obtained from its definition, as $$\label{infgen-right} X_{\Omega}(\MM{l}) := \frac{d}{ds}\Big|_{s=0}\Big(\MM{l}\circ\eta(s)\Big) = T\MM{l} \circ X \,,$$ in which the vector field $X \in \mathfrak{X}(\Omega)$ is tangent to the curve of diffeomorphisms $\eta _s$ at the identity $s = 0$. Thus, pairing the map $\MM{J}_\Omega$ with the vector field $X \in \mathfrak{X}(\Omega)$ yields $$\begin{aligned} \left\langle \MM{J}_\Omega (\MM{l}, \MM{\pi} ), X \right\rangle & = -\,\langle\, \MM{\pi} \cdot {\mathrm{d}}\MM{l} \,, X \,\rangle \\& = -\,\int_S \pi_kl_{k,j}X_j (\MM{x}) \,{\mathrm{d}}V(\MM{x}) \\& = -\,\left\langle (\MM{l}, \MM{\pi}), T\MM{l}\cdot X \right\rangle \\& = -\,\left\langle (\MM{l}, \MM{\pi}), X_{\Omega}(\MM{l}) \right\rangle \,,\end{aligned}$$ where $\left\langle \,\cdot\,, \,\cdot\,\right\rangle:\,T ^{\ast}_{\MM{l}}\Omega\times T_{\MM{l}}\Omega\mapsto\mathbb{R}$ is the $L^2$ pairing of an element of $T ^{\ast}_{\MM{l}}\Omega $ (a one-form density) with an element of $T_{\MM{l}}\Omega $ (a vector field).\ Consequently, the Clebsch map (\[momentum map\]) satisfies the defining relation (\[momentummapdef\]) to be a momentum map, $$\label{momentummap-JOmega} \MM{J}(\MM{l}, \MM{\pi}) = -\,\MM{\pi} \cdot {\mathrm{d}}\MM{l} \,,$$ with the $L^2$ pairing of the one-form density $-\,\MM{\pi} \cdot {\mathrm{d}}\MM{l}$ with the vector field $X$. Being the cotangent lift of the action of $\operatorname{Diff} (\Omega)$, the momentum map $\mathbf{J}_\Omega$ in (\[rightmommap\]) is equivariant and Poisson. That is, substituting the canonical Poisson bracket into relation yields the Lie-Poisson bracket on the space of $\MM{m}$’s. See, for example, Holm & Kupershmidt (1983) and Marsden & Weinstein (1983) for more explanation, discussion and applications. The momentum map property of the Clebsch representation guarantees that the canonically conjugate variables $(\MM{l},\MM{\pi})$ may be eliminated in favour of the spatial momentum $\MM{m}$. Before its momentum map property was understood, the use of the Clebsch representation to eliminate the canonical variables in favour of Eulerian fluid variables was a tantalising mystery (Seliger & Whitham, 1968). Note that the right action of $\operatorname{Diff}(\Omega)$ on the inverse map is not a symmetry. In fact, as we shall see, the right action of $\operatorname{Diff}(\Omega)$ on the inverse map generates the fluid motion itself. Elimination theorem ------------------- Eliminating the canonically conjugate variables $(\MM{l}, \MM{\pi})$ produces an equation of motion for $\MM{m}=\delta{\ell}/\delta\MM{u}$, which is constructed in the proof of the following theorem: The labels $\MM{l}$ and their conjugate momenta $\MM{\pi}$ may be eliminated from the equations arising from the variational principle (\[clebsch\]) to obtain the weak form of the following equation of motion for $\delta{\ell}/\delta\MM{u}$: $${\frac{\partial }{\partial t}}{\frac{\delta \ell}{\delta \MM{u}}} + \operatorname{ad}^*_{\MM{u}}{\frac{\delta \ell}{\delta \MM{u}}} =0,$$ where $$\operatorname{ad}^*_{\MM{u}}\MM{m} = \nabla\cdot(\MM{u}\MM{m}) + (\nabla\MM{u})^T\cdot \MM{m}$$ is defined by $$\langle \operatorname{ad}^*_{\MM{u}}\MM{m},\MM{w}\rangle = -\langle \MM{m},\operatorname{ad}_{\MM{u}}\MM{w}\rangle = \langle \MM{m},(\MM{u}\cdot\nabla)\MM{w} - (\MM{w}\cdot\nabla)\MM{u}\rangle,$$ and $\langle\cdot,\cdot\rangle$ is the $L^2$ inner-product. This equation of motion for ${\delta\ell}/{\delta\MM{u}}$ is the Euler-Poincaré equation for the diffeomorphism group (EPDiff) (Holm *et al.*, 1998). Take the time-derivative of the inner product of ${\delta\ell}/{\delta\MM{u}}$ with a time-independent vector field $\MM{w}$: $$\begin{aligned} {\frac{d }{d t}}\Bigg\langle {\frac{\delta \ell}{\delta \MM{u}}},\MM{w}\Bigg\rangle & = & {\frac{d }{d t}}\Bigg\langle -(\nabla\MM{l})^T\cdot\MM{\pi} , \MM{w} \Bigg\rangle = {\frac{d }{d t}}\Bigg\langle -\MM{\pi},(\MM{w}\cdot\nabla)\MM{l}\Bigg\rangle, \\ & = & \Bigg\langle \nabla\cdot(\MM{u}\MM{\pi}), (\MM{w}\cdot\nabla)\MM{l}\Bigg\rangle + \Bigg\langle \MM{\pi}, (\MM{w}\cdot\nabla)(\MM{u}\cdot\nabla)\MM{l}\Bigg\rangle , \\ & = & \Bigg\langle \MM{\pi}, -(\MM{u}\cdot\nabla)(\MM{w}\cdot\nabla)\MM{l}+ (\MM{w}\cdot\nabla)(\MM{u}\cdot\nabla)\MM{l}\Bigg\rangle = \Bigg\langle \MM{\pi},-\left(\operatorname{ad}_{\MM{u}}\MM{w}\cdot\nabla\right)\MM{l} \Bigg\rangle , \\ &=& \Bigg\langle -(\nabla\MM{l})^T\cdot\MM{\pi},\operatorname{ad}_{\MM{u}}\MM{w} \Bigg\rangle = \Bigg\langle {\frac{\delta \ell}{\delta \MM{u}}},\operatorname{ad}_{\MM{u}}\MM{w}\Bigg\rangle = -\,\Bigg\langle \operatorname{ad}^*_{\MM{u}}{\frac{\delta \ell}{\delta \MM{u}}},\MM{w}\Bigg\rangle,\end{aligned}$$ which is the (weak form of the) EPDiff equation. Example: EPDiff($H^1$) ---------------------- To give a concrete example, consider EPDiff with $\ell[\MM{u}]$ being the $H^1_\lambda$-norm for $\MM{u}$. This is the $n$-dimensional Camassa-Holm (CH) equation (Camassa & Holm, 1993; Holm *et al.*, 1998; Holm & Marsden, 2004), which has applications in computational anatomy (Holm *et al.*, 2004; Miller *et al.*, 2002). This system has the reduced Lagrangian $$\ell[\MM{u}] = \int_{\Omega} \frac{1}{2}\,\left(|\MM{u}|^2+\lambda^2|\nabla\MM{u}|^2\right){\mathrm{d}}V(\MM{x}) = \int_{\Omega} \frac{1}{2}\,\left(u_iu_i+\lambda^2u_{i,j}u_{i,j}\right){\mathrm{d}}V(\MM{x}) = \frac{1}{2}\|\MM{u}\|^2_{H^1_\lambda}.$$ The EPDiff equation amounts to $${\frac{\partial \MM{m}}{\partial t}} + (\MM{u}\cdot\nabla)\MM{m} + (\nabla\MM{u})^T\cdot\MM{m} + \MM{m}\nabla\cdot\MM{u} = 0, \qquad \MM{m} = (1-\lambda^2\nabla^2)\MM{u} \,.$$ When $n=1$, these reduce to the Camassa-Holm (CH) equation, $$m_t + um_x + 2mu_x = 0,\qquad m = u - \lambda^2u_{xx} \,.$$ Advected quantities ------------------- To construct more general fluid equations we shall include advected quantities $a$ whose flow-rules are defined by $$a_t + \mathcal{L}_{\MM{u}}a = 0,$$ where $\mathcal{L}_{\MM{u}}$ is the Lie derivative. Such advected variables typically arise in the potential energy or the thermodynamic internal energy of an ideal fluid. For example, advected scalars $s$ (as in salinity) satisfy $${\frac{\partial }{\partial t}}s + \mathcal{L}_{\MM{u}}s=0, \quad \textrm{\emph{i.e.}} \quad s_t + (\MM{u}\cdot\nabla)s = 0,$$ and advected densities $\rho\,{\mathrm{d}}V$ satisfy, $${\frac{\partial }{\partial t}}(\rho\,{\mathrm{d}}V) + \mathcal{L}_{\MM{u}}(\rho\,{\mathrm{d}}V)=0, \quad \textrm{\emph{i.e.}} \quad \rho_t+\nabla\cdot(\rho\MM{u}) = 0. \,. \label{advecD}$$ A more extensive list of different types of advected quantity is given in Holm *et al.* (1998). We write the reduced Lagrangian $\ell$ as a functional of the Eulerian fluid variables $\MM{u}$ and $a$, and add further constraints to the action $S$ to account for their advection relations, $$\label{principle with advected qs} S = \int \ell[\MM{u},a]\,{\mathrm{d}}t + \int {\mathrm{d}}t\int_{\Omega}\MM{\pi}\cdot(\MM{l}_t+ (\MM{u}\cdot\nabla)\MM{l}) + \phi(a_t+\mathcal{L}_{\MM{u}}a) \,{\mathrm{d}}V(\MM{x}).$$ The Euler-Lagrange equations, which follow from the stationarity condition $\delta S=0$, are $$\begin{aligned} \delta\MM{u}:&& {\frac{\delta \ell}{\delta \MM{u}}} + (\nabla\MM{l})^T\cdot\MM{\pi} + \phi\diamond a = 0, \label{EL advected 1}\\ \delta\MM{\pi}:&& \MM{l}_t + (\MM{u}\cdot\nabla)\MM{l} = 0, \nonumber\\ \delta\MM{l}:&& -\MM{\pi}_t - \nabla\cdot(\MM{u}\MM{\pi}) = 0, \label{EL advected 3}\\ \delta\phi:&& a_t + \mathcal{L}_{\MM{u}}a = 0, \nonumber \\ \delta{a}:&& -\phi_t-\mathcal{L}_{\MM{u}}\phi + {\frac{\delta \ell}{\delta a}} = 0, \label{EL advected 5}\end{aligned}$$ where the diamond operator ($\diamond$) is defined as the dual of the Lie derivative operation $\mathcal{L}_{\MM{u}}$ with respect to the $L^2$ pairing. Explicitly, under integration by parts, $$\int_\Omega (\phi\diamond a)\cdot\MM{u}\,{\mathrm{d}}{V}(\MM{x}) = -\int_\Omega (\phi\mathcal{L}_{\MM{u}}a)\,{\mathrm{d}}{V}(\MM{x}).$$ The map to the spatial momentum in equation (\[EL advected 1\]) $${\frac{\delta \ell}{\delta \MM{u}}} =: \MM{m} = -\,\pi_A\nabla l^A -\, \phi\diamond a \,,$$ is again a momentum map, this time for the semidirect-product action of the diffeomorphisms on $\Omega\times V^*$. Again the momentum map property allows the canonical variables to be eliminated in favour of the Eulerian quantities. As a result, eliminating the variables $\MM{l}$, $\MM{\pi}$ and $\phi$ leads to the Euler-Poincaré equation with advected quantities $a$. The labels $\MM{l}$, their conjugate momenta $\MM{\pi}$ and the conjugate momentum $(\phi)$ to the advected quantities $(a)$ may be eliminated from equations (\[EL advected 1\]-\[EL advected 5\]) to obtain the weak form of the Euler-Poincaré equation with advected quantities: $${\frac{\partial }{\partial t}}{\frac{\delta \ell}{\delta \MM{u}}} + \operatorname{ad}^*_{\MM{u}}{\frac{\delta \ell}{\delta \MM{u}}} = a\diamond{\frac{\delta \ell}{\delta a}}, \qquad a_t + \mathcal{L}_{\MM{u}}a = 0.$$ Take the time-derivative of the inner product of ${\delta\ell}/{\delta\MM{u}}$ with a function of $\MM{w}$: $$\begin{aligned} {\frac{d }{d t}}\Bigg\langle {\frac{\delta \ell}{\delta \MM{u}}},\MM{w}\Bigg\rangle & = & {\frac{d }{d t}}\Bigg\langle -(\nabla\MM{l})^T\cdot\MM{\pi} - \phi\diamond a , \MM{w} \Bigg\rangle = {\frac{d }{d t}}\Bigg\langle -\MM{\pi},(\MM{w}\cdot\nabla)\MM{l}\Bigg\rangle + {\frac{d }{d t}}\Bigg\langle \phi,\mathcal{L}_{\MM{w}}a\Bigg\rangle \\ & = & \Bigg\langle \nabla\cdot(\MM{u}\MM{\pi}), (\MM{w}\cdot\nabla)\MM{l}\Bigg\rangle + \Bigg\langle \MM{\pi}, (\MM{w}\cdot\nabla)(\MM{u}\cdot\nabla)\MM{l}\Bigg\rangle + \Bigg\langle {\frac{\delta \ell}{\delta a}}\diamond a,\MM{w}\Bigg\rangle\\ & & \quad + \Bigg\langle -{\frac{\delta \ell}{\delta a}}-\mathcal{L}_{\MM{u}}\phi, \mathcal{L}_{\MM{w}}a\Bigg\rangle + \Bigg\langle \phi,-\mathcal{L}_{\MM{w}}\mathcal{L}_{\MM{u}}a\Bigg\rangle , \\ & = & \Bigg\langle \MM{\pi}, -(\MM{u}\cdot\nabla)(\MM{w}\cdot\nabla)\MM{l}+ (\MM{w}\cdot\nabla)(\MM{u}\cdot\nabla)\MM{l}\Bigg\rangle + \Bigg\langle {\frac{\delta \ell}{\delta a}}\diamond a,\MM{w}\Bigg\rangle \\ & & \quad +\Bigg\langle\phi,\mathcal{L}_{\MM{u}}\mathcal{L}_{\MM{w}}a -\mathcal{L}_{\MM{w}}\mathcal{L}_{\MM{u}}a \Bigg\rangle , \\ & =&\Bigg\langle \MM{\pi},-\left(\operatorname{ad}_{\MM{u}}\MM{w}\cdot\nabla\right)\MM{l} \Bigg\rangle + \Bigg\langle \phi,\mathcal{L}_{\operatorname{ad}_{\MM{u}}\MM{w}}a \Bigg\rangle + \Bigg\langle {\frac{\delta \ell}{\delta a}}\diamond a,\MM{w}\Bigg\rangle, \\ &=& \Bigg\langle -(\nabla\MM{l})^T\cdot\MM{\pi} -\phi\diamond a,\operatorname{ad}_{\MM{u}}\MM{w} \Bigg\rangle + \Bigg\langle {\frac{\delta \ell}{\delta a}}\diamond a,\MM{w}\Bigg\rangle , \\ & = & \Bigg\langle {\frac{\delta \ell}{\delta \MM{u}}},\operatorname{ad}_{\MM{u}}\MM{w}\Bigg\rangle + \Bigg\langle {\frac{\delta \ell}{\delta a}}\diamond a,\MM{w}\Bigg\rangle = \Bigg\langle -\operatorname{ad}^*_{\MM{u}}{\frac{\delta \ell}{\delta \MM{u}}} + {\frac{\delta \ell}{\delta a}}\diamond a,\MM{w}\Bigg\rangle.\end{aligned}$$ These Euler-Poincaré equations with advected quantities cover all conservative fluid equations which describe the advection of material. For a large collection of examples, see Holm *et al.* (1998). Example: Incompressible Euler equations --------------------------------------- As an example, consider the reduced Lagrangian for the incompressible Euler equations $$\ell[\MM{u},\rho,p] = \int_\Omega \frac{\rho|\MM{u}|^2}{2} + p(1-\rho)\,{\mathrm{d}}V(\MM{x}) \,.$$ Here $\rho(\MM{x},t)$ is the ratio of the local fluid density to the average density over $\Omega$; this is governed by the continuity equation (\[advecD\]). The pressure $p$ is a Lagrange multiplier that fixes the incompressibility constraint $\rho=1$. The variational derivatives in this case are $${\frac{\delta \ell}{\delta \MM{u}}} = \rho\MM{u}, \qquad {\frac{\delta \ell}{\delta \rho}} = \frac{|\MM{u}|^2}{2}-p,\qquad {\frac{\delta \ell}{\delta p}} =1-\rho\,.$$ Consequently, the Euler-Poincaré equations become $$\begin{aligned} (\rho\MM{u})_t + (\MM{u}\cdot\nabla)(\rho\MM{u}) + \rho\MM{u}(\nabla\cdot\MM{u}) + \rho(\nabla\MM{u})^T\cdot \MM{u} &=& \rho\nabla\left(\frac{|\MM{u}|^2}{2} -p\right), \\ \rho_t+\nabla\cdot(\rho\MM{u}) &=& 0, \\ \qquad \rho&=&1,\end{aligned}$$ and rearrangement gives the Euler fluid equations, $$\MM{u}_t + (\MM{u}\cdot\nabla)\MM{u} = -\nabla p, \qquad \nabla\cdot\MM{u}=0.$$ Inverse map multisymplectic formulation for EPDiff($H^1$) {#inverse map EPDiff} ========================================================= As we now have a canonical variational principle for fluid dynamics *via* the inverse map, one may obtain its multisymplectic formulation by extending the phase space so that the Lagrangian is affine in the space and time derivatives. In this section we show how to do this for EPDiff($H^1$) as discussed in the previous section. Affine Lagrangian for EPDiff($H^1$) ----------------------------------- After introducing the inverse map constraint, the Lagrangian becomes $$L = \frac{1}{2}u_iu_i + \frac{\lambda^2}{2}u_{i,j}u_{i,j} + \pi_k\left(l_{k,t} + u_jl_{k,j}\right).$$ Any high-order derivatives and nonlinear functions of first-order derivatives must now be removed from the Lagrangian to make it affine. We introduce a tensor variable $$W_{ij} = u_{i,j}\,;$$ this relationship may be enforced by using Lagrange multipliers. However, it turns out that the multipliers can be eliminated and the Lagrangian becomes $$\label{epmslag} L = \frac{1}{2}u_iu_i - \frac{\lambda^2}{2} W_{ij}W_{ij} +\lambda^2W_{ij}u_{i,j}+ \pi_k\left(l_{k,t} + u_jl_{k,j}\right),$$ which is now affine in the space and time derivatives of $\MM{u}$, $W$, $\MM{l}$ and $\MM{\pi}$. Multisymplectic structure ------------------------- The Euler-Lagrange equations for the affine Lagrangian (\[epmslag\]) are $$\begin{aligned} \delta u_i:&& u_i - \lambda^2W_{ij,j}+ \pi_k l_{k,i} = 0, \\ \delta l_k:&& -\pi_{k,t} - (\pi_k u_j)_{,j} = 0. \\ \delta \pi_k:&& l_{k,t} + u_j l_{k,j} = 0, \\ \delta W_{ij}:&& -\lambda^2 W_{ij} + \lambda^2 u_{i,j} = 0 .\end{aligned}$$ These equations possess the following multisymplectic structure as in equation (\[Eul-Lag-eqns\]): $$\begin{pmatrix} 0 & \pi_k\partial_i & & -\lambda^2\partial_j \\ -\pi_k\partial_i & 0 & -\partial_t-u_j\partial_j & 0 \\ 0 & \partial_t+u_j\partial_j & 0 & 0 \\ \lambda^2\partial_j & 0 & 0 & 0 \\ \end{pmatrix} \begin{pmatrix} u_i \\ l_k \\ \pi_k \\ W_{ij}\\ \end{pmatrix} = \nabla H,$$ where $\partial_t=\partial/\partial t,\ \partial_i=\partial/\partial x_i$, and $$H = -\left(\frac{1}{2}u_iu_i -\frac{\lambda^2}{2}W_{ij}W_{ij}\right) = -\left(\frac{1}{2}|\MM{u}|^2 -\frac{\lambda^2}{2}|W|^2\right).$$ One-form quasi-conservation law ------------------------------- For our multisymplectic formulation of EPDiff($H^1$), the independent variables are $$q^j = x_j, \quad j=1,\ldots n, \qquad q^{n+1} = t,$$ and the dependent variables are $$\begin{aligned} z^i = u_i, \qquad z^{n + k} = l_k, \qquad z^{2n + k} = \pi_k,\qquad z^{(i+2)n+j}=W_{ij}\,,\end{aligned}$$ where $i,j$ and $k$ range from $1$ to $n$. Comparing (\[epmslag\]) with (\[mslag\]) gives the following non-zero components $L^{\alpha}_j$: $$L^j_i = \lambda^2W_{ij}, \qquad L_{n+k}^j = \pi_ku_j, \qquad L_{n+k}^{n+1} = \pi_k, \qquad i,j,k=1,\ldots,n.$$ Therefore the one-form quasi-conservation law amounts to $$\label{epqcl} \left(\pi_k{\mathrm{d}}l_k\right)_{,t}+\left(\lambda^2W_{ij}{\mathrm{d}}u_i+\pi_ku_j{\mathrm{d}}l_k\right)_{,j}={\mathrm{d}}L.$$ The exterior derivative of this expression yields the structural conservation law $$\label{epsympcl} \left({\mathrm{d}}\pi_k\wedge{\mathrm{d}}l_k\right)_{,t}+\left(\lambda^2{\mathrm{d}}W_{ij}\wedge{\mathrm{d}}u_i+u_j{\mathrm{d}}\pi_k\wedge{\mathrm{d}}l_k+\pi_k{\mathrm{d}}u_j\wedge{\mathrm{d}}l_k\right)_{,j}=0.$$ Conservation of energy ---------------------- For EPDiff($H^1$), the ${\mathrm{d}}t$-component of the pullback of the one-form conservation law (\[epqcl\]) gives $$\left(\pi_kl_{k,t} - L\right)_{,t} +\left( \lambda^2W_{ij}u_{i,t} + \pi_ku_jl_{k,t} \right)_{,j} = 0.$$ In terms of $\MM{u}$ and its derivatives, this amounts to $$\left(u_im_i-\frac{1}{2}u_iu_i-\frac{\lambda^2}{2}u_{i,j}u_{i,j}\right)_{,t} +\left(\lambda^2u_{i,j}u_{i,t}+u_iu_jm_i\right)_{,j} = 0,$$ where $$m_i=u_i-\lambda^2u_{i,kk}.$$ This is the energy conservation law for EPDiff($H^1$). Conservation of momentum ------------------------ Similarly, the conservation law that is associated with translations in the $x_i$-direction is $$\left(\pi_kl_{k,i}\right)_{,t} +\left( \lambda^2 W_{kj}u_{k,i} + \pi_ku_jl_{k,i} - \delta_{ij}L \right)_{,j}=0,$$ which amounts to the momentum conservation law $$m_{i,t}+\left(\lambda^2u_{k,i}u_{k,j}-u_jm_i-\delta_{ij}\left(\frac{1}{2}u_ku_k+\frac{\lambda^2}{2}u_{k,l}u_{k,l}\right)\right)_{,j}.$$ Conservation of vorticity ------------------------- Next, consider the coefficient of each ${\mathrm{d}}x_r\wedge{\mathrm{d}}x_s$ in the pull-back of the structural (two-form) conservation law (\[epsympcl\]). This is $$\left(\pi_{k,r}l_{k,s}-\pi_{k,s}l_{k,r}\right)_{,t} +\left(\lambda^2\left(W_{ij,r}u_{i,s}-W_{ij,s}u_{i,r}\right) +u_j\left(\pi_{k,r}l_{k,s}-\pi_{k,s}l_{k,r}\right) +\pi_k\left(u_{j,r}l_{k,s}-u_{j,s}l_{k,r}\right)\right)_{,j}=0,$$ which amounts to $$\left(m_{r,s}-m_{s,r}\right)_{,t}+\left(\lambda^2\left(u_{i,s}u_{i,jr}-u_{i,r}u_{i,js}\right)+(u_jm_r)_{,s}-(u_jm_s)_{,r}\right)_{,j}=0.$$ One can regard this as a vorticity conservation law for EPDiff($H^1$); it is a differential consequence of the momentum conservation law. Particle relabelling symmetry ----------------------------- As we discussed in Section \[inverse map sec\], fluid equations in general, and EPDiff in particular, are invariant under relabelling of particles. In the context of the inverse map variables, relabelling is accomplished by the action of the diffeomorphism group $\operatorname{Diff}(\Omega)$ defined by $$\MM{l}\mapsto \eta\circ\MM{l}\equiv\eta(\MM{l}), \qquad \eta\in\operatorname{Diff}(\Omega).$$ The corresponding infinitesimal action of the vector fields $\mathfrak{X}(\Omega)$ is then $$\MM{l} \mapsto \MM{\xi}\circ\MM{l}\equiv\MM{\xi}(\MM{l}), \qquad \MM{\xi}\in\mathfrak{X}(\Omega),$$ and the cotangent lift of this action is $$(\MM{\pi},\MM{l}) \mapsto \left(-(\nabla\MM{\xi}(\MM{l}))^T\cdot\MM{\pi} ,\MM{\xi}(\MM{l})\right).$$ To obtain the symmetry generator (\[X\]), we extend the above action to first derivatives as follows: $$\begin{aligned} X &=& \xi_k(\MM{l}){\frac{\partial }{\partial l_k}} + (\xi_k(\MM{l}))_{,t}{\frac{\partial }{\partial l_{k,t}}} + (\xi_k(\MM{l}))_{,i}{\frac{\partial }{\partial l_{k,i}}} \\ & & - \pi_k{\frac{\partial \xi_k(\MM{l})}{\partial l_j}}{\frac{\partial }{\partial \pi_j}} - \left(\pi_k{\frac{\partial \xi_k(\MM{l})}{\partial l_j}}\right)_{,t}{\frac{\partial }{\partial \pi_{j,t}}} - \left(\pi_k{\frac{\partial \xi_k(\MM{l})}{\partial l_j}}\right)_{,i}{\frac{\partial }{\partial \pi_{j,i}}}.\end{aligned}$$ The relabelling symmetries are variational, because $$XL = \pi_k(\xi_k(\MM{l}))_{,t} + \pi_ku_i(\xi_k(\MM{l}))_{,i} - \pi_k{\frac{\partial \xi_k(\MM{l})}{\partial l_j}}\left(l_{j,t}+u_il_{j,i} \right)=0.$$ Noether’s theorem then gives the conservation law $$\left(\pi_k\xi_k(\MM{l})\right)_{,t}+\left(\pi_ku_j\xi_k(\MM{l})\right)_{,j}=0.$$ A conservation law exists for each element $\MM{\xi}$ of $\mathfrak{X}(\Omega)$, so particle relabelling generates an infinite space of conservation laws. Circulation theorem ------------------- To see how the particle relabelling conservation laws relate to conservation of circulation, note that if $\rho$ is any density that satisfies $$\rho_{,t} + (\rho u_j)_{,j} = 0,$$ then $$\left( \frac{\pi_k\xi_k(\MM{l})}{\rho}\right)_{,t} + u_j \left( \frac{\pi_k\xi_k(\MM{l})}{\rho}\right)_{,j}=0.$$ If we pick a loop $C(t)$ which is advected with the flow, then $${\frac{d }{d t}}\oint_{C(t)} \frac{\pi_k\xi_k(\MM{l})}{\rho}{\mathrm{d}}x = 0.$$ For a vector field $\MM{\xi}$ which is tangent to the loop at time $0$, and satisfies $|\MM{\xi}|=1$ on the loop, then $$\MM{\xi}{\mathrm{d}}x = (\nabla\MM{l})\cdot{\mathrm{d}}{\MM{x}}$$ for all times $t$, and one finds $$\label{circulation} {\frac{d }{d t}}\oint_{C(t)}\frac{\MM{\pi}\cdot(\nabla\MM{l})}{\rho} \cdot{\mathrm{d}}\MM{x}=0,$$ The momentum formula (\[momentum map\]) gives $${\frac{d }{d t}}\oint_{C(t)}\frac{(1-\lambda^2\nabla^2)\MM{u}}{\rho}\cdot{\mathrm{d}}\MM{x}= {\frac{d }{d t}}\oint_{C(t)}\frac{\MM{m}}{\rho}\cdot{\mathrm{d}}\MM{x}=0,$$ which is the circulation theorem for EPDiff. Inverse map multisymplectic formulation for Euler-Poincaré equation with advected quantities {#advected} ============================================================================================ To extend this method to more general equations with advected quantities is very simple: take the Lagrangian obtained from equation (\[inverse map sec\]) and add variables to represent higher-order derivatives. For the sake of brevity we shall compute one example, the incompressible Euler equations, and briefly discuss the implications for the circulation theorem. Multisymplectic form of incompressible Euler equations ------------------------------------------------------ We start with the reduced Lagrangian $$\ell[\MM{u},p,\rho] = \int_{\Omega}\frac{1}{2}\rho u_iu_i + p(1-\rho){\mathrm{d}}V(\MM{x}),$$ where $p$ is the pressure and $\rho$ is the relative density, and add dynamical constraints to form the Lagrangian: $$L = \frac{1}{2}\rho u_iu_i + p(1-\rho) + \pi_k\left(l_{k,t}+u_il_{k,i}\right) +\phi\left(\rho_{,t}+(\rho u_i)_{,i}\right).$$ This Lagrangian is already affine in the first-order derivatives, so the Euler-Lagrange equations are automatically multisymplectic in these variables: $$\begin{pmatrix} 0 & 0 & \pi_k\partial_i & 0 & -\rho\partial_i & 0 \\ 0 & 0 & 0 & 0 & -\partial_t -u_i\partial_i & 0 \\ -\pi_k\partial_i & 0 & 0 & -\partial_t-u_i\partial_i & 0 & 0 \\ 0 & 0 & \partial_t+u_i\partial_i & 0 & 0 & 0 \\ \rho\partial_i & \partial_t+u_i\partial_i & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{pmatrix} \begin{pmatrix} u_i \\ \rho \\ l_k \\ \pi_k \\ \phi \\ p \\ \end{pmatrix} = \nabla H,$$ where the quantity $$H = -\left(\frac{1}{2}\rho u_iu_i + p(1-\rho)\right)$$ is negative of the Hamiltonian density. Circulation theorem for advected quantities ------------------------------------------- The conservation law for particle-relabelling follows exactly as in Section \[inverse map EPDiff\], and we obtain equation (\[circulation\]) as before. The difference is that now the momentum formula (momentum map) is $$\MM{m} = {\frac{\partial \ell}{\partial \MM{u}}} = -\pi_k\nabla l_k -\phi\diamond a$$ and so one obtains $${\frac{d }{d t}}\oint_{C(t)}\frac{\MM{m}}{\rho}\cdot{\mathrm{d}}\MM{x}= \oint_{C(t)}\frac{1}{\rho}{\frac{\partial \ell}{\partial a}}\diamond a\cdot {\mathrm{d}}\MM{x}.$$ For the incompressible Euler equations, $a$ is the relative density $\rho$, so $${\frac{\partial \ell}{\partial a}}\diamond a = \rho\nabla{\frac{\partial \ell}{\partial \rho}},$$ which leads to the circulation theorem $${\frac{d }{d t}}\oint_{C(t)}\frac{\MM{m}}{\rho}\cdot{\mathrm{d}}\MM{x}=\oint_{C(t)}\nabla {\frac{\partial \ell}{\partial \rho}}\cdot{\mathrm{d}}\MM{x}=0.$$ A note on multisymplectic integrators {#numerics} ===================================== In this section we discuss briefly how to produce multisymplectic numerical integrators, using the inverse map formulation given in this paper. We note in particular that the multisymplectic method will satisfy a discrete form of the particle-relabelling symmetry and hence we will obtain a method that has discrete conservation laws for $-\MM{\pi}\cdot\nabla\MM{l}$. Variational integrators ----------------------- A multisymplectic integrator for a PDE is a numerical method which preserves a discrete conservation law for the two-form $\kappa$ given in equation (\[kappa\]) (Bridges & Reich, 2001). As described in (Hydon, 2005), a discrete variational principle with a Lagrangian that is affine in first-order differences automatically leads to a set of difference equations which are multisymplectic. This now makes it very simple to construct multisymplectic integrators for fluid dynamics using the inverse map formulation: one simply replaces the spatial and time integrals in the action with numerical quadratures, replaces the first-order derivatives by differences, and takes variations following the standard variational integrator approach (Lew *et al.*, 2003). Whilst the method will preserve the discrete conservation law for the two-form $\kappa$, the one-form quasi-conservation law will not be preserved in general, and hence the other conservation laws will not be exactly preserved. Discrete relabelling symmetry ----------------------------- As $\MM{\pi}$ and $\MM{l}$ are still continuous in the discretised equations, the multisymplectic integrator will have a discrete particle-relabelling symmetry analogous to the one given in Section \[inverse map EPDiff\], with the only difference being the discretisation of the cotangent lift. Following the variational integrator programme described in Lew *et al.* (2003), the discrete form of Noether’s theorem will give rise to discrete conservation laws for the multisymplectic method. Remapping labels ---------------- If this approach is to be applied to numerical solutions with intense vorticity then one needs to address the problem that eventually the numerical discretisation of the labels $\MM{l}$ will become very poor due to tangling, and hence the approximation to the momentum $$\label{mom} \MM{m} = -(\nabla\MM{l})^T\MM{\pi} = - \pi_k\nabla l_k \,,$$ will degrade with time. One possible approach would be to apply discrete particle-relabelling, mapping the labels back to the Eulerian grid in such a way that the momentum (\[mom\]) stays fixed. This transformation is exactly the relabelling given in Section \[inverse map EPDiff\]. Numerically, one could construct a transformation (using a generating function for example) which satisfies $$\MM{l} \mapsto \MM{X} + \mathcal{O}(\Delta x^p,\Delta t^p), \qquad \MM{\pi}(\nabla\MM{l})\mapsto \MM{\pi}(\nabla\MM{l}) + \mathcal{O}(\Delta x^p,\Delta t^p),$$ where $p$ is the order of the method. For instance, one might use a variational discretisation of the relabelling transformation, which is generated by a symplectic vector field whose Hamiltonian is $\MM{\pi}\cdot\MM{\xi}(\MM{l})=\pi_k\xi_k(\MM{l})$. In this way, one may still retain some of the conservative properties of the method. Summary and Outlook {#summary} =================== Summary ------- This paper describes a multisymplectic formulation of Euler-Poincaré equations (which are, in essence, fluid dynamical equations with a particle-relabelling symmetry). We have used the inverse map to obtain a canonical variational principle, following Holm and Kupershmidt (1983). As noted in Hydon (2005), a multisymplectic formulation can be obtained by choosing variables such that the Lagrangian at most linear in the first-order derivatives, and contains no higher-order derivatives. We have shown how to construct the multisymplectic formulation for the Euler-Poincaré equations for diffeomorphisms, using the example of the EPDiff($H^1$) equations, and how to extend the method to the Euler-Poincaré equations with advected quantities. These equations encompass many fluid systems, including incompressible Euler, shallow-water, Euler-alpha, Green-Naghdi, perfect complex fluids, inviscid magnetohydrodynamics, *etc.* The techniques of Hydon (2005) have led to conservation laws for these systems, including the usual multisymplectic conservation laws for energy and momentum plus an infinite set of conservation laws which arise from the particle-relabelling symmetry of fluid dynamics. We have highlighted the connection between these latter conservation laws and Kelvin’s circulation theorem, and showed that multisymplectic integrators based on this formulation will have discrete conservation laws associated with this symmetry. Outlook ------- In the last section of this paper we have discussed the possibility of developing multisymplectic integrators for fluids using this framework. It is undoubtedly simple to construct such integrators, but the issue of accuracy with time arises whenever the flow is strongly mixing and numerical errors make the label field $\MM{l}$ very noisy. A discretisation of the relabelling map discussed in this paper could provide a way to prevent this problem whilst retaining some of the geometric properties of the method. These ideas may aid the future development of integrators that have conservation laws for vorticity and circulation, which are desirable for numerical weather prediction and other applications. In a different direction, we believe that multisymplectic integrators would be especially apt for applications of EPDiff to template-matching in computational anatomy (Holm *et al*., 2004). The matching problem is an initial-final value problem. In such problems, space and time may be treated on an equal footing, just as in the multisymplectic formulation. Acknowledgements ---------------- The work of DDH was partially supported by the Royal Society of London Wolfson Award and the US Department of Energy Office of Science ASCR. Bridges, T. J. Multi-symplectic structures and wave propagation. [*Math. Proc. Camb. Philos. Soc.*]{}, 121:147–190, 1997. Bridges, T. J. & Reich, S. Multi-symplectic integrators: numerical schemes for [H]{}amiltonian [PDE]{}s that preserve symplecticity. [*Phys. Lett.*]{}, 284A:184–193, 2001. Camassa, R. & Holm, D. D. An integrable shallow-water equation with peaked solitons. [*Physical Review Letters*]{}, 71:1661–1664, 1993. Holm, D. D. & Kupershmidt, B. [P]{}oisson brackets and [C]{}lebsch representations for magnetohydrodynamics, multifluid plasmas, and elasticity. [*Physica D*]{}, 6:347–363, 1983. Holm, D. D. & Marsden, J. E. Momentum maps and measure valued solutions (peakons, filaments, and sheets) of the [E]{}uler-[P]{}oincaré equations for the diffeomorphism group. In J. E. Marsden & T. S. Ratiu, editors, [*The Breadth of Symplectic and [P]{}oisson Geometry: A Festshrift for Alan Weinstein*]{}, Birkhäuser, Boston, MA., 2004, pp. 203–235. http://arxiv.org/abs/nlin.CD/0312048. Holm, D. D., Marsden, J. E. & Ratiu, T. S., The Hamiltonian Structure of Continuum Mechanics in Material, Inverse Material, Spatial and Convective Representations. In [*Hamiltonian Structure and Lyapunov Stability for Ideal Continuum Dynamics*]{}, Univ. Montreal Press, 1986, pp. 1–124. Holm, D. D., Marsden, J. E., & Ratiu, T. S. The [E]{}uler–[P]{}oincaré equations and semidirect products with applications to continuum theories. [*Adv. in Math.*]{}, 137:1–81, 1998. http://arxiv.org/abs/chao-dyn/9801015. Holm, D. D., Rananather, J. T., Trouvé, A., & Younes, L. Soliton dynamics in computational anatomy. [*NeuroImage*]{}, 23:170–178, 2004. http://arxiv.org/abs/nlin.SI/0411014. Hydon, P. E. Multisymplectic conservation laws for differential and differential-difference equations. [*Proc. Roy. Soc. LOnd. A*]{}, 461:1627–1637, 2005. Lew, A., Marsden, J. E., Ortiz, M. & West, M. An overview of variational integrators. In L. P. Franca, T. E. Tezduyar & A. Masud, editors, [*Finite Element Methods: 1970s and Beyond*]{}. CIMNE, Barcelona, Spain, 2004, pp. 98–115. Lin, C. C. Liquid Helium. In [*Proceedings of the Enrico Fermi International School of Physics, Course XXI*]{}, G. Careri, editor, Academic Press, New York, 1963, pp. 93–146. Marsden, J. E. & Weinstein, A. Coadjoint orbits, vortices, and [C]{}lebsch variables for incompressible fluids. [*Physica D*]{}, 7:305–323, 1983. Miller, M. I., Trouvé, A. & Younes, L. On the metrics and [E]{}uler-[L]{}agrange equations of computational anatomy. [*Ann. Rev. Biomed. Engrg.*]{}, 4:375–405, 2002. Seliger, R. L. & Whitham, G. B. Variational principles in continuum mechanics. [*Proc. Roy. Soc. Lond. A*]{} 305: 1–25, 1968. Serrin, J. Mathematical principles of classical fluid mechanics. In S. Flügge & C. Truesdell, editors, [*Handbuch der Physik*]{} VIII/I. Springer-Verlag, Berlin, 1959, pp. 125–263. [^1]: This is only done for clarity and the equations are easily extended to the case when the domain $\Omega$ is a curved manifold. [^2]: As we discuss later, this right action contrasts with fluid particle relabelling, which arises by the left action of the diffeomorphisms on the inverse map.

--- abstract: 'In this paper, we propose a novel image set representation and classification method by maximizing the margin of image sets. The margin of an image set is defined as the difference of the distance to its nearest image set from different classes and the distance to its nearest image set of the same class. By modeling the image sets by using both their image samples and their affine hull models, and maximizing the margins of the images sets, the image set representation parameter learning problem is formulated as an minimization problem, which is further optimized by an expectation—maximization (EM) strategy with accelerated proximal gradient (APG) optimization in an iterative algorithm. To classify a given test image set, we assign it to the class which could provide the largest margin. Experiments on two applications of video-sequence-based face recognition demonstrate that the proposed method significantly outperforms state-of-the-art image set classification methods in terms of both effectiveness and efficiency.' author: - 'Jim Jing-Yan Wang, Majed Alzahrani and Xin Gao' title: |  \ **Large Margin Image Set Representation and Classification [^1] [^2]** --- Introduction ============ visual information processing and understanding is based on single images. The classification of a visual target, such as a human face, is also conducted based on a single image, where each training or test sample is an individual image [@Zhao2003399; @JonathonPhillips20001090; @wang2012scimate; @wang2012adaptive; @zhou2010region]. With the rapid development of video technologies, sequences of images is more commonly available than single images. Consequently, the visual target classification task could be improved from image-based to image-set-based. The image set classification problem has been proposed and studied recently [@MMD2008; @Chu20111567; @Fan2006177; @Kim2007; @Chin2006461; @Wang20091161; @MMD2008; @Arandjelovic2005581; @sun2012unsupervised]. In image set classification, each sample is a set of images instead of one single image, and each class is represented by one or more training samples. The classification problem is to assign a given test sample to one of the known classes. For example, in image-set-based face recognition problem, each sample is a set of facial images with different poses, illuminations, and expressions. Compared to the traditional single image classification, image set classification has the potential of achieving higher accuracy, because image sets usually contain more information than single images. Even if images in the image set are of low quality, we could still exploit the temporal relationship and the complementarity between the images to improve the classification accuracy. Although image set classification has proposed a novel and promising scheme for the visual classification problem, it has also brought challenges to the machine learning and computer vision communities. Traditional single-image-based representation and classification methods are not suitable for this problem, such as principal component analysis (PCA) [@Wold198737], support vector machine (SVM) [@SVM2013], and K-nearest neighbors (KNN) [@KNN2013]. To handle the problem of image set classification, a number of methods have been proposed. In [@Kim2007], Kim et al. developed the discriminant canonical correlations (DCC) for image set classification, by proposing a linear discriminant function to simultaneously maximize the canonical correlations of within-class sets and minimize the canonical correlations of between-class sets. The classification is done by transforming the image sets using the discriminant function and comparing them by the canonical correlations. In [@MMD2008], Wang et al. formulated the image set classification problem as the computation of manifold-manifold distance (MMD), i.e., calculating the distance between nonlinear manifolds, each of represented one image set. In [@MDA2009], Wang et al. presented the manifold discriminant analysis (MDA) for the image set classification problem, by modeling each image set as a manifold and formulating the problem as a classification-oriented multi-manifold learning problem. Cevikalp and Triggs later introduced the affine hull-based image set distance (AHISD) for image set based face recognition [@AHISD2010], by representing images as points in a linear or affine feature space and characterizing each image set by a convex geometric region spanned by its feature points. Set dissimilarity was measured by geometric distances between convex models. Hu et al. proposed to represent each image set as both the image samples of the set and their affine hull model [@Hu2011; @Hu2012], and introduced a novel between-set distance called sparse approximated nearest point (SANP) distance for the image set classification. In [@Wang2012], Wang et al. modeled the image set with its natural second-order statistic, i.e. covariance matrix (COV), and proposed a novel discriminative learning approach to image set classification based on the covariance matrix. Among all these methods, affine subspace-based methods, including AHISD [@AHISD2010] and SANP [@Hu2011; @Hu2012], have shown their advantage over the other methods. However, all these methods are unsupervised ones, ignoring the class labels of the images sets in the training set. Moreover, most image-set-based classification methods are under the framework of pairwise image set comparison [@MMD2008; @MDA2009; @AHISD2010; @Hu2011; @Hu2012]. A similar approach has been successfully adopted on pairwise comparison on individual samples by Mu et al. [@mu2013local], which exploits abundant discriminative information for each local manifold. In set classification case, a test image set is compared to all the training image set one by one, and then the nearest neighbor rule is utilized to decide which class the test image set belongs to. The disadvantage of this strategy lays in the following two folds: - When the training image set number is large, this strategy is quite time-consuming. - When a pair of image sets are compared, all other image sets are ignored, and thus the global structure of the image set dataset is ignored. To overcome these issues, in this paper, we propose a novel image set representation and classification method. Similarly to SANP [@Hu2011; @Hu2012], we also use the image samples of an image set and their affine hull model to represent the image set. To utilize the class labels of each image set, inspired by large margin framework for feature selection [@sun2010local], we propose to maximize the margin of each image set. Based on this representation and its corresponding pairwise distance measure, we define two types of nearest neighboring image sets for each image set — the nearest neighbor from the same class and the nearest neighbor from different classes. The margin of a image set is defined as the difference between its distances to nearest miss and nearest hit, and the representation parameter is learned by maximizing the margins of the image sets. To classify a test image set, we assign it to the class which could achieve the largest margin for it. The contributions of the proposed Large Margin Image Set (LaMa-IS) representation and classification method are of three folds: 1. Using the class labels, we define the margin of the image sets, such that the discriminative ability can be improved in a supervised manner. 2. The global structure of the image sets can also be explored by searching the nearest hit and nearest miss from the entire database for each images set. 3. To classify a test image set, we only need to compare it to every class instead of every training image set, which could reduce the time complexity of the online classification procedure significantly, especially when the number of training image sets is much larger than the number of classes. The rest of the paper is organized as follows: in Section \[sec:Method\], we propose the novel LaMa-IS algorithm; in Section \[sec:experiment\], the experiment results on several image-set-based face recognition problems are given; and finally in Section \[sec:conclusion\], we draw conclusions. Methods {#sec:Method} ======= In this section we will introduce the proposed LaMa-IS method for image set representation and classification. Objective Function ------------------ Suppose we have a database of image sets denoted as $\{(X_i,y_i)_{i=1}^N\}$, where $X_i$ is the data matrix of the $i$-th image set, and $y_i\in \{1,\cdots,C\}$ is its corresponding class label. In the data matrix $X_i = [{{\textbf{x}}}_{i,1},\cdots,{{\textbf{x}}}_{i,N_i}]\in \mathbb{R}^{D\times N_i}$, the $n$-th column, ${{\textbf{x}}}_{i,n}\in \mathbb{R}^D$ is the $D$-dimensional visual feature vector of the $n$-th image of the $i$-th image set, and $N_i$ is the number of images in $i$-th image set. Note that the feature vector for an image can be the original pixel values or some other visual features extracted from the image, such as local binary pattern (LBP) [@LBP2013]. To represent an image set, two linear model has been employed to approximate the structure of the image set following [@Hu2011; @Hu2012]: - Using the images in the image set, we can model the $i$-th image set as an linear combination of the images in the $i$-th set as $$\begin{aligned} {{\textbf{x}}}= \sum_{n=1}^{N_i} {{\textbf{x}}}_{i,n} \alpha_{i,n} = X_i {{\boldsymbol{\alpha}}}_i, \end{aligned}$$ where ${{\boldsymbol{\alpha}}}_i=[\alpha_{i,1},\cdots,\alpha_{i,N_i}]^\top\in \mathbb{R}^{N_i}$ is the linear combination coefficient vector. - We can also use the affine hull model to represent the image set, using the image mean and the orthonormal bases of the $i$-th image, which is represented as $$\begin{aligned} {{\textbf{x}}}= {{\boldsymbol{\mu}}}_i + U_i {{\textbf{v}}}_i, \end{aligned}$$ where ${{\boldsymbol{\mu}}}_i=\frac{1}{N_i}\sum_{n=1}^{N_i} {{\textbf{x}}}_{i,n}$ is the image mean, the columns of $U_i$ are the orthonormal bases obtained from the SVD of the centered $X_i$, and ${{\textbf{v}}}_i$ is the coefficient vector of $U_i$. In this paper, we try to represent the image sets using both models mentioned above simultaneously, by solving the parameters ${{\boldsymbol{\alpha}}}_i$ and ${{\textbf{v}}}_i$. The representation error is given by the squared $l_2$ norm distance between these two models as $$\begin{aligned} \mathcal{R}_{{{\textbf{v}}}_i,{{\boldsymbol{\alpha}}}_i}=||({{\boldsymbol{\mu}}}_i + U_i {{\textbf{v}}}_i) - X_i {{\boldsymbol{\alpha}}}_i||^2_2, \end{aligned}$$ where ${{\boldsymbol{\alpha}}}_i$ and ${{\textbf{v}}}_i$ are the parameters for the two models, respectively. To compare a pair of image sets, we only use the second model and compute the squared $l_2$ norm distance between them suggested by [@Hu2011] as $$\begin{aligned} \mathcal{D}_{{{\textbf{v}}}_i,{{\textbf{v}}}_j}=||({{\boldsymbol{\mu}}}_i + U_i {{\textbf{v}}}_i) - ({{\boldsymbol{\mu}}}_j + U_j {{\textbf{v}}}_j)||^2_2. \end{aligned}$$ Given the defined distance function $\mathcal{D}_{{{\textbf{v}}}_i,{{\textbf{v}}}_j}$, we can find two nearest neighboring sets for each set $X_i$, one from the same class as $X_i$ (called nearest hit or $\mathcal{H}_i$) and the other from different classes (called nearest miss or $\mathcal{M}_i$), defined as $$\begin{aligned} \mathcal{H}_i = \underset{j:y_j = y_i,j\neq i}{argmin} \mathcal{D}_{{{\textbf{v}}}_i,{{\textbf{v}}}_j},\\ \mathcal{M}_i = \underset{j:y_j \neq y_i,j\neq i}{argmin} \mathcal{D}_{{{\textbf{v}}}_i,{{\textbf{v}}}_j}. \end{aligned}$$ The margin of the $i$-th set is then defined as the difference of the distances between $\mathcal{M}_i $ and $\mathcal{H}_i$, as $$\begin{aligned} \rho_i= \mathcal{D}_{{{\textbf{v}}}_i,{{\textbf{v}}}_{\mathcal{M}_i}} - \mathcal{D}_{{{\textbf{v}}}_i,{{\textbf{v}}}_{\mathcal{H}_i}}, \end{aligned}$$ where $\mathcal{H}_i^{y_i}$ is the nearest set from the same class of the $i$-th set, and $\mathcal{M}_i^{y_i}$ is the nearest one form a different class. The main difficulty here is that ${{\textbf{v}}}_i$ is a variable to be solved when we compute the margin. Thus, it is impossible to directly find the nearest neighbors. To overcome this problem, following the principles of the expectation-maximization (EM) algorithm, we develop a probabilistic model where the nearest neighbors of a given set are treated as hidden variables. The probabilities of the $j$-th set being the nearest miss or hit of the $i$-th set is denoted as $P(j=\mathcal{M}_i|\{{{\textbf{v}}}_i\})$ and $P(j=\mathcal{H}_i|\{{{\textbf{v}}}_i\})$, respectively, and they are estimated via the standard kernel density estimation: $$\label{equ:probality} \begin{aligned} &P(j=\mathcal{H}_i|\{{{\textbf{v}}}_i\})= \left\{\begin{matrix} \frac{K_{{{\textbf{v}}}_i,{{\textbf{v}}}_j}} {\sum_{k:y_k=y_i} K_{{{\textbf{v}}}_i,{{\textbf{v}}}_k}}, & if~ y_j = y_i,~and~j\neq i \\ 0, &else \end{matrix}\right.\\ &P(j=\mathcal{M}_i|\{{{\textbf{v}}}_i\})= \left\{\begin{matrix} \frac{K_{{{\textbf{v}}}_i,{{\textbf{v}}}_j}} {\sum_{k:y_k\neq y_i} K_{{{\textbf{v}}}_i,{{\textbf{v}}}_k}}, & if~ y_j \neq y_i,~and~j\neq i \\ 0 &else \end{matrix}\right. \end{aligned}$$ where $K_{{{\textbf{v}}}_i,{{\textbf{v}}}_j}=exp(-\frac{\mathcal{D}_{{{\textbf{v}}}_i,{{\textbf{v}}}_j}}{2\sigma^2})$ is the Gaussian kernel function, and $\sigma$ is the band-width parameter. Subsequently the probabilistic margin is defined as $$\begin{aligned} \rho_i=& \sum_{j=1}^N P(j=\mathcal{M}_i|\{{{\textbf{v}}}_i\}) \times \mathcal{D}_{{{\textbf{v}}}_i,{{\textbf{v}}}_j}\\ &- \sum_{j=1}^N P(j=\mathcal{H}_i|\{{{\textbf{v}}}_i\})\times \mathcal{D}_{{{\textbf{v}}}_i,{{\textbf{v}}}_j} \end{aligned}$$ Apparently, the margin of each set should be maximized. Considering the representation error to be minimized and the margin to be maximized simultaneously, we have the following objective function regarding the parameters ${{\textbf{v}}}_i$ and ${{\boldsymbol{\alpha}}}_i$ as follows $$\begin{aligned} \label{equ:objective} \underset{\{{{\boldsymbol{\alpha}}}_i,{{\textbf{v}}}_i\}}{min} \sum_{i=1}^N & \left \{ \mathcal{R}_{{{\textbf{v}}}_i,{{\boldsymbol{\alpha}}}_i} + \lambda \|{{\boldsymbol{\alpha}}}_i\|_1 \vphantom{ \left. - \sum_{j=1}^N P(j=\mathcal{M}_i|\{{{\textbf{v}}}_i\}) \times \mathcal{D}_{{{\textbf{v}}}_i,{{\textbf{v}}}_j} \right] } \right . \\ & +\gamma \left [ \sum_{j=1}^N P(j=\mathcal{H}_i|\{{{\textbf{v}}}_i\})\times \mathcal{D}_{{{\textbf{v}}}_i,{{\textbf{v}}}_j} \right . \\ & \left. \left. - \sum_{j=1}^N P(j=\mathcal{M}_i|\{{{\textbf{v}}}_i\}) \times \mathcal{D}_{{{\textbf{v}}}_i,{{\textbf{v}}}_j} \right] \right \}, \end{aligned}$$ where the $\|{{\boldsymbol{\alpha}}}_i\|_1$ is the $l_1$ norm based sparse term on ${{\boldsymbol{\alpha}}}_i$, and $\lambda$ and $\gamma$ are the trade-off weights, which are set by cross-validation. Optimization ------------ We adopt the EM framework to optimize the objective function in (\[equ:objective\]) in an iterative algorithm. The algorithm is composed of two iterative steps: the E-step and the M-step. ### E-step In the E-step, we compute the probabilities of $P(j=\mathcal{H}_i|\{{{\textbf{v}}}_i\})$ and $P(j=\mathcal{M}_i|\{{{\textbf{v}}}_i\})$ based on $\{{{\textbf{v}}}_i\}$ estimated in the previous iteration, as in (\[equ:probality\]). ### M-Step In the M-Step, we try to optimize (\[equ:objective\]) by fixing the probabilities. By denoting $P_{ij}=P(j=\mathcal{H}_i|\{{{\textbf{v}}}_i\}) - P(j=\mathcal{M}_i|\{{{\textbf{v}}}_i\})$, the objective function in (\[equ:objective\]) is reduced to $$\label{equ:objective1} \begin{aligned} \underset{\{{{\boldsymbol{\alpha}}}_i,{{\textbf{v}}}_i\}}{min} &\sum_{i=1}^N \left( \mathcal{R}_{{{\textbf{v}}}_i,{{\boldsymbol{\alpha}}}_i} + \lambda \|{{\boldsymbol{\alpha}}}_i\|_1 + \gamma \sum_{j=1}^N P_{ij} \mathcal{D}_{{{\textbf{v}}}_i,{{\textbf{v}}}_j} \right). \end{aligned}$$ We optimize the representation parameters ${{\boldsymbol{\alpha}}}_i$ and ${{\textbf{v}}}_i$ for each set one by one. To optimize ${{\boldsymbol{\alpha}}}_i$ and ${{\textbf{v}}}_i$, we fix all the remaining representation parameters ${{\boldsymbol{\alpha}}}_j$ and ${{\textbf{v}}}_j(j \neq i)$. Using the property $\mathcal{D}_{{{\textbf{v}}}_i,{{\textbf{v}}}_i}=0$ and $\mathcal{D}_{{{\textbf{v}}}_i,{{\textbf{v}}}_j}=\mathcal{D}_{{{\textbf{v}}}_j,{{\textbf{v}}}_i}$, we rewrite the optimization of objective of (\[equ:objective1\]) with respect to only ${{\boldsymbol{\alpha}}}_i$ and ${{\textbf{v}}}_i$ as follows: $$\label{equ:objective2} \underset{{{\boldsymbol{\alpha}}}_i,{{\textbf{v}}}_i}{min}~ \left \{ \begin{aligned} &H({{\boldsymbol{\alpha}}}_i,{{\textbf{v}}}_i)\\ &= \mathcal{R}_{{{\textbf{v}}}_i,{{\boldsymbol{\alpha}}}_i} + \lambda \|{{\boldsymbol{\alpha}}}_i\|_1\\ &~~~~~~~+ \gamma \left ( \sum_{j:j\neq i} P_{ij} \mathcal{D}_{{{\textbf{v}}}_i,{{\textbf{v}}}_j} + \sum_{j:j\neq i} P_{ji} \mathcal{D}_{{{\textbf{v}}}_j,{{\textbf{v}}}_i} \right ) \\ &= \mathcal{R}_{{{\textbf{v}}}_i,{{\boldsymbol{\alpha}}}_i} + \lambda \|{{\boldsymbol{\alpha}}}_i\|_1+ \gamma \sum_{j:j\neq i} (P_{ij}+P_{ji}) \mathcal{D}_{{{\textbf{v}}}_i,{{\textbf{v}}}_j} \\ &=f({{\boldsymbol{\alpha}}}_i,{{\textbf{v}}}_i) + g({{\boldsymbol{\alpha}}}_i) \end{aligned} \right \}$$ The objective function $H({{\boldsymbol{\alpha}}}_i,{{\textbf{v}}}_i)$ in (\[equ:objective2\]) is a composite model consisting of a smooth function $f({{\boldsymbol{\alpha}}}_i,{{\textbf{v}}}_i) = \mathcal{R}_{{{\textbf{v}}}_i,{{\boldsymbol{\alpha}}}_i} + \gamma \sum_{j:j\neq i} (P_{ij}+P_{ji}) \mathcal{D}_{{{\textbf{v}}}_i,{{\textbf{v}}}_j}$ and a non-smooth function $g(\alpha_i) = \lambda \|{{\boldsymbol{\alpha}}}_i\|_1$. To solve the optimization problem in (\[equ:objective2\]), we employ the accelerated proximal gradient (APG) algorithm introduced in [@Beck2009]. In the $t$-th iteration, to obtain the new solution ${{\boldsymbol{\alpha}}}_i^t$ and ${{\textbf{v}}}_i^t$, we solve the following proximal regularization problem based on the previous the solution ${{\boldsymbol{\alpha}}}_i^{t-1}$ and ${{\textbf{v}}}_i^{t-1}$: $$\label{equ:objective3} {{\boldsymbol{\alpha}}}^t_i,{{\textbf{v}}}_i^t= \arg\min_{{{\boldsymbol{\alpha}}}_i,{{\textbf{v}}}_i} \left \{ \begin{aligned} &Q_{L^t}({{\boldsymbol{\alpha}}}_i,{{\boldsymbol{\beta}}}_i^t,{{\textbf{v}}}_i,{{\textbf{u}}}_i^t)\\ &= \frac{L^t}{2} \left \| {{\boldsymbol{\alpha}}}_i - \left ( {{\boldsymbol{\beta}}}_i^t - \frac{1}{L^t} \nabla f_{{{\boldsymbol{\beta}}}_i^t} \right ) \right \|^2_2\\ &+ \frac{L^t}{2} \left \| {{\textbf{v}}}_i - \left ( {{\textbf{u}}}_i^t - \frac{1}{L^t} \nabla f_{{{\textbf{u}}}_i^t} \right ) \right \|^2_2\\ &+ g({{\boldsymbol{\alpha}}}_i) \end{aligned} \right \},$$ where $$\label{equ:beta} \begin{aligned} &{{\boldsymbol{\beta}}}_i^{t}={{\boldsymbol{\alpha}}}_i^{t-1}+ \frac{\tau^{t-1}-1}{\tau^t}\left( {{\boldsymbol{\alpha}}}^{t-1}_i-{{\boldsymbol{\alpha}}}^{t-2}_i \right ),\\ &{{\textbf{u}}}_i^{t}={{\textbf{v}}}_i^{t-1}+ \frac{\tau^{t-1}-1}{\tau^t}\left( {{\textbf{v}}}^{t-1}_i-{{\textbf{v}}}^{t-2}_i \right ), \end{aligned}$$ with $\tau^t= \frac{1+ \sqrt{1+4 {\tau^{t-1}}^2} }{2} $ being the initial approximation of the next solution for ${{\boldsymbol{\alpha}}}^t_i$, $L^t = \eta^{i^t}L^{t-1}$ being the step size related to the Lipschitz constant, and $i^t$ being the smallest nonnegative integers such that $H({{\boldsymbol{\alpha}}}_i^{t-1},{{\textbf{v}}}_i^{t-1})\leq Q_{L^t}({{\boldsymbol{\alpha}}}_i^{t-1},{{\boldsymbol{\beta}}}_i^t, {{\textbf{v}}}_i^{t-1},{{\textbf{u}}}_i^t)$, and $$\begin{aligned} \nabla f_{{{\boldsymbol{\beta}}}_i^t} =& \left. \frac{\partial f({{\textbf{v}}}_i,{{\boldsymbol{\alpha}}}_i)}{\partial {{\boldsymbol{\alpha}}}_i} \right |_{{{\boldsymbol{\alpha}}}_i={{\boldsymbol{\beta}}}_i^t} =-2 X_i^\top (\mu_i+U_iv_i - X_i {{\boldsymbol{\beta}}}_i^t),\\ \nabla f_{{{\textbf{u}}}_i^t} =& \left. \frac{\partial f({{\textbf{v}}}_i,{{\boldsymbol{\alpha}}}_i)}{\partial {{\textbf{v}}}_i} \right |_{{{\textbf{v}}}_i={{\textbf{u}}}_i^t}\\ =& 2 \left[1+\gamma \sum_{j:j\neq i} (P_{ij}+P_{ji})\right] U_i^\top (U_i {{\textbf{u}}}_i^t+{{\boldsymbol{\mu}}}_i)\\ &-2U_i X_i {{\boldsymbol{\alpha}}}_i - 2 \gamma \sum_{j:j\neq i} (P_{ij}+P_{ji}) ({{\boldsymbol{\mu}}}_j-U_j {{\textbf{u}}}_j^t). \end{aligned}$$ It could be proved that when $g({{\boldsymbol{\alpha}}}_i)$ is given as $g({{\boldsymbol{\alpha}}}_i)=\lambda \|{{\boldsymbol{\alpha}}}_i\|_1$, the optimal solution of (\[equ:objective3\]) can be obtained by $$\label{equ:alpha} \begin{aligned} {{\boldsymbol{\alpha}}}^t_i &= \tau_{ \frac{\lambda}{L^t}} \left( {{\boldsymbol{\beta}}}_i^t - \frac{1}{L^t} \nabla f_{{{\boldsymbol{\beta}}}_i^t} \right),\\ {{\textbf{v}}}^t_i &= {{\textbf{u}}}_i^t - \frac{1}{L^t} \nabla f_{{{\textbf{u}}}_i^t}, \end{aligned}$$ where $\tau_{\varepsilon}(w)= \left( |w|-\varepsilon \right)_+ sgn(w) $ is the soft-thresholding operators. The final iterative representation algorithm for LaMa-IS method is summarized in Algorithm \[alg:LaMaIS-Rep\]. **Input**: Training Image Sets $\{(X_i,y_i)\}_{i=1}^N$; **Input**: Maximum iteration number $T$; Initialize the representation parameters $\{({{\boldsymbol{\alpha}}}_i^0,{{\textbf{v}}}_i^0)\}_{i=1}^N$, $\{({{\boldsymbol{\alpha}}}_i^{-1},{{\textbf{v}}}_i^{-1})\}_{i=1}^N$ and $\tau^0$; Update the probabilities of $P(j=\mathcal{H}_i|\{{{\textbf{v}}}_i^{t-1}\})$ and $P(j=\mathcal{M}_i|\{{{\textbf{v}}}_i^{t-1}\})$ as in (\[equ:probality\]); Update the approximated initial representation parameters ${{\boldsymbol{\beta}}}_i^{t}$ and ${{\textbf{u}}}_i^{t}$ as in (\[equ:beta\]). Update the representation parameters ${{\boldsymbol{\alpha}}}_i^{t}$ and ${{\textbf{v}}}_i^{t}$ as in (\[equ:alpha\]). **Output**: Optimal solution $\{({{\boldsymbol{\alpha}}}_i^{T},{{\textbf{v}}}_i^{T})\}_{i=1}^N$. Classification -------------- Given a test image set $X_k$, we first assume that it belongs to class $y\in\{1,\cdots,C\}$, and then compute the distance to the sets of class $y$ as $$\begin{aligned} \mathcal{E}_y= \underset{{{\boldsymbol{\alpha}}}_k,{{\textbf{v}}}_k}{\min} & \left \{ \mathcal{R}_{{{\boldsymbol{\alpha}}}_k,{{\textbf{v}}}_k} + \lambda \|{{\boldsymbol{\alpha}}}_k\|_1 \vphantom{ \left . \sum_{j=1}^N P(i=\mathcal{M}_k^{y}|\{{{\textbf{v}}}_i\},{{\textbf{v}}}_k)\times \mathcal{D}_{{{\textbf{v}}}_k,{{\textbf{v}}}_i}~ \right ] } \right.\\ &+ \gamma \left [ \sum_{j=1}^N P(i=\mathcal{H}_k|\{{{\textbf{v}}}_i\},{{\textbf{v}}}_k,y_k=y)\times \mathcal{D}_{{{\textbf{v}}}_k,{{\textbf{v}}}_i} \right .\\ & - \left . \left . \sum_{j=1}^N P(i=\mathcal{M}_k|\{{{\textbf{v}}}_i\},{{\textbf{v}}}_k,y_k=y)\times \mathcal{D}_{{{\textbf{v}}}_k,{{\textbf{v}}}_i}~ \right ] \right\}, \end{aligned}$$ where $P(i=\mathcal{H}_k|\{{{\textbf{v}}}_i\},{{\textbf{v}}}_k,y_k=y)$ is the probability of $X_i$ being the nearest hit of $X_i$ conditional on $y_k=y$. This problem can also be solved by an EM algorithm with the APG optimization. We then assign a label $y_k$ to $X_k$ as follows: $$\begin{aligned} y_k= \underset{y\in \{1,\cdots,C\}}{argmin}~ \mathcal{E}_y \end{aligned}$$ Time Complexity --------------- Given the iteration number $T$, in the off-line procedure of our method, the presentation parameters are updated one by one for $T$ times. Since there are $N$ training image sets, the training time complexity is $O(N\times T)$. In the on-line classification procedure, the test image set is compared to each class, and in each comparison, the representation parameter of the test image set is updated for $T$ times, and the representation parameters of the training image set are fixed. Since the number of classes is $C$, the time complexity for the on-line classification is $O(C\times T)$. In contrast, the SANP algorithm does not have a training procedure. The test image set is compared to all the training image sets. In each comparison, the parameter of both the test image and training image set are updated, thus the time complexity is $O(2\times N\times T)$. Usually $C\ll N$, thus the time complexity is reduced significantly compared to SANP. Experiments {#sec:experiment} =========== We evaluated the proposed method on two comprehensive face image set classification tasks where the goal is to conduct video-based face recognition. Dataset and Setup ----------------- To evaluate our method, we used two real-world face video sequence datasets: - **Conference Face Set**: We collected a human face video sequence dataset from an international academic conference of 5 days. The videos of 32 different conference participants’ faces are captured during the oral session, poster session, reception banquet and some other casual scenes with an handheld camera. The face images extracted from the frames of this dataset cover large variations in illumination, head pose and facial expression, making the face recognition task from these videos quite challenging. For example, the oral presenters’ faces were captured during both oral presentation procedure when light was off and Q&A procedure when light was on. Moreover, the participants’ expressions were quite serious when discussing academic problems, while much more relaxed during the reception banquet. All the images are taken from the same coordinates, but not always of the same sample size. The videos of 32 participants were first captured and then split into totally 507 continuous video sequences to construct the dataset. The number of video sequences for each participant varies from 12 to 19. The entire dataset was further divided into a training set (328 sequences) and an independent test set (179 sequences). The statistic information of the dataset is summarized in Table \[tab:ConfData\]. The frames of each sequence are considered as an image set, and the face region is firstly extracted, then the local binary pattern (LBP) features are extracted from the face region, and finally the LBP feature vectors are combined to construct the data matrix of an image set. Sets Sample Number Participant Number Samples for Each Participant -------------- --------------- -------------------- ------------------------------ Training Set 328 32 8 $\sim$ 13 Test Set 179 32 3 $\sim$ 7 Entire Set 507 32 12 $\sim$ 19 - **YouTube Face Set**: We also used a large-scale face image set database — YouTube Celebrities database [@Kim2008]. It is the largest video database proposed for video face tracking and recognition. In this database, 1910 video sequences of 47 celebrities (actors, actresses and politicians) were collected from YouTube. The number of sequences for each celebrity varies from 17 to 108. The number of frames of each sequence varies from 8 to 400, and most frames are of low resolution and highly compressed. This database is more challenging than the other image set databases, due to the large variations in poses, illuminations and expressions. We also randomly split the database into a training set (1275 sequences) and an independent test set (635 sequences). The statistics of the training set and test set is given in Table \[tab:YoutubeData\]. The face area is cropped from each frame and scaled to size of $20\times 20$, and the pixel values are used as the visual features after histogram equalization. Sets Sample Number Person Number Samples for Each Person -------------- --------------- --------------- ------------------------- Training Set 1910 47 11 $\sim$ 72 Test Set 635 47 6 $\sim$26 Entire Set 1910 47 17 $\sim$ 108 To conduct the experiments, we first performed the cross validation to the training sets to select the optimal parameters. The 8-fold cross validation and 10-fold cross validation were performed to the conference face database and the YouTube face database, respectively. Using the parameters learned by the training set, we classified the image sets in the independent test sets. The classification accuracies are reported as the performance measure. Results ------- \ We compared our method against the state-of-the-art image set classification methods, including DCC [@Kim2008], MMD [@MMD2008], MDA [@MDA2009], AHISD [@AHISD2010], COV [@Wang2012], and SANP [@Hu2011; @Hu2012]. The performance of different methods on the conference face database is shown in Figure \[fig:FigConf\]. In Figure \[fig:FigConf\] (a), the boxplots of the accuracies of the 8-fold cross validation is shown, while in Figure \[fig:FigConf\] (b), the accuracies of evaluation on the independent test set is given. It is obvious that LaMa-IS outperforms all other methods on both the training and test sets. The better performance of LaMa-IS is mainly due to the usage of both the image set class labels and the exploration of the global structure of the image set database. MDA also matches the image samples against its neighbors of the same set and different sets, which is another strategy of the large margin framework. However, we defined the margin at the image set level, instead of the individual image sample level. Thus although MDA also archives good classification accuracy, LaMa-IS performs significantly better than MDA. SANP and LaMa-IS both use the same two models to represent the image sets, but SANP only focuses on the comparison of a pair of image sets, whereas LaMa-IS learns the representations for all the training image sets in a discriminant manner. It turns out that tuning the representation parameters coherently for all the image sets by using the class labels is not a trivial task, which can significantly improve the performance. This can be verified by the outperformance of LaMa-IS over SANP. COV represents each image set as a covariance matrix and also utilizes the class labels to learn the representation parameters. However, its performance is inferior to both LaMa-IS and SANP, which means that the representation model of using both image samples and affine hull model is more effective than the covariance matrix, especially when the training sample number is not large. \ \ Figure \[fig:FigYoutube\] shows the performance of different methods on the YouTube face set. It can be seen that LaMa-IS also outperforms all other methods, on both the training set and test set. Different from the results in Figure \[fig:FigConf\], COV outperforms SANP slightly. The possible reason is that the YouTube database provides more image sets and thus provides sufficient discriminant information to be utilized by COV. We further compared the computation time of different methods on the YouTube database for training and testing (classification of one image set), and reported the results in Figure \[fig:FigYoutube\] (c). It could be seen that the training procedure for MDA, MMD and DCC is quite time consuming, and at the same time, the accuracy of these three methods is not satisfactory. SANP, on the other hand, has quite high time cost on the on-line testing process, but requires no time on training as it does not require a training step. Its performance is also much better than MDA, MMD and DCC. Comparing with SANP, the proposed LaMa-IS method requires an off-line training procedure, which does not consume much time, but can boost the classification performance significantly on this large scale database. Additionally, its on-line classification procedure is also computationally efficient. Conclusion {#sec:conclusion} ========== In this paper, we have proposed a novel large-margin-based image set representation and classification method, LaMa-IS. To represent an image set, LaMa-IS encodes information from both the image samples in the set and their affine hull models. We defined the margin of an image set as the difference of the distance to its nearest image set from different classes and the distance to its nearest image set of the same class. The maximum margin is optimized by an expectation—maximization (EM) strategy with accelerated proximal gradient (APG) optimization in an iterative algorithm. In the classification procedure, LaMa-IS compares a test image set to every class, instead of every image set, making it a computationally efficient algorithm for large-scale applications. Experimental results on two comprehensive face image set classification tasks demonstrate that the proposed method significantly outperforms the state-of-the-art methods in terms of both effectiveness and efficiency. [10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{} W. Zhao, R. Chellappa, P. Phillips, and A. Rosenfeld, “Face recognition: A literature survey,” *ACM Computing Surveys*, vol. 35, no. 4, pp. 399–458, 2003. P. Jonathon Phillips, H. Moon, S. Rizvi, and P. Rauss, “The feret evaluation methodology for face-recognition algorithms,” *IEEE Transactions on Pattern Analysis and Machine Intelligence*, vol. 22, no. 10, pp. 1090–1104, 2000. Y. Wang, W. Jiang, and G. Agrawal, “Scimate: A novel mapreduce-like framework for multiple scientific data formats,” in *Cluster, Cloud and Grid Computing (CCGrid), 2012 12th IEEE/ACM International Symposium on*.1em plus 0.5em minus 0.4emIEEE, 2012, pp. 443–450. J.-Y. Wang, I. Almasri, and X. Gao, “Adaptive graph regularized nonnegative matrix factorization via feature selection,” in *Pattern Recognition (ICPR), 2012 21st International Conference on*.1em plus 0.5em minus 0.4emIEEE, 2012, pp. 963–966. Y. Zhou, L. Li, T. Zhao, and H. Zhang, “Region-based high-level semantics extraction with cedd,” in *Network Infrastructure and Digital Content, 2010 2nd IEEE International Conference on*.1em plus 0.5em minus 0.4emIEEE, 2010, pp. 404–408. R. Wang, S. Shan, X. Chen, and W. Gao, “Manifold-manifold distance with application to face recognition based on image set,” in *26th IEEE Conference on Computer Vision and Pattern Recognition, CVPR*, 2008. W.-S. Chu, J.-C. Chen, and J.-J. Lien, “Kernel discriminant transformation for image set-based face recognition,” *Pattern Recognition*, vol. 44, no. 8, pp. 1567–1580, 2011. W. Fan and D.-Y. Yeung, “Face recognition with image sets using hierarchically extracted exemplars from appearance manifolds,” in *FGR 2006: Proceedings of the 7th International Conference on Automatic Face and Gesture Recognition*, vol. 2006, 2006, pp. 177–182. T.-K. Kim, J. Kittler, and R. Cipolla, “[Discriminative learning and recognition of image set classes using canonical correlations]{},” *[IEEE Transactions on Pattern Analysis and Machine Intelligence]{}*, vol. [29]{}, no. [6]{}, pp. [1005–1018]{}, [2007]{}. T.-J. Chin, K. Schindler, and D. Suter, “Incremental kernel svd for face recognition with image sets,” in *FGR 2006: Proceedings of the 7th International Conference on Automatic Face and Gesture Recognition*, vol. 2006, 2006, pp. 461–466. T. Wang and P. Shi, “Kernel grassmannian distances and discriminant analysis for face recognition from image sets,” *Pattern Recognition Letters*, vol. 30, no. 13, pp. 1161–1165, 2009. O. Arandjelović, G. Shakhnarovich, J. Fisher, R. Cipolla, and T. Darrell, “Face recognition with image sets using manifold density divergence,” in *Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition*, vol. 1, 2005, pp. 581–588. Q. Sun, P. Wu, Y. Wu, M. Guo, and J. Lu, “Unsupervised multi-level non-negative matrix factorization model: Binary data case.” *Journal of Information Security*, vol. 3, no. 4, 2012. S. Wold, K. Esbensen, and P. Geladi, “Principal component analysis,” *Chemometrics and Intelligent Laboratory Systems*, vol. 2, no. 1-3, pp. 37–52, 1987. C. Cortes and V. Vapnik, “Support-vector networks,” *Machine Learning*, vol. 20, no. 3, pp. 273–297, 1995. S. A. Dudani, “Distance-weighted k-nearest-neighbor rule.” *IEEE Transactions on Systems, Man and Cybernetics*, vol. SMC-6, no. 4, pp. 325–327, 1976. R. Wang and X. Chen, “[Manifold Discriminant Analysis]{},” in *CVPR: 2009 IEEE Conference on Computer Vision and Pattern Recognition, VOLS 1-4*, [2009]{}, pp. [429–436]{}. H. Cevikalp and B. Triggs, “[Face Recognition Based on Image Sets]{},” in *2010 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)*, [2010]{}, pp. [2567–2573]{}. Y. Hu, A. S. Mian, and R. Owens, “[Sparse Approximated Nearest Points for Image Set Classification]{},” in *[2011 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)]{}*, [2011]{}, pp. [121–128]{}. ——, “[Face Recognition Using Sparse Approximated Nearest Points between Image Sets]{},” *[IEEE Transactions on Pattern Analysis and Machine Intelligence]{}*, vol. [34]{}, no. [10]{}, pp. [1992–2004]{}, [2012]{}. R. Wang, H. Guo, L. S. Davis, and Q. Dai, “[Covariance Discriminative Learning: A Natural and Efficient Approach to Image Set Classification]{},” in *[2012 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)]{}*, [2012]{}, pp. [2496–2503]{}. Y. Mu, W. Ding, and D. Tao, “Local discriminative distance metrics ensemble learning,” *Pattern Recognition*, vol. 46, no. 8, pp. 2337–2349, 2013. Y. Sun, S. Todorovic, and S. Goodison, “Local-learning-based feature selection for high-dimensional data analysis,” *Pattern Analysis and Machine Intelligence, IEEE Transactions on*, vol. 32, no. 9, pp. 1610–1626, 2010. T. Ahonen, A. Hadid, and M. Pietikäinen, “Face description with local binary patterns: Application to face recognition,” *IEEE Transactions on Pattern Analysis and Machine Intelligence*, vol. 28, no. 12, pp. 2037–2041, 2006. A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” *SIAM Journal on Imaging Sciences*, vol. 2, no. 1, pp. 183–202, Mar. 2009. M. Kim, S. Kumar, V. Pavlovic, and H. Rowley, “Face tracking and recognition with visual constraints in real-world videos,” in *26th IEEE Conference on Computer Vision and Pattern Recognition, CVPR*, 2008. [^1]: Jim Jing-Yan Wang is with the University at Buffalo, The State University of New York, Buffalo, NY 14203, USA. [^2]: Majed Alzahrani and Xin Gao are with the Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia.

--- abstract: 'Let $p_{1}<p_2<\dots <p_{\nu}<\cdots$ be the sequence of prime numbers and let $m$ be a positive integer. We give a strong asymptotic formula for the distribution of the set of integers having prime factorizations of the form $p_{m^{k_1}}p_{m^{k_{2}}} \cdots p_{m^{k_{n}}}$ with $k_{1}\le k_{2}\le \dots \le k_{n}$. Such integers originate in various combinatorial counting problems; when $m=2$, they arise as Matula numbers of certain rooted trees.' address: 'Department of Mathematics, Ghent University, Krijgslaan 281 Gebouw S22, B 9000 Gent, Belgium' author: - Hans Vernaeve - Jasson Vindas - Andreas Weiermann title: Asymptotic distribution of integers with certain prime factorizations --- Introduction ============ Let $\left\{p_\nu\right\}_{\nu=1}^{\infty}$ be the sequence of all prime numbers arranged in increasing order and let $m>1$ be a fixed positive integer. We shall consider the class of integers only admitting prime factors from the subsequence $\left\{p_{m^{k}}\right\}_{k=0}^{\infty}$, that is, the set $$\label{Matulaeq1} A_{m}=\left\{p_{m^{k_1}}p_{m^{k_{2}}} \cdots p_{m^{k_{n}}}\in\mathbb{N}:\ 0\leq k_{1}\le k_{2}\leq \cdots \le k_{n}\right\}\:.$$ The aim of this article is to provide an asymptotic formula for the distribution of $A_{m}$, namely, the following counting function $$M_{2,m}(x)=\underset{n\in A_{m}}{\sum_{n\le x}} 1\: .$$ The function $M_{2,2}$ arises in various interesting combinatorial counting problems; particularly, in connection with rooted trees. In 1968 Matula gave an enumeration of (non-planar) rooted trees by prime factorization [@Matula1968], the so-called Matula numbers. Number theoretic aspects of this rooted tree coding have been investigated in detail in [@b-t; @g-ivic1996]. Such numbers may be used to deduce many intrinsic properties of rooted trees [@Deutsch; @g-ivic1994; @g-y]. The set $A_{2}$ in fact corresponds to a class of Matula numbers. In Section \[rooted trees\] we review Matula coding of rooted trees and give the interpretation of $M_{2,2}$ as the counting function of rooted trees with height less or equal to 2, under Matula’s enumeration. It is worth mentioning that the significance of Matula numbers comes from applications in organic chemistry, as they can be employed to develop efficient nomenclatures for representing molecules of a variety of organic compounds (cf. [@elk1989; @elk1990; @elk1994; @elk2011; @g-ivic-elk1993]). As explained in Section \[rooted trees\], $M_{2,2}$ might also be regarded as a “transfinite counting function” for the ordinal $\omega^{\omega}$ in a certain complexity norm [@Weiermann2010]. In [@Weiermann2010] Weiermann found the weak asymptotics of the counting function $M_{2,2}$. Using a Tauberian theorem by Kohlbecker for partitions [@kohlbecker1958], he showed that $$\label{Matulaeq3} \log M_{2,2}(x)\sim \pi \sqrt{\frac{2\log x}{3\log 2}}\,.$$ The asymptotic relation (\[Matulaeq3\]) resembles the one obtained by Hardy and Ramanujan in 1917 for the celebrated (unrestricted) partition function, $$\label{Matulaeq4} \log p(n)\sim \pi \sqrt{\frac{2n}{3}}\:,$$ which they [@hardy-ramanujan1918], and independently Uspensky [@Uspensky1920], greatly refined later to $$\label{Matulaeq5} p(n)\sim \frac{e^{\pi \sqrt{\frac{2n}{3}}}}{(4\sqrt{3})n}\: .$$ Naturally, the transition from (\[Matulaeq4\]) to (\[Matulaeq5\]) consists in finding missing asymptotic terms. The problem we address here is of similar nature. We shall fill the gap between (\[Matulaeq3\]) and the strong asymptotics by exhibiting hidden lower order terms in the approximation (\[Matulaeq3\]), as stated in the following theorem. \[Matulath1\] The function $M_{2,m}$ has asymptotic behavior $$\label{Matulaeq6} M_{2,m}(x)\sim \frac{e^{K_m}\sqrt{3}\log m}{2 \pi^{2}\log 2}(\log x)^{\frac{\log\left(\frac\pi {\sqrt{6\log m}}\right)}{2\log m}} \exp\left(\pi\sqrt{\frac{2\log x}{3\log m}} - \frac {(\log\log x)^{2}}{8\log m}\right)\: ,$$ where $$K_m=\frac{1}{2\log m}\left((\log\log m)^2+\gamma^{2}-2\gamma\log\log m-\frac{\pi^{2}}{6}-\log^2\left(\frac\pi{\sqrt{6\log m}}\right)\right)- C_{2,m}\:,$$ $\gamma$ is the Euler-Mascheroni constant, and $C_{2,m}$ is given by the convergent series $$C_{2,m}=\sum_{k=1}^\infty \left(\log\log p_{m^k} - \log k - \log\log m - \frac{\log k}{k \log m} - \frac{\log\log m}{k \log m}\right)\:.$$ We will provide a proof of Theorem \[Matulath1\] in Section \[proof\]. The proof is based on Ingham’s method from [@Ingham]; however, it turns out that Ingham’s original theorem for partitions [@Ingham Thm. 2] is not directly applicable to our context. In Section \[Ingham theorem\], we shall slightly extend his result. It is likely that such an extension of Ingham’s theorem might be useful for treating partition problems other than the one dealt with in this article. Two counting problems and $M_{2,m}$ {#rooted trees} =================================== Rooted trees {#Matula Numbers} ------------ Matula’s coding of (non-planar) rooted trees in terms of prime factorizations provides a bijection between such trees and the positive integers. The same rooted tree enumeration was rediscovered by Göbel in [@gobel1980]. It is defined as follows. If we denote the trivial one-vertex tree by $\bullet$, then its Matula number is $n(\bullet):=1$. Inductively, if $T_1$, $T_2$, …, $T_l$ are trees and $T$ is given as $$\begin{tikzpicture}[thick,auto,baseline] \coordinate[vertex] (r) at (0,0); \coordinate[tree,label=left:$T_1$] (n1) at (-1,1); \coordinate[tree,label=left:$T_2$] (n2) at (-.5,1.5); \coordinate[tree,label=right:$T_{l-1}$] (n3) at (.5,1.5); \coordinate[tree,label=right:$T_{l}$] (n4) at (1,1); \coordinate[label=right:$\dots$] (dots) at (-.4,1.5); \draw (r) to (n1); \draw (r) to (n2); \draw (r) to (n3); \draw (r) to (n4); \end{tikzpicture}$$ then its Matula number is defined as $n(T) := p_{n(T_1)}\cdots p_{n(T_l)}$. If $T_{1,k}$ is the tree of height one with $k$ nodes above the root, then $n(T_{1,k})= p_1^k = 2^k$. If $T$ has height two, then $$n(T) = p_{n(T_{1,k_1})}p_{n(T_{1,k_2})} \cdots p_{n(T_{1,k_{\nu-1}})} p_{n(T_{1,k_\nu})} = p_{2^{k_1}}p_{2^{k_{2}}}\cdots p_{2^{k_{\nu-1}}} p_{2^{k_\nu}}\:,$$ where the $j$-th node connected to the root carries a tree $T_{1,k_j}$. $$\begin{tikzpicture}[thick,auto,baseline] \coordinate[vertex] (r) at (0,0); \coordinate[vertex,label=left:$T_{1,k_1}$] (n1) at (-1.2,.7); % \coordinate[vertex] (n11) at (-1.5,1.2); \coordinate[vertex,label=left:$T_{1,k_2}$] (n2) at (-.5,1.5); \coordinate[vertex] (n21) at (-.5,2); \coordinate[vertex,label=right:$T_{1,k_{\nu-1}}$] (n3) at (.5,1.5); \coordinate[vertex] (n31) at (.25,1.9); \coordinate[vertex] (n32) at (.5,2); \coordinate[vertex] (n33) at (.8,1.9); \coordinate[vertex,label=right:$T_{1,k_\nu}$] (n4) at (1.2,.7); \coordinate[vertex] (n41) at (.95,1.15); \coordinate[vertex] (n42) at (1.2,1.2); \coordinate[vertex] (n43) at (1.45,1.15); \coordinate[label=right:$\dots$] (dots) at (-.4,1.5); \draw (r) to (n1); \draw (r) to (n2); \draw (r) to (n3); \draw (r) to (n4); % \draw (n1) to (n11); \draw (n2) to (n21); \draw (n3) to (n31); \draw (n3) to (n32); \draw (n3) to (n33); \draw (n4) to (n41); \draw (n4) to (n42); \draw (n4) to (n43); \end{tikzpicture}$$ It is then clear that Matula coding gives a bijection between the set of rooted trees with height equal to 1 or 2 and the set $A_{2}$ defined in (\[Matulaeq1\]). Consequently, $M_{2,2}(x)$ counts the number of Matula numbers corresponding to trees with $0<\text{height}(T)\le 2$, that are below $x$, i.e., $$\underset{\text{height}(T)\le 2}{\sum_{n(T)\le x}1}=M_{2,2}(x)+1\:.$$ Thus, this rooted tree counting function has also asymptotics (\[Matulaeq6\]). Ordinal counting functions {#ordinals} -------------------------- It might seem surprising at first sight that the counting function $M_{2,2}$ is related to studying asymptotic properties of transfinite ordinals. Since transfinite ordinals rarely show up in a number-theoretic context we will explain some features of this connection in informal and general terms. The rest of the paper will not depend on the exposition given in this subsection, but it might be useful as a source of inspiration for further study. In naive set theory ordinals generalize the ordering of the natural numbers $0<1<2<\cdots$ by continuing beyond the first limit point $\omega$ like $0<1<2<\cdots<\omega+1<\omega+2<\cdots$. This process can be continued beyond the next limit $\omega+\omega$ like $0<1<2<\cdots<\omega+1<\omega+2<\cdots< \omega\cdot 2 +1<\omega\cdot 2 +2<\cdots$ and by iteration like $0<1<2<\cdots<\omega+1<\omega+2<\cdots< \omega\cdot 2 +1<\omega\cdot 2+2 <\cdots<\omega\cdot 3<\cdots<\omega\cdot n<\cdots$. At a certain moment we reach the first limit of limits $\omega\cdot \omega$ and, again by iteration, we reach limits of limits of limits and in the limit of this counting we reach $\omega^\omega$ (an ordinal which – as will become clear soon – is of relevance to $M_{2,m}$). The ordinal $\omega^\omega$ is not at all frightening since it appears as the order type of the polynomials in $\Nat[x]$ under eventual domination or as the order type of the multisets of natural numbers. There is of course no bound in counting through the ordinals and by further counting we reach $\omega^{\omega^\omega}, \omega^{\omega^{\omega^\omega}},\ldots$, but the higher we go the more complicated the description becomes. Slight extensions could still be dealt with by combinatorial means (which can still be formalized in Peano arithmetic) and stronger extensions will require from some moment onwards basic set theoretic machinery. There is still some nice and accessible visualization of the ordinals less than $\varepsilon_0$, which is the limit of the finite powers of $\omega$ showing up in the sequence $\omega,\omega^\omega,\omega^{\omega^\omega}, \omega^{\omega^{\omega^\omega}},\ldots$. For this we consider a subclass of Hardy’s orders of infinity. Let ${\mathcal E}$ be the class of unary functions $f:\Nat\to\Nat$ such that 1. the function $c_0$ is an element of ${\mathcal E}$ where $c_0(x)=x$ and 2. with two functions $f,g$ in ${\mathcal E}$ also the function $h$ is in ${\mathcal E}$, where $h(x)=x^{f(x)}+g(x)$. On ${\mathcal E}$ we define the ordering of eventual domination as usual by $f\prec g$ if and only if there exists a non-negative integer $k$ such that $f(x)<g(x)$ for all $x\geq k$. The structure $\langle {\mathcal E},\prec \rangle $ is isomorphic with $\langle \{\alpha:\alpha<\varepsilon_0\},< \rangle $ and so we can identify both structures. If we also write $id=c_{0}$ for the identity function on $\Nat$, then the isomorphism maps $\omega$ to $id$, $\omega^\omega$ to $id^{id}$, $\omega^{\omega^\omega}$ to $id^{id^{id}}$, etc. The ordinal $\varepsilon_0$ is the proof-theoretic ordinal of first order Peano arithmetic $PA$. $PA$ proves (after an appropriate formalization of the context) the scheme of transfinite induction for all strict initial segments of $\varepsilon_0$ but not the scheme of transfinite induction for the full segment up to $\varepsilon_0$. For treating ${\mathcal E}$ in the context of arithmetic we need a specific (easily definable) coding of the elements of ${\mathcal E}$ into the natural numbers. One of the standard devices for achieving this is provided by associating to the elements of ${\mathcal E}$ their canonical counterparts in the finite non-planar rooted trees. Such a bijection $t$ can defined recursively as follows. First, $t(c_0):=\bullet$. Every $f$ can be written as $f=id^{g_1}+\cdots+id^{g_n}$, then let $$t(f):= \(r) at (0,0); (n1) at (-1,1); (n2) at (-.5,1.5); (n3) at (.5,1.5); (n4) at (1,1); (dots) at (-.4,1.5); (r) to (n1); (r) to (n2); (r) to (n3); (r) to (n4); .$$ By this identification we can – using Matula’s coding – canonically associate to $f\in {\mathcal E}$ its Gödel number $\lceil f\rceil:=n(t(f))$. This coding has been used explicitly by Troelstra and Schwichtenberg in [@Troelstra p. 320, Def. 10.1.5]. In this context we arrive at the following interpretation $$1+M_{2,2}(x)=\#\{f\in {\mathcal E}: f\prec id^{id}\wedge \lceil f\rceil\leq x\}\: .$$ For coding a larger segment of ordinals Schütte [@Schutte Sec. V.8] used a related coding $Nr$ which when restricted to ${\mathcal E}$ has the property that $1+M_{2,4}(x)$ is the number of ordinals $\alpha$ below $\omega^\omega$ such that $Nr(\alpha)\leq x$. Until now, the study of ordinal counting functions has found applications to logical limit laws for ordinals and to phase transitions for Gödel incompleteness results (it seems very interesting and intriguing to find additional applications). A further discussion of phase transitions will be beyond the scope of this exposition, but we want to include an intriguing example for a zero-one law (see [@burris] for an account on logical limit laws). As usual, we use $\models$ for the satisfaction relation from model theory. Let $\varphi$ be a sentence in the language of linear orders. Let $$\delta_\varphi:=\lim_{x\to\infty}\frac{\#\{f\in {\mathcal E}: f\prec id^{id} \wedge \lceil f\rceil\leq x \wedge \langle \{g\in {\mathcal E}:g\prec f \},\prec \rangle \models \varphi \}} {\#\{f\in {\mathcal E}: f\prec id^{id} \wedge \lceil f\rceil\leq x\}}\: .$$ Then $\delta_\varphi$ exists and either $\delta_\varphi=1$ or $\delta_\varphi=0.$ A proof of this and similar results has been obtained in [@Weiermann2012] by an analysis of the asymptotic behavior of $M_{2,2}$ and related counting functions. At the beginning of this subsection it has been indicated that ordinals might provide a source of inspiration for further research and we will now indicate some possible options. For $g\in {\mathcal E}$, let $c_g(x):={\#\{f\in {\mathcal E}: f\prec g \wedge \lceil f\rceil\leq x\}}$. For various choices of $g$ some preliminary results on weak asymptotics for $c_g$ have been obtained in [@Weiermann2010]. Moreover, strong asymptotics for $c_{{id^k}}$ can be obtained by elementary means. We believe that the methods of this paper will allow one to provide strong asymptotics for $c_{id^{id^k}}$ for any given fixed $k$. A strong asymptotic formula for $c_{id^{id^{id}}}$ (which would resemble something like multiplicative double partitions) seems however to require new methods. A general challenge would be then to provide a general theorem on strong asymptotics for $c_g$ for any fixed $g$ and for analogous functions emerging from the Schütte coding. An extension of Ingham’s theorem for unrestricted partitions {#Ingham theorem} ============================================================ As mentioned in the Introduction, we need an extension of Ingham’s theorem for strong asymptotics of partition functions. The extension will follow from a complex Tauberian theorem for large asymptotic behavior of the Laplace transform, also due to Ingham [@Ingham Thm. $1'$]. Let $0<\lambda_{0}<\lambda_1<\dots<\lambda_k\to \infty$ be a sequence of real numbers and let $$N(u)= \sum_{\lambda_k\le u}1$$ be its counting function. Consider the additive semigroup $\Lambda$ generated by $\left\{\lambda_k\right\}_{k=0}^{\infty}$, i.e., $$\Lambda = \{r\in\mathbb{R}: r=\sum_{k=0}^l n_k\lambda_k,\ n_k\in\mathbb{N} \}\:.$$ For $r\in\Lambda$, the partition function $p(r)$ is defined as the number of ways of writing $r$ as $r=\sum_{k=0}^l n_k\lambda_k$. We further set $$P(u)= \underset{r\in\Lambda}{\sum_{r\le u}} p(r)\:.$$ The following theorem obtains the asymptotic behavior of $P(u)$ if one knows a certain average asymptotic behavior for $N(u)$. It slightly extends that of Ingham by allowing an extra term of the form $B\log^2 u$ in the asymptotic expansion (\[eqN\]). As usual, $\zeta$ stands for the Riemann zeta function and $\Gamma$ for the Euler Gamma function. The constant $\gamma_{1}$ denotes the Stieltjes constant, that is, $$\gamma_{1}=\lim_{n\to\infty} \sum_{k=1}^n \frac{\log k}k-\frac{\log^2 n}2\: .$$ \[main-partition-thm\] Suppose that $$\label{eqN} \int_0^u \frac{N(t)}t\,dt = \frac{A}{\alpha}u^\alpha + B\log^2 u + C \log u + D +o(1)\: ,$$ with $\alpha,A>0$. Then $$\label{Matulaeq3.2} P(u) \sim \left(\frac{1-\beta}{2\pi}\right)^{\frac12} e^{D'} M^{-(C+\frac12)} u^{C-\beta C-\frac\beta2}\exp\left(\frac{Mu^\beta}{\beta}+ B\log^2\left(\frac{u^{1-\beta}}{M}\right)\right)\:,$$ where $$\beta=\frac\alpha{\alpha+1},\quad M=(A\alpha\Gamma(\alpha+1)\zeta(\alpha+1))^{\frac1{\alpha+1}},\quad D'= D+ \left(\frac{\pi^{2}}{6}-2\gamma_1 -\gamma^2\right)B.$$ In order to deduce Theorem \[main-partition-thm\] from Ingham’s Tauberian theorem, we proceed to find the asymptotic behavior of the Laplace-Stieltjes transform of $P$. Set $$F(s)=\sum_{r\in \Lambda}e^{-sr}=\int_{0}^{\infty}e^{-su}dP(u)$$ and $$f(s)=s\int_{0}^{\infty}\frac{N(u)}{e^{su}-1}\:du\:.$$ The generating function identity $F(s)=e^{f(s)}$ is well-known. \[Ingham-extl1\] If $(\ref{eqN})$ holds, then, as $\sigma\to 0^{+}$, $$\sigma\int_0^\infty \frac{N(u)}{e^{\sigma u}-1}\,du = \frac{A\Gamma(\alpha+1)\zeta(\alpha+1)}{\sigma^\alpha} + B\log^2 \sigma - C\log \sigma + D' + o(1)\: .$$ Here $$\label{Matulaeq3.3} D'= D+ \left(\frac{\pi^{2}}{6}-2\gamma_1 -\gamma^2\right)B\:.$$ We employ standard Schwartz distribution calculus in our manipulations. It might also be possible to give a classical proof along the lines of that of [@Korevaarbook Thm. IV.23.1]. For Schwartz distributions, we follow the notation exactly as in [@estrada-kanwal2002 Chap. 2]. Taking distributional derivative in (\[eqN\]), we obtain $$\begin{aligned} \frac{N(\lambda u)}{u}=&A\lambda^{\alpha}u_{+}^{\alpha-1}+(B\log^{2}\lambda+C\log\lambda+D)\delta(u)+2B\operatorname*{Pf}\left(\frac{H(u)\log u}{u}\right)\\ & +(2B\log\lambda+C)\operatorname*{Pf}\left(\frac{H(u)}{u}\right)+o(1)\:, \ \ \ \lambda\to\infty\:,\end{aligned}$$ distributionally in the space of tempered distributions $\mathcal{S}'$ (cf. [@estrada-kanwal2002 Sec. 3.9], [@p-s-v Sec. 2.5]), where $\delta$ stands for the Dirac delta distribution, $H$ is the Heaviside function, and $\operatorname*{Pf}$ denotes regularization via Hadamard finite part [@estrada-kanwal2002 Sec. 2.4]. Testing this asymptotic expansion at the test function $\psi(u)=u/(e^u-1)$, setting $\sigma=1/\lambda$, and taking into account the well-known formula $$\label{eqRiemann} \Gamma(s+1)\zeta(s+1)=\int_{0}^{\infty}\frac{u^{s}}{e^{u}-1}\: du\: , \ \ \ \Re e\:s>0\:,$$ we obtain $$\sigma\int_0^\infty \frac{N(u)}{e^{\sigma u}-1}\,du=\frac{A\Gamma(\alpha+1)\zeta(\alpha+1)}{\sigma^\alpha} + B\log^2 \sigma - \left(C+2BK\right)\log \sigma + D' + o(1)\: ,$$ where $D'=D+2BK'+CK$, and the constants $K$ and $K'$ are given by the Hadamard finite part at $0$ of the integrals $$K=\operatorname*{F.p.}\int_{0}^{\infty}\frac{du}{e^{u}-1} \quad \mbox{and} \quad K'=\operatorname*{F.p.}\int_{0}^{\infty}\frac{\log u}{e^{u}-1}\:du\:.$$ Hence, it remains to evaluate these two constants. We will do so by inspecting the Laurent expansion at $s=0$ of the analytic continuation of (\[eqRiemann\]). In fact, the classical procedure of Marcel Riesz [@estrada-kanwal2002; @gelfand-shilovVol1] yields the analytic continuation of (\[eqRiemann\]) to $\mathbb{C}\setminus\left\{0,-1,-2,\dots\right\}$ as the finite part integral of the right hand side. By employing the Gelfand-Shilov Laurent expansion at $s=0$ for the distribution $u_{+}^{s-1}$ [@gelfand-shilovVol1 p. 87], we conclude $$\operatorname*{F.p.}\int_{0}^{\infty}\frac{u^{s}}{e^{u}-1}\: du= \frac{1}{s}+\sum_{n=0}^{\infty}\frac{s^{n}}{n!}\operatorname*{F.p.}\int_{0}^{\infty}\frac{\log^{n}u}{e^{u}-1}\:du\,, \quad 0<|s|<1\:.$$ On the other hand, since $$\zeta(s+1)=\frac{1}{s}+\gamma-\gamma_{1} s+\cdots\quad \mbox{and} \quad \Gamma(s+1)=1-\gamma s+\left(\frac{\gamma^{2}}{2}+\frac{\pi^{2}}{12}\right)s^{2}+\cdots\:,$$ we have $$\Gamma(s+1)\zeta(s+1)=\frac{1}{s} +\left(\frac{\pi^{2}}{12}- \gamma_{1}-\frac{\gamma^{2}}{2}\right)s+ \cdots\:, \quad 0<|s|<1\:,$$ and therefore $K=0$ and $K'=\pi^{2}/12- \gamma_{1}-\gamma^{2}/2$. We can now apply Ingham’s Tauberian theorem [@Ingham Thm. $1'$] to the generating function $F(s)$ with $$\varphi(s):= \frac{A\Gamma(\alpha+1)\zeta(\alpha+1)}{s^\alpha} \quad\text{and}\quad \chi(s):= e^{D'} s^{-C}e^{B\log^{2}s}\:.$$ Indeed, in view of Lemma \[Ingham-extl1\], $$F(\sigma)\sim \chi(\sigma)e^{\varphi(\sigma)}, \quad \sigma\to0^{+}\:,$$ and the quoted theorem of Ingham immediately implies that $$P(u)\sim \frac{\chi(\sigma(u))\exp\left(\varphi(\sigma(u))+u\sigma(u)\right)}{\sqrt{2\pi\sigma^{2}(u)\varphi''(\sigma(u))}}, \quad u\to\infty\:,$$ where $\sigma(u)$ is the inverse function of $-\varphi'$, i.e., $\sigma(u)=M u^{-\frac1{\alpha+1}}$ with $M=(A\alpha\Gamma(\alpha+1)\zeta(\alpha+1))^{\frac1{\alpha+1}}$. Setting $\beta=\alpha/(\alpha+1)$, we have $$\begin{aligned} e^{\varphi(\sigma(u))+ u\sigma(u)} &= e^{\frac{Mu^\beta}\beta}, \\ \frac{\chi(\sigma(u))}{\sigma(u)\sqrt{2\pi \varphi''(\sigma(u))}} &= \left(\frac{1-\beta}{2\pi}\right)^{\frac12} e^{D'} M^{-(C+\frac12)} u^{C-\beta C-\frac\beta2}e^{B\log^2(u^{1-\beta}/M)}\,.\end{aligned}$$ whence $(\ref{Matulaeq3.2})$ follows. In the sequel, we will only use the case $\alpha=1$ of Theorem \[main-partition-thm\], which we state in the next corollary for the sake of convenience. \[cor-partition-thm\] Suppose that $$\int_0^u \frac{N(t)}t\,dt = Au + B\log^2 u + C\log u + D + o(1)\:.$$ Then $$P(u) \sim \frac{e^{D'+B\log^2\left(\pi\sqrt{\frac A6}\right)}}{2\sqrt \pi} {\left(\pi\sqrt{\frac A6}\right)}^{-(C+\frac12)} u^{\frac C2 - \frac14 -B\log\left(\pi\sqrt{\frac A6}\right)}\exp\left(\pi\sqrt{\frac{2Au}3}+ \frac B4 \log^2 u\right)\:,$$ where $D'$ is given by (\[Matulaeq3.3\]). The original problem {#proof} ==================== We now proceed to give a proof of Theorem \[Matulath1\]. We translate our original problem into an additive partition problem. Consider $\lambda_k = \log p_{m^k}$. Then, with the notation of the preceding section, $$1+M_{2,m}(e^u) %\sum_{\scriptstyle\sum_{j=1}^m n_j \log p_{l^j}\le u\atop n_j\ge 0} 1 = \underset{r\in\Lambda}{\sum_{r\le u}} p(r) = P(u)\: .$$ Of course, here $p(r)=1$ for each $r\in \Lambda=\left\{\sum_{k=0}^{l}n_{k}\log p_{m^{k}}:\:n_{k}\in\mathbb{N} \right\}$. Thus, we are interested in the average asymptotics of the counting function $$N(u)=\sum_{\log p_{m^k}\le u} 1\, .$$ Observe that $$\label{integral-N-over-t} \int_0^u \frac{N(t)}t\,dt% = N(u)\log u - \int_0^u \log t\, dN(t) = N(u)\log u - \sum_{\log p_{m^k}\le u}\log\log p_{m^k}\:.$$ We first need to estimate $\log\log p_{m^k}$. \[Matulal2\] $$\label{loglogplk} \log\log p_{m^k} = \log k + \log\log m + \frac{\log k}{k \log m} + \frac{\log\log m}{k \log m} + O\left(\frac{\log^{2} k}{k^2}\right).$$ Using the prime number theorem, Cipolla [@Cipolla] found in 1902 an asymptotic formula for $p_n$. Employing just two terms in the expansion, we have $$p_n = n\log n + O(n\log\log n).$$ This leads to $$\log p_n = \log n + \log\log n + O\left(\frac{\log\log n}{\log n}\right)$$ and $$\log\log p_n = \log\log n + \frac{\log\log n}{\log n} + O\left(\frac{(\log\log n)^2}{\log^2 n}\right)\:.$$ Thus, for $n=m^k$, we obtain the required formula. Lemma \[Matulal2\] immediately yields: $$\label{C2l} C_{2,m}=\sum_{k=1}^\infty \left(\log(\log p_{m^k}) - \log k - \log\log m - \frac{\log k}{k \log m} - \frac{\log\log m}{k \log m}\right)$$ converges. Next, \[avg-asymptotics-of-N\] $$\label{Matulaeq4.4} \int_0^u \frac{N(t)}{t}\,dt =\frac{u}{\log m} - \frac{\log^2 u}{2\log m} + \frac{1}{2}\log u +D_m + o(1)\: ,$$ where $$D_m = \frac{(\log\log m)^2}{2\log m} + \log\left(\frac{1}{\log 2}\sqrt{\frac{\log m}{2\pi}}\right)- C_{2,m} - \frac{\gamma_1}{\log m} - \frac{\log\log m}{\log m}\gamma\:.$$ We first notice that $$p_{m^k}\le e^u %\mbox{ iff } l^m\le \pi(e^u) \mbox{ if and only if } k\le y:= \frac{\log(\pi(e^u))}{\log m}\:,$$ where $\pi(x)$ is the distribution of the prime numbers. By the prime number theorem, $$y %= \frac{1}{\log l}\log \left(\frac{e^u}{u} + O\left(\frac{e^u}{u^2}\right)\right) = \frac u {\log m} - \frac{\log u}{\log m} + O\left(\frac 1 u\right)\:.$$ Thus $N(u)=\left\lfloor y \right\rfloor+1$ and, combining equations , and , $$\begin{aligned} \int_0^u \frac{N(t)}{t}\,dt &= (\left\lfloor y \right\rfloor+1) \log u -\log\log 2-\sum_{k=1}^{\left\lfloor y \right\rfloor} \log(\log p_{m^k}) \\ &=(\left\lfloor y \right\rfloor +1)\log u - \log\log2- C_{2,m} - \log(\left\lfloor y \right\rfloor !) - (\log\log m)\left\lfloor y \right\rfloor \\ & \ \ \ - \frac{1}{\log m}\sum_{k=1}^{\left\lfloor y \right\rfloor} \frac{\log k}k - \frac{\log\log m}{\log m}\sum_{k=1}^{\left\lfloor y \right\rfloor} \frac{1}k + o(1)\:.\end{aligned}$$ Using Stirling’s formula and the defining formulas for $\gamma$ and $\gamma_1$ $$\log(n!) = n\log n - n + \frac{1}{2}\log n + \log(\sqrt{2\pi}) + o(1)\:,$$ $$\gamma = \sum_{k=1}^n \frac{1}{k} - \log n + o(1) \quad\text{and}\quad \gamma_1 = \sum_{k=1}^n \frac{\log k}k - \frac{\log^2 n}2 + o(1)\:,$$ we have $$\begin{aligned} \int_0^u \frac{N(t)}{t}\,dt &=\left\lfloor y \right\rfloor \log \left(\frac{u}{\left\lfloor y \right\rfloor \log m}\right) +\log u+ \left\lfloor y \right\rfloor - \frac{\log^2\left\lfloor y \right\rfloor}{2\log m} - \left(\frac{1}{2} + \frac{\log\log m}{\log m}\right)\log\left\lfloor y \right\rfloor\\ & \ \ \ -\log\log 2- C_{2,m} - \log(\sqrt{2\pi}) - \frac{\gamma_1}{\log m} - \frac{\log\log m}{\log m}\gamma + o(1)\:.\end{aligned}$$ Since $$\left\lfloor y \right\rfloor \log \left(\frac{u}{\left\lfloor y \right\rfloor\log m}\right) =\left\lfloor y \right\rfloor \log \left(1 + \frac{u-\left\lfloor y \right\rfloor \log m}{\left\lfloor y \right\rfloor \log m}\right) = \frac{u-\left\lfloor y \right\rfloor \log m}{\log m} + O\left(\frac{\log^{2}u} {u}\right)$$ and $$\log\left\lfloor y \right\rfloor= \log (y+ O(1)) = \log y + O\left(\frac1y\right) = \log u - \log\log m+ O\left(\frac{\log u}u\right),$$ we obtain (\[Matulaeq4.4\]), as required. The asymptotic formula (\[Matulaeq6\]) follows by combining Lemma \[avg-asymptotics-of-N\] and Corollary \[cor-partition-thm\] after a straightforward calculation. The proof of Theorem \[Matulath1\] is complete. [99]{} R. de la Bretèche, G. Tenenbaum, *Sur certaines équations fonctionnelles arithmétiques,* Ann. Inst. Fourier (Grenoble) **50** (2000), 1445–1505. S. N. Burris, *Number theoretic density and logical limit laws,* Mathematical Surveys and Monographs, 86, American Mathematical Society, Providence, RI, 2001. M. Cipolla, *La determinazione assintotica dell’ $n^{imo}$ numero primo,* Napoli Rend. **3** (1902), 132–166. E. Deutsch, *Rooted tree statistics from Matula numbers,* Discrete Appl. Math. **160** (2012), 2314–2322. S. B. Elk, *A problem with the application of Matula’s method of prime numbers and rooted trees for canonical nomenclatures of alkanes,* Graph theory notes (New York) **18** (1989), 40–43. S. B. Elk, *A canonical ordering of polybenzenes and polymantanes using a prime number factorization technique,* J. Math. Chem. **4** (1990), 55–68. S. B. Elk, *Expansion of Matula numbers to heteroatoms and to ring compounds,* J. Chem. Inf. Comput. Sci. **35** 1995, 233–236. S. B. Elk, *The Structure-Nomenclature Cycle of Chemistry,* in: Mathematical Chemistry Monographs, vol. 11, Univ. Kragujevac, Kragujevac, 2011. R. Estrada, R. P. Kanwal, *A distributional approach to asymptotics. Theory and applications,* Birkhäuser Boston, Inc., Boston, MA, 2002. I. M. Gel’fand, G. E. Shilov, *Generalized functions. Vol. I: Properties and operations,* Academic Press, New York-London, 1964. F. Göbel, *On a 1-1-correspondence between rooted trees and natural numbers,* J. Combin. Theory B **29** (1980), 141–143. I. Gutman, A. Ivić, *Graphs with maximal and minimal Matula numbers,* Bull. Acad. Serbe des Sciences et des Arts Sér. Math. **18** (1994), 65–74. I. Gutman, A. Ivić, *On Matula numbers,* Discrete Mathematics **150** (1996), 131–142. I. Gutman, A. Ivić, S. B. Elk, *Matula numbers for coding chemical structures and some of their properties,* J. Serb. Chem. Soc. **58** (1993), 193–201. I. Gutman, Y. N. Yeh, *Deducing properties of trees from their Matula numbers,* Publ. Inst. Math. (Beograd) (N.S.) **53**(67) (1993), 17–22. G. H. Hardy, S. Ramanujan, *Asymptotic formulae in combinatory analysis,* Proc. London Math. Soc. **17** (1918), 75–115. A. E. Ingham, *A Tauberian theorem for partitions*, Ann. of Math. (2) **42** (1941), 1075–1090. E. E. Kohlbecker, *Weak asymptotic properties of partitions,* Trans. Amer. Math. Soc. **88** (1958), 346–365. J. Korevaar, *Tauberian theory. A Century of developments*, Grundlehren der Mathematischen Wissenschaften, 329, Springer-Verlag, Berlin, 2004. D. W.  Matula, *A natural rooted tree enumeration by prime factorization,* Siam Rev. **10** (1968), 273. S. Pilipović, B. Stanković, J. Vindas, *Asymptotic behavior of generalized functions,* Series on Analysis, Applications and Computation, 5, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. K. Schütte, *Proof Theory*, Grundlehren der Mathematischen Wissenschaften, 225, Springer-Verlag, Berlin-New York, 1977. A. S. Troelstra, H. Schwichtenberg, *Basic proof theory*, Cambridge Tracts in Theoretical Computer Science, 43, Cambridge University Press, Cambridge, 2000. J. V. Uspensky, *Asymptotic expressions of numerical functions occurring in problems of partitions of numbers,* Bull. Acad. Sci. de Russie **14** (1920), 199–218. A. Weiermann, *Analytic combinatorics of the transfinite: a unifying Tauberian perspective,* in: Logic Colloquium 2007, pp. 238–267, Lect. Notes Log., 35, Assoc. Symbol. Logic, La Jolla, CA, 2010. A. Weiermann, A. R. Woods, *Some natural zero one laws for ordinals below $\varepsilon_0$,* in: Cooper, S. Barry (ed.) et al., How the world computes. Turing centenary conference and 8th conference on computability in Europe, CiE 2012, Cambridge, UK, June 18–23, 2012. Proceedings. Berling. Springer Lecture Notes in Computer Science 7318, pp. 723-732.

--- abstract: 'We have studied the energy gap of the 1D AF-Heisenberg model in the presence of both uniform ($H$) and staggered ($h$) magnetic fields using the exact diagonalization technique. We have found that the opening of the gap in the presence of a staggered field scales with $h^{\nu}$, where $\nu=\nu(H)$ is the critical exponent and depends on the uniform field. With respect to the range of the staggered magnetic field, we have identified two regimes through which the $H$-dependence of the real critical exponent $\nu(H)$ can be numerically calculated. Our numerical results are in good agreement with the results obtained by theoretical approaches.' address: 'Institute for Advanced Studies in Basic Sciences, Zanjan 45195-1159, Iran' author: - 'S. Mahdavifar' title: ' Scaling behavior of the energy gap of spin-$\frac{1}{2}$ AF-Heisenberg chain in both uniform and staggered fields ' --- Introduction ============ The effect of external magnetic fields in the quantum properties of low-dimensional magnets has been of much interest in recent years. Experimental and theoretical studies of these systems have revealed a plethora of quantum flactuation phenomena, not usally observed in higher dimensions. The magnetization processes in antiferromagnetic (AF) spin chains and ladders have been under intensive investigation using novel numerical techniques. The progress in the experimental front is achived by introduction of high-field neutron scattering studies and synthesis of magnetic quasi-one dimensional systems such as the spin-$\frac{1}{2}$ antiferromagnet Cu benzoate [@dender; @asano1; @asano2] and $Yb_{4}As_{3}$ [@oshikawa1; @shiba; @kohgi]. Due to these developments we can now observe the effect of a staggered magnetic field (or even more complicated interactions) on the low energy behavior of a one-dimentional quantum model in the laboratory. There exist different mechanisms for generating a staggered field in a real magnet [@oshikawa2; @wang; @sato]. In Cu benzoate the alternating crystal axes is the source of such a field. Dender et.al. [@dender] showed that an effective staggered field can be generated by the alternating g-tensor. Theoreticaly, Afflec et.al. [@oshikawa2] have studied how an effective staggered field is generated by Dzyaloshinskii-Moriya (DM) interaction if the crystal symmetry is sufficiently low. They showed that in the presence of DM interaction along the AF chain, an applied uniform field $\overrightarrow{H}$ generates an effective staggered field $\overrightarrow{h}$. Ignoring small residual anisotropies, they obtained an effective hamiltonian where a one-dimensional Heisenberg AF chain is placed in perpendicular uniform ($H$) and staggered ($h$) fields $$\hat{H}= \sum_{j} [J \overrightarrow{S}_{j}. \overrightarrow{S}_{j+1}-H S_{j}^{x}+h (-1)^{j} S_{j}^{z}]\label{Hamiltonian}$$ It is expected [@oshikawa2; @alcaraz1] that the staggered field induces an excitation gap in the $S=\frac{1}{2}$ Heisenberg antiferromagnetic (AF) chain, which should be otherwise gapless. Such as excitation gap caused by the staggered field is indeed found in real magnets [@dender; @kohgi; @feyerherm]. In the absence of the staggered magnetic field ($h=0$) and the uniform magnetic field ($H=0$), the spectrum is gapless. In the ground state, the system is in the spin-fluid phase, where the decay of correlations fallow a power low. When a uniform magnetic field is applied the spectrum of the system remains gapless until a critical field $H_{c}=2 J$, is reached. Here a phase transition of the Pokrovsky-Talapov type [@pokrovsky] occurs and the ground state becomes a completely ordered ferromagnetic state [@griffiths]. Since the uniform magnetic field does not destroy the exact integrability of the Heisenberg model, the eigenspectra is exactly solvable. Applying a staggered magnetic field, the integrability is lost. The application of a staggered magnetic field when $H=0$, produces an antiferromagnetically ordered (Neel order) ground-state and induces a gap in the spectrum of the model. Heisenberg model in both staggered and uniform fields has been recently studied [@lou] using density matrix renormalization group (DMRG). It is shown that bound midgap states generally exist in open boundary AF-Heisenberg chains. The gap and midgap energies in the thermodynamic limit are obtained by extrapolating numerical results of small chain sizes up to 200 sites. It is revealed that some of the gap and midgap energies for the half-integer spin chains fit well to a scaling function derived from the quantum Sine-Gordon model, but other low energy excitations do not fit equally well. In this paper, we present the numerical results obtained on the low-energy states of the 1D AF-Heisenberg model in both uniform and staggered fields using an exact diagonalization technique for finite systems. We calculate the spin gap as a function of applied staggered field in the presence of small uniform field ($0\leq H<0.1$). With respect to the range of the staggered magnetic field, we show that there are two regimes in which we can compute the real critical exponent of the energy gap and it is important to note to which one of these regimes the numerical data are related. In Sec. II we discuss the scaling behavior of the gap using the available limiting behaviors. The leading exponent of the staggered field $h$, depends on $H$ boath in finite size and thermodynamic limit. In Sec. III, we explain how, in certain limits the numerical calculations may produce incorrect result for the critical exponent. We apply a perturbative approach[@langari] to find the correct critical exponent in the small-$x$ ($x=N h^{\nu(H)}$) regime. In Sec. IV, we increase the scaling parameter $x$ and find the correct critical exponent in the large-$x$ regime. Finally, the summary and discussion are presented in Sec. V. The Scaling Behavior of the Gap =============================== In the high field neutron-scattering experiment on Cu benzoate [@dender], which is a quasi-one dimensional $S=\frac{1}{2}$ antiferromagnet, the magnetic field induces a gap in excitation spectrum of the magnet. The observed gap is proportional to $H_{0}^{0.65}$, where $H_{0}$ is the magnitude of the applied field. This exponent of about $\frac{2}{3}$ describing the field dependence of the gap obtained in different experiments [@feyerherm; @kohgi] identify the source of this gap as the staggered field. Using bosonization techniques, Affleck et.al showed that, the gap scales as $$\begin{aligned} \Delta(h, H) \sim h^{\nu(H)}, \label{e5}\end{aligned}$$ where $\nu(H)$ is the critical exponent of the gap and when $H$ is stricly $0$, $\nu(H=0)=\frac{2}{3}$. The $H$-dependence of the exponent $\nu(H)$ is studied numerically in Ref.\[15\]. Their approach is based on the $\eta$-exponent, defined through the static structure factor of the model in the absence of a staggered field ($h=0$). They show that there is a relation between the critical exponent of the gap and $\eta$-exponent. Then by computing the $\eta$-exponent of the structure factor of the model, they predict the $H$-dependence of $\nu(H)$. Similarly, In an interesting recent work [@chernyshev], the effect of an external field on the gap of the 2D AF Heisenberg model with DM interaction has been studied. It is shown that the effect of the external field on the gap can be predicted by investigating the on-site magnetization of the model. Here we study the evolution of the gap, using the conformal estimates of the small perturbation $h\ll1$, and the finite size scaling estimates of the energy eigenvalues of the small chains in the presence of the staggered field ($h\neq0$). We argue that there are two regimes in which the real critical exponent can be numerically calculated and it is very important to note to which one of these regimes the numerical data are related. Let us rewrite the Hamiltonian (1) in the form $$\begin{aligned} \hat{H}&=&\hat{H}_{0}+V \nonumber \\ \hat{H}_{0}&=& \sum_{j} [J \overrightarrow{S}_{j}. \overrightarrow{S}_{j+1}-H S_{j}^{x}] \nonumber \\ V&=&h\sum_{j} (-1)^{j} S_{j}^{z}, \label{efh6}\end{aligned}$$ where $\hat{H}_{0}$ is exactly solvable by the Bethe ansatz and the staggered field $h\ll1$ is very small. For a small perturbation V, we can use conformal estimates. The large distance asymptotic of the correlation function of the model in the absence of the staggered field $(h=0)$ is obtained [@bogoliubovn] as $$\begin{aligned} \langle S_{j}^{z} S_{j+n}^{z}\rangle \sim \frac{(-1)^{n}}{n^{\alpha(H)}}.\end{aligned}$$ Where $\alpha(H)$ is a function of the uniform ($H$) field and is found using the Bethe ansatz as $$\begin{aligned} \alpha(H) \sim 1-\frac{1}{2 \ln(1/H)},~~~~~~~~H \rightarrow 0\end{aligned}$$ where $\alpha(0)=1$ and $\alpha(2)=\frac{1}{2}$. By investigating the perturbed action for the model and performing an infinitesimal renormalization group with a scale $\lambda$, one can show that the staggered magnetic field scales as $h'=h \lambda^{2-\frac{\alpha(H)}{2}}$ which leads that the energy gap scales as Eq.(\[e5\]) by critical exponent $$\begin{aligned} \nu(H)=\frac{2}{4-\alpha(H)}, \label{gafnew}\end{aligned}$$ which is also obtained with the bosonization technique in Ref.\[7\]. For example, in the absence of a uniform magnetic field, $\Delta\sim h^{\frac{2}{3}}$, in agreement with the bosonization and experimental results. Since, by increasing the uniform field $H$, $\alpha(H)$ decreases, thus we can conclude that the critical gap exponent $\nu(H)$ drops with increasing uniform field $H$. To make a numerical check on effect of the uniform field on the energy gap we have implemented the modified Lanczos algorithm [@grosso] on finite-size chains ($N=12, 14, ..., 24$) using periodic boundary conditions to calculate the energy gap. We have computed the energy gap for different chain lengths in the cases of the uniform fields $0\leq H<0.1$. The energy gap as a function of the chain length ($N$), uniform ($H$) and staggered ($h$) fields is defined as $$\begin{aligned} \Delta (N, h, H)=E_{1}(N, h, H)-E_{0}(N, h, H),\end{aligned}$$ where $E_{0}$ is the ground state energy and $E_{1}$ is the first excited state. In the absence of staggered field ($h=0$) , the spectrum of the AF Heisenberg model is gapless up to $H=2 J$. The gap vanishes in the thermodynamic limit proportional to the inverse of the chain length [@fledderjohann2] $$\lim_{N\rightarrow\infty}\Delta(N, h=0, H){\longrightarrow} \frac{A(H)}{N}. \label{gaf}$$ The coefficient A is known exactly from the Bethe ansatz solution [@klumper] and also can be computed in principle by the methods of conformal invariance and finite-size scaling [@alcaraz2; @woynarovich; @alcaraz3]. When the staggered field is applied, a non zero gap develops. Thus the staggered field $h_{c}=0$ is a critical point for our model. In general, the critical point of an infinite system is defined, in the Hamiltonian formulation, as the value of $h$, $h_{c}$, at which the mass gap $\Delta(h, H)$ vanishes as Eq.(\[e5\]). With our Lanczos scheme we can compute $\Delta(N, h, H)$, which approaches $\Delta(h, H)$ when N is large. The natural measure of the deviation of the finite system from the infinite one is $\frac{L}{L_{0}}$, where $L$ is the linear dimension of the finite system ($L=N a$, a is the lattice spacing) and $L_{0}$ is the correlation length of the infinite system ($L_{0}=\xi a$). Thus, we assume that $\Delta(N, h, H)$ depends on $h$ through $\frac{L}{L_{0}}$ as $$\Delta(N, h, H)\sim f(\frac{L}{L_{0}})=f(x),$$ where $x=N h^{\nu(H)}$ is a scaling parameter, and $f(x)$ is the scaling function. As expected, this equation behaves in the combined limit $$N\longrightarrow \infty, ~~~~~h\longrightarrow 0~~~~~( x\gg1)$$ as Eq.($\ref{e5}$), thus we assume the asymptotic form of the scaling function $f(x)$ $$f(x) \sim x^{\phi}. \label{phi}$$ In addition, we need a factor to cancel the N-dependence of $f(x)$ as $N\longrightarrow\infty$. This factor must be of the form $N^{-1}$. Thus, we have $$\Delta(N, h, H)\sim N^{-1} f(x)\label{dgap},$$ If we multiply both sides of Eq.(\[dgap\]) by $N$ we get $$\lim_{N \to \infty (x \gg 1)} N \Delta(N, h, H) \sim x. \label{ngap}$$ Eq.(\[ngap\]) shows that the large-$x$ behavior of $N \Delta(N, h, H)$ is linear in $x$ where the scaling exponent of the energy gap is $\nu(H)$. ![ The value of scaling function $g_{1}(N, H)$ at the fixed uniform field $H=0$, versus the chain length $N=12, 14, ..., 24$. The best fit is obtained by $\nu(H=0)=2.05\pm0.01$. In the inset, the function $g_{1}(N, H)$ is plotted versus N at the uniform field $H=0.05$ and the best fit is obtained by $\nu(H=0.05)=2.33\pm0.01$. Data for different staggered fields $0.001\le h \le 0.005$ fall exactly on each other.[]{data-label="fig1"}](fig1.eps){width="8cm"} Small-x Regime ============== Since the scaling of the gap can only be observed in the thermodynamic limit and for very small value of $h$, we compute the energy gap of the model for several values of staggered field $0.001 \leq h \leq 0.01$ and different chain lengths $N=12, 14, ..., 24$ for fixed uniform fields $0\leq H<0.1$. We have plotted the values of $N \Delta(N, h, H)$ versus $N h^{\nu(H)}$ for $H=0, 0.03, 0.05, 0.07, 0.09$. The results have been computed on a chain with the periodic boundary conditions. According to Eq.(\[ngap\]), we have found from our numerical results that the linear behavior is very well satisfied by $\nu(H)\cong2.0$ independent of H. This is very far from the correct value of critical exponent $\nu(H)\leq\frac{2}{3}$ (Eq.($\ref{gafnew}$)). Note that the horizontal axes values in the small-$x$ regime are limited to very small values of $x=N h^{\nu(H)}<0.0024$. Thus, we are not allowed to obtain the real scaling exponent of the gap which exists in the thermodynamic limit ($N\longrightarrow \infty$ or $x\gg1$). When $x$ is small, or in other words $h$ is very small, we might be away from the thermodynamic behavior to observe the correct scaling. For very small $h$ in the finite size systems ($N\sim24$) the value of $x$ will be small ($x\ll1$), which avoid us to get the information on the large-$x$ behavior of the scaling function $f(x)$. In this case, the values of the energy gap coming from a finite system are basically representing the perturbative behavior [@langari], which we reproduce for the convenience . We start from the Hamiltonian Eq.($\ref{efh6}$). The energy eigenstates of the $\hat{H}_{0}$ carry momentum $p=0$ or $p=\pi$ $$\begin{aligned} T\mid \psi_{n}(h=0, H)\rangle=\pm \mid \psi_{n}(h=0, H)\rangle,\end{aligned}$$ where, T is translation operator and $\{\mid \psi_{n}(h=0, H)\rangle\}$ are eigenstates of unperturbed Hamiltonian $\hat{H}_{0}$. Since the operator $\sum_{j} (-1)^{j}S_{j}^{z}$ changes the momentum of the state by $\pi$, we obtain that $$\begin{aligned} \langle \psi_{n}(0, H) \mid V \mid \psi_{n}(0, H)\rangle=0\end{aligned}$$ Thus, the gap can be rewritten in the following form $$\begin{aligned} \Delta(N, h, H)&=&\Delta(N, 0, H)+g_1(N, H) h^2+ \nonumber \\ &\cdots&+g_n(N, H) h^{2n}, \label{pertexp}\end{aligned}$$ where $n$ is an integer. It is a good approximate to neglect the effect of higher order terms for $h\leq 0.01$. Because the second-order perturbation correction is not zero in the staggered field, the leading nonzero term is $h^{2}$. If the small-$x$ behavior of the scaling function is defined as $f(x)\sim x^{\phi_{s}}$, we find that $$\begin{aligned} \nu(H) \phi_{s}=2 \label{nu}.\end{aligned}$$ This shows that in the small-$x$ regime, $N\Delta(N, h, H)$ is a linear function of $x^{\frac{2}{\nu(H)}}$. This is in agreement with our data in small-$x$ regime, where $\phi_{s}=1$ and according to Eq.($\ref{nu}$), the value of $\nu(H)$ is found to be $\nu(H)=2.0$. Let us consider the large-$N$ behavior of $g_{1}(N, H)$ at fixed $H$ as $$\begin{aligned} \lim_{N\rightarrow\infty} g_1(N, H) \simeq a_1(H) N^{\mu(H)}. \label{e2}\end{aligned}$$ This leads to $$\begin{aligned} \Delta(N, h, H)\simeq \frac{A(H)}{N}(1+b_{1}(H) N^{\mu(H)+1} h^{2}),\label{e24}\end{aligned}$$ where $b_{1}(H)$ is a constant (at fixed $H$). We can write Eq. ($\ref{e24}$) in terms of the scaling variable $x$ $$\begin{aligned} N \Delta(N, h, H)\simeq A(H)(1+N^{\mu(H)+1-\frac{2}{\nu(H)}} x^{\frac{2}{\nu(H)}})\end{aligned}$$ For large-$N$ limit this equation should be independent of $N$, leading to the relation between $\mu(H)$ and $\nu(H)$ as $$\begin{aligned} \nu(H)=\frac{2}{\mu(H)+1}\label{s26}\end{aligned}$$ The above arguments propose to look for the large-$N$ behavior of $g_{1}(N, H)$. To determine the $\mu(H)$ exponent, we have plotted in Fig.1 the following expression versus $N$ $$\begin{aligned} g_{1}(N, H)\simeq\frac{\Delta(N, h, H)-\Delta(N, 0, H)}{h^{2}}\end{aligned}$$ for fixed values of staggered field $h$ ($0.001\leq h \leq 0.005$), and different sizes, $N=12, 14, ..., 24$ at the uniform field $H=0$. We found the best fit to our data for $\mu(H=0)=2.04\pm0.01$. The inset in Fig.1 shows the $g_{1}(N, H)$ versus $N$ at fixed $H=0.05$. In this case, the best fit, found for $\mu(H=0.5)=2.33\pm0.01$. Our data for different $h$ values, fall perfectly on each other, which shows that our results for $g_{1}(N, H)$ in fixed uniform field $H$, are independent of the staggered field $h$ as expected. By using Eq.($\ref{s26}$) we have found, $\nu(H=0)=0.66\pm0.01$ and $\nu(H=0.05)=0.60\pm0.01$. We have also implemented our numerical tool to calculate the critical exponent $\nu(H)$ at $H=0.03, 0.07, 0.09$. The results have been presented in Table I. ![ Difference between the two lowest energy levels and the ground state energy as a function of the staggered magnetic field for finite chain length $N=24$ and $H=0$ in the region $0.01\leq h \leq 0.4$.[]{data-label="fig2"}](fig2.eps){width="8cm"} ![ The product of energy gap and chain length ($N \Delta$) versus $N h^{\nu(H)}$ at the uniform field $H=0$. In the range of staggered field $0.04 \leq h \leq 0.4$, linear behavior is obtained by choosing $\nu(H=0)=0.66$ for all different chain lengths $N=18, 20, 22, 24$. In the inset, $N \Delta$ is plotted for uniform field $H=0.05$. The linear behavior is obtained by $\nu(H=0.05)=0.63$. Data for different chain lengths fall on each other.[]{data-label="fig3"}](fig3.eps){width="8cm"} ![ A graph of the critical exponent $\nu$ versus uniform field $H$. Both critical exponents $\nu_{s}$ (squares) and $\nu_{l}$ (circles) start at the known value $\frac{2}{3}$ and then drop with the increasing uniform field $H$. In the inset, we have plotted the critical exponent $\nu$ which is obtained directly by extrapolating the numerical results of the energy gap in the range of the staggered field $0.001\leq h \leq 0.01$. []{data-label="fig4"}](fig4.eps){width="8cm"} \[table2\] [cccc]{} $H$                   & $\mu$                  & $\nu_{S}$                  & $\nu_{L}$\ $0.0$                   & $2.05$                  &$0.65$                  &$0.66$\ $0.03$                   &$2.20$                  &$0.63$                  &$0.64$\ $0.05$                   &$2.33$                  &$0.60$                  &$0.63$\ $0.07$                   &$2.58$                  &$0.56$                  &$0.62$\ $0.09$                   &$2.80$                  &$0.53$                  &$0.60$\ large-x regime ============== In our numerical calculations because of memory issues, we were limited to consider the maximum chain length $N=24$. Therefore the value of $x$ cannot be increased by increasing the size of chain. The problem appears if the calculation is done by density matrix renormalization group (DMRG) method. In that case, we may extend the calculation to larger sizes, $N\sim200$, which cannot increase $x$ much larger than one (for $0.001<h<0.01$). On the other hand we may increase the staggered field for increasing $x$. But we should note that, in general there is usually level crossing between the energy levels in finite size systems. Which can change the behavior of the gap [@dmitriev] and lead to incommensurate effects. As an example, the dependences on the excitation energies of the three lowest levels on the staggered field $h$ ($0.01\leq h \leq0.4$) are shown in Fig.2 for $N=24$ and $H=0$. From this figure, it can be seen that the two lowest excited states do not cross each other in this case. This means that we can increase the value of $x$ by increasing $h$ up to $h=0.4$. Since that the regime where we can observe the right scaling of the gap is in very small value of $h$, we have restricted our numerical computations to upper limit of staggered field $h=0.4$. In this respect, we have performed some numerical computations on the Hamiltonian Eq.($\ref{Hamiltonian}$) for large-$x$ values, and the results are plotted in Fig.3. In this figure the range of the staggered field is $0.04\leq h \leq 0.4$, which causes that we get large-$x$ values for the considered chain lengths $N=18, 20, 22, 24$ and $ H=0$. The inset in Fig.3 shows $N\Delta(N, h, H)$ versus $x=N h^{\nu(H)}$ at fixed uniform field $H=0.05$. In this case, we have obtained, $\nu(H=0)=0.66$ and $\nu(H=0.05)=0.63$. It should be mentioned that if we choose another value for $\nu$ the plot will not be linear in $x$. Also, our data for different size chains fall perfectly on each other, which is expected from the scaling behavior. We have extended our numerical computations to consider values of the uniform field in the small region $0\leq H <0.1$. The results have been presented in Table I. We have listed $\mu$, the resulting $\nu_{s}$ that is obtained from perturbative approach, and result of the large-$x$ regime $\nu_{l}$ for different values of the uniform field $H$. In Fig.4 we have plotted the critical exponent $\nu(H)$ versus the uniform field $H$. As it is clearly seen from this figure, $\nu_{s}$ and $\nu_{l}$ start at the known value $\frac{2}{3}$ and then drop with the increasing uniform field $H$. The exponents are in good agreement with each other and show good agreement with the exponents derived in the field theoretical approach (Eq.($\ref{gafnew}$)). The inset in Fig.4 shows the H-dependence of the critical exponent $\nu$ which is obtained directly by extrapolating the numerical results of the energy gap in the range of the staggered field $0.001\leq h \leq 0.01$ . It is clearly seen from this figure that the behavior of the gap in finite systems is different from its behavior at the thermodynamic limit, and with respect to the range of the staggered magnetic field the behavior of the gap deviates from the predicted scaling behavior. On the other hand, Fouet et.al studied [@fouet] the gap-induced by the staggered field $h$ at the saturation uniform field $H_{c}=2 J$. Using field theoretical arguments, they found that the gap scales as $\Delta(h, H_{c})\sim h^{4/5}$. By applying the DMRG method for system sizes up to $N=100$, they have also computed the exponent of the energy gap $0.81$. We have extended our numerical computations to consider the uniform field at the saturation value $H_{c}$. In this case, we have obtained, $\nu_{s}(H_{c})=0.78$ and $\nu_{l}(H_{c})=0.82$ in good agreement with the Foet results. conclusions =========== In this paper, we have studied the energy gap of the 1D AF-Heisenberg model in the presence of both uniform ($H$) and staggered ($h$) magnetic fields using the exact diagonalization technique. We have implemented the modified Lanczos method to obtain the excited state energies with the same accuracy as the ground state one. We have been limited to a maximum of $N = 24$, because of memory considerations. We have shown, if the energy gap in the thermodynamic limit had been obtained by extrapolating the numerical results for finite systems, then the behavior of the gap may deviate from the predicted scaling behavior. This deviation depends on the range of the staggered magnetic field ($h$). We have found in the range of very small values of the staggered magnetic field ($0.001\leq h \leq 0.01$), the values of the energy gap coming from a finite system basically represent the perturbative behavior. We have shown that in this range of the staggered magnetic field, we are not allowed to read the scaling exponent of the energy gap directly from the extrapolated numerical results. We have applied a general finite size scaling procedure for investigating the $H$-dependence of the critical exponent of the gap. We have identified two regimes through which the real critical exponent $\nu(H)$ can be numerically calculated. To find the correct exponent of the gap in small-$x$ regime ($x=N h^{\nu(H)}\ll1$), we have used the scaling behavior of the coefficient of the leading term in the perturbation expansion, which is introduced by authors in Ref.\[25\]. In the large-$x$ regime using the standard finite size scaling Eq.(\[ngap\]), we have computed the correct critical exponent. On the other hand, using the conformal estimates of the small perturbation ($h\ll1$), we have found the $H$-dependence of the critical exponent (Eq.(\[gafnew\])) from the theoretical point of view. Our numerical results in both regimes are in well agreement with the results obtained by the theoretical and numerical approaches. Acknowledgments =============== I would like to thank A. Langari, M. R. H. Khajehpour, G. I. Japaridze, J. Abouie and M. Kohandel for insightful comments and fruitful discussions that led to an improvement of this work. [99]{} D. C. Dender, P. R. Hammar, D. H. Reich, C. Broholm and G. Aeppli, Phys. Rev. Lett. [**79**]{}, 1750 (1997). T. Asano, H. Nojiri, Y. Inagaki, J. P. Boucher, T. Sakon, Y. Ajiro and M. Motokawa, Phys. Rev. Lett. [**84**]{}, 5880 (2000). T. Asano, H. Nojiri, W. Hiegemoto, A. Koda, R. Kadono and Y. Ajiro, J. Phys. Soc. Jpn. [**72**]{}, 594 (2002). M. Oshikawa, K. Ueda, H. Aoki, A. Ochiai and M. Kohgi, J. Phys. Soc. Jpn. [**68**]{}, 3181 (1999). H. Shiba, K. Ueda and O. Sakai, J. Phys. Soc. Jpn. [**69**]{}, 1493 (2000). M. Kohgi, K. Iwasa, J. M. Mignot, B. Fak, P. Gegenwart, M. Lang, A. Ochiai, H. Aoki and T. Suzuki, Phys. Rev. Lett. [**86**]{}, 2439 (2001). M. Oshikawa and I. Affleck, Phys. Rev. Lett. [**79**]{}, 2883 (1997); Phys. Rev. [**B 60**]{}, 1038 (1999). Y. -J. Wang, F. H. L. Essler, M. Fabrizio and A. A. Nersesyan, Phys. Rev. [**B 66**]{}, 024412 (2002). M. Sato and M. Oshikawa, Phys. Rev. [**B 69**]{}, 054406 (2004) F. C. Alcaraz and A. L. Malvazzi, J. Phys. [**A 28**]{}, 1521 (1995). R. Feyerherm, S. Abens, D. Gunther, T. Ishida, M. Meibner, M. Meschke, T. Nogami and M. Steiner, J. Phys. Condens. Matter [**12**]{}, 8495 (200). V. L. Pokrovsky and A. L. Talapov, Soc. Phys. JETP [**51**]{}, 134 (1980). R. B. Griffiths, Phys. Rev. [**133**]{}, A768 (1964). J. Lou, C. Chen, J. Zhao, X. Wang, T. Xiang, Z. Su and L. Yu, Phys. Rev. Lett. [**94**]{}, 217207 (2005). A. Fledderjohann, M. Karbach and K.-H. Mutter, Eur. Phys. J. [**B 7**]{}, 225 (1999). A. L. Chernyshev, Phys. Rev. [**B 72**]{}, 174414 (2005). N. M. Bogoliubov, A. G. Izergin and V. E. Korepin, Nucl. Phys. [**B 275**]{}, 687 (1986). G. Grosso and L. Martinelli, Phys. Rev. [**B 51**]{}, 13033 (1995). A. Fledderjohann, M. Karbach and K.-H. Mutter, Eur. Phys. J. [**B 5**]{}, 487 (1998). A. Klumper, Eur. Phys. J. [**B 5**]{}, 677 (1998), and references therein. F. C. Alcaraz, M. N. Barber, M. T. Batchelor, Phys. Rev. Lett. [**58**]{}, 771 (1987). F. Woynarovich, H. -P. Eckle, J. Phys. [**A** ]{}: Math. Gen. 20, [**L97**]{} (1987). F. C. Alcaraz, M. N. Barber, M. T. Batchelor, Ann. Phys. [**182**]{}, 280 (1988). D. V. Dmitriev, V. Ya. Krivnov, A. A. Ovchinikov, and A. Langari, JETP [**95**]{}, 538 (2002). A. Langari and S. Mahdavifar, Phys. Rev. [**B 73**]{}, 54410 (2006). J. -B. Fouet, O. Tchernyshyov, and F. Mila, Phys. Rev. [**B 70**]{}, 174427 (2004).

--- abstract: 'We prove the theorem: The second order quasi-linear differential operator as a second rank divergence free tensor in the equation of motion for gravitation could always be derived from the trace of the Bianchi derivative of the fourth rank tensor, which is a homogeneous polynomial in curvatures. The existence of such a tensor for each term in the polynomial Lagrangian is a new characterization of the Lovelock gravity.' author: - 'Naresh Dadhich[^1]' title: Characterization of the Lovelock gravity by Bianchi derivative --- The Bianchi differential identity involving the purely anti-symmetric derivative of a derivative ($D^2=0$) was famously interpreted by John Wheeler [@mtw] as the statement of the fact, [*boundary of boundary is zero*]{}. The familiar examples of it are curl of gradient and divergence of curl being zero. The former signifies a scalar while the latter a vector field. However its contraction in the two cases is vacuous and hence does not lead to a non-trivial statement$^{[1]}$. When we go beyond vector to a tensor field, it becomes interesting.\ Gravity is the universal force which means it links to all particles unmindful of their mass being non-zero or zero. It is its linkage to massless particles which leads to the profound realization that it could only be described by curvature of spacetime [@nd1; @nd2]. This means the dynamics of gravity resides in spacetime curvature which must fully and entirely determine it. That is the gravitational dynamics must follow from the curvature which is described by the fourth rank Riemann curvature tensor (it is defined as $A^i_{;lk} - A^i_{;kl} = R^i{}_{mlk}A^m$, a generalized “curl”). It involves second and first derivatives of the spacetime metric, $g_{ab}$. The Bianchi identity is given by the anti-symmetric derivative of $R^i{}_{mlk}$, $$R^i{}_{m[lk;n]} = 0 .$$ If the gravitational dynamics has to follow from the curvature, it has to follow from this identity which is the only available geometric relation. The only thing we can do to it is to contract on the available indices which does lead, unlike for scalar and vector, to a non-vacuous relation, $$G^{a}{}_{b;a} = 0, ~~~~ G_{ab} = R_{ab} - {1\over2}Rg_{ab}$$ where $R_{ab}$ is the Ricci tensor, the contraction of Riemann, while $R$ is the trace of Ricci. Now the trace (contraction) of the Bianchi identity yields a non-trivial differential identity from which we can deduce the following relation $$G_{ab} = \kappa T_{ab} - \Lambda g_{ab}, ~~~ T^{a}{}_{b;a} = 0$$\ where $T_{ab}$ is the second rank symmetric tensor with vanishing divergence and $\kappa$ and $\Lambda$ are constants. The left hand side of the equation is a second order differential operator on the metric $g_{ab}$. For this equation to describe dynamics of gravity, the tensor $T_{ab}$ should describe the source/charge for gravity which should also be universal. It should be shared by all particles and hence $T_{ab}$ should represent energy momentum distribution. Thus we obtain the Einstein equation for gravitation which entirely follows from the spacetime curvature. We have however two constants of which one $\kappa$ is to be determined by experimentally measuring the strength of the force and is identified with Newton’s constant, $\kappa = -8 \pi G/c^2$. Why is there new constant $\Lambda$ which though arises in the equation as naturally as the energy momentum tensor, $T_{ab}$? It is perhaps because of the absence of fixed spacetime background which exists for the rest of physics and the new constant may be a signature of this fact. It is the universal character of gravity which makes spacetime itself dynamic. The force free state would however be characterized by homogeneity and isotropy of space and homogeneity of time which will in general be described by spacetime of constant curvature and not necessarily of zero curvature. The new constant $\Lambda$ is the measure of the constant curvature of spacetime and it identifies the most general spacetime for force free state. It may in some deep and fundamental sense be related to the basic structure of spacetime.\ We also know that the complete contraction of Riemann gives the scalar curvature, $R$, the Einstein-Hilbert Lagrangian which on variation leads to the divergence free Einstein tensor, $G_{ab}$, and subsequently the Einstein equation. We thus have the two distinct but equivalent derivations for the gravitational dynamics. The former is simply driven by the geometry while the latter is in the spirit of every dynamical equation following from an action. It is always possible to write an action constructed from Riemann curvature for higher derivative gravity and derive the corresponding equation of motion. Similarly, is it possible to derive an analogue of $G_{ab}$, a divergence free differential operator from the Bianchi derivative of the higher order curvature polynomial? This is the question we wish to address and show that the answer is in affirmative. It would give yet another characterization of the Lovelock gravity.\ We believe that gravitational dynamics to follow from the spacetime curvature should be a general principle which should be true in general for higher order theories as well. So comes the question of going beyond the linear order in Riemann. Let us consider the quadratic tensor, $${\mbox{$\mathcal{R}$}}_{abcd} = R_{abmn}R_{cd}{}^{mn} + \alpha R_[{a}{}^{m}R_{b]mcd} + \beta R R_{abcd} \label{gb1}$$ where $\alpha, \beta$ are constants. We consider the Bianchi derivative, ${\mbox{$\mathcal{R}$}}_{ab[cd;e]}$ which on contraction gives $$g^{ac} g^{bd} {\mbox{$\mathcal{R}$}}_{ab[cd;e]} = (-2{\mbox{$\mathcal{R}$}}_e{}^c + {\mbox{$\mathcal{R}$}} \delta_e{}^c)_{;c} \label{bd1}$$ where ${\mbox{$\mathcal{R}$}}_{ac} = g^{bd} {\mbox{$\mathcal{R}$}}_{abcd}$ and $ {\mbox{$\mathcal{R}$}} = g^{ab} {\mbox{$\mathcal{R}$}}_{ab}$. It turns out that for $\alpha=4$ and $\beta=1$, we obtain $$\begin{aligned} {\mbox{$\mathcal{R}$}}^{cd}{}{}_{[cd;e]}&=& (-2{\mbox{$\mathcal{R}$}}_e{}^c + {\mbox{$\mathcal{R}$}} \delta_e{}^c)_{;c} \nonumber \\ &=& (-H_e{}^c + {1\over2} L_{GB} \delta_e{}^c)_{;c}. \end{aligned}$$ That is $${\mbox{$\mathcal{R}$}}^{cd}{}{}_{[cd;e]} - {1\over2} (L_{GB} \delta_e{}^c)_{;c} = -H_e{}^c_{;c} = 0.\label{bd2}$$ The tensor $H_{ab}$ is divergence free, $H_a{}^b_{;b} = 0$, and is given by $$\begin{aligned} H_{ab}& =& 2(RR_{ab} - 2R_{a}{}^{m}R_{bm} - 2R^{mn}R_{ambn}\nonumber \\ &+& R_{a}{}^{mnl}R_{bmnl}) - \frac{1}{2} L_{GB} g_{ab}.\end{aligned}$$ It results from the variation of the well-known Gauss-Bonnet Lagrangian $L_{GB} = R_{abcd}^2 - 4R_{ab}^2 + R^2$ where we have written $R_{abcd}^2 = R_{abcd}R^{abcd}$. That is, we can write $$H_{ab} = 2{\mbox{$\mathcal{R}$}}_{ab} - \frac{1}{2}{\mbox{$\mathcal{R}$}} g_{ab} \label{hab}$$ where ${\mbox{$\mathcal{R}$}} = L_{GB}$.\ Though $H_{ab}$ can be written in terms of ${\mbox{$\mathcal{R}$}}_{abcd}$ but it doesn’t follow directly from it as $G_{ab}$ does from $R_{abcd}$. We note that Bianchi derivative vanishes only for the Riemann curvature signifying the fact it can be written as a generalized curl of a vector. No other tensor will have vanishing Bianchi derivative. However to find the analogue of $G_{ab}$, we donot require vanishing of Bianchi derivative of the quadratic tensor ${\mbox{$\mathcal{R}$}}_{abcd}$ but instead vanishing of its trace would suffice. We see that even the trace of Bianchi derivative does not vanish but is equal to ${1\over2} (L_{GB} \delta_e{}^c)_{;c}$. It suggests that the curvature polynomial should also include term involving its own trace which would make no contribution in the linear case. Before we do that, let us write ${\mbox{$\mathcal{R}$}}_{abcd}$ and $H_{ab}$ for a general order $n$ in the Lovelock polynomial  [@lov]. In the context of higher derivative gravity theories, a unified scheme of writing Lagrangian is given  [@paddy] as an invariant product, $Q^{abcd}R_{abcd}$ where $Q^{abcd}$ has the same symmetry properties as the Riemann tensor. It is constructed from metric and Riemann curvature and has vanishing divergence, $Q^{abcd}{}{}{}{}_{;c}=0$. For the Lovelock Lagrangian, it is required to be a homogeneous function of the Riemann curvature. Since ${\mbox{$\mathcal{R}$}}_{abcd}$ is also a homogeneous polynomial in Riemann curvature for a given order, it is hence possible to write $${\mbox{$\mathcal{R}$}}_{abcd} = Q_{ab}{}^{mn}R_{mncd} \label{new}$$ where $Q_{abcd}$ is in general given for $n$-th order by  [@paddy], $$Q^{ab}{}_{cd} = \delta^{aba_1b_1...a_nb_n}_{cdc_1d_1...c_nd_n}R_{a_1b_1}{}^{c_1d_1}...R_{a_nb_n}{}^{c_nd_n}. \label{new1}$$ Let us verify this for the quadratic Gauss-Bonnet case where $Q_{abcd}$ could explicitly be written as  [@paddy], $$Q_{abcd} = R_{abcd} - 2R_{a[c}g_{d]b} + 2R_{b[c}g_{d]a} + Rg_{a[c}g_{d]b}.$$ It is easy to see that when it is substituted in Eqn. (\[new\]), we obtain the Gauss-Bonnet ${\mbox{$\mathcal{R}$}}_{abcd}$ as given in Eqn. (\[gb1\]) with $\alpha=4, \beta=1$. We could thus write ${\mbox{$\mathcal{R}$}}_{abcd}$ by using Eqs (\[new1\]) and (\[new\]) for any term in the Lovelock polynomial and the corresponding $H_{ab}$ for the $n$-th polynomial will be given by $$H_{ab} = n{\mbox{$\mathcal{R}$}}_{ab} - \frac{1}{2} {\mbox{$\mathcal{R}$}} g_{ab} \label{genH}$$ which will for the linear case $n=1$ be the Einstein tensor, $G_{ab}$. In this case, the trace of the Bianchi derivative vanishes because the Riemann tensor satisfies the Bianchi identity. However $H_{ab}$ as defined above is the analogue of the Einstein tensor for $n$-th order polynomial and hence the generalized Einstein tensor is divergence free. As trace of $R_{abcd}$ is the Einstein-Hilbert Lagrangian similarly trace of ${\mbox{$\mathcal{R}$}}_{abcd}$ gives the Gauss-Bonnet and higher order Lovelock Lagrangian. Note that Eqn. (\[bd1\]) holds good in general for any order $n$ of the polynomial ${\mbox{$\mathcal{R}$}}_{abcd}$. We can thus write in general for Eqn. (\[bd2\]), $${\mbox{$\mathcal{R}$}}^{cd}{}{}_{[cd;e]} - \frac{n-1}{n}({\mbox{$\mathcal{R}$}} \delta_e{}^c)_{;c} = -\frac{2}{n}{}{} H_e{}^c_{;c} = 0 \label{bd3}$$ where ${\mbox{$\mathcal{R}$}}$ is the corresponding $n$-th order Lagrangian. For $n=1$, the second term on the left vanishes indicating the Bianchi identity and $H_{ab}=G_{ab}$. For $n>1$, the trace of Bianchi derivative does not vanish unless the trace ${\mbox{$\mathcal{R}$}}$ is also included as indicated by the second term which then leads to the required divergence free tensor, $H_{ab}$. Apart from the quadratic Gauss-Bonnet case, we have also verified the above relation for $n=3$ cubic polynomial. It is interesting that the order of polynomial appears in the definition of the corresponding curvature polynomial tensor yielding the second rank symmetric divergence free tensor.\ We now turn to redefining the curvature polynomial which also includes its trace such that trace of the Bianchi derivative vanishes directly giving the required differential operator in terms of divergence free $H_{ab}$. We thus write $${\mbox{$\mathcal{F}$}}_{abcd} = {\mbox{$\mathcal{R}$}}_{abcd} - \frac{n-1}{n(d-1)(d-2)} {\mbox{$\mathcal{R}$}} (g_{ac}g_{bd} - g_{ad}g_{bc}) \label{F}$$ where $n$ being the order of the polynomial and $d$ the dimension of spacetime under consideration. Then $$-\frac{n}{2} {\mbox{$\mathcal{F}$}}^{cd}{}{}_{[cd;e]} = H_e{}^c_{;c} = 0$$ which verifies Eqn. (\[genH\]). In terms of ${\mbox{$\mathcal{F}$}}_{abcd}$, $H_{ab}$ is given by $$n({\mbox{$\mathcal{F}$}}_{ab} -\frac{1}{2} {\mbox{$\mathcal{F}$}} g_{ab}) = H_{ab}$$ where ${\mbox{$\mathcal{F}$}}_{ab} = g^{cd} {\mbox{$\mathcal{F}$}}_{acbd}$ and ${\mbox{$\mathcal{F}$}} = g^{ab} {\mbox{$\mathcal{F}$}}_{ab}$. It is interesting to note that ${\mbox{$\mathcal{F}$}}_{ab}$ is the analogue of the Ricci, $R_{ab}$ and $H_{ab}$ of the Einstein, $G_{ab}$ with the order $n$ of the polynomial ${\mbox{$\mathcal{R}$}}_{abcd}$ which is the analogue of the Riemann, $R_{abcd}$. Thus $H_{ab}$ is obtained from ${\mbox{$\mathcal{F}$}}_{abcd}$ in the same manner as $G_{ab}$ is from $R_{abcd}$, and for $n=1$, ${\mbox{$\mathcal{F}$}}_{abcd} = {\mbox{$\mathcal{R}$}}_{abcd} = R_{abcd}$. All this however happens only for the specific Lovelock coefficients in the polynomial and thereby determining the Lovelock polynomial.\ Note that $H_{ab}$ defines the general conserved (Einstein) tensor for any order $n$ of the homogeneous curvature polynomial and let us take its trace which would give $${\mbox{$\mathcal{F}$}} = \frac{d-2n}{n(d-2)} {\mbox{$\mathcal{R}$}}.$$ For the usual Einstein gravity, $n=1$, it is ${\mbox{$\mathcal{F}$}} = {\mbox{$\mathcal{R}$}}$ with $d>2$ and the Einstein tensor vanishes for $d=2$. Thus we have the general result that $d>2n$ for $n$-th order polynomial and $H_{ab}$ vanishes for $d=2n$. It is the variation of ${\mbox{$\mathcal{R}$}}$ (and so also ${\mbox{$\mathcal{F}$}}$) gives $H_{ab}$ which vanishes for $d=2n$ but not ${\mbox{$\mathcal{R}$}}$. If we take ${\mbox{$\mathcal{F}$}}$ which is proportional to the Lovelock Lagrangian ${\mbox{$\mathcal{R}$}}$, then the Lagrangian itself vanishes for $d=2n$. It is remarkable that the existence of an analogue of the Einstein tensor in general - a conserved tensor which is non-zero only for $d>2n$, uniquely determines the Lovelock polynomial.\ By redefining the curvature polynomial we have been able to derive the divergence free differential operator for the gravitational dynamics. This is what we had set out to do and it could be stated as follows:\ [**Theorem:**]{} [*The second order quasi-linear differential operator as a second rank divergence free tensor in the equation of motion for gravitation could always be derived from the trace of the Bianchi derivative of the fourth rank tensor, ${\mbox{$\mathcal{F}$}}_{abcd}$, which is a homogeneous polynomial in curvatures. The trace of the curvature polynomial is proportional to the corresponding term in the Lovelock action and corresponding to each term in the Lovelock Lagrangian, there exists a fourth rank tensor which is a new characterization of the Lovelock Lagrangian.* ]{}\ It is the requirement of quasi-linearity (linear in second derivative) of the equation of motion which singled out the Lovelock polynomial with specific coefficients. In our case it is replaced by the requirement that the fourth rank tensor which is homogeneous in curvatures yields a divergence free second rank tensor through the trace of its Bianchi derivative. This automatically ensures the quasi-linearity of the equation. For non Lovelock action, there would not exist a fourth rank tensor with this property.\ In a sense, the Riemann curvature could be looked upon as a [*Bianchi potential*]{} giving the Einstein tensor for the Einstein gravity while ${\mbox{$\mathcal{F}$}}_{abcd}$ is the [*Bianchi potential*]{} giving $H_{ab}$ for the Lovelock dynamics. Thus each term in the Lovelock polynomial has a potential tensor ${\mbox{$\mathcal{F}$}}_{abcd}$ given by Eqs (\[new\]), (\[new1\]) and (\[F\]). It is remarkable that there exists [*Bianchi potential*]{} for each term in the Lovelock Lagrangian. Non existence of potential characterizes all other actions like $f(R)$ gravity. The existence of [*Bianchi potential*]{} thus becomes a distinguishing feature for the Lovelock action. In this context it may be noted that the requirement that both Palitini and metric action give the same equation of motion also picks up the Lovelock action  [@jab]. We have thus three distinct properties (namely quasi-linearity of equation of motion, equivalence of Palitini and metric formulation and existence of Bianchi potential) which characterize the Lovelock action. I believe that there should exist a thread knitting them and it would be interesting to probe that.\ However, the relevance of the order in the Lagrangian for the gravitational dynamics depends upon the spacetime dimension. For instance, for $d<5$, the quadratic Gauss-Bonnet term makes no contribution to the equation of motion and similarly the cubic term becomes relevant only for $d>6$. We would however like to emphasize that for $d>4$, $H_{ab}$ must be included along with $G_{ab}$ for the classical dynamics of gravitation. In the ultra violet limt of the theory signifying super strong gravitational field, it is pertinent to include higher order curvature effects. If we continue to have a well defined evolution of the field, the equation must be quasi-linear. This will inevitably and uniquely lead to Lovelock polynomial and higher dimension. Further one loop correction in string theory generates the Gauss-Bonnet term [@st]. That is, strong field limit of classical gravity and one loop quantum correction seem to share the same Gauss-Bonnet ground. It could therefore be envisioned that the classical limit to quantum gravity is perhaps via the Lovelock gravity and the relevant order in the polynomial (loop correction) being given by the spacetime dimension under consideration. The Gauss-Bonnet and higher orders may therefore represent a intermediary state between the classical Einstein gravity and quantum gravity [@dad1].\ Gravity is an inherently self interactive force and the self interaction could only be evaluated by successive iterations. The Einstein gravity is self interactive but it contains only the first iteration through the square of first derivative in Riemann curvature. The question is how do we stop at the first iteration? The second iteration would ask for a quadratic polynomial in Riemann curvature which should give the corresponding term in the equation of motion. Thus the quadratic tensor, ${\mbox{$\mathcal{R}$}}_{abcd}$ as given in Eqn. (\[gb1\]) with specific coefficients, will alone meet the requirement (quasi-linearity of the differential operator) for inclusion of the second iteration. However its effect in the equation of motion can be felt only for dimension, $d>4$, and hence we have to go to higher dimension for physical realization of the second iteration of self interaction [@dad1; @dad2]. It is remarkable that even classical dynamics of gravity asks for dimension $>4$. As two and three dimensions were not big enough for free propagation of gravity, similarly four dimension is not big enough to fully accommodate self interaction dynamics of gravity. Then the most pertinent question is where does this chain end? Let us envision the AdS/CFT-like scenario where the $3$-brane ($(3+1)$-spacetime) forms the boundary enclosing the higher dimensional bulk spacetime. If matter fields remain confined to $3$-brane, the bulk would then be free of matter and hence it would be maximally symmetric (homogeneous and isotropic in space and homogeneous in time). It is then a spacetime of constant curvature, dS/AdS, with vanishing Weyl curvature. There is no free gravity to propagate any further and hence the chain stops at the second level at least in this particular construction. Whether this is a generic setting or not is however an open question? It may be noted here that gravitational equation with inclusion of both $G_{ab}$ and $H_{ab}$ for empty space in higher dimension admits dS/AdS as solution [@dad1].\ In what dimension should matter live is to be entirely determined by the matter dynamics. If the matter fields are gauge vector fields described by $2$-form, the conformal invariance (universal scale change, $g_{ab}\to f^2 g_{ab}$) of their dynamics will dictate that they live in four dimension. This general principle is always obeyed by the matter field dynamics unless a scale in terms of mass etc. is introduced by spontaneous symmetry breaking. The symmetry breaking is indicative of theory being incomplete and it is hoped that the complete theory would restore the conformal invariance. It is therefore reasonable to take that matter remains confined to $3$-brane. This view is also supported by the string theory where open strings have their endpoints on the brane indicating residence of matter there.\ The most fundamental question is how do we know there exist higher dimensions and if so why do we not see them? The existence of dimension can only be probed by a physical interaction. All our probes are matter field forces like electromagnetic, which remain, as argued above, confined to $3$-brane and hence they cannot fathom higher dimension. Since gravity is universal and hence it cannot be confined entirely to the brane and can propagate in higher dimension. Above we have argued quite convincingly that there is a strong case for higher dimension for physical realization of the second iteration of self interaction. It is a purely classical motivation (we have also elsewhere [@dad1; @dad2] given a couple of more classical arguments) while higher dimension is a natural arena for string theory. The only way higher dimension could thus be probed is by a very high energy pure gravitational experiment. This is what we have not yet been able to do and hence the question remains open.\ We had set out to establish the general principle that gravitational dynamics resides in spacetime curvature and hence it should always and entirely be driven by spacetime geometry. This we have shown by deriving the quasi-linear differential operator for the equation of motion for the Lovelock gravity. In particular, we have found a new geometric characterization of the Lovelock gravity in existence of [*Bianchi potential*]{} for each term in the polynomial. This is indeed an interesting general property.\ [*Acknowledgment:*]{} I wish to thank T Padmanabhan and Dawood Kothawala for useful discussion and the latter also for Eqn. (\[genH\]). [9]{} C. W. Misner, K. S. Thorne and J. A. Wheeler (1973) Gravitation (Freeman). N. Dadhich, [*Subtle is the Gravity*]{}, gr-qc/0102009. N.Dadhich, [*Universalization as a Physical guiding Principle*]{}, gr-qc/0311028. N. Dadhich (2007) [*On the Gauss Bonnet Gravity*]{}, Mathematical Physics, (Proceedings of the 12th Regional Conference on Mathematical Physics), Eds., M.J. Islam, F. Hussain, A. Qadir, Riazuddin and Hamid Saleem, World Scientific, 331 (hep-th/0509126). N. Dadhich (2007) [*Formation of a Black Hole from an AdS Spacetime*]{}, Theoretical High Energy Physics (Proceedings of the International Workshop on Theoretical High Energy Physics), Ed. A. Misra, American Institute of Physics, 101 (arXiv:0705.2490). D. Lovelock (1971) Jour. Math. Phys. [**12**]{}, 498. A. Mukhopadhyay and T. Padmanabhan (2006) Phys. Rev. [**D74**]{}, 124023. O.  Exirifard and M. M.  Sheikh-Jabbari (2008) Phys. Lett. [**B661**]{}, 158. B. Zwiebach (1985) Phys. Lett. [**B156**]{}, 315. [^1]: Electronic address:nkd@iucaa.ernet.in

--- abstract: 'We report a femtosecond response in photoinduced magnetization rotation in the ferromagnetic semiconductor GaMnAs, which allows for detection of a four-state magnetic memory at the femtosecond time scale. The temporal profile of this cooperative magnetization rotation exhibits a discontinuity that reveals two distinct temporal regimes, marked by the transition from a highly non-equilibrium, carrier-mediated regime within the first 200 fs, to a thermal, lattice-heating picosecond regime.' author: - 'J. Wang' - 'I. Cotoros' - 'X. Liu' - 'J. Chovan' - 'J. K. Furdyna' - 'I. E. Perakis' - 'D. S. Chemla' title: | Memory Effect in the Photoinduced Femtosecond Rotation of Magnetization\ in the Ferromagnetic Semiconductor GaMnAs --- ![(Color online) Static magnetic memory. (a)-(b): Sweeping a slightly tilted [*B*]{} field (5$^o$ from the [*Z*]{}-axis and 33$^o$ from the [*X*]{}-axis) up (dashed line) and down (solid line) leads to consecutive 90$^o$ magnetization switchings between the [*XZ*]{} and [*YZ*]{} planes, manifesting as a “major" hysteresis loop in the Hall magneto-resistivity. (c)-(d): “Minor" hysteresis loop with [*B*]{} field sweeping in the vicinity of 0T. The magnetic memory state X$-$(0) or Y$+$(0) is parallel to one of the easy axis directions in the [*XY*]{} plane. ](Fig1f.eps) \[dep\] Magnetic materials displaying [*carrier-mediated*]{} ferromagnetic order offer fascinating opportunities for non-thermal, potentially [*femtosecond*]{} manipulation of magnetism. A model system of such materials is Mn-doped III-V ferromagnetic semiconductors that have received a lot of attention lately [@ohno1998]. On the one hand, their magnetic properties display a strong response to excitation with light or electrical gate and current via carrier density tuning [@Koshiharaetal97PRL; @ohnoetal2000; @wangetalPRL2007]. On the other hand, the strong coupling ($\sim$1 eV in GaMnAs) between carriers (holes) and Mn ions inherent in carrier-mediated ferromagnetism could enable a [*femtosecond*]{} cooperative magnetic response induced by photoexcited carriers. Indeed, the existence of a very early non-equilibrium, non-thermal femtosecond regime of collective spin rotation in (III,Mn)Vs has been predicted theoretically [@ChovanetalPRL2006]. In addition, a coherent driving mechanism for femtosecond spin rotation via [*virtual*]{} excitations has also been recently demonstrated in antiferro- and ferri-magnets [@coherent]. Nevertheless, all prior studies of photoexcited magnetization rotation in ferromagnetic (III,Mn)Vs showed dynamics on the few picosecond timescale, which accesses the quasi-equilibrium, quasi-thermal, lattice-heating regime [@psRotationGaMnAs]. Up to now in these materials, the main observation on the femtosecond time scale has been photoinduced demagnetization [@wangetalPRL2005; @wangetalreview2006; @wang2008; @Cywinski_PRB07]. Custom-designed (III,Mn)V hetero- and nano-structures show rich magnetic memory effects. One prominent example is GaMnAs-based four-state magnetic memory, where “giant” magneto-optical and magneto-transport effects allow for ultrasensitive magnetic memory readout [@fourstate]. However, all detection schemes demonstrated so far have been static measurements. Achieving an understanding of collective magnetic phenomena on the femtosecond time scale is critical for terahertz detection of magnetic memory and therefore essential for developing realistic “spintronic” devices and large-scale functional systems. In this Letter, we report on photoinduced [*femtosecond*]{} collective magnetization rotation that allows for femtosecond detection of magnetic memory in GaMnAs. Our time-resolved magneto-optical Kerr effect (MOKE) technique directly reveals a photoinduced four-state magnetic hysteresis via a quasi-instantaneous magnetization rotation. We observe for the first time a distinct initial temporal regime of the magnetization rotation within the first $\sim$200 fs, during the photoexcitation and highly non-equilibrium, non-thermal carrier redistribution times. We attribute the existence of such a regime to a [*carrier-mediated*]{} effective magnetic field pulse, arising without assistance from either lattice heating or demagnetization. The main sample studied was grown by low-temperature molecular beam epitaxy (MBE), and consisted of a 73-nm Ga$_{0.925}$Mn$_{0.075}$As layer on a 10 nm GaAs buffer layer and a semi-insulating GaAs \[100\] substrate. The Curie temperature and hole density were 77 K and $3 \times 10^{20}$ cm$^{-3}$, respectively. As shown in Fig. 1, our structure exhibits a four-state magnetic memory functionality. By sweeping an external magnetic field B aligned nearly perpendicularly to the sample normal, with small components in both the [*X*]{} and [*Y*]{} directions in the sample plane, one can sequentially access four magnetic states, X$+\rightarrow$Y$-\rightarrow$X$-\rightarrow$Y$+$, via abrupt 90$^o$ magnetization ($\mathbf{M}$) switchings between the [*XZ*]{} and [*YZ*]{} planes \[Fig. 1(a)\]. In these magnetic states, $\mathbf{M}$ aligns along a direction arising as a combination of the external B field and the anisotropy fields, which point along the in-plane easy axes \[100\] and \[010\]. The multistep magnetic switchings manifest themselves as abrupt jumps in the four-state hysteresis in the Hall magneto-resistivity $\rho _{Hall}$ \[Fig. 1(b)\] (planar Hall effect [@fourstate]). The continuous slopes of $\rho _{Hall}$ indicate a coherent out-of-plane $\mathbf{M}$ rotation during the perpendicular magnetization reversal (anomalous Hall effect [@ohno1998]). Fig. 1(c)-(d) show the B scans in the vicinity of 0T, with the field turning points between the coercivity fields, i.e., $B_{c1}<\left|B\right|<B_{c2}$. This leads to a “minor” hysteresis loop, accessesing two magnetic memory states at $B=$0T: X$-$(0) and Y$+$(0). We now turn to the transient magnetic phenonmena. We performed time-resolved MOKE spectroscopy [@wangetalreview2006] using 100 fs laser pulses. The linearly polarized ($\sim$12 degree from the crystal axis \[100\]) UV pump beam was chosen at 3.1 eV, with peak fluence $\sim$ 10$\mu$J/cm$^2$. A NIR beam at 1.55 eV, kept nearly perpendicular to the sample ($\sim$ 0.65 degree from the normal), was used as probe. The signal measured in this polar geometry reflects the out-of-plane magnetization component, M$_z$. ![(Color) Photoinduced femtosecond four-state magnetic hysteresis. (a) B field scans of $\triangle \theta _{k}$ at 5K for time delays $\triangle t=$ -1 ps, 600 fs, and 3.3 ps. The traces are vertically offset for clarity. Inset (left): temporal profiles of normalized Kerr ($\theta _{k}$) and ellipticity ($\eta _{k}$) angle changes at 1.0T; Inset (right): static magnetization curve at 5K ($\sim$4 mrad), measured in the same experimental condition (but without the pump pulse). (b) Temporal profiles of photoinduced $\triangle \theta _{k}$ for the four magnetic states. Shaded area: pump–probe cross–correlation.[]{data-label="mag-dep"}](Fig2f.eps) Fig. 2(a) shows the B field scan traces of the photoinduced change, $\Delta\theta_K$, in the Kerr rotation angle at three time delays, $\triangle t=$ -1 ps, 600 fs, and 3.3 ps. The magnetic origin of this femtosecond MOKE response [@KoopmansetAl00PRL] was confirmed by control measurements showing a complete overlap of the pump–induced rotation ($\theta _{k}$) and ellipticity ($\eta _{k}$) changes \[left inset, Fig. 2(a)\]. $\Delta\theta_K$ is negligible at $\triangle t=$-1 ps. However, a mere $\triangle t=$600 fs after photoexcitation, a clear photoinduced four-state magnetic hysteresis is observed in the magnetic field dependence of $\Delta\theta_K$ (and therefore $\Delta M_z$), with four abrupt switchings at $\left|B_{c1}\right|=$0.074T and $\left|B_{c1}\right|=$0.33T due to the magnetic memory effects. As marked by the arrows in Fig. 2(a), the four magnetic states X$+$, X$-$, Y$-$, Y$+$ for $\left|B\right|=$0.2T give different photoinduced $\Delta\theta_K$. It is critical to note that the steady-state MOKE curve, i.e. $\theta_K$ without pump field, doesn’t show any sign of magnetic switching or memory behavior \[right inset, Fig. 2(a)\]; these arise from the pump photoexcitation. The B field scans also show a saturation behavior at $\left|B\right|>$0.6T, to be discussed later. We note that the photo–induced hysteresis loops at $\triangle t=$3.3 ps and 600 fs sustain similar shapes, with only slightly larger amplitudes at 3.3 ps. This observation confirms that the dynamic magnetic processes responsible for the abrupt switchings occur on a femtosecond time scale. Fig. 2(b) shows the photoinduced $\Delta\theta_K$ dynamics for the four initial states X$+$, X$-$, Y$-$, and Y$+$. An extremely fast $\Delta\theta_K$ develops within 200 fs, with magnitude and sign that distinctly differ, depending on the initially prepared state, consistent with Fig.2 (a). The substantial difference in $\Delta\theta_K$ under the same B field - for instance between the X+ and Y+ states - shows that the magnetic dynamics is not due to simple demagnetization [@BeaurepaireetalPRL96; @wangetalPRL2005; @wangetalreview2006]. The photoinduced dynamics of the zero–B field memory states \[Fig. 1(c)\] elucidates the salient features of the femtosecond magnetic processes. Fig. 3(a) shows the temporal profiles of the photo-induced $\Delta\theta_K$ for X$-$(0) and Y$+$(0) initial states. Since the initial magnetization vector lies within the sample plane, $\Delta \theta _{k}$ in the first 200 fs reveals an out–of–plane spin rotation, with negligible contribution from demagnetization. More intriguingly, the $\mathbf{M}$ in X$-$ and Y$+$ initial states rotates to different [*Z*]{}-axis directions, as illustrated in Fig. 3(b). This leads to opposite signs of the photoinduced signals and is responsible for the four-state magnetic switchings. Furthermore, the observation of an initial discontinuity in the temporal profiles of the $\mathbf{M}$ tilt reveals [*two distinct temporal regimes*]{}, marked in Fig. 3(a): a substantial magnetization rotation concludes after the first 200 fs and is followed by a [*much slower*]{} rotation change afterwards (over 100’s of ps). ![(Color online) (a) Photo-induced $\Delta \theta _{k}$ for two in-plane magnetic memory states, shown together with the pump-probe cross-correlation (shaded). The opposite, out–of–plane $\mathbf{M}$ rotations for the X$-$(0) and Y$+$(0) are illustrated in (b).[]{data-label="power"}](Fig3f.eps) We now discuss the origin of the observed femtosecond magnetization rotation. In the previously held picture of light–induced magnetization rotation in ferromagnets, the photoexcitation alters the anisotropy fields via quasi-equilibrium mechanisms, such as heating of the lattice (magneto-crystalline anisotropy) or heating of the spins (shape anisotropy) [@heating]. Since the in-plane magnetic memory states of Fig. 1(c) have negligible shape anisotropy, a significant B field within the standard picture can only occur on a time scale of several picoseconds via the lattice heating mechanism. However, it has been shown theoretically [@ChovanetalPRL2006; @ChovanetalPRB2008] that the Mn spin in GaMnAs can respond quasi-instantaneously to a femtosecond effective magnetic field pulse generated by hole spins via nonlinear optical processes assisted by interactions. This light–induced B field pulse may be thought of as a femtosecond modification of the magnetic anisotropy fields. In the realistic system, one needs to also treat microscopically the transient magnetic anisotropy changes, due to the complex valence bands and highly non–thermal hole populations in the femtosecond regime, which drastically affect the photoexcited carrier spin. Due to the [*hole-mediated*]{} effective exchange interaction between Mn spins, the anisotropy fields in GaMnAs result from the coupling of several valence bands by the [*spin-orbit interaction*]{} and depend on the transient hole distribution and coherences between different bands [@ChovanetalPRB2008]. In the static case, recent experimental [@anisotropy-exp] and theoretical [@anisotropy-the] investigations have shown that increasing the hole density significantly reduces the cubic anisotropy (K$_c$) along the \[100\] direction, while enhancing the uniaxial anisotropy (K$_u$) along \[1-10\]. One therefore expects that the photoexcited hole population turns on an effective magnetic field pulse ($\Delta B_{c}$) along the \[1-10\] direction \[Fig. 4(a)\]. This photo-triggered $\Delta B_{c}$ then exerts a spin torque on the $\mathbf M$ vector, $\Delta \overrightarrow B_{c}\times \overrightarrow { M}$, and pulls it away from the sample plane. The directions of these spin torques for the X$-$(0) and Y$+$(0) states are opposite, leading to different $\mathbf M$ rotation paths \[Fig. 3\]. Since this mechanism is mediated by the the non–thermal holes, the appearance of $\Delta B_{c}$ is quasi-instantaneous, limited only by the pulse duration of $\sim$100 fs [@ChovanetalPRB2008]. This femtosecond magnetic anisotropy contribution from the non–thermal photoexcited carriers should be contrasted to the quasi–thermal contribution, arising from, e.g., the transient lattice temperature elevation on the picosecond time scale [@psRotationGaMnAs]. Next we turn to the origin of the discontinuity that reveals the [*two temporal regimes*]{} in the collective magnetization rotation \[Fig. 3\]. The quick termination of the initial magnetization tilt implies that the effective $\Delta B_{c}$ pulse induced by the photoexcitation decays within the first hundreds of femtoseconds. The photoexcitation of a large (as compared to the ground state anisotropy field) $\Delta B_{c}$ requires an extensive [*non-thermal*]{} distribution of transient holes in [*the high momentum states*]{} of the valence band [@anisotropy-the]. This is due to the large spin anisotropy of these hole states, empty in the unexcited sample, via their strong spin-orbit interaction. In our experiment, immediately following photoexcitation at 3.1 eV, a large density of transient holes distribute themselves over almost half of the Brillouin zone along the L\[111\] direction. The Mn–hole spin exchange interaction is also believed to be enhanced along \[111\] due to strong p-d orbital hybridization [@BurchPRB2004]. Consequently, these photoexcited holes contribute strongly to the magnetic anisotropy fields. The following rapid relaxation and thermalization of the high momentum holes, due to carrier-carrier and carrier-phonon scattering, reduce $\Delta B_{c}$ within a few hundred femtoseconds. The subsequent picosecond magnetization rotation process arises from the change in magnetic anisotropy induced by the lattice temperature elevation. Our results reveal a complex scenario of collective spin rotation, marked by the transition from a non-equilibrium, carrier-mediated regime ($<$200 fs) to a thermal, lattice-heating regime on the ps time scale. ![(Color online) (a) Schematics of the photoexcited carrier–induced anisotropy field $\Delta B_{c}$. (b) Simulations of $\Delta M_{z}/M_0$ for the two magnetic memory states. Parameters used in the calculation are $K_{c}=1.198\cdot 10^{-2}meV$, $K_{u}=0.373\cdot 10^{-2}meV$, $K_{3}=0.746\cdot 10^{-2}meV$, $T_{1}=330$fs and $3\% $ of photoexcited carriers. (c) Schematics of the photoinduced M$_{z}$ for the X$-$, X$+$, Y$-$ and Y$+$ states at $\left|B\right|=$0.2T. []{data-label="temperature"}](Fig4f.eps) We modelled the transient anisotropy phenomenologically by deriving $\Delta B_{c}$ from the magnetic free energy, $$E_{anis}=-{K_{c}\over S^{4}} S_{x}^{2}S_{y}^{2}+{K_{u}\over S^{2}} S_{x}^{2}+{K_{3}\over S^{2}} S_{z}^{2},$$ describing cubic (K$_c$) and uniaxial (K$_u$) contributions, and added a time–dependent modification of $K_{c}/K_{u}$ due to the strongly anisotropic photoexcited hole states [@ChovanetalPRB2008]. The corresponding contribution to the Mn spin equation of motion is $\partial _{t}{\bf S}={\bf S}\times {\bf H}_{anis}$, where ${\bf H}_{anis}=- {\partial E_{anis}\over \partial {\bf S}}$. The light–induced change in the magnitude of $K_{c}/K_{u}$ increases during the pulse and then decreases with the energy relaxation time (T$_1$) of the high–momentum photoexcited holes. The results of our calculation are shown in Fig. 4(b), which gives a similar time dependence of the normalized $\Delta M_{z}$, with magnitude $\sim$ 0.4$\%$ of the total magnetization M$_0$ ($\sim$4 mrad at 5K), which compares well with the experiment. Finally, Fig. 4(c) illustrates the femtosecond detection of the four-state magnetic memory shown in Fig. 2. By incorporating both the photo-induced rotation (red arrows) and the demagnetization (green arrows) effects, we can visualize the different M$_z$ changes for the four magnetic states, consistent with our observation. Demagnetization results in the high field saturation behaviour observed in Fig. 2(a). For $\left|B\right|>0.60T$, $\mathbf M$ is aligned mostly along the sample normal. Then the photo-induced signals arise from the decrease in the $\mathbf M$ amplitude, which is more or less constant with respect to the field. In conclusion, we report on the femtosecond magnetic response of photoinduced magnetization rotation in GaMnAs, which allows for femtosecond detection of four-state magnetic memory. Our observations unequivocally identify a [*non-thermal, carrier-mediated*]{} mechanism of magnetization rotation, relevant only in the [*femtosecond*]{} regime, without assistance of either lattice heating or demagnetization. This femtosecond cooperative magnetic phenomenon may represent an as-yet-undiscovered universal principle in all carrier-mediated ferromagnetic materials - a class of rapidly emerging “multi-functional" materials with significant potential for future applications, e.g., the oxides with promise of far above room temperature Curie temperature. This work was supported by the Office of Basic Energy Sciences of the US Department of Energy under Contract No. DE-AC02-05CH11231, by the National Science Foundation DMR-0603752, and by the EU STREP program HYSWITCH. [10]{} bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} H. Ohno, Science [**281**]{}, 951 (1998). ., ****, (). H. Ohno [*et al*]{}., Nature [**408**]{}, 944 (2000). ., ****, (). ., ****, (); ****, (). A. V. Kimel [*et al*]{}., Nature [**435**]{}, 655 (2005); F. Hansteen [*et al*]{}, Phys. Rev. Lett. 95, 047402 (2005). J. Qi, [*et al*]{}., App. Phys. Lett. [**91**]{}, 112506 (2007); D. M. Wang [*et al*]{}., Phys. Rev. B [**75**]{}, 233308 (2007); Y. Hashimoto [*et al*]{}., Phys. Rev. Lett. [**100**]{}, 067202 (2008) ., ****, (). ., ****, (). J. Wang [*et al*]{}., arXiv:0804.0451 (2008). L. Cywi[ń]{}ski [*et al*]{}., ****, (). H. Tang [*et al*]{}., Phys. Rev. Lett. [**90**]{}, 107201 (2003). ., ****, (). E. Beaurepaire [*et al*]{}., Phys. Rev. Lett. [**76**]{}, 4250 (1996). M. V. Kampen [*et al*]{}, Phys. Rev. Lett. [**88**]{}, 227201 (2002); M. Vomir [*et al*]{}, Phys. Rev. Lett. 94, 237601 (2005). J. Chovan and I. E. Perakis, Phys. Rev. B. 77, 085321 (2008); and unpublished. X. Liu [*et al*]{}., Phys. Rev. B. 71, 035307 (2005); K. Hamaya [*et al*]{}., Phys. Rev. B 74, 045201 (2006) T. Dietl [*et al*]{}., Phys. Rev. B. 63, 195205 (2001); M. Abolfath [*et al*]{}., Phys. Rev. B 63, 055418 (2001) K. S. Burch [*et al*]{}., Phys. Rev. B 70, 205208 (2004)

--- abstract: 'We propose the generalized uncertainty principle (GUP) with an additional term of quadratic momentum motivated by string theory and black hole physics as a quantum mechanical framework for the minimal length uncertainty at the Planck scale. We demonstrate that the GUP parameter, $\beta_0$, could be best constrained by the the gravitational waves observations; GW170817 event. Also, we suggest another proposal based on the modified dispersion relations (MDRs) in order to calculate the difference between the group velocity of gravitons and that of photons. We conclude that the upper bound reads $\beta_0 \simeq 10^{60}$. Utilizing features of the UV/IR correspondence and the obvious similarities between GUP (including non-gravitating and gravitating impacts on Heisenberg uncertainty principle) and the discrepancy between the theoretical and the observed cosmological constant $\Lambda$ (apparently manifesting gravitational influences on the vacuum energy density), known as [*catastrophe of non-gravitating vacuum*]{}, we suggest a possible solution for this long-standing physical problem, $\Lambda \simeq 10^{-47}~$GeV$^4/\hbar^3 c^3$.' author: - Abdel Magied Diab - Abdel Nasser Tawfik bibliography: - 'upperBoundofGUPParameterV1.bib' title: A Possible Solution of the Cosmological Constant Problem based on Minimal Length Uncertainty and GW170817 and PLANCK Observations --- @pre@post Introduction {#intro} ============ The cosmological constant, $\Lambda$, an essential ingredient of the theory of general relativity (GR) [@Einstein1917As], was guided by the idea that the evolution of the Universe should be static [@Tawfik:2011mw; @Tawfik:2008cd]. This model was subsequently refuted and accordingly the $\Lambda$-term was abandoned from the Einstein field equation (EFE), especially after the confirmation of the celebrated Hubble obervations in 1929 [@Hubble:1929ig], which also have verified the consequences of Friedmann solutions for EFE with vanishing $\Lambda$ [@Friedman:1922kd]. Nearly immediate after publishing GR, a matter-free solution for EFE with finite $\Lambda$-term was obtained by de Sitter [@deSitter:1917zz]. Later on when it has been realised that the Einstein [*static*]{} Universe was found unstable for small perturbations [@Mulryne:2005ef; @Wu:2009ah; @delCampo:2011mq], it was argued that the inclusion of the $\Lambda$-term remarkably contributes to the stability and simultaniously supports the expansion of the Universe, especially that the initial singularity of Friedmann-Lem$\hat{\mbox{a}}$itre-Robertson-Walker (FLRW) models could be improved, as well [@Weinberg1972AA; @Misner1984B]. Furthermore, the observations of type-Ia high redshift supernovae in late ninteeth of the last century [@Riess:1998cb; @Perlmutter:1998np] indicated that the expanding Universe is also accelerating, especially at a small $\Lambda$-value, which obviously contributes to the cosmic negative pressure [@Garriga:1999bf; @Martel:1997vi]. With this regard, we recall that the cosmological constant can be related to the vacuum energy density, $\rho$, as $\Lambda=8\pi G \rho/c^2$, where $c$ is the speed of light in vacuum and $G$ is the gravitational constant. In 2018, the PLANCK observations have provided us with a precise estimation of $\Lambda$, namely $\Lambda_{\mbox{Planck}} \simeq 10^{-47}$GeV$^4/\hbar^3 c^3$ [@Aghanim:2018eyx]. When comparing this tiny value with the theoretical estimation based on quantum field theory in weakly- or non-gravitating vacuum, $\Lambda_{\mbox{QFT}} \simeq 10^{74}$GeV$^4/\hbar^3 c^3$, there is, at least, a $121$-orders-of-magnitude-difference to be fixed [@Adler:1995vd; @Weinberg:1988cp; @Zeldovich:1968ehl]. The disagreement between both values is one of the greatest mysteries in physics and known as the cosmological constant problem or [*catastrophe of non-gravitating vacuum*]{}. Here, we present an attempt to solve this problem. To this end, we utilize the generalized uncertainty principle (GUP), which is an extended version of Heisenberg uncertainty principle (HUP), where a correction term encompassing the gravitational impacts is added, and thus an alternative quantum gravity approach emerges [@Tawfik:2014zca; @Tawfik:2015rva]. To summarize, the present attempt is motivated by the similarity of GUP (including non-gravitating and gravitating impacts on HUP) and the disagreement between theoretical and observed estimations for $\Lambda$ (manifesting gravitational influences on the vacuum energy density) and by the remarkable impacts of $\Lambda$ on early and late evolution of the Universe [@Tawfik:2019jsa; @Tawfik:2011mw; @Tawfik:2008cd]. So far, there are various quantum gravity approaches presenting quantum descriptions for different physical phenomena in presence of gravitational fields to be achnowledged, here [@Tawfik:2014zca; @Tawfik:2015rva]. The GUP offers a quantum mechanical framework for a potential minimal length uncertainty in terms of the Planck scale [@Tawfik:2017syy; @Tawfik:2016uhs; @Dahab:2014tda; @Ali:2013ma]. The minimal length uncertainty, as proposed by GUP, exhibits some features of the UV/IR correspondence [@Maldacena:1997re; @Gubser:1998bc; @Witten:1998qj], which has been performed in viewpoint of local quantum field theory. Thus, it is argued that the UV/IR correspondence is relevant to revealing several aspects of short-distance physics, such as, the cosmological constant problem [@Weinberg:1988cp; @Banks:2000fe; @Cohen:1998zx; @ArkaniHamed:2000eg]. Therefore, a precise estimation of the minimal length uncertainty strongly depends on the proposed upper bound of the GUP parameter, $\beta_0$ [@Dahab:2014tda; @Tawfik:2013uza]. Various ratings for the upper bound of $\beta_0$ have been proposed, for example, by comparing quantum gravity corrections to various quantum phenomena with electroweak [@Das:2008kaa; @Das:2009hs] and astronomical [@Scardigli:2014qka; @Feng:2016tyt] observations. Accordingly, $\beta_0$ ranges between $10^{33}$ to $10^{78}$ [@Scardigli:2014qka; @Feng:2016tyt; @Walker:2018muw]. As a preamble of the present study, we present a novel estimation for $\beta_0$ from the binary neutron stars merger, the gravitational wave event GW170817 reported by the Laser Interferometer Gravitational-Wave Observatory (LIGO) and the Advanced Virgo collaborations [@TheLIGOScientific:2017qsa]. With this regard, there are different efforts based on the features of the UV/IR correspondence in order to interpret the $\Lambda$ problem [@Chang:2001bm; @Chang:2011jj; @Miao:2013wua; @Shababi:2017zrt; @Vagenas:2019wzd] with Liouville theorem in the classical limit [@Fityo:2008zz; @Chang:2001bm; @Wang:2010ct]. Having a novel estimation of $\beta_0$, a solution of the $\Lambda$ problem, [*catastrophe of non-gravitating vacuum*]{}, could be best proposed. The present paper is organized as follows. Section \[MDRGUP\] reviews the basic concepts of the GUP approach with quadratic momentum. The associated modifications of the energy-momentum dispersion relations related to GR and rainbow gravity are also outlined in this section. In section \[GUPparameter\], we show that the dimensionless GUP parameter, $\beta_o$, could be, for instance, constrained to the gravitational wave event GW170817. Section \[LamdaProblem\] is devoted to calculating the vacuum energy density of states and shows how this contributes to understanding the cosmological constant problem with an quantum gravity approach, the GUP. The final conclusions are outlined in section \[conclusion\]. Generalized Uncertainty Principle and Modified Dispersion Relations \[MDRGUP\] ============================================================================== Several approaches to the quantum gravity, such as GUP, predict a minimal length uncertainties that could be related to the Planck scale [@Tawfik:2015rva; @Tawfik:2014zca]. There were various laboratory experiments conducted to examine the GUP effects [@Bawaj:2014cda; @Marin:2013pga; @Pikovski:2011zk; @Khodadi:2018kqp]. In this section, we focus the discussion on GUP with a quadratic momentum uncertainty [@Tawfik:2015rva; @Tawfik:2014zca]. This version of GUP was obtained from black hole physics [@Gross:1987kza] and supported by [*gedanken*]{} experiments [@Maggiore:1993zu], which have been proposed Kempf, Mangano, and Mann (KMM), [@Kempf:1994su] x p , \[GUPuncertainty\] where $\Delta x$ and $\Delta p$ are the uncertainties in position and momentum, respectively. The GUP parameter can be exressed as $\beta = \beta_0 (\ell_p/\hbar)^2 = \beta_0/ (M_p c)^2$, where $\beta_0$ is a dimensionless parameter, $\ell_p=1.977 \times 10^{-16}~$GeV$^{-1}$ is the Planck length, and $M_p= 1.22 \times 10^{19}~$GeV$/c^2$ is the Planck mass. Equation (\[GUPuncertainty\]) implies the existence of a minimum length uncertainty, which is related to the Planck scale, $\Delta x_{\mbox{min}} \approx \hbar \sqrt{\beta} =\ell_p \sqrt{\beta_0}$. It should be noticed that the minimum length uncertainty exhibits features of the UV/IR correspondence [@Maldacena:1997re; @Gubser:1998bc; @Witten:1998qj]. $\Delta x$ is obviously proportional to $\Delta p$, where large $\Delta p$ (UV) becomes proportional to large $\Delta x$ (IR). Equation (\[GUPuncertainty\]) is a noncommutative relation; $[\hat{x}_i,\; \hat{p}_j] = \delta_{ij} i \hbar [1+\beta p^2]$, where both position and momentum operators can be defined as \_i = \_[0i]{}, \_j= \_[0j]{} (1+p\^2), where $\hat{x}_{0i}$ and $\hat{p}_{0j}$ are corresponding operators obtained from the canonical commutation relations $[\hat{x}_{0i},\; \hat{p}_{0j}]=\delta_{ij} i \hbar,$ and $p^2= g_{ij} p^{0i} \; p^{0j}$. We can now construct the modified dispersion relation (MDR) due to quadratic GUP. We start with the background metric in GR gravitational spacetime ds\^2 =g\_ dx\^ dx\^= g\_[00]{} c\^2 dt\^2 + g\_[ij]{} dx\^i dx\^j, with $g_{\mu \nu}$ is the Minkowski spacetime metric tensor $(-,+,+,+)$. Accordingly, the modified four-momentum squared is given by p\_p\^= g\_ p\^p\^&=& g\_[00]{} (p\^0)\^2 + g\_[ij]{} p\^[0i]{} p\^[0j]{} (1+p\^2)\ &=& -(p\^0)\^2 + p\^2 + 2 p\^2 p\^2. \[modifyMomentum\] Comparing this with the conventional dispersion relation, $p_\mu p^\mu = - m^2c^2$, the time component of the momentum can then be written as (p\^0)\^2 &=& m\^2c\^2 + p\^2 (1+p\^2). The energy of the particle $\omega$ can be defined as $\omega/c = - \zeta_\mu p^\mu = - g_{\mu \nu} \zeta^\mu p^\nu$, where the killing vector is given as $\zeta^\mu = (1,0,0,0)$. Therefore, the energy of the particle could be expressed as $\omega=-g_{00} c (p^0)=c (p^0)$ and the modified dispersion relation in GR gravity reads \^2 = m\^2 c\^4 + p\^2 c\^2 (1+ 2p\^2). \[MDRrel\] For $\beta \rightarrow 0$, the standard dispersion can be obtained. The rainbow gravity generalizes the MDR in doubly special relativity to curved spacetime [@magueijo2004gravity], where the geometry spacetime is explored by a test particle with energy $\omega$ [@Magueijo:2001cr; @Magueijo:2002am], \^2 f\_1 ()\^2 - (pc)\^2 f\_2 ()\^2 = (mc\^2)\^2, where $\omega_p$ is the Planck energy and $f_1 (\omega/\omega_p)$ and $f_2 (\omega/\omega_p)$ are known as the rainbow functions which are model-depending. The rainbow functions can be defined as [@AmelinoCamelia:1996pj; @AmelinoCamelia:1997gz], f\_1 (/\_p) &=& 1, f\_2 (/\_p) = , \[Rainbowfuncs\] where $\eta$ and $n$ are free positive parameters. It was argued that for the logarithmic corrections of black hole entropy [@Tawfik:2015kga], the integer $n$ is limited as $n=1,2$ [@Gangopadhyay:2016rpl]. Therefore, it would be eligible to assume that $n=2$. Thus, the MDR for rainbow gravity with GUP can be written as, \^2 = . \[MDRrain\] Again, as $\beta \rightarrow 0$, Eq. (\[MDRrain\]) goes back to the standard dispersion relation. We have constructed two different MDRs for quadratic GUP, namely Eqs (\[MDRrel\]) and (\[MDRrain\]) in GR and rainbow gravity, respectively. Bounds on GUP parameter from GW170817 shall be outlined in the section that follows. Bounds on GUP parameter from GW170817 {#GUPparameter} ===================================== Instead of violating Lorentz invariance [@Tawfik:2012hz], we intend to investigate the speed of the graviton from the GW170817 event. To this end, we use MDRs obtained from the quadratic GUP approaches, section \[MDRGUP\]. Thus, defining an upper bound on the dimensionless GUP parameter $\beta_0$ for given bounds on mass and energy of the graviton, where $m_g\lesssim 4.4 \times 10^{-22}~$eV$/c^2$ and $\omega = 8.5 \times 10^{-13}~$eV, respectively, plays an essential role. Assuming that the gravitational waves propagate as free waves, we could, therefore, determine the speed of the mediator, that of the graviton, from the group velocity of the accompanying wavefront, i.e. $v_g = \partial \omega/\partial p$, where $\omega$ and $p$ are the energy and momentum of the graviton, respectively [@Mirshekari:2011yq]. The idea is that the group velocity of the graviton can be simply deduced from the MDRs, Eqs. (\[MDRrel\]) for the GR gravity and (\[MDRrain\]) and the rainbow gravity, in presence and then in absence of the GUP impacts, which have been discussed in section \[MDRGUP\]. Accordingly, Eq. (\[MDRrel\]) implies that the group velocity reads v\_g = = (1+ 4 p\^2). \[vgMDR1\] The unmodified momentum $p$ in terms of the modified parameters up to $\mathcal{O} (\beta)$, can be expressed as $p=a+b \beta$, where $a$ and $b$ are arbitrary parameters. By substituting this expression into Eq. (\[MDRrel\]), we find that $p^2= (\omega_g/c)^2 - m^2 c^2 $. Thus, Eq. (\[vgMDR1\]) can be rewritten as v\_g = c { \^[1/2]{} +4 \^[3/2]{} }, where $\omega_g$ is the energy of the graviton. It is obvious that for $\beta \rightarrow 0 $, i.e. in absence of GUP impacts, the group velocity reads v\_g = c . Then, the difference between the speed of photon (light) and that of graviton without GUP impacts is given as | v| = | c-v\_g| = c | ( )\^2 | 1.34 10\^[-19]{}c. \[vDr\] Although the small difference obtained, we are - in the gravitational waves epoch - technically able to measure even a such tiny difference! In light of this, we could use the results associated with the GW170817 event, such as the graviton velocity, in order to set an upper bound on the GUP parameter, $\beta_0$. For a massless graviton, the difference between the speed of photons (light) and that of the gravitons in presence of the GUP impacts reads |v\_| &=&| 4| = | 4\_0 ()\^2| 1.95 10 \^[-80]{} \_0 c. \[vMDR\] Thus, the upper bound on the dimensionless parameter, $\beta_0$, of the quadratic GUP can be simply deduced from Eqs. (\[vDr\]) and (\[vMDR\]), \_0 8.89 10\^[60]{}. \[MDRbeta1\] The group velocity of the graviton due to MDR and rainbow gravity when applying the quadratic GUP approach, Eq. (\[MDRrain\]), can be expressed as v\_g = = () . Similarly, one can for a massless graviton express the conventional momentum in terms of the GUP parameter. In order of $\mathcal{O}(\beta)$, we get c p = \_g . \[pcRainbow\] The unmodified momentum can be expressed in GUP-terms up to $\mathcal{O}(\beta)$; $p=a_0 + a_1 \beta$, where $a_0$ and $a_1$ are arbitrary parameters. Nevertheless, the investigation of the speed of the graviton from the GW150914 observations [@Abbott:2016blz] specifies the rainbow gravity parameter, $ \eta (\omega_g/\omega_p)^2\leq 3.3\times 10^{-21}$ [@Gwak:2016wmg]. Accordingly, Eq. (\[pcRainbow\]) can be reduced to $c p=\omega_g (1- \beta \omega_g^2/c^2)$ and the group velocity of the massless graviton becomes v\_g = c . Then, the difference between the speed of photons and that of the gravitons reads |v\_| &=&| 5 | 2.43 10\^[-80]{} \_0 c. \[vgRainbow\] By comparing Eqs. (\[vgRainbow\]) and (\[vDr\]), the upper bound of the GUP parameter $\beta_0$ can be estimated as \_0 5.5 10\^[60]{}. \[Rainbowbeta1\] It is obvious that both results, Eqs. (\[MDRbeta1\]) and (\[Rainbowbeta1\]), are very close to each other; $\beta_0 \lesssim 10^{60}$. The improved upper bound of $\beta_0$ is very similar to the ones reported in refs. [@Scardigli:2014qka; @Feng:2016tyt], which - as well - are depending on astronomical observations. The present results are based on mergers of spinning neutron stars. Thus, it is believed that more accurate observations, the more precise shall be $\beta_0$. Having set a upper bound on the GUP parameter and counting on the spoken similarities between GUP and the catastrophe of non-gravitating vacuum, we can now propose a possible solution of the cosmological constant problem. A Possible Solution of the Cosmological Constant Problem {#LamdaProblem} ======================================================== The cosmological constant can be given as $\Lambda = 3 H_0^2 \Omega_\Lambda$, where $H_0$ and $\Omega_\Lambda$ are the Hubble parameter and the dark energy density, respectively [@Carroll:2000fy]. On the other hand, the origin of the catastrophe of non-gravitating vacuum would be understood from the disproportion of the value of $\Lambda$ in the theoretical calculations, while this is apparently impacting the GW observations [@Sahni:2002kh]. From the most updated PLANCK observations, the values of $\Omega_\Lambda = 0.6889 \pm 0.0056$ and $H_0 = 67.66 \pm 0.42~$Km $\cdot$ s$^{-1}$ $\cdot$ Mpc$^{-1}$ [@Aghanim:2018eyx]. Then, the vacuum energy density &=& () \_= \_, \[VacuEnergy\] where the scale of the visible light, $\ell_0= c/H_0 \simeq 1.368 \times 10^{23}~$Km [@Aghanim:2018eyx]. Therefore, one can use Eq. (\[VacuEnergy\]) to esiamte the vacuum energy density in order of $10^{-47}~$GeV$^4/(\hbar^3c^3)$. In quantum field theory, the cosmological constant is to be calculated from sum over the vacuum fluctuation energies corresponding to all particle momentum states [@Carroll:2000fy]. For a massless particle, we obtain d\^3 (\_p /2) 9.6010\^[74]{} \^4/ (\^3 c\^3). \[QFTlamda\] This is clearly infinite integral. But, it is usually cut off, at the Planck scale, $\mu_p = \hbar/\ell_p$. We assume $\omega_p$ is the vacuum energy of quantum harmonic state $\hbar \omega_p = [p^2c^2+m_g^2c^4]^{1/2}$. To propose a possible solution of the cosmological constant problem, it is initially needed to determine the number of states in the phase space volume taking into account GUP, Eq. (\[GUPuncertainty\]). An analogy can be found in Liouville theorem in the classical limit. We need to make sure that the size of each quantum mechanical state in phase space volume is depending on the modified momentum $p$, especially when taking GUP into consideration, Eq. (\[GUPuncertainty\]). In other words, the number of quantum states in the phase space volume is assumed not depending on time. In the classical limit, the relation of the quantum commutation relations and the Poisson brackets is given as $[\hat{A}, \hat{B}] = i\hbar \{A, B\}$. Details on the Poisson bracket in D-Dimensions are outlined in appendix \[LiouvilleTheorem\]. Consequently, the modified density of states implies different implications on quantum field theory, such as, the cosmological constant problem. In D-dimensional spherical coordinate systems, the density of states in momentum space is given as [@Fityo:2008zz; @Chang:2001bm; @Wang:2010ct] , where $V$ is the volume of space. It should be noticed that in quantum mechanics, the number of quantum stated per unit volume is given as $V/(2\pi \hbar)^D$. Therefore, for Liouville theorem, the weight factor in 3-D dimension reads [@Fityo:2008zz; @Chang:2001bm; @Wang:2010ct] (review appendix \[LiouvilleTheorem\]) . \[densityStates\] In quantum field theory, the modification in the quantum number of state of the phase space volume should have consequences on different quantum phenomena, such as, the cosmological constant problem and the black body radiation. At finite weight factor of GUP, the sum over all momentum states per unit volume of the phase space modifies the vacuum energy density. The cosmological constant, on the other hand, is determined by summing over the vacuum fluctuations, the energies, corresponding to a particular momentum state \_ (m) &=& d\^3 (p\^2) (\_p /2) = For a massless particle, the vacuum energy density, which is directly related to $\Lambda$, reads \_(m=0) &=&   dp = = 1.78 10\^[-48]{} \^4/(\^3 c\^3). \[GUPLamda\] The agreement between the observed value of the cosmological constant, $\Lambda \simeq 10^{-47}~$GeV$^4/\hbar^3 c^3$, and our calculations based on quantum gravity approach, Eq. (\[GUPLamda\]), is very convincing. We conclude that the connection between the estimated upper bound on $\beta_0$, Eqs. (\[vgRainbow\]) and (\[vDr\]), from GW170817 event [@TheLIGOScientific:2017qsa] and the most updated observations of the PLANCK collaboration [@Aghanim:2018eyx] for the cosmological constant $\Lambda$, Eq. (\[QFTlamda\]), and our estimated value of $\Lambda(m=0)$, Eq. (\[GUPLamda\]), gives an interpretation for the cosmological constant problem in presence of the minimal length uncertainty. Conclusions \[conclusion\] =========================== In the present study, we have proposed the generalized uncertainty principle (GUP) with an addition term of quadratic momentum, from which we have driven the modified dispersion relations for GR and rainbow gravity, Eq. (\[MDRrel\]) and Eq. (\[MDRrain\]), respectively. Counting on the similarities between GUP (manifesting gravitational impacts on HUP) and the likely origin of the great discrepancy between the theoretical and observed values of the cosmological constant that in the gravitational impacts on the vacuum energy density, the present study suggests a possible solution for the long-standing cosmological constant problem ([*catastrophe of non-gravitating vacuum*]{}) that $\Lambda \simeq 10^{-47}~$GeV$^4/\hbar^3 c^3$. We have assumed that the gravitational waves propagate as a free wave. Therefore, we could drive the group velocity in terms of the GUP parameter $\beta_0$ for GR and rainbow gravity, Eq. (\[MDRbeta1\]) and Eq. (\[Rainbowbeta1\]), respectively. Moreover, we have used recent results on gravitational waves, the binary neutron stars merger, GW170817 event, in order to determine the speed of the gravitons. Then, we have calculated the difference between the speed of gravitons and that of (photons) light, at finite and visnishing GUP parameter. We have shown that the upper bound on the dimensionless GUP parameter, $\beta \sim 10^{60}$, is merely constrained by such a speed difference. We have concluded that the speed of graviton is directly related to the GUP approach utilized in. The cosmological constant problem, which is stemming from the large discrepancy between the QFT-based calculations and the cosmological observations, is tagged as $\Lambda_{QFT}/\Lambda_{exp} \sim 10^{121}$. This quite large ratio can be interpreted by features of the UV/IR correspondence and the impacts of gravity. For the earlier, the large $\Delta x$ (IR) corresponds to a large $\Delta p$ (UV) in scale of Planck momentum. For the later, the GUP approach, for instance, Eq. (\[GUPuncertainty\]), plays an essential role. We have assumed that in calculating the density of states where GUP approach is taken into account, a possible solution of the cosmological constant problem, Eq. (\[densityStates\]), can be proposed. At Planck scale, the resulting density of the states seems to impact the vacuum energy density of each quantum state, Eq. (\[GUPLamda\]). A refined value of the cosmological constant we have obtained for a novel upper bound on $\beta_0$, which - in turn - was determined from the GW170817 observations. Finally, the possible matching between the estimation of the upper bound on the GUP parameter deduced from the gravitational waves, GW170817 event, and the one estimated from the PLANCK 2018 observations seems to support the conclusion about the great importance of constructing a theory for quantum gravity. This likely helps in explaining various still-mysterious phenomena in physics. Algebra of quantum mechanical commutators and Poisson brackets \[LiouvilleTheorem\] =================================================================================== For a binary set of anticommutative functions on position and momentum, for instance, in D-dimensions, the Poisson bracket expresses their binary operation { F(x\_1, x\_D; p\_1, p\_D ), G(x\_1, x\_D; p\_1, p\_D )} &=&\ ( - ) { x\_i, p\_j } &+& { x\_i, x\_j }. During a time duration, $\delta t$, the Hamilton’s equations of motion for position and momentum can be given as x\_i\^= x\_i + x\_i, p\_i\^= p\_i + p\_i, where, x\_i, &=& { x\_i, H} t = { x\_i, p\_j } + { x\_i, x\_j} ,\ p\_i, &=& { p\_i, H} t = - { x\_i, p\_j } , where $H\equiv H(x,p;t)$ is the Hamiltonian, itself. The estimation of the change in the phase space volume during the time evolution requires to determine the Jacobain of the transformation from $(x_1, \cdots x_D;\; p_1, \cdots p_D)$ to $(x_1^\prime, \cdots x_D^\prime;\; p_1^\prime, \cdots p_D^\prime)$, i.e. d\^Dx\^ d\^D p\^= , where $\mathcal{J}$ is the Jacobain of the transformation, which can be expressed as &=& = 1 + ( + ) t. The general notations of position and momentum brackets lead to following algebraic relations {x\_i p\_i} = f\_[ij]{} (x, p), {x\_i, x\_j} = g\_[ij]{}(x,p), {p\_i,p\_j}= h\_[ij]{}(p). Thus, the Jacobain of the transformation is given as [@Fityo:2008zz] = \_[i=1]{}\^D f\_[ii]{}(x,p) = 1+ \_[i=1]{}\^D (f\_[ii]{} (x,p) - 1). \[jacobian\] Therefore the invariant phase space in D-dimension reads . Finally, the quantum density of states can be determined from .

--- abstract: 'We introduce higher-order support varieties for pairs of modules over a commutative local complete intersection ring, and give a complete description of which varieties occur as such support varieties. In the context of a group algebra of a finite elementary abelian group, we also prove a higher-order Avrunin-Scott-type theorem, linking higher-order support varieties and higher-order rank varieties for pairs of modules.' address: - | Petter Andreas Bergh\ Institutt for matematiske fag\ NTNU\ N-7491 Trondheim\ Norway - | David A. Jorgensen\ Department of mathematics\ University of Texas at Arlington\ Arlington\ TX 76019\ USA author: - 'Petter Andreas Bergh & David A. Jorgensen' title: 'Realizability and the Avrunin-Scott theorem for higher-order support varieties' --- [^1] Introduction {#sec:intro} ============ Support varieties for modules over commutative local complete intersections were introduced in [@Avramov] and [@AvramovBuchweitz], inspired by the cohomological varieties of modules over group algebras of finite groups. These geometric invariants encode several homological properties of the modules. For example, the dimension of the variety of a module equals its complexity. In particular, a module has finite projective dimension if and only if its support variety is trivial. In this paper, we define higher-order support varieties for pairs of modules over complete intersections. These varieties are defined in terms of Grassmann varieties of subspaces of the canonical vector space associated to the defining regular sequence of the complete intersection. Thus, for a fixed dimension $d$, the support varieties of order $d$ are subsets of the Grassmann variety of $d$-dimensional subspaces of the canonical vector space, under a Pl" ucker embedding into $\mathbb P^{{c \choose d}-1}$. For $d=1$, we recover the classical support varieties: the varieties of order $1$ are precisely the projectivizations of the support varieties defined in [@AvramovBuchweitz]. We show that several of the results that hold for classical support varieties also hold for the higher-order varieties. Among these is the realizability result: we give a complete description of the closed subsets of the Grassmann variety that occur as higher-order support varieties. We also prove a higher-order Avrunin-Scott result for group algebras of finite elementary abelian groups. Namely, we extend the notion of $r$-rank varieties from [@CarlsonFriedlanderPevtsova] to higher-order rank varieties of pairs of modules and show that these varieties are isomorphic to the higher-order support varieties. In Section 2 we give our definition of higher-order support varieties, and prove some of their elementary properties. In particular, we show that they are well-defined, independent of the choice of corresponding intermediate complete intersection, and are in fact closed subsets of the Grassmann variety. In Section 3 we discuss the realizability question, and in Section 4 we prove the higher-order Avrunin-Scott result. Higher-order support varieties {#sec:hdsv} ============================== In this section and the next, we fix a regular local ring $(Q, {\operatorname{\mathfrak{n}}\nolimits}, k)$ and an ideal $I$ generated by a regular sequence of length $c$ contained in ${\operatorname{\mathfrak{n}}\nolimits}^2$. We denote by $R$ the complete intersection ring $$R = Q/I,$$ and by $V$ the $k$-vector space $$V=I/{\operatorname{\mathfrak{n}}\nolimits}I.$$ For an element $f\in I$, we let $\overline f$ denote its image in $V$. If the codimension of the complete intersection $R=Q/I$ is at least 2, then $V$ has dimension at least 2, and it makes sense to consider subspaces $W$ of $V$. Each such subspace has many corresponding complete intersections, in the following sense: if $W$ is a subspace of $V$, then choosing preimages in $I$ of a basis of $W$ we obtain another regular sequence [@BrunsHerzog Theorem 2.1.2(c,d)], and the ideal $J\subseteq I$ it generates. We thus get natural projections of complete intersections $Q\to Q/J\to R$. We call $Q/J$ a *complete intersection intermediate to $Q$ and $R$*, or when the context is clear, simply an *intermediate complete intersection*. We now give our definition of higher-order support variety. We fix a basis of $V$, and let ${\operatorname{G}\nolimits}_d(V)$ denote the Grassmann variety of $d$th order subspaces of $V$ under the Pl" ucker embedding into $\mathbb P^{{c \choose d}-1}$ with respect to the chosen basis of $V$. We set $$V_R^d(M,N)=\{p_W\in{\operatorname{G}\nolimits}_d(V)\mid {\operatorname{Ext}\nolimits}_{Q/J}^i(M,N)\ne 0 \text{ for infinitely many $i$}\},$$ where $W$ is a $d$th order subspace of $V$, $p_W$ is the corresponding point in the Grassmann variety ${\operatorname{G}\nolimits}_d(V)$, and $Q/J$ is an intermediate complete intersection corresponding to $W$. We also define ${\operatorname{V}\nolimits}_R^d(M)={\operatorname{V}\nolimits}_R^d(M,k)$. We note that ${\operatorname{V}\nolimits}_R^1(M,N)$ is the projectivization of the affine support variety ${\operatorname{V}\nolimits}_R(M,N)$ defined in [@AvramovBuchweitz]. There are two aspects of the definition which warrant further discussion. 1. \[independent\] The definition is independent of the chosen intermediate complete intersection $Q/J$ corresponding to $W$, and 2. \[closed\] ${\operatorname{V}\nolimits}_R^d(M,N)$ is a closed set in $G_d(V)$. We next give proofs of these two statements. Let $Q/J$ and $Q/J'$ be two complete intersections intermediate to $Q$ and $R$. The condition that $$(J+{\operatorname{\mathfrak{n}}\nolimits}I)/{\operatorname{\mathfrak{n}}\nolimits}I=(J'+{\operatorname{\mathfrak{n}}\nolimits}I)/{\operatorname{\mathfrak{n}}\nolimits}I$$ in $V$ defines an equivalence relation on the set of such intermediate complete intersections. The following result addresses (\[independent\]) above. Suppose that $Q/J$ and $Q/J'$ are equivalent complete intersections intermediate to $Q$ and $R$, that is, $(J+{\operatorname{\mathfrak{n}}\nolimits}I)/{\operatorname{\mathfrak{n}}\nolimits}I=(J'+{\operatorname{\mathfrak{n}}\nolimits}I)/{\operatorname{\mathfrak{n}}\nolimits}I$ in $V$. Then for all finitely generated $R$-modules $M$ and $N$ one has ${\operatorname{Ext}\nolimits}_{Q/J}^i(M,N)=0$ for all $i\gg 0$ if and only if ${\operatorname{Ext}\nolimits}_{Q/J'}^i(M,N)=0$ for all $i\gg 0$. Let $W=(J+{\operatorname{\mathfrak{n}}\nolimits}I)/{\operatorname{\mathfrak{n}}\nolimits}I$ and consider the natural map of $k$-vector spaces $\varphi_{J}:J/{\operatorname{\mathfrak{n}}\nolimits}J\to W\subseteq V$ defined by $f+{\operatorname{\mathfrak{n}}\nolimits}J \mapsto f+{\operatorname{\mathfrak{n}}\nolimits}I$. This is an isomorphism: it is onto by construction, and one-to-one since $J\cap {\operatorname{\mathfrak{n}}\nolimits}I={\operatorname{\mathfrak{n}}\nolimits}J$. The condition that $(J+{\operatorname{\mathfrak{n}}\nolimits}I)/{\operatorname{\mathfrak{n}}\nolimits}I=(J'+{\operatorname{\mathfrak{n}}\nolimits}I)/{\operatorname{\mathfrak{n}}\nolimits}I$ is equivalent to $\varphi_J(J/{\operatorname{\mathfrak{n}}\nolimits}J)=\varphi_{J'}(J'/{\operatorname{\mathfrak{n}}\nolimits}J')$. By [@BerghJorgensen Proposition 3.2], one has the equality $\varphi_J({\operatorname{V}\nolimits}_{Q/J}(M,N))=\varphi_{J'}({\operatorname{V}\nolimits}_{Q/J'}(M,N))$, where ${\operatorname{V}\nolimits}_{Q/J}(M,N)$ denotes the affine support variety of $M$ and $N$ over the complete intersection $Q/J$. By [@AvramovBuchweitz Proposition 2.4(1) and Theorem 2.5] one has that ${\operatorname{Ext}\nolimits}_{Q/J}^i(M,N)=0$ for all $i\gg 0$ if and only if ${\operatorname{V}\nolimits}_{Q/J}(M,N)=\{0\}$. The same holds over $Q/J'$, and thus the result follows by the injectivity of $\varphi_{J}$. Next, we address the second point in the remark. \[proposition:closed\] For all finitely generated $R$-modules $M$ and $N$ one has that ${\operatorname{V}\nolimits}_R^d(M,N)$ is a closed set in $G_d(V)$. This result follows from an incidence correspondence (see, for example, [@Harris Example 6.14]), as we now describe. Set $$\Gamma =\{(p_W,x)\in{\operatorname{G}\nolimits}_d(V)\times {\operatorname{G}\nolimits}_1(V) \mid x\in W\cap{\operatorname{V}\nolimits}_R^1(M,N)\}.$$ Since $\Gamma$ is an incidence correspondence, it is a closed subset of the product space ${\operatorname{G}\nolimits}_d(V)\times {\operatorname{G}\nolimits}_1(V)$. We have the two natural projections $$\xymatrixrowsep{2pc} \xymatrixcolsep{0pc} \xymatrix { & {\operatorname{G}\nolimits}_d(V)\times {\operatorname{G}\nolimits}_1(V) \ar[dl]_\pi \ar[dr]^{\pi'}& \\ {\operatorname{G}\nolimits}_d(V) & & {\operatorname{G}\nolimits}_1(V) }$$ Now by classical results from elimination theory (see, for example, [@Eisenbud Theorem 14.1]), the image of $\Gamma$ under $\pi$ is closed in ${\operatorname{G}\nolimits}_d(V)$. It suffices now to show that $\pi(\Gamma)={\operatorname{V}\nolimits}_R^d(M,N)$. We have $p_W\in\pi(\Gamma)$ if and only if $x\in{\operatorname{V}\nolimits}_R^1(M,N)$ for some $x\in W$. This is equivalent to ${\operatorname{Ext}\nolimits}_{Q/(f)}^i(M,N)\ne 0$ for infinitely many $i$ and for some $f\in I$ with $\overline f=x\in W$. By [@AvramovBuchweitz Proposition 2.4(1) and Theorem 2.5], this condition is the same as ${\operatorname{Ext}\nolimits}_{Q/J}^i(M,N)\ne 0$ for infinitely many $i$. By definition, this happens if and only if $p_W\in{\operatorname{V}\nolimits}_R^d(M,N)$. \(1) Let ${\operatorname{\mathcal{T}}\nolimits}=\{(p_W,x)\in {\operatorname{G}\nolimits}_d(V)\times {\operatorname{G}\nolimits}_1(V) \mid x\in W\}$. Then the map $\tau:{\operatorname{\mathcal{T}}\nolimits}\to G_d(V)$ given by $\tau(p_W,x)=p_W$ is the tautological bundle over the Grassmann variety ${\operatorname{G}\nolimits}_d(V)$. For $\Gamma$ as in the proof of Proposition \[proposition:closed\], we have $\Gamma\subseteq{\operatorname{\mathcal{T}}\nolimits}$, and $\tau(\Gamma)={\operatorname{V}\nolimits}_R^d(M,N)$. Thus ${\operatorname{V}\nolimits}_R^d(M,N)$ may be interpreted as the image under the tautological bundle of the fiber of ${\operatorname{V}\nolimits}_R^1(M,N)$ in ${\operatorname{\mathcal{T}}\nolimits}$. \(2) In the definition of ${\operatorname{V}\nolimits}_R^d(M,N)$, a specific basis of $V$ was chosen. We remark that the definition is independent of the choice of basis, in the sense that if another basis of $V$ is chosen, then the two higher-order support varieties are isomorphic. Indeed, this is true for the first order affine varieties ${\operatorname{V}\nolimits}_R(M,N)$ by [@AvramovBuchweitz Remark 2.3]. It then follows that the same is true for the projectivizations ${\operatorname{V}\nolimits}_R^1(M,N)$, namely, there is an automorphism $\xi:{\operatorname{G}\nolimits}_1(V) \to {\operatorname{G}\nolimits}_1(V)$ such that if ${\operatorname{V}\nolimits}_R^1(M,N)$ is the support variety with respect to the first basis, and ${\operatorname{V}\nolimits}_R^1(M,N)'$ is the support variety with respect to the second, then $\xi({\operatorname{V}\nolimits}_R^1(M,N))={\operatorname{V}\nolimits}_R^1(M,N)'$. The general result for the higher-order support varieties follows from the incidence correspondence from the proof above. We now give basic properties of higher-order support varieties, akin to those of the one-dimensional affine support varieties. \[thm:props\] The following hold for finitely generated $R$-modules $M$ and $N$. 1. ${\operatorname{V}\nolimits}_R^d(k)={\operatorname{G}\nolimits}_d(V)$. 2. ${\operatorname{V}\nolimits}_R^d(M,N)={\operatorname{V}\nolimits}_R^d(N,M)$. For $d=1$, we moreover have ${\operatorname{V}\nolimits}_R^1(M,N)={\operatorname{V}\nolimits}_R^1(M)\cap{\operatorname{V}\nolimits}_R^1(N)$. 3. ${\operatorname{V}\nolimits}_R^d(M,M)={\operatorname{V}\nolimits}_R^d(k,M)={\operatorname{V}\nolimits}_R^d(M)$. 4. If $M'$ is a syzygy of $M$ and $N'$ is a syzygy of $N$, then ${\operatorname{V}\nolimits}_R^d(M,N)={\operatorname{V}\nolimits}_R^d(M',N')$. \[syzygy\] 5. If $0\to M_1\to M_2\to M_3\to 0$ and $0\to N_1\to N_2\to N_3\to 0$ are short exact sequences of finitely generated $R$-modules, then for $\{h,i,j\}=\{1,2,3\}$ there are inclusions $${\operatorname{V}\nolimits}_R^d(M_h,N)\subseteq{\operatorname{V}\nolimits}_R^d(M_i,N)\cup{\operatorname{V}\nolimits}_R^d(M_j,N);$$ $${\operatorname{V}\nolimits}_R^d(M,N_h)\subseteq{\operatorname{V}\nolimits}_R^d(M,N_i)\cup{\operatorname{V}\nolimits}_R^d(M,N_j).$$ 6. If $M$ is Cohen-Macaulay of codimension $m$, then $${\operatorname{V}\nolimits}_R^d(M)={\operatorname{V}\nolimits}_R^d({\operatorname{Ext}\nolimits}_R^m(M,R)).$$ In particular, if $M$ is a maximal Cohen-Macaulay $R$-module, then ${\operatorname{V}\nolimits}_R^d(M)={\operatorname{V}\nolimits}_R^d({\operatorname{Hom}\nolimits}_R(M,R))$. 7. \[regseq\] If $x_1,\dots,x_d$ is an $M$-regular sequence, then $${\operatorname{V}\nolimits}_R^d(M)={\operatorname{V}\nolimits}_R^d(M/(x_1,\dots,x_d)M).$$ The proof of properties (1)–(7) for the affine one-dimensional support varieties ${\operatorname{V}\nolimits}_R(M,N)$ are given in [@AvramovBuchweitz] (see also [@BerghJorgensen].) Since ${\operatorname{V}\nolimits}_R^1(M,N)$ is simply the projectivization of ${\operatorname{V}\nolimits}_R(M,N)$, the same properties also hold for these varieties. Finally, properties (1)–(7) for $d>1$ follow from the $d=1$ case, as we now indicate. For a subset $X$ of ${\operatorname{G}\nolimits}_1(V)$, we let $$\Gamma(X)=\{(p_W,x)\in{\operatorname{G}\nolimits}_d(V)\times{\operatorname{G}\nolimits}_1(V)\mid x\in W\cap X\}.$$ The proofs make repeated use of the fact that ${\operatorname{V}\nolimits}_R^d(M,N)=\pi(\Gamma({\operatorname{V}\nolimits}_R^1(M,N)))$, where $\pi$ is as in the proof of Proposition \[proposition:closed\]. For example, for (1) we have ${\operatorname{V}\nolimits}_R^d(k)=\pi(\Gamma({\operatorname{V}\nolimits}_R^1(k)))=\pi({\operatorname{G}\nolimits}_1(V))={\operatorname{G}\nolimits}_d(V)$. For (2), we use the fact that ${\operatorname{V}\nolimits}_R^1(M,N)={\operatorname{V}\nolimits}_R^1(M)\cap{\operatorname{V}\nolimits}_R^1(N)={\operatorname{V}\nolimits}_R^1(N,M)$. Therefore ${\operatorname{V}\nolimits}_R^d(M,N)=\pi\left(\Gamma\left({\operatorname{V}\nolimits}_R^1(M,N)\right)\right)=\pi\left(\Gamma\left({\operatorname{V}\nolimits}_R^1(N,M)\right)\right) ={\operatorname{V}\nolimits}_R^d(N,M)$ To prove (3), we use the equalities ${\operatorname{V}\nolimits}_R^d(M,M)=\pi(\Gamma({\operatorname{V}\nolimits}_R^1(M,M)))=\pi(\Gamma({\operatorname{V}\nolimits}_R^1(k,M)))={\operatorname{V}\nolimits}_R^d(k,M)$. The remaining equality and (4) are proved similarly. To prove (5), we use the fact that for subsets $X$ and $Y$ of ${\operatorname{G}\nolimits}_1(V)$ one has $\Gamma(X\cup Y)=\Gamma(X)\cup\Gamma(Y)$. (We also use the fact that $\pi$ preserves unions, and both $\pi$ and $\Gamma$ preserve containment.) Therefore $$\begin{aligned} {\operatorname{V}\nolimits}_R^d(M_h,N)&=\pi(\Gamma({\operatorname{V}\nolimits}_R^1(M_h,N)))\\ &\subseteq\pi(\Gamma({\operatorname{V}\nolimits}_R^1(M_i,N)\cup{\operatorname{V}\nolimits}_R^1(M_j,N)))\\ &=\pi(\Gamma({\operatorname{V}\nolimits}_R^1(M_i,N)))\cup\pi(\Gamma(V_R^1(M_j,N)))\\ &={\operatorname{V}\nolimits}_R^d(M_i,N)\cup{\operatorname{V}\nolimits}_R^d(M_j,N).\end{aligned}$$ The proofs of (6) and (7) are analogous to the proofs of [@AvramovBuchweitz Theorem 5.6(10)] and [@AvramovIyengar 7.4] (see also [@BerghJorgensen Theorem 2.2(7) and (8)].) We can extend Proposition 2.4(1) of [@AvramovBuchweitz], to a sort of generalized Dade’s Lemma, in the projective context. Fix $1\le d\le c$. Then ${\operatorname{Ext}\nolimits}_R^i(M,N)=0$ for all $i\gg 0$ if and only if ${\operatorname{V}\nolimits}_R^d(M,N)=\emptyset$. By [@AvramovBuchweitz Proposition 2.4(1) and Theorem 2.5], ${\operatorname{Ext}\nolimits}_R^i(M,N)=0$ for all $i\gg 0$ if and only if ${\operatorname{V}\nolimits}_R^1(M,N)=\emptyset$. The latter holds if and only if $\Gamma=\Gamma({\operatorname{V}\nolimits}_R^1(M,N))=\emptyset$, which in turn holds if and only if ${\operatorname{V}\nolimits}_R^d(M,N)=\pi(\Gamma)=\emptyset$, where $\Gamma$ and $\pi$ are from the proof of Proposition \[proposition:closed\]. Realizability ============= In this section we give a complete description of which closed subsets of ${\operatorname{G}\nolimits}_d(V)$ can possibly occur as the $d$th order support variety ${\operatorname{V}\nolimits}_R^d(M,N)$ of a pair of finitely generated $R$-modules $(M,N)$. The basis of the description is the following result in the first order case. \[dimensionone\] Every closed subset of ${\operatorname{G}\nolimits}_1(V)$ is the support variety of some finitely generated $R$-module. Specifically, if $Z$ is a closed subset of ${\operatorname{G}\nolimits}_1(V)$, then there exists a finitely generated $R$-module $M$ such that $Z={\operatorname{V}\nolimits}_R^1(M,k)$. This is well-known in the affine case, see, for example, [@Bergh]. Since every closed set in ${\operatorname{G}\nolimits}_1(V)$ is the projectivization of a cone in $V$, and ${\operatorname{V}\nolimits}_R^1(M,N)$ is the projectivization of ${\operatorname{V}\nolimits}_R(M,N)$, the result follows. The framework of the proof of Proposition \[proposition:closed\] allows us to complete the description of realizable higher-order varieties. Recall that $\pi$ denotes the projection map ${\operatorname{G}\nolimits}_d(V)\times{\operatorname{G}\nolimits}_1(V)\to{\operatorname{G}\nolimits}_d(V)$. \[dimensiond\] For a closed subset $Z$ of ${\operatorname{G}\nolimits}_1(V)$, set $$\Gamma(Z)=\{(p_W,x)\in{\operatorname{G}\nolimits}_d(V)\times{\operatorname{G}\nolimits}_1(V) \mid x\in W\cap Z\}.$$ Let $Y$ be a closed subset of ${\operatorname{G}\nolimits}_d(V)$. Then $Y={\operatorname{V}\nolimits}_R^d(M,N)$ for a pair of finitely generated $R$-modules $(M,N)$ if and only if $Y=\pi(\Gamma(Z))$ for some closed subset $Z$ of ${\operatorname{G}\nolimits}_1(V)$. Suppose that $Y={\operatorname{V}\nolimits}_R^d(M,N)$ for a pair of finitely generated $R$-modules $(M,N)$. Then the proof of Proposition \[proposition:closed\] shows that $Y=\pi(\Gamma({\operatorname{V}\nolimits}_R^1(M,N)))$. Conversely, suppose that $Y=\pi(\Gamma(Z))$ for some closed subset $Z$ of ${\operatorname{G}\nolimits}_1(V)$. Then Theorem \[dimensionone\] shows that $Z={\operatorname{V}\nolimits}_R^1(M,N)$ for some pair of finitely generated $R$-modules $(M,N)$. Thus $Y=\pi(\Gamma({\operatorname{V}\nolimits}_R^1(M,N)))={\operatorname{V}\nolimits}_R^d(M,N)$, again from the proof of Proposition \[proposition:closed\]. Theorem \[dimensiond\] shows that, in contrast to first order support varieties, the realizability of varieties in ${\operatorname{G}\nolimits}_d(V)$ for $d>1$ as $d$th order support varieties of a pair of finitely generated $R$-modules is more restrictive. Indeed, consider a smallest nontrivial first order support variety ${\operatorname{V}\nolimits}_R^1(M,N)$, namely, one consisting of a single point $x$. Then ${\operatorname{V}\nolimits}_R^d(M,N)$ consists of all $d$-dimensional planes in $V$ containing $x$. Changing the basis of $V$ if necessary, we can assume that $x=(1,0,\dots,0)\in{\operatorname{G}\nolimits}_1(V)$. Then there is an obvious bijective correspondence between $d$-dimensional subspaces of $V$ containing $x$, and $(d-1)$-dimensional subspaces of a $(c-1)$-dimensional $k$-vector space. Thus $\dim{\operatorname{V}\nolimits}_R^d(M,N)=\dim {\operatorname{G}\nolimits}_{d-1}(k^{c-1})=(d-1)(c-d)$. In particular, we have $\dim{\operatorname{V}\nolimits}_R^{c-1}(M,N)=c-2$, which is of codimension one in ${\operatorname{G}\nolimits}_{c-1}(V)$, and this is when ${\operatorname{V}\nolimits}_R^1(M,N)$ is nontrivially as small as possible. The following example illustrates the previous discussion. Let $k$ be a field (of arbitrary characteristic), and $Q=k[[x_1,\dots,x_c]]$. Then $Q$ is a regular local ring with maximal ideal ${\operatorname{\mathfrak{n}}\nolimits}=(x_1,\dots,x_c)$. For $I=(x_1^2,\dots,x_c^2)$, the quotient ring $R=Q/I$ is a codimension $c$ complete intersection. Let $M=R/(x_1)$. Then it is not hard to show that relative to the basis $\overline {x_1^2},\dots,\overline {x_c^2}$ of $V=I/{\operatorname{\mathfrak{n}}\nolimits}I$, the one-dimensional support variety of $M$ is ${\operatorname{V}\nolimits}^1_R(M,k)=\{(1,0,\dots,0)\}$. Thus we have $\dim{\operatorname{V}\nolimits}_R^d(M,k)=(d-1)(c-d)$, for $1\le d\le c-1$. Higher-order rank varieties and a higher-order Avrunin-Scott Theorem ==================================================================== In this final section we consider complete intersections of a special form, namely, those which arise as the group algebra $kE$ of a finite elementary abelian $p$-group $E$, where $k$ has characteristic $p$. In this case one has $$kE\cong k[x_1,\dots,x_c]/(x_1^p,\dots,x_c^p).$$ Note that by assigning $\deg x_i=1$ for $1\le i \le c$, the $k$-algebra $kE$ is standard-graded. We let $kE_1$ denote the degree-one component of $kE$; this is a $k$-vector space of dimension $c$. For any linear form $u$ of $kE_1$ one has $u^p=0$, and thus the subalgebra $k[u]$ of $kE$ generated by $u$ is isomorphic to $k[x]/(x^p)$ (for $x$ an indeterminate). Since $k[u]$ is a principal ideal ring, every finitely generated $k[u]$-module is a direct sum of a free module and a torsion module. Recall from [@Carlson] that the *rank variety* ${\operatorname{W}\nolimits}_E(M)$ of a $kE$-module $M$ is the set of those linear forms $u\in kE_1$ such that the torsion part of $M$ as a $k[u]$-module is nonzero. It was conjectured by Carlson [@Carlson] and subsequently proven by Avrunin and Scott [@AvruninScott] that the rank variety and the group cohomological support variety ${\operatorname{V}\nolimits}_{kE}(M)$ of a $kE$-module agree. Recall that $I$ denotes the ideal $(x_1^p,\dots,x_c^p)$, and $V$ the $k$-vector space $I/{\operatorname{\mathfrak{n}}\nolimits}I$, where ${\operatorname{\mathfrak{n}}\nolimits}$ is the maximal ideal $(x_1,\dots,x_c)$. We now want to show that the classical Avrunin-Scott theorem mentioned above is a special case of a more general result involving the higher-order varieties. We generalize the definition of $d$th order rank varieties from [@CarlsonFriedlanderPevtsova] (which they call $d$-rank varieties) to $d$th order rank varieties ${\operatorname{W}\nolimits}^d_E(M,N)$ of pairs of modules $(M,N)$. Fix a basis of $kE_1$, and consider the Grassmann variety ${\operatorname{G}\nolimits}_d(kE_1)$ of $d$-dimensional subspaces of $kE_1$ under the Pl" ucker embedding into $\mathbb P^{{c \choose d}-1}$ with respect to the chosen basis. We set $${\operatorname{W}\nolimits}^d_E(M,N)=\{p_W\in {\operatorname{G}\nolimits}_d(kE_1) \mid {\operatorname{Ext}\nolimits}^i_{k[W]}(M,N)\ne 0 \text{ for infinitely many $i$}\}$$ where ${\operatorname{G}\nolimits}_d(kE_1)$ is the Grassmann variety of $d$-dimensional subspaces of $kE_1$, $p_W$ is the point in ${\operatorname{G}\nolimits}_d(kE_1)$ corresponding to the $d$-dimensional subspace $W$, and $k[W]$ is the subalgebra of $kE$ generated by $W$. Consider the Frobenius isomorphism $\Phi:k \to k$ given by $\Phi(a)=a^p$. We have a $\Phi$-equivariant isomorphism of $k$-vector spaces $$\alpha: kE_1 \to V$$ defined as follows. For $u\in kE_1$, we choose a preimage $\widetilde u$ in $Q$, and then we set $\alpha(u)=\widetilde u^p+{\operatorname{\mathfrak{n}}\nolimits}I\in V$. It is clear that $\alpha$ is a $\Phi$-equivariant homomorphism of $k$-vector spaces, which is defined independent of the choice of preimage. Since $k$ is algebraically closed, it contains $p$th roots, and therefore $\alpha$ is onto. Since $\dim kE_1=\dim V$, $\alpha$ is also one-to-one. Taking as a basis for $V$ the image under $\alpha$ of the chosen basis of $kE_1$, we obtain an induced $\Phi$-equivariant isomorphism of Grassmann varieties $$\beta:{\operatorname{G}\nolimits}_d(kE_1) \to {\operatorname{G}\nolimits}_d(V)$$ with respect to these bases. Specifically, let $p_W$ be a point in ${\operatorname{G}\nolimits}_d(kE_1)$, and $W$ the associated $d$-dimensional subspace of $kE_1$. Let $\widetilde W^p$ denote the ideal of $Q$ generated by the $p$th powers of linear preimages in $Q$ of a basis of $W$. Then $\beta(p_W)$ is the point in ${\operatorname{G}\nolimits}_d(V)$ (with respect to the chosen basis of $V$) corresponding to the subspace $\widetilde W^p+{\operatorname{\mathfrak{n}}\nolimits}I/{\operatorname{\mathfrak{n}}\nolimits}I$. \[ASthm\] Given finitely generated $kE$-modules $M$ and $N$, one has $$\beta({\operatorname{W}\nolimits}_E^d(M,N))={\operatorname{V}\nolimits}^d_{kE}(M,N).$$ The proof relies on the following lemma, which is a statement extracted from the proof of [@Avramov Theorem (7.5)]. For completeness we include the proof here. For any non-zero linear form $u\in kE_1$ we choose a preimage $\widetilde u$ in $Q=k[x_1,\dots,x_c]$, which is also a linear form, and define a homomorphism from $\mu :k[u]\to Q/(\widetilde u^p)$ by sending $u$ to $\widetilde u+(\widetilde u^p)$. Note that $Q/(\widetilde u^p)$ is free when regarded as module over $k[u]$ via $\mu$. We have the commutative diagram of ring homomorphisms $$\xymatrixrowsep{2pc} \xymatrixcolsep{1.9pc} \xymatrix{ & Q/(\widetilde u^p) \ar[d]\\ k[u] \ar@{^{(}->}[r] \ar[ur]^\mu & kE }$$ where the vertical map is the natural projection. In particular, the action of $k[u]$ on a $kE$-module $M$ factors through $\mu$. Let $M$ be a finitely generated $kE$-module. Then $M$ has finite projective dimension over $k[u]$ if and only if it has finite projective dimension over $Q/(\widetilde u^p)$. The proof follows part of that of [@Avramov Theorem (7.5)]. Suppose that $M$ has finite projective dimension over $Q/(\widetilde u^p)$. Since $Q/(\widetilde u^p)$ is free over $k[u]$ any free resolution of $M$ over $Q/(\widetilde u^p)$ is also one of $M$ over $k[u]$. Thus $M$ has a finite free resolution over $k[u]$. Conversely, suppose $M$ is free as a $k[u]$-module. Let $F$ be a minimal free resolution of $M$ over $Q/(\widetilde u^p)$. Since $F$ is also a free resolution of $M$ over $k[u]$ and ${\operatorname{Tor}\nolimits}_i^{k[u]}(M,k)=0$ for all $i>0$, we see that $F\otimes_{k[u]}k$ is a minimal free resolution of $M \otimes_{k[u]}k$ over $Q/(\widetilde u^p)\otimes_{k[u]}k\cong Q/(\widetilde u)$. Since $Q/(\widetilde u)$ is regular and $F\otimes_{k[u]}k$ is a minimal, we must have that $F_c\otimes_{k[u]}k=0$, and this implies $F_c=0$. Thus $F$ is a finite free resolution, and so $M$ has finite projective dimension over $Q/(\widetilde u)$. We now give a proof of Theorem \[ASthm\]. Suppose that $p_W\in{\operatorname{W}\nolimits}_E^d(M,N)$. Then by definition there exist infinitely many nonzero ${\operatorname{Ext}\nolimits}^i_{k[W]}(M,N)$. Therefore, by Dade’s Lemma, there exist infinitely nonzero ${\operatorname{Ext}\nolimits}_{k[u]}^i(M,N)$ for some linear form $u\in W$. Thus both $M$ and $N$ have infinite projective dimension over $k[u]$. Therefore, by the lemma, both $M$ and $N$ have infinite projective dimension over $Q/(\widetilde u^p)$, and so it follows from [@AvramovBuchweitz Proposition 5.12] that there exist infinitely many nonzero ${\operatorname{Ext}\nolimits}^i_{Q/(\widetilde u^p)}(M,N)$. This implies that there exist infinitely many nonzero ${\operatorname{Ext}\nolimits}^i_{Q/(\widetilde W^p)}(M,N)$, where $\widetilde W^p$ represents the ideal generated by the $p$th powers of linear preimages in $Q$ of a basis of $W$. This gives $\beta(p_W)\in{\operatorname{V}\nolimits}^d_{kE}(M,N)$. For the reverse containment we just retrace our steps, noting that any $f\in I$ is equivalent mod ${\operatorname{\mathfrak{n}}\nolimits}I$ to an element of the form $a_1x_1^p+\cdots+a_cx_c^p=(\sqrt[p]{a_1}x_1+\cdots+\sqrt[p]{a_c}x_c)^p$, $a_i\in k$, and hence it is clear how to employ the previous lemma. [CFP]{} L.L. Avramov, *Modules of finite virtual projective dimension*, Invent. Math. 96 (1989), no. 1, 71–101. L.L. Avramov, R.-O. Buchweitz, *Support varieties and cohomology over complete intersections*, Invent. Math. 142 (2000), no. 2, 285–318. L.L. Avramov, S. B. Iyengar, *Constructing modules with prescribed cohomological support*, Illinois J. Math. 51 (2007), no. 1, 1–20. G.S. Avrunin, L.L.  Scott *Quillen stratification for modules*, Invent. Math. 66 (1982), no. 2, 277–286. P.A. Bergh *On support varieties for modules over complete intersections*, Proc. Amer. Math. Soc. 135 (2007), no. 12, 3795–3803. P.A. Bergh, D.A. Jorgensen *Support varieties over complete intersections made easy*, preprint. W. Bruns, J. Herzog *Cohen-Macaulay Rings*, Cambridge studies in advanced mathematics 39, Cambridge University Press, Cambridge, 1993. J.F. Carlson, *The varieties and the cohomology ring of a module*, J. Algebra 85 (1983), no. 1, 104–143. J.F. Carlson, E.M. Friedlander, J. Pevtsova *Representations of elementary abelian $p$-groups and bundles on Grassmannians*, Adv. Math. 229 (2012), no. 5, 2985–3051. D. Eisenbud, *Commutative algebra, with a view toward algebraic geometry*, Graduate Texts in Mathematics 150, Springer-Verlag, New York, 1995. J. Harris, *Algebraic Geometry, a First Course* Graduate Texts in Mathematics 133, Springer-Verlag, New York, 1992. [^1]: Part of this work was done while we were visiting the Mittag-Leffler Institute in February and March 2015. We would like to thank the organizers of the Representation Theory program.

--- abstract: 'We study the phase diagram of a Dirac semimetal in a magnetic field at a nonzero charge density. It is shown that there exists a critical value of the chemical potential at which a first-order phase transition takes place. At subcritical values of the chemical potential the ground state is a gapped state with a dynamically generated Dirac mass and a broken chiral symmetry. The supercritical phase is the normal (gapless) phase with a nontrivial chiral structure: it is a Weyl semimetal with a pair of Weyl nodes for each of the original Dirac points. The nodes are separated by a dynamically induced chiral shift. The direction of the chiral shift coincides with that of the magnetic field and its magnitude is determined by the quasiparticle charge density, the strength of the magnetic field, and the strength of the interaction. The rearrangement of the Fermi surface accompanying this phase transition is described.' author: - 'E. V. Gorbar' - 'V. A. Miransky' - 'I. A. Shovkovy' date: 'July 26, 2013' title: Engineering Weyl nodes in Dirac semimetals by a magnetic field --- Introduction {#1} ============ During past decades, a remarkable overlap of such seemingly different areas in physics as condensed matter and relativistic physics took place (for a recent review, see Ref. ). It was especially clearly manifested in studying graphene.[@graphene] Such well known phenomena as the Klein paradox, the dynamics of a supercritical charge, and the dynamical generation of a Dirac mass in a magnetic field (magnetic catalysis), revealed in relativistic field theory, were first observed in studies of graphene (see Refs. ). In the present paper, we will consider manifestations in condensed matter of another relativistic phenomenon: the rearrangement of the Fermi surface in three dimensional relativistic matter in a magnetic field. The original motivation for studying this phenomenon was its possible realization in magnetars and pulsars, and in heavy ion collisions.[@Gorbar:2009bm; @Gorbar:2011ya] But as we discuss below, it could also be relevant for such new materials as Dirac and Weyl semimetals. Dirac and Weyl semimetals possess low-energy quasiparticles near the Fermi surface, which are described by the Dirac and Weyl equation, respectively.[@review] As was established long ago, an example of a semimetal whose low-energy effective theory includes three dimensional Dirac fermions is yielded by bismuth (for reviews, see Refs. ). On the other hand, examples of the realization of the Weyl semimetals have been considered only recently.[@Wan; @Burkov1; @Burkov2] Weyl semimetals, which are three dimensional analogs of graphene, present a new class of materials with nontrivial topological properties.[@Volovik] Since their electronic states in the vicinity of Weyl nodes have a definite chirality, this leads to quite unique transport and electromagnetic properties of these materials. The most interesting signatures of Dirac and Weyl semimetals discussed in the literature [@Burkov1; @1210.6352; @Franz; @Carbotte; @Basar; @Abanin; @Landsteiner] are connected with different nondissipative transport phenomena intimately related to the axial anomaly.[@anomaly] Many of them were previously suggested in studies of heavy ion collisions (for a review, see Ref. ). In this paper, we will consider a different signature of Dirac semimetals: a dynamical rearrangement of their Fermi surfaces in a magnetic field. As we show below, this rearrangement is quite spectacular: a Dirac semimetal is transformed into a Weyl one. The resulting Weyl semimetal has a pair of Weyl nodes for each of the original Dirac points. Each pair of the nodes is separated by a dynamically induced (axial) vector $2\mathbf{b}$, whose direction coincides with the direction of the magnetic field. The magnitude of the vector $\mathbf{b}$ is determined by the quasiparticle charge density, the strength of the magnetic field, and the strength of the interaction. This phenomenon of the dynamical transformation of Dirac into Weyl semimetals is a condensed matter analog of the previously studied dynamical generation of the chiral shift parameter in magnetized relativistic matter in Refs. . This paper is organized as follows. In Sec. \[section2\] we introduce the model and set up the notations. The gap equation for the fermion propagator in the model is derived in Sec. \[section3\]. We show that, at a nonzero charge density, a pair of Weyl nodes necessarily arises in the normal phase of a Dirac metal as soon as a magnetic field is turned on. In Sec. \[section4\] a perturbative solution of the gap equation describing the normal phase of the model is analyzed. A nonperturbative solution with a dynamical gap that spontaneously breaks the chiral symmetry is analyzed in Sec. \[section5\]. A phase transition between the normal phase and the phase with chiral symmetry breaking is revealed and described. In Sec. \[section6\] we compare the dynamics in Dirac semimetals and graphene. A deep connection of the normal phase of a Dirac metal in a magnetic field with the quantum Hall state with the filling factor $\nu = 2$ in graphene is pointed out. The discussion of the results and conclusions is given in Sec. \[section7\]. For convenience, throughout this paper, we set $\hbar=1$. Model {#section2} ===== As stated in the Introduction, the main goal of this paper is to show that a dynamical transformation of Dirac semimetals into Weyl ones can be achieved by applying an external magnetic field to the former. It is convenient, however, to start our discussion from writing down the general form of the low-energy Hamiltonian for a Weyl semimetal, $$H^{\rm (W)}=H^{\rm (W)}_0+H_{\rm int}, \label{Hamiltonian-model-Weyl}$$ where $$H^{\rm (W)}_0=\int d^3r \left[\,v_F \psi^{\dagger} (\mathbf{r})\left( \begin{array}{cc} \bm{\sigma}\cdot(-i\bm{\nabla}-\mathbf{b}) & 0\\ 0 & -\bm{\sigma}\cdot(-i\bm{\nabla}+\mathbf{b}) \end{array} \right)\psi(\mathbf{r})-\mu_{0}\, \psi^{\dagger} (\mathbf{r})\psi(\mathbf{r}) \right] \label{free-Hamiltonian}$$ is the Hamiltonian of the free theory, which describes two Weyl nodes of opposite (as required by the Nielsen–Ninomiya theorem [@NN]) chirality separated by vector $2\mathbf{b}$ in momentum space. In the rest of this paper, following the terminology of Refs. , we will call $\mathbf{b}$ the chiral shift parameter. There are two reasons for choosing this terminology. First, as Eq. (\[free-Hamiltonian\]) implies, vector $\mathbf{b}$ shifts the positions of Weyl nodes from the origin in the momentum space and, secondly, the shift has opposite signs for fermions of different chiralities. The other notations are: $v_F$ is the Fermi velocity, $\mu_{0}$ is the chemical potential, and $\bm{\sigma}=(\sigma_x,\sigma_y,\sigma_z)$ are Pauli matrices associated with the conduction-valence band degrees of freedom in a generic low-energy model.[@Burkov3] Based on the similarity of the latter to the spin matrices in the relativistic Dirac equation, we will call them pseudospin matrices. The interaction part of the Hamiltonian describes the Coulomb interaction, i.e., $$H_{\rm int} = \frac{1}{2}\int d^3rd^3r^{\prime}\,\psi^{\dagger}(\mathbf{r})\psi(\mathbf{r})U(\mathbf{r}-\mathbf{r}^{\prime}) \psi^{\dagger}(\mathbf{r}^{\prime})\psi(\mathbf{r}^{\prime}). \label{int-Hamiltonian}$$ In order to present our results in the most transparent way, in this study we will utilize a simpler model with a contact four-fermion interaction, $$U(\mathbf{r}) = \frac{e^2}{\kappa |\mathbf{r}|} \rightarrow g\, \delta^3(\mathbf{r}), \label{model-interaction}$$ where $\kappa$ is a dielectric constant and $g$ is a dimensionful coupling constant. As we argue in Sec. \[section7\], such a model interaction should at least be sufficient for a qualitative description of the effect of the dynamical generation of the chiral shift parameter by a magnetic field in Dirac semimetals. Before proceeding further with the analysis, we find it very convenient to introduce the four-dimensional Dirac matrices in the chiral representation: $$\gamma^0 = \left( \begin{array}{cc} 0 & -I\\ -I & 0 \end{array} \right),\qquad \bm{\gamma} = \left( \begin{array}{cc} 0& \bm{\sigma} \\ - \bm{\sigma} & 0 \end{array} \right), \label{Dirac-matrices}$$ where $I$ is the two-dimensional unit matrix, and rewrite our model Hamiltonian in a relativistic form, $$H^{\rm (W)} = \int d^3 r\, \bar{\psi} (\mathbf{r})\left[ -i v_F (\bm{\gamma}\cdot \bm{\nabla})-(\mathbf{b}\cdot \bm{\gamma})\gamma^5-\mu_{0}\gamma^0 \right]\psi(\mathbf{r}) +\frac{g}{2}\int d^3r\,\rho(\mathbf{r})\rho(\mathbf{r}), \label{free-Hamiltonian-Weyl-rel}$$ where, by definition, $\bar{\psi} \equiv \psi^{\dagger}\gamma^0$ is the Dirac conjugate spinor field, $\rho(\mathbf{r})\equiv \bar{\psi}(\mathbf{r}) \gamma^0 \psi (\mathbf{r})$ is the charge density operator, and the matrix $\gamma^5$ is $$\gamma^5 \equiv i\gamma^0\gamma^1\gamma^2\gamma^3 = \left( \begin{array}{cc} I & 0\\ 0 & -I \end{array} \right),$$ where, as is clear from the first term in the free Hamiltonian (\[free-Hamiltonian\]), the eigenvalues of $\gamma^5$ correspond to the node degrees of freedom. The low-energy Hamiltonian of a Dirac semimetal corresponds to a special case in Eq. (\[free-Hamiltonian-Weyl-rel\]) when $\mathbf{b}=0$, i.e., $$H^{\rm (D)} = H^{\rm (W)}\Big|_{\mathbf{b} = 0}. \label{free-Hamiltonian-Dirac}$$ Unlike Weyl semimetals, Dirac semimetals are invariant under time reversal symmetry. Note that, for the clarity of presentation, in this paper we consider only the chiral limit when the bare Dirac mass term $m_0\bar{\psi} (\mathbf{r})\psi(\mathbf{r})$ is absent. In the presence of an external magnetic field, one should replace $\bm{\nabla} \to \bm{\nabla}+ie\mathbf{A}/c$, where $\mathbf{A}$ is the vector potential and $c$ is the speed of light. Thus, the Hamiltonian of the Dirac semimetal model in an external magnetic field has the following form: $$H^{\rm (D)}_{\rm mag} = \int d^3 r\, \bar{\psi} (\mathbf{r})\left[ -i v_F \left( \bm{\gamma}\cdot(\bm{\nabla}+ie\mathbf{A}/c) \right)-\mu_{0}\gamma^0 \right]\psi(\mathbf{r}) +\frac{g}{2}\int d^3r\,\rho(\mathbf{r})\rho(\mathbf{r}). \label{Hamiltonian-Dirac-magnetic}$$ Note that both this Hamiltonian and the Weyl semimetal Hamiltonian (\[free-Hamiltonian-Weyl-rel\]) are invariant under the chiral $U(1)_{+}\times U(1)_{-}$ symmetry, where $+$ and $-$ correspond to the node states with $+1$ and $-1$ eigenvalues of the $\gamma_5$ matrix, respectively. The currents connected with the $U(1)_{+}$ and $U(1)_{-}$ symmetries are anomalous. However, because these symmetries are Abelian, one can introduce conserved charges for them.[@Jackiw] Gap equation {#section3} ============ In this section, we will derive the gap equation for the fermion propagator in the Dirac semimetal model (\[Hamiltonian-Dirac-magnetic\]) and show that at a nonzero charge density, a nonzero $\mathbf{b}$ [*necessarily*]{} arises in the normal phase as soon as a magnetic field is turned on. In model (\[Hamiltonian-Dirac-magnetic\]), we easily find the following free fermion propagator: $$iS^{-1}(u,u^\prime) = \left[(i\partial_t+\mu_0)\gamma^0 -v_F(\bm{\pi}\cdot\bm{\gamma})\right]\delta^{4}(u- u^\prime), \label{sinverse}$$ where $u=(t,\mathbf{r})$ and $\bm{\pi} \equiv -i \bm{\nabla} + e\mathbf{A}/c$ is the canonical momentum. In the rest of the paper, we will choose the vector potential in the Landau gauge, $\mathbf{A}= (0, x B,0)$, where $B$ is a strength of the external magnetic field pointing in the $z$ direction. An ansatz for the full fermion propagator can be written in the following form (we will see that this ansatz is consistent with the Schwinger–Dyson equation for the fermion propagator in the mean-field approximation): $$iG^{-1}(u,u^\prime)= \Big[(i\partial_t+\mu )\gamma^0 - v_F(\bm{\pi}\cdot\bm{\gamma}) +\gamma^0(\bm{\tilde{\mu}}\cdot\bm{\gamma})\gamma^5 +v_F (\mathbf{b} \cdot \bm{\gamma})\gamma^5 -m\Big]\delta^{4}(u- u^\prime). \label{ginverse}$$ This propagator contains dynamical parameters $\tilde{\bm{\mu}}$, $\mathbf{b}$, and $m$ that are absent at tree level in Eq. (\[sinverse\]). Here $m$ plays the role of the dynamical Dirac mass and $\mathbf{b}$ is the chiral shift.[@Gorbar:2009bm; @Gorbar:2011ya] By taking into account the Dirac structure of the $\tilde{\bm{\mu}}$ term, we see that it is related to the anomalous magnetic moment $\mu_{\rm an}$ (associated with the pseudospin) as follows: $\tilde{\bm{\mu}}\equiv \mu_{\rm an}\mathbf{B}$. It should be also emphasized that the dynamical parameter $\mu$ in the full propagator may differ from its tree-level counterpart $\mu_0$ \[see Eq. (\[gap-mu-text\]) below\]. In order to determine the values of these dynamical parameters, we will use the Schwinger–Dyson (gap) equation for the fermion propagator in the mean-field approximation, i.e., $$iG^{-1}(u,u^\prime) = iS^{-1}(u,u^\prime) - g \left\{ \gamma^0 G(u,u) \gamma^0 - \gamma^0\, \mbox{tr}[\gamma^0G(u,u)]\right\} \delta^{4}(u- u^\prime). \label{gap}$$ The first term in the curly brackets describes the exchange (Fock) interaction and the last term presents the direct (Hartree) interaction. Separating different Dirac structures in the gap equation, we arrive at the following set of equations: $$\begin{aligned} \mu - \mu_0 &=& -\frac{3}{4}\, g \, \langle j^{0}\rangle , \label{gap-mu-text} \\ \mathbf{b} &=& \frac{g}{4v_F } \, \langle\mathbf{j}_5\rangle , \label{gap-Delta-text} \\ m &=& - \frac{g}{4} \, \langle \bar{\psi}\psi\rangle , \label{gap-m-text}\\ \bm{\tilde{\mu}} &=& \frac{g}{4} \, \langle \bm{\Sigma} \rangle . \label{gap-tilde-mu-text}\end{aligned}$$ The fermion charge density, the axial current density, the chiral condensate, and the anomalous magnetic moment condensate on the right-hand side of the above equations are determined through the full fermion propagator as follows: $$\begin{aligned} \langle j^{0}\rangle &\equiv &-{\,\mbox{tr}}\left[\gamma^0G(u,u)\right] , \label{density-text}\\ \langle \mathbf{j}_5\rangle &\equiv &-{\,\mbox{tr}}\left[\bm{\gamma}\,\gamma^5G(u,u)\right], \label{axial-current-text}\\ \langle \bar{\psi}\psi\rangle&\equiv & -{\,\mbox{tr}}\left[G(u,u)\right] , \label{chi-condensate-text}\\ \langle \bm{\Sigma} \rangle &\equiv &-{\,\mbox{tr}}\left[ \gamma^0\bm{\gamma}\,\gamma^5G(u,u)\right] . \label{pseudospin-condensate-text}\end{aligned}$$ Note that the right-hand sides in Eqs. (\[chi-condensate-text\]) and (\[pseudospin-condensate-text\]) differ from those in Eqs. (\[density-text\]) and (\[axial-current-text\]) by the inclusion of an additional $\gamma^0$ matrix inside the trace. Since, according to Eq. (\[Dirac-matrices\]), the $\gamma^0$ matrix mixes quasiparticle states from different Weyl nodes, we conclude that while $\langle j^{0}\rangle$ and $\langle \mathbf{j}_5\rangle$ describe the charge density and the axial current density, the chiral condensate $\langle \bar{\psi}\psi\rangle$ and the anomalous magnetic moment condensate $\langle \bm{\Sigma} \rangle$ describe internode coherent effects. As is known,[@Vilenkin; @Metlitski:2005pr; @Newman] in the presence of a fermion charge density and a magnetic field, the axial current $\langle \mathbf{j}_5\rangle$ is generated even in the free theory. Therefore, according to Eq. (\[gap-Delta-text\]), the chiral shift $\mathbf{b}$ is induced already in the lowest order of the perturbation theory. As a result, a Dirac semimetal is [*necessarily*]{} transformed into a Weyl one, as soon as an external magnetic field is applied to the system (see also a discussion in Sec. \[section4\]). In order to derive the propagator $G(u,u^\prime)$ in the Landau-level representation, we invert $G^{-1}(u,u^\prime)$ in Eq. (\[ginverse\]) by using the approach described in Appendix A of Ref. . For our purposes here, the expression for the propagator in the coincidence limit $u^\prime\to u$ is sufficient \[cf. Eq. (A26) in Ref. \]: $$G(u,u)= \frac{i}{2\pi l^2}\sum_{n=0}^{\infty} \int\frac{d\omega d k^{3}}{(2\pi)^2} \frac{{\cal K}_{n}^{-}{\cal P}_{-}+{\cal K}_{n}^{+}{\cal P}_{+}\theta(n-1)}{U_n}, \label{full-propagator}$$ where ${\cal P}_{\pm}\equiv \frac12 \left(1\pm i s_\perp \gamma^1\gamma^2\right)$ are the pseudospin projectors, $l=\sqrt{c/|eB|}$ is the magnetic length, and $s_\perp={\,\mbox{sgn}}(eB)$. Also, by definition, $\theta(n-1)=1$ for $n\geq 1$ and $\theta(n-1)=0$ for $n < 1$. The functions ${\cal K}_{n}^{\pm}$ and $U_n$ with $n\geq 0$ are given by $$\begin{aligned} {\cal K}_{n}^{\pm}&=& \left[\left(\omega+\mu \mp s_{\perp}v_F b\right)\gamma^0 \pm s_{\perp}\tilde\mu + m -v_Fk^{3}\gamma^3\right]\Big\{ (\omega+\mu)^2 + \tilde{\mu}^2 - m^2 - (v_F b)^2 - (v_Fk^{3})^2 - 2nv^2_F|eB|/c \nonumber \\ && \mp 2s_{\perp}\left[\tilde\mu (\omega+\mu)+v_F b m \right]\gamma^0 \pm 2s_{\perp} (\tilde\mu + v_F b\gamma^0)v_Fk^{3} \gamma^3\Big\} \label{K_n^pm}\end{aligned}$$ and $$U_n =\left[(\omega+\mu)^2 + \tilde{\mu}^2 - m^2 - (v_F b)^2 - (v_Fk^{3})^2 -2nv^2_F|e B |/c\right]^2 - 4\left[\left(\tilde{\mu}\,(\omega+\mu) + v_F b m\right)^2 +(v_Fk^{3})^2\left((v_F b)^2-\tilde{\mu}^2\right)\right], \label{U_n}$$ where we took into account that the only nonvanishing components of the axial vectors $\mathbf{b}$ and $\bm{\tilde{\mu}}$ are the longitudinal projections $b$ and $\tilde{\mu}$ on the direction of the magnetic field. Note that the zeros of the function $U_n$ determine the dispersion relations of quasiparticles. Perturbative solution {#section4} ===================== In order to obtain the leading order perturbative solution to the gap equations, we can use the free propagator on the right-hand side of Eqs. (\[density-text\]) through (\[pseudospin-condensate-text\]), i.e., $$S(u,u)= \frac{i}{2\pi l^2}\sum_{n=0}^{\infty} \int\frac{d\omega d k^{3}}{(2\pi)^2} \frac{ \left[\left(\omega+\mu_{0}\right)\gamma^0 -v_Fk^{3}\gamma^3\right] \left[{\cal P}_{-}+{\cal P}_{+}\theta(n-1)\right]}{(\omega+\mu_{0})^2 - (v_F k^{3})^2 -2nv^2_F|eB|/c}. \label{full-free-propagator}$$ Note that unlike the high Landau levels with $n\geq 1$, where both spin projectors ${\cal P}_+$ and ${\cal P}_-$ contribute, the lowest Landau level (LLL) with $n=0$ contains only one projector ${\cal P}_-$. The reason for this is well known. The Atiyah-Singer theorem connects the number of the zero energy modes (which are completely pseudospin polarized) of the two-dimensional part of the Dirac operator to the total flux of the magnetic field through the corresponding plane. This theorem states that LLL is topologically protected (for a discussion of the Atiyah-Singer theorem in the context of condensed matter physics, see Ref. ). By making use of Eq. (\[full-free-propagator\]), we straightforwardly calculate the zeroth order result for the charge density, $$\langle j^{0}\rangle_0=\frac{\mu_0}{2v_F(\pi l)^2} +\frac{{\,\mbox{sgn}}(\mu_0)}{v_F(\pi l)^2}\sum_{n=1}^{\infty} \sqrt{\mu_0^2-2nv^2_F|eB|/c}\,\, \theta\left(\mu_0^2-2nv^2_F|eB|/c\right), \label{perturbative-charge}$$ and the axial current density $$\langle \mathbf{j}_5\rangle_0 = -\frac{e\mathbf{B}\mu_0}{2\pi^2 v_Fc}. \label{perturbative-axial-current}$$ As to the chiral condensate and the anomalous magnetic moment condensate, they vanish, i.e., $\langle \bar{\psi}\psi\rangle_0 = 0$ and $\langle\bm{\Sigma} \rangle_0 = 0$. This is not surprising because both of them break the chiral $U(1)_{+} \times U(1)_{-}$ symmetry of the Dirac semimetal Hamiltonian (\[Hamiltonian-Dirac-magnetic\]). Then, taking into account Eqs. (\[gap-m-text\]) and (\[gap-tilde-mu-text\]), we conclude that both the Dirac mass $m$ and the parameter $\tilde{\bm{\mu}}$ are zero in the perturbation theory. After taking into account the gap equations (\[gap-mu-text\]) and (\[gap-Delta-text\]), the results in Eqs. (\[perturbative-charge\]) and (\[perturbative-axial-current\]) imply that there is a perturbative renormalization of the chemical potential and a dynamical generation of the chiral shift. Of special interest is the result for the axial current density given by Eq. (\[perturbative-axial-current\]). This is generated already in the free theory and known in the literature as the topological contribution.[@Son:2004tq; @Metlitski:2005pr] Its topological origin is related to the following fact: in the free theory, only the LLL contributes to both the axial current and the axial anomaly. By combining Eqs. (\[gap-Delta-text\]) and (\[perturbative-axial-current\]), we find $$\mathbf{b} = - \frac{ge\mathbf{B}\mu_0}{8\pi^2 v_F^2 c}. \label{bvalue}$$ This is our principal result, which reflects the simple fact that at $\mu_0 \neq 0$ (i.e., nonzero charge density), the absence of the chiral shift is not protected by any additional symmetries in the normal phase of a Dirac metal in a magnetic field. Indeed, in the presence of a homogeneous magnetic field pointing in the $z$ direction, the rotational $SO(3)$ symmetry in the model is explicitly broken down to the $SO(2)$ symmetry of rotations around the $z$ axis. The dynamically generated chiral shift parameter $\mathbf{b}$ also points in the same direction and does not break the leftover $SO(2)$ symmetry. The same is true for the discrete symmetries: while the parity ${\cal P}$ is preserved, all other discrete symmetries, charge conjugation ${\cal C}$, time reversal ${\cal T}$, ${\cal CP}$, ${\cal CT}$, $PT$, and ${\cal CPT}$, are broken. Last but not least, the chiral shift does not break the chiral $U(1)_{+}\times U(1)_{-}$ symmetry considered in Sec. \[section2\]. This implies that the dynamical chiral shift is necessarily generated in the normal phase of a Dirac semimetal in a magnetic field, and the latter is transformed into a Weyl semimetal. In the phase with a dynamically generated chiral shift $\mathbf{b}$, the quasiparticle dispersion relations, i.e., $\omega_{n,\sigma} = -\mu + E_{n,\sigma}$, are determined by the following Landau-level energies (recall that we assume that the magnetic field points in the $z$ direction): $$\begin{aligned} E_{0, \sigma} &=& v_F \left( s_\perp b +\sigma k^3\right) ,\qquad n=0, \\ E_{n,\sigma} &=& \pm v_F \sqrt{\left(s_{\perp}b+\sigma k^3\right)^2+2n|eB|/c},\qquad n\geq 1,\end{aligned}$$ where $\sigma=\pm$ corresponds to different Weyl nodes. The corresponding dispersion relations are shown graphically in Fig. \[fig:Fermi\]. Note a qualitatively different character (compared to higher Landau levels) of the dispersion relations in the LLL given by the straight lines whose signs of slope correlate with the Weyl nodes. Of course, this correlation is due to the complete polarization of quasiparticle pseudospins in the LLL. As seen from Fig. \[fig:Fermi\], the effect of the chiral shift is not only to shift the relative position of the Weyl nodes in momentum space, but also to induce a chiral asymmetry of the Fermi surface.[@Gorbar:2009bm; @Gorbar:2011ya] ![Dispersion relations of the quasiparticles from different Weyl nodes and their Fermi surfaces. The two dispersion relations are the mirror images of each other.[]{data-label="fig:Fermi"}](Fig_Fermi_node1.eps "fig:"){width=".45\textwidth"} ![Dispersion relations of the quasiparticles from different Weyl nodes and their Fermi surfaces. The two dispersion relations are the mirror images of each other.[]{data-label="fig:Fermi"}](Fig_Fermi_node2.eps "fig:"){width=".45\textwidth"} Nonperturbative solution: Phase transition {#section5} ========================================== The magnetic catalysis phenomenon, which takes place for planar as well as three-dimensional relativistic charged fermions because of the dimensional reduction,[@magCat; @magCat3+1] implies that at vanishing $\mu_0$, the ground state in the model at hand is characterized by a nonzero Dirac mass $m$ that spontaneously breaks the chiral symmetry. (For the corresponding studies in graphene, see Ref. .) Such a vacuum state can withstand a finite stress due to a nonzero chemical potential. However, as we discuss below, when $\mu_0$ exceeds a certain critical value $\mu_{\rm cr}$, the chiral symmetry restoration and a new ground state are expected. The new state is characterized by a nonvanishing chiral shift parameter $\mathbf{b}$ and a nonzero axial current in the direction of the magnetic field. Since no symmetry of the theory is broken, this state is the [*normal*]{} phase of the magnetized matter that happens to have a rather rich chiral structure. This phase was considered in the previous section. Let us describe this transition in more detail. The value of the dynamical Dirac mass $m$ in the vacuum state can be easily calculated following the same approach as in Ref. . At weak coupling, in particular, we can use the following expression for the chiral condensate: $$\langle \bar{\psi}\psi\rangle \simeq -\frac{m}{4\pi^2 v_F} \left(\Lambda^2 +\frac{1}{l^2}\ln\frac{v_F^2}{\pi m^2 l^2} \right) , \label{condensate3+1}$$ obtained in the limit of a small mass (which is consistent with the weak coupling approximation), using the gauge invariant proper-time regularization. Here the ultraviolet momentum cutoff $\Lambda$ can be related, for example, to the value of lattice spacing $a$ as follows: $\Lambda \simeq \pi/a$. Finally, by taking into account gap equation (\[gap-m-text\]), we arrive at the solution for the dynamical mass, $$m \simeq \frac{v_F}{\sqrt{\pi}l}\exp\left(-\frac{8 \pi^2 v_F l^2}{g}+\frac{(\Lambda l)^2}{2}\right). \label{DiracMass}$$ This zero-temperature, nonperturbative solution exists for $\mu_0< m$. The free energies of the two types of states, i.e., the nonperturbative state with a dynamically generated Dirac mass (and no chiral shift) and the perturbative state with a nonzero chiral shift (and no Dirac mass) become equal at about $\mu_0\simeq m/\sqrt{2}$. This is analogous to the Clogston relation in superconductivity.[@Clogston] At the critical value $\mu_{\rm cr} \simeq m/\sqrt{2}$, a first order phase transition takes place. Indeed, both these solutions coexist at $\mu_0< m$, and while for $\mu_0 < \mu_{\rm cr}$ the nonperturbative (gapped) phase with a chiral condensate is more stable, the normal (gapless) phase becomes more stable at $\mu_0 > \mu_{\rm cr}$. Note that at $\mu_0 < m$ the chemical potential is irrelevant in the gapped phase: the charge density is absent there. On the other hand, at any nonzero chemical potential, there is a nonzero charge density in the normal (gapless) phase. Therefore, at $\mu_{\rm cr} \simeq m/\sqrt{2}$, there is a phase transition with a jump in the charge density, which is a clear manifestation of a first order phase transition. Dirac semimetals vs. graphene in a magnetic field {#section6} ================================================= It is instructive to compare the states in the magnetized Dirac semimetals and graphene. First of all, we would like to point out that the chiral shift is a three-dimensional analog of the Haldane mass,[@Haldane; @NSR] which plays an important role in the dynamics of the quantum Hall effect in graphene. Indeed, in the formalism of the four-component Dirac fields in graphene, the Haldane mass condensate is described by the same vacuum expectation value as that of the axial current in three dimensions:[@Gorbar:2008hu] $$\langle \bar{\psi}\gamma^{3}\gamma^{5}\psi\rangle = -{\,\mbox{tr}}\left[\gamma^3\,\gamma^5G(u,u)\right]$$ for a magnetic field pointing in the $z$ direction, which is orthogonal to the graphene plane; cf. Eq. (\[axial-current-text\]). Moreover, similar to the solution with the chiral shift, the solution with the Haldane mass (with the same sign for both spin-up and spin-down quasiparticles) describes the normal phase: it is a singlet with respect to the $SU(4)$ symmetry, which is a graphene analog of the chiral group in Dirac and Weyl semimetals. Also, in graphene, there is a phase transition similar to that described in the previous section. It happens when the LLL is completely filled.[@Gorbar:2008hu] In other words, the quantum Hall state with the filling factor $\nu = 2$ in graphene is associated with the normal phase containing the Haldane mass. As is well known, the Haldane mass leads to the Chern-Simons term in an external electromagnetic field.[@Haldane; @NSR] This feature reflects a topological nature of the state with the filling factor $\nu = 2$ in a graphene. As was recently shown in Ref. , the chiral shift term $$\bar{\psi}(\mathbf{b} \cdot \bm{\gamma})\gamma^5\psi$$ leads to an induced Chern-Simons term of the form $\frac{1}{2}b_\mu\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}A_\nu$ in Weyl semimetals \[here $b_\mu$ is a four-dimensional vector $(0,\mathbf{b})$\]. Therefore it should also be generated in Dirac semimetals in a magnetic field. Note, however, the following principle difference between them: while in Weyl semimetals the chiral shift $\mathbf{b}$ is present in the free Hamiltonian, it is dynamically generated in the normal phase of Dirac semimetals in a magnetic field \[see Eq. (\[bvalue\])\]. Like in graphene, the generation of the Chern-Simons term implies a topological nature of the normal state in this material.[@footnote2] Discussion {#section7} ========== In the present paper, we considered manifestations of a relativistic phenomenon, the rearrangement of the Fermi surface in three-dimensional matter in a magnetic field, in a Dirac semimetal. It was shown that its normal phase at a nonzero charge density and in a magnetic field has a nontrivial chiral structure: it is a Weyl semimetal with a pair of Weyl nodes for each of the original Dirac points. The nodes are separated by a dynamically induced chiral shift that is directed along the magnetic field. The phase transition between the normal phase and the phase with chiral symmetry breaking is revealed, and the rearrangement of the Fermi surface accompanying this phase transition is described.[@SekineNomura] Although we studied a simple model with a contact four-fermion interaction, we believe that the present qualitative results apply equally well to more realistic models. The studies in the relativistic Nambu-Jona-Lasinio (NJL) model (with a contact interaction) on the one hand [@Gorbar:2009bm; @Gorbar:2011ya] and in QED on the other [@Gorbar:2013upa] strongly support the validity of this statement. Namely, the dynamical generation of the chiral shift in a magnetic field and at a nonzero fermion density is a universal phenomenon. In the present study we analyzed the simplest model: it is isotropic (in the absence of the magnetic field), has no gap (i.e., no bare Dirac mass), and no Zeeman term for spin. In real materials, such as bismuth,[@Falkovsky; @Edelman] the presence of an anisotropy, a gap, and the Zeeman term for spin should be taken into account. This task, as well as the generalization of the study to the case of nonzero temperature, is outside the scope of the present paper and will be considered elsewhere. Here we just want to mention that the effects of a bare Dirac mass term and temperature were studied in the NJL model in Ref. . It was shown there that the chirality remains a good approximate quantum number even for massive fermions in the vicinity of the Fermi surface, provided the mass is sufficiently small, i.e., $m\ll\mu$. As for the temperature effects,[@Gorbar:2011ya] an interesting feature of the chiral shift is that it is insensitive to the temperature when $T \ll \mu$, and [*increases*]{} with temperature $T \gtrsim \mu$. A natural extension of this work would be the study of the phase diagram of a Weyl semimetal in a magnetic field and a nonzero charge density. Because of a built-in chiral shift in the Hamiltonian of such a material, one may expect that its phase with chiral symmetry breaking, realized at zero (or low) density, should be inhomogeneous, possibly, a Larkin-Ovchinnikov-Fulde-Ferrel (LOFF) like one (cf. Refs. ). We expect that a first order phase transition between the normal phase and the phase with chiral symmetry breaking will take place in that case too. [*Note added in proof.*]{} Recently, an experimental observation of the transition from a Dirac semimetal to a Weyl semimetal in a magnetic field was reported.[@1307.6990] We thank V. P. Gusynin for useful remarks. The work of E.V.G. was supported partially by the European FP7 program, Grant No. SIMTECH 246937, SFFR of Ukraine, Grant No. F53.2/028, and Grant STCU No. 5716-2 “Development of Graphene Technologies and Investigation of Graphene-based Nanostructures for Nanoelectronics and Optoelectronics". The work of V.A.M. was supported by the Natural Sciences and Engineering Research Council of Canada. The work of I.A.S. was supported in part by the U.S. National Science Foundation under Grant No. PHY-0969844. [99]{} O. Vafek and A. Vishwanath, arXiv:1306.2272 \[cond-mat.mes-hall\]. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grogorieva, and A. A. Firsov, Science, [**306**]{}, 666 (2004). A. F. Young and P. Kim, Nature Physics [**5**]{}, 222 (2009). Y. Wang, D. Wong, A. V. Shytov, V. W. Brar, C. Sangkook, Q. Wu, H. Z. Tsai, W. Regan, A. Zettl, R. K. Kawakami, S. G. Louie, L. S. Levitov, and M. F. Crommie, Science [**340**]{}, 734 (2013). , edited by A. Niemi, F. Wilczek, E. Ardonne, and H. Hansson, Phys. Scr. [**T146**]{} (IOP Publishing, Bristol, UK, 2012) E. V. Gorbar, V. A. Miransky, and I. A. Shovkovy, Phys. Rev. C [**80**]{}, 032801(R) (2009). E. V. Gorbar, V. A. Miransky, and I. A. Shovkovy, Phys. Rev. D [**83**]{}, 085003 (2011). L. A. Fal’kovskii, Sov. Phys. Usp. [**11**]{}, 1 (1968) \[Usp. Fiz. Nauk [**94**]{}, 1 (1968)\]. V. S. Edel’man, Adv. Phys. [**25**]{}, 555 (1976); Sov. Phys. Usp. [**20**]{}, 819 (1977) \[Usp. Fiz. Nauk [**123**]{}, 257 (1977)\]. X. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, Phys. Rev. B [**83**]{}, 205101 (2011). A. A. Burkov and L. Balents, Phys. Rev. Lett. [**107**]{}, 127205 (2011). Y. Chen, D. L. Bergman, and A. A. Burkov, Phys. Rev. B [**88**]{}, 125110 (2013). G. E. Volovik, [*The Universe in a Helium Droplet*]{} (Claredon Press, Oxford, 2003). P. Goswami and S. Tewari, arXiv:1210.6352 \[cond-mat.mes-hall\]. M. M. Vazifeh and M. Franz, arXiv:1303.5784 \[cond-mat.mes-hall\]. P. E. C. Ashby and J. P. Carbotte, Phys. Rev. B [**87**]{}, 245131 (2013). G. Basar, D. E. Kharzeev, and Ho-Ung Yee, arXiv:1305.6338 \[hep-ph\]. S. A. Parameswaran, T. Grover, D. A. Abanin, D. A. Pesin, and A. Vishwanath, arXiv:1306.1234 \[cond-mat.str-el\]. K. Landsteiner, arXiv:1306.4932 \[hep-th\]. S. L. Adler, Phys. Rev.  [**177**]{}, 2426 (1969); J. S. Bell and R. Jackiw, Nuovo Cimento A [**60**]{}, 47 (1969). D. E. Kharzeev, K. Landsteiner, A. Schmitt, and Ho-Ung Yee, Lect. Notes Phys. [**871**]{}, 1 (2013). H. B. Nielsen and M. Ninomiya, Phys. Lett. B [**130**]{}, 389 (1983). A. A. Zyuzin and A. A. Burkov, Phys. Rev. B [**86**]{}, 115133 (2012). R. Jackiw, “Field Theoretical Investigations in Current Algebra" in S. B. Treiman, R. Jackiw, B. Zumino, and E. Witten, [*Current Algebra and Anomalies*]{} (Princeton University Press, Princeton, New Jersey, 1985) p. 136. A. Vilenkin, Phys. Rev. D [**22**]{}, 3080 (1980). M. A. Metlitski and A. R. Zhitnitsky, Phys. Rev. D [**72**]{}, 045011 (2005). G. M. Newman and D. T. Son, Phys. Rev. D [**73**]{}, 045006 (2006). M. I. Katsnelson, [*Graphene: Carbon in Two Dimensions*]{} (Cambridge University Press, Cambridge, UK, 2012). D. T. Son and A. R. Zhitnitsky, Phys. Rev.  D [**70**]{}, 074018 (2004). , Phys. Rev. Lett.  [**73**]{}, 3499 (1994); Phys. Rev.  D [**52**]{}, 4718 (1995). V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Phys. Lett. B [**349**]{}, 477 (1995); Nucl. Phys. B [**462**]{}, 249 (1996). D. V. Khveshchenko, Phys. Rev. Lett. [**87**]{}, [206401]{} (2001); E. V. Gorbar, V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Phys. Rev.  B [**66**]{}, 045108 (2002); V. P. Gusynin, V. A. Miransky, S. G. Sharapov, and I. A. Shovkovy, Phys. Rev. B [**74**]{}, [195429]{} (2006); I. F. Herbut, Phys. Rev. Lett. [**97**]{}, [146401]{} (2006); Phys. Rev. B [**75**]{}, [165411]{} (2007); [*ibid.*]{} [**76**]{}, [085432]{} (2007); M. Ezawa, J. Phys. Soc. Jpn. [**76**]{}, [094701]{} (2007). A. M. Clogston, Phys. Rev. Lett. [**9**]{}, 266 (1962). F. D. M. Haldane, Phys. Rev. Lett. [**61**]{}, 2015 (1988). In relativistic physics, such a mass was considered in A. J. Niemi and G. W. Semenoff, Phys. Rev. Lett.  [**51**]{}, 2077 (1983) and A. N. Redlich, Phys. Rev. Lett.  [**52**]{}, 18 (1984). E. V. Gorbar, V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, Phys. Rev. B [**78**]{}, 085437 (2008). A. G. Grushin, Phys. Rev. D [**86**]{}, 045001 (2012). The generation of the Chern-Simons term in these materials is intimately connected with their anisotropy. A possibility of the generation of an induced Chern-Simons term in a relativistic isotropic hot matter was considered in A. N. Redlich and L. C. R. Wijewardhana, Phys. Rev. Lett.  [**54**]{}, 970 (1985). A different type of the phase diagram in Dirac and Weyl semimetals was studied in A. Sekine and K. Nomura, J. Phys. Soc. Jpn. [**82**]{}, 033702 (2013). E. V. Gorbar, V. A. Miransky, I. A. Shovkovy, and X. Wang, Phys. Rev. D [**88**]{}, 025043 (2013). K.-Y. Yang, Y.-M. Lu, and Y. Ran, Phys. Rev. B [**84**]{}, 075129 (2011). H. Wei, S.-P. Chao, and V. Aji, Phys. Rev. Lett. [**109**]{}, 196403 (2012). Z. Wang and S.-C. Zhang, Phys. Rev. B [**87**]{}, 161107(R) (2013). H.-J. Kim, K.-S. Kim, J.-F. Wang, M. Sasaki, N. Satoh, A. Ohnishi, M. Kitaura, M. Yang, and L. Li, arXiv:1307.6990 \[cond-mat.str-el\].

--- abstract: 'Superlinear scaling in cities, which appears in sociological quantities such as economic productivity and creative output relative to urban population size, has been observed but not been given a satisfactory theoretical explanation. Here we provide a network model for the superlinear relationship between population size and innovation found in cities, with a reasonable range for the exponent.' author: - Samuel Arbesman - 'Jon M. Kleinberg' - 'Steven H. Strogatz' bibliography: - 'cityscaling.bib' title: Superlinear Scaling for Innovation in Cities --- Introduction ============ It has been known for nearly a hundred years that living things obey scaling relationships. Max Kleiber first recognized that the metabolic rates of different mammals scale according to their masses raised to a $3/4$-power [@ARB:Kle32]. More recently, Geoffrey West and his colleagues have provided a theoretical explanation for this scaling law, as well as for many other allometric laws found in biology [@ARB:Wes97; @ARB:Whi2006]. Their theory is based upon the fractal branching networks (such as circulatory systems) found in all living things, whose function is to convey energy and nutrients to all parts of the organism. They argue that the larger the organism, the more efficient the system that can be constructed to provide energy, thereby yielding the observed sublinear exponent of $3/4$. More recently, West and his team examined a variety of properties of cities. They found that cities, which have long been compared to living things [@ARB:Zipf49; @ARB:Jacobs01; @ARB:Aristotle98], obey scaling relationships as well [@ARB:Bet2007]. Similar to living things, cities have economics of scale, yielding sublinear scaling for such quantities as the number of gas stations within a city as a function of its population. In other words, you need fewer gas stations per person, in a bigger city. Examples of such scaling laws are shown in the upper portion of table \[table\_1\]. On the other hand, cities also exhibit superlinear scaling, which appears in relation to sociological quantities. As shown in the lower part of table \[table\_1\], properties of cities related to economic productivity and creative output have exponents that are all found to cluster between 1 and 1.5, with the mean around 1.2. Thus, the productivity *per person* increases as a city gets larger. However, this superlinear scaling has not been given a satisfactory mathematical explanation. [l d]{} Urban Indicators ($y$) & ()\ Gasoline stations & 0.77\ Gasoline sales & 0.79\ Length of electrical cables & 0.87\ Road surface & 0.83\ New patents & 1.27\ Inventors & 1.25\ Private R & D employment & 1.34\ ÔSupercreativeÕ employment & 1.15\ R & D establishments & 1.19\ R& D employment & 1.26\ Total wages & 1.12\ Total bank deposits & 1.08\ GDP & 1.15\ \[table\_1\] Here we suggest a theoretical explanation for the superlinear relationship between population size and innovation found in cities, with a reasonable range for the exponent. Due to the sociological nature of the variables being measured, it is natural to use a network model of a city, since it is reasonable to assume that network effects must underlie the superlinear scaling, as West and his colleagues have suggested [@ARB:Bet2007; @ARB:Bet2008]. For this, we draw on a recent class of models that derives superlinear scaling and [*densification properties*]{} from hierarchically organized networks [@ARB:Les2005], adapting these models to the present question of productivity. Model and Results ================= We first assume that all social interactions and relationships are arranged in a hierarchical tree structure [@ARB:Kle2001; @ARB:Les2005; @ARB:Wat2002]. Picture a binary tree, or in general, a tree where each branch splits into $b$ new branches. For example, in a city, each person is in a household, and there are many households on a block, and many blocks in a neighborhood, and so forth. Or the grouping could be based on your family tree, or corporations, or many other ways to group individuals. While in reality each individual belongs to many independent hierarchies [@ARB:Wat2002], here we simplify it as a single hierarchy, with branching number $b \geq 2$. We define the [*distance*]{} $d$ between two individuals in this hierarchy to be the height of their lowest common ancestor. We view the total system as a city, meaning that a city of population $N$ represents a single tree that contains $N$ leaves. On top of the tree structure, which serves to determine the social distance among nodes, a random graph is placed showing the social connections — who actually knows whom. ![\[fig\_1\]Network representation of the social structure of a city’s inhabitants. For example, if we are determining distance from individual A, then B is at distance 1, while G is at a distance of 3.](fig_1){width="3"} Our modeling strategy is to use these social connections as the basis for a city’s productivity: at a high level, we assume that each interaction between a pair of people contributes to the overall productivity, in a way that depends on the distance $d$ as measured within the hierarchy. More concretely, our procedure for generating networks will produce a directed graph, and we will account for the productivity benefits of each edge $(v,w)$ by allocating it to $v$’s overall productivity. This allocation to $v$ rather than to both nodes is essentially for purposes of analysis, since we will be focusing in the end on the total population’s productivity rather than any one individual’s, and for determining this total we will see that it does not matter to which individual we allocate the benefits of the edge $(v,w)$. The total creative productivity of the city is defined to be the sum of the productivities of each individual, and so we first consider how to compute individual productivities. To calculate the total productivity of a single person, three separate effects must be considered: (1) the probability of connecting to an individual at distance $d$; (2) the number of available people at distance $d$; and (3) the creative output that is obtained by linking to a single person at distance $d$. Multiplying these together gives the productivity due to one person linking to all of his collaborators at distance $d$, as seen below: $$\left [ \dfrac{\text{\# contacts at }d}{\text{\# people at }d} \right ] \left [ \text{\# people at }d \right ] \left [ \dfrac{\text{output at }d}{\text{contacts at }d} \right ]$$ By summing this term over all distances, the total creative contribution of a single individual is obtained. The functional form of each term in the above recipe for calculating the productivity of a single individual is discussed below. Taking the first term, the social connections between collaborators are constructed such that the likelihood of forming a connection at a certain social distance drops off exponentially fast with distance [@ARB:Kle2001; @ARB:Les2005; @ARB:Wat2002]. That is, the probability of a connection being made between nodes of a social distance $d$ (where $d$ is the height of the first common internal node) is assumed proportional to $b^{-\alpha d}$, where $\alpha$ is a tunable parameter greater than or equal to zero. It is natural that the connection probability should decay with social distance, but why exponentially? We have assumed that the social network tree is self-similar at all levels (values of $d$). Since the tree is self-similar, it makes sense to have the function also be self-similar (scale-free) with respect to the value of $d$, and doing this yields an exponential function (this assumption is relaxed in the next section). Since at each increase in $d$ there are exponentially more potential contacts to interact with, we multiply the above function by a second term, $b^{d}$, which means that as we increase $d$, while the likelihood of making a connection decays, there are exponentially more contacts to make. To keep things simple, we suppose connections are only made between residents of the city (connections outside a city are viewed as contributing less directly to the city’s total productivity, and are ignored). Lastly, the usefulness of a social connection within a city is assumed to vary with its social distance. For example, one could assume that there is a productivity benefit as social distance increases. This can be explained as being due to the fact that individuals that are socially distant are exposed to different ideas and experiences, and that collaboration between two more socially distant individuals is more productive than interaction between ones that are closer. However, the value of a social connection is left open, and simply assumed to be proportional to $b^{\beta d}$, where $\beta$ is a tunable parameter that can hold any value (even negative values, allowing the value of a connection to decrease with distance). An exponential function is reasonable here as well, if we assume that a connection’s innovation potential depends on the number of individuals that lie between the two endpoints of the connection in social space. This assumption is also relaxed in the next section. The total productivity of the social connections within an $N$-person city is now a random variable equal to the sum of all the individual productivities. Its expectation $P(N)$ is given by $$P(N)=N \sum_{d=1}^{\log_b N} b^{-\alpha d}b^d b^{\beta d}$$ In summary, the first term, $b^{-\alpha d}$, is the probability of connecting at distance $d$. The second term, $b^{d}$, is the number of nodes at distance $d$. And the final term, $b^{\beta d}$, is productivity per connection. So, when these are multiplied together and then summed for each distance, they yield the expected productivity of one node in the full network. When multiplied by $N$ we get the productivity for the entire network. This can be summed exactly since it is a finite geometric series, and we get the following solution: $$P(N) = N \left ( \dfrac{b^{\beta + 1}}{b^{\beta + 1} - b^{\alpha}} N^{\beta - \alpha + 1} - 1 \right )$$ For large values of $N$, we find $P(N)$ is proportional to $N^{\beta-\alpha+2}$, if $\alpha < 1 + \beta$. On the other hand, if $\alpha > 1 + \beta$, the function $P(N)$ becomes linear, because the geometric series converges to a constant as $N$ becomes large: $$P(N)=N \sum_{d=1}^{\log_b N} {(b^{\beta - \alpha + 1})^d} \xrightarrow[N \gg 1] {} N \dfrac{b^{\beta - \alpha + 1}}{1 - b^{\beta - \alpha + 1}}$$ This is found to be in good agreement with numerical evaluation of the above summation, as can be seen in Figure \[fig\_2\]. The growth of the productivity function with distance is not essential. The key is, rather, that as a city increases in size, it is more likely to contain socially distant contacts. Having $\alpha < 1$ means that the expected number of contacts scales superlinearly [@ARB:Les2005]. Correspondingly, even if $\beta$ is slightly negative (meaning that more distant connections are less productive), as long as $\beta - \alpha + 1 > 0$, the densification of the social network due to the increasing size of the city means that the productivity will grow superlinearly. For the special case when $\beta$ is zero (all connections are equally beneficial), the exponent $\beta - \alpha + 2$ must lie strictly between 1 and 2, which is where all the measured exponents for urban innovation lie. This is because we are assuming that $0 < \alpha < 1 + \beta$ to get superlinear behavior, which means that $\alpha$ is between 0 and 1. ![\[fig\_2\]Simulation and fit for $P(N)$. The points show the value of $P(N)$, calculated for $\beta = 0.3$ and $\alpha = 1.1$. The least-squares approximation of the exponent is 1.205, and the expected value is $\beta - \alpha + 2 = 1.2$, for large values of $N$.](fig_2){width="3"} Of course, other parameter relationships are also capable of yielding the expected range of superlinear exponents. For $\beta > 0$, there is an exponentially decreasing probability of connecting to someone at a social distance $d$ away, but connections at this distance confer a productivity benefit that is exponentially increasing in $d$. In general, this model is a reasonable explanation for the values observed within cities related to productivity and innovation, and can be fit properly to explain the superlinear exponents observed within cities. Expansion of the Analysis ========================= The assumptions of exponentials for the three functions that make up the sum discussed above are stringent ones. What happens if we relax these assumptions? Using a numerical simulation, each of the components of the sum can be modified, and we can graph the resulting scaling relationship and see if it remains superlinear. And in fact, the model is robust under a variety of situations. For example, instead of using an exponential for the creative benefit function, if we use a ÔlinearithmicÕ function ($d \ln{d}$), the resulting function asymptotically approaches a superlinear function, as seen in Figure \[fig\_3\]. A similar superlinear result can be obtained by replacing the function for the number of nodes at distance $d$ with a linearithmic function and leaving the other two functions exponential (by doing this, we are implicitly changing the structure of the social distance tree, such that the number of nodes no longer grows exponentially with distance). ![\[fig\_3\]A ’linearithmic’ function. Using the function $b^{-\alpha d} b^{d} (d \ln{d})$ as the term within the sum for $P(N)$ (where $\alpha = 0.6$), the resulting function mimics a power law, with an exponent of 1.48.](fig_3){width="3"} In fact, even if all three functions are linear, the sum still grows superlinearly with $N$, as seen in Figure \[fig\_4\]. Indeed, if the function is proportional to $d^3$, using the Euler-MacLaurin summation, we find that $P(N) \approx N (\log_b{N})^4$, which grows a bit faster than linearly. ![\[fig\_4\]Linear Functions. Using the function $(50 - \alpha d) (1 + \beta d) (2 d)$ as the term within the sum for $P(N)$ (where $\alpha = 0.4$ and $\beta = 0.1$), the resulting function still mimics a power law, with an exponent of 1.13.](fig_4){width="3"} However, if the average productivity per node grows with $N$ but only at the rate $\log{N}$, then the rate of growth is only slightly superlinear, mimicking a power law exponent of 1.05, as seen in Figure \[fig\_5\]. Slightly faster than logarithmic growth for the summation appears to be required for superlinear growth of $P(N)$. ![\[fig\_5\]The function $P(N) = N \log_b{N}$ mimics a power law, with an exponent of 1.05.](fig_5){width="3"} What can be seen is that using fairly loose assumptions, superlinear growth can be obtained. Notably, these functions need not be power laws. They can simply be superlinear functions (such as $P(N) \approx N (\log_b{N})^x$), that mimic power laws. It is possible that this could be true of the observed city productivity data as well — that is, it is possible that the productivity functions observed are superlinear, but not necessarily power laws. Further measurement could help resolve this question. Discussion ========== Ultimately, the heart of the model is the relationship between long-distance ties and productivity in large cities. These long-distance ties, which are prevalent in a higher proportion when there is a larger population, provide the potential for productive social interactions. Granovetter’s classic paper ‘The Strength of Weak Ties’ considers this explicitly [@ARB:Gra73]. As part of his study, Granovetter examined the structure of Boston’s West End and its inability to organize against a neighborhood urban renewal project, which included the large-scale destruction of buildings to make room for new residential high-rises [@ARB:Col2005]. While the West End contained many strong ties, since most individuals had been lifelong residents of the area, these strong ties often resulted in cliques, where everyone was connected within a single group. Crucially, however, Granovetter argues that there were few, if any, ties [*between*]{} these tightly-knit local cliques. Since personal ties are generally necessary for information spread and organizational ability (or as Granovetter put it, “people rarely act on mass-media information unless it is also transmitted through personal ties,” [@ARB:Gra73], p. 1374), the inhabitants of the West End would have had a great deal of difficulty in organizing their opposition to the municipal project. In contrast, if there had been interaction throughout the social hierarchy, such as between communities within the neighborhood, the outcome might have been much different. Along these lines, Charlestown, a similar Boston neighborhood, was able to successfully organize against urban renewal. Granovetter argues that there was a rich interconnection between different communities, allowing for wider coordination. Alexander has similarly argued that rich interconnectivity between communities creates better cities [@ARB:Alex65]. Of course, any good model must be testable in order for it to rise above the level of a pleasant story. Additional work by the first author [@ARB:Arb2008] has indirectly attempted to determine what the value of $\beta$ is (it seems to be close to zero). Beyond this, given a network of social interaction for a city, its hierarchical social structure could be determined [@ARB:Cla2008], to see if it conforms to the type of growth with distance that is discussed above. This has not yet been done, but it should be feasible, given the relevant datasets. Furthermore, it would be interesting to consider models as well as empirical data that consider interactions at scales larger than within a single city, such as between cities or within entire geographical regions. In addition though, there are other possible explanations for this superlinear scaling in cities. For example, it could be that larger cities have a larger proportion of more highly educated individuals, which is enough to yield increased productivity per capita. Or it could be that larger cities simply have a greater transient population, which provides more fodder for different ways of thinking about the world, yielding a higher rate of productivity per individual. By distinguishing between our model based on social interaction and other competing models, we can get a better sense of how good our model is. But how can this be done? While cities do exhibit superlinear scaling for a variety of quantities, many cities do not lie exactly on the predicted curve for a given property, based on a curve of best fit. For example, some cities will produce more patents than expected, while others will produce far fewer than expected, given their population. By looking at the pattern of social interaction in the underperforming cities as compared to the overperforming cities, we can determine how reasonable our model is. And by examining how well other models can predict this type of variation, as opposed to ours, we can determine what is the likeliest explanation for superlinear scaling within cities. Nonetheless, as argued above, the presence of socially distant ties within a single city can be a powerful force. By using simple assumptions about social interactions, we gain a useful tool in understanding the mathematical behavior of innovation and productivity in cities. Acknowledgments =============== We thank Geoffrey West, Luis Bettencourt, Stephen Ellner, and Adam Siepel for helpful discussions. Research supported in part by National Science Foundation grant DMS-0412757 to S.H.S.

--- title: 'Measurement of $A_{\Gamma}$' --- The measurement of the charm CP violation observable $A_{\Gamma}$ using of $pp$ collisions at $\sqrt{s}=7$ TeV recorded by the LHCb detector in 2011 is presented. This new result is the most accurate to date. Introduction ============ CP violation in charm meson decays is expected to be small in the Standard Model (SM) and any significant enhancement would be a signal of New Physics (NP). Thus far no CP violation has been unambiguously observed in the charm system. The CP violation observable $A_{\Gamma}$ is defined as the asymmetry of the effective lifetimes of and decaying to the same CP eigenstate, or , $$A_{\Gamma} = \frac{\hat{\Gamma}(\Dz\to\Kp\Km) - \hat{\Gamma}(\Dzb\to\Kp\Km)}{\hat{\Gamma}(\Dz\to\Kp\Km) + \hat{\Gamma}(\Dzb\to\Kp\Km)} \approx \frac{A_{m}+A_{d}}{2}y\cos\phi-x\sin\phi,$$ where $A_{m}$ and $A_{d}$ are the asymmetries due to CP violation in mixing and decay respectively, $\phi$ is the interference phase between mixing and decay and $x$ and $y$ are the charm mixing parameters. In the Standard Model [$A_{\Gamma}$]{} is expected to be small[@Lenz]($\sim$10$^{-4}$) and roughly independent of the final state. New Physics (NP) models may introduce larger CP violation and some final state dependence of the phase $\phi$ leading to a difference in [$A_{\Gamma}$]{} between the and final states[@Sokoloff], $$\Delta A_{\Gamma} = A_{\Gamma}(KK) - A_{\Gamma}(\pi\pi) = \Delta A_{D} y\cos\phi + (A_{M}+A_{D})y\Delta\cos\phi - x\Delta\sin\phi.$$ The experimental status of the measurement of [$A_{\Gamma}$]{}, including the Heavy Flavour Averaging Group (HFAG)[@HFAG] average and excluding the results presented here, is shown in Fig. \[fig:agamma\]. ![Experimental status of [$A_{\Gamma}$]{}.[]{data-label="fig:agamma"}](a_gamma_14may12.pdf){width="48.00000%"} Presented here are new results for the measurement of using 1 fb$^{-1}$ of $pp$ collisions at a centre of mass energy of 7 TeV recorded by the LHCb detector in 2011[@paper]. Analysis Method =============== The mean lifetimes of the and are extracted via a fit to their decay times. The data to be fitted is broken into eight subsets. The splits are motivated by the two detector magnet polarities with which data was taken and two separate data-taking periods to account for know differences in detector alignment and calibration . Finally the and candidates have been fitted separately. The initial flavour of the is determined by searching for the decay $\Dstarp\to\Dz{\HepParticle{\pi}{s}{+} }$ where the charge on the pion indicates the flavour. Due to the small $Q$ value of this decay the pion is referred to as slow. The procedure is carried out in two stages. In the first the mass and the difference between the and masses ([$\Delta m$ ]{}) are fitted simultaneously. This allows for the separation of the signal and background components and the determination of the background probability density functions in the subsequent fits. Example mass and [$\Delta m$ ]{}fit results for the final state can be see in Fig. \[fig:massfit\]. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Fit of the mass (left) and [$\Delta m$ ]{}(right) for subset of data containing $\Dzb\to\Kp{}\Km$ candidates with magnet polarity down for the earlier run period.[]{data-label="fig:massfit"}](Massfit_D0bar_KK_log.pdf "fig:"){width="48.00000%"} ![Fit of the mass (left) and [$\Delta m$ ]{}(right) for subset of data containing $\Dzb\to\Kp{}\Km$ candidates with magnet polarity down for the earlier run period.[]{data-label="fig:massfit"}](Deltamfit_D0bar_KK_log.pdf "fig:"){width="48.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The second stage fits decay times and the natural logarithm of the impact parameter $\chi^{2}$ (). Those candidates originating from decays (secondary) have longer measured lifetimes than those originating at the primary vertex (prompt) as the has not been reconstructed. It is therefore necessary to separate these in the fit to avoid biasing the lifetime measurement. This is done using the variable. Due to the flight distance of the the impact parameter of the is larger than those of prompt candidates as shown in Fig. \[fig:secondary\]. Example fits are in Fig. \[fig:timefit\]. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![The separation of prompt (left) and secondary (right) decays by considering their impact parameters.[]{data-label="fig:secondary"}](prompt.pdf "fig:"){width="30.00000%"} ![The separation of prompt (left) and secondary (right) decays by considering their impact parameters.[]{data-label="fig:secondary"}](secondary.pdf "fig:"){width="30.00000%"} ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Fit of the (left) and decay time (right) for data subset containing the $\Dzb\to\Kp{}\Km$ candidates with magnet polarity down for the earlier run period.[]{data-label="fig:timefit"}](LogIPfit_D0bar_KK_log.pdf "fig:"){width="48.00000%"} ![Fit of the (left) and decay time (right) for data subset containing the $\Dzb\to\Kp{}\Km$ candidates with magnet polarity down for the earlier run period.[]{data-label="fig:timefit"}](Timefit_D0bar_KK_log.pdf "fig:"){width="48.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Lifetime biases due to the acceptance of the trigger and selections are corrected for using the “swimming” method. The primary vertex is moved along the direction of its flight and the trigger rerun to find the point in lifetime at which the candidate changes from being rejected to accepted. One can thus construct an acceptance function in lifetime for each event as shown in Fig. \[fig:swimming\]. An average acceptance function for the whole data set can then be constructed and folded in to the fit. For a complete description see [@Vava]. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![The swimming method. The primary vertex is ‘swum’ along the direction (from left to right). The trigger is rerun for each position and the lifetime of the candidate at which it becomes accepted by the trigger is found (middle).[]{data-label="fig:swimming"}](swim1.pdf "fig:"){width="30.00000%"} ![The swimming method. The primary vertex is ‘swum’ along the direction (from left to right). The trigger is rerun for each position and the lifetime of the candidate at which it becomes accepted by the trigger is found (middle).[]{data-label="fig:swimming"}](swim2.pdf "fig:"){width="30.00000%"} ![The swimming method. The primary vertex is ‘swum’ along the direction (from left to right). The trigger is rerun for each position and the lifetime of the candidate at which it becomes accepted by the trigger is found (middle).[]{data-label="fig:swimming"}](swim3.pdf "fig:"){width="30.00000%"} ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Summary of systematic uncertainties =================================== The systematic uncertainties of the method are evaluated through a mixture of studies on simplified simulated data and variations to the fit. Table \[tab:systematics\] summarises the results of these studies. Additionally some extra considerations such as detector resolution and track reconstruction efficiency (amongst others) are looked at and found to have a negligible effect on the resultant measurement. The data are also split into bins of various kinematic variables (for example momentum $p$, transverse momentum $p_{T}$ and flight direction) and no systematic variation in the result is found. The dominant systematic uncertainty comes from the acceptance function. This includes the uncertainty of the turning point positions determined by the swimming method and their subsequent utilisation in the fit procedure. Effect ()$\times 10^{-3}$ ()$\times 10^{-3}$ ------------------------ -------------------- -------------------- Mis-reconstructed bkg. $\pm 0.02$ $\pm 0.00$ Charm from [ ]{} $\pm 0.07$ $\pm0.07$ Other backgrounds $\pm0.02$ $\pm0.04$ Acceptance function $\pm0.09$ $\pm0.11$ Total $\pm0.12$ $\pm0.14$ : Summary of the systematic uncertainties on the measurement of for the two final states.[]{data-label="tab:systematics"} Results ======= The results of the measurement for the and final states are: $$A_{\Gamma}(KK) = (0.35\pm0.62_{stat}\pm0.12_{syst})\times 10^{-3}$$ $$A_{\Gamma}(\pi\pi) = (0.33\pm1.06_{stat}\pm0.14_{syst})\times 10^{-3}$$ The two numbers show no CP violation within the experimental uncertainty and are consistent with each other. They show a considerable improvement in accuracy over previous results. At the same time a complimentary measurement of was made on the same data using an alternative method by which the time evolution of the ratio of [ ]{}and [ ]{}yields was examined. The two methods yielded consistent results. Analysis of the 2 fb$^{-1}$ 2012 data set is to follow which will increase the precision further. Acknowledgements ================ We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC (China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We also acknowledge the support received from the ERC under FP7. The Tier1 computing centres are supported by IN2P3 (France), KIT and BMBF (Germany), INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom). We are thankful for the computing resources put at our disposal by Yandex LLC (Russia), as well as to the communities behind the multiple open source software packages that we depend on. [99]{} M. Bobrowski, A. Lenz, J. Riedl, and J. Rohrwild, *How large can the SM contribution to CP violation in $\Dz-\Dzb$ mixing be?*, JHEP **03** (2010) 009. A. L. Kagan and M. D. Sokoloff, *On indirect CP violation and implications for $D^0-\bar{D}^0$ and $B^0_s-\bar{B}^0_s$ mixing*, Phys. Rev. **D80** (2009) 076008. Heavy Flavor Averaging Group, Y. Amhis *et al.*, *Averages of b-hadron, c-hadron, and $\tau$-lepton properties as of early 2012*, **arXiv:1207.1158**,updated results and plots available at **http://www.slac.stanford.edu/xorg/hfag/**. LHCb collaboration, R. Aaij *et al.*, *Measurements of indirect CP asymmetries in $D^0\to K^-K^+$ and $D^0\to\pi^-\pi^+$ decays*, submitted to Phys. Rev. Lett., **arXiv:1310.7201**. V. Gligorov *et al.*, *Swimming: A data driven acceptance correction algorithm*, J. Phys. Conf. Ser. **396** (2012) 022016.

--- abstract: 'We present the results from a multiwavelength study of the flaring activity in HBL, 1ES 1959+650, during January 2015-June 2016. The source underwent significant flux enhancements showing two major outbursts (March 2015 and October 2015) in optical, UV, X-rays and gamma-rays. Normally, HBLs are not very active but 1ES 1959+650 has shown exceptional outburst activity across the whole electromagnetic spectrum (EMS). We used the data from Fermi-LAT, Swift-XRT $\&$ UVOT and optical data from Mt. Abu InfraRed Observatory (MIRO) along with archival data from Steward Observatory to look for possible connections between emissions at different energies and the nature of variability during flaring state. During October 2015 outburst, thirteen nights of optical follow-up observations showed brightest and the faintest nightly averaged V-band magnitudes as 14.45(0.03) and 14.85(0.02), respectively. In optical, the source showed a hint of optical intra-night variability during the outburst. A significant short-term variability in optical during MJD 57344 to MJD 57365 and in gamma-rays during MJD 57360 and MJD 57365 was also noticed. Multiwavelength study suggests the flaring activity at all frequencies to be correlated in general, albeit with diverse flare durations. We estimated the strength of the magnetic field as 4.21 G using the time-lag between optical and UV bands as synchrotron cooling time scale (2.34 hrs). The upper limits on the sizes of both the emission regions, gamma-ray and optical, are estimated to be of the order of $10^{16}$cm using shortest variability time scales. The quasi-simultaneous flux enhancements in 15 GHz and VHE gamma-ray emissions indicates to a fresh injection of plasma into the jet, which interacts with a standing sub-mm core resulting in co-spatial emissions across the EMS. The complex and prolonged behavior of the second outburst in October 2015 is discussed in detail.' author: - 'Navpreet Kaur$^{1, 2}$, S. Chandra$^3$, Kiran S Baliyan$^1$, Sameer$^{1,4}$, & S. Ganesh$^1$' bibliography: - 'reference.bib' title: 'Multi-wavelength study of flaring activity in HBL 1ES 1959+650 during 2015-16 ' --- Introduction ============ Blazars are a sub-class of Active Galactic Nuclei (AGN), with a relativistic jet pointed at small angles ($<$ 15$^{\circ}$) to our line of sight [@urry1995]. The emission in blazars is mostly dominated by the highly variable non-thermal continuum flux; the variability time scale ranging from tens of minutes to a few years, across the whole electromagnetic spectrum (EMS) [@bland1979; @wagner1995; @fan2005; @fan2009]. Their spectral energy distribution (SED) has two characteristic broad peaks implying two different emission processes at work, namely, the synchrotron process, from radio to UV/X-ray energies [@urrymush1982] and the inverse-Compton (IC) process in which high energy emission (X-ray to TeV $\gamma$-rays) is produced via up-scattering of low energy seed photons by the relativistic electrons that gave rise to synchrotron emission. The origin(s) of the seed photons are still under debate [@baliyan2005; @bottcher2005]. According to the leptonic scenario, the IC photons may either be generated by the up-scattering of the synchrotron photons by the same population of leptons ($e^-/e^+$) [@konigl1981; @marschergear1985; @ghis-tavec2009] under Synchrotron Self Comptonization (SSC) process or the photons from the external regions, e.g., torus, accretion disk, line emitting regions, etc., serving as seeds for the up-scattering to higher energies, under External Comptonization (EC) [@bottcher2007] process. On the other hand, in hadronic models, the high energy emission in blazars is mainly produced by the proton synchrotron and pion decay in the jet plasma [@mannheim1989]. Blazars comprise the two kinds of objects : 1.) Flat Spectrum Radio Quasars (FSRQ), identified by emission lines in the optical/UV spectra, and 2.) BL Lac objects, identified by the extremely weak lines or a featureless optical/UV continuum [@stickel1993]. The classification based on the broad-band SEDs divides the BL Lac objects into three sub-categories, namely; high energy peaked BL Lac objects (HBLs; $\nu_{s} \textgreater 10^{15} $ Hz ), Intermediate energy peaked BL Lac objects (IBLs; $ 10^{14} Hz \textless \nu_{s} \textless 10^{15} $ Hz) and Low energy peaked BL Lac objects (LBLs; $ \nu_{s} \textless 10^{14} $ Hz). The high energy emission in the HBLs is generally well explained by the SSC models, with a possible EC component in a few exceptional flaring states [@bottcher2007]. The blazars being extremely variable across the EMS, their variability can serve as a tool to understand AGN structure and emission processes, as their central engines are too compact to be resolvable [@ciprini2003; @marscher2008]. The HBLs generally are less variable than LBLs [@jannuzi1994] but some of them are very active with flares and outbursts detected almost over the complete accessible EM spectrum, ranging from the radio to TeV $\gamma$-rays [@acciari2011; @furniss2015]. While almost all the TeV flares are witnessed to have a counterpart in optical and X-rays, barring some orphan flares, the GeV energy region might show weaker activity. Blazars also show significant polarization in optical [@chandra2011 and references there-in] and radio wavelengths which is a measure of the alignment and the strength of the magnetic field. The changes in the degree of optical polarization (DP) and position angle (PA) are commonly seen during the flares in the blazars. Such rapid variations in DP and PA during a flare have been modeled for many sources (e.g., for 1ES 1011+496 - @aleksic2016, Mrk 421 - @zhang2015, for 3C 279 - @kiehlmann2016 [@hayashida2012; @abdoPol2010]). The outbursts in blazars are mostly thought to be a manifestation of the shock formation and their movement down the jet [@orienti2015; @marscher2010], internal inhomogeneities and their interaction with shocks, re-collimation of shocks downstream the jet causing re-acceleration [@spada2001] or, a new population of the relativistic plasma injected into the jet. In spite of the considerable efforts until now, none of the proposed models are able to explain blazar phenomena. Our understanding of the geometry of the jet, the emission processes responsible for different flaring activities and the behavior of the objects during their quiescent phase are limited by the sample size and the scarcity of simultaneous data over a broad energy range. Therefore, there is need for extensive multi-wavelength studies on a large sample of blazars to enable a comprehensive understanding of the emission processes, in general. The HBL 1ES 1959+650, redshift z=0.048 [@perlman1996], was first detected in radio band using NRAO Green Bank Telescope [@gregorycondon1991] and observed in X-rays during the slew survey by the Einstein Imaging Proportional Counter [@elvis1992]. The first TeV detection of this source was reported by the Seven Telescope Array group in 1999 [@nishiyama1991]. This source was identified as an optical BL Lac object by @schachter1993. Later, a bright ($m_{R}$ = 14.9) elliptical galaxy was confirmed by @scarpa2000 as the host. The source has undergone various outburst stages, including intense activity at very high energies (GeV-TeV). @krawczynski2004 reported an “orphan” flare at VHE during an outburst in 2002, in a multi-wavelength campaign (WHIPPLE and HEGRA for TeV, RXTE for X-rays, Boltwood and Abastumani observatory for optical, UMRAO for radio 14.5 GHz) from May 18 - August 14, 2002. The authors reported a correlation between the $\gamma$-ray and X-ray fluxes but during orphan TeV flare, no enhancement in X-ray flux was seen and there was no correlation between optical and X-ray/$\gamma$-ray emissions. @bottcher2005 explained 2002 orphan TeV flare using a hadronic synchrotron mirror model in which the orphan TeV photons originated from the interaction of relativistic protons with an external photon field supplied by synchrotron radiation reflected off a dilute reflector. Another intense flaring activity was seen during 2012 April- June covered in a multi wavelength campaign by @aliu2014. During the outburst, 1ES 1959+650 emitted enhanced flux in gamma-rays without any significant simultaneous rise at X-ray energies. The authors proposed a reflected emission model to explain elevated $\gamma$-ray flux, via pion production with very high energy protons (10-100 TeV). 1ES 1959+650 was reported in unprecedented high flux state across all the energies (gamma-ray, X-ray, UV, Optical and radio) during 2015 which extended to 2016 as well. The source underwent two major outbursts; first in X-rays during March 2015 and the second one during October 2015 with exceptionally large count rates (more than 20 counts/sec) in X-rays [@AtelXray2015], making it only the third TeV source with such high X-ray count rate after Mrk 421 and Mrk 501. Such outburst activities provide an opportunity to study the underlying physical processes responsible for emission at different energies. Recently, @kapanadze2016 reported prolonged X-ray activity in 1ES1959+650 during their 6-month coverage of 2015 October outburst (MJD 57235-57410) along with enhanced $\gamma$-ray (0.3-100 GeV) flux at several epochs. They also claimed an orphan $\gamma-$ray flare at MJD 57314 with no enhancement at other energy regimes, albeit with a rider that due to sparse data, a rapid X-ray flare might have occurred. To understand the overall behavior of 1ES 1959+650, we carried out an extensive multi-wavelength study of the outburst activities in 1ES 1959+650 from 2015 January-2016 June (MJD 57040-57570). Optical follow-up observations from Mt Abu InfraRed Observatory (MIRO) were carried out for the source when it exhibited high flux state in X-ray, GeV and TeV energies. This paper is organized as follows. Section 2 describes the observations and data analysis techniques in optical, X-ray and Gamma-rays. Detailed analysis of the light curves and the results are discussed in Section 3. Section 4 presents a brief summary of the work presented in this paper. Multi-wavelength Observations and Data Reduction \[sec:sec2\] ============================================================= Multi-wavelength data from various resources, space-borne observatories, namely, Fermi ($\gamma$-rays), Swift (UV, optical & X-rays) and the ground-based facility, MIRO (optical/IR) are used in this study. We have also used publicly available archival data from Steward Observatory, Arizona [@smith2009]in optical and radio data at 15 GHz from OVRAO [@richard2011] to discuss various outburst episodes. We briefly summarize the data analysis techniques used for the data from aforementioned resources in the following. Gamma-ray : Fermi-LAT ---------------------- FERMI Large Area Telescope (Fermi-LAT) is a primary instrument on board the Fermi satellite [@atwood2009]. The LAT has an unprecedented sensitivity in the $\gamma$-ray band (20 MeV - 300 GeV) and scans the entire sky in approximately 3 hrs except for few extremely high priority specific pointing mode observations where the observations are taken for 30 minutes for the prioritized sources. It provides a multi-dimensional data base of location, energy and time for each detected event. We analyzed 1ES 1959+650 Fermi-LAT data from January 01, 2015 (MJD 57023) to June 31, 2016 (MJD 57550) using standard recommended procedure by making use of the latest ScienceTools (version v10r0p5). The photon class events lying within the region of interest (ROI) of 10$^{\circ}$ , zenith angle $\textless$ 100$^{\circ}$ , within the energy range of 0.1 - 300 GeV are extracted using “gtselect” tool. We discarded the data when the rocking angle of the spacecraft was greater than 52$^{\circ}$ to avoid any photon contamination from the Earth’s limb. An unbinned likelihood analysis was performed using gtlike tool with the help of input source model covering a region of 20$^{\circ}$ around the source position, generated using 3rd FGL catalog [@acero2015]. A maximum likelihood analysis using gtlike has been used to reconstruct the source energy spectrum. The background model was constructed using third Fermi LAT catalog $(gll_psc_v16.fit)$ that contains 36 gamma-ray sources lying within ROI &lt;12$^{\circ}$, as well as diffuse emission with no extended sources within this region. We have made use of a log-parabolic model and a power law model for the sources with significant and without spectral curvature, respectively. The source (1ES 1959+650) spectral parameters within 3$^{\circ}$ were kept free during spectral fitting, while sources outside of the aforementioned range were held fixed as in the 3FGL catalog. The Galactic diffuse emission and the isotropic emission component was modeled using ${\textit gll\_iem\_v06.fits}$ and $iso\_P8R2\_SOURCE\_v6\_v06.txt$, respectively. Fermi-LAT data was reduced using a Python based package called Enrico [@sanchezdiel2013]. A time binning of 2 days was used to extract the source light curve. To look for the details of post-outburst activity in the source, we used 2.5 day binned data set. X-ray, UV, optical : Swift-XRT/UVOT ----------------------------------- We have made use of around 95 observation IDs observed by the instruments XRT and UVOT onboard Swift, during January 01, 2015 to December 31, 2015. The [*heasoft*]{} (version 6.17) package along-with the recently updated calibration database (2016 January 21 for XRT & 2016 March 05 for UVOT) is used for the analysis of the above-mentioned data. The [*xrtpipeline*]{} tool provided freely as a part of [*heasoft*]{} package, with default parameters are used to extract the cleaned events files. This source, being very bright, is mostly observed in WT mode. The typical full frame count rates for WT mode observations are always less than 25 c/s which will be pile-up free as recommended by the instrument team at University of Leicester, UK [^1] (pile-up occurs for rate $\ge$ 100). Following this, the pile up corrections are not performed for WT mode data. For this case, a circular area of 27 pixels (equivalently $\sim$ 63“) centered at the position of the 1ES 1959+650 is used as the source region. The background region was extracted as a concentric annulus with inner and outer radii of 80 pixels and 120 pixels, keeping the average half width of annular region at 100 pixel (recommended for a proper background subtraction in WT mode). The PC mode observations are corrected for pile up using the prescriptions suggested by aforementioned team [^2]. The innermost circular area with radius 10” is excluded (chosen for a highly piled up observations) from the source region of 70“ circle around the source. The background in this case is used as an annular region around the source, with inner and outer radii of 150” and 350", respectively. The clean events files are then used to extract the products (spectrum & light-curves) for the source and background regions using the [*xselect*]{} tool. The spectra thus obtained are then fitted with an absorbed log-parabola model with nH value fixed to the galactic value (1.07 $\times$ 10$^{21}$ cm$^{-2}$) using [*xspec*]{} (version 12.9.0), a standard tool for X-ray spectral fitting provided as a part of [*heasoft*]{} package. Using the BACKSCAL keyword in WT mode, source and background spectrum files were edited to the proper values, before importing to the fitting tool, to avoid the wrong background subtraction during the fit. The log-parabola was chosen instead of the commonly used absorbed power-law which was giving very poor fit ($\chi_{\nu}$ $\ge$ 1.9). The absorbed broken power-law model was providing similar fit as that using curvature model. We prefer log-parabola because it provides natural turn-over in the spectrum instead of a sudden break as given by broken power-law models. The background subtracted count rates extracted for energy band 0.3-10.0 keV are used to generate the light-curves (see Fig. \[fig:fig1\]). The unabsorbed fluxes in 0.3-10.0 keV band are also estimated by adding component “cflux" and fitting after freezing the normalization. The fluxes thus obtained are also used for timing analysis in this paper. The UVOT data analysis is done in a similar fashion as adopted in @chandra2015. The snapshots observations in the filters V (5468 Å), B (4392 Å), U (3465 Å), UVW1 (2600 Å), UVM2 (2246 Å), and UVW2 (1928 Å), for all the OBsIDs, were integrated with the [*uvotimsum*]{} task and analyzed using the [*uvotsource*]{} task, with a source region of 5”, while the background was extracted from an annular region centered on 1ES 1959+650 with external and internal radii of 40” and 7”, respectively. The observed magnitudes from all OBsID are then corrected for extinction according to the model described in @cardelli1989. A tool, developed in-house, using R-platform[^3], is used to perform the required reddening corrections. The corrected fluxes are then used for timing analysis in the following section \[sec:sec3\] (See Fig. \[fig:fig1\] for light curve). Optical observations : MIRO --------------------------- Following an alert [@Atel8193] of an enhanced $\gamma$-ray activity in 1ES 1959+650 on October 20, 2015 (MJD 57315), we made optical photometric observations using two telescope facilities at Mt. Abu InfraRed Observatory (MIRO) i.e., 1.2 m & 0.5 m telescopes. The 1.2 m telescope is equipped with LN2-cooled CCD (1296 $\times$ 1152 pixels; pixel size = 22 $\micron$) at its f/13.2 Cassegrain focus, whereas, a thermo-electrically cooled (T $\sim$-80$^{\circ}$C) iKon ANDOR CCD (2048 $\times$ 2048; pixel size =25 $\micron$) is used as backend instrument for 0.5 m telescope. The dark current in both systems is negligible. The observations were carried out using BVRI Johnsons-Cousins filters for total thirteen nights during October 23 to December 13, 2015. High temporal resolution (exposure time of $\approx$ 40-50 seconds) data with the optimum signal-to-noise ratio (S/N $\textgreater$ 5) are obtained. The flat field images were taken during twilight and dawn on daily basis whereas bias frames were taken whenever telescopes were slewing to change the source. During 2015 November- December, we also monitored the source to look for intra-night variability (INV). The data were reduced using standard data reduction procedures using IRAF package (Image Reduction and Analysis Facility) and locally developed pipelines [@chandra2011; @Nav2017]. On each night, master bias and master flat frames were generated by combining all bias and flat field images, respectively. The science images were then corrected with bias and flat field and aperture photometry was performed on the source as well as on comparison stars using DAOPHOT package. The source magnitudes thus obtained were calibrated using two comparison stars 4 and 6 [@villata1998] having similar brightness as that of the source. Table \[tab:table1\] gives the details of photometry data obtained from MIRO where column 1 and 2 represent the date and MJD of observations, third and fourth column are nightly averaged R- and V-band magnitudes along with photometric errors, column 5 shows the telescopes used. In order to look for INV in 1ES 1959+650, we monitored the source for more than an hour on 9 nights during October - December 2015. Supplementary data : Optical (Steward Observatory); Radio(OVRO) --------------------------------------------------------------- We have used optical photometry and polarimetry data available online from Steward Observatory [^4] during October 2015 to look for the polarization behavior during its flaring state. We have also utilized the publicly available radio data from Owens Valley Radio Observatory (OVRO)[^5] at 15 GHz frequency to checkfor any correlated activity with other data used here. Results and Discussion {#sec:sec3} ====================== Multi-wavelength light curves (MWLC) are constructed using analyzed data as described in the previous section (section 2) and are shown in Figure \[fig:fig1\] where X-axis represents the time in MJD (Modified Julian Day) and Y-axis shows respective flux or magnitude values at various energies. In Figure \[fig:fig1\] (from the top), the first panel: Fermi-LAT $\gamma$-ray (0.1 to 300 GeV) flux, second panel: Swift-XRT flux at three X-ray energy bands i.e., (0.3 - 3.0 keV: X1 band), (3.0 - 10.0 keV : X2 band), and (0.3 - 10.0 keV : X3 band), third panel: Swift-UVOT UV (UVW1-band) light-curve, fourth panel: Swift-UVOT, MIRO and Steward Observatory V-band optical light-curve, fifth panel: OVRO 15 GHz radio light-curve. Multi-wavelength light curve {#sec:sec3.1} ---------------------------- As can be noticed from the Figure \[fig:fig1\], the light curves across all the energy bands (Gamma-rays – X-rays – UV – Optical – Radio) appear very complex in nature, especially during the two major outbursts, with a number of flares, sub-flares with diverse rates and periods of quiescence appearing through out the entire electromagnetic spectrum (EMS). Such random trends are typical in blazar light-curves [@Chatterjee2012]. The shape and length of flares in the light-curve along with polarization information tell about the emission mechanisms at work, strength of the magnetic fields, etc., in the jet. A rising trend in MWLC corresponds to the acceleration of the relativistic particles as a dominant process while a declining trend indicates to their subsequent cooling. The flux at GeV energies (0.1 - 300 GeV) for 1ES 1959+650 remained in low state (F$_\gamma$ $\approx$ 1.3 $\times$ 10$^{-08}$ ph  cm$^{-2}$ s$^{-1}$) most of the time super-imposed by several mini-flares during 2015-16. It should, however, be noted that it is more than twice the average flux level reported in 3FGL catalogue (F$_\gamma$ $=$ 5.83 $\pm$ 0.18 $\times 10^{-09} ph\ cm^{-2} s^{-1}$). During the October 2015 outburst, the source showed flux level as high as 3.8 $\times$ 10$^{-07}$ ph cm$^{-2}$ s$^{-1}$ with 2-day binning, highest ever reported for this source in the (0.1-300 GeV) range. @kapanadze2016 (hereafter, K16) reported $\gamma$-ray flux as 1.2$\times$ 10$^{-07}$ ph cm$^{-2}$ s$^{-1}$ at this epoch in the (0.1-100) GeV range with a 3 day binning. The difference in maximum flux level might have arisen due to different binning, energy range covered (0.3-100 GeV) and analysis method used in K16. Several sub-flares, with varying rise/fall rates, have been noticed before and after the onset of major outbursts - March 2015 (outburst 1) and October 2015 (outburst 2) which is extended to 2016 June. On the other hand, X-ray flux was seen to behave erratically during the whole period with significantly enhanced flux when the source was in outburst in $\gamma-ray$, UV, optical and radio. A flare beginning at MJD 57070 (outburst 1) appears truncated due to lack of data in X-ray; UV and optical met the similar fate while a clear flare is noticed in $\gamma$-ray light-curve, peaking at about MJD 57107 (F $\approx$ 3.2 $\times$ 10$^{-07}$ ph cm$^{-2}$ s$^{-1}$). The trend shows that a complete data set could have led to correlated flaring in all these energy bands. In soft X-ray band (0.3-3.0 keV; X1 band) counts varied from 3-13 cts sec$^{-1}$, but for X2-band (3.0-10.0 keV; relatively hard flux state), the counts remained between 1-2 cts sec$^{-1}$. During the outburst phase, the total number of counts in X-ray reached as high as 10 - 20 cts sec$^{-1}$. However, K16 report maximum count rate as 22.95 in 0.3 - 2.0 keV range at MJD 57382.8, same as reported in @K16_ATel2015. Such high X-ray count rate makes it only the third TeV source after Mrk 501 & Mrk 421. The UV light curve (3rd panel from the top) showed significant variation during outburst period. A consistently high flux is seen starting around MJD 57245 (2015 August 10: onset of outburst 2), with signatures of various flares spanning a few days, super-imposed over an already high flux. Almost similar trend is seen in optical V-band light curve but with less flux modulations superimposed on the increased flux level. These elevated flux levels, almost twice the base value, continued for more than 200 days due to flaring activity. The radio light curve at 15 GHz shows slowly decreasing flux up to MJD 57184 after which it starts increasing again, showing strong flare and then crossing 2.5 Jy level at about MJD 57304 (2015 October 9: outburst 2). After this, radio flux drops sharply, reaching lowest value in the whole duration, just when optical degree of polarization and position angle had undergone rapid changes (cf Figure \[fig:fig2\], right panel). Though there is very limited data on DP and PA, these rapid variations, are significant as they are closely followed by a major flare in gamma-rays. Subsequently, optical, UV and X-ray emissions show their peak in their respective light-curves. The details of the second outburst will be discussed in the context of $\gamma$-rays, in the next section. Apart from these major flaring activities, several mini-flares spanning a few days in $\gamma$-rays can be seen in the light-curve, e.g., flare peaking at about MJD 57070 (flux drops from (1.7- 0.2) $\times10^{-7} ph cm^{-2}s^1$), 57152 (with simultaneously enhanced emissions at radio, UV & X-rays), 57191 (no data in X-ray/UV/Optical but 15GHz flux is significantly enhanced), 57340 etc. The detailed investigation of the light-curve reveals twin peaks in $\gamma$ $-$ rays, spanning almost 6 days each, just after the major outburst 2 peak. Both the flares are at almost the same flux level as that of the outburst 2. In the following section, we will discuss various features in the light-curve. MWLC: Outburst 1 (March 2015) ----------------------------- The duration of March 2015 outburst in $\gamma$-ray flux was roughly of 29 days (MJD 57094 - 57123) with a peak flux value reaching $F_{\gamma} $ = 3.25$\times$ 10$^{-07}$ ph cm$^{-2}$ s$^{-1}$ on MJD 57109. During the peak outburst, source brightened by more than 10 times the quiescent state flux value ($F_{\gamma} \textgreater$ 0.3 $\times$ 10$^{-07}$ ph cm$^{-2}$ s$^{-1}$). The outburst is temporally almost symmetric with rise time and decay time as 15 and 14 days, respectively. The outburst has several pre-burst flares contributing to it with diverse rates. Similarly, outburst in X-ray has a duration of about 17 days, peaking around MJD 57091 with 10.1 counts $s^{-1}$. The rising rate for X-rays is 4.4 $\pm$ 1.61 $\times$ 10$^{-12}$ ergs cm$^{-2}$ s$^{-1}$ with a rather sharp decline. However, it is quite possible that it could have been truncated due to lack of the data, the peak still to occur, just like what we see in UV and optical bands. In that case, all the emissions are likely to peak simultaneously, indicating to having almost same origin. Notice the similar enhancement in the form of a mild outburst in radio band lasting for about 14 days with a sharp decay. It appears that March 2015 outburst activity first started in radio band, then in X-rays, UV, optical and later in $\gamma$-ray. However, there are some pre-outburst flares seen in optical, UV and gamma-rays as well. The $\gamma$-ray outburst starts when X-ray emission has almost peaked and UV/optical are in the process of reaching peak values (Figure \[fig:fig1\]). X-ray peak leads UV/optical and $\gamma$-ray ones by about 7 and 8 days, respectively. Since 1ES 1959+650 is an HBL, X-ray emission (dominated by soft X-rays) is expected to come from synchrotron process which is also responsible for UV and optical emissions. High energy particles giving rise to X-ray emission cool faster and hence X-ray emission leads optical/UV. However, 7-days lag between X-ray and optical/UV appears to be a bit long and perhaps emission regions are also aligned differently to LOS. It could also be that UV and optical peaks could not be seen due to truncated data. In SSC, the $\gamma-ray$ emission is produced by IC process where synchrotron seed photons take time to travel to high-energy ($\gamma-ray$) emission region, where these are up-scattered [@sokolovMarscher2004]. It is possible that the emissions are generated when the emitting plasma passed through a standing or slowly moving shock down the jet and below the sub-mm core. @jorstad2001 reported that $\gamma$-ray flares were associated with ejection of super-luminal radio knots into the jet, which initiates flare in radio and $\gamma$-ray flare followed. All the four flares in the light-curve upto MJD 57220 rise slowly while decay very fast. The variations during the outbursts clearly follow a common trend across all the energies, indicating that acceleration timescales for electrons are longer than their cooling timescales. MWLC: Outburst 2 (October 2015- June 2016) ------------------------------------------ The build up to the second major outburst (MJD 57285 - 57370 or upto 57570) started much earlier, due to lack of data in other bands (X-ray, UV and optical) we consider the period from MJD 57225, which marks significant enhancement in the fluxes in all energy bands, including 15GHz radio band as shown in Figure \[fig:fig2\]. While flux in optical, UV and $\gamma-$ray starts gradually, flux in radio and X-ray energy bands rise sharply, reaching a plateau at about MJD 57246 where even $\gamma-ray$ flux has also increased significantly, while optical and UV are still rising. These two bands reach plateau when radio and $\gamma-$ray fluxes have already peaked. The rise and fall rates for $\gamma-$rays during 2nd outburst were estimated as - 9.01 $\pm$ 2.21 $\times$ 10$^{-09}$ ergs cm$^{-2}$ s$^{-1}$ (MJD 57285 - 57317) and -3.31 $\pm$ 1.08 $\times$ 10$^{-09}$ ergs cm$^{-2}$ s$^{-1}$ (MJD 57317 - 57326). For X-rays the estimated rise rate is 3.44 $\pm$0.82 $\times$ 10$^{-12}$ ergs cm$^{-2}$ s$^{-1}$ (MJD 57249 - 57382). The main outburst showed sudden flux enhancement, supported by sub-flares, spanning 41 days in $\gamma$-rays (MJD 57285 - 57326) reaching its peak value on MJD 57316.8 (2015 October 22). A prolonged, erratic flaring activity (with more than twice the average flux) with duration of about 145 days in X-ray flux is clearly seen, in which a flare peaks around MJD 57382 with largest ever 20 cts s$^{-1}$, followed by four other significant flares, each of them detected with &gt;10 cts s$^{-1}$. Note that we have selected X-ray data-sets to estimate flare duration when the source count rate was above 5 cts s$^{-1}$. Let us discuss the major flares in the light-curve where $\gamma-ray$ flux is normally above $ 10^{-7} ph cm^{-2}s^{-1}$. The flare at MJD 57237 decays by 1$\times 10^{-7} ph cm^{-2}s^{-1}$ in 2d, rising again by about 1.2$\times 10^{-7} ph cm^{-2}s^{-1}$ to 1.7$\times 10^{-7} ph cm^{-2}s^{-1}$. The flux in two other flares at MJD 57247.2 and 57262 rise by more than two-fold to about 2.0$\times 10^{-7} ph cm^{-2}s^{-1}$ in 2d. The pre-outburst flux of flare at MJD 57297 changes from 0.7$\times 10^{-7} ph cm^{-2}s^{-1}$ to 2.14$\times 10^{-7} ph cm^{-2}s^{-1}$ and rises further to 2.4$\times 10^{-7} ph cm^{-2}s^{-1}$ at MJD 57302. It is followed by peak flux (0.28Jy) in radio at MJD 57305 and a 26$\sigma$ detection of VHE by VERITAS [@mukherjee2015_atel_8148], accompanied by a significant enhancement in optical, UV and X-ray flux. After two more flares contributing to the flux, October 2015 outburst reaches its peak flux 3.75$\times 10^{-7} ph cm^{-2}s^{-1}$ at MJD 57316.88 (2015 October 22) in the (0.1-300 GeV) energy range. The peak in $\gamma-$ray flux is followed by delayed peaks in optical, UV, and X-ray bands showing significant variability at about MJD 57322. However, radio flux is at its lowest after a sharp decay. It should be noticed that this major flare in $\gamma-$rays happened just after rapid changes in optical polarization and position angle followed by peak in radio flux. All these activities signal that the emissions were correlated and were perhaps caused by passage of a blob through the sub-mm core as around the same time significant VHE emission and noticeable radio emission [@mukherjee2015_atel_8148; @RATAN_Atel2015] was detected . We, therefore, feel that the occurrence of an orphan flare in the (0.3-100 Gev) $\gamma-$ray range is doubtful at this epoch, as claimed by @kapanadze2016, while discussing the X-ray flux variations with other bands. After the major flare, the $\gamma-$ray flux drops to 0.7$\times 10^{-7} ph cm^{-2}s^{-1}$ in 10d, rising again to 2.7$\times 10^{-7} ph cm^{-2}s^{-1}$ in 2d. This flare at MJD 57328 is followed by X-ray (K16), UV and optical. The $\gamma-$ray flare at MJD 57342 is accompanied by E1-event in X-ray as reported in K16 (we do not have that data as it was a ToO observation) with 20 cts/second, followed by first of twin peak flare in optical, discussed later and flare in UV. FACT also reported 3$\sigma$ detection of VHE emission at this epoch. The first twin $\gamma-$ray flare at MJD 57360, flux 3.5$\times 10^{-7} ph cm^{-2}s^{-1}$, is followed by, after one day, second optical twin peak and preceded by E3 event in X-ray (19 cts/s; K16). Radio emission is also enhanced with enhancement in UV flux. All the emissions appear to be nicely correlated. After this, there is break in our $\gamma-$ray data, but there are significant flares in optical, UV and X-ray. In fact this was also ToO slot as reported in K16, where a $\gamma-$ray flare is accompanied with E2 event in X-ray along with enhanced flux in optical and UV. A nicely correlated flare in radio, optical, UV, X-ray and $\gamma-$ray fluxes is noticed peaking at MJD 57421, in the light-curve followed by a Swift data break upto MJD 57510. Just before that, a large outburst occurs (flux change (1 - 2.2) $\times 10^{-7} ph cm^{-2}s^{-1}$) at MJD 57506, decaying part of which is captured by all other bands. A clean flare in UV, optical, followed by X-ray and $\gamma-$ray flares at MJD 57540, with 0.25 mag brightness in V and a change in UV flux 2.5-3.5 mJy is seen. A number of flares contribute to the last outburst in $\gamma-$ray flux centered at MJD 57544.72 when flux increases almost three fold to 3.34 $\times 10^{-7} ph cm^{-2}s^{-1}$ , the outburst lasting about 25 days. X-ray counts increase from 5 to 15, radio also shows correlated enhancement in the flux while UV and optical are decreasing. As evident from the 15GHz radio light-curve, the source was active much before the enhancement started in X-ray/UV/optical emissions, i.e., around MJD 57185 and exhibited slow rise, peaking around MJD 57305 with a flux value of 0.28 $\pm$ 0.02 Jy, and a sharp fall reaching to half of its peak flux in 14 days, i.e., 0.17 $\pm$ 0.02 Jy on MJD 57319, extinguishing the 134 days of activity in radio. This sudden "shut off” of activity in radio happened just 7-days prior to the highest peak in $\gamma$-ray flux and just after the rapid changes in the optical polarization (see, Figure \[fig:fig2\]) had taken place. @Lahteenmaki2003 [@jorstad2001] also noticed flaring of radio emission at 37 GHz just before a flare in $\gamma$-ray flux occurred. With a break in the data, enhanced flux, by a factor of two ($F_{radio} = $ 0.250 $\pm$ 0.002 Jy), is noticed in 15GHz radio band. Therefore, across the whole spectrum, an outburst with significant high flux levels and slow rising/decaying trend spanning over a few months (long-term variations) is noticed along with flux variations lasting for a few days (short-term variations). A rapid variability in $\gamma$-rays with clear mini-flares lasting over a few days is seen, in general, but not always, showing slow rise and fast decrease, with 2-day binned data-set. A slow rising trend in flares suggests fast cooling of electrons and a less stochastic acceleration process [@kapanadze2016]. A prolonged activity in X-ray band showing chaotic behavior in the light-curve is noticed, when the source was peaking at GeV, $\gamma-$ray energies. UV and optical peaks were seen to occur later as compared to X-rays. ### Correlation in the flux variations It is clear from Figures \[fig:fig1\] & \[fig:fig2\] that while flux starts rising first at lower energies (V, UV, X-ray) during outburst 2, it peaks first at $\gamma$-rays followed by X-ray, UV & V bands. To check if the variations in these bands are correlated, we used discrete correlation function (DCF) which was first introduced by [@Edelson1988] and generalized by [@Hufnagel1992] to include a better error estimate. A brief description of the method is given by @Tornikoski1994 and @Hufnagel1992. A variant of DCF is the zDCF [@Alexander2014] which corrects for various biases of the DCF method by employing equal population binning and Fishers z-transform. Lags have been computed based on maximum likelihood criterion satisfying one sigma confidence interval. Figure \[fig:fig3\] shows three columns consisting of three panels each. Top two panels show light curves at two different energies while bottom panel shows discrete correlation between them. In the correlation study on the present data sets, we notice that due to multiple overlapping flares in almost all the bands, and particularly erratic, elevated flux in X-rays, the correlations between various fluxes are not very strong. We could not get a clear correlation with X-ray vis-a-vis other bands. However, as is reflected by light-curves also, UV and optical fluxes are correlated (figure3, column 3), with UV leading optical by about a few hours. Similarly, $\gamma$-ray versus V and UV are correlated with $\gamma-$ray leading by 20 and 18 days, respectively. It can be clearly seen that events in $\gamma-$rays are fairly correlated with events in UV, optical and $\gamma-$ray variations. Therefore, the high energy $\gamma-$ray emission was followed by emissions at lower frequencies, X-ray, UV and optical, in general. However, there are instances when $\gamma-$ray emission lags behind low energy emissions, which can be explained based on light-travel time arguments and/or differently aligned emissions regions with respect to LOS. In case of HBLs, low energy emissions (IR to soft X-rays) are produced by the relativistic electrons as synchrotron radiation while the high energy emission is expected to be generated through Inverse Compton (IC) process under which synchrotron photons are up-scattered by the same population (synchrotron) of electrons which produced them. Generally, it is the most preferred scenario known as one-zone SSC that is capable of explaining SED of the high energy peaked blazars (HBL) and has been used in other studies [@Bottacini2010 and references there-in]. Since significant lags were noticed between high energy $\gamma-$rays and other low energy emissions, opacity effects of the turbulent medium inside the jet could be responsible. In this case, the higher energy emission would occur first followed by low energy emission, as longer wavelengths are more susceptible to opacity effects compared to shorter ones. The second explanation could be if the emissions are generated in different regions and/or are aligned differently to the line of sight of an observer. For example, if emission region emitting at higher energy is oriented closer to observer’s line of sight as compared to other regions emitting at lower frequencies, then high energy emission would be more strongly Doppler boosted (due to relativistic effects) and will show faster variations in the light-curve [@MK97; @Finke2008]. The low-energy emitting region being slightly away from observer’s line of sight, would show delayed emission. This could be the reason that an activity is first seen at higher energies followed by lower energies and still appears to be correlated. Nevertheless, the origin and prolonged activity in X-rays, as mentioned above, is still intriguing. The lack of connection between optical/UV and X-rays, very low degree of polarization (random), longer cooling timescales in X-ray light curves as noticed for Mrk 421 [@balokovic2016] suggest contributions from multiple emission regions. @raiteri2015 suggested a complex UV and X-ray behavior for PG 1553+113 using multi-wavelength WEBT campaign data. A recent study by @Cavaliere2017 suggested the presence of an extra keV synchrotron component during particle progressive acceleration along with canonical optical to GeV emissions. It would, therefore, be very interesting to look for the processes and regions leading to the origin of X-rays and their relationship with other energy-bands during outburst and longer quiescent states to understand emission mechanisms at work inside the jet. ### Synchrotron cooling time scale and magnetic field strength In the earlier section, we noticed that during second major outburst, the optical emission was delayed by a few hours with respect to the emission in UV band. Now, since 1ES 1959+650 is an HBL, both the emissions are generated by synchrotron process in which electrons are accelerated to the relativistic velocities, which then cool down, radiating at various frequencies- higher frequency emission being emitted first due to faster cooling rate. Therefore, the time lag between emissions at two frequencies can be taken as the difference in the radiative cooling time scales of the population of electrons emitting at those frequencies [@UrryCM1997; @baliyan1996]. We can, therefore use the delay between the UV and optical emissions as cooling timescale of the synchrotron electrons and estimate the magnetic field. Using the expression for cooling time scale [@UrryCM1997], $$\label{eq:2} t_{lag} \approx t_{cool} = 2.0 \times 10^{4} \sqrt{\frac {\delta}{(1+z)}} B^{-3/2} (\nu_{15}^{-1/2} ) sec$$ where, B is magnetic field in Gauss, $\nu_{15}$ is the frequency in $10^{15}$ Hz, $\delta$ $=$ 15 [@MK97] and $t_{lag}$ = 2.34 hrs. This gives us a magnetic field estimate of 4.21 G which is on the higher side in such systems. ### Estimation of the sizes of emission regions {#sec:sec3.1.1} The central regions of the AGN are very compact in size and can not be resolved by any existing facility. The variability property provides a tool to explore those deeper regions. We have noticed that there are several flares with short time scales. The shortest time scale of variation provides an upper limit to the size of emission region at that particular waveband, based on the arguments of light-travel time. In $\gamma-$rays, doubling time scale is the time period when flux doubles its initial value with more than 3$\sigma$ significance and we used it to estimate the upper bound of an emission region, $R_{\gamma}$ [@saito2015]. Based on the light-travel causality relation, we estimated R from, $$\label{eq:3} R \textless \frac{\delta c \tau_{d}}{(1+z)}$$ where, R is the radius of the emission region, $\delta$ is taken as 40 [@aliu2014] for $\gamma-$rays, $\tau_{d}$ = 1 day is the flux doubling timescale and z= 0.048, redshift of the source. The $\gamma-$ray emission size is estimated as $R_{\gamma} \textless 9.89 \times 10^{16}$ cm. For the optical emission region, we used variability timescale of 4.5 hrs as a characteristic timescale and $\delta =15$, to estimate the upper limit to the size of the emission region as $R_{op}$ $\leq$ 3.71 x $10^{16}$ cm. Thus $\gamma-$ray emission region is of the same order but about 3 times larger in size than optical region size. The emission region sizes obtained suggest that the $\gamma-ray$ and optical emission might be co-spatial in nature. However, one has to be cautious as due to the larger bin size in case of $\gamma-$rays as compared to optical, it is difficult to get an accurate value for doubling time scale. The location of the $\gamma-$ray production site is not very well known for all the sources, in general. But, for a few sources many authors reported location of $\gamma-$ray emission region based on VLBI and high energy (GeV to TeV) simultaneous data-sets [@Agudo2011; @marscher2014]. Based on these observations, respective models predict that the $\gamma-$rays are produced close to the standing radio core (at 1-10pc) as $\gamma-$rays cannot escape from the vicinity of black hole due to photon absorption effects. To determine the distance of the $\gamma-$ray emitting region from the central source, we need to know the opening angle close to the base of the jet, Doppler factor ($\delta$), flux doubling time scale ($\tau_{d}$) and redshift (z). The jet opening angle for blazars is generally less than 1 degree due to small viewing angle [@Jorstad2005], in general. Therefore, using the jet opening angle close to 1 degree, which is the upper limit of opening angle for BL Lacs [@Pushkarev_iX_2012], we estimate the distance to the location of $\gamma-$ray emission from central engine, using following equation as, $$d= \frac {\delta c \tau_{d}} {(1+z)\theta_j}$$ The location of high energy $\gamma-$ray emission region from central SMBH is estimated as d = 1.72 pc. The result indicates that the location of $\gamma-$ray production site is close to the standing shock (sub-mm core) inside the jet. Since the jet opening angle for BL Lacs is difficult to measure due to several reasons, for eg., jet bending at parsec scale or faint emission where the jet bends etc. [@rector2003], and the different values are reported by many authors [@rector2003; @Jorstad2005; @Lister2011], hence, the results should be considered with a caution. Twin flares during October 2015 outburst: Optical/$\gamma$-ray light-curve {#sec:sec3.1.2} -------------------------------------------------------------------------- Figure \[fig:fig2\] (left side; top to bottom) shows $\gamma$-ray to radio band light-curve with optical polarization for 1ES 1959+650 during MJD 57250 - 57460 constructed using data from space-based instruments (LAT/XRT/UVOT), from MIRO and Steward Observatory (R band converted to V-band as given in @Tagliaferri2003). On a careful look, a twin peak structure is noticed in optical as well as in $\gamma$-ray light-curve during outburst and post-outburst phase, respectively. Optical twin flares peak around 2015 November 18, MJD 57344 (R = 14.45 $\pm$ 0.02 mag) and December 5, MJD 57361 (R = 14.52 $\pm$ 0.02 mag); whereas $\gamma$-ray twin flares were recorded around 2015 November 30, MJD 57360 and December 5, MJD 57365.5, both showing similar flux levels. It appears that the emission in $\gamma$-rays and optical are correlated with each other peaking almost at same time. From the onset of outburst, 1ES 1959+650 brightened gradually during October 2015 with V = 14.85$\pm$ 0.02 mag on MJD 57306 (2015 October 11). On October 13, it brightened by 0.08 mag within a day, decaying by 0.06 mag during MJD 57309 and MJD 57310. On the later date, it brightened by 0.07 mag in about four hours, to 14.76 $\pm$0.02 mag. After that, the source went into low-flux state and started dimming, reaching 14.81 (0.02) mag on MJD 57311 (October 16, 2015). However, on MJD 57344 (November 17, 2015) our observations detected the source in its brightest level during whole 2015 with 14.45 $\pm$0.02 mag. The twin peaks in optical have been reported in a few HBLs [@sokolovMarscher2004]. The flux enhancement in the sources like blazars is well explained by shock models, kink models etc. Here a propagating shock in a random magnetic field plasma [@bland1979; @marschergear1985] hits the Mach disk which leads to the formation of double structured feature in the light-curve. HBLs 1ES 0229+200, 1ES 0502+675, 1ES 2344+514 etc. are seen with such double peaks in optical, with or without periodicity present in their light curves [@kapanadze2010]. On the other hand, in the shock-in-jet model, at the shock front, two regions contribute to the emitted radiation - emission from forward and reverse shocks which could be responsible for short duration twin structures in the optical light curve. On the other hand, in sufficiently magnetized environments, kink instabilities can efficiently convert the magnetic energy into bulk kinetic and thermal energy in the jet . When the shock propagates down the jet through kinky nodes, it illuminates them along its path before getting dissipated as described by @zhang2016 using relativistic magneto hydrodynamic simulations. DP and PA change during October 2015 outburst --------------------------------------------- It is interesting to see how the polarization behaved during multi-frequency outburst during October 2015 in 1ES 1959+650. The Steward observatory optical polarization and position angle (DP and PA) data are plotted in the bottom two panels of Figure \[fig:fig2\] and, with more clarity, in Fig.\[fig:fig2\] (on the right) along with R-band data for October 11 - 16, 2015 (MJD 57306 - 57312). During October 12 -13, 2015 DP along with R-band brightness increased sharply (DP: 0.3% to 2.5%, R mag: 14.37 to 14.31) followed by 10 degree change in PA to 153 degree . While brightness and DP remained at higher levels (DP: 3.13 - 1.57, R-mag: 14.3 -14.37), PA, after successive rotations by 20 degree on next two days, settled down around a value of 120 degree. DP is maximum when the source is brightest during these observations. Variations in DP and R-band magnitudes are also seen at intra-night time scales as well. On October 15, 2015, 1ES 1959+650 brightened by 0.07 mag in R (about $3\sigma$) in about 4.8 hrs, while DP changed by 0.64% $(> 8\sigma)$ within 3.6 hrs. Here, we discuss multi-wavelength flare patterns with changes in optical polarization features. Figure \[fig:fig2\] clearly shows that DP and PA significantly changed during October 2015 outburst when the source was in very high flux state across the whole electromagnetic spectrum (EMS). While the source was slowly brightening in optical, X-ray and $\gamma$-ray, the degree of polarization changed by 2.8 % within 4 days i.e., from MJD 57307 (DP: 0.3 %) to MJD 57310 (DP: 3.0 %). This increase in DP is followed by a change in the position angle of polarization by almost 80 degrees within six days duration. The 15 GHz radio flux reached its peak and then decreased sharply during these changes in optical polarization features, after which flux in $\gamma$-rays, followed by optical, UV and X-rays peaked. Perhaps, all this coincided with injection of fresh plasma in the jet which led to flaring in radio, gamma-rays and other bands. The situation well suits for the case of emission feature moving down the jet, interacting with the standing shock which results in compression of plasma, alignment of magnetic field resulting in polarization changes and acceleration of charged particles. These physical processes lead to enhanced emission at all the frequencies and increased degree of polarization [@marscher2014]. Unfortunately, the polarization data does not cover the domain of flares reaching their peaks. Injection of a new component in the jet --------------------------------------- In order to understand the multi-frequency connection of the major outburst activity in 1ES 1959+650 during October 2015, 15 GHz OVRO data and Astronomers Telegrams (ATels) on TeV activity reported during this outburst are considered. During MJD 57303-57304, the onset of TeV activity [@mukherjee2015_atel_8148] showing much harder spectra, $\alpha_{(0.2-7) TeV}$ = 2.5, was followed by a significant enhancement in 15 GHz radio flux, $F_{radio}$ = 0.28 $\pm$ 0.02 Jy) and $\gamma$-ray $(> 100 MeV ; F_{\gamma} = (2.27 \pm 0.68) \times$ 10$^{-07}$ ph cm$^{-2}$ s$^{-1})$ . After six days of flux rise in radio and gamma-rays, the source showed historically highest X-ray fluxes on MJD 57311.99 i.e., (10.13 $\pm$ 0.10) cts/s in (0.3-10.0) keV and (7.34 $\pm$ 0.09) cts/s in (0.3-3.0) keV. A quasi-simultaneous enhancement in GeV, $F_{\gamma} = (2.86 \pm 0.67)$ $\times$ 10$^{-07}$ ph cm$^{-2}$ s$^{-1}$ on MJD 57313; $F_{\gamma} = (3.77 \pm 0.75)$ $\times$ 10$^{-07}$ ph cm$^{-2}$ s$^{-1}$ on MJD 57316.86 and (7.99 $\pm$ 0.06) cts/s in X-ray (0.3-10.0) keV on MJD 57314.28 were reported. The source was detected with the brightest ever in X-ray flux on MJD 57326.48 with exceptionally high count rates i.e., (11.81 $\pm$ 0.09) cts/s in (0.3-10.0) keV and (9.12 $\pm$ 0.08) cts/s in (0.3-3.0) keV. The source was dimming in GeV ($F_{\gamma} = (2.05 \pm 0.54$) $\times$ 10$^{-07}$ ph cm$^{-2}$ s$^{-1}$ on MJD 57347.80) but brightening up in optical, showing V= 14.94 $\pm$ 0.03 mag on MJD 57347.91. We also noticed flux enhancement in radio by almost 0.1 Jy, showing $F_{radio}$ = (0.275 $\pm$ 0.003 Jy) on MJD 57357, followed by GeV activity on MJD 57365.20 ($F_{\gamma} = (3.73 \pm 0.72$) $\times$ 10$^{-07}$ ph cm$^{-2}$ s$^{-1}$). The almost continuous enhancement in flux during October 2015 outburst, at different frequencies, suggests a scenario of continuous injection of fresh relativistic particles near the origin of the jet, that accelerates the particles (electrons) up to relativistic energies, which, in turn cool down by emitting higher energy photons followed by the emissions at lower frequencies. This can also explain a mild bluer when brighter color seen in the present study. However, color remains same with time, within errors, except during flaring when it shows a BWB behavior, typical of HBLs. A quasi-simultaneous enhancement in the high energy $\gamma$-ray and radio flux, perhaps, may be associated with the presence of a standing shock feature or core. The emission in $\gamma-$rays is supposed to arise from the seed photons in the acceleration and collimation zone inside the jet, where all particles (here, electrons) spiral in helical magnetic field and emit in VHE gamma rays [@marscher2008]. The radio emission is opaque upstream the stationary core due to synchrotron self absorption effects, and becomes transparent only when the emission feature interacts with the radio-core. During October 2015 outburst, correlated flux in radio and TeV energies is noticed using publicly available data and a few published reports. A quasi-simultaneous flux variability is also noticed by @tagliaferri2008 and @hayashida2008 along-with many other authors who reported a correlated activity between highly variable strong $\gamma-$rays and radio frequencies [@ramakrishnan2015] with an evidence of the ejection of a super-luminal knot from radio core using VLBI analysis [@jorstad2001; @schinzel2012; @jorstad2015]. Their study suggests that if the origin of seed photons responsible for enhanced gamma-rays is from the radio core (mm-core), then it would be due to the interaction of the moving shock with the standing core. Summary and conclusions ======================= We presented the analysis of the multi-wavelength data from Fermi and Swift, available publicly, MIRO and Steward Observatory optical data for HBL 1ES 1959+650 during the year 2015-2016, covering two major outbursts. We also made use of available optical polarization data to discuss October 2015 outburst in detail. The source was active in all the energy bands and showed significant flux enhancements during most of the period covered in this study. It is worth noting that 1ES 1959+650 exhibited highest ever flux in $\gamma-$rays and more than 20 counts per second in X-ray emission, making it third source after Mrk421, Mrk501 having such high counts. MIRO data showed the source in its brightest state (V = 14.45 $\pm$ 0.03) during 2015. A mild indication of the optical intra-night variability is seen on one of the nights during October 2015 when the source brightens by 0.07 mag in 4.5 hrs. Also, the source exhibited short-term variability (over a few days) with a significant ($>$ 0.3 mag in V band) variability amplitude. The first outburst (March 2015) was characterized by the emission in $\gamma-$rays, perhaps, to follow those at lower frequencies with origin from synchrotron radiation, which is explainable via SSC mechanism. The second outburst (October 2015) was rather complex with the 15GHz radio emission peaking first, just around the time optical polarization features changed rapidly. It was followed by the peaks in $\gamma-$ray flux, X-ray, UV and optical emissions. UV and optical variations were delayed by about 20 days with respect to those in $\gamma-$rays. It appears that various emission regions were aligned differently resulting in varying Doppler boosting of the flux and variability time scales. However, since $\gamma-$ray flux peaked just after radio, followed by emissions at lower frequencies, the processes could be related to the injection of fresh plasma in the jet. We estimated magnetic field strength (B $=$ 4.21 G) by using time lag between UV and optical emission as synchrotron cooling timescale. The emission region sizes for $\gamma-$ray and optical were estimated using shortest time scales of variability which were found to be of the order of $\approx \ 10^{16}$ cm. The $\gamma-$ray emission region appears to be located at a distance of 1.72 pc from central SMBH, which is close to the standing shock feature in the jet. As per the Astronomers’ Telegrams (quasi-) simultaneous enhancement in radio and TeV flux were noticed during MJD 57303-57304, indicating to the injection of a new component in the jet which propagates down the jet interacting with standing conical shock (or radio core). It leads to quasi-simultaneous emissions at almost all the frequencies. The long term multi-frequency study suggests a mild bluer-when-brighter trend in flux, which is relatively stronger during flaring. Acknowledgements {#acknowledgements .unnumbered} ================ This work is supported by the Department of Space, Government of India. Data from the Steward Observatory spectro-polarimetric monitoring project were used. This program is supported by Fermi Guest Investigator grants NNX08AW56G, NNX09AU10G, NNX12AO93G, and NNX15AU81G. This research made use of Enrico, a community-developed Python package to simplify Fermi-LAT analysis [@sanchezdiel2013]. This research has made use of data from the OVRO 40-m monitoring program [@richard2011] which is supported in part by NASA grants NNX08AW31G, NNX11A043G, and NNX14AQ89G and NSF grants AST-0808050 and AST-1109911. {width="18cm" height="14cm"} {width="8.5cm" height="10cm"} {width="8.5cm" height="10cm"} {width="5cm" height="10cm"} {width="5cm" height="10cm"} {width="5cm" height="10cm"} [ccccccccccc]{} 11-Oct-2015 &57306.30 &– &– &– &– &0.74 &0.06 &124.0 &2.3 &Steward\ &57306.70 &14.39 &0.03 &– &– &– &– &– &– &Steward\ &57306.81 &14.39 &0.03 &– &– &– &– &– &– &Steward\ 12-Oct-2015 &57307.09 &– &– &– &– &0.37 &0.06 &155.4 &4.4 &Steward\ &57307.21 &– &– &– &– &0.60 &0.06 &152.1 &2.7 &Steward\ &57307.59 &14.38 &0.02 &– &– &– &– &– &– &Steward\ &57307.71 &14.39 &0.02 &– &– &– &– &– &– &Steward\ 13-Oct-2015 &57308.08 &– &– &– &– &2.52 &0.10 &153.6 &1.1 &Steward\ &57308.24 &– &– &– &– &2.51 &0.05 &163.4 &0.5 &Steward\ &57308.74 &14.31 &0.02 &– &– &– &– &– &– &Steward\ 14-Oct-2015 &57309.09 &– &– &– &– &1.98 &0.05 &149.8 &0.7 &Steward\ &57309.21 &– &– &– &– &2.18 &0.04 &141.0 &0.6 &Steward\ &57309.59 &14.30 &0.02 &– &– &– &– &– &– &Steward\ &57309.71 &14.31 &0.02 &– &– &– &– &– &– &Steward\ 15-Oct-2015 &57310.08 &– &– &– &– &3.13 &0.05 &124.1 &0.4 &Steward\ &57310.23 &– &– &– &– &2.49 &0.07 &122.2 &0.8 &Steward\ &57310.59 &14.37 &0.03 &– &– &– &– &– &– &Steward\ &57310.78 &14.30 &0.02 &– &– &– &– &– &– &Steward\ 16-Oct-2015 &57311.09 &– &– &– &– &1.71 &0.10 &118.9 &1.7 &Steward\ &57311.21 &– &– &– &– &1.57 &0.05 &123.1 &0.9 &Steward\ &57311.71 &14.35 &0.02 &– &– &– &– &– &– &Steward\ 23-Oct-2015 &57318 &14.39 &0.02 &14.80 &0.03 &– &– &– &– &MIRO (1.2m)\ 17-Nov-2015 &57344 &13.99 &0.02 &14.48 &0.02 &– &– &– &– &MIRO (1.2m)\ 18-Nov-2015 &57345 &14.02 &0.02 &14.45 &0.01 &– &– &– &– &MIRO (1.2m)\ 29-Nov-2015 &57356 &14.19 &0.01 &– &– &– &– &– &– &MIRO (1.2m)\ 30-Nov-2015 &57357 &14.23 &0.01 &– &– &– &– &– &– &MIRO (1.2m)\ 04-Dec-2015 &57361 &14.06 &0.02 & 14.56 &0.01 &– &– &– &– &MIRO (0.5m)\ 05-Dec-2015 &57362 &14.14 &0.01 & 14.57 &0.03 &– &– &– &– &MIRO (0.5m)\ 06-Dec-2015 &57363 &14.20 &0.02 & 14.61 &0.05 &– &– &– &– &MIRO (0.5m)\ 07-Dec-2015 &57364 &14.18 &0.02 & 14.58 &0.04 &– &– &– &– &MIRO (0.5m)\ 10-Dec-2015 &57367 &14.07 &0.08 & – &– &– &– &– &– &MIRO (1.2m)\ 11-Dec-2015 &57368 &14.14 &0.01 & – &– &– &– &– &– &MIRO (1.2m)\ 12-Dec-2015 &57369 &14.11 &0.03 & – &– &– &– &– &– &MIRO (1.2m)\ 13-Dec-2015 &57370 &14.12 &0.03 & – &– &– &– &– &– &MIRO (1.2m)\ [^1]: http://www.swift.ac.uk/analysis/xrt/xselect.php [^2]: http://www.swift.ac.uk/analysis/xrt/pileup.php [^3]: https://www.r-project.org/ [^4]: http://james.as.arizona.edu/ psmith/Fermi/ [^5]: http://www.astro.caltech.edu/ovroblazars/index.php?page=home

--- abstract: 'A real Bott manifold is the total space of a sequence of $\R P^1$ bundles starting with a point, where each $\R P^1$ bundle is the projectivization of a Whitney sum of two real line bundles. A real Bott manifold is a real toric manifold which admits a flat riemannian metric. An upper triangular $(0,1)$ matrix with zero diagonal entries uniquely determines such a sequence of $\R P^1$ bundles but different matrices may produce diffeomorphic real Bott manifolds. In this paper we determine when two such matrices produce diffeomorphic real Bott manifolds. The argument also proves that any graded ring isomorphism between the cohomology rings of real Bott manifolds with $\Z/2$ coefficients is induced by an affine diffeomorphism between the real Bott manifolds. In particular, this implies the main theorem of [@ka-ma08] which asserts that two real Bott manifolds are diffeomorphic if and only if their cohomology rings with $\Z/2$ coefficients are isomorphic as graded rings. We also prove that the decomposition of a real Bott manifold into a product of indecomposable real Bott manifolds is unique up to permutations of the indecomposable factors.' address: 'Department of Mathematics, Osaka City University, Sumiyoshi-ku, Osaka 558-8585, Japan.' author: - Mikiya Masuda title: Classification of real Bott manifolds --- [^1] Introduction ============ A [*real Bott tower*]{} of height $n$, which is a real analogue of a Bott tower introduced in [@gr-ka94], is a sequence of $\R P^1$ bundles $$\label{tower} M_n\stackrel{\R P^1}\longrightarrow M_{n-1}\stackrel{\R P^1}\longrightarrow \cdots\stackrel{\R P^1}\longrightarrow M_1 \stackrel{\R P^1}\longrightarrow M_0=\{\textrm{a point}\}$$ such that $M_j\to M_{j-1}$ for $j=1,\dots,n$ is the projective bundle of the Whitney sum of a real line bundle $L_{j-1}$ and the trivial real line bundle over $M_{j-1}$, and we call $M_n$ a [*real Bott manifold*]{}. A real Bott manifold naturally supports an action of an elementary abelian 2-group and provides an example of a real toric manifold which admits a flat riemannian metric invariant under the action. Conversely, it is shown in [@ka-ma08] that a real toric manifold which admits a flat riemannian metric invariant under an action of an elementary abelian 2-group is a real Bott manifold. Real line bundles are classified by their first Stiefel-Whitney classes as is well-known and $H^1(M_{j-1};\Z/2)$, where $\Z/2=\{0,1\}$, is isomorphic to $(\Z/2)^{j-1}$ through a canonical basis, so the line bundle $L_{j-1}$ is determined by a vector $A_j$ in $(\Z/2)^{j-1}$. We regard $A_j$ as a column vector in $(\Z/2)^n$ by adding zero’s and form an $n\times n$ matrix $A$ by putting $A_j$ as the $j$-th column. This gives a bijective correspondence between the set of real Bott towers of height $n$ and the set $\T(n)$ of $n\times n$ upper triangular $(0,1)$ matrices with zero diagonal entries. Because of this reason, we may denote the real Bott manifold $M_n$ by $M(A)$. Although $M(A)$ is determined by the matrix $A$, it happens that two different matrices in $\T(n)$ produce (affinely) diffeomorphic real Bott manifolds. In this paper we introduce three operations on $\T(n)$ and say that two elements in $\T(n)$ are [*Bott equivalent*]{} if one is transformed to the other through a sequence of the three operations. Our first main result is the following. \[main\] The following are equivalent for $A,B$ in $\T(n)$: 1. $A$ and $B$ are Bott equivalent. 2. $M(A)$ and $M(B)$ are affinely diffeomorphic. 3. $H^*(M(A);\Z/2)$ and $H^*(M(B);\Z/2)$ are isomorphic as graded rings. Moreover, any graded ring isomorphism from $H^*(M(A);\Z/2)$ to $H^*(M(B);\Z/2))$ is induced by an affine diffeomorphism from $M(B)$ to $M(A)$. In particular, we obtain the following main theorem of [@ka-ma08]. \[maincoro\] Two real Bott manifolds are diffeomorphic if and only if their cohomology rings with $\Z/2$ coefficients are isomorphic as graded rings. It is asked in [@ka-ma08] whether Corollary \[maincoro\] holds for any real toric manifolds but a counterexample is given in [@masu08]. We say that a real Bott manifold is *indecomposable* if it is not diffeomorphic to a product of more than one real Bott manifolds. Using Corollary \[maincoro\] together with an idea used to prove Theorem \[main\], we are able to prove our second main result. \[main1\] The decomposition of a real Bott manifold into a product of indecomposable real Bott manifolds is unique up to permutations of the indecomposable factors. In particular, we have \[main1coro\] Let $M$ and $M'$ be real Bott manifolds. If $S^1\times M$ and $S^1\times M'$ are diffeomorphic, then $M$ and $M'$ are diffeomorphic. It would be interesting to ask whether Theorem \[main1\] and Corollary \[main1coro\] hold for any real toric manifolds. The author learned from Y. Kamishima that Corollary \[main1coro\] can also be obtained from the method developed in [@ka-na08] and [@nazr08] and that the cancellation property above fails to hold for general compact flat riemannian manifolds, see [@char65-1]. This paper is organized as follows. In Section \[sect:rbott\] we describe $M(A)$ and its cohomology rings explicitly in terms of the matrix $A$. In Section \[sect:matrix\] we introduce the three operations on $\T(n)$. To each operation we associate an affine diffeomorphism between real Bott manifolds in Section \[sect:affine\], which implies the implication (1) $\Rightarrow$ (2) in Theorem \[main\]. The implication (2) $\Rightarrow$ (3) is trivial. In Section \[sect:cohom\] we prove the latter statement in Theorem \[main\]. The argument also establishes the implication (3) $\Rightarrow$ (1). In the proof we introduce a notion of eigen-element and eigen-space in the first cohomology group of a real Bott manifold using the multiplicative structure of the cohomology ring and they play an important role on the analysis of isomorphisms between cohomology rings. Using this notion, we prove Theorem \[main1\] in Section \[sect:decom\]. Real Bott manifolds and their cohomology rings {#sect:rbott} ============================================== As mentioned in the Introduction, a real Bott manifold $M(A)$ of dimension $n$ is associated to a matrix $A\in\T(n)$. In this section we give an explicit description of $M(A)$ and its cohomology ring. We set up some notation. Let $S^1$ denote the unit circle consisting of complex numbers with unit length. For elements $z\in S^1$ and $a\in \Z/2$ we use the following notation $$z(a):=\begin{cases} z \quad&\text{if $a=0$}\\ \bar z\quad&\text{if $a=1$}. \end{cases}$$ For a matrix $A$ we denote by $A^i_j$ the $(i,j)$ entry of $A$ and by $A^i$ (resp. $A_j$) the $i$-th row (resp. $j$-th column) of $A$. Now we take $A$ from $\T(n)$ and define involutions $a_i$’s on $T^n:=(S^1)^n$ by $$\label{ai} a_i(z_1,\dots,z_n):=(z_1,\dots,z_{i-1},-z_i,z_{i+1}(A^i_{i+1}),\dots, z_n(A^i_n))$$ for $i=1,\dots,n$. These involutions $a_i$’s commute with each other and generate an elementary abelian 2-group of rank $n$, denoted by $G(A)$. The action of $G(A)$ on $T^n$ is free and the orbit space is the desired real Bott manifold $M(A)$. $M(A)$ is a flat riemannian manifold. In fact, Euclidean motions $s_i$’s $(i=1,\dots,n)$ on $\R^n$ defined by $$s_i(u_1,\dots,u_n):=(u_1,\dots,u_{i-1}, u_i+\frac{1}{2}, (-1)^{A_{i+1}^i}u_{i+1},\dots, (-1)^{A_{n}^i}u_n)$$ generate a crystallographic group $\Gamma(A)$, where the subgroup generated by $s_1^2,\dots,s_n^2$ consists of all translations by $\Z^n$, and the action of $\G(A)$ on $\R^n$ is free and the orbit space $\R^n/\G(A)$ agrees with $M(A)$ through an identification $\R/\Z$ with $S^1$ via an exponential map $u\to \exp(2\pi\sqrt{-1}u)$. $M(A)$ admits an action of an elementary abelian 2-group defined by $(u_1,\dots,u_n)\to (\pm u_1,\dots,\pm u_n)$ and this action preserves the flat riemannian metric on $M(A)$. Let $G_k$ $(k=1,\dots,n)$ be a subgroup of $G(A)$ generated by $a_1,\dots,a_k$. Needless to say $G_n=G(A)$. Let $T^k:=(S^1)^k$ be a product of first $k$-factors in $T^n=(S^1)^n$. Then $G_k$ acts on $T^k$ by restricting the action of $G_k$ on $T^n$ to $T^k$ and the orbit space $T^k/G_k$ is a real Bott manifold of dimension $k$. Natural projections $T^k\to T^{k-1}$ for $k=1,\dots,n$ produce a real Bott tower $$M(A)=T^n/G_n\to T^{n-1}/G_{n-1} \to \dots\to T^1/G_1\to \text{\{a point\}}.$$ The graded ring structure of $H^*(M(A);\Z/2)$ can be described explicitly in terms of the matrix $A$. We shall recall it. For a homomorphism $\lambda\colon G(A)\to \Z_2=\{\pm 1\}$ we denote by $\R(\lambda)$ the real one-dimensional $G(A)$-module associated with $\lambda$. Then the orbit space of $T^n\times \R(\lambda)$ by the diagonal action of $G(A)$, denoted by $L(\lambda)$, defines a real line bundle over $M(A)$ with the first projection. Let $\lambda_j\colon G(A)\to \Z_2$ $(j=1,\dots,n)$ be a homomorphism sending $a_i$ to $-1$ for $i=j$ and $1$ for $i\not=j$, and we set $$x_j=w_1(L(\lambda_j))$$ where $w_1$ denotes the first Stiefel-Whitney class. \[cohoA\] As a graded ring $$H^*(M(A);\Z/2)=\Z/2[x_1,\dots,x_n]/(x_j^2=x_j\sum_{i=1}^nA^i_jx_i\mid j=1,\dots,n).$$ Let $B$ be another element of $\T(n)$. Since $M(A)=T^n/G(A)$ and $M(B)=T^n/G(B)$, an affine automorphism $\f$ of $T^n$ together with a group isomorphism $\phi\colon G(B)\to G(A)$ induces an affine diffeomorphism $f\colon M(B)\to M(A)$ if $\f$ is $\phi$-equivariant, i.e., $\f(gz)=\phi(g)\f(z)$ for $g\in G(B)$ and $z\in T^n$. Since the actions of $G(A)$ and $G(B)$ on $T^n$ are free, the isomorphism $\phi$ will be uniquely determined by $\f$ if it exists. We shall use $b_i$ and $y_j$ for $M(B)$ in place of $a_i$ and $x_j$ for $M(A)$. \[f\*\] If $\phi(b_i)=\prod_{j=1}^na_j^{F^i_j}$ with $F^i_j\in \Z/2$, then $f^*(x_j)=\sum_{i=1}^nF^i_jy_i$. A map $T^n\times \R(\lambda\circ\phi)\to T^n\times \R(\lambda)$ sending $(z,u)$ to $(\f(z),u)$ induces a bundle map $L(\lambda\circ\phi) \to L(\lambda)$ covering $f\colon M(B)\to M(A)$. Since $(\lambda_j\circ\phi)(b_i)=F^i_j$, this implies the lemma. Three matrix operations {#sect:matrix} ======================= In this section we introduce three operations on matrices used in later sections to analyze when $M(A)$ and $M(B)$ (resp. $H^*(M(A);\Z/2)$ and $H^*(M(B);\Z/2)$) are diffeomorphic (resp. isomorphic) for $A,B\in \T(n)$. In the following $A$ will denote an element of $\T(n)$. [**1st operation (Op1).**]{} For a permutation matrix $S$ of size $n$ we define $$\Phi_S(A):=SAS^{-1}.$$ To be more precise, there is a permutation $\sigma$ on a set $\{1,\dots,n\}$ such that $S^i_j=1$ if $i=\sigma(j)$ and $S^i_j=0$ otherwise. We note that if we set $B=\Phi_S(A)$, then $SA=BS$ and $$\label{SA=BA} A^i_j=(SA)^{\sigma(i)}_j=(BS)^{\sigma(i)}_j=B^{\sigma(i)}_{\sigma(j)}.$$ $\Phi_S(A)$ may not be in $\T(n)$ but we will perform the operation $\Phi_S$ on $A$ only when $\Phi_S(A)$ stays in $\T(n)$. [**2nd operation (Op2).**]{} For $k\in \{1,\dots,n\}$ we define a square matrix $\Phi^k(A)$ of size $n$ by $$\label{2nd} \Phi^k(A)_j:=A_j+A^k_jA_k\quad\text{for $j=1,\dots,n$}.$$ $\Phi^k(A)$ stays in $\T(n)$ and since the diagonal entries of $A$ are all zero and we are working over $\Z/2$, the composition $\Phi^k\circ \Phi^k$ is the identity; so $\Phi^k$ is bijective on $\T(n)$. [**3rd operation (Op3).**]{} Let $I$ be a subset of $\{1,\dots,n\}$ such that $A_i=A_j$ for $i,j\in I$ and $A_i\not=A_j$ for $i\in I$ and $j\notin I$. Since the diagonal entries of $A$ are all zero, the condition $A_i=A_j$ for $i,j\in I$ implies that $A^i_j=0$ for $i,j\in I$. Let $C=(C^i_k)_{i,k\in I}$ with $C^i_k\in \Z/2$ be an invertible matrix of size $|I|$. Then we define a square matrix $\Phi^I_C(A)$ of size $n$ by $$\label{3rd} \Phi^I_C(A)^i_j:=\begin{cases} \sum_{k\in I}C^i_kA^k_j\quad&\text{$(i\in I)$}\\ A^i_j\quad&\text{$(i\notin I)$}. \end{cases}$$ $\Phi^I_C(A)$ stays in $\T(n)$ and since $C$ is invertible, $\Phi^I_C$ is bijective on $\T(n)$. We say that two elements in $\T(n)$ are [*Bott equivalent*]{} if one is transformed to the other through a sequence of the three operations (Op1), (Op2) and (Op3). $\T(2)$ has two elements and they are not Bott equivalent. $\T(3)$ has $2^3=8$ elements and they are classified into four Bott equivalence classes as follows: 1. The zero matrix of size $3$ 2. ${\tiny \begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\end{pmatrix}\quad \begin{pmatrix} 0 & 0 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0\end{pmatrix}\quad \begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0\end{pmatrix}\quad \begin{pmatrix} 0 & 0 & 1\\ 0 & 0 & 1\\ 0 & 0 & 0\end{pmatrix} }$ 3. ${\tiny \begin{pmatrix} 0 & 1 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0\end{pmatrix} }$ 4. ${\tiny \begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0\end{pmatrix}\quad \begin{pmatrix} 0 & 1 & 1\\ 0 & 0 & 1\\ 0 & 0 & 0\end{pmatrix} }$ $\T(4)$ has $2^6=64$ elements and one can check that it has twelve Bott equivalence classes, see [@ka-ma08] and [@nazr08]. Furthermore, $\T(5)$ has $2^{10}=1024$ elements and one can check that it has $54$ Bott equivalence classes. The author learned from Admi Nazra that he classified real Bott manifolds of dimension 5 up to diffeomorhism from a different viewpoint (see [@ka-na08], [@nazr08]) and found the 54 Bott equivalence classes in $\T(5)$. The author does not know the number of Bott equivalence classes in $\T(n)$ for $n\ge 6$ although it is in between $2^{(n-2)(n-3)/2}$ and $2^{n(n-1)/2}$ (see Example \[Deltan\] below). Let $\T_k(n)$ $(1\le k\le n-1)$ be a subset of $\T(n)$ such that $A\in \T(n)$ is in $\T_k(n)$ if and only if $A$ has exactly $k$ non-zero columns. There is only one Bott equivalence class in $\T_1(n)$ and the corresponding real Bott manifold is the product of a Klein bottle and $(\R P^1)^{n-2}$. $\T_2(3)$ has two Bott equivalence classes represented by $${\tiny \begin{pmatrix} 0 & 1 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0\end{pmatrix}}\quad \text{}\quad {\tiny \begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0\end{pmatrix} }$$ But $\T_2(n)$ for $n\ge 4$ has four Bott equivalence classes; two of them are represented by $n\times n$ matrices with the above $3\times 3$ matrices at the right-low corner and $0$ in others, and the other two are represented by $n\times n$ matrices with the following $4\times 4$ matrices at the right-low corner and $0$ in others $${\tiny \begin{pmatrix} 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\end{pmatrix}}\quad \text{}\quad {\tiny \begin{pmatrix} 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{pmatrix} }$$ \[Deltan\] Let $\Delta(n)$ be a subset of $\T(n)$ such that $A\in \T(n)$ is in $\Delta(n)$ if and only if $A^{i}_{i+1}=1$ for $i=1,\dots,n-1$. Only the operation (Op2) is available on $\Delta(n)$ and one can change $(i,i+2)$ entry into $0$ for $i=1,\dots,n-2$ using the operation, so that $A$ is Bott equivalent to a matrix $\bA$ of this form [$$\label{reduced} \bA=\begin{pmatrix} 0&1&0&\bA^1_4&\bA^1_5&\dots&\bA^1_{n-1}&\bA^1_n\\ 0&0&1&0&\bA^2_5&\dots&\bA^2_{n-1}&\bA^2_n\\ \vdots&\vdots& \ddots &\ddots &\ddots&\ddots &\vdots & \vdots\\ 0& 0& \dots&0&1 &0 & \bA^{n-4}_{n-1} & \bA^{n-4}_n\\ 0& 0& \dots&0&0 &1 & 0 & \bA^{n-3}_n\\ 0& 0& \dots&0&0 &0 & 1 & 0\\ 0& 0& \dots&0&0 &0 & 0 & 1\\ 0& 0& \dots&0&0 &0 & 0 & 0 \end{pmatrix}$$]{} $\bA$ is uniquely determined by $A$ and two elements $A,B\in \Delta(n)$ are Bott equivalent if and only if $\bA=\bB$. Therefore there are exactly $2^{(n-2)(n-3)/2}$ Bott equivalent classes in $\Delta(n)$ for $n\ge 2$. As remarked above $\Phi_S(A)$ may not stay in $\T(n)$. This awkwardness can be resolved if we consider the union of $\Phi_S(\T(n))$ over all permutation matrices $S$. The three operations above preserve the union and are bijective on it. This union is a natural object. In fact, it is shown in [@ma-pa08 Lemma 3.3] that a square matrix $A$ of size $n$ with entries in $\Z/2$ lies in the union if and only if all principal minors of $A+E$ (even the determinant of $A+E$ itself) are one in $\Z/2$ where $E$ denotes the identity matrix of size $n$. Affine diffeomorphisms {#sect:affine} ====================== In this section we associate an affine diffeomorphism between real Bott manifolds to each operation introduced in the previous section, and prove the implication $(1)\Rightarrow (2)$ in Theorem \[main\], that is If $A, B\in \T(n)$ are Bott equivalent, then the associated real Bott manifolds $M(A)$ and $M(B)$ are affinely diffeomorphic. We set $B=\Phi_S(A), \Phi^k(A), \Phi^I_C(A)$ respectively for the three operations introduced in the previous section. In order to prove the proposition above, it suffices to find a group isomorphism $\phi\colon G(B)\to G(A)$ and a $\phi$-equivariant affine automorphism $\f$ of $T^n$ which induces an affine diffeomorphism from $M(B)$ to $M(A)$. [*The case of the operation (Op1).*]{} Let $S$ and $\sigma$ be as before. We define a group isomorphism $\phi_S\colon G(B)\to G(A)$ by $$\label{phiS} \phi_S(b_{\sigma(i)}):=a_{i}$$ and an affine automorphism $\f_S$ of $T^n$ by $$\f_S(z_1,\dots,z_n):=(z_{\sigma(1)},\dots,z_{\sigma(n)}).$$ Then it follows from (applied to $b_{\sigma(i)}$) that the $j$-th component of $\f_S(b_{\sigma(i)}(z))$ is $z_{\sigma(j)}(B^{\sigma(i)}_{\sigma(j)})$ for $j\not=i$ and $-z_{\sigma(i)}$ for $j=i$ while that of $a_i(\f_S(z))$ is $z_{\sigma(j)}(A^i_j)$ for $j\not=i$ and $-z_{\sigma(i)}$ for $j=i$. Since $A^i_j=B^{\sigma(i)}_{\sigma(j)}$ by , this shows that $\f_S$ is $\phi_S$-equivariant. It follows from Lemma \[f\*\] and that the affine diffeomorphism $f_S\colon M(B)\to M(A)$ induced from $\f_S$ satisfies $$\label{op1coho} f_S^*(x_j)=y_{\sigma(j)}\quad \text{for $j=1,\dots,n$.}$$ [*The case of the operation (Op2).*]{} We define a group isomorphism $\phi^k\colon G(B)\to G(A)$ by $$\label{phik} \phi^k(b_i):=a_ia_k^{A^i_k} $$ and an affine automorphism $\f^k$ of $T^n$ by $$\f^k(z_1,\dots,z_n):=(z_1,\dots,z_{k-1},\sqrt{-1}z_k,z_{k+1},\dots,z_n).$$ We shall check that $\f^k$ is $\phi^k$-equivariant, i.e., $$\label{fkb} \f^k(b_i(z))= a_ia_k^{A^i_k}(\f^k(z)).$$ The identity is obvious when $i=k$ because $A^k_k=0$ and $B^k_j=A^k_j$ for any $j$ by . Suppose $i\not=k$. Then the $j$-th component of the left hand side of is given by $$\begin{cases}z_j(B^i_j)\quad&\text{for $j\not=i,k$},\\ -z_i \quad&\text{for $j=i$},\\ \sqrt{-1}(z_k(B^i_k)) \quad&\text{for $j=k$}, \end{cases}$$ while that of the right hand side of is given by $$\begin{cases}z_j(A^i_j+A^k_jA^i_k)\quad&\text{for $j\not=i,k$},\\ -z_i(A^k_iA^i_k) \quad&\text{for $j=i$},\\ (-1)^{A^i_k}(\sqrt{-1}z_k)(A^i_k) \quad&\text{for $j=k$}. \end{cases}$$ Since $B^i_j=A^i_j+A^k_jA^i_k$ by , the $j$-th components above agree for $j\not=i,k$. They also agree for $j=i$ because either $A^k_i$ or $A^i_k$ is zero. We note that $B^i_k=A^i_k$ by , and the $k$-th components above are both $\sqrt{-1}z_k$ when $B^i_k=A^i_k=0$ and $\sqrt{-1}\bar{z_k}$ when $B^i_k=A^i_k=1$. Thus the $j$-th components above agree for any $j$. Since $A^i_k=B^i_k$ for any $i$, it follows from Lemma \[f\*\] and that the affine diffeomorphism $f^k\colon M(B)\to M(A)$ induced from $\f^k$ satisfies $$\label{op2coho} (f^k)^*(x_j)=y_j\quad\text{for $j\not=k$}, \quad \quad (f^k)^*(x_k)=y_k+\sum_{i=1}^nB^i_ky_i.$$ [*The case of the operation (Op3).*]{} The homomorphism $\operatorname{GL}(m;\Z)\to \operatorname{GL}(m;\Z/2)$ induced from the surjective homomorphism $\Z\to \Z/2$ is known (and easily proved) to be surjective. We take a lift of the matrix $C=(C^i_k)_{i,k\in I}$ to $\operatorname{GL}(|I|,\Z)$ and denote the lift by $\tC$. Then we define a group isomorphism $\phi^I_C\colon G(B)\to G(A)$ by $$\label{phiIC} \phi^I_C(b_i):=\begin{cases} \prod_{k\in I}a_k^{C^i_k}\quad& \text{for $i\in I$},\\ a_i\quad&\text{for $i\notin I$,} \end{cases}$$ and the $j$-th component of an affine automorphism $\f^I_{\tC}$ of $T^n$ by $$\label{fIC} \f^I_{\tC}(z)_j:=\begin{cases} \prod_{\ell\in I}z_\ell^{\tC^\ell_j} \quad&\text{for $j\in I$},\\ z_j\quad&\text{for $j\notin I$.} \end{cases}$$ We shall check that $\f^I_{\tC}$ is $\phi^I_C$-equivariant. To simplify notation we abbreviate $\f^I_{\tC}$ and $\phi^I_C$ as $\f$ and $\phi$ respectively. What we prove is the identity $$\label{fb=bf} \f(b_i(z))_j=\phi(b_i)\f(z)_j.$$ We distinguish four cases. [*Case 1.*]{} The case where $i,j\in I$. As remarked in the definition of (Op3), $A^k_\ell=0$ whenever $k,\ell\in I$, so $B^i_\ell=0$ for any $\ell\in I$ by . It follows from and that $$\f(b_i(z))_j=(-z_i)^{\tC^i_j} \prod_{\ell\in I,\ell\not=i}z_\ell(B^i_\ell)^{\tC^\ell_j} =(-1)^{C^i_j}\prod_{\ell\in I}z_\ell^{\tC^\ell_j}$$ while $$\phi(b_i)\f(z)_j=(\prod_{k\in I}a_k^{C^i_k})\f(z)_j =(-1)^{C^i_j}\prod_{\ell\in I}z_\ell^{\tC^\ell_j}.$$ [*Case 2.*]{} The case where $i\in I$ but $j\notin I$. In this case we have $$\f(b_i(z))_j=z_j(B^i_j)$$ while $$\phi(b_i)\f(z)_j =z_j(\sum_{k\in I}C^i_kA^k_j)=z_j(B^i_j)$$ where the last identity follows from . [*Case 3.*]{} The case where $i\notin I$ but $j\in I$. In this case we have $$\f(b_i(z))_j=\prod_{\ell\in I}z_\ell(B^i_\ell)^{\tC^\ell_j}$$ while $$\phi(b_i)\f(z)_j =(\prod_{\ell\in I}z_\ell^{\tC^\ell_j})(A^i_j) =\prod_{\ell\in I}z_\ell(A^i_j)^{\tC^\ell_j}.$$ Since $B^i_\ell=A^i_\ell$ for $i\notin I$ by , the above verifies . [*Case 4.*]{} The case where $i,j\notin I$. In this case $$\f(b_i(z))_j=z_j(B^i_j)$$ while $$\phi(b_i)\f(z)_j=z_j(A^i_j).$$ Since $B^i_j=A^i_j$ for $i\notin I$ by , the above verifies . It follows from Lemma \[f\*\] and that the affine diffeomorphism $f^I_C\colon M(B)\to M(A)$ induced from $\f^I_C$ satisfies $$\label{op3coho} (f^I_C)^*(x_j)=\begin{cases}\sum_{i\in I}C^i_jy_i \quad&\text{for $j\in I$,}\\ y_j\quad&\text{for $j\notin I$.} \end{cases}$$ Cohomology isomorphisms {#sect:cohom} ======================= In this section we prove the latter statement in Theorem \[main\] and the implication (3) $\Rightarrow$ (1) at the same time, i.e. the purpose of this section is to prove the following. \[MAMBcoho\] Any isomorphism $H^*(M(A);\Z/2)\to H^*(M(B);\Z/2)$ is induced from a composition of affine diffeomorphisms corresponding to the three operations (Op1), (Op2) and (Op3), and if $H^*(M(A);\Z/2)$ and $H^*(M(B);\Z/2)$ are isomorphic as graded rings, then $A$ and $B$ are Bott equivalent. We introduce a notion and prepare a lemma. Remember that $$\label{HMA} H^*(M(A);\Z/2)=\Z/2[x_1,\dots,x_n]/(x_j^2=x_j\sum_{i=1}^nA^i_jx_i\mid j=1,\dots,n).$$ One easily sees that products $x_{i_1}\dots x_{i_q}$ $(1\le i_1<\dots<i_q\le n)$ form a basis of $H^q(M(A);\Z/2)$ as a vector space over $\Z/2$ so that the dimension of $H^q(M(A);\Z/2)$ is $\binom{n}{q}$ (see [@ma-pa08 Lemma 5.3]). We set $$\label{alphaj} \alpha_j=\sum_{i=1}^nA^i_jx_i\quad\text{for $j=1,\dots,n$}$$ where $\alpha_1=0$ since $A$ is an upper triangular matrix with zero diagonal entries. Then the relations in are written as $$\label{alpha} x_j^2=\alpha_jx_j \quad\text{for $j=1,\dots,n$.}$$ Motivated by this identity we introduce the following notion. We call an element $\alpha\in H^1(M(A);\Z/2)$ an [*eigen-element*]{} of $H^*(M(A);\Z/2)$ if there exists $x\in H^1(M(A);\Z/2)$ such that $x^2=\alpha x$, $x\not=0$ and $x\not=\alpha$. The set of all elements $x\in H^1(M(A);\Z/2)$ satisfying the equation $x^2=\alpha x$ is a vector subspace of $H^1(M(A);\Z/2)$ which we call the [*eigen-space*]{} of $\alpha$ and denote by $\EA(\alpha)$. We also introduce a notation $\tEA(\alpha)$ which is the quotient of $\EA(\alpha)$ by the subspace spanned by $\alpha$, and call it the [*reduced eigen-space*]{} of $\alpha$. Eigen-elements and (reduced) eigen-spaces are invariants preserved under graded ring isomorphisms. By $\alpha_j$’s are eigen-elements of $H^*(M(A);\Z/2)$ and the following lemma shows that these are the only eigen-elements. \[EAa\] If $\alpha$ is an eigen-element of $H^*(M(A);\Z/2)$, then $\alpha=\alpha_j$ for some $j$ and the eigen-space $\EA(\alpha)$ of $\alpha$ is generated by $\alpha$ and $x_i$’s with $\alpha_i=\alpha$. By the definition of eigen-element there exists a non-zero element $x\in H^1(M(A);\Z/2)$ different from $\alpha$ such that $x^2=\alpha x$. Since both $x$ and $x+\alpha$ are non-zero, there exist $i$ and $j$ such that $x=x_i+p_i$ and $x+\alpha=x_j+q_j$ where $p_i$ is a linear combination of $x_1,\dots,x_{i-1}$ and $q_j$ is a linear combination of $x_1,\dots,x_{j-1}$. Then $$x_ix_j+x_iq_j+x_jp_i+p_iq_j=0$$ because $x(x+\alpha)=0$. As remarked above, products $x_{i_1}x_{i_2}$ $(1\le i_1<i_2\le n)$ form a basis of $H^2(M(A);\Z/2)$, so $i$ must be equal to $j$ for the identity above to hold. Then as $x_j^2=x_j\alpha_j$, it follows from the identity above that $\alpha_j=q_j+p_i$ (and $p_iq_j=0$). This implies that $\alpha=\alpha_j$, proving the former statement of the lemma. We express a non-zero element $x\in \EA(\alpha)$ as $\sum_{i=1}^nc_ix_i$ $(c_i\in \Z/2)$ and let $m$ be the maximum number among $i$’s with $c_i\not=0$. *Case 1.* The case where $x_m$ appears when we express $\alpha$ as a linear combination of $x_1,\dots,x_n$. We express $x(x+\alpha)$ as a linear combination of the basis elements $x_{i_1}x_{i_2}$ $(1\le i_1<i_2\le n)$. Since $x_m$ appears in both $x$ and $\alpha$, it does not appear in $x+\alpha$. Therefore the term in $x(x+\alpha)$ which contains $x_m$ is $x_m(x+\alpha)$ and it must vanish because $x(x+\alpha)=0$. Therefore $x=\alpha$. *Case 2.* The case where $x_m$ does not appear in the linear expression of $\alpha$. In this case, the term in $x(x+\alpha)$ which contains $x_m$ is $x_m(x_m+\alpha)=x_m(\alpha_m+\alpha)$ since $x_m^2=\alpha_m x_m$, and it must vanish because $x(x+\alpha)=0$. Therefore $\alpha_m=\alpha$. The sum $x+x_m$ is again an element of $\EA(\alpha)$. If $x\not=x_m$ (equivalently $x+x_m$ is non-zero), then the same argument applied to $x+x_m$ shows that there exists $m_1(\not=m)$ such that $\alpha_{m_1}=\alpha$ and $x+x_m+x_{m_1}$ is again an element of $\EA(\alpha)$. Repeating this argument, $x$ ends up with a linear combination of $x_i$’s with $\alpha_i=\alpha$. With this preparation we shall prove Proposition \[MAMBcoho\]. Let $B$ be another element of $\T(n)$. We denote the canonical basis of $H^*(M(B);\Z/2)$ by $y_1,\dots,y_n$ and the elements in $H^1(M(B);\Z/2)$ corresponding to $\alpha_j$’s by $\beta_j$’s, i.e., $\beta_j=\sum_{i=1}B^i_jy_i$ for $j=1,\dots,n$. Let $\varphi\colon H^*(M(A);\Z/2)\to H^*(M(B);\Z/2)$ be a graded ring isomorphism. It preserves the eigen-elements and (reduced) eigen-spaces. In the following we shall show that we can change $\varphi$ into the identity map by composing isomorphisms induced from affine diffeomorphisms corresponding to the three operations (Op1), (Op2) and (Op3). Through the operation (Op1) we may assume that $\varphi(\alpha_j)=\beta_j$ for any $j$ because of . Then $\varphi$ restricts to an isomorphism $\EA(\alpha_j) \to \EB(\beta_j)$ between eigen-spaces and induces an isomorphism $\tEA(\alpha_j)\to \tEB(\beta_j)$ between reduced eigen-spaces. Let $\alpha$ (resp. $\beta$) stand for $\alpha_j$ (resp. $\beta_j$) and suppose that $\varphi(\alpha)=\beta$. Let $I$ be a subset of $\{1,\dots,n\}$ such that $\alpha_i=\alpha$ if and only if $i\in I$. We denote the image of $x_i$ (resp. $y_i$) in $\tEA(\alpha)$ (resp. $\tEB(\beta)$) by $\bar x_i$ (resp. $\bar y_i$). The $\bar x_i$’s (resp. $\bar y_i$’s) for $i\in I$ form a basis of $\tEA(\alpha)$ (resp. $\tEB(\beta)$) by Lemma \[EAa\], so if we express $\varphi(\bar x_j)= \sum_{i\in I}C^i_j\bar y_i$ with $C^i_j\in \Z/2$, then the matrix $C=(C^i_j)_{i,j\in I}$ is invertible. Therefore, through the operation (Op3), we may assume that $C$ is the identity matrix because of . This means that we may assume that $\varphi(x_j)=y_j$ or $y_j+\beta_j$ for each $j=1,\dots,n$. Finally through the operation (Op2), we may assume that $\varphi(x_j)=y_j$ for any $j$ because of and hence $A=B$ (and $\varphi$ is the identity) because $\varphi(\alpha_j) =\beta_j$, $\alpha_j=\sum_{i=1}^nA^i_jx_i$ and $\beta_j=\sum_{i=1}^nB^i_jy_i$ for any $j$, proving the proposition. Unique decomposition of real Bott manifolds {#sect:decom} =========================================== We say that a real Bott manifold is *indecomposable* if it is not diffeomorphic to a product of more than one real Bott manifolds. The purpose of this section is to prove Theorem \[main1\] in the Introduction, that is \[bdeco\] The decomposition of a real Bott manifold into a product of indecomposable real Bott manifolds is unique up to permutations of the indecomposable factors. Namely, if $\prod_{i=1}^k M_i$ is diffeomorphic to $\prod_{j=1}^\ell N_j$ where $M_i$ and $N_j$ are indecomposable real Bott manifolds, then $k=\ell$ and there is a permutation $\sigma$ on $\{1,\dots,k=\ell\}$ such that $M_i$ is diffeomorphic to $N_{\sigma(i)}$ for $i=1,\dots,k$. $H^*(\prod_{i=1}^kM_i;\Z/2)=\bigotimes_{i=1}^kH^*(M_i;\Z/2)$ by Künneth formula and the diffeomorphism types of real Bott manifolds are detected by cohomology rings with $\Z/2$ coefficient by Corollary \[maincoro\], so the theorem above reduces to a problem on the decomposition of a cohomology ring into tensor products over $\Z/2$. We call a graded ring over $\Z/2$ a [*Bott ring*]{} of rank $n$ if it is isomorphic to the cohomology ring with $\Z/2$ coefficient of a real Bott manifold of dimension $n$. Let $\MH$ be a Bott ring of rank $n$, so it has an expression $$\label{MH} \MH=\Z/2[x_1,\dots,x_n]/(x_j^2=x_j\sum_{i=1}^nA^i_jx_i\mid j=1,\dots,n)$$ with $A\in\T(n)$. The eigen-elements of $\MH$ are $$\label{eigenj} \text{$\alpha_j=\sum_{i=1}^nA^i_jx_i$\quad $(j=1,\dots,n)$.}$$ We denote by $\MH^q$ the degree $q$ part of $\MH$ and define $$\begin{split} N(\MH)&:=\{ x\in \MH^1\mid x^2=0\},\text{ and}\\ S(\MH)&:=\{x\in \MH^1\backslash\{0\}\mid \exists \bx\in\MH^1\backslash\{0\} \ \text{with $x\bx=0$ and $\bx\not=x$}\}. \end{split}$$ In terms of eigen-elements and eigen-spaces, $N(\MH)$ is the eigen-space of the zero eigen-element. Also, if we write $\bx=x+\alpha$ with $\alpha\in\MH^1$, then $x\bx=0$ means that $x^2=\alpha x$; so $S(\MH)$ with the zero element added is the union of eigen-spaces of all non-zero eigen-elements in $\MH$. The latter statement in Lemma \[EAa\] shows that the eigen-element $\alpha$ is uniquely determined by $x$, hence so is $\bx$. $N(\MH)=\MH^1$ if and only if $A$ in is the zero matrix. Unless $N(\MH)=\MH^1$, $S(\MH)\not=\emptyset$. \[MHS\] The graded subring $\MH_S$ of a Bott ring $\MH$ generated by $S(\MH)$ is a Bott ring. The isomorphism class of $\MH$ does not change through the three operations (Op1), (Op2) and (Op3). Through (Op1) we may assume that the first $\ell$ columns of the matrix $A$ in are all zero but none of the remaining columns is zero. If the maximum number of linearly independent vectors in the first $\ell$ rows of $A$ is $m$, then we may assume that the first $\ell-m$ rows are zero by applying the operation (Op3) to the first $\ell$ columns. Then $\MH_S$ is the Bott ring associated with the $(n-\ell+m)\times(n-\ell+m)$ submatrix of $A$ at the right-low corner of $A$. Let $\MH_S$ be as in Lemma \[MHS\] and let $V$ be a subspace of $N(\MH)$ complementary to $N(\MH)\cap \MH_S^1$. The dimension of $V$ is $\ell-m$ in the proof of Lemma \[MHS\]. The graded subalgebra of $\MH$ generated by $V$ is an exterior algebra $\Lambda(V)$, so $$\label{MH0} \MH=\Lambda(V)\otimes \MH_S.$$ We say that a Bott ring $\MH$ is *semisimple* if $\MH$ is generated by $S(\MH)$. Clearly $\MH_S$ is semisimple and $\MH$ is semisimple if and only if $\MH=\MH_S$. \[YMA\] Let $\MH$ be a Bott ring. If $\MH=\bigotimes_{i=1}^r\MH_i$ with Bott subrings $\MH_i$’s of $\MH$, then $S(\MH)=\coprod_{i=1}^rS(\MH_i)$. Therefore $\MH$ is semisimple if and only if all $\MH_i$’s are semisimple. Let $x\in S(\MH)$ and write $x=\sum_{i=1}^r y_i$ and $\bx= \sum_{i=1}^r z_i$ with $y_i,z_i\in \MH_i$. Since $x\bx=0$, we have $$\text{$y_iz_j+y_jz_i=0$ for all $i\not=j$. }$$ Suppose that $y_i\not=0$ and $z_j\not=0$ for some $i\not=j$. Then $y_i=z_i$ and $y_j=z_j$ to satisfy the equations above. This shows that $x=\bx$, which contradicts the fact that $x\in S(\MH)$. Therefore $x=y_i$ and $\bx=z_i$ for some $i$, proving the lemma. Recall that a Bott ring $\MH$ has a decomposition $\Lambda(V)\otimes \MH_S$ in . \[semis\] If $\MH$ has another decomposition $\Lambda(U)\otimes \MS$ where $U$ is a subspace of $N(\MH)$ and $\MS$ is a semisimple subring of $\MH$, then $\dim U=\dim V$ and $\MS=\MH_S$. Since both $S(\Lambda(U))$ abd $S(\Lambda(V))$ are empty, $S(\MH)=S(\MS)$ by Lemma \[YMA\] and this implies the corollary. \[factor\] Let $\MH=\bigotimes_{i=1}^r\MH_i$ be as in Lemma \[YMA\] and $\pi_i\colon \MH\to \MH_i$ be the projection. Let $\ML$ be a semisimple Bott ring and let $\psi\colon \ML\to \MH$ be a graded ring monomorphism. If the composition $\pi_i\circ \psi\colon \ML\to \MH_i$ is an isomorphism for some $i$, then $\psi(\ML)=\MH_i$. Let $y\in S(\ML)$. Then $\psi(y)\in S(\MH)$ because $\psi$ is a graded ring monomorphism, and it is actually in $S(\MH_i)$ by Lemma \[YMA\] since $(\pi_i\circ \psi)(y)\not=0$. This shows that $\psi(S(\ML))\subset S(\MH_i)$ but since $\pi_i\circ\psi$ is an isomorphism, the inclusion should be the equality. Therefore $\psi(\ML)=\MH_i$ because $\ML$ and $\MH_i$ are both semisimple. We say that a semisimple Bott ring is *simple* if it is not isomorphic to the tensor product (over $\Z/2)$ of more than one semisimple Bott rings, in other words, a simple Bott ring is a Bott ring isomorphic to the cohomology ring (with $\Z/2$ coefficient) of an indecomposable real Bott manifold different from $S^1$. A Bott ring isomorphic to the cohomology ring of the Klein bottle with $\Z/2$ coefficient is simple and we call it especially a *Klein ring*. If an element $x\in S(\MH)$ satisfies $(x+\bx)^2=0$, then the subring generated by $x$ and $\bx$ is a Klein ring and we call such a pair $\{x,\bx\}$ a *Klein pair*. We note that $x$ and $\bx$ have the same eigen-element and $\{x,\bx\}$ is a Klein pair if and only if the eigen-element of $x$ and $\bx$, that is $x+\bx$, lies in $N(\MH)$. \[klein\] If $S(\MH)\not=\emptyset$, then a Klein pair exists in $\MH$ and the quotient of $\MH$ by the ideal generated by a Klein pair is again a Bott ring. Let $\MH$ be of the form . The assumption $S(\MH)\not=\emptyset$ is equivalent to $A$ being non-zero as remarked before. As in the proof of Lemm \[MHS\], we may assume through the operation (Op1) that the first $\ell$ columns of $A$ are zero and none of the remaining columns is zero. Then $x_1,\dots,x_{\ell}$ are elements of $N(\MH)$ and the eigen-element $\alpha_{\ell+1}$ of $x_{\ell+1}$ is a linear combination of $x_1,\dots,x_{\ell}$, so $\alpha_{\ell+1}$ lies in $N(\MH)$ which means that $\{x_{\ell+1},\bx_{\ell+1}\}$ is a Klein pair. If $\{x,\bx\}$ is a Klein pair, then the eigen-element of $x$ is non-zero and belongs to $N(\MH)$, so through the operation (Op1) we may assume that it is $\alpha_{\ell+1}$. Then, applying the operation (Op3) to the eigen-space of $\alpha_{\ell+1}$, we may assume $x=x_{\ell+1}$. We further may assume $\alpha_{\ell+1}=x_\ell$ by applying the operation (Op3) to $N(\MH)$. The quotient ring of $\MH$ by the ideal generated by the Klein pair $\{x,\bx\}$ is then nothing but to take $x_\ell=x_{\ell+1}=0$ in $\MH$, so it is a Bott ring associated with a $(n-2)\times(n-2)$ matrix obtained from $A$ by deleting $\ell$-th and $\ell+1$-st columns and rows. Now we are in a position to prove the unique decomposition of a semisimple Bott ring into a tensor product of simple Bott rings. \[simpl\] Let ${\MC}_i$ $(i=1,\dots,p)$ and ${\MD}_j$ $(j=1,\dots,q)$ be simple Bott rings. If there exists a graded ring isomorphism $$\label{MA} \vf\colon \bigotimes_{i=1}^p\MC_i\to \bigotimes_{j=1}^q \MD_j,$$ then $p=q$ and $\vf$ preserves the factors, i.e. there is a permutation $\rho$ on $\{1,\dots,p=q\}$ such that $\vf(\MC_i)=\MD_{\rho(i)}$ for $i=1,\dots,p$. We set $\MC=\bigotimes_{i=1}^p\MC_i$ and $\MD=\bigotimes_{j=1}^q\MD_j$. If either $\MC$ or $\MD$ is simple (i.e. $p=1$ or $q=1$), then both of them must be simple and the proposition is trivial. In the sequel we will assume that both $\MC$ and $\MD$ are not simple (so that $p\ge 2$ and $q\ge 2$), and prove the proposition by induction on the rank of $\MC$, that is, $\dim \MC^1$. If $\vf(\MC_i)=\MD_j$ for some $i$ and $j$, say $\vf(\MC_p)=\MD_q$, then we factorize them so that $\vf$ induces an isomorphism $\bar \vf\colon \bigotimes_{i=1}^{p-1}\MC_i\to \bigotimes_{j=1}^{q-1} \MD_j$. By the induction assumption, we conclude $p=q$ and may assume that $\bar \vf(\MC_i)=\MD_i$ for $i=1,\dots,p-1$ if necessary by permuting the suffixes of $\MD_j$’s. Then it follows from Lemma \[factor\] that $\vf(\MC_i)=\MD_{i}$ for $i=1,\dots,p-1$. This together with $\vf(\MC_p)=\MD_q$ where $p=q$ proves the statement in the lemma. In the sequel, it suffices to show that $\vf(\MC_i)=\MD_j$ for some $i$ and $j$ when we have an isomorphism $\vf$ in the proposition. *Case 1.* The case where some $\MC_i$ or $\MD_j$ is a Klein ring. We may assume that $\MC_p$ is a Klein ring without loss of generality. Let $\{x,\bx\}$ be a Klein pair in $\MC_p$. Its image by $\vf$ sits in some $\MD_j$ by Lemma \[YMA\] and we may assume that it sits in $\MD_q$. If $\MD_q$ is also a Klein ring, then $\vf(\MC_p)=\MD_q$. Therefore we may assume that $\MD_q$ is not a Klein ring in the following. Our isomorphism $\vf$ induces an isomorphism $$\bar \vf \colon \MC/(x,\bx)=\bigotimes_{i=1}^{p-1}\MC_i\cong \bigotimes_{j=1}^{q-1}\MD_j\otimes (\MD_q/(\vf(x),\vf(\bx)))$$ where $(u,v)$ denotes the ideal generated by the elements $u$ and $v$ and $\MD_q/(\vf(x),\vf(\bx))$ is a Bott ring by Lemma \[klein\]. Since $\operatorname{rank}(\MC/(x,\bx))=\operatorname{rank}\MC-2$, it follows from the induction assumption that $p-1\ge q$ and we may assume that $\bar \vf(\MC_i)=\MD_i$ for $i=1,\dots,q-1$ and $\bar \vf(\otimes_{i=q}^{p-1}\MC_i)=\MD_q/(\vf(x),\vf(\bx))$, in particular, $\bar \vf(\MC_1)=\MD_1$ as $q\ge 2$. Then, it follows from Lemma \[factor\] that $\vf(\MC_1)=\MD_1$. *Case 2.* The case where none of $\MC_i$’s and $\MD_j$’s is a Klein ring. Let $\{x,\bx\}$ be a Klein pair of $\MC_p$ and we may assume that its image by $\vf$ sits in $\MD_q$ as before. Then $\vf$ induces an isomorphism $$\bar \vf\colon \MC/(x,\bx)=\bigotimes_{i=1}^{p-1}\MC_i\otimes (\MC_p/(x,\bx)) \to \bigotimes_{j=1}^{q-1}\MD_j\otimes (\MD_q/(\vf(x),\vf(\bx))),$$ where the quotients $\MC_p/(x,\bx)$ and $\MD_q/(\vf(x),\vf(\bx))$ are both Bott rings by Lemma \[klein\]. The induction assumption can be applied to this situation as before. If $\bar \vf(\MC_i)=\MD_j$ for some $1\le i\le p-1$ and $1\le j\le q-1$, then $\vf(\MC_i)=\MD_j$ by Lemma \[factor\]. If $\bar \vf(\bigotimes_{i=1}^{p-1}\MC_i)=\MD_q/(\vf(x),\vf(\bx))$ and $\bar \vf(\MC_p/(x,\bx))=\bigotimes_{j=1}^{q-1}\MD_j$, then $\vf$ restricts to an isomorphism $$(\bigotimes_{i=1}^{p-1}\MC_i)\otimes\langle x,\bx\rangle\to \MD_q$$ where $\langle x,\bx\rangle$ denotes the Klein ring generated by $x$ and $\bx$, and this contradicts the fact that $\MD_q$ is simple as $p\ge 2$. Now Theorem \[bdeco\] follows from Corollaries \[maincoro\], \[semis\] and Proposition \[simpl\]. [**Acknowledgment**]{}. I would like to thank Y. Kamishima for communications which stimulated this research and A. Nazra for informing me of his classification of real Bott manifolds of dimension $\le 5$ up to diffeomorphism. [19]{} L. S. Charlap, *Compact flat Riemannian manifolds: I*, Ann. of Math. 81 (1965), 15–30. M. Grossberg and Y. Karshon, *Bott towers, complete integrability, and the extended character of representations*, Duke Math. J 76 (1994), 23–58. Y. Kamishima and M. Masuda, *Cohomological rigidity of real Bott manifolds*, preprint, arXiv:0807.4263. Y. Kamishima and A. Nazra, *Seifert fibered structure and rigidity on real Bott towers*, in preparation. M. Masuda, *Cohomological non-rigidity of generalized real Bott manifolds of height 2*, preprint, arXiv:0809.2215. M. Masuda and T. Panov, *Semifree circle actions, Bott towers, and quasitoric manifolds*, Sbornik Math. (to appear), arXiv:math.AT/0607094. A. Nazra, *Real Bott tower*, Tokyo Metropolitan University, Master Thesis 2008. [^1]: The author was partially supported by Grant-in-Aid for Scientific Research 19204007

--- abstract: 'The two time-dependent Schrödinger equations in a potential $V(s,u)$, $u$ denoting time, can be interpreted geometrically as a moving interacting curves whose Fermi-Walker phase density is given by $-(\partial V/\partial s)$. The Manakov model appears as two moving interacting curves using extended da Rios system and two Hasimoto transformations.' address: | $^{1}$ Institute of Electronics, Bulgarian Academy of Sciences, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria\ $^{2}$ Université de Cergy-Pontoise, 2 avenue, A. Chauvin, F-95302, Cergy-Pontoise Cedex, France\ $^{3}$Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria author: - 'N. A. KOSTOV$^{1}$, R. DANDOLOFF$^{2}$, V. S. GERDJIKOV$^{3}$ and G. G. GRAHOVSKI$^{2,3}$' title: THE MANAKOV SYSTEM AS TWO MOVING INTERACTING CURVES --- Introduction ============ In recent years, there has been a large interest in the applications of the Frenet-Serret equations [@e60; @RogSchief] for a space curve in various contexts, and interesting connections between geometry and integrable nonlinear evolution equations have been revealed. The subject of how space curves evolve in time is of great interest and has been investigated by many authors. Hasimoto [@h72] showed that the evolution of a thin vortex filament regarded as a moving space curve can be mapped to the nonlinear Schrödinger equation (NLSE): $$\begin{aligned} \label{eq:1} i \Psi_{u}+\Psi_{ss}+\frac{1}{2}|\Psi|^{2}\Psi=0,\end{aligned}$$ Here, u and s are time and space variables, respectively, subscripts denote partial derivatives. Lamb [@L77] used Hasimoto transformation to connect other motions of curves to the mKdV and sine-Gordon equations. Langer and Perline [@lr91] showed that the dynamics of non-stretching vortex filament in $\bbbr^{3}$ leads to the NLS hierarchy. Motions of curves on $S^{2}$ and $S^{3}$ were considered by Doliwa and Santini [@DoSa]. Lakshmanan [@Laks] interpreted the dynamics of a nonlinear string of fixed length in $\bbbr^{3}$ through the consideration of the motion of an arbitrary rigid body along it, deriving the AKNS spectral problem without spectral parameter. Recently, Nakayama [@Nakaya] showed that the defocusing nonlinear Schrödinger equation, the Regge-Lund equation, a coupled system of KdV equations and their hyperbolic type arise from motions of curves on hyperbola in the Minkowski space. Recently the connection between motion of space or plane curves and integrable equations has drawn wide interest and many results have been obtained [@FokGel; @dodd; @ciel; @ChouQu; @LanPer; @H02; @CaIv]. Preliminaries {#sec:2} ============= The Manakov model {#sec:2.1} ----------------- Time-dependent Schrödinger equation in potential $V(s,u)$ $$\begin{aligned} i\Psi_{u}+\Psi_{ss}-V(s,u)\Psi=0,\end{aligned}$$ goes into the NLS eq. (\[eq:1\]) if the potential $V(s,u)=-1/2 |\psi(s,u)|^2$. Similarly, a set of two time-dependent Schrödinger equations: $$\begin{aligned} i\Psi_{1,u}+\Psi_{1,ss}-V(s,u)\Psi_{1}=0,\quad i\Psi_{2,u}+\Psi_{2,ss}-V(s,u)\Psi_{2}=0,\end{aligned}$$ where $V(s,u)=-|\Psi_{1}|^{2} -|\Psi_{2}|^{2}$, can be viewed as the Manakov system: $$\begin{aligned} &&i\Psi_{1,u}+\Psi_{1,ss}+(|\Psi_{1}|^{2} +|\Psi_{2}|^{2})\Psi_{1}=0,\label{ManakovSys1}\\ &&i\Psi_{2,u}+\Psi_{2,ss}+(|\Psi_{1}|^{2}+|\Psi_{2}|^{2})\Psi_{2}=0. \label{ManakovSys2}\end{aligned}$$ It is convenient to use two Hasimoto transformations [@h72] $$\begin{aligned} \Psi_{i}=\kappa_{i}(s,u) \exp\left[i\int^{s}\tau_{i}(s',u)d s'\right],\quad i=1,2,\end{aligned}$$ in Eqs. (\[ManakovSys1\]), (\[ManakovSys2\]). Equating imaginary and real parts, this leads to the coupled partial differential equations (extended daRios system [@r1906]) $$\begin{aligned} &&\kappa_{i,u}=-(\kappa_{i}\tau_{i})_{s}-\kappa_{i,s}\tau_{i},\quad i=1,2 ,\label{daRios1}\\ &&\tau_{i,u}=\left[\frac{\kappa_{i,ss}}{\kappa_{i}}-\tau_{i}^{2}\right]_{s}-V(s,u)_{s}, \label{daRios2}\end{aligned}$$ where $$\begin{aligned} V(s,u)=-|\Psi_{1}|^{2} -|\Psi_{2}|^{2}=-\kappa_{1}^{2}-\kappa_{2}^2.\end{aligned}$$ Soliton curves {#sec:2.2} -------------- A three dimensional space curve is described in parametric form by a position vectors ${\bf r}_{i}(s), i=1,2$, where s is the arclength. Let ${\bf t}_{i}=(\partial {\bf r}_{i}/\partial s), i=1,2$ be the two unit tangent vectors along the two curves. At a given instant of time the triads of unit orthonormal vectors $({\bf t}_{i},{\bf n}_{i},{\bf b}_{i})$, where ${\bf n}_{i}$ and ${\bf b}_{i}$ denote the normals and binormals, respectively, satisfy the Frenet-Serret equations for two curves $$\begin{aligned} \label{FSeq} {\bf t}_{i,s}=\kappa_{i} {\bf n_{i}},\quad {\bf n}_{i,s}=-\kappa_{i} {\bf t}_{i}+ \tau_{i}{\bf b}_{i},\quad {\bf b}_{i,s}=-\tau_{i} {\bf n}_{i},\quad i=1,2 ,\end{aligned}$$ $\kappa_{i}$ and $\tau_{i}$ denote, respectively the two curvatures and torsions of the curves. A moving curves are described by $r_{i}(s,u)$, where u denote time. The temporal evolution of two triads corresponding to a given value $s$ can be written in the general form as $$\begin{aligned} {\bf t}_{i,u}=g_{i} {\bf n}_{i}+h_{i} {\bf b}_{i},\quad {\bf n}_{i,u}=-g_{i} {\bf t}_{i} + \tau^{0}_{i}{\bf b}_{i},\quad {\bf b}_{i,u}=-h_{i} {\bf t}_{i} - \tau^{0}_{i}{\bf n}_{i},\end{aligned}$$ where the coefficients $g_{i}$,$h_{i}$ and $\tau_{i}^{0}$, as well as $\kappa_{i}$ and $\tau_{i}$, are functions of $s$ and $u$. $$\begin{aligned} \left(% \begin{array}{c} {\bf t}_{i} \\ {\bf n}_{i} \\ {\bf b}_{i} \\ \end{array}% \right)_{s} =\left(% \begin{array}{ccc} 0 & \kappa_{i} & 0 \\ -\kappa_{i} & 0 & \tau_{i} \\ 0 & -\tau_{i} & 0 \\ \end{array}% \right) \left(% \begin{array}{c} {\bf t}_{i} \\ {\bf n}_{i} \\ {\bf b}_{i} \\ \end{array}% \right),\quad \left(% \begin{array}{c} {\bf t}_{i} \\ {\bf n}_{i} \\ {\bf b}_{i} \\ \end{array}% \right)_{u} =\left(% \begin{array}{ccc} 0 & g_{i} & h_{i} \\ -g_{i} & 0 & \tau^{0}_{i} \\ -h_{i} & -\tau^{0}_{i} & 0 \\ \end{array}% \right) \left(% \begin{array}{c} {\bf t}_{i} \\ {\bf n}_{i} \\ {\bf b}_{i} \\ \end{array}% \right).\nonumber\end{aligned}$$ Introduce $$\begin{aligned} L_{i} =\left(% \begin{array}{ccc} 0 & \kappa_{i} & 0 \\ -\kappa_{i} & 0 & \tau_{i} \\ 0 & -\tau_{i} & 0 \\ \end{array}% \right),\qquad M_{i} =\left(% \begin{array}{ccc} 0 & g_{i} & h_{i} \\ -g_{i} & 0 & \tau^{0}_{i} \\ -h_{i} & -\tau^{0}_{i} & 0 \\ \end{array}% \right)\end{aligned}$$ and $\Delta {\bf t}_{i} \equiv ({\bf t}_{i,su}-{\bf t}_{i,us})$, $ \Delta {\bf n}_{i} \equiv ({\bf n}_{i,su}-{\bf n}_{i,us})$, and $\Delta {\bf b}_{i} \equiv ({\bf b}_{i,su}-{\bf b}_{i,us}) $. Then $$\begin{aligned} \left(% \begin{array}{c} \Delta {\bf t}_{i} \\ \Delta {\bf n}_{i} \\ \Delta {\bf b}_{i} \\ \end{array}% \right) =&&\left( \partial_{s} M_{i}-\partial_{u} L_{i}+[L_{i},M_{i}] \right) \left(% \begin{array}{c} {\bf t}_{i} \\ {\bf n}_{i} \\ {\bf b}_{i} \\ \end{array}% \right)\nonumber\\&& = \left(% \begin{array}{ccc} 0 & \alpha^{1}_{i} & \alpha^{2}_{i} \\ -\alpha^{1}_{i} & 0 & \alpha^{3}_{i} \\ - \alpha^{2}_{i} & -\alpha^{3}_{i} & 0 \\ \end{array}% \right) \left(% \begin{array}{c} {\bf t}_{i} \\ {\bf n}_{i} \\ {\bf b}_{i} \\ \end{array}% \right),\end{aligned}$$ where $$\begin{aligned} \alpha^{1}_{i}=\kappa_{i,u}g_{i,s}+h_{i}\tau_{i},\, \alpha^{2}_{i}=-h_{i,s}+\kappa_{i}\tau_{i}^{0}-g_{i}\tau_{i},\, \alpha^{3}_{i}=\tau_{i,u}-\tau_{i,s}-\kappa_{i} h_{i}.\end{aligned}$$ $$\begin{aligned} \label{KappaTau} \kappa_{i,u}=g_{i,s}-h_{i}\tau_{i},\qquad \tau_{i}^{0}=(h_{i,s}+g_{i}\tau_{i})/\kappa_{i},\end{aligned}$$ A generic curve evolution must satisfy the geometric constraints $$\begin{aligned} \Delta {\bf t}_{i}\cdot( \Delta {\bf n}_{i}\times \Delta {\bf b}_{i})=0, \label{constraint}\end{aligned}$$ i.e., $\Delta {\bf t}_{i}$, $ \Delta {\bf n}_{i}$ and $ \Delta {\bf b}_{i}$ must remain coplanar vectors under time involution. Further, since Eq. (\[constraint\]) is automatically satisfied for $\Delta {\bf t}_{i}=0$, we see that $\Delta {\bf n}_{i}$ and $\Delta {\bf b}_{i}$ need not necessarily vanish. In addition, we see from (\[constraint\]) that $\Delta {\bf t}_{i}=0$ implies $\alpha_{i}^{1}=\alpha_{i}^{2}=0$, so that $$\begin{aligned} \Delta {\bf n}_{i}=\alpha_{i}^{3} \Delta {\bf b}_{i},\quad \Delta {\bf b}_{i}=\alpha_{i}^{3} \Delta {\bf n}_{i}\quad g_{i}=-\kappa_{i} \tau_{i},\qquad h_{i}=\kappa_{i,s}.\end{aligned}$$ Substituting these in the second equation in (\[KappaTau\]) gives $$\begin{aligned} \tau_{i}^{0}=\left[\frac{\kappa_{i,ss}}{\kappa_{i}}-\tau_{i}^{2}\right],\end{aligned}$$ hence Eq. (\[daRios1\]) yields $(\tau_{i,u}-\tau^{0}_{i,s})=-V(s,u)_{s}=(\kappa_{1}^{2}+\kappa_{2}^2)_{s}$. Next there is an underlying angle anholonomy [@bd99; @bbd90] or ’Fermi-Walker phase’ $\delta\Phi^{FW}=(\tau_{i,u}-\tau^{0}_{i,s}) dsdu$ with respect to its original orientation, when $s$ and $u$ change along an infinitesimal closed path of area $dsdu$. Integration of the extended da Rios system ========================================== The coupled nonlinear equations (\[daRios1\]),(\[daRios2\]) constitute the extended Da Rios system as derived in [@r1906] . The solutions of (\[daRios1\]),(\[daRios2\]) with $\kappa(\xi)$ and $\tau=\tau(\xi)$, where $\xi=s-c\,u$ are simple. On substitution, we obtain $$\begin{aligned} &&c\,\kappa_{i,\xi}=2\kappa_{i,\xi}\tau_{i}+\kappa_{i}\tau_{i,\xi},\quad \xi=s-cu, i=1,2 , \\&& -c \tau_{i,\xi}=\left[ -\tau_{i}^2+\frac{\kappa_{i,\xi\xi}}{\kappa_{i}}+\kappa_{1}^{2}+\kappa_{2}^{2} \right]_{\xi}, \qquad\tau_{i}=\frac{c}{2}.\end{aligned}$$ where we use the boundary condition $\kappa_{i}\rightarrow 0, i=1,2$ as $s\rightarrow \infty$. Hence $\kappa_{i}$ obey the nonlinear oscillator equations $$\begin{aligned} \kappa_{i,\xi\xi}+ \left(\sum_{j=1}^{2}\kappa_{j}^{2}\right)\kappa_{i}=a_{i}\kappa_{i},\quad i=1,2. \label{oscillator7}\end{aligned}$$ where $a_{i}, i=1,2$ are arbitrary constants. [**Example 1**]{} One soliton solutions of the Manakov system are given by $$\begin{aligned} &&\Psi_{1}=\sqrt{2a}\,\epsilon_{1}\mbox{e}^{i(\frac{1}{2} c(s-s_{0})+(a-\frac{1}{4} c^2) u)}\mbox{sech}(\sqrt{a}(s-s_{0}-ct))\\ &&\Psi_{2}=\sqrt{2a}\,\epsilon_{2}\mbox{e}^{i(\frac{1}{2} c(s-s_{0})+(a-\frac{1}{4} c^2) u)}\mbox{sech}(\sqrt{a}(s-s_{0}-ct)),\end{aligned}$$ and $|\epsilon_{1}|^{2}+|\epsilon_{2}|^{2}=1$ We first note that for Manakov system, the expressions for the curvatures $\kappa_{i}, i=1,2$ and the torsions $\tau_{i}, i=1,2$ for the moving curves corresponding to a one soliton solutions of the Manakov system are given by $$\begin{aligned} \kappa^2=\kappa_{1}^{2}+\kappa_{2}^{2}=\sqrt{2a}\,\mbox{sech}(\sqrt{a}(s-s_{0}-ct)),\quad \tau_{1}=\tau_{2}=\frac{1}{2}c.\end{aligned}$$ and $$\begin{aligned} \kappa_{1}=\sqrt{2a}\,\epsilon_{1}\mbox{sech}(\sqrt{a}(s-s_{0}-ct)),\quad \kappa_{2}=\sqrt{2a}\,\epsilon_{2}\mbox{sech}(\sqrt{a}(s-s_{0}-ct)).\nonumber\end{aligned}$$ ![Two curves (\[oneSolManSys\]) of one soliton solution of Manakov system, $\epsilon_{1}=\sqrt{2}/\sqrt{3}$, $\epsilon_{2}=1/\sqrt{3}$ []{data-label="fig1"}](solit.eps){width="10cm" height="8cm"} [**Example 2**]{} One special solution of Manakov system is written by $$\begin{aligned} \kappa_{1}=C_{1}\mbox{cn}(\alpha \xi,k),\qquad \kappa_{2}=C_{2}\mbox{cn}(\alpha \xi,k),\end{aligned}$$ where $$\begin{aligned} \alpha^2=\frac{a_{1}}{2k^2-1},\quad C_{1}^{2}+C_{2}^{2}= 2\alpha^2 k^2, \quad a_{1}=a_{2}=a,\end{aligned}$$ In the limit $k\rightarrow 1$ we obtain the well known [*Manakov*]{} soliton solution $$\begin{aligned} &&\Psi_{1}=\frac{ \sqrt{2a} \epsilon_{1} \exp\left\{i\left(\frac{1}{2}c(s-s_{0})+(a-\frac{1}{4} c^{2})u\right) \right\} } {\mbox{ch}(\sqrt{a} (s-s_{0}-ct)) },\nonumber \\ &&\Psi_{2}=\frac{ \sqrt{2a} \epsilon_{2} \exp\left\{i\left(\frac{1}{2}c(s-s_{0})+(a-\frac{1}{4} c^{2})u\right) \right\} } {\mbox{ch}(\sqrt{a} (s-s_{0}-ct)) }. \nonumber\end{aligned}$$ Here we introduce the following notations $$\begin{aligned} |\epsilon_{1}|^2+|\epsilon_{2}|^2=1, \quad \zeta_{1}=\frac{1}{2} c+i\sqrt{a} =\xi_{1}+i\eta_{1},\end{aligned}$$ where $s_{0}$ is the position of soliton, $(\epsilon_{1},\epsilon_{2})$ are the components of polarization vector. The real part of $\zeta_{1}$ i.e. $c/2$ gives us the soliton velocity while the imaginary part of $\zeta_{1}$, i.e. $\sqrt{2a}$, gives the soliton amplitude and width. [**Example 3**]{} Integrating (\[FSeq\]) for two unit tangent vectors along the curves ${\bf t}_{i}=(\partial {\bf r}_{i}/\partial s), i=1,2$ for position vectors ${\bf r}_{i}(s), i=1,2$ we obtain $$\begin{aligned} {\bf r}_{j}=\left(\begin{array}{c} \frac{s}{2}-\frac{\epsilon_{j}}{\epsilon_{j}^{2}+\frac{1}{2} c} \tanh{\left(\epsilon_{j}(s-cu)\right)} \\ -\frac{\epsilon_{j}}{\epsilon_{j}^{2}+\frac{1}{2} c} \mbox{sech}{\left(\epsilon_{j}(x-cu)\right)} \cos{\left(\frac{1}{2}cs+(\epsilon_{j}^{2}-\frac{1}{4} c^2)u \right)} \\ -\frac{\epsilon_{j}}{\epsilon_{j}^{2}+\frac{1}{2} c} \mbox{sech}{\left(\epsilon_{j}(x-cu)\right)} \sin{\left(\frac{1}{2}cs+(\epsilon_{j}^{2}-\frac{1}{4} c^2)u \right)} \\ \end{array}\right),\quad j=1,2 , \label{oneSolManSys}\end{aligned}$$ and $\epsilon_{1}=\cos{\alpha},\,\epsilon_{2}=\sin{\alpha}$, where $\alpha$ is arbitrary positive number. [**Example 4**]{} Let $u(x)=6\wp(\xi+\omega^{\prime})$ be the two-gap Lamé potential with simple periodic spectrum (see for example [@ek94]) $$\lambda_{0}=-\sqrt{3g_{2}},\quad \lambda_{1}=-3e_{0}, \quad \lambda_{2}=-3e_{1}, \quad \lambda_{3}=-3e_{2}, \quad \lambda_{4}=\sqrt{ 3g_{2}}.$$ and the corresponding Hermite polynomial have the form $$F(\wp(\xi+\omega^{\prime}),\lambda)=\lambda^{2}- 3\wp(\xi+\omega^{\prime})\lambda+ 9\wp^{2}(\xi+\omega^{\prime})-\frac{9}{4}g_{2} . \label{HerPol}$$ Consider the genus $2$ nonlinear anisotropic oscillator (\[oscillator7\]) with Hamiltonian $$H=\frac{1}{2}(p_{1}^{2}+p_{2}^{2})+\frac{1}{4}(\kappa_{1}^{2}+\kappa_{2}^{2})^{2}- \frac{1}{2}(a_{1}\kappa_{1}^{2}+a_{2}\kappa_{2}^{2}),$$ where $(\kappa_{i},p_{i})$, $i=1,2$ are canonical variables with $p_{i}=\kappa_{i,x}$ and $a_{1},a_{2}$ are arbitrary constants. The simple solutions of these system are given in terms of Hermite polynomial $$\kappa_1^2=2\frac{F(x,\tilde{\lambda}_{1})} {\tilde{\lambda}_{2}-\tilde{\lambda}% _{1}} ,\quad \kappa_2^2=2\frac{F(x,\tilde{\lambda}_{2})} {\tilde{\lambda}_{1}-% \tilde{\lambda}_{2}} ,$$ Let us list the corresponding solutions [**(A)**]{} Periodic solutions in terms of single Jacobian elliptic function The nonlinear anisotropic oscillator admits the following solutions: $$\begin{aligned} \kappa_1 = C_1 \mbox{sn}(\alpha \xi, k), \qquad \kappa_2 = C_2 \mbox{cn}(\alpha \xi, k). \label{onegap}\end{aligned}$$ Here the amplitudes $C_1$, $C_2$ and the temporal pulse-width $1/\alpha$ are defined by the parameters $a_{1}$ and $a_{2}$ as follows: $$\begin{aligned} \alpha^2 k^2 = a_{2}-a_{1} , \quad C^{2}_{1} = a_{2} + \alpha^2 - 2\alpha^2 k^2 , \quad C^{2}_{2} = a_{1} + \alpha^{2}+\alpha^2 k^2 ,\end{aligned}$$ where $0 < k < 1$. Following our spectral method it is clear, that the solutions (\[onegap\]) are associated with eigenvalues $\lambda_2 = - e_2$ and $\lambda_3 = - e_3$ of one – gap Lamé potential. [**(B)**]{} Periodic solutions in terms of products of Jacobian elliptic functions Another solution is defined by [@ft89] $$\begin{aligned} \kappa_1 = C \mbox{dn}(\alpha \xi, k) \mbox{sn}(\alpha \xi, k), \qquad \kappa_2 = C \mbox{dn}(\alpha \xi ,k ) \mbox{cn}(\alpha \xi, k), \label{Flor}\end{aligned}$$ where $\mbox{sn}$,$\mbox{cn}$, $\mbox{dn}$ are the standard Jacobian elliptic functions, $k$ is the modulus of the elliptic functions $ 0 < k < 1$. The wave characteristic parameters: amplitude $C$, temporal pulse-width $1/\alpha$ and $k$ are related to the physical parameters and, $k$ through the following dispersion relations $$\begin{aligned} C^{2} = \frac{2}{5} (4a_{2} - a_{1}) , \quad \alpha^{2} = \frac{1}{15}(4a_{2}-a_{1}) , \quad k^{2} = \frac {5 (a_{2}-a_{1})} {4a_{2}-a_{1}} .\end{aligned}$$ We have found the following solutions of the nonlinear oscillator [@ku92] $$\begin{aligned} \kappa_1 = C\alpha^2 k^2 \mbox{cn}(\alpha \xi,k)\mbox{sn}(\alpha \xi,k), \quad \kappa_2 = C\alpha^2 \mbox{dn}^2 (\alpha \xi, k) + C_{1} \label{UzKos}\end{aligned}$$ where $C$, $C_1$, $\alpha$ and $k$ are expressed through parameters $a_{1}$ and $a_{2}$ by the following relations $$\begin{aligned} C^2 & = & \frac {18} {a_{2}-a_{1}}, \quad \alpha^2 = \frac {1}{10} \left( 2 a_{2}-3a_{1}+ \sqrt{\frac{5}{3}(a_{2}^{2}-a_{1}^2) } \right) \nonumber \\ k^2 & = & \frac {2 \sqrt { \frac{5}{3} (a_{2}^{2}-a_{1}^{2})}} {\sqrt{\frac{5}{3} (a_{2}^{2}-a_{1}^{2})}+2a_{2}-3a_{1}},\quad C_1 = \frac {C}{30} (4a_{1}-a_{2}),.\end{aligned}$$ [**(C)**]{} Periodic solutions associated with the two-gap Treibich-Verdier potentials. Below we construct the two periodic solutions associated with the Treibich-Verdier potential. Let us consider the potential $$u(x)=6\wp(\xi+\omega^{\prime})+2{\frac{(e_1-e_2)(e_1-e_3)}{\wp(\xi+\omega^{% \prime})-e_1}}, \label{tv4}$$ and construct the solution in terms of Lamé polynomials associated with the eigenvalues $\tilde{\lambda}_1,\tilde{\lambda}_2$, $\tilde{\lambda}_1 > \tilde{\lambda}_2$ $$\begin{aligned} \tilde{\lambda}_1=e_2+2e_1+2\sqrt{(e_1-e_2)(7e_1+2e_2)}, \\ \tilde{\lambda}_2=e_3+2e_1+2\sqrt{(e_1-e_3)(7e_1+2e_3)}. \nonumber \label{zz}\end{aligned}$$ The finite and real solutions $q_1,q_2$ have the form $$\begin{aligned} \kappa_1= C_{1}\mbox{sn}(\xi,k)\mbox{dn}(\xi,k) +C_{2}\mbox{sd}(\xi,k) ,\, \kappa_2= C_{3}\mbox{cn}(\xi,k)\mbox{dn}(\xi,k) +C_{4}\mbox{cd}(z,k), \nonumber\end{aligned}$$ where $C_{i}$, $i=1,\ldots 4$ are constants and have important geometrical interpretation [@ek94] and $\mbox{sd}$, $\mbox{cd}$, are standard Jacobian elliptic functions. The concrete expressions in terms of $k,\tilde{ \lambda}_{1},\tilde{\lambda_{2}}$ are given in [@ceek95; @chr:eil:eno:kos] In a similar way we can find the elliptic solution associated with the eigenvalues $$\begin{aligned} \tilde{\lambda}_1&=&e_2+2e_1+2\sqrt{(e_1-e_2)(7e_1+2e_2)},\quad \tilde{% \lambda}_2=-6e_1, \label{zz2}\end{aligned}$$ We have $$\begin{aligned} \kappa_1=\tilde{C}_{1}\mbox{dn}^{2}(\xi,k),\qquad \kappa_2= C_{1}\mbox{sn}(\xi,k)\mbox{dn}(\xi,k) +C_{2}\mbox{sd}(\xi,k) , \label{ee3}\end{aligned}$$ where $\tilde{C}_{1}, C_{1}, C_{2}$ are given in [@ceek95; @chr:eil:eno:kos]. The general formula for elliptic solutions of genus $2$ nonlinear anisotropic oscillator is given by [@ceek95] $$\begin{aligned} \kappa_1^2&=&{\frac{1}{\tilde{\lambda}_2-\tilde{\lambda}_1}} \left(2\tilde{\lambda}_1^2+2\tilde{\lambda}_1\sum_{i=1}^N \wp(\xi-x_i) \right. \nonumber \\ &&+\left.6\sum_{1\leq i< j\leq N}\wp(\xi-x_i)\wp(\xi-x_j)-{\frac{Ng_2}{4}}+ \sum_{1\leq i< j\leq 5}\lambda_i\lambda_j\right), \nonumber \\ \kappa_2^2&=&{\frac{1}{\tilde{\lambda}_1-\tilde{\lambda}_2}} \left(2\tilde{\lambda}_2^2+2\tilde{\lambda}_2\sum_{i=1}^N \wp(\xi-x_i) \right. \nonumber \\ &&+\left.6\sum_{1\leq i< j\leq N}\wp(\xi-x_i)\wp(\xi-x_j)-{\frac{Ng_2}{4}}+ \sum_{1\leq i< j\leq 5}\lambda_i\lambda_j\right), \nonumber\end{aligned}$$ where $x_{i}$ are solutions of equations $\sum_{i\neq j}\wp'(x_{i}-x_{j})=0, j=1,\ldots, N$. Extended da Rios-Betchov system =============================== Following Betchov we can derive the system of equations, which may be reduced to those for a two fictitious gases with negative pressures accompanied with two complicated nonlinear dispersive stresses. Introducing four new variables $\rho_{1}=\kappa_{1}^{2}$, $\rho_{1}=\kappa_{1}^{2}$, $u_{1}=2\tau_{1}$, $u_{2}=2 \tau_{2}$ using extended Da Rios system (\[daRios1\]), (\[daRios2\]) we obtain $$\begin{aligned} \label{Betchov} &&\frac{\partial\rho_{1}}{\partial u}+\frac{\partial (\rho_{1}u_{1})}{\partial s}=0,\qquad \frac{\partial\rho_{2}}{\partial u}+\frac{\partial(\rho_{2}u_{2})}{\partial s}=0, \nonumber \\&& \frac{\partial(\rho_{1} u_{1})}{\partial u}+\frac{\partial }{\partial s}\left[\rho_{1} u_{1}^2-(\rho_{1}^2+\rho_{2}^2)-\rho_{1}\frac{\partial^{2}}{\partial s^2}(\log \rho_{1})\right]=0,\nonumber \\&& \frac{\partial(\rho_{2} u_{2})}{\partial u}+\frac{\partial }{\partial s}\left[\rho_{2} u_{2}^2-(\rho_{1}^2+\rho_{2}^2)-\rho_{2}\frac{\partial^{2}}{\partial s^2}(\log \rho_{2})\right]=0. \nonumber\end{aligned}$$ HF system is gauge equivalent to Manakov system =============================================== The vector nonlinear Schrödinger equation is associated with type ${\bf A.III}$ symmetric space ${\rm SU(n+1)}/{\rm S(U(1)}\otimes {\rm U(n)})$. The special case $n=2$ of such symmetric space is associated with the famous Manakov system [@ma74]. Let us first fix the notations and the normalizations of the basis of ${\frak g} $. By $\Delta _+ $ ($\Delta _- $) we shall denote the set of positive (negative) roots of the algebra with respect to some ordering in the root space. By $\{E_{\alpha }, H_i\} $, $\alpha \in \Delta $, $i=1 \dots r $ we denote the Cartan–Weyl basis of ${\frak g} $ with the standard commutation relations [@Helg]. Here $H_i$ are Cartan generators dual to the basis vectors $e_i$ in the root space. The root system is invariant under the action of the Weyl group ${\frak W}({\frak g}) $ of the simple Lie algebra ${\frak g} $ [@Helg]. Let us now consider the gauge equivalent systems. The notion of gauge equivalence allows us to associate with the vector nonlinear Schrödinger equation an equivalent equation solvable by the ISM for the gauge equivalent linear problem [@ForKu*83]: $$\begin{aligned} \label{eq:2.3} \tilde{L}\tilde{\psi }(x,t,\lambda )= \left(i{d \over dx}-\lambda {\cal S}(x,t) \right) \tilde{\psi }(x,t,\lambda )=0, \nonumber\\ \tilde{M}\tilde{\psi }(x,t,\lambda )= \left(i{d \over dt}-\lambda^2 {\cal S}-\lambda {\cal S}_{x} {\cal S}(x,t) \right) \tilde{\psi }(x,t,\lambda )=0,\end{aligned}$$ where $$\begin{aligned} \label{eq:2.4} &&\tilde{\psi }(x,t,\lambda ) = \psi _0^{-1}\psi (x,t,\lambda ), \quad{\cal S}(x,t)=\sum_{\alpha=1}^{r}(S_{\alpha}E_{\alpha}+S_{\alpha}^{*}E_{-\alpha}) +\sum_{j=1}^{r} S_{j} H_{j} ,\nonumber\\&& {\cal S}(x,t) = \mbox{Ad}_{\hat{\psi }_0} J \equiv \psi _0^{-1}J\psi _0(x,t), \qquad J= \sum_{s=1}^n H_s,\end{aligned}$$ and $\psi _0=\psi (x,t,0) $ is the Jost solution at $\lambda =0 $. The zero-curvature condition $[\tilde{L},\tilde{M}]=0 $ is equivalent to $i {\cal S}_t-[{\cal S},{\cal S}_{xx}]=0.$ with ${\cal S}^2=I_{n}$. Conclusions =========== In this paper the Manakov model is interpreted as two moving interacting curves. We derive new extended Da Rios system and obtain the soliton, one-, and two-phase periodic solution of two thin vortex filaments in an incompressible inviscid fluid. The solution was explicitly given in terms of Weierstrass and Jacobian elliptic functions. Acknowledgements {#acknowledgements .unnumbered} ================ The present work is supported by the National Science Foundation of Bulgaria, contract No F-1410. The work of one of us GGG is supported also by the Bulgarian National Scientific Foundation Young Scientists scholarship for the project “Solitons, differential geometry and biophysical models”. [9]{} L.P. Eisenhart, [*A treatise on the differential geometry of curves and surfaces*]{}, Dover, New York, 1960. W. K. Schief and C. Rogers, [*Binormal motion of curves of constant curvature and torsion. Generation of soliton surfaces.*]{} Proc. R. Soc. Lond. A, [**455**]{} (1999) 3163-3188;\ C. Rogers and W. K. Schief, [*Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory*]{}, Cambridge Univ. Press, Cambridge (2002). H. Hasimoto, A soliton on a vortex filament, [*J.Fluid Mech.*]{} [**51**]{} (1972) 477–485. G. L. Lamb, Jr., Solitons on moving space curves, [*J. Math. Phys.* ]{} [**18**]{} (1977) 1654–1661. J. Langer, R. Perline , Poisson geometry of the filament equation, [*J. Nonlinear Sci.*]{}, 1991, [**1**]{}, N 1, (1991) 71–93. A. Doliwa, P. M. Santini, An elementary geometric characterization of the integrable motions of a curve, [*Phys. Lett. A*]{}, [**185**]{}, (1994) 373–384. M. Lakshmanan, Rigid body motions, space curves, prolongation structures, fiber bundles, and solitons, [*J. Math. Phys.*]{}, [**20**]{}, (1979) 1667–1672. K. Nakayama, Motion of curves in hyperboloid in the Minkowski space, [*J. Phys. Soc. Jpn.*]{}, [**67**]{}, (1998) 3031–3037. A. S. Fokas, I. M. Gelfand, Surfaces on Lie groups, on Lie algebras, and their integrability, [*Comm. Math. Phys.*]{} [**177**]{} (1996), 203–220;\ A. S. Fokas, I. M. Gelfand, F. Finkel and Q. M. Liu, A formula for constructing infinitely many surfaces on Lie algebras and integrable equations, [*Selecta Math. (New ser.)*]{} [**6**]{} (2000) 347–375. R. K. Dodd, Soliton immersions, [*Commun. Math. Phys.*]{} [**197**]{} (1998) 641–665. Sym, A.: Soliton surfaces and their applications (soliton geometry from spectral problems), In: [*Geometrical aspects of the Einstein equations and integrable systems.*]{} (Lect. Notes Phys. [**239**]{}), Berlin–Heidelberg– New York: Springer, 1985, pp. 154–231. Pohlmeyer, K., Integrable Hamiltonian systems and interactions through quadratic constraints, [*Commun. Math. Phys.*]{} [**46**]{} (1976) 207 – 221. J. Cieśliński. A generalized formula for integrable classes of surfaces in Lie algebras. [*J. Math. Phys.*]{} [**38**]{} (1997) 4255–4272. Kai-Seng Chou, Changzheng Qu, Integrable equations arising from motions of plane curves, [*Physica D*]{} [**162**]{} (2002) 9-–33. J. Langer, R. Perline, Geometric Realizations of Fordy-Kulish Nonlinear Schrödinger Systems, [*Pacific J. Math.*]{} [**195**]{} (2000) 157–178. H. J. Shin, Vortex filament motion under the localized induction approximation in terms of Weierstrass elliptic functions, [*Phys. Rev.*]{} [**E 65**]{}, (2002) 036317. A. Calini, T. Ivey, Connecting geometry, topology and spectra for finite-gap NLS potentials, [*Physica D*]{} [**152-153**]{} (2001), 9-19;A. Calini, T. Ivey, Finite-gap solutions of the Vortex Filament Equation: Genus one solutions and symmetric solutions, [*Journal of Nonlinear Science*]{} [**15**]{}, (2005), 321–361. L. S. Da Rios, Sul moto d’un liquido indefinito con un filetto vorticoso, [*Rend. Circ. Mat. Palermo*]{} [**22**]{} (1906) 117–135. R. Balakrishnan, R. Dandoloff, The Schrödinger equation as a moving curve, [*Phys. Lett. A*]{} [**260**]{} (1999) 62–67. R. Balakrishnan, A. R.  Bishop, R. Dandoloff, Geometric phase in the classical continuous antiferromagnetic Heisenberg spin chain , [*Phys.Rev. Lett.*]{} [**64**]{} (1990) 2107-2110. Enolskii, V., Kostov N., On the geometry of elliptic solitons, [*Acta Applicandae Math.*]{} [**36**]{} (1994) 57–86. M. Florjanzyk, R. Tremblay, Periodic and solitary waves in bimodal optical fibers, [*Phys. Lett. A*]{} [**25**]{} (1989) 34–36. N. Kostov, M. Uzunov, New kinds of periodical waves in birefringent optical fibers, [*Opt.Commun.*]{} [**89**]{} (1992) 389–392. P.L. Christiansen, J.C. Eilbeck, V.Z. Enolskii, N.A. Kostov, Quasi-periodic solutions of the coupled nonlinear Schrodinger equations, [*Proc. Royal Soc. London A*]{} [**451**]{} (1995) 685–700. Christiansen, P., Eilbeck, J., Enolskii, V., Kostov, N., Quasi-periodic and periodic solutions for Manakov type systems of coupled nonlinear Schrodinger equations, [*Proc. Royal Soc. London A.*]{} [**456**]{} (2000) 2263–2281. Manakov S. V., On the theory of two-dimensional stationary self-focusing of electromagnetic waves, [*Sov.Phys. JETP*]{} [**38**]{} (1974) 248–253. S. Helgasson, [*Differential geometry, Lie groups and symmetric spaces*]{}, AMS Providence, 2001. Fordy A. P., Kulish P. P., Nonlinear Schrodinger Equations and Simple Lie Algebras, [*Commun. Math. Phys.*]{} [**89**]{} (1983) 427–443.

--- abstract: 'We study the phase behavior of bowl-shaped particles using computer simulations. These particles were found experimentally to form a meta-stable worm-like fluid phase in which the bowl-shaped particles have a strong tendency to stack on top of each other \[M.Marechal *et al*, Nano Letters **10**, 1907 (2010)\]. In this work, we show that the transition from the low-density fluid to the worm-like phase has an interesting effect on the equation of state. The simulation results also show that the worm-like fluid phase transforms spontaneously into a columnar phase for bowls that are sufficiently deep. Furthermore, we describe the phase behavior as obtained from free energy calculations employing Monte Carlo simulations. The columnar phase is stable for bowl shapes ranging from infinitely thin bowls to surprisingly shallow bowls. Aside from a large region of stability for the columnar phase, the phase diagram features four novel crystal phases and a region where the stable fluid contains worm-like stacks.' author: - Matthieu Marechal - Marjolein Dijkstra bibliography: - 'bowls.bib' title: 'Phase behavior and structure of colloidal bowl-shaped particles: simulations' --- Introduction ============ The concept of a mesogenic particle in the form of a bowl is relatively old in the molecular liquid crystal community. Such molecules are expected to form a columnar phase, which can be ferroelectric, i.e., a phase with a net electric dipole moment, when the particles possess a permanent dipole moment. Ferroelectric phases have potential applications for optical and electronic devices. In fact, crystalline (as opposed to liquid crystalline) ferroelectrics are already applied in sensors, electromechanical devices and non-volatile memory [@FerroApp]. A columnar ferroelectric phase may have the advantage over a crystal, that grain boundaries and other defects anneal out faster due to the partially fluid nature of the columnar phase. In reality, columnar phases of conventional disc-like particles often exhibit many defects, as flat thin discs can diffuse out of a column and columns can split up. The presence of these defects limits their potential use for industrial applications [@simulation-bowls]. Less defects are expected in a columnar phase of bowl-shaped mesogens, where particles are supposed to be more confined in the lateral directions. A whole variety of bowl-like molecules have already been synthesized and investigated experimentally [@Sawamura2002; @simpson2004; @xu1993rbl; @malthete1987icc]. In addition, buckybowlic molecules, *i.e.* fragments of $C_{60}$ whose dangling bonds have been saturated with hydrogen atoms, have been shown to crystallize in a columnar fashion [@Rabideau1996; @Forkey1997; @Matsuo2004; @Sakurai2005; @Kawase2006]. However, the number of theoretical studies is very limited as it is difficult to model the complicated particle shape in theory and simulations. In a recent simulation study, the attractive-repulsive Gay-Berne potential generalized to bowl-shaped particles has been used to investigate the stacking of bowl-like mesogens as a function of temperature [@simulation-bowls]. The authors reported a nematic phase and a columnar phase. This columnar phase did not exhibit overall ferroelectric order, although polar regions were found. In another very recent simulation study [@Cinacchi2010] of hard contact lenses (infinitely thin, shallow bowls), a new type of fluid phase was found in which the particles cluster on a spherical surface for bowls which are not too shallow. No columnar phase was found since the focus was on rather shallow bowls at a relatively low densities. Recently, a procedure has been developed to synthesize bowl-shaped colloidal particles [@Carmen]. This method starts with the preparation of highly uniform oil-in-water emulsion droplets. Subsequently, the droplets were used as templates around which a solid shell with tunable thickness is grown. In the next step of the synthesis, the oil in the droplets is dissolved and finally, during drying, the shells collapse into hemispherical double-walled bowls. In addition to these larger, more easily imaged colloids, a whole variety of bowl-shaped nanoparticles and smaller colloids have been synthesized and characterized [@Charnay2003; @Wang2004; @Liu2005; @Jagadeesan2008; @Love2002; @Hosein2007], and possible applications of these systems have been put forward. We also note that recently hemispherical particles were synthesized at an air-solution interface [@higuchi] and on a substrate [@Xia]. These hemispherical particles are intended to be used as microlense arrays, but they can also serve as a new type of shape-anisotropic colloidal particle. In our simulations, we model the particles as the solid of revolution of a crescent (see Fig. \[fig:particles\]a). The diameter $\sigma$ of the particle and the thickness $D$ are defined as indicated in Fig. \[fig:particles\]a. We define the shape parameter of the bowls by a reduced thickness $D/\sigma$, such that the model reduces to infinitely thin hemispherical surfaces for $D/\sigma=0$ and to solid hemispheres for $D/\sigma=0.5$. The advantages of this simple model is that it interpolates continuously between an infinitely thin bowl and a hemispherical solid particle (the two colloidal model systems, which we discussed above), and that we can derive an algorithm that tests for overlaps between pairs of bowls, which is a prerequisite for Monte Carlo simulations of hard-core systems. In a recent combined experimental and simulation study (for which we performed the simulations), the phase behavior of repulsive bowl-shaped colloids was investigated [@Marechal2010bowls]. The colloids were shown to form a worm-like fluid phase, in which the particles form long curved stacks running in random directions. By comparing the distribution of stack lengths, the simulation model was shown to describe the colloidal particles well. No evidence of columnar ordering was found in the experiments and in simulations of bowls with corresponding deepness, which was explained by the glassy behavior of the particles preventing rearrangements. The phase behavior of the model particles is expected to also describe other repulsive bowl shaped particles well, provided that the dimensions of the simulation particle are chosen such that the diameter of a stack and the inter-particle distance in the stack are the same as for the particles to be modeled. In this work, we expand the simulation results on the hard bowl-shaped particles. First, we elaborate on the model for the collapsed shells; the overlap algorithm is described in the appendix. Also, the (free energy) methods are explained in more detail than in Ref. [@Marechal2010bowls]. In the results section, we study the properties of the isotropic phase. We investigate the nature and the location of the transition between the homogeneous fluid phase and the fluid phase that contains the worm-like stacks. Furthermore, we show the packing diagram and the phase diagram with a tentative homogeneous–to–worm-like fluid transition line. In the last section we summarize and discuss the results. ![ (a) The theoretical model of the colloidal bowl is the solid of revolution of a crescent around the axis indicated by the dashed line. The thickness of the double-walled bowl is denoted by $D$ and the diameter of the bowl by $\sigma$. (b) The bowls are defined using two spheres of radii $R_1$ and $R_2$, that are a distance of $L$ apart. The direction vector, $\mathbf{u}_i$ and the reference point of the particle, $\mathbf{r}_i$, (the dot in the center of the smaller sphere) are indicated. \[fig:particles\]](particles){width="45.00000%"} Methods ======= Model ----- We describe the model that we use to represent the bowls in more detail. Consider a sphere with a radius $R_1$ at the origin and a second sphere with radius $R_2>R_1$ at position $-L\mathbf{u}_i$, where $\mathbf{u}_i$ is the unit vector denoting the orientation of the bowl and $L>0$. The bowl is represented by that part of the sphere with radius $R_1$ that has no overlap with the larger sphere, see Fig. \[fig:particles\]b. We have chosen the values for $L$ and $R_2$ such that the bowls are hemispherical (see appendix for explicit expressions for $L$ and $R_2$). We define the thickness of the bowls by $D=L-(R_2-R_1)$, such that the model reduces to the surface of a hemisphere for $D=0$ and to a solid hemisphere for $D=R_1$. The volume of the particle is $\frac{\pi}{4}\, D\, (\sigma^2 - D\sigma + \frac{2}{3} D^2)$, where $\sigma\equiv 2R_1$ is our unit of length. The algorithm to determine overlap between our bowls is described in the appendix. Fluid phase ----------- We employ standard $NPT$ MC simulations to obtain the equation of state (EOS) for the fluid phase. In addition, we obtain the compressibility by measuring the fluctuations in the volume: $$\frac{\langle V^2\rangle - \langle V\rangle^2}{\langle V\rangle}=\frac{k_B T}{\rho} \, \left(\frac{\partial\rho}{\partial P}\right)_T,$$ where $\rho=N/V$ is the number density and the derivative of the pressure is taken at constant temperature is denoted by the subscript $T$. We determine the free energy at density $\rho_1$ by integrating the EOS from reference density $\rho_0$ to $\rho_1$: $$\frac{F(\rho_1)}{N}=\mu(\rho_0)-\frac{P(\rho_0)}{\rho_0} + \int_{\rho_0}^{\rho_1} \frac{P(\rho)}{\rho^2} d \rho\label{eqn:widom}$$ where the chemical potential $\mu(\rho_0)$ is determined using the Widom particle insertion method [@widom], and $P(\rho_0)$ is determined by a local fit to the EOS. To investigate the structure of the fluid phase, we measure the positional correlation function [@Veerman_Frenkel], $$g_c(z)=\frac{1}{N \rho A_\text{col}} \langle \sum_{i=1}^N \sum_{j=1}^{N_\text{col}(i)} \delta(\mathbf{r}_{ij} \cdot \mathbf{u}_i-z) \rangle\label{eqn:gcz},$$ where the sum over $j$ runs over $N_\text{col}(i)$ particles in a column of radius $\sigma/2$ with orientation $u_i$ centered around particle $i$, and where the area of the column is denoted by $A_\text{col}=\pi \sigma^2/4$. At sufficiently high pressure the particles stack on top of each other to form disordered worm-like piles which resemble the stacks observed in the experiments [@Marechal2010bowls]. As the stacks have a strong tendency to buckle, we cannot use $g_c(z)$ to determine the length of the stacks. We therefore determine the stack size distribution using a cluster criterion. Particle $i$ and $j$ belong to the same cluster if $$\begin{aligned} |\mathbf{r}_{ij} + (\zeta D/2 + \sigma/4) (\mathbf{u}_j-\mathbf{u}_i)| & < & \sigma/2 \quad\text{and} \nonumber\\ \mathbf{u}_i \cdot \mathbf{u}_j & > & 0, \label{eqn:cluster_crit}\end{aligned}$$ and where the first condition has to be satisfied for $\zeta=-1$, $0$ or $1$ and $\mathbf{r}_{ij}=\mathbf{r}_j-\mathbf{r}_i$, with $\mathbf{r}_i$ denoting the center of the sphere with radius $R_1$ of particle $i$, see Fig. \[fig:particles\]b. If both conditions are satisfied, particle $j$ is just above ($\zeta=1$) or below ($\zeta=-1$) particle $i$ in the stack, or, when the stack is curved, particle $j$ can be next to particle $i$ ($\zeta=0$). We now define the cluster distribution as the fraction of particles that belongs to a cluster of size $n$: $\mathcal{P}_\text{stack}(n)\equiv n N_n/N$, where $N_n$ is the number of clusters of size $n$. We checked that the cluster size distribution does not depend sensitively to the choice of parameters in Eq. (\[eqn:cluster\_crit\]). Columnar phases --------------- We also perform $NPT$ Monte Carlo simulations of the columnar phase using a rectangular simulation box with varying box lengths in order to relax the inter-particle distance in the $z$ direction, along the columns, independently from the lattice constant in the horizontal direction. The difference between the free energy of the columnar phase at a certain density and the free energy of the fluid phase at a lower density is determined using a thermodynamic integration technique [@Bates_Frenkel]. We apply a potential which couples a particle to its column: $$\Phi_\mathrm{hex}(\mathbf{r}^N,\lambda)=\lambda\sum_{i=1}^N\cos(2 \pi N_x x_i/L_x)\sin(\pi N_y y_i/L_y), \label{eqn:col_pot}$$ where $x_i$ and $y_i$ are the $x$ and $y$ components respectively of $\mathbf{r}_i$, $N_\alpha$ is the number of columns in the $\alpha$ direction and $L_\alpha$ is the size of the box in the $\alpha$ direction. In our simulations, we calculate Eq. (\[eqn:col\_pot\]) while fixing the center of mass. To do so efficiently, we first calculate all four combinations $$\lambda\sum_{i=1}^N \mathrm{trig1}(2 \pi N_x x_i/L_x) \mathrm{trig2}(\pi N_y y_i/L_y)\label{eqn:col_gen_trigs}$$ for $\mathrm{trig1}=\cos,\sin$ and $\mathrm{trig2}=\cos,\sin$. The change in these four expressions upon displacement of a single particle while keeping the center of mass fixed can be expressed in terms of single particle properties and the previous values of the expressions by using some basic trigonometry. In this way, $\Phi_\mathrm{hex}(\mathbf{r}^N,\lambda)$, which is Eq. \[eqn:col\_gen\_trigs\] for $\mathrm{trig1}=\cos$ and $\mathrm{trig2}=\sin$, can be calculated without performing the full summation over all particles in Eq. (\[eqn:col\_gen\_trigs\]) every time we displace a particle. Unfortunately, this calculation requires the evaluation of many more trigonometric functions than the simple expression (\[eqn:col\_pot\]), but the extra computation time is negligible compared to the overlap check. In addition to this positional potential, we also constrain the direction of the particle, using the potential $$\Phi_\text{ang}(\mathbf{u}^N,\lambda)=\lambda' \sum_{i=1}^N u_{i,z},\label{eqn:col_pot_ang}$$ where we used $\lambda'=0.1\lambda$ and where $u_{i,z}$ is the $z$ component of $\mathbf{u}_i$. The thermodynamic integration path from the columnar phase to the fluid is as follows: We start from the columnar phase at a certain density $\rho_2$. Subsequently, we slowly turn on the two potentials, *i.e.* we increase $\lambda$ from 0 to $\lambda_\mathrm{max}$. Next, we integrate the equation of state to go from $\rho_2$ to $\rho_1$, while keeping $\lambda=\lambda_\text{max}$ fixed. During this step the columnar phase will only be stable below the coexistence density, if $\lambda_\text{max}$ is sufficiently high. We find that $\lambda_\text{max}=20k_BT$ suffices to guarantee stability of the columnar phase. Finally, fixing the density $\rho_1$, we gradually turn off the potentials, while integrating over $\lambda$ from $\lambda_\text{max}$ to 0. During this last step, the columnar phase melts continuously, provided that the density $\rho_1$ is low enough and that $\lambda$ is high enough to prevent melting during the density integration step. The resulting free energy difference between the columnar phase and fluid phase is given by $$\begin{gathered} F_\text{col}(\rho_2)-F_\text{fluid}(\rho_1)=\\\int_0^{\lambda_\text{max}} \big\langle \Phi_\text{hex}(\mathbf{r}^N,\lambda)/\lambda+\Phi_\text{ang}(\mathbf{u}^N,\lambda)/\lambda\big\rangle\big|_{\rho=\rho_1}+\\ \int_{\rho_1}^{\rho_2} d\rho\left.\frac{NP(\rho)}{\rho^2}\right|_{\lambda=\lambda_\text{max}} \\ -\int_0^{\lambda_\text{max}} \big\langle \Phi_\text{hex}(\mathbf{r}^N,\lambda)/\lambda+\Phi_\text{ang}(\mathbf{u}^N,\lambda)/\lambda\big\rangle\big|_{\rho=\rho_2}\end{gathered}$$ The positional potential (\[eqn:col\_pot\]) is designed to stabilize a hexagonal array of columns, but, strictly speaking, it does not have the hexagonal symmetry of the columnar phase, since it is not invariant under a 60 degrees rotation of the whole system around a lattice position. However, we find that replacing Eq. (\[eqn:col\_pot\]) by a positional potential that does have this symmetry, does not have a significant effect on the free energy difference. A second type of columnar phase can be constructed by flipping half of the bowls. In this way we obtain alternating vertical sheets (*i.e.* rows of columns) of bowls that point upwards and sheets of bowls that point downwards, we will refer to this phase as the inverted columnar phase. We calculate the free energy of this phase using the method described above, with the modification that the angular potential now reads, $$\Phi_\text{ang}(\mathbf{u}^N,\lambda)=\lambda' \sum_i u_{i,z}^2. \label{eqn:col_pot_ang2}$$ This potential could also have been used for the non-inverted columnar phase, and we have found that the result of the free energy integration for the columnar phase is the same whether we use Eq. (\[eqn:col\_pot\_ang2\]) or Eq. (\[eqn:col\_pot\_ang\]). Crystals {#sec:cryst} -------- ### Packing As the crystal phases of the bowls are not known a priori, we developed a novel pressure annealing method to obtain the possible crystal phases [@PhysRevLettSSS], which we named after the thermal annealing technique commonly used to find energy minima. Fully variable box shape $NPT$ simulations were performed on system of only 2-6 particles. By construction, the final configuration of such a simulation is a crystal, where the unit cell is the simulation box. One cycle of such a simulation consists of the following steps: We start at a pressure of $10 k_B T/\sigma^3$. Subsequently, we run a series of simulations, where the pressure increases by a factor of ten each run: $P\sigma^3/k_B T=10,100,\ldots, 10^6$. At the highest pressure ($10^6 k_B T/\sigma^3$) we measure the density and angular order parameters, $S_1\equiv\|\langle \mathbf{u}_i\rangle\|$ and $S_2\equiv \lambda_2$, where $\lambda_2$ is the highest eigenvalue of the matrix whose components are $Q_{\alpha_\beta}=\frac{3}{2} \langle u_{i\alpha} u_{i\beta} \rangle - \frac{1}{2}\delta_{\alpha\beta}$, where $\alpha,\beta=x,y,z$. We store the density if it is the highest density found so far for these values of $S_1$ and $S_2$. We ran 1000 of such cycles for each aspect ratio, which is enough to visit each crystal phase multiple times. After completing the simulations, we tried to determine the lattice parameters of the resulting crystal by hand. Although this last step is not necessary, it is convenient to have analytical expressions for the lattice vectors and the density. The pressure annealing runs were performed for $D/\sigma=0.1,0.15,\ldots,0.5$. For many of the crystals, we were not able to find analytical expressions for the lattice parameters. For these crystals, we obtain the densities of the close packed crystals for intermittent values of $D$ by averaging the density in single simulation runs at a pressure of $10^6 k_BT/\sigma^3$. The initial configurations for the value of $D$ of interest were obtained from the final configurations of the pressure annealing simulations for another value of $D$ by one of the following two methods, depending on whether we needed to decrease or increase $D$: When decreasing $L$ no overlaps are created so the final configuration of the simulation for the previous value of $L$ can be used as initial configuration. On the other hand, increasing $L$ results in an overlap, which is removed by scaling the system uniformly. Subsequently, the pressure is stepwise increased from 1000$k_BT/\sigma^3$ to $10^6 k_BT/\sigma^3$, by multiplying by 10 each step. ### Free energies We calculate the free energy of the various crystal phases by thermodynamic integration using the Einstein crystal as a reference state [@FrenkelSmit]. The Einstein integration scheme that we employ here is similar to the one that was used to calculate the free energies of crystals of dumbbells in Ref. [@dumbbell_article]. We briefly sketch the integration scheme here and discuss the modifications that we applied. We couple both the positions and the direction of the particles with a coupling strength $\lambda$, such that for $\lambda\rightarrow \infty$, the particles are in a perfect crystalline configuration. First, we integrate $\partial F/\partial\lambda$ over $\lambda$ from zero to a large but finite value for $\lambda$. Subsequently we replace the hard-core particle–particle interaction potential by a soft interaction, where we can tune the softness of the potential by the interaction strength $\gamma$. We integrate over $\partial F/\partial \gamma$ from a system with essentially hard-core interaction (high $\gamma=\gamma_\text{max}$), to an ideal Einstein crystal ($\gamma=0$). Some minor alterations to the scheme of Ref. [@dumbbell_article] were introduced, which were necessary, because of the different shape of the particle. For the coupling of the orientation of bowl $i$, *i.e.*, $\mathbf{u}_i$, to an aligning field, we have to take into account that the bowls have no up down symmetry, while the dumbbells are symmetric under $\mathbf{u}_i\rightarrow -\mathbf{u}_i$. The potential energy function that achieves the usual harmonic coupling of the particles to their lattice positions, as well as the new angular coupling, reads: $$\begin{gathered} \beta U({\bf r}^N,{\bf u}^N;\lambda) =\\ \lambda \sum_{i=1}^{N} ({\bf r}_i-{\bf r}_{0,i})^2/\sigma^2 + \sum_{i=1}^N \lambda (1-\cos(\theta_{i0})) , \label{eqn:einstein}\end{gathered}$$ where ${\bf r}_i$ and ${\bf u}_i$ denote, respectively, the center-of-mass position and orientation of bowl $i$ and ${\bf r}_{0,i}$ the lattice site of particle $i$, $\theta_{i0}$ is the angle between $\mathbf{u}_i$ and the ideal tilt vector of particle $i$, and $\beta=1/k_BT$. The Helmholtz free energy [@dumbbell_article] of the noninteracting Einstein crystal is modified accordingly, but the only modification is the integral over the angular coordinates: $$J(\lambda)=\int_{-1}^1 e^{\lambda (x-1)} dx=\frac{1-e^{-2\lambda}}{\lambda}.$$ Although the shape of the bowls is more complex than that of the dumbbell, we can still use a rather simple form for the pairwise soft potential interaction: $$\beta U_{\mathrm{soft}}({\bf r}^N,{\bf u}^N;\gamma)=\sum_{i<j} \beta \varphi({\bf r}_{i}-{\bf r}_{j},\mathbf{u}_i,\mathbf{u}_j,\gamma)$$ with $$\begin{gathered} \beta\varphi({\bf r}_{j}-{\bf r}_{i},\mathbf{u}_i,\mathbf{u}_j,\gamma) =\\ \left\{\begin{array}{cc} \gamma (1-A (r_{ij}'/\sigma_\text{max})^2) & \text{if $i$ and $j$ overlap}\\ 0 & \mathrm{otherwise}\end{array} \right. , \label{eqn:softpot_part}\end{gathered}$$ where $r_{ij}'\equiv | \mathbf{r}_j-\mathbf{r}_i+\frac{\sigma-D}{2}(\mathbf{u}_i-\mathbf{u}_j)|$ *i.e.* the distance between the “centers” of bowl $i$ and bowl $j$, $\sigma_\text{max}$ is the maximal $r_{ij}'$ for which the particles overlap: $\sigma_\text{max}^2=\sigma^2+(\sigma-D)^2$, $A$ is an adjustable parameter that is kept fixed during the simulation at a value $A=0.5$, and $\gamma$ is the integration parameter. It was shown in Ref. [@Fortini_soft_pot] that in order to minimize the error and maximize the efficiency of the free energy calculation, the potential must decrease as a function of $r$ and must exhibit a discontinuity at $r$ such that both the amount of overlap and the number of overlaps decrease upon increasing $\gamma$. Here, we have chosen this particular form of the potential because it can be evaluated very efficiently in a simulation, although it does not describe the amount of overlap between bowls $i$ and $j$ very accurately. We checked that adding a term that tries to describe the angular behavior of the amount of overlap does not significantly change our results of the free energy calculations. Also, we checked that by employing the usual Einstein integration method (*i.e.* only hard-core interactions) at a relatively low density we obtained the same result as by using the method of Fortini [*et al.*]{}[@Fortini_soft_pot]. Finally, we set the maximum interaction strength $\gamma_\text{max}$ to 200. We perform variable box shape NPT simulations [@Parrinello] to obtain the equation of state for varying $D$. In these simulations not only the edge length changes, but also the angles between the edges are allowed to change. We employ the averaged configurations in the Einstein crystal thermodynamic integration. We calculate the free energy as a function of density by integrating the EOS from a reference density to the density of interest: $$F(\rho_1^*)= F(\rho_0) +\int_{\rho_0}^{\rho_1} d\rho \left\langle \frac{N P(\rho)}{\rho^2}\right\rangle \label{eqn:int_eos}$$ Results ======= ![ The final configuration obtained from simulations at $P\sigma^3/k_B T=50$ and $D=0.3\sigma$ The colors denote different stacks.\[fig:snapshots\_worms\]](worms2.png){width="45.00000%"} Stacks ------ We perform standard Monte Carlo simulations in the isobaric-isothermal ensemble (NPT). Fig. \[fig:snapshots\_worms\] shows a typical configuration of bowl-shaped particles with $D = 0.3~\sigma$ at $P\sigma^3/k_BT=50$, displaying stacking behavior typical for the worm-like phase. ![The equation of state for bowl-shape particles with $D=0.1\sigma$, reduced pressure $P^*=\beta P\sigma^3$ (left axis), and the reduced compressibility $\frac{1}{\rho}\,\frac{\partial \rho}{\partial P^*}$ on a log scale (right axis) as a function of packing fraction $\phi$. The points are data obtained from $NPT$ simulations. The solid line is a fit to the pressure; the dashed line is the corresponding reduced compressibility, $\frac{1}{\rho}\big(\frac{\partial \mathsf{fit}(\rho)}{\partial \rho}\big)^{-1}$. \[fig:eos\]](compr_lab){width="49.00000%"} The equation of state (EOS) of the fluid is somewhat peculiar: the pressure as a function of density is not always convex for all densities, although the compressibility does decrease monotonically with packing fraction $\phi$ for $D=0.1\sigma$, see Fig. \[fig:eos\], where the packing fraction is defined as $\phi=\frac{\pi D}{4}(\sigma^2-D\sigma+\frac{2}{3}D^2)N/V$. This behavior persist for all $D\leq0.2\sigma$, but for $D\geq 0.25\sigma$ the pressure is always convex. We investigate the origin of these peculiarities using $g_c(z)$, the positional correlation function along the director of a particle, which includes only the particles in a column around a particle, as defined in Eq. (\[eqn:gcz\]). ![The pair correlation function, $g_c(z)$, of a fluid of bowl-shaped particles with $D=0.2\sigma$ as a function of the dimensionless inter-particle distance $z/\sigma$ along the axis of a reference bowl for various reduced pressures $P^*\equiv\beta P\sigma^3$. Only particles within a cylinder of diameter $\sigma$ around the bowl are considered, as indicated by the subscript ‘c’. We show typical two-particle configurations that contribute to $g_c(z)$ for $z/\sigma=-0.5,-0.2,0.2,0.4$ and $1$, where the filled bowls denote the reference particle, and the open bowls with thick outlines denote the other particle. \[fig:gcz\]](rdf_c_exp){width="49.00000%"} As can be seen from $g_c(z)$ in Fig. \[fig:gcz\], the structure of the fluid changes dramatically as the pressure is increased. At $P^*\equiv\beta P\sigma^3 =1$, the correlation function is typical for a low density isotropic fluid of hemispherical particles; no effect of the dent of the particles is found at low densities. The only peculiar feature of $g_c(z)$ for $P^*=1$ is that it is not symmetric around zero, but this is caused by our choice of reference point on the particle (see Fig. \[fig:particles\]b), which is located below the particle if the particle points upwards. In contrast, at $P^*=10$ $g_c(z)$ already shows strong structural correlations. Most noteworthy is the peak at $z=D$, that shows that the fluid is forming short stacks of aligned particles. Also, note that the value of $g_c(z)$ is nonzero around $z=0$. This is caused by pairs of bowls that align anti-parallel and form a sphere-like object, as depicted in Fig. \[fig:gcz\]. Finally, at $P^*=50$ and higher, long worm-like stacks are fully formed and $g_c(z)$ shows multiple peaks at $z=D n$ for both positive and negative integer values of $n$. Furthermore, at these pressures, there are no sphere-like pairs, as can be observed from the value of $g_c(0)$. The formation of stacks explains the peculiar behavior of the pressure: At low densities, the bowls rotate freely, which means that the pressure will be dominated by the rotationally averaged excluded volume. The excluded volume of two particles that are not aligned is nonzero, even for $D=0$, and gives rise to the convex pressure which is typical for repulsive particles. As the density increases and the bowls start to form stacks, the available volume increases, and the pressure increases less than expected, which can even cause the EOS to be concave. At even higher densities the worm-like stacks are fully formed, and the pressure is again a convex function of density for $D>0$, dominated by the excluded volume of locally aligned bowls. The excluded volume of completely aligned infinitely thin bowls is zero, and, therefore, the pressure increases almost linearly with density for $D=0$ when the stacks are fully formed. ![The probability, $\mathcal{P}_\text{stack}(n)$, to find a particle in a stack of size $n$ for $D/\sigma=0.2,0.3$ and $0.4$ and $P\sigma^3/k_BT=50$ \[fig:clust\]](clusters){width="49.00000%"} To quantify the length of the stacks we calculated the stack distribution as shown in Fig. \[fig:clust\]. As can be seen from the figure, the length of the stacks is strongly dependent on $D/\sigma$. However, we have found that above a certain threshold pressure the distribution of stacks is nearly independent of pressure. We investigated whether the worm-like stacks could spontaneously reorient to form a columnar phase. We increased the pressure in small steps of 1 $k_BT/\sigma^3$ from well below the fluid–columnar transition to very high pressures, where the system was essentially jammed. At each pressure, we ran the simulation for $4\cdot 10^6$ Monte Carlo cycles, where a cycle consists of $N$ particle and volume moves. These simulations show that the bowls with a thickness $D\geq 0.25 \sigma$ always remained arrested in the worm-like phase, which is similar to the experimental observations [@Marechal2010bowls]. However, for $D/\sigma=0.1$ and $0.2$, we find that the system eventually transforms into a columnar phase in the simulations (see Fig. \[fig:snapshot\_col\]). This might be explained by the fact that the isotropic-to-columnar transition occurs at lower packing fractions for deeper bowls (smaller $D$), which facilitates the rearrangements of the particles into stacks and the alignment of the stacks into the columnar phase. ![ The final configuration of a simulation of bowls with $L=0.1D$ at $P\sigma^3/k_B T=38$. The gray values denote different columns. \[fig:snapshot\_col\]](columnar){width="45.00000%"} Packing ------- ![The various crystal phases that were considered as possible stable structures. Five of these were found using the pressure annealing method: X, IX, B, IB and IX’. X, IX, B and IB are densely packed structures for $D \lesssim 0.5\,\sigma$ and fcc${}^2$ and IX’ are densely packed crystal structures for (nearly) hemispherical bowls ($D \simeq 0.5\,\sigma$). \[fig:crystals\]](crystals6){width="49.00000%"} We found six candidate crystal structures, denoted X,IX,IX’,B,IB and fcc${}^2$, using the pressure annealing method. Snapshots of a few unit cells of these crystal phases are shown in Fig. \[fig:crystals\]. We will describe these crystal structures using the order parameters $S_1$, that measures alignment of the particles, and the nematic order parameter ($S_2$), that is nonzero for both parallel and anti-parallel configurations. Crystal structure X has $S_1\simeq 1$ and $S_2\simeq 1$, and the particles are stacked head to toe in columns. The lattice vectors are $$\begin{array}{c} \displaystyle \mathbf{a_1}=\sigma \hat{x} \qquad \mathbf{a_2}=D \hat{z} \\[1.0em] \displaystyle \mathbf{a_3}=\frac{\sigma}{2}\hat{x} + \frac{1}{2} \sqrt{\sigma^2-D^2+2\sigma \sqrt{\sigma^2-D^2} }\;\hat{y}+\frac{D}{2} \hat{z}, \end{array}$$ and the density is $$\rho\sigma^3=\left[\frac{D \sigma}{2} \sqrt{\sigma^2-D^2+2\sigma \sqrt{\sigma^2-D^2} }\right]^{-1}.$$ The order parameters of the second crystal structure, are $S_1\simeq 0$ and $S_2\simeq 1$, which is caused by the fact that half of the particles point upwards, and the other half downwards. Further investigation shows that there are two phases with $S_2 \simeq 1$ and $S_1 \simeq0$: one at low $D$ (IX) and one at $D\simeq \sigma/2$ (IX’). The structure within the columns of the first (IX) of these two structures is the same as for the X structure, but one half of these columns are upside down, like in the inverted columnar phase (in fact, the IX crystal melts into the inverted columnar phase). The lattice vectors of crystal structure IX are $$\begin{array}{c} \displaystyle \mathbf{a_1}=\sigma \hat{x} \qquad \mathbf{a_2}=D \hat{z} \\[1.0em] \displaystyle \mathbf{a_3}=\frac{\sigma}{2}\hat{x} + \frac{1}{2} \sqrt{3\sigma^2-4 D^2}\;\hat{y}, \end{array}\label{eqn:IXa}$$ and the density is $$\rho\sigma^3=\left[\frac{D \sigma}{2} \sqrt{3\sigma^2-4 D^2}\right]^{-1}.\label{eqn:IXd}$$ The columns in the IX crystal are arranged in such a way that the rims of the bowls can interdigitate. The IX’ crystal can be obtained from the IX phase at $D=\sigma/2$ by shifting every other layer by some distance perpendicular to the columns, such that the particles in these layers fit into the gaps in the layers below or above. In this way a higher density than Eq. (\[eqn:IXd\]) is achieved. The columns of the third crystal phase (B) resemble braids with alternating tilt direction of the particles within each column. Because of this tilt $S_1$ and $S_2$ have values between 0 and 1, that depend on $D$. Furthermore, the inverted braids structure (IB), that has $0<S_2<1$ and $S_1=0$, can be obtained by flipping one half of the columns of the braid-like phase (B) upside down. These braid-like columns piece together in such a way that the particles are interdigitated. In other words, this phase is related to the B phase in exactly the same way as the IX phase is related to the X phase. ![Packing diagram: maximum packing fraction ($\phi$) of various crystal phases as a function of the thickness ($D$) of the bowls. The points are the results of the pressure annealing simulations. The thin dot-dashed lines are obtained from the pressure annealing results by slowly increasing or decreasing $D$ as described in Sec. \[sec:cryst\], except for the IX phase (thin dashed line with open squares) and the X phase (thin solid line with filled squares), for which the packing fraction can be expressed analytically. The thick lines denote the packing fractions of the perfect hexagonal columnar phase (col) and the paired fcc phase (fcc${}^2$). Any points that lie below these lines are expected to be thermodynamically unstable (see text). \[fig:packing\]](packing){width="49.00000%"} Finally, in the paired face-centered-cubic (fcc${}^2$) phase, pairs of hemispheres form sphere-like objects that can rotate freely and that are located at the lattice positions of an fcc crystal. The density at close packing is $2\sqrt{2}/\sigma^3$, *i.e.* twice the density of fcc. In Fig. \[fig:packing\] the results of the pressure annealing method are shown, along with the known packing fraction of the perfect hexagonal columnar phase (col). Since the columnar phase has positional degrees of freedom and the fcc${}^2$ phase has rotational degrees of freedom, we expect these phases to have a higher entropy (lower free energy) than any crystal phase with the same or lower maximum packing fraction whose degrees of freedom have all been frozen out. Therefore, any crystal structure with a packing fraction below the thick lines in Fig. \[fig:packing\] is most likely thermodynamically unstable. At first, we were unable to find the fcc${}^2$ using the pressure annealing method as described in Sec. \[sec:cryst\]. However, if we increase the pressure slowly to 100$k_BT/\sigma^3$ in simulations of 12 particles, we did observe the fcc${}^2$ phase for hemispherical particles ($D=\sigma/2$). In these simulations at finite pressure, it is important to constrain the length of all box vectors such that they remain larger than say $1.5\sigma$. Otherwise the box will become extremely elongated, such that the particles can interact primarily with their own images. When a particle interacts with it is neighbors, the Gibbs free energy $G=F+PV$ decreases, because the volume decreases without any decrease in entropy due to restricted translational motion (if a particle moves, its image moves as well, so a particle translation will never cause overlap of the particle with its image). The decrease in Gibbs free energy is of course an extreme finite size effect, which should be avoided if we wish to predict the equilibrium phase behavior. For the pressure annealing simulations at very high pressures, these effects are not important, because the entropy term in the Gibbs free energy is small compared to $PV$. We did not attempt to find the columnar phase using the modified pressure annealing method, as we were only interested in finding candidate crystal structures. Furthermore, the columnar phase was already found in more standard simulations with a larger number of particles. Free energies ------------- In order to determine the regions of the stability of the fluid, the columnar phase and the six crystal phases, we calculated the free energies of all phases as explained in the Methods section. The results of the reference free energy calculations are shown in Tbls. \[tbl:col\_fs\] and \[tbl:fs\]. We find that the columnar phase with all the particles pointing in the same direction is more stable than the inverted columnar phase, where half of the columns are upside down. However, the free energy difference between the two phases is only $0.013\pm 0.002 k_B T$ per particle at $\phi=0.5193$ and $D=0.3\sigma$. Based on this small free energy difference we do not expect polar ordering to occur spontaneously. Similar conclusions, based on direct simulations, were already drawn in Ref. [@simulation-bowls]. ![Dimensionless free energies $\beta F\, \sigma^3/V$ for hard bowls with $L=0.3\sigma$ and the fluid-columnar, columnar-IX and IX-IB coexistences, which were calculated using common tangent constructions. The columnar phases is denoted “col”. The irrelevant free energy offset is defined in such a way that the free energy of the ideal gas reads $\beta F/V=\rho (\log(\rho\sigma^3)-1)$. The free energies of the various phases are so close, that they are almost indistinguishable. \[fig:f\_of\_rho\]](f_cumb){width="49.00000%"} The densely-packed crystal structures in Fig. \[fig:crystals\] at $D \lesssim 0.3$, the worm-like fluid phase (Fig. \[fig:snapshots\_worms\]) and the columnar phase (Fig. \[fig:snapshot\_col\]) show striking similarity in the local structure: in all these phases the bowls are stacked on top of each other, such that (part of) one bowl fits into the dent of another bowl. As a result, the free energies and pressures of the various phases, are often almost indistinguishable near coexistence. For this reason it was sometimes difficult to determine the coexistence densities for $D<0.3\sigma$. Exemplary free energy curves for the various stable phases consisting of bowls with $D=0.3\sigma$ are shown in Fig. \[fig:f\_of\_rho\]. Phase diagram ------------- In Fig. \[fig:ph-dia\_shells\], we show the phase diagram in the packing fraction $\phi$ - thickness $D/\sigma$ representation. The packing fraction is defined as $\phi=\frac{\pi D}{4}(\sigma^2-D\sigma+\frac{2}{3}D^2)N/V$. For $D/\sigma\leq0.3$, we find an isotropic-to-columnar phase transition at intermediate densities, which resembles the phase diagram of thin hard discs [@Veerman_Frenkel]. However, the fluid-columnar-crystal triple point for discs is at a thickness-to-diameter ratio of about $L/\sigma \sim 0.2-0.3$, while in our case the triple point is at about $D/\sigma \sim 0.3-0.4$. The shape of the bowls stabilizes the columnar phase compared to the fluid and the crystal phase. We find four stable crystal phases IX, IB, IX’ and fcc${}^2$, while we had six candidate crystals. The two phases that were not stable are the X and B crystals, which are very similar to the stable IX and IB crystals respectively, except that X and B have considerable lower close packing densities. Therefore, one could have expected these phases to be unstable. On the other hand, we observe from the phase diagram, that IX is stable at intermediate densities for $0.25\sigma < D< 0.45\sigma$, while IB packs better than IX. In other words, stability can not be inferred from small differences in packing densities. ![Phase diagram in the packing fraction ($\phi$) versus thickness ($D$) representation. The light gray areas are coexistence areas, while the state points in the dark gray area are inaccessible since they lie above the close packing line. IX, IB, IX’ and fcc${}^2$ denote the crystals as shown in Fig. \[fig:crystals\], “F” is the fluid and “col” is the columnar phase. The lines are a guide to the eye. Worm-like stacks were found in the area marked “worms” bounded from below by the dashed line. This line denotes the probability to find a particle in a cluster that consists of more than two particles, $\mathcal{P}_\text{stack}(n>2)=1/2$. \[fig:ph-dia\_shells\]](ph-dia){width="49.00000%"} [0.49]{}[@\*[7]{}[c@]{}]{} $D/\sigma$ & phase 1 & phase 2 & $\rho_1\sigma^3$ & $\rho_2\sigma^3$ & $\beta P \sigma^3$ & $\mu^*$\ 0 & fluid & col & 4.083 & 4.824 & 26.11 & 15.22\ \ [0.49]{}[@\*[7]{}[c@]{}]{} $D/\sigma$ & phase 1 & phase 2 & $\phi_1$ & $\phi_2$ & $\beta P d^3$ & $\mu^*$\ 0.1 & fluid & col & 0.2778 & 0.3297 & 26.35 & 15.59\ 0.1 & col & IX & 0.8095 & 0.8104 & $2.7\!\cdot\!10^3$ & -\ 0.2 & fluid & col & 0.4096 & 0.4688 & 27.23 & 16.68\ 0.2 & col & IX & 0.7021 & 0.7108 & 325 & -\ 0.3 & fluid & col & 0.5286 & 0.5472 & 49.52 & 26.13\ 0.3 & col & IX & 0.6864 & 0.6944 & 281.4 & 91.03\ 0.3 & IX & IB & 0.6117 & 0.6226 & 110.9 & 44.92\ 0.4 & fluid & IB & 0.6098 & 0.6455 & 105.9 & 51.06\ 0.45 & fluid & IB & 0.6026 & 0.6545 & 87.92 & 46.90\ 0.5 & fluid & fcc${}^2$ & 0.4878 & 0.5383 & 28.34 & 22.10\ 0.5 & fcc${}^2$ & IX’ & 0.6870 & 0.7278 & 139.2 & 67.36\ Almost all coexistence densities were calculated by employing the common tangent construction to the free energy curves, except for the col–IX coexistence at $D=0.1\sigma$ and $0.2\sigma$. At these values of $D$ the transition occurs at very high pressures, while the free energy of the columnar phase is calculated at the fluid–col transition, which occurs at a low pressure. To get a value for the free energy of the columnar phase we would have to integrate the equation of state up to these high pressures, accumulating integration errors. Furthermore, we expect the coexistence to be rather thin, which would further complicate the calculation. So, instead we just ran long variable box shape $NPT$ simulations to see at which pressure the IX phase melts into the inverted columnar phase. As the free energy difference between the inverted columnar phase and the columnar phase is small, we assume that this is the coexistence pressure for the col–IX transition, although technically it is only a lower bound. The density of the columnar phase at this pressure is determined using a local fit of the equation of state. All coexistences are tabulated in Tbl. \[tbl:coexes\]. We draw a tentative line in the phase diagram to mark the transition from a structureless fluid to a worm-like fluid *i.e.* a fluid with many stacks. In a dense but structureless fluid, stacks of size 2 are quite probable, but larger stacks occur far less frequently. We calculate the probability to find a particle in a stack that contains more than 2 particles $\mathcal{P}_\text{stack}(n>2)=1-\mathcal{P}_\text{stack}(1)-\mathcal{P}_\text{stack}(2)$ and define the worm-like phase by the criterion $\mathcal{P}_\text{stack}(n>2)\geq 1/2$ in Fig. \[fig:ph-dia\_shells\]. We do not imply that the transition to the worm-like phase is a true thermodynamic phase transition; the transition is rather continuous. The type of stacks in the fluid changes from worm-like for $D=0.3\sigma$ to something resembling the columns in the braid-like crystals B and IB (see Fig. \[fig:crystals\]) for $D=0.4\sigma$. Therefore, the region of stability worm-like phase was ended at $D=0.35\sigma$, where there are similar amounts of braid-like and worm-like stacks. Summary and discussion ====================== We have studied the phase behavior of hard bowls in Monte Carlo simulations. We find that the bowls have a strong tendency to form stacks, but the stacks are bent and not aligned. We measured the equation of state and the compressibility in Monte Carlo $NPT$ simulations. The pressure we obtained from these simulations is concave for some range of densities for deep bowls. This is due to the increase in free volume when large stacks form. Using $g_c(z)$, the pair correlation function along the direction vector, we showed that the concavity of the pressure coincides with a dramatic change in structure from a homogeneous fluid to the worm-like fluid. We measured the three-dimensional stack length distribution in the simulations. When the pressure is increased slowly, the deep bowls spontaneously order into a columnar phase in our simulations. This poses severe restrictions on the thickness of future bowl-like mesogens (molecular or colloidal), which are designed to easily order into a globally aligned lyotropic columnar phase. We determined the phase diagram using free energy calculations for a particle shape ranging from an infinitely thin bowl to a solid hemisphere. We find that the columnar phase is stable for $D\leq0.3\sigma$ at intermediate packing fractions. In addition, we show using free energy calculations that the stable columnar phase possesses polar order. However, the free energy penalty for flipping columns upside down is very small, which makes it hard to achieve complete polar ordering in a spontaneously formed columnar phase of bowls. Acknowledgments {#acknowledgments .unnumbered} =============== We thank Rob Kortschot, Ahmet Demirörs and Arnout Imhof for useful discussions. Financial support is acknowledged from an NWO-VICI grant and from the High Potential Programme of Utrecht University. Overlap algorithm ================= The overlap algorithm for our bowls checks whether the surfaces of two bowls intersect. Fig. \[fig:particles\] shows that the surface of the bowl consists of two parts. Part $p$ of the surface contains the part of the surface of the sphere of radius $R_p$, within an angle $\theta_p$ from the $z$-axis, where $p=1$ denotes the smaller sphere and the larger sphere is labeled $p=2$. We set $\theta_1=\pi/2$, to get a hemispherical outer surface. The edges of both surfaces have to coincide, such that our particles have a closed surface. Using this restriction $L$, $\theta_2$ and $R_2$ can all be expressed in terms of the radius of the smaller sphere, $R_1$, and the thickness of the bowl $D$, in the following way: $$\begin{aligned} R_2 & = & R_1+\frac{D^2}{2(R_1-D)}\\ \theta_2 & = & \arcsin(R_1/R_2)\\ L & = & R_2 \cos(\theta_2).\end{aligned}$$ Overlap occurs if either of the two parts of the surface of a bowl overlaps with either of the two parts of another bowl. So we have to check four pairs of infinitely thin (and not necessarily hemispherical) bowls, labeled $i$ and $j$, for overlap. The algorithm for two such surfaces that are equal in shape was already implemented by He and Siders [@He1990UFOs] as part of their overlap algorithm for their “UFO” particles, which are defined as the intersection between two spheres. An equivalent overlap algorithm was used by Cinacchi and Duijneveldt [@Cinacchi2010] to simulate infinitely thin contact lense-like particles, but the overlap algorithm was not described explicitly. We can not use one of these algorithms, since the two parts of the surface of our particle are unequal in shape. Therefore, we implemented a slightly different version of the overlap algorithm, which we describe in the remainder of this section. In our overlap algorithm, the existence of a overlap or intersection between two infinitely thin bowls is checked in three steps. - First, we check whether the full surfaces of the spheres intersect, i.e. $|R_i-R_j|<r_{ij}\equiv|\mathbf{r}_j-\mathbf{r}_i|<R_i+R_j$. If this intersection does not exist, there is no overlap, otherwise we proceed to the next step. - Secondly, we determine the intersection of the surface of each sphere with the other bowl. The intersection of bowl $i$ with the sphere of bowl $j$ exists if $$|\omega_{ij}+\zeta \phi_{ij}| < \theta_i \label{eqn:sphere_bowl}\\$$ for $\zeta=1$ or $-1$, where $$\begin{aligned} \cos(\phi_{ij})& =& \frac{R_i^2-R^2_j+r_{ij}^2}{2 r_{ij} R_i}\ \mathrm{and}\\ \cos(\omega_{ij})&=& \frac{\mathbf{u}_i \cdot \mathbf{r}_{ij}}{r_{ij}}.\end{aligned}$$ see Fig. \[fig:calc\_shells\]a. ![ The relevant lengths and angles which are used in the first and second steps (a) and in the third step (b) of the overlap algorithm. Shown are bowl $i$ and (part of) the sphere of bowl $j$ (a), the arcs of $i$ and $j$ and the circular intersection of the spheres of $i$ and $j$ (b). In (a) $\mathbf{r}_{ij}$ lies in the plane, while the plane of view in (b) is perpendicular to $\mathbf{r}_{ij}$. In this case, the sphere of particle $j$ overlaps with bowl $i$, but the arcs do not overlap, so particle $i$ and particle $j$ do not overlap. \[fig:calc\_shells\]](calcs){width="49.00000%"} This intersection is an arc, which is part of the circle that is the intersection between the two spheres. If in fact this arc is a full circle and the other particle has a nonzero intersection, the particles overlap. This is the case when Eq. (\[eqn:sphere\_bowl\]) holds for $\zeta=1$ *and* $\zeta=-1$. If, on the contrary, either of the two arcs does not exist, there is no overlap. Otherwise, if both arcs exist, but neither of them is a full circle, proceed to the next step. - Finally, if the two arcs overlap there is overlap, otherwise the particles do not overlap. The arcs overlap if $$|\alpha_{ij}|<|\gamma_i|+|\gamma_j|,\label{eqn:arc_overlap}$$ where $$\begin{aligned} \cos(\alpha_{ij})&=&\frac{ \mathbf{n}_i^\perp \cdot \mathbf{n}_j^\perp}{|\mathbf{n}_i^\perp| |\mathbf{n}_j^\perp|} \\ \cos(\gamma_i)&=& \frac{\cos(\theta_i)-\cos(\phi_{ij})\cos(\omega_{ij})}{\sin(\phi_{ij})\sin(\omega_{ij})},\end{aligned}$$ where $\mathbf{n}_i^\perp=\mathbf{n}_i-(\mathbf{r}_{ij}\cdot\mathbf{n}_i)\mathbf{r}_{ij}/r^2_{ij}$ and the expressions for $\gamma_j$ and $\mathbf{n}_j^\perp$ are equal to the expressions for $\gamma_i$ and $\mathbf{n}_i^\perp$ with $i$ and $j$ interchanged. The arcs together with the relevant angles are drawn in Fig. \[fig:calc\_shells\]b. The inequalities (\[eqn:sphere\_bowl\]) and (\[eqn:arc\_overlap\]) are expressed in cosines and sines using some simple trigonometry. In this way no inverse cosines need to be calculated during the overlap algorithm. For $D=0.5\sigma$ the bottom surface is a disk rather than an infinitely thin bowl. So the overlap check consists of bowl–bowl, bowl–disc and disc–disc overlap checks. For brevity, we will not write down the bowl–disk overlap algorithm, but it can be implemented in a similar way as the algorithm for bowl–bowl overlap described above. The disk–disk overlap algorithm was already implemented by Eppenga and Frenkel [@Eppenga].

--- abstract: 'The ALHAMBRA survey aims to cover 4 square degrees using a system of 20 contiguous, equal width, medium-band filters spanning the range 3500 $\AA$ to 9700 $\AA$ plus the standard JHKs filters. Here we analyze deep near-IR number counts of one of our fields (ALH08) for which we have a relatively large area (0.5 square degrees) and faint photometry (J=22.4, H=21.3 and K=20.0 at the 50% of recovery efficiency for point-like sources). We find that the logarithmic gradient of the galaxy counts undergoes a distinct change to a flatter slope in each band: from 0.44 at $[17.0,18.5]$ to 0.34 at $[19.5,22.0]$ for the J band; for the H band 0.46 at $[15.5,18.0]$ to 0.36 at $[19.0,21.0]$, and in Ks the change is from 0.53 in the range $[15.0,17.0]$ to 0.33 in the interval $[18.0,20.0]$. These observations together with faint optical counts are used to constrain models that include density and luminosity evolution of the local type-dependent luminosity functions. Our models imply a decline in the space density of evolved early-type galaxies with increasing redshift, such that only 30$\%$ - 50$\%$ of the bulk of the present day red-ellipticals was already in place at $z\sim1$.' author: - 'D. Cristóbal-Hornillos,J. A. L. Aguerri, M. Moles, J. Perea, F. J. Castander, T. Broadhurst, E.  J. Alfaro, N. Benítez, J. Cabrera-Caño, J. Cepa, M. Cerviño, A. Fernández-Soto, R. M. González Delgado, C. Husillos, L. Infante , I. Márquez, V. J. Martínez, J. Masegosa, A. del Olmo, F. Prada, J. M. Quintana, and S. F. Sánchez' title: 'Near-IR Galaxy Counts and Evolution from the Wide-Field ALHAMBRA survey[^1]' --- Introduction ============ It is well understood that the stellar masses of galaxies are better examined with near-IR (NIR) observations compared to shorter wavelengths mainly because the near-IR light is relatively less affected by recent episodes of star formation and by internal dust extinction. Moreover, the K-corrections are also smaller and better constrained in the NIR and hence massive high redshift objects are relatively prominent in the NIR. Despite this relative insensitivity to luminosity evolution and the effects of dust, the hope of using the NIR counts to constrain the cosmological parameters has not proved feasible because evolution in the space density of galaxies was soon understood to be of comparable significance for the NIR counts as the cosmological curvature [@1992Natur.355...55B]. Disentangling the effects of cosmology from evolution is not straightforward even in the NIR, and now it has become more appropriate to turn the question around and make use of the impressive constraints on the cosmological parameters from WMAP [@2003ApJS..148..175S], and type Ia supernovae [@1998AJ....116.1009R; @1999ApJ...517..565P], in order to measure more carefully the rate of evolution . In addition, imaging in the NIR has progressed well with fully cryogenic wide-field imagers now available on several 4m class telescopes. We also have at our disposal now much better estimates of the luminosity functions of different classes of galaxies from the large local redshift surveys in particular the SDSS [@2003ApJ...592..819B; @2003AJ....125.1682N], or 2dFGRS [@2001MNRAS.326..255C]. The evolution of the luminosity functions has been addressed making use of spectroscopic redshift surveys. However, the results of those studies differ due to the limitations in terms of low statistic, or small fields probed which lead to uncertainties due to the large scale structure. In a mild (20-30%) evolution in the number density of massive objects since $z\sim1$ is found. Also a roughly constant number density for red galaxies to $z=0.8$ is found in [@2002ApJ...571..136I]. Using COMBO-17 photometric redshift information , and the rest-frame color bimodality at each redshift, [@2004ApJ...608..752B] with a sample of $\sim$25000 galaxies, concluded that the stellar mass in red-galaxies has increase in a factor 2-3 from $z\sim1$ to the present. Combining spectroscopic with photometric redshift data point to an increase in a factor of $\sim2.7$ for the density of red bulge-dominated galaxies between $z=1$ and $z=0.6$. [@2007ApJ...665..265F] found a different evolution since $z\sim1$ in number densities in the red and blue galaxy population, being constant for the blue galaxies, while the number density of the red galaxies increases in a factor 3. Wide field imaging with larger covered area, and greater numbers selected to uniform faint limits is complementary to the redshift surveys in examining statistical models proposed for evolution. In practical terms it is most useful to combine faint NIR counts with deep blue counts when examining models of evolution to contrast the effects of luminosity and density evolution which affect these two spectral ranges in different ways. In [@2000MNRAS.311..707M; @2001MNRAS.323..795M] the authors use non-evolving models with a higher $\phi^*$ normalization in the B-band even if this leads to an over-prediction of bright galaxies than is observed. pointed out that both the optical and NIR counts present an excess over the no-evolution models, finding passive evolution models more suitable to match the distributions. The authors emphasize nevertheless their disappointment with the fact that in the passive evolution models the faint number counts are dominated by early-type galaxies, whereas the real data show that spiral and Sd/Irr galaxies are the main contributors to the faint counts even in the K-band. A characteristic feature of the NIR galaxy counts reported in several works is the change of slope at $17 \leq Ks \leq 18$. This distinctive flattening is not observed in the B-band counts. This effect has been interpreted in terms of a change in the dominant galaxy population, becoming increasingly dominated by an intrinsically bluer population [@1993ApJ...415L...9G; @2006ApJ...639..644E]. In the model described in [@2003ApJ...595...71C] a delay in the formation of the bulk of the early-type galaxies to $z_{form}<2$ and the presence of a dwarf star-forming population are invoked to match the Ks-band counts. A similar dwarf star-forming population at $z>1$, that is not present at lower redshift was found compatible in [@1996Natur.383..236M; @2001MNRAS.323..795M] but that work uses a $q_0=0.5, \Lambda=0.0$ cosmology, requiring some revision. Alternatively, an increase of $\phi^*$ for late-type galaxies, driven via mergers, can produce similar results without introducing an ad hoc population that is unseen in the local LFs [@2006ApJ...639..644E]. In any case, a low $z_{form}\sim1.5$ for the ellipticals remains necessary to generate a significant decrease in the number of red galaxies and to account for the distinctive break in the NIR count slope at Ks$\sim$17.5. Here we use the NIR data from the first completed ALHAMBRA field, hereafter ALH08 (details of the project can be found in [@MOLES2008] and http://www.iaa.es/alhambra). The limiting magnitudes (at S/N=5 in an aperture diameter of 2$\times$FWHM) reached in the 3 NIR bands are in mean for the eight frames in ALH8: J=22.6, H=21.5 and Ks=20.1 with a 0.3 rms (Vega system), and the total area covered amounts to $\sim 0.5$ square degrees. The completed survey will cover 8 independent fields with a total area of 4 square degrees. The ALHAMBRA survey occupies a middle ground in terms of the product of depth and area in all three standard NIR filters. The bright end of the counts is well constrained by our relatively large area allowing a careful examination of the location and size of the break in the count-slopes in J,H and Ks at fainter magnitudes. We have paid special attention to S/G separation, which at the intermediate magnitude range, is effectively achieved using optical-NIR color indices by combining our ALH08 NIR data with the corresponding Sloan DR5 data. Unless specified otherwise, all magnitudes here are presented in the Vega system, and the favored cosmological model, with H$_{0} = 70$ , $\Omega_{M} = 0.3$, $\Omega_{\Lambda} = 0.7$ was adopted through this paper. Observing Strategy and Data Reduction ===================================== The ALHAMBRA survey is collecting data in 23 optical-NIR filters using the Calar Alto 3.5m telescope with the cameras LAICA for the 20 optical filters, and OMEGA2000 for the NIR, JHKs filters [@MOLES2008; @BENITEZ2008]. In this paper we discuss the galaxy number counts in the J, H and Ks bands computed in the completed ALH08 pointing. Due to the OMEGA2000 and LAICA geometries, two parallel strips of $\sim1$ degree $\times 0.25$ degrees are acquired in each of the eight ALHAMBRA fields. Each of these strips is covered by four OMEGA2000 pointings. Fig. \[Fig:f08p11\_KS\] shows the central part of the processed image for one such pointing in ALH08. The OMEGA2000 camera has a focal plane array of type HAWAII-2 by Rockwell with 2048 x 2048 pixels. The plate scale is 0.45 arcsec/pixel, giving a full field of view of $\sim$236 arcmin$^2$. The JHKs images were taken using a dither pattern of 20 positions, with single images time exposures of 80 s in J, 60 s in H and 46 s in Ks band (obtained using respectively 16, 20 and 23 software co-adds). The total exposure time was 5 ks in each of the three filters. Due to the high quantity of raw data images we have implemented a dedicated reduction pipeline to process the images. This code will be presented and discussed in a forthcoming technical paper. The use of the reduction pipeline guaranties the homogeneity of the process, and allows us to perform a first automatic analysis of the resulting images and to verify the quality of the products in real time. The magnitude at the 50% of recovery efficiency for point-like sources in the eight final images corresponds in average to J$=22.4\pm0.24$, H$=21.3\pm0.14$ and Ks$=20.0\pm0.13$ in the Vega system. The galaxy number counts have been computed in the high signal to noise region of the final images, covering a total of $\sim$1600 arcmin$^2$, or 0.45 sq deg. Flat-fielding and sky subtraction --------------------------------- Firstly, the individual images of each observing run were dark-subtracted, and flat-fielded by super-flat images constructed with the science images in each filter. In the case of NIR imaging it is specially important to remove properly the high sky level that is changing in short timescales. The sky structure of each image was removed with the XDIMSUM package [@1995ApJ...450..512S] with a sky image constructed with the median of the 6 images closest in acquisition time, which in the case of the J, H, Ks filters correspond to timescales of 480, 360, 276 s respectively. During this process cosmic ray masks for each individual image were also created. SExtractor was used with the preliminary sky subtracted images to compute the number of detected objects (above a given S/N and with ellipticity lower than a certain limit), and to make a robust estimate of the FWHM of each image. This information together with the sky level variation was used to automatically remove low S/N images, lying outside the survey requirements, and also to identify exposures presenting problems such as telescope trailing. Ghost and linear pattern masks ------------------------------ The readout of the detector array produces ghost images coming from bright stars. As those spurious effects are replicated in all the readout channels (separated by 128 pixels) of the detector’s quadrant where the real bright source is located, it was possible to mask them before image combination. Linear patterns produced by moving objects were located in the images using two different approaches: i) Objects with high ellipticity and pixel area were identified from the individual image SExtractor catalogs. ii) Linear patterns split in multiple spots and structures were located using the Hough transform. Linear patterns detected by any of the two methods were masked. Image combination ----------------- Using XDIMSUM, the images were combined masking the cosmic rays, bad pixels and linear patterns. Those preliminary co-added images were used to create object and bright cores masks. Object masks were used to cover the sources in the second iteration of the XDIMSUM sky subtraction procedure. Bright cores masks operate when constructing an improved version of the cosmic ray masks. At this point, the individual images have been dark current corrected, flat fielded and sky subtracted. Each individual image has an associated mask containing the bad pixels, cosmic rays, linear patterns and ghost images. To perform the final combination of the processed images we used SWARP [@2002ASPC..281..228B], a code that combines the images, correcting at the same time the geometrical distortions in the individual images using the information stored in WCS headers. The astrometric calibration of each individual image and the update of its WCS headers was done by the pipeline using an automatic module. During the SWARP combination the extinction variations among the images were also corrected. More technical details on the image combination are given in appendix \[Ap:Combination\]. Photometric calibration ----------------------- For the photometric calibration we used the 2MASS catalogs [@2003tmc..book.....C]. After having combined all the images of each pointing, the objects in common with the 2MASS catalogs were found and those with higher signal to noise ratio selected to compute a zero point offset. In Fig. \[Fig:2mass2\] an example of the photometric calibration for one of the pointings in ALH08 is shown. The histogram with the rms in the photometric calibration for the ALH08 pointings in the three NIR bands is shown in Fig. \[Fig:2mass3\]. The mean value for the rms is $0.028\pm0.006$ mags, and a mean of 36 calibration sources have been used in each frame. We have not found any appreciable color related trend. The calibrated images were inspected for the possible presence of ring patterns that would have been produced by pupil ghosts. We computed for each band, and for all the sources used in the calibration process of the different final frames, the difference between our photometry and the 2MASS photometry as a function of the radial distance to the nominal field center. We found a significant effect, over 0.02 mags only in the J-band images, as it is shown in Fig. \[Fig:radial\]. This effect was corrected by fitting the pupil ghost, using the mscpupil task in the IRAF MSCRED package [@2002adaa.conf..309V], and removing this pupil image from the flat-fields and individual images. The final resulting radial differences are shown for the J band in Fig. \[Fig:radial2\] where it clearly appears that all the systematics was corrected well below the $1\sigma$ level. Galaxy Number Counts ==================== The steps to compute robust counts to the 50% detection level of the images are similar to those described in [@2003ApJ...595...71C] and in [@2006ApJ...639..644E]. In the next section we detail how the best set of SExtractor parameters was estimated as a compromise between optimizing the depth at the 50% completeness level while keeping low the number of spurious sources. Completeness corrections ------------------------ In order to compute the corrections that should be applied to the faint part of the galaxy counts we have performed a set of Monte Carlo simulations where real sources from the images were injected back to the same science image at different positions. The completeness correction to be applied depends on the surface brightness profile of the source. To account for this we computed a different correction function for sources in three effective radius (Re) intervals. Those Re intervals in pixels for the simulations were chosen from the histogram of the Fig. \[Fig:rerec\] (top panel): $Re\le1.75$, $1.75 < Re \le 2.25$ and $Re > 2.25$. In Fig. \[Fig:rerec\] (bottom panel) it is shown how the Re recovered decreases as the magnitude goes fainter (see below for a description of how these values were obtained). For the simulations, bright sources were selected from the image in each Re interval. These sources were artificially dimmed to the 0.5 magnitude bin under study and injected back on the image randomly. In each iteration 40 sources were simulated, computing the recovering fraction and the robust mean (trimming the 10% on either side of the distribution) of the SExtractor desired output parameters differences between the dimmed input sources and the recovered ones. Using 9 of these iterations in each magnitude bin, we estimated the mean and rms of the recovered fraction, and SExtractor parameters differences over all the meaningful magnitude range. In Fig. \[Fig:completeness\] the completeness correction for the three Re ranges is shown for one of the pointings in the J band. It can be argued that with that method unrealistic pseudo-artificial sources could be produced. Whereas this could be true, the goal of this procedure is to parameterize the recovering efficiency on the basis of the source and image characteristics without any physical consideration, which will be implicitly taken into account when performing the corrections on the real data. As a validation of the procedure, the same simulations have been performed using real NICMOS F110W sources from the HDF-S Flanking Fields. The NICMOS images were resampled and convolved with a suitable Gaussian kernel to match the pixel scale and FWHM of the OMEGA2000 image under study. Initial sources were taken in the interval \[m-1,m+1\] (being m the magnitude under study), which produces a realistic $mag-R_e$ relation. As can be seen from the Fig. \[Fig:completeness\] the results in the completeness correction are the same that using dimmed sources from the image. In the top panel of Fig. \[Fig:profprof\] we show the depth to the 50% and 80% of recovering efficiency, computed using a linear interpolation in the magnitude vs. efficiency data. The figure indicates that a decreasing of the detection thresholds, in order to get a fainter level at the 50% of recovery efficiency limit, will not improve by much the limit at the 80% of completeness, which seems to reach a plateau. Moreover, as will be seen from the reliability plots, it produces a significant increase of detected spurious sources, as we explain below. Detection reliability --------------------- To accurately compute the galaxy number counts it is important to establish the reliability of the detections at the faint magnitude end. To find out the optimum way to evaluate that reliability we have studied the performance of three different methods. The first approach was to create artificial sky images with the same rms and background distribution than the real ones. The ratio of spurious versus real detections was computed running SExtractor over the science and the artificial sky images. We have also inspected the performance of the method used in [@2003ApJ...595...71C]. Basically half exposure time images were created from two complementary sets of the data. The detections were performed in the total time image and the source fluxes measured in the half time images using the SExtractor double image mode for the same automatic apertures. Those created sources showing a magnitude difference greater than $3\sigma$ were considered spurious. The last method and the one that, at the end, produces the best results, consisted in constructing sky images using a similar combination procedure that the used to create the science images: combining the unregistered processed images with subtracted background using an artificial dither pattern. The major difficulty here was to remove the extended sources that could bias the sky even when doing trimmed mean (discarding 20% of the pixels at each side). We have confirmed that SExtractor does locate these smooth deviations over the sky rms when filtering is used. To avoid this we multiplied those sky images by -1 and used them as real sky images. In the Fig. \[Fig:spuvsspu\] we show a comparison between the spurious rate at the 50% of detection efficiency computed over the ’real sky’ and the artificial sky. A good agreement between both methods is observed, indicating that the use of artificial images to compute the ratio of spurious to total detections is adequate. Being artificial images faster to construct we made use of them to estimate the detection reliability in each pointing. The bottom panel of Fig. \[Fig:profprof\] shows the magnitude reached at the 50% of detection efficiency versus the spurious to total ratio at the same magnitude bin. From the figure we can see that in order to reach a deep 50% detection limiting magnitude, maintaining at the same time the number of spurious detections below 20%, the optimum SExtractor DETECT\_THRESH-FILTER combinations are: DETECT\_THRESH=1.2 or 1.4 without filtering, or DETECT\_THRESH=0.8 using a filtering with a Gaussian kernel of size similar to the image FWHM. In the top panel of Fig. \[Fig:profprof\] it is shown that those combinations reach roughly the same magnitude at the 80% of recovery efficiency. We have decided to use the latter filter-DETECT\_THRESH combination because, as can be outlined from the bottom panel in Fig. \[Fig:magrec\], the differences between the input and the recovered AUTO magnitudes in the simulations are close to zero in all the magnitude range up to the magnitude at the 80% of recovery efficiency. Close to the magnitude at the 50% of completeness the recovered magnitude appears to be $\sim0.1-0.2$ mag brighter than in the previous bin. This effect could be due to the fact than only those sources that suffer from noise brightening were found by SExtractor and used to compute the input-output magnitude differences, producing a bias towards brighter recovered objects. Following those results, the number counts in the following sections will be computed and corrected up to the magnitude of 80% of recovery efficiency for point-like sources, avoiding any possible systematics due to a magnitude shift at the 50% completeness bin. As pointed before, the photometry of the sources was obtained using MAG\_AUTO. Simulations indicate no significant differences between simulated and recovered magnitudes (at the 80% recovery efficiency) for the used values of the SExtractor parameters FILTER and DETECT\_THRESH. Using the same kind of simulations we computed the rms of the recovered MAG\_AUTO values in each bin to characterize the photometric error, the results are shown in Fig. \[Fig:MagError\]. Star-galaxy separation ---------------------- In [@2003ApJ...595...71C] it was shown that fainter than Ks=17.0 the correction due to stars is $<0.06$ dex. However, given the lower Galactic latitude of the ALH08 field, a higher number of contaminating stars is expected. A correct star subtraction is relevant in the intermediate magnitude range. At bright magnitudes stars can be easily separated from galaxies using a compactness criteria. In contrast, at intermediate magnitudes the star/galaxy separation is more demanding because many galaxies are barely resolved with small apparent sizes compared with the FWHM and pixel resolution, as can be seen from the diagram shown in Fig. \[Fig:peakiso\]. The viability of star/galaxy separation using the SExtractor neural network has also been analyzed. For this purpose, using the same Monte Carlo method explained before, bright stars and galaxies from the images have been artificially dimmed to each magnitude bin and their SExtractor stellarity parameter recovered. In Fig. \[Fig:classstar1\] it clearly appears how, in one final frame with FWHM=1.1, the input and recovered CLASS\_STAR differences are less than 0.1 up to Ks=18.0. But, in the next bin Ks$>18.5$, the CLASS\_STAR of the dimmed stars and galaxies could lead to some misclassification. Nevertheless, as can be seen from the histogram in the bottom panel of Fig. \[Fig:classstar1\], a non negligible number of objects start to populate the range from 0.4 to 0.8 in CLASS\_STAR at magnitudes fainter than 17 in Ks, and the selection of the CLASS\_STAR cut off might bias the star counts estimates. An alternative way to perform the star/galaxy separation makes use simply of color-color diagrams. [@1997ApJ...476...12H] have established a reliable star/galaxy separation using the B-I vs I-K colors. Here we make use of the SDSS DR5 data for the ALH08 field to proceed with this separation using the g-r vs J-Ks colors shown in Fig. \[Fig:sgsep\]. The star counts are corrected by the ratio of the Sloan/Alhambra completeness factors in the corresponding filter shown in Fig. \[Fig:classstar\]. The star counts using Sloan-Alhambra colors were computed to magnitude 19.5,18.5 and 18.0 respectively in the J,H and Ks band, where the Sloan/Alhambra completeness is $>$0.5. For the fainter star counts we used a 0-slope extrapolation where the models of star counts in the Galaxy are flatter (see Fig. \[Fig:starcor\]). In Fig. \[Fig:classstar\] bottom panel, the correction to the $\log(N)$ galaxy counts is presented, showing that fainter than the magnitudes up to where the color-color separation could be performed the correction in the three bands is $<0.08$ dex. To check the validity of the star counts computed using the color-color approach, in Fig. \[Fig:starcor\] those are compared with the [@1996AA...305..125R] Galactic star counts models. The lines correspond to star counts in the Galaxy for galactic coordinates (l=100,b=-45), very close to the coordinates of the Alhambra-08 field (l=99, b=-44). From this figure it appears that it is possible to accurately remove the stars from the galaxy counts using the color-color diagram. This was the method we have finally used to do the star/galaxy separation. Finally is noteworthy to mention that, star/galaxy separation will be done more accurately when the photometry in the 23 Alhambra bands is available, as the system will provide a sort of low resolution spectroscopy for each object. Results ======= The corrected galaxy number counts have been computed for the ALH08 field in the three standard NIR filters in a consistent way in the sense that they have been estimated following the same scheme for the three bands. The number count data, computed and corrected up to the magnitude of 80% of recovery efficiency for point-like sources, are presented in the Tabs. \[Tab:cJ\], \[Tab:cH\] and \[Tab:cK\]. The error in the number counts for each pointings are the sum in quadrature of the rms in the estimation of the completeness corrections, with the contribution of Poisson noise and galaxy clustering calculated for each magnitude bin following eq. \[Eq:error\] [@1997ApJ...476...12H], that includes the angular correlation function. $$\label{Eq:error} \sigma_i^2= N_i(m)+5.3\left(\frac{r_0}{r_{\star}}\right)\Omega_i^{(1-\gamma)/2}N_i^2(m)$$ being $N_i$ the raw counts in the pointing, $r_0=7.1$ Mpc, $\gamma=1.77$ and $5\log(r_{\star})=m-M_{\star}-25$. $M_{\star}$ is set to -22.95,-23.69 and -23.93 for the J, H and Ks filters. The final uncertainty in the combined counts per square degree and magnitude is given by: $$\sigma(m)= \frac{\sqrt{\sum\sigma_i^2}}{\Omega\Delta m}$$ [llllll]{} 13.00 & 43.0 & 1.00 & $1.31^{+0.49}_{-1.31}$ & $1.30^{+0.73}_{-1.30}$ & 0.441\ 13.50 & 31.0 & 1.00 & $0.92^{+0.73}_{-0.92}$ & $0.95^{+0.96}_{-0.95}$ & 0.441\ 14.00 & 53.0 & 1.00 & $1.13^{+0.66}_{-1.13}$ & $1.08^{+0.90}_{-1.08}$ & 0.441\ 14.50 & 75.0 & 1.00 & $1.63^{+0.36}_{-1.63}$ & $1.64^{+0.61}_{-1.64}$ & 0.441\ 15.00 & 118.0 & 1.00 & $1.88^{+0.28}_{-1.07}$ & $1.89^{+0.49}_{-1.89}$ & 0.441\ 15.50 & 129.0 & 1.00 & $2.00^{+0.23}_{-0.54}$ & $2.00^{+0.41}_{-2.00}$ & 0.441\ 16.00 & 175.0 & 1.00 & $2.11^{+0.22}_{-0.47}$ & $2.11^{+0.34}_{-2.11}$ & 0.441\ 16.50 & 242.0 & 1.00 & $2.55^{+0.10}_{-0.13}$ & $2.56^{+0.24}_{-0.55}$ & 0.441\ 17.00 & 309.0 & 1.00 & $2.77^{+0.07}_{-0.08}$ & $2.77^{+0.15}_{-0.23}$ & 0.441\ 17.50 & 436.0 & 1.00 & $2.91^{+0.06}_{-0.07}$ & $2.91^{+0.09}_{-0.12}$ & 0.441\ 18.00 & 634.0 & 1.00 & $3.22^{+0.04}_{-0.04}$ & $3.22^{+0.11}_{-0.14}$ & 0.441\ 18.50 & 832.0 & 1.00 & $3.40^{+0.03}_{-0.03}$ & $3.40^{+0.09}_{-0.11}$ & 0.441\ 19.00 & 1128.0 & 1.07 & $3.63^{+0.02}_{-0.02}$ & $3.64^{+0.06}_{-0.07}$ & 0.441\ 19.50 & 1615.0 & 1.08 & $3.80^{+0.02}_{-0.02}$ & $3.80^{+0.05}_{-0.05}$ & 0.441\ 20.00 & 2303.0 & 1.09 & $3.99^{+0.06}_{-0.08}$ & $3.99^{+0.08}_{-0.10}$ & 0.441\ 20.50 & 3285.0 & 1.12 & $4.18^{+0.04}_{-0.05}$ & $4.18^{+0.06}_{-0.07}$ & 0.441\ 21.00 & 4455.0 & 1.17 & $4.34^{+0.03}_{-0.03}$ & $4.34^{+0.05}_{-0.05}$ & 0.441\ 21.50 & 5030.0 & 1.25 & $4.50^{+0.02}_{-0.03}$ & $4.50^{+0.03}_{-0.03}$ & 0.381\ 22.00 & 1739.0 & 1.52 & $4.67^{+0.03}_{-0.03}$ & $4.67^{+0.01}_{-0.02}$ & 0.109\ [llllll]{} 12.00 & 22.0 & 1.00 & $0.97^{+0.64}_{-0.97}$ & $0.98^{+0.98}_{-0.98}$ & 0.444\ 12.50 & 40.0 & 1.00 & $1.62^{+0.30}_{-1.62}$ & $1.60^{+0.47}_{-1.60}$ & 0.444\ 13.00 & 39.0 & 1.00 & $1.19^{+0.56}_{-1.19}$ & $1.19^{+0.79}_{-1.19}$ & 0.444\ 13.50 & 53.0 & 1.00 & $0.52^{+1.19}_{-0.52}$ & $0.55^{+1.44}_{-0.55}$ & 0.444\ 14.00 & 97.0 & 1.00 & $1.85^{+0.28}_{-0.95}$ & $1.86^{+0.49}_{-1.86}$ & 0.444\ 14.50 & 106.0 & 1.00 & $1.94^{+0.24}_{-0.61}$ & $1.94^{+0.54}_{-1.94}$ & 0.444\ 15.00 & 131.0 & 1.00 & $1.81^{+0.33}_{-1.81}$ & $1.81^{+0.59}_{-1.81}$ & 0.444\ 15.50 & 223.0 & 1.00 & $2.39^{+0.14}_{-0.21}$ & $2.39^{+0.25}_{-0.63}$ & 0.444\ 16.00 & 288.0 & 1.00 & $2.74^{+0.07}_{-0.09}$ & $2.74^{+0.16}_{-0.25}$ & 0.444\ 16.50 & 363.0 & 1.00 & $2.88^{+0.06}_{-0.07}$ & $2.88^{+0.12}_{-0.17}$ & 0.444\ 17.00 & 541.0 & 1.00 & $3.10^{+0.04}_{-0.05}$ & $3.10^{+0.06}_{-0.07}$ & 0.444\ 17.50 & 788.0 & 1.00 & $3.37^{+0.03}_{-0.03}$ & $3.37^{+0.10}_{-0.14}$ & 0.444\ 18.00 & 1074.0 & 1.00 & $3.56^{+0.02}_{-0.02}$ & $3.57^{+0.06}_{-0.07}$ & 0.444\ 18.50 & 1512.0 & 1.05 & $3.76^{+0.02}_{-0.02}$ & $3.76^{+0.06}_{-0.07}$ & 0.444\ 19.00 & 2110.0 & 1.05 & $3.94^{+0.07}_{-0.08}$ & $3.94^{+0.08}_{-0.10}$ & 0.444\ 19.50 & 3138.0 & 1.07 & $4.14^{+0.04}_{-0.05}$ & $4.14^{+0.05}_{-0.06}$ & 0.444\ 20.00 & 4456.0 & 1.12 & $4.32^{+0.03}_{-0.03}$ & $4.32^{+0.04}_{-0.05}$ & 0.444\ 20.50 & 5879.0 & 1.23 & $4.49^{+0.02}_{-0.02}$ & $4.49^{+0.04}_{-0.05}$ & 0.444\ 21.00 & 2578.0 & 1.48 & $4.65^{+0.02}_{-0.02}$ & $4.65^{+0.02}_{-0.02}$ & 0.165\ [llllll]{} 12.00 & 25.0 & 1.00 & $1.14^{+0.54}_{-1.14}$ & $1.13^{+0.74}_{-1.13}$ & 0.440\ 12.50 & 44.0 & 1.00 & $1.61^{+0.32}_{-1.61}$ & $1.60^{+0.52}_{-1.60}$ & 0.440\ 13.00 & 41.0 & 1.00 & $...$ & $...$ & 0.440\ 13.50 & 78.0 & 1.00 & $1.89^{+0.24}_{-0.55}$ & $1.90^{+0.32}_{-1.90}$ & 0.440\ 14.00 & 101.0 & 1.00 & $2.03^{+0.20}_{-0.39}$ & $2.03^{+0.47}_{-2.03}$ & 0.440\ 14.50 & 141.0 & 1.00 & $2.21^{+0.16}_{-0.26}$ & $2.22^{+0.32}_{-2.22}$ & 0.440\ 15.00 & 188.0 & 1.00 & $2.41^{+0.12}_{-0.17}$ & $2.41^{+0.26}_{-0.73}$ & 0.440\ 15.50 & 291.0 & 1.00 & $2.74^{+0.07}_{-0.09}$ & $2.74^{+0.14}_{-0.21}$ & 0.440\ 16.00 & 375.0 & 1.00 & $2.96^{+0.05}_{-0.06}$ & $2.96^{+0.11}_{-0.15}$ & 0.440\ 16.50 & 585.0 & 1.00 & $3.22^{+0.03}_{-0.04}$ & $3.22^{+0.08}_{-0.09}$ & 0.440\ 17.00 & 940.0 & 1.00 & $3.49^{+0.02}_{-0.02}$ & $3.49^{+0.06}_{-0.07}$ & 0.440\ 17.50 & 1307.0 & 1.03 & $3.67^{+0.02}_{-0.02}$ & $3.67^{+0.05}_{-0.06}$ & 0.440\ 18.00 & 1836.0 & 1.04 & $3.86^{+0.01}_{-0.02}$ & $3.86^{+0.06}_{-0.07}$ & 0.440\ 18.50 & 2614.0 & 1.05 & $4.04^{+0.05}_{-0.06}$ & $4.04^{+0.07}_{-0.08}$ & 0.440\ 19.00 & 3698.0 & 1.11 & $4.24^{+0.04}_{-0.04}$ & $4.24^{+0.06}_{-0.07}$ & 0.440\ 19.50 & 4477.0 & 1.37 & $4.42^{+0.03}_{-0.03}$ & $4.42^{+0.04}_{-0.04}$ & 0.440\ 20.00 & 574.0 & 1.55 & $4.50^{+0.04}_{-0.05}$ & $4.50^{+0.02}_{-0.02}$ & 0.054\ In the Figs. \[Fig:countsJ\], \[Fig:countsH\], and \[Fig:countsKS\] the Alhambra counts in the J,H, and Ks bands are plotted together with the computed number counts from other surveys. Let us first point out the general aspect of those counts, leaving more detailed considerations for the next section. Our corrected J band galaxy counts are in overall agreement with those computed in other works [@1999ApJ...520..469T; @2007MNRAS.378..429F], the bright end of the results presented by [@2001PASJ...53...25M], and the ELAIS-N2 data from [@2000ApJ...540..593V]. Nevertheless, our counts are lower than those given by [@1998ApJ...505...50B] at magnitudes $J>21.5$. The Alhambra counts in the H band are in good correspondence with the published data by [@2001AJ....121..598M; @2006MNRAS.371.1601F] and with the bright part of the data from , after applying an offset of -0.215 mag. This offset was calculated following the -0.065 calibration difference among Las Campanas Infrared Survey (LCIRS, [@2002ApJ...570...54C]) reported in , and the systematic of -0.28 magnitude difference between LCIRS and 2MASS magnitudes reported in [@2006MNRAS.371.1601F]. However, the faint end ($H>20$) of our data is significantly above the faint number count data from [@1998ApJ...503L..19Y] and [@2006MNRAS.370.1257M], obtained from NICMOS observations. Regarding the Ks filter, for which there are numerous number counts studies, our galaxy counts are in good agreement with most of the published data, as can be appreciated in the Fig. \[Fig:countsKS\]. The measured slopes ------------------- We have found, as in previous studies, that the slope of the galaxy number counts displays a clear change at Ks$\sim$17.3 (see Fig. \[Fig:slopes1\]). However, in the present work we also have found this change of slope in the J and H band galaxy counts at J$\sim$19.1 and H$\sim$ 18.2. The slopes of the bright end and faint end counts in the different bands were measured using the least squares method. The values that we have calculated and the magnitude ranges used for the fits are listed in Tab. \[Tab:slopes\]. The results show that the faint slopes for the three NIR filters are the same within $2\sigma$, whereas the bright slope is steeper in the Ks band. As is described in the fact that the photometric error increases at fainter magnitudes, and that the differential galaxy number counts also rises towards lower fluxes, lead to a stepper observed slope at the faint end. This effect is related with the Eddington bias [@1913MNRAS..74....5E]. To investigate if the computed magnitude errors could bias our slope estimates we have done the following study. First, in each filter we use the exponential grow function fit to the magnitude errors (Fig. \[Fig:MagError\]). Then we extended our computed number counts one magnitude fainter using the corresponding slope in Tab. \[Tab:slopes\], simulating a Gaussian decay for fainter magnitudes. The parameter $\sigma$ for the Gaussian is the value of the exponential fit to the error values for point-like sources at one magnitude fainter that the last bin given in Tabs. \[Tab:cJ\], \[Tab:cH\] and \[Tab:cK\] ($\sigma=0.35,0.41$ and 0.44 at respectively J=23.0, H=22.0 and Ks=21.0). Using the fitted magnitude errors for point-like objects ($\sigma(m)$), and assuming that real distribution is close to the extended number counts ($N(m)$), we simulate the bias produced by the photometric error on the observed counts, by convolving the $N(m)$ with $\sigma(m)$ using the eq. \[Eq:convol\]. The results indicate an increase in the slope less than 0.015 in the case of J and H filters, and 0.03 in the Ks band, meaning that the original distributions would have a faint end slope of 0.33,0.34 and 0.30 in the J,H, and Ks filters in the ranges given in Tab. \[Tab:slopes\]. $$\label{Eq:convol} N(m_{obs})=\int N(m)\frac{1}{\sqrt{2\pi \sigma}} e^{\frac{(m-m_{obs})^2}{2\sigma^2(m)}}dm$$ The increase of the slope is not observed when we simulate this bias directly over the observed count distribution. This is due to the fact that fainter than the 80% completeness magnitude bin there is a fast decrease of the number of detected sources, and that in this bin the typical photometric error ($\sigma_m=0.23-0.26$) for the dominant point-like sources, makes that the bias do not significantly affect to the previous bins. In this case the slopes after applying the bias are lower than the computed from the observed number counts ($0.33 \pm 0.01$, $0.33 \pm 0.02$, and $0.30 \pm 0.02$ in J,H and Ks). This result would suggest that the original distribution slopes at the faint end would be higher than the ones given in Tab. \[Tab:slopes\]. We have used the combined results from this paragraph and the previous one to increase the uncertainty in the slope, leaving the values computed directly from the observed number counts as satisfactory estimates of the real distribution slope at the faint end. [lllll]{} J & \[17.0,18.5\] & 0.44$\pm$0.04 & \[19.5,22.0\] & 0.34$\pm$0.01\ H & \[15.5,18.0\] & 0.46$\pm$0.02 & \[19.0,21.0\] & 0.36$\pm$0.02\ Ks & \[15.0,17.0\] & 0.53$\pm$0.02 & \[18.0,20.0\] & 0.33$\pm$0.03\ Our results for the Ks band are in good agreement with other K-band surveys which also report a similar change of slope in the galaxy counts in the range 17.0-18.0 . At the bright part our slopes are close to the values measured in (see Tab. \[Tab:surveysK\]), while other references point to a steeper bright slope . This could be due to the fact that the last authors could extend the power-law fit to brighter magnitudes due to their larger surveyed area, whereas our fit is closer to the magnitude where the break is found which would lead to a decrease in the slope if the break transition is smooth. In the faint part of the Ks counts a slope of $0.33$ found in this work is in agreement with . The value of the faint slope from the ALHAMBRA data is however steeper than the value reported in some surveys covering smaller areas, where the fitted range extends to fainter magnitudes, [@1993ApJ...415L...9G; @1997ApJ...475..445M; @2001PASJ...53...25M]. Here the differences might be due to cosmic variance. A larger than $2\sigma$ disagreement is found with the faint slope of [@2003ApJ...595...71C] fitted in the range \[17.5,19.5\], although their value increase to 0.29 when the fit interval is extended to Ks=21.0. give a higher value for the faint slope, however due to the brighter limiting magnitude of their number counts they could established a break or the beginning of a roll-over in the interval Ks=\[16.5,17.0\]. [llllllll]{} Gardner93 &5688 & 14.5 & \[10.0,16.0\] & 0.67 & \[18.0,23.0\] & 0.23 & K’\ Gardner93 &582 & 16.75 & — & — & — & — & —\ Gardner93 &167.7 & 18.75 & — & — & — & — & —\ Gardner93 &16.5 & 22.5 & — & — & — & — & —\ Glazebrook94 & 552 & 16.5 & — & — & — & — & K\ Djorgovski95 & 3 & 23.5 & — & — & \[20.0,23.5\] & 0.32$\pm$0.02 & K\ McLeod95 & 22.5,2.0 & 19.5,21.25 & — & — & — & — & Ks\ Gardner96 &35424 & 15.75 & $<$16.0 & 0.63$\pm$0.01 & — & — & K\ Moustakas97 &2.0 & 24.0 & — & — & \[18.0,23.0\] & 0.23$\pm$0.02 & K\ Huang97 &29628 & 16.0 & \[12.0,16.0\] & 0.689$\pm$0.013 & — & — & K’\ Minezaki98 &181,2.21& 19.1,21.2 & — & — & \[18.25,18-75\] & 0.28 & K’\ Bershady98 &1.5 & 24.00 & — & — & $>$18.5 & 0.36 & K\ Szokoly98 & 2185 & 16.5 & \[14.5,16.5\] & 0.50$\pm0.03$ & —& —& Ks\ Saracco99 &20 & 22.25 & — & — & \[17.25,22.5\] & 0.38 & Ks\ V[ä]{}is[ä]{}nen00 &3492,2088 & 16.75,17.75 & \[15.0,18.0\] & 0.40-0.45 & — & — & K\ Martini01 &180,51 & 17.0,18.0 & \[14.0,18.0\] & 0.54 & — & — & K\ Daddi00 & 701,447 & 18.5,19.0 & \[14.0,17.5\] & 0.53$\pm0.02$ & $>$17.5 & 0.32$\pm0.02$ & Ks\ K[ü]{}mmel00 &3348 & 17.25 & \[10.5,17.0\] & 0.56$\pm$0.01 & \[16.5,17.5\] & 0.41 & K\ Maihara01 &4 & 25.25 & — & — & $>$20.1 & 0.23 & K’\ Saracco01 &13.6 & 22.75 & — & — & $>$19 & 0.28 & Ks\ Huang01 &720 & 19.5 & $<$16.5 & 0.64 & $>$17 & 0.36 & K’\ Cimatti02 & 52 & 19.75 & — & — & — & — & Ks\ Cristobal03 &180,50 & 20.0,21.0 & \[15.5,17.5\] & 0.54 & \[17.5,19.5\] & 0.25 & Ks\ Iovino05 & 414 & 20.75 & \[15.75,18.0\]& 0.47$\pm$0.23 & \[18.0,21.25\] & 0.29$\pm$0.08 & Ks\ Elston06 & 25560 & 19.2 & — & — & — & — & Ks\ Imai07 &750,306 & 18.625,19.375 & $<$18.00 & 0.32$\pm$0.06 & $>$19.00 & 0.32$\pm$0.06 & Ks\ Feulner07 &925 & 20.75 & — & — & — & — & K\ This work & 1584,194 &19.5,20.0 & \[15.0,17.0\] & 0.53$\pm$0.02 & \[18.0,20.0\] & 0.33$\pm$0.03 & Ks\ [llllllll]{} Bershady98 & 1.5 & 24.5 & — & — & $>$19.5 & 0.35 & J\ Teplitz99 & 180 & 21.75 & — & — & — & — & J\ Saracco99 & 20 & 23.75 & — & — & \[18.0,24.0\]& 0.36 & J\ V[ä]{}is[ä]{}nen00 & 2520,1275 & 18.25,19.25 & \[17.0,19.5\] & 0.40-0.45 & — & — & J\ Martini01 & 180,27 & 18.5,20.5 & \[16.0,20.5\] & 0.54 & — & — & J\ Maihara01 & 4 & 26.25 & — & — & \[21.1,25.1\]& 0.23 & J\ Saracco01 & 13.6 & 24.25 & — & — & $>$20 & 0.34 & J\ Iovino05 & 391 & 22.25 & \[17.25,22.25\] & 0.39$\pm0.06$ & — & — & J\ Feulner07 & 925 & 22.25 & — & — &— & — & J\ Imai07 & 750,306 & 19.625,20.375 & \[17.0,19.5\] & 0.39$\pm0.02$ & $>$19.5 & 0.30$\pm0.03$ & J\ This work & 1588,392 &21.0,22.0 & \[17.0,18.5\] & 0.44$\pm$0.04 & \[19.5,22.0\] & 0.34$\pm$0.01 & J\ [llllllll]{} Teplitz98 & 35.4 & 22.8 & — & — & — & — & F160W\ Yan98 & 8.7,2.9 & 23.5,24.5 & — & — & \[20,24.5\] &0.315$\pm$0.02 & F160W\ Thompson99 & 0.7 & 27.4 & — & — & — & — & F160W\ Martini01 & 180,80 & 18.0,19.0 & $<$19 & 0.47 & — & — & H\ Chen02 & 1408 & 20.8 & $<$19 & 0.45$\pm$0.01 & $>$19 &0.27$\pm$0.01 & H\ Moy03 & 97.2,619 & 20.5,19.8 & — & — & — & — & H\ Metcalfe06 & 49 & 22.9 & — & — & — & — & H\ Metcalfe06-Nicmos & 0.90 & 27.2 & — & — & — & — & F160W\ Frith06 & 1080 & 17.75 & — & — & — & — & H\ This work & 1598,594 &20.5,21.0 & \[15.5,18.0\] & 0.46$\pm$0.02 & \[19.0,21.0\] & 0.36$\pm$0.01 & H\ In the H and J bands there are fewer works reporting count-slope values. Our results, given in Tab. \[Tab:slopes\], show similar slopes for the J and H filter at the bright and faint ends. As can be seen in Tabs. \[Tab:surveysJ\] and \[Tab:surveysH\] our result at the bright end are in good agreement with the bright-end slope values to H=19 given in [@2001AJ....121..598M] and [@2002ApJ...570...54C], and in the J filter with the results in [@2000ApJ...540..593V] and . At the faint end, only the slope values in [@2001PASJ...53...25M] in the J filter, estimated in an area of 4 arcsec$^2$, and [@2002ApJ...570...54C] have a significant discrepancy. Comparison with Models of Evolution =================================== Historically, the galaxy number counts have been used to examine parameters of the cosmological model and to test different galaxy evolution scenarios. Now that the cosmological parameters are fixed using other methods the consequences derived from the galaxy counts for the galaxy evolution have become more precise. The ALHAMBRA computed counts in the three NIR bands provide a good dataset which, when combined with other optical data and independent determinations of the local luminosity functions, allow evolution to be examined, in particular the still uncertain question of the formation and evolutionary history of early type galaxies. In this section we compare our counts with semi-analytic predictions from number-count models, following the recipes given in [@1998PASP..110..291G], which trace back the redshift evolution of the galaxy Spectral Energy Distribution (SED) of different galaxy classes. The SEDs have been computed using the codes of [@2003MNRAS.344.1000B]. We apply dust attenuation following [@2006ApJ...639..644E], $\tau_B=0.6$ that corresponds to $\tau_V=0.4$ if the attenuation follows a $\propto \lambda^{-2}$ power law. We apply dust extinction either directly and by the same amount to all the galaxies using [@2003MNRAS.344.1000B] codes, following the prescription given in [@2000ApJ...539..718C], or by using the luminosity dependent extinction law proposed in [@1991ApJ...383L..37W]. The parameters we use to characterize four different galaxy types are given in Tab. \[Tab:galparams\]. In the LF parameterization for the different bands, $M^{\star}$ is changed according to the rest-frame colors of the evolved SED (from $z_f$ to $z=0$), whereas $\alpha$ and $\phi^{\star}$ are assumed to be the same in all filters. In the first step we compare the ALHAMBRA NIR counts with the prediction obtained using the model proposed in [@2003ApJ...595...71C]. The extinction correction was applied directly to the SEDs, as an entry parameter in the code described in [@2003MNRAS.344.1000B] using $\tau_V=1.2$ for stars younger than $10^7$yr , and $\mu=0.3$ as the fraction of it coming from an ambient contribution which affects the old stars too. The parameters used in the local luminosity function are $M^{\star}=-24.07$, $\alpha=-1.00$, $\phi^{\star}=4.94\times 10^{-3}$ calculated in [@1997ApJ...480L..99G] ([@2001MNRAS.326..255C] provide the parameters for the $\Lambda$-cosmology), which were transformed to take into account the presence of different galactic types in the local LF adopting the galaxy mixing E/S0=28%, Sab/Sbc=47%, Scd=13%. The model also adds a dwarf star-forming population, characterized by an stellar population of age 1Gyr at all redshifts, and a steeper slope LF ($M^{\star}=-23.12$, $\alpha=-1.5$, $\phi^{\star}=0.96\times10^{-3}$) given in [@1998PASP..110..291G]. The formation redshifts are $z_f=2.0$ for the E/S0 and intermediate-type disk galaxies and $z_f=1.0$ for the Scd, although the formation redshift for the Scd could be $z_f=4.0$ or higher without modifying the total counts in NIR, that at magnitudes fainter than Ks=21.5 are dominated by the dwarf star-forming population. Due to the disappearance of the red-galaxy population at $z>2.0$ this model reproduces the change of slope in the Ks band observed in the present data (see Fig. \[Fig:countsKS\]). However, as it was discussed in [@2006ApJ...639..644E], this model fails to simultaneously reproduce the blue band counts as can be seen in Fig. \[Fig:countsB\], where the predicted counts are compared with some B-band galaxy counts from the literature. The number counts models that we consider now, below, include some modifications which aim to simultaneously reproduce the counts in both the NIR filters and blue filters. For these, the local type dependent luminosity functions were computed from Sloan data in [@2003AJ....125.1682N], as shown in Tab. \[Tab:lumfunc\]. [lcccc]{} E/S0 & Single star pop. & — & 1 & Salpeter\ early Sp & Exponential &4& 1 & Salpeter\ late Sp & Exponential & 7& $2/5$ & Salpeter\ Im & Constant & .. & $1/5$ & Salpeter\ [lccc]{} E/S0 & -21.53 & 1.61& -0.83\ early Sp & -21.08 & 3.26& -1.15\ late Sp & -21.08 & 1.48& -0.71\ Im & -20.78 & 0.37& -1.90\ We consider first the two models proposed in [@2006ApJ...639..644E]. In the first model a $\phi^{\star}$ evolution $\propto (1+z)^2$, driven via mergers, is considered for the spiral and irregular galaxies. This evolution in $\phi^{\star}$ is compensated by the evolution in $M^{\star}$ to conserve the luminosity density. Is it important to take in mind that these models calculate the galaxy number counts tracing back the evolution of the stellar populations to $z=0$, so the intrinsic brightening with $z$ of the stellar populations must be added to the $M^{\star}$ evolution when is compared with luminosity functions computed at higher redshifts. The formation redshift for the majority of the ellipticals and intermediate type disk galaxies in this model was set to 1.5. In the second model, the low formation redshift for the early spiral galaxies could be set to $z_f=4$, avoiding at the same time an unreasonable high number of late type galaxies at high-z, using the merger parameterization $\phi^*\propto \exp{[(-Q/\beta)((1+z)^{-\beta}-1)]}$ given in [@1992Natur.355...55B]. The value of $\beta=1+(2q_0)^{0.6}/2$ was set to 1.53 using $q_0=-0.55$. A value of $Q=1$ was used as in [@2006ApJ...639..644E]. The extinction correction, which is important in the blue bands, was set to $\tau_B=0.6$ with the prescription given in [@1991ApJ...383L..37W]. As is shown in Figs. \[Fig:countsJ\] and \[Fig:countsH\] these two models, that fit the B (Fig. \[Fig:countsB\]) and Ks (Fig. \[Fig:countsKS\]) galaxy counts, overestimate the slope variation at J$\sim$19 in the Alhambra counts, and at H$\sim$18 in data from other surveys. In order to explain the change of slope in the NIR galaxy counts, the population of red Elliptical galaxies has to decrease with the redshift. Although a model taking into account only the stellar evolution with look back time fits the blue band counts (see Fig. \[Fig:countsB\]), this model over-predicts the faint counts in the NIR bands as can be seen from Fig. \[Fig:countsKS\]. Due to the red color of the slope change only the Elliptical population parameterized with a short burst of star-formation can play this role. Fig. \[Fig:colorev\] shows the evolution with redshift of the J-H and J-Ks colors for an Elliptical galaxy and an early spiral formed at z=4 (with stellar populations according to the parameters in Tab. \[Tab:galparams\], and reddening in the spectra applied directly from the [@2003MNRAS.344.1000B] code using $\tau_V=0.8$ and $\mu=0.5$). The estimated colors of the slope change from the fits given in Tab. \[Tab:slopes\] are J-H$\sim0.97\pm 0.03$ and J-Ks$\sim1.84\pm 0.03$, providing evidence that the Elliptical population at z$\sim$1 must be the responsible of the turn down of the NIR counts. In the next model we introduce number-density evolution of the elliptical population parameterized using $\phi^{\star}\propto (1+z)^{-2}$. The formation redshifts is set to $z_f=4.0$ for all galaxy types, being in more agreement with the evolved red galaxies found at $z>2$ [@2003ApJ...587L..83V; @2004ApJ...617..746D]. The evolution in the early type densities was not accompanied by an evolution in $M^{\star}$, arguing that a substantial number of ellipticals formed in spiral-spiral mergers as expected for hierarchical galaxy formation. For the early spiral galaxies no number-density evolution was considered, and the density evolution parameterization in [@2006ApJ...639..644E] $\phi^{\star}\propto (1+z)^{2}$ was used for the two later type galaxies. The reddening in the spectra was applied directly from the [@2003MNRAS.344.1000B] code using $\tau_V=0.8$ and $\mu=0.5$. This model, as can be seen in the Figs. \[Fig:countsB2\], \[Fig:countsJ2\], \[Fig:countsH2\], and \[Fig:countsKS2\] fits the galaxy counts in the optical and NIR filters, reproducing the feature of the slope turn down in the three NIR filters. In order to avoid an unreasonable high number of late type galaxies at high-z, the simple merger evolution $\phi^*\propto \exp{[(-Q/\beta)((1+z)^{-\beta}-1)]}$ given in [@1992Natur.355...55B] could be used with similar results. In this case, we used Q=-3 to parameterize the decrease in number of E/S0 galaxies, and Q=1 to produce the required merger rate in the late spirals and irregulars. The number evolution given by these parameterizations are displayed in Fig \[Fig:phievol\], showing that at $z\sim1$ the implied number density of E/S0 galaxies is only $\sim1/4$ of the present day $\phi^*(0)$. Density evolution in the early type galaxies was observed in previous works studying the type-dependent LF evolution . found an increase in $\phi^{\star}$ of an order of magnitude for the early type galaxies from $z\sim1.2$ to $z=0$, that is over the $\phi^{\star} \propto (1+z)^{-2}$ simulated here. However, they used the spectra of a present day Sa type galaxy to separate the different galaxies, which leads to an over-estimation of number-density evolution for the Early-type group. In [@2007ApJ...669..184A] it is shown that evolution in the fraction of the stellar mass locked in massive early-type galaxies is produced in the interval $0.7<z<1.7$. A model in which $\phi^*$ for the Elliptical galaxies is constant to $z\sim 0.6$ and then evolve as $\phi^{\star} \propto (0.4+z)^{-2}$ for higher redshifts also produces a good fit to the optical and NIR counts (see Figs. \[Fig:countsB2\], \[Fig:countsJ2\], \[Fig:countsH2\], and \[Fig:countsKS2\]). In this model the population of red elliptical galaxies has doubled since $z=1$, in good agreement with the increase of a factor of $\sim 2$ in the number evolution of red galaxies given in [@2004ApJ...608..752B]. Finally, we have implemented a simple recipe to simulate downsizing in the elliptical population by maintaining $\phi^\star$ constant with redshift for the LF of bright galaxies. The results are compatible with our NIR galaxy counts and B-band counts from the literature in the case that $\phi^\star$ is constant with redshift for red-ellipticals brighter than $M^\star-0.7$ ($\sim-22.0$ in the Sloan $r'$ band in AB system), decreasing the number densities for the bulk of the ellipticals as $\phi^{\star}\propto (1+z)^{-2}$. Color analysis ============== More information about the evolution of the galaxy populations could be obtained from color histograms. The separate number counts in each band at the magnitude ranges that we are sampling are less sensitive to the formation redshift (for values $zf>=4$) or the e-folding timescale of the star formation than color histograms. Figure  \[Fig:tracs\] shows the color-magnitude diagram builded with the ALHAMBRA data through the filter centered at 6130 $\AA$ (F613) and Ks. The modelled evolutionary tracks for the 4 galaxy spectra considered in Tab. \[Tab:galparams\] are also displayed. Models with no evolution (top panel) and passive evolution (bottom panel ) have been considered. As can be appreciated the evolved spectra produce better match to the data than the no evolved version, principally at the faint blue end which is better described by models considering passive evolution in the late Sp and Irr spectra. In Fig. \[Fig:chisto\] it is shown the F613-Ks color histogram for different Ks magnitude bins. We have used an e-folding timescale $\tau=0.7$ Gyr to describe the Elliptical galaxies in those plots, this longer timescale produce better fits to the red end of the color histograms than an instantaneous star forming event which over-predict the number of red galaxies. Both models produce similar results when fitting the galaxy counts in individual optical and NIR bands. The simulated histograms correspond to models where the population of E/S0 galaxies decrease with redshift, the population of spirals stay constant, and the late type galaxies increase as as $\phi^{\star}\propto (1+z)^{2}$ conserving the luminosity densities. The population of early spiral galaxies have been divided in two classes: one remain as in Tab. \[Tab:galparams\], whereas the amount of extinction have been doubled for the other. This try to avoid the fact that the discretization of the actual galaxy population in four classes tend to produce sharp histograms. As can be appreciated the models reproduce the overall shape of the data for bright Ks magnitudes. Nevertheless, at faint Ks magnitudes the models predict higher values in the color range $3\le F613-Ks\le5$. As could be inferred from Fig. \[Fig:chisto\] to obtain a better match to the data the number densities of early spiral galaxies formed in a shorter time-scale has to decrease with $z$, such fading of the spirals will lead to an under-prediction of the blue-band number counts unless that the number of star-forming increases at higher redshift. In Fig. \[Fig:chisto2\] the F613-Ks histograms for the Ks bins: \[16.5,17.5\], \[17.5,18.5\] show a better concordance with the observed data. Those histograms correspond to a models where the number densities for the early spirals decrease with $z$ as $\phi^{\star}\propto (1+z)^{-1}$, the late type spirals number density remain constant with redshift, and number densities of Irr galaxies increase $\propto (1+z)^{3}$. The luminosity density in not conserved within any galaxy class. Similar results could be obtained using $\phi^*\propto \exp{[(-Q/\beta)((1+z)^{-\beta}-1)]}$ [@1992Natur.355...55B], with Q=-1, and Q=3 to describe the number evolution of early spiral and Irr galaxies. This number evolution formulation avoid a step increase of Irr galaxies at high redshift. However in Fig. \[Fig:chisto2\] a shortage of red galaxies is seen at about F613-Ks$\sim5$, this could be due to the fact that we only use a discrete number of galaxy parameterizations, for example is well know the existence of dusty starburst with the same red colors than the passive extremely red objects . Covering a wider range in galaxy internal extinction or formation timescale will tend to smear the bi-modality present in the simulated histograms. As could be seen in Fig. \[Fig:models2\] the number counts produced by this model in B + NIR filters also produce good fits to the observed data points. In this models the number counts at faint magnitudes will be dominated by the star-forming galaxies. The number count slope at faint magnitudes will be $-0.4(\alpha +1)$ [@2003RMxAC..16..203B], being $\alpha$ the slope of the dominant luminosity function for $M<<M^\star$. With this parameterization the slope of the number counts tend to $0.36$ at the fainter end. Summary ======= We have presented galaxy counts in the J,H, and Ks filters covering an area of 0.45 square degrees and an average 50% detection efficiency depth of J$\sim22.4$, H$\sim21.3$ and Ks$\sim20.0$ (Vega system). The depth reached, and the precision of the counts over a range of five magnitudes makes the data valuable for examining the change of the count slope reported in the Ks filter, and to extend this examination to the J and H bands. We find that a change in slope occur in each of the NIR bands in the range J=\[18.5,19.5\], H=\[18.0,19.0\] and Ks=\[17.0,18.0\]. The NIR colors where the break in the galaxy counts are found imply that this change is related to the population of red galaxies at z$\sim$1. We have compared our number counts results with predictions from a wide range of number count models, concluding that in order to reproduce the described changes in the NIR slopes, a decrease in bulk of the population of red elliptical galaxies is needed. Good fits to the B-band and NIR counts are obtained with a parameterization for the number evolution of the elliptical population as $\phi^{\star}\propto (1+z)^{-2}$ with no accompanying evolution in $M^{\star}$, corresponding to evolution in which the majority of ellipticals formed in spiral-spiral mergers. Performing a color analysis show that also the population of early spirals has to decrease at higher redshift in order to describe the color distribution in r-Ks. Models using the parameterization of [@1992Natur.355...55B] $\phi^*\propto \exp{[(-Q/\beta)((1+z)^{-\beta}-1)]}$, with Q=-3, Q=-1, and Q=3 to describe the number evolution of ellipticals, early spiral and Irr galaxies, with no number density evolution for late-spiral systems, produce good fits to the observed distribution, avoiding at the same time a high number of young systems at high $z$. A good match to the optical and NIR data is also obtained if the population of red-galaxies in the models remain constant to $z\sim0.6$ and afterwards its number density decrease as $\phi^{\star} \propto (0.4+z)^{-2}$, or if the number density of red-ellipticals is constant with redshift for galaxies brighter than $M^\star-0.7$ ($\sim-22.0$ in the Sloan $r'$ band in AB system), decreasing as $\phi^{\star}\propto (1+z)^{-2}$ for the bulk of the ellipticals. Alhambra is processing the data obtained in 20 medium-band optical and 3 NIR filters reaching high quality photometric redshift measurements ($\Delta z/(1+z)\le0.03$). Also an accurate classification by Spectral Energy distribution will be acquired. Those data will allow for the study of the evolution of the different galaxy types to $z\sim1$, which will complement the results given in this article, disentangling what populations contribute to the number counts at different redshift intervals. Also the study of number counts for red galaxy populations, passive EROS or BzK [@2004ApJ...617..746D] galaxies will constrain the formation redshift and formation timescale for massive Elliptical galaxies. The authors acknowledge support from the Spanish Ministerio de Educación y Ciencia through grants AYA2002-12685-E, AYA2003-08729-C02-01, AYA2003-0128, AYA2007-67965-C03-01, AYA2004-20014-E, AYA2004-02703, AYA2004-05395, AYA2005-06816, AYA2005-07789, AYA2006-14056, and from the [*Junta de Andalucía*]{}, TIC114, TIC101 and [*Proyecto de Excelencia*]{} FQM-1392. NB, JALA, MC, and AFS acknowledge support from the MEC [*Ramón y Cajal*]{} Programme. NB acknowledges support from the EU IRG-017288. This work has made use of software designed at TERAPIX. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation This publication makes use of data from the Sloan Digital Sky Survey. Funding for the Sloan Digital Sky Survey (SDSS) has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS Web site is http://www.sdss.org/. The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, University of Pittsburgh, Princeton University, the United States Naval Observatory, and the University of Washington. Individual image combination {#Ap:Combination} ============================ As mentioned in the text, to combine the processed images we used SWARP [@2002ASPC..281..228B]. With this software, individual images were projected into subsections of the final frame using the inverse mapping, in which each output pixel center was associated to a position in the input image at which it is interpolated. With this schema, the code corrected at the same time the geometrical distortions in the individual images using the astrometric information stored in the image headers. Estimating the relative transparency ------------------------------------ Using a filtered version of the SExtractor catalogs computed over the sky subtracted images, an accurate estimate of the relative transparency was computed by tracing the high S/N objects in all the images. The relative transparency values were used inside SWARP to scale the individual images to the same flux level in order to uniformize the zero points in the outer dither areas, this allowed to use 2MASS catalogs to calibrate the ALHAMBRA NIR photometry in the final images. Astrometry calibration ---------------------- When computing the resampled version of the individual frames, SWARP uses the WCS information stored in the headers. In order to obtain a better matching of the individual images, firstly the pipeline calculated the external astrometrical solution for a reference image. The individual images were then calibrated internally with respect to it, thus obtaining the equatorial coordinates from the reference image. In this paper the individual image with better transparency in a given pointing was used as reference. However, to get a better internal astrometry between the different filters, after completing a pointing in the 23 ALHAMBRA filters, the images with better FWHM and transparency in a set of selected optical filters, are combined to produce a deep image that will be used afterwards as reference. We have determined that the USNO-B1.0 catalog [@2003AJ....125..984M] provides an adequate number of objects to perform a high quality external astrometric calibration. We used our own code to match the sources with brighter apparent magnitudes in the reference image with those in the USNO-B1.0. Once a meaningful number of pairs was identified, the CCMAP IRAF task was used in two iterations to acquire the required astrometric solution with a 2nd order polynomial. A histogram of the external astrometric solution rms and the number of objects used for the final images of ALH08 is shown in Fig. \[Fig:wcs\]. The median external astrometry rms is $0.12^{\prime\prime}\pm 0.01$ in RA and $0.11^{\prime\prime}\pm 0.01$ DEC. Having calibrated a reference image, the rest of the individual images were calibrated internally. The median internal rms in the astrometric solution for the OMEGA2000 data used in this paper is $0.06^{\prime\prime}$ in RA and DEC using a median of 160 objects, as shown in Fig \[Fig:wcs\]. Image co-adding --------------- Swarp allows the user to choose among several interpolation functions for inverse mapping. For selecting the more appropriate kernel we analyzed the resulting final image FWHM and its pixel-to-pixel correlation. Tab. \[correlations\] shows the correlation values, between adjacent pixels and for pixel pairs separated by 2 pixels, in the final images obtained using different available interpolation functions. As can be seen in the table, the bilinear function produce a higher correlation which translate into an underestimation of the flux errors. Using the Lanczos-3 function the FWHM of the final image was improved by $\sim0.05^{\prime\prime}$ compared with the nearest neighbor interpolation, whereas the auto-correlation at 1 pixel remain acceptable $\sim0.16$ (when the images are combined using the average). The Lanczos-4 function did not decrease substantially nor the FWHM neither the correlation at 1 pixel, on the contrary it produced large artifacts at the bad pixels and image borders, so finally we have decided to take the Lanczos-3 function. [cccccc]{}\ 1pix & +0.017 & +0.186 & +0.112 & +0.074 & +0.068\ 2pix & -0.035 & -0.013 & -0.072 & -0.059 & -0.101\ \ \ \ 1pix & +0.035 & +0.274 & +0.177 & +0.156 & +0.130\ 2pix & -0.030 & -0.018 & -0.069 & -0.110 & -0.083\ \[correlations\] Abraham, R. G., et al.  2007, , 669, 184 Aguerri, J. A. L., & Trujillo, I. 2002, , 333, 633 Bell, E. F., et al. 2004, , 608, 752 Benitez et al. 2008, submitted Bertin, E., & Arnouts, S. 1996, , 117, 393 Bertin, E., Mellier, Y., Radovich, M., Missonnier, G., Didelon, P., & Morin, B. 2002, Astronomical Data Analysis Software and Systems XI, 281, 228 Bershady, M. A., Lowenthal, J. D., & Koo, D. C. 1998, , 505, 50 Bershady, M. A. 2003, Revista Mexicana de Astronomia y Astrofisica Conference Series, 16, 203 Blanton, M. R., et al.  2003, , 592, 819 Broadhurst, T. J., Ellis, R. S., & Glazebrook, K. 1992, , 355, 55 Bruzual, G., & Charlot, S. 2003, , 344, 1000 Capak, P., et al. 2004, , 127, 180 Charlot, S., & Fall, S. M. 2000, , 539, 718 Chen, H.-W., et al. 2002, , 570, 54 Cimatti, A., et al. 2002, , 392, 395 Crist[ó]{}bal-Hornillos, D., Balcells, M., Prieto, M., Guzm[á]{}n, R., Gallego, J., Cardiel, N., Serrano, [Á]{}., & Pell[ó]{}, R. 2003, , 595, 71 Cole, S., et al. 2001, , 326, 255 Covey, K. R., et al.  2007, , 134, 2398 Cutri, R. M., et al.  :’a 2003, The IRSA 2MASS All-Sky Point Source Catalog, NASA/IPAC Infrared Science Archive. http://irsa.ipac.caltech.edu/applications/Gator/, Daddi, E., Cimatti, A., Pozzetti, L., Hoekstra, H., R[ö]{}ttgering, H. J. A., Renzini, A., Zamorani, G., & Mannucci, F. 2000, , 361, 535 Daddi, E., Cimatti, A., Renzini, A., Fontana, A., Mignoli, M., Pozzetti, L., Tozzi, P., & Zamorani, G. 2004, , 617, 746 Djorgovski, S., et al. 1995, , 438, L13 Eddington, A. S. 1913, , 74, 5 Eliche-Moral, M. C., Balcells, M., Prieto, M., Garc[í]{}a-Dab[ó]{}, C. E., Erwin, P., & Crist[ó]{}bal-Hornillos, D. 2006, , 639, 644 Elston, R. J., et al.  2006, , 639, 816 Faber, S. M., et al.  2007, , 665, 265 Feulner, G., Goranova, Y., Hopp, U., Gabasch, A., Bender, R., Botzler, C. S., & Drory, N. 2007, , 378, 429 Finlator, K., et al.  2000, , 120, 2615 Fontana, A., et al. 2004, , 424, 23 Fried, J. W., et al.  2001, , 367, 788 Frith, W. J., Metcalfe, N., & Shanks, T. 2006, , 371, 1601 Gardner, J. P., Cowie, L. L., & Wainscoat, R. J. 1993, , 415, L9 Gardner, J. P., Sharples, R. M., Carrasco, B. E., & Frenk, C. S. 1996, , 282, L1 Gardner, J. P., Sharples, R. M., Frenk, C. S., & Carrasco, B. E. 1997, , 480, L99 Gardner, J. P. 1998, , 110, 291 Giallongo, E., Salimbeni, S., Menci, N., Zamorani, G., Fontana, A., Dickinson, M., Cristiani, S., & Pozzetti, L. 2005, , 622, 116 Glazebrook, K., Peacock, J. A., Collins, C. A., & Miller, L. 1994, , 266, 65 Huang, J.-S., Cowie, L. L., Gardner, J. P., Hu, E. M., Songaila, A., & Wainscoat, R. J. 1997, , 476, 12 Huang, J.-S., et al.  2001, , 368, 787 Ilbert, O., et al. 2006, , 453, 809 Im, M., et al. 2002, , 571, 136 Imai, K., Matsuhara, H., Oyabu, S., Wada, T., Takagi, T., Fujishiro, N., Hanami, H., & Pearson, C. P. 2007, , 133, 2418 Iovino, A., et al. 2005, , 442, 423 Kauffmann, G., Charlot, S., & White, S. D. M. 1996, , 283, L117 K[ü]{}mmel, M. W., & Wagner, S. J. 2000, , 353, 867 K[ü]{}mmel, M. W., & Wagner, S. J. 2001, , 370, 384 McCracken, H. J., Metcalfe, N., Shanks, T., Campos, A., Gardner, J. P., & Fong, R. 2000, , 311, 707 McLeod, B. A., Bernstein, G. M., Rieke, M. J., Tollestrup, E. V., & Fazio, G. G. 1995, , 96, 117 Maihara, T., et al.  2001, , 53, 25 Martini, P. 2001, , 121, 598 Metcalfe, N., Shanks, T., Campos, A., Fong, R., & Gardner, J. P. 1996, , 383, 236 Metcalfe, N., Shanks, T., Campos, A., McCracken, H. J., & Fong, R. 2001, , 323, 795 Metcalfe, N., Shanks, T., Weilbacher, P. M., McCracken, H. J., Fong, R., & Thompson, D. 2006, , 370, 1257 Minezaki, T., Kobayashi, Y., Yoshii, Y., & Peterson, B. A. 1998, , 494, 111 Moles et al. 2008, AJ, in press Monet, D. G., et al. 2003, , 125, 984 Moustakas, L. A., Davis, M., Graham, J. R., Silk, J., Peterson, B. A., & Yoshii, Y. 1997, , 475, 445 Moy, E., Barmby, P., Rigopoulou, D., Huang, J.-S., Willner, S. P., & Fazio, G. G. 2003, , 403, 493 Nakamura, O., Fukugita, M., Yasuda, N., Loveday, J., Brinkmann, J., Schneider, D. P., Shimasaku, K., & SubbaRao, M. 2003, , 125, 1682 Perlmutter, S., et al. 1999, , 517, 565 Riess, A. G., et al.  1998, , 116, 1009 Robin, A. C., Haywood, M., Creze, M., Ojha, D. K., & Bienayme, O. 1996, , 305, 125 Saracco, P., D’Odorico, S., Moorwood, A., Buzzoni, A., Cuby, J.-G., & Lidman, C. 1999, , 349, 751 Saracco, P., Giallongo, E., Cristiani, S., D’Odorico, S., Fontana, A., Iovino, A., Poli, F., & Vanzella, E. 2001, , 375, 1 Spergel, D. N., et al.  2003, , 148, 175 Stanford, S. A., Eisenhardt, P. R. M., & Dickinson, M. 1995, , 450, 512 Szokoly, G. P., Subbarao, M. U., Connolly, A. J., & Mobasher, B. 1998, , 492, 452 Teerikorpi, P. 2004, , 424, 73 Teplitz, H. I., McLean, I. S., & Malkan, M. A. 1999, , 520, 469 V[ä]{}is[ä]{}nen, P., Tollestrup, E. V., Willner, S. P., & Cohen, M.  2000, , 540, 593 Valdes, F. G. 2002, Automated Data Analysis in Astronomy, 309 van Dokkum, P. G., et al. 2003, , 587, L83 Volonteri, M., Saracco, P., Chincarini, G., & Bolzonella, M. 2000, , 362, 487 Wang, B. 1991, , 383, L37 Wolf, C., Meisenheimer, K., Rix, H.-W., Borch, A., Dye, S., & Kleinheinrich, M. 2003, , 401, 73 Yan, L., McCarthy, P. J., Storrie-Lombardi, L. J., & Weymann, R. J. 1998, , 503, L19 [^1]: [Based on observations collected at the German-Spanish Astronomical Center, Calar Alto, jointly operated by the Max-Planck-Institut für Astronomie Heidelberg and the Instituto de Astrofísica de Andalucía (CSIC).]{}

--- abstract: | For each ${\small b\in\left( 0,\,\infty\right) }$ we intend to generate a decreasing sequence of subsets $\left( \mathcal{Y}_{b}^{\left( n\right) }\right) \subset Y_{\mathrm{conc}}$ depending on $b$ such that whenever $n\in\mathbb{N}$, then $\mathcal{A}\cap\mathcal{Y}_{b}^{\left( n\right) }% $ is dense in $\mathcal{Y}_{b}^{\left( n\right) }$ and the following four sets $\mathcal{Y}_{b}^{\left( n\right) }$, $\mathcal{Y}_{b}^{\left( n\right) }\backslash\left( \mathcal{A}\cap\mathcal{Y}_{b}^{\left( n\right) }\right) $, $\mathcal{A}\cap\mathcal{Y}_{b}^{\left( n\right) }$ and $\mathcal{Y}_{\mathrm{conc}}$ are pairwise equinumerous. Among others we also show that if $f$ is any measurable function on a measure space $\left( \Omega,\mathcal{F},\lambda\right) $ and $p\in\left[ 1,\infty\right) $ is an arbitrary number then the quantities $\left\Vert f\right\Vert _{L^{p}}$ and $\sup_{\Phi\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}}\left( \Phi\left( 1\right) \right) ^{-1}\left\Vert \Phi\circ\left\vert f\right\vert \right\Vert _{L^{p}}$ are equivalent, in the sense that they are both either finite or infinite at the same time. address: | Institute of Mathematics\ University of Miskolc\ H-3515 Miskolc–Egyetemváros\ Hungary author: - 'N. K. Agbeko' date: 'March 24th, 2006' title: 'Bijections and metric spaces induced by some collective properties of concave Young-functions' --- Introduction ============ We know that concave functions play major roles in many branches of mathematics for instance probability theory ([@BURK1973], [@GARS1973], [@MOGY1981], say), interpolation theory (cf. [@TRIEB1978], say), weighted norm inequalities (cf. [@GARFRAN1985], say), and functions spaces (cf. [@SINN2002], say), as well as in many other branches of sciences. In the line of [@BURK1973], [@GARS1973] and [@MOGY1981], the present author also obtained in martingale theory some results in connection with certain collective properties or behaviors of concave Young-functions (cf. [@AGB1986], [@AGB1989]). The study presented in [@AGB2005] was mainly motivated by the question why strictly concave functions possess so many properties, worth to be characterized using appropriate tools that await to be discovered. We say that a function $\Phi:\left[ 0,\,\infty\right) \rightarrow\left[ 0,\,\infty\right) $ belongs to the set $\mathcal{Y}_{\mathrm{conc}}$ (and is referred to as a concave Young-function) if and only if it admits the integral representation$$\Phi\left( x\right) =\int\nolimits_{0}^{x}\varphi\left( t\right) dt, \label{id1}%$$ (where $\varphi:\left( 0,\,\infty\right) \rightarrow\left( 0,\,\infty \right) $ is a right-continuous and decreasing function such that it is integrable on every finite interval $\left( 0,\,x\right) $) and $\Phi\left( \infty\right) =\infty$. It is worth to note that every function in $\mathcal{Y}_{\mathrm{conc}}$ is strictly concave. We will remind some results obtained so far in [@AGB2005]. We shall say that a concave Young-function $\Phi$ satisfies the *density-level property* if $A_{\Phi}\left( \infty\right) <\infty$, where $A_{\Phi}\left( \infty\right) :=\int\nolimits_{1}^{\infty}% \frac{\varphi\left( t\right) }{t}dt$. All the concave Young-functions possessing the density-level property will be grouped in a set $\mathcal{A}$. In Theorems \[theo1\] and \[theo2\] (cf. [@AGB2005]), we showed that the composition of any two concave Young-functions satisfies the density-level property if and only if at least one of them satisfies it. These two theorems show that concave Young-functions with the density-level property behave like left and right ideal with respect to the composition operation. We also proved ([@AGB2005], Lemma 5, page 12) that if $\Phi\in \mathcal{Y}_{\mathrm{conc}}$, then there are constants $C_{\Phi}>0$ and $B_{\Phi}\geq0$ such that$$A_{\Phi}\left( \infty\right) -B_{\Phi}\leq% %TCIMACRO{\dint _{0}^{\infty}}% %BeginExpansion {\displaystyle\int_{0}^{\infty}} %EndExpansion \frac{\Phi\left( t\right) }{\left( t+1\right) ^{2}}dt\leq C_{\Phi}% +A_{\Phi}\left( \infty\right) .$$ This led us to the idea to search for a Lebesgue measure (described here below) with respect to which every concave Young-function turns out to be square integrable ([@AGB2005], Lemma 6, page 13), i.e. $\mathcal{Y}% _{\mathrm{conc}}\subset L^{2}:=L^{2}\left( \left[ 0,~\infty\right) ,~\mathcal{M},~\mu\right) $, where $\mathcal{M}$ is a $\sigma$-algebra (of $\left[ 0,~\infty\right) $) containing the Borel sets and $\mu :\mathcal{M}\rightarrow\left[ 0,~\infty\right) $ is a Lebesgue measure defined by $\mu\left( \left[ 0,~x\right) \right) =\frac{1}{3}\left( 1-\frac{1}{\left( x+1\right) ^{3}}\right) $ for all $x\in\left[ 0,~\infty\right) $. The mapping $d:L^{2}\times L^{2}\rightarrow\left[ 0,~\infty\right) $, defined by$$\operatorname*{d}\left( f,g\right) =\sqrt{% %TCIMACRO{\dint _{\left[ 0,~\infty\right) }}% %BeginExpansion {\displaystyle\int_{\left[ 0,~\infty\right) }} %EndExpansion \left( f-g\right) ^{2}d\mu}=\sqrt{% %TCIMACRO{\dint _{0}^{\infty}}% %BeginExpansion {\displaystyle\int_{0}^{\infty}} %EndExpansion \frac{\left( f\left( x\right) -g\left( x\right) \right) ^{2}}{\left( x+1\right) ^{4}}dx}, \label{dist}%$$ is known to be a semi-metric. Further on, we proved in ([@AGB2005], Theorem 8, page 16) that $\mathcal{A}$ is a dense set in $\mathcal{Y}_{\mathrm{conc}}$. Throughout this communication $\Phi_{\operatorname{id}}$ will denote the identity function defined on the half line $\left[ 0,\text{ }\infty\right) $ and we write $\left\Vert \Phi\right\Vert :=\sqrt{% %TCIMACRO{\dint _{\left[ 0,\text{ }\infty\right) }}% %BeginExpansion {\displaystyle\int_{\left[ 0,\text{ }\infty\right) }} %EndExpansion \Phi^{2}d\mu}$ whenever $\Phi\in\mathcal{Y}_{\mathrm{conc}}$. We intend to generate a decreasing sequence of subsets $\left( \mathcal{Y}% _{b}^{\left( n\right) }\right) \subset\mathcal{Y}_{\mathrm{conc}}$ depending on $b$ such that whenever $n\in\mathbb{N}$, then $\mathcal{A}% \cap\mathcal{Y}_{b}^{\left( n\right) }$ is dense in $\mathcal{Y}% _{b}^{\left( n\right) }$ and the following four sets $\mathcal{Y}% _{b}^{\left( n\right) }$, $\mathcal{Y}_{b}^{\left( n\right) }% \backslash\left( \mathcal{A}\cap\mathcal{Y}_{b}^{\left( n\right) }\right) $, $\mathcal{A}\cap\mathcal{Y}_{b}^{\left( n\right) }$ and $\mathcal{Y}% _{\mathrm{conc}}$ are pairwise equinumerous. We shall also prove that the two pairs $\left( \mathcal{Z}^{\ast\left( n\right) },\operatorname*{dist}% \right) $ and $\left( \mathcal{Z}^{\left( n\right) },\operatorname*{dist}% \right) $ are metric spaces, where $\mathcal{Z}^{\ast\left( n\right) }=\left\{ \mathcal{Y}_{b}^{\left( n\right) }:{\small b\in\left( 0,\,\infty\right) }\right\} $ and $\mathcal{Z}^{\left( n\right) }=\left\{ \mathcal{A}_{b}^{\left( n\right) }:{\small b\in\left( 0,\,\infty\right) }\right\} $ for each $n\in\mathbb{N}$ and the distance between any two sets $\mathcal{F}$ and $\mathcal{G}$ in $\mathcal{Y}_{\mathrm{conc}}$ being defined by $$\begin{aligned} \operatorname*{dist}\left( \mathcal{F},\mathcal{G}\right) & :=\sup\left\{ \inf\left\{ \operatorname{d}\left( \Phi,\Psi\right) :\Psi\in\mathcal{G}% \right\} :\Phi\in\mathcal{F}\right\} \\ & =\sup\left\{ \inf\left\{ \operatorname{d}\left( \Phi,\Psi\right) :\Phi\in\mathcal{F}\right\} :\Psi\in\mathcal{G}\right\} .\end{aligned}$$ We show in the last section that if $f$ is any measurable function on a measure space $\left( \Omega,\mathcal{F},\lambda\right) $ and $p\in\left[ 1,\infty\right) $ is an arbitrary number then the quantities $\left\Vert f\right\Vert _{L^{p}}$ and $\sup_{\Phi\in\widetilde{\mathcal{Y}_{\mathrm{conc}% }}}\left( \Phi\left( 1\right) \right) ^{-1}\left\Vert \Phi\circ\left\vert f\right\vert \right\Vert _{L^{p}}$ are equivalent, in the sense that they are both either finite or infinite at the same time, where $\widetilde {\mathcal{Y}_{\mathrm{conc}}}$ is a proper subset of $\mathcal{Y}% _{\mathrm{conc}}$.We then use this subset to express the value of $\left\Vert f\right\Vert _{L^{p}}$ whenever $\left\Vert f\right\Vert _{L^{p}}<\infty$. Bijections between subsets of $\mathcal{Y}_{\mathrm{conc}}$ =========================================================== We first anticipate that there are as many elements in each of the sets $\mathcal{A}$ and $\mathcal{Y}_{\mathrm{conc}}\backslash\mathcal{A}$ as there exist in $\mathcal{Y}_{\mathrm{conc}}$, showing how broad the set of concave Young-functions possessing the density-level property and its complement really are. \[theo1\]The sets $\mathcal{A}$, $\mathcal{Y}_{\mathrm{conc}}$ and $\mathcal{Y}_{\mathrm{conc}}\backslash\mathcal{A}$ are pairwise equinumerous. We first show that there is a bijection between $\mathcal{A}$ and $\mathcal{Y}_{\mathrm{conc}}$. In fact, since $\mathcal{A}$ is a proper subset of $\mathcal{Y}_{\mathrm{conc}}$ there is an injection from $\mathcal{A}$ to $\mathcal{Y}_{\mathrm{conc}}$, as a matter of fact, the identity mapping from $\mathcal{A}$ into $\mathcal{Y}_{\mathrm{conc}}$ will do. Fix any number $\alpha\in\left( 0,\,1\right) $ and define the mapping $S_{\alpha }:\mathcal{Y}_{\mathrm{conc}}\rightarrow\mathcal{A}$ by $S_{\alpha}\left( \Phi\right) =\Phi^{\alpha}$. We point out that this mapping exists in virtue of Theorem 2 in [@AGB2005]. It is not hard to see that $S_{\alpha}$ is an injection. Then the Schröder-Bernstein theorem entails that there exists a bijection between $\mathcal{A}$ and $\mathcal{Y}_{\mathrm{conc}}$. To complete the proof it is enough to show that there is a bijection between $\mathcal{A}$ and $\mathcal{Y}_{\mathrm{conc}}\backslash\mathcal{A}$. In fact, fix arbitrarily some $\Phi\in\mathcal{Y}_{\mathrm{conc}}\backslash\mathcal{A}$ and define the function $h_{\Phi}:\mathcal{A}\rightarrow\mathcal{Y}_{\mathrm{conc}% }\backslash\mathcal{A}$ by $h_{\Phi}\left( \Delta\right) =\Delta+\Phi$. Obviously, $h_{\Phi}$ is an injection. Now, fix any $\Delta\in\mathcal{A}$ and define the function $f_{\Delta}:\mathcal{Y}_{\mathrm{conc}}\backslash \mathcal{A}\rightarrow\mathcal{A}$ by $f_{\Delta}\left( \Phi\right) =\Delta\circ\Phi$. We point out that this function always exists due to Theorem \[theo2\] in [@AGB2005]. It is not difficult to show that $f_{\Delta}$ is an injection if we take into account that $\Delta$ is an invertible function. Consequently, the Schröder-Bernstein theorem guarantees the existence of a bijection between $\mathcal{A}$ and $\mathcal{Y}_{\mathrm{conc}}\backslash\mathcal{A}$. Therefore, we can conclude on the validity of the argument. Write $\mathcal{A}_{b}:=\left\{ \Phi\in\mathcal{A}:\Phi\left( b\right) =b\right\} $ and $\mathcal{Y}_{b}:=\left\{ \Phi\in\mathcal{Y}_{\mathrm{conc}% }:\Phi\left( b\right) =b\right\} $ for every number $b\in\left( 0,\,\infty\right) $. Let us denote by $\mathcal{Z}:=\left\{ \mathcal{A}_{b}:b\in\left( 0,\,\infty\right) \right\} $ and $\mathcal{Z}^{\ast}:=\left\{ \mathcal{Y}_{b}:b\in\left( 0,\,\infty\right) \right\} $. It is obvious that $\mathcal{A}_{b}\subset\mathcal{Y}_{b}$ for every number $b\in\left( 0,\,\infty\right) $ and $\mathcal{Z}\cap\mathcal{Z}^{\ast }=\varnothing$. \[lem1\]For every number $b\in\left( 0,\,\infty\right) $ the identities $\mathcal{A}_{b}=\left\{ \frac{b\Phi}{\Phi\left( b\right) }:\Phi \in\mathcal{A}\right\} $ and $\mathcal{Y}_{b}=\left\{ \frac{b\Phi}% {\Phi\left( b\right) }:\Phi\in\mathcal{Y}_{\mathrm{conc}}\right\} $ hold true. Pick any function $\Psi\in\mathcal{A}_{b}$. Then $\Psi\in\mathcal{A}$ and $\Psi\left( b\right) =b$, so that $\Psi=\frac{b\Psi}{\Psi\left( b\right) }\in\left\{ \frac{b\Phi}{\Phi\left( b\right) }:\Phi\in\mathcal{A}\right\} $, i.e. $\mathcal{A}_{b}\subset\left\{ \frac{b\Phi}{\Phi\left( b\right) }:\Phi\in\mathcal{A}\right\} $. To show the reverse inclusion consider any function $\Psi\in\left\{ \frac{b\Phi}{\Phi\left( b\right) }:\Phi \in\mathcal{A}\right\} $. Then necessarily there must exist some $\Phi \in\mathcal{A}$ such that $\Psi=\frac{b\Phi}{\Phi\left( b\right) }$. It is obvious that $\Psi\in\mathcal{A}$ and $\Psi\left( b\right) =b$, i.e. $\Psi\in\mathcal{A}_{b}$. Hence, $\left\{ \frac{b\Phi}{\Phi\left( b\right) }:\Phi\in\mathcal{A}\right\} \subset\mathcal{A}_{b}$. These two inclusions yield that $\mathcal{A}_{b}=\left\{ \frac{b\Phi}{\Phi\left( b\right) }% :\Phi\in\mathcal{A}\right\} $. The proof of identity $\mathcal{Y}% _{b}=\left\{ \frac{b\Phi}{\Phi\left( b\right) }:\Phi\in\mathcal{Y}% _{\mathrm{conc}}\right\} $ can be similarly carried out. \[def1\]A proper subset $\mathcal{G}$ of $\mathcal{A}$ is said to be maximally bounded if each of the sets $\mathcal{G}$ and $\mathcal{A}% \backslash\mathcal{G}$ is equinumerous with $\mathcal{A}$, i.e. there is a bijection between $\mathcal{A}$ and $\mathcal{G}$, and $\operatorname*{diam}% (\mathcal{G})<\infty$, where $\operatorname*{diam}(\mathcal{G}):=\sup\left\{ \operatorname*{d}\left( \Phi_{1},\Phi_{2}\right) :\Phi_{1},~\Phi_{2}% \in\mathcal{G}\right\} $ is the diameter of $\mathcal{G}$. We note that Definition \[def1\] makes sense for the two reasons here below. On the one hand we assert that $\operatorname*{diam}(\mathcal{A})=\sup\left\{ \operatorname*{d}\left( \Phi_{1},\Phi_{2}\right) :\Phi_{1},~\Phi_{2}% \in\mathcal{A}\right\} =\infty$. In fact, fix some $\Phi\in\mathcal{A}$ and define a sequence $\left( \Phi_{n}\right) \subset\mathcal{Y}% _{\mathrm{conc}}$ by $\Phi_{2n}=4n\Phi$ and $\Phi_{2n-1}=\left( 2n-1\right) \Phi$, $n\in\mathbb{N}$. It is clear that $\left( \Phi_{n}\right) \subset\mathcal{A}$ and $\operatorname*{d}\left( \Phi_{2n},\Phi _{2n-1}\right) =\left( 2n+1\right) \left\Vert \Phi\right\Vert $, $n\in\mathbb{N}$. Hence, $\operatorname*{diam}(\mathcal{A})=\infty$. On the other hand the set $\left\{ \left( \Phi\left( 1\right) \right) ^{-1}\Phi:\Phi\in\mathcal{Y}_{\mathrm{conc}}\right\} $ is of finite diameter. In fact for any $\Phi$, $\Psi\in\mathcal{Y}_{\mathrm{conc}}$ we have, via Lemma \[lem3\] in [@AGB2005], that $$\operatorname*{d}\left( \left( \Phi\left( 1\right) \right) ^{-1}% \Phi,\left( \Psi\left( 1\right) \right) ^{-1}\Psi\right) \leq\left\Vert \left( \Phi\left( 1\right) \right) ^{-1}\Phi\right\Vert +\left\Vert \left( \Psi\left( 1\right) \right) ^{-1}\Psi\right\Vert \leq2\left\Vert S\right\Vert <\infty.$$ Let us define two relations $\mathrm{\bot}\subset\mathcal{A}\times\mathcal{A}% $ and $\mathrm{\bot}^{\ast}\subset\mathcal{Y}_{\mathrm{conc}}\times \mathcal{Y}_{\mathrm{conc}}$ as follows: 1. We say that $\left( \Phi,\Psi\right) \in\mathrm{\bot}$, where $\left( \Phi,\Psi\right) \in\mathcal{A}\times\mathcal{A}$, (and write $\Phi \mathrm{\bot}\Psi$) if and only if there is some constant $c\in\left( 0,\,\infty\right) $ such that $\Psi\left( x\right) =c\Phi\left( x\right) $ for all $x\in\left( 0,\,\infty\right) $. 2. We say that $\left( \Phi,\Psi\right) \in\mathrm{\bot}^{\ast}$, where $\left( \Phi,\Psi\right) \in\mathcal{Y}_{\mathrm{conc}}\times\mathcal{Y}% _{\mathrm{conc}}$, (and write $\Phi\mathrm{\bot}^{\ast}\Psi$) if and only if there is some constant $c\in\left( 0,\,\infty\right) $ such that $\Psi\left( x\right) =c\Phi\left( x\right) $ for all $x\in\left( 0,\,\infty\right) $. It is not hard to see that $\mathrm{\bot}$ and $\mathrm{\bot}^{\ast}$ are equivalence relations on $\mathcal{A}$ and $\mathcal{Y}_{\mathrm{conc}}$ respectively, i.e. they are reflexive, symmetric and transitive. Their corresponding equivalence classes are respectively$$\begin{aligned} p_{\mathrm{\bot}}\left( \Psi\right) & :=\left\{ \Phi:\Phi\in \mathcal{A}\text{ and }\Phi\mathrm{\bot}\Psi\right\} ,\text{ }\Psi \in\mathcal{A}\\ p_{\mathrm{\bot}^{\ast}}\left( \Delta\right) & :=\left\{ \Phi:\Phi \in\mathcal{Y}_{\mathrm{conc}}\text{ and }\Phi\mathrm{\bot}^{\ast}% \Delta\right\} ,\text{ }\Delta\in\mathcal{Y}_{\mathrm{conc}}%\end{aligned}$$ and their respective induced factor (or quotient) sets can be given by$$\begin{aligned} \mathcal{A}/\mathrm{\bot} & :=\left\{ \mathcal{C}:\mathcal{C}% \subset\mathcal{A}\text{ and }\mathcal{C}=p_{\mathrm{\bot}}\left( \Psi\right) \text{ for some }\Psi\in\mathcal{A}\right\} ,\\ \mathcal{Y}_{\mathrm{conc}}/\mathrm{\bot}^{\ast} & :=\left\{ \mathcal{C}% :\mathcal{C}\subset\mathcal{Y}_{\mathrm{conc}}\text{ and }\mathcal{C}% =p_{\mathrm{\bot}^{\ast}}\left( \Delta\right) \text{ for some }\Delta \in\mathcal{Y}_{\mathrm{conc}}\right\}\end{aligned}$$ One can easily verify that for all $\Psi\in\mathcal{A}$ and $\Delta \in\mathcal{Y}_{\mathrm{conc}}$ the equivalence classes $p_{\mathrm{\bot}% }\left( \Psi\right) $ and $p_{\mathrm{\bot}^{\ast}}\left( \Delta\right) $ are of continuum size or magnitude. \[theo2\]Let $b\in\left( 0,\,\infty\right) $ be any fixed number.**Part I.** Define the mapping $f:\mathcal{A}% \rightarrow\mathcal{A}_{b}$ by $f\left( \Phi\right) =\frac{b}{\Phi\left( b\right) }\Phi$. Then there is a unique mapping $g:\mathcal{A}/\mathrm{\bot }\rightarrow\mathcal{A}_{b}$ for which the diagram$$\xymatrix{\mathcal{A} \ar[r]^{p_{\mathrm{\bot}}} \ar[dr]_f & \mathcal{A}/\mathrm{\bot} \ar[d]^g\\ ~ & \mathbf{b} } \label{com3}%$$ commutes *(*i.e. $f=g\circ p_{\mathrm{\bot}}$*)* and moreover, the mapping $g$ is a bijection.**Part II.** Define the mapping $f^{\ast}:\mathcal{Y}_{\mathrm{conc}}\rightarrow\mathcal{Y}_{b}$ by $f^{\ast }\left( \Delta\right) =\frac{b}{\Delta\left( b\right) }\Delta$. Then there is a unique mapping $g^{\ast}:\mathcal{Y}_{\mathrm{conc}}/\mathrm{\bot}% ^{\ast}\rightarrow\mathcal{Y}_{b}$ for which the diagram$$\xymatrix{\mathcal{Y}_{\mathrm{conc}} \ar[r]^{p_{\mathrm{\bot}^{\ast}}} \ar[dr]_{f^{\ast}} & \mathcal{Y}_{\mathrm{conc}}/\mathrm{\bot}^{\ast} \ar[d]^{g^{\ast}}\\ ~ & \mathbf{b}^{\ast} } \label{com4}%$$ commutes *(*i.e. $f^{\ast}=g^{\ast}\circ p_{\mathrm{\bot}^{\ast}}% $*)* and moreover, the mapping $g^{\ast}$ is a bijection. We point out that the proof of Theorem \[theo2\] is obvious. \[prop1\]Let $b\in\left( 0,\,\infty\right) $ be an arbitrarily fixed number.**Part I.** There is a bijection between $\mathcal{Y}_{b}$ and $\mathcal{Y}_{\mathrm{conc}}$.**Part II.** There is a bijection between $\mathcal{A}_{b}$ and $\mathcal{A}$. We shall only show the first part because the other case can be similarly proved. To this end, write $\mathcal{Y}_{bb}:=\left\{ b\Phi:\Phi \in\mathcal{Y}_{\mathrm{conc}}\right\} $. We note that $\mathcal{Y}_{bb}$ and $\mathcal{Y}_{\mathrm{conc}}$ are equinumerous for the reasons that $\mathcal{Y}_{bb}\subset\mathcal{Y}_{\mathrm{conc}}$ and the function $F:\mathcal{Y}_{\mathrm{conc}}\rightarrow\mathcal{Y}_{bb}$, defined by $F\left( \Phi\right) =b\Phi$, can be easily shown to be an injection. Thus it will be enough to prove that $\mathcal{Y}_{bb}$ and $\mathcal{Y}_{b}$ are equinumerous. In fact, consider the function $Q:\mathcal{Y}_{b}\rightarrow \mathcal{Y}_{bb}$ defined by $Q\left( \frac{b}{\Phi\left( b\right) }% \Phi\right) =b\Phi$. We shall just point out that function $Q$ can be easily shown to be a bijection, which ends the proof. \[cor1\]Let $b\in\left( 0,\,\infty\right) $ be arbitrary. Then the following six sets $\mathcal{A}$, $\mathcal{A}_{b}$, $\mathcal{Y}_{b}$, $\mathcal{A}/\mathrm{\bot}$, $\mathcal{Y}_{\mathrm{conc}}/\mathrm{\bot}^{\ast }$and $\mathcal{Y}_{\mathrm{conc}}$ are equinumerous. We note that $\mathcal{A}$ and $\mathcal{Y}_{\mathrm{conc}}$ are equinumerous (by Theorem \[theo1\]) and, by Theorem 2, $\mathcal{A}/\mathrm{\bot}$ and $\mathcal{A}_{b}$ are equinumerous. On the other hand $\mathcal{A}$ and $\mathcal{A}_{b}$ are equinumerous as well as $\mathcal{Y}_{b}$ and $\mathcal{Y}_{\mathrm{conc}}$ are (by Proposition \[prop1\]). Thus $\mathcal{A}_{b}$ and $\mathcal{Y}_{\mathrm{conc}}$ are equinumerous. Therefore, as $\mathcal{Y}_{b}$ and $\mathcal{Y}_{\mathrm{conc}}/\mathrm{\bot }^{\ast}$ are equinumerous (by Theorem 2), we can conclude on the validity of the argument. \[rem1\]Let $b_{1}$ and $b_{2}\in\left( 0,\,\infty\right) $ be two arbitrary distinct numbers. Then $\mathcal{A}_{b_{1}}\cap\mathcal{A}_{b_{2}}$ and $\mathcal{Y}_{b_{1}}\cap\mathcal{Y}_{b_{2}}$ are empty sets. \[rem2\]Let $b_{1}$ and $b_{2}\in\left( 0,\,\infty\right) $ be two arbitrary distinct numbers. Then $\mathcal{A}_{b_{1}}\cup\mathcal{A}_{b_{2}% }\notin\mathcal{Z}$ and $\mathcal{Y}_{b_{1}}\cup\mathcal{Y}_{b_{2}}% \notin\mathcal{Z}^{\ast}$. \[rem3\]Fix arbitrarily a number $b\in\left( 0,\,\infty\right) $. Then it is easily seen that the function $h_{b}:\left[ 0,~\infty\right) \rightarrow\left[ 0,~\infty\right) $, defined by $h_{b}\left( x\right) =x+b$, is square integrable with respect to measure $\mu$ and, moreover, $C_{b}:=% %TCIMACRO{\dint _{0}^{\infty}}% %BeginExpansion {\displaystyle\int_{0}^{\infty}} %EndExpansion \frac{\left( h_{b}\left( x\right) \right) ^{2}}{\left( x+1\right) ^{4}% }dx=\frac{1}{3}\left( b^{2}+b+1\right) <\infty$. \[rem4\]If $\Phi\in\mathcal{Y}_{b}$, then $\Phi\left( x\right) \leq h_{b}\left( x\right) $ for all $x\in\left[ 0,\,\infty\right) $. Fix any $\Phi\in\mathcal{Y}_{b}$. As $\Phi$ is a concave function its graph must lie below the tangent of equation $y=\varphi\left( b\right) \left( x-b\right) +b$ at point $\left( b,b\right) $ since $\Phi\left( b\right) =b$. Consequently, for all $x\in\left[ 0,\,\infty\right) $ we have:$$\begin{aligned} \Phi\left( x\right) & \leq\varphi\left( b\right) \left( x-b\right) +b\leq\varphi\left( b\right) x+b=b\varphi\left( b\right) \frac{x}{b}+b\\ & \leq\Phi\left( b\right) \frac{x}{b}+b=h_{b}\left( x\right) .\end{aligned}$$ \[prop2\]Let $b\in\left( 0,\,\infty\right) $ be any number. Then $\mathcal{Y}_{b}$ is of finite diameter. Let $b\in\left( 0,\,\infty\right) $ be the source of $\mathcal{Y}_{b}% \in\mathcal{Z}^{\ast}$. We need to prove that $\mathcal{Y}_{b}$ has a finite diameter. In fact, consider two arbitrary functions $\Phi_{1}$, $\Phi_{2}% \in\mathcal{Y}_{b}$. Then$$\operatorname{d}\left( \Phi_{1},\Phi_{2}\right) =\left\Vert \Phi_{1}% -\Phi_{2}\right\Vert \leq\left\Vert \Phi_{1}\right\Vert +\left\Vert \Phi _{2}\right\Vert \leq\sqrt{2C_{b}}\text{,}%$$ via Remarks \[rem4\] and \[rem3\]. Therefore,$$\operatorname*{diam}\left( \mathcal{Y}_{b}\right) :=\sup\left\{ \operatorname{d}\left( \Phi_{1},\Phi_{2}\right) :\Phi_{1},~\Phi_{2}% \in\mathcal{Y}_{b}\right\} \leq\sqrt{2C_{b}}<\infty.$$ \[theo3\]Let $b\in\left( 0,\,\infty\right) $ be any number. Then $\mathcal{Y}_{b}$ is maximally bounded. We just point out that the proof follows from the conjunction of both Propositions \[prop2\] and \[prop1\]. In the sequel $H_{\left[ 0,~1\right] }$ will stand for the collection of all finite sequences $\left( t_{1},~\ldots,~t_{k}\right) \subset\left[ 0,~1\right] $ such that $t_{1}+\ldots+t_{k}=1$. For any fixed $b\in\left( 0,\,\infty\right) $ and every counting number $n\in\mathbb{N}$ write $\operatorname*{X}\limits_{i=1}^{n}\mathcal{A}_{b}$ (resp. $\operatorname*{X}\limits_{i=1}^{n}\mathcal{Y}_{b}$) for the $n$-fold Descartes product of $\mathcal{A}_{b}$ (resp. $\mathcal{Y}_{b}$). For $n=1$ let us set $\mathcal{A}_{b}^{\left( 1\right) }=\mathcal{A}_{b}$, $\mathcal{Y}_{b}^{\left( 1\right) }=\mathcal{Y}_{b}$ and whenever $n\geq2$, write $\mathcal{Y}_{b}^{CO\left( n\right) }=\left\{ \Delta_{1}\circ \Delta_{2}\circ\ldots\circ\Delta_{n}:\left( \Delta_{1},\Delta_{2}% ,\ldots,\Delta_{n}\right) \in\operatorname*{X}\limits_{i=1}^{n}% \mathcal{Y}_{b}\right\} $, $$\mathcal{A}_{b}^{CO\left( n\right) }=\left\{ \Phi_{1}\circ\ldots\circ \Phi_{n}:\left( \Phi_{1},\ldots,\Phi_{n}\right) \in\operatorname*{X}% \limits_{i=1}^{n}\mathcal{Y}_{b}\text{ and }\Phi_{j}\in\mathcal{A}_{b}\text{ for some index }j\right\} ,$$ $\mathcal{Y}_{b}^{\left( n\right) }=\left\{ \sum_{i=1}^{k}t_{i}\Delta _{i}:\Delta_{1},~\Delta_{2},~\ldots,~\Delta_{k}\in\mathcal{Y}_{b}^{CO\left( n\right) },~\left( t_{1},~\ldots~t_{k}\right) \in H_{\left[ 0,~1\right] }\right\} $, $\mathcal{A}_{b}^{\left( n\right) }=\left\{ \sum_{i=1}% ^{k}t_{i}\Phi_{i}:\Phi_{1},~\Phi_{2},~\ldots,~\Phi_{k}\in\mathcal{A}% _{b}^{CO\left( n\right) },~\left( t_{1},~\ldots~t_{k}\right) \in H_{\left[ 0,~1\right] }\right\} $. Further, for $n=1$ write $\mathcal{Z}^{\left( 1\right) }=\mathcal{Z}$, $\mathcal{Z}^{\ast\left( 1\right) }=\mathcal{Z}^{\ast}$ and, for $n\in\mathbb{N}\backslash\left\{ 1\right\} $ write $\mathcal{Z}^{\left( n\right) }:=\left\{ \mathcal{A}_{b}^{\left( n\right) }:b\in\left( 0,\,\infty\right) \right\} $ and $\mathcal{Z}^{\ast\left( n\right) }:=\left\{ \mathcal{Y}_{b}^{\left( n\right) }:b\in\left( 0,\,\infty \right) \right\} $. \[rem5\]For any pair of numbers $n\in\mathbb{N}$ and $b\in\left( 0,\,\infty\right) $ the set $\mathcal{A}_{b}^{\left( n\right) }$ is a proper subset of $\mathcal{Y}_{b}^{\left( n\right) }$. \[rem6\]For any pair of numbers $n\in\mathbb{N}$ and $b\in\left( 0,\,\infty\right) $ we have $\mathcal{A}_{b}^{\left( n\right) }% \subset\mathcal{A}_{b}^{\left( 1\right) }=\mathcal{A}_{b}$. We point out that Remark \[rem6\] is a direct consequent of Theorem 2 in [@AGB2005], page 6. \[rem7\]Let $b\in\left( 0,\,\infty\right) $, $n\in\mathbb{N}$ and $k\geq n$ be arbitrary numbers. Then *(1)* $\Phi_{1}\circ\Phi_{2}% \circ\ldots\circ\Phi_{k}\in\mathcal{A}_{b}^{CO\left( n\right) }$ whenever $\Phi_{1},~\Phi_{2},~\ldots,~\Phi_{k}\in\mathcal{Y}_{b}^{\left( 1\right) }$ and $\Phi_{j}\in\mathcal{A}_{b}^{\left( 1\right) }$ for some index $j\in\left\{ 1,~\ldots,~k\right\} $*(2)* $\Delta_{1}\circ \Delta_{2}\circ\ldots\circ\Delta_{k}\in\mathcal{Y}_{b}^{CO\left( n\right) }$ whenever $\Delta_{1},~\Delta_{2},~\ldots,~\Delta_{k}\in\mathcal{Y}% _{b}^{\left( 1\right) }$. Note that $\Phi_{1}\circ\Phi_{2}\circ\ldots\circ\Phi_{k}=\Phi_{1}\circ\Phi _{2}\circ\ldots\circ\Phi_{n-1}\circ\Psi_{1}$ and $\Delta_{1}\circ\Delta _{2}\circ\ldots\circ\Delta_{n-1}\circ\Psi_{2}$, where $\Psi_{1}=\Phi_{n}% \circ\Phi_{n+1}\circ\ldots\circ\Phi_{k}$ and $\Psi_{2}=\Delta_{n}\circ \Delta_{n+1}\circ\ldots\circ\Delta_{k}$. From this simple observation the result easily follows. From Remark \[rem7\] the following result can be easily derived, since it implies that $\mathcal{A}_{b}^{CO\left( n+1\right) }$ is a proper subset of $\mathcal{A}_{b}^{CO\left( n\right) }$ and, $\mathcal{Y}_{b}^{CO\left( n+1\right) }$ is also a proper subset of $\mathcal{Y}_{b}^{CO\left( n\right) }$. \[lem2\]Let $b\in\left( 0,\,\infty\right) $ and $n\in\mathbb{N}$ be arbitrary numbers. Then the following two assertions are valid.*(1)* The set $\mathcal{A}_{b}^{\left( n+1\right) }$ is a proper subset of $\mathcal{A}_{b}^{\left( n\right) }$.*(2)* The set $\mathcal{Y}_{b}^{\left( n+1\right) }$ is a proper subset of $\mathcal{Y}% _{b}^{\left( n\right) }$. \[theo4\]For any fixed pair of numbers $n\in\mathbb{N}$ and $b\in\left( 0,\,\infty\right) $, the two sets $\mathcal{A}_{b}^{\left( n\right) }$ and $\mathcal{A}_{b}$ are equinumerous. Throughout the proof we shall fix any counting number $n\in\mathbb{N}$. We first note that the identity function $I_{\operatorname{id}}:\mathcal{A}% _{b}^{\left( n\right) }\rightarrow\mathcal{A}_{b}$ is an injection, since $\mathcal{A}_{b}^{\left( n\right) }\subset\mathcal{A}_{b}$. Next, pick any $\Delta\in\mathcal{A}_{b}$ and define the function $f_{\Delta}:\mathcal{A}% _{b}\rightarrow\mathcal{A}_{b}^{\left( n\right) }$ by $f_{\Delta}\left( \Phi\right) =\underset{\left( n-1\right) \text{-fold}}{\underbrace {\Delta\circ\ldots\circ\Delta}}\circ\Phi$. We show that $f_{\Delta}$ is an injection. In fact, let $\Phi_{1}$, $\Phi_{2}\in\mathcal{A}_{b}$ be arbitrary and assume that $f_{\Delta}\left( \Phi_{1}\right) =f_{\Delta}\left( \Phi_{2}\right) $. Then taking into account that $\Delta$ is an invertible function we can easily deduce that $\Phi_{1}=\Phi_{2}$, i.e. $f_{\Delta}$ is an injection. Therefore, the Schröder-Bernstein theorem entails that there is a bijection between $\mathcal{A}_{b}$ and $\mathcal{A}_{b}^{\left( n\right) }$. This was to be proved. \[prop3\]For any pair of numbers $n\in\mathbb{N}$ and $b\in\left( 0,~\infty\right) $ the sets $\mathcal{A}_{b}^{\left( n\right) }$ and $\mathcal{Y}_{b}^{\left( n\right) }\backslash\mathcal{A}_{b}^{\left( n\right) }$ are equinumerous. Let $\Phi\in\mathcal{Y}_{b}^{\left( n\right) }\backslash\mathcal{A}% _{b}^{\left( n\right) }$ and $\left( \alpha,\beta\right) \in H_{\left[ 0,~1\right] }$ be arbitrarily fixed. Define the function $h_{\Phi}^{\left( \alpha,\beta\right) }:\mathcal{A}_{b}^{\left( n\right) }\rightarrow \mathcal{Y}_{b}^{\left( n\right) }\backslash\mathcal{A}_{b}^{\left( n\right) }$ by $h_{\Phi}^{\left( \alpha,\beta\right) }\left( \Delta\right) =\alpha\Delta+\beta\Phi$. It is clear that $h_{\Phi}^{\left( \alpha,\beta\right) }$ is actually an injection. Now, fix any $\Delta \in\mathcal{A}_{b}^{\left( n\right) }$ and define the function $f_{\Delta }:\mathcal{Y}_{b}^{\left( n\right) }\backslash\mathcal{A}_{b}^{\left( n\right) }\rightarrow\mathcal{A}_{b}^{\left( n\right) }$ by $f_{\Delta }\left( \Phi\right) =\Delta\circ\Phi$. We note that this function always exists because of the inclusion $\mathcal{A}_{b}^{\left( n\right) }% \subset\mathcal{A}$ and Theorem \[theo2\] in [@AGB2005]. Here too we can easily check that $f_{\Delta}$ is an injection. Therefore, The Schröder-Bernstein theorem yields the result to be proven. For any pair of numbers $n\in\mathbb{N}$ and $b\in\left( 0,~\infty\right) $ the following five sets $\mathcal{Y}_{b}^{\left( n\right) }$, $\mathcal{A}% _{b}^{\left( n\right) }$, $\mathcal{Y}_{b}^{\left( n\right) }% \backslash\mathcal{A}_{b}^{\left( n\right) }$, $\mathcal{A}$ and $\mathcal{Y}_{\mathrm{conc}}$ are pairwise equinumerous. The metrization of sets $\mathcal{Z}^{\left( n\right) }$ and $\mathcal{Z}^{\ast\left( n\right) }$ ===================================================================================================== We shall only deal with the metrization of sets $\mathcal{Z}$ and $\mathcal{Z}^{\ast}$ since all the results in this section can be easily extended to the sets $\mathcal{Z}^{\left( n\right) }$ and $\mathcal{Z}% ^{\ast\left( n\right) }$. Whenever $\Phi\in\mathcal{Y}_{\mathrm{conc}}$ write $G_{\Phi}:=\left\{ \left( x,\Phi\left( x\right) \right) :x\in\left( 0,~\infty\right) \right\} $ for the graph of $\Phi$ on $\left( 0,~\infty\right) $ and $G_{\Phi}^{a||b}:=\left\{ \left( x,\Phi\left( x\right) \right) :x\in\left[ a,~b\right] \right\} $ for the graph of $\Phi$ on the interval $\left[ a,~b\right] $ where $a<b$ are any non-negative numbers. \[rem8\]Let $b_{1}$ and $b_{2}\in\left( 0,\,\infty\right) $ be two arbitrary distinct numbers. If $b_{1}<b_{2}$, then the following two assertions hold true:*(1)* For all $\Phi_{1}\in\mathcal{A}% _{b_{1}}$ and $\Phi_{2}\in\mathcal{A}_{b_{2}}$ the inequality $\Phi _{1}\left( b_{2}\right) <\Phi_{2}\left( b_{1}\right) $ holds.$\emph{(2)}$ For all $\Phi_{1}\in\mathcal{Y}_{b_{1}}$ and $\Phi_{2}% \in\mathcal{Y}_{b_{2}}$ the inequality $\Phi_{1}\left( b_{2}\right) <\Phi_{2}\left( b_{1}\right) $ holds. Suppose that $b_{1}<b_{2}$ and fix arbitrarily two functions $\Phi_{1}% \in\mathcal{Y}_{b_{1}}$ and $\Phi_{2}\in\mathcal{Y}_{b_{2}}$. Obviously, $\Phi_{1}$ must hit $\Phi_{\operatorname{id}}$ *prior to* $\Phi_{2}$. Hence, $G_{\Phi_{1}}^{b_{1}||b_{2}}$ lies below $G_{\Phi_{2}}^{b_{1}||b_{2}}$. But since $G_{\Phi_{1}}^{b_{1}||\infty}$ lies above the graph of the line of equation $y=b_{1}$ in the interval $\left( b_{1},\,\infty\right) $, we have as an aftermath that $\Phi_{1}\left( b_{1}\right) <\Phi_{1}\left( b_{2}\right) <\Phi_{2}\left( b_{1}\right) $. To end the proof we note that assertion (2) can be similarly shown. The binary relations $\prec$ and $\preceq$ , defined on $\mathcal{Z}$ respectively by $\mathcal{A}_{b_{1}}\prec\mathcal{A}_{b_{2}}$ if and only if $\Phi_{1}\left( b_{2}\right) <\Phi_{2}\left( b_{1}\right) $ for all pairs $\left( \Phi_{1},\Phi_{2}\right) \in\mathcal{A}_{b_{1}}\times\mathcal{A}% _{b_{2}}$, and by $\mathcal{A}_{b_{1}}\preceq\mathcal{A}_{b_{2}}$ if and only if $\mathcal{A}_{b_{1}}\prec\mathcal{A}_{b_{2}}$ or $\mathcal{A}_{b_{1}% }=\mathcal{A}_{b_{2}}$. We point out that The binary relations $\prec$ and $\preceq$ can be similarly defined on $\mathcal{Z}^{\ast}$. We point out that the law of trichotomy is valid on $\left( \mathcal{Z}% ,\preceq\right) $ and $\left( \mathcal{Z}^{\ast},\preceq\right) $, i.e. whenever $\left( \mathcal{A}_{b_{1}},\mathcal{A}_{b_{2}}\right) \in\mathcal{Z}\times\mathcal{Z}$ or $\left( \mathcal{A}_{b_{1}}% ,\mathcal{A}_{b_{2}}\right) \in\mathcal{Z}^{\ast}\times\mathcal{Z}^{\ast}$, then precisely one of the following holds: $\mathcal{A}_{b_{1}}=\mathcal{A}% _{b_{2}}$, $\mathcal{A}_{b_{1}}\prec\mathcal{A}_{b_{2}}$, $\mathcal{A}_{b_{2}% }\prec\mathcal{A}_{b_{1}}$. Hence, we can easily check that $\left( \mathcal{Z},\preceq\right) $ and $\left( \mathcal{Z}^{\ast},\preceq\right) $ are chains, i.e. they are totally ordered sets. \[theo5\]The functions $f_{1}:\left( 0,\,\infty\right) \rightarrow \mathcal{Z}$ and $f_{2}:\left( 0,\,\infty\right) \rightarrow\mathcal{Z}% ^{\ast}$, defined respectively by $f_{1}\left( p\right) =\mathcal{A}_{p}$ and $f_{2}\left( p\right) =\mathcal{Y}_{p}$, are order preserving bijections. We show that the function $f_{1}:\left( 0,\,\infty\right) \rightarrow \mathcal{Z}$, $f_{1}\left( p\right) =\mathcal{A}_{p}$, is an order preserving bijection. In fact, it is not hard to see via Remark \[rem1\] that $f_{1}$ is an injection. Now pick any element $\mathcal{C}\in\mathcal{Z}% $. Obviously, there must exist some number $p\in\left( 0,\,\infty\right) $ such that $\mathcal{C}=\mathcal{A}_{p}=f_{1}\left( p\right) $, i.e. $f_{1}$ is a surjection. Consequently, $f_{1}$ is a bijection. To end the proof of this part we simply point out that the bijection $f_{1}$ is order preserving in virtue of Remark \[rem5111\]. Finally, we note that we can similarly prove that $f_{2}$ is also an order preserving bijection. Since the sets $\left( \mathcal{Z},\preceq\right) $ and $\left( \mathcal{Z}^{\ast},\preceq\right) $ are chains it is natural to look for a metric on them. We shall do this in the following two results. But before that let us recall the definitions of some distances known in the literature (cf. [@KUR1966], say). If $\Phi\in\mathcal{Y}_{\mathrm{conc}}$ is any function and $\mathcal{F}$, $\mathcal{G}\subset\mathcal{Y}_{\mathrm{conc}}$ are arbitrary non-empty subsets, then we define the distance from the point $\Phi$ to the set $\mathcal{G}$ by $\rho\left( \Phi,\mathcal{G}\right) :=\inf\left\{ \operatorname{d}\left( \Phi,\Psi\right) :\Psi\in \mathcal{G}\right\} =\inf\left\{ \operatorname{d}\left( \Psi,\Phi\right) :\Psi\in\mathcal{G}\right\} =\rho\left( \mathcal{G},\Phi\right) $ and the distance between the two sets $\mathcal{F}$ and $\mathcal{G}$ by $$\begin{aligned} \operatorname*{dist}\left( \mathcal{F},\mathcal{G}\right) & :=\sup\left\{ \inf\left\{ \operatorname{d}\left( \Phi,\Psi\right) :\Psi\in\mathcal{G}% \right\} :\Phi\in\mathcal{F}\right\} \\ & =\sup\left\{ \inf\left\{ \operatorname{d}\left( \Phi,\Psi\right) :\Phi\in\mathcal{F}\right\} :\Psi\in\mathcal{G}\right\} .\end{aligned}$$ First we find sufficient conditions for which the distance from a point to a subset (both in $\mathcal{Y}_{\mathrm{conc}}$) should be positive, in order to guarantee that the distance between two sets in $\mathcal{Y}_{\mathrm{conc}}$ have sense. \[lem3\]Let $b_{1}$ and $b_{2}\in\left( 0,\,\infty\right) $ be two arbitrary distinct numbers. Then $\rho\left( \mathcal{Y}_{b_{1}},\Phi _{2}\right) >0$ and $\rho\left( \mathcal{A}_{b_{1}},\Phi_{2}\right) >0$ whenever $\Phi_{2}\in\mathcal{Y}_{b_{2}}$. It is enough to show that $\rho\left( \mathcal{A}_{b_{1}},\Phi_{2}\right) >0$ whenever $\Phi_{2}\in\mathcal{Y}_{b_{2}}$. In fact, suppose in the contrary that $\rho\left( \mathcal{A}_{b_{1}},\Phi_{2}\right) =0$ for some $\Phi_{2}\in\mathcal{Y}_{b_{2}}$. Then there can be extracted some sequence $\left( \Delta_{n}\right) \subset\mathcal{A}_{b_{1}}$ such that $\operatorname{d}\left( \Delta_{n},\Phi_{2}\right) >\operatorname{d}\left( \Delta_{n+1},\Phi_{2}\right) $, $n\in\mathbb{N}$, and $\lim_{n\rightarrow \infty}\operatorname{d}\left( \Delta_{n},\Phi_{2}\right) =\rho\left( \mathcal{A}_{b_{1}},\Phi_{2}\right) =0$. We point out that this can be done because of the definition of the infimum. For each $n\in\mathbb{N}$ let us set $\Gamma_{n}:=\inf_{k\geq n}\left( \Delta_{k}-\Phi_{2}\right) ^{2}% $. Clearly, $\left( \Gamma_{n}\right) $ is a non-decreasing sequence of measurable functions with its corresponding sequence of integrals $\left( \int_{\left[ 0,\text{ }\infty\right) }\Gamma_{n}d\mu\right) $ been bounded above by $C_{b_{1}}+C_{b_{2}}<\infty$, see Remark \[rem3\]. Then by the Beppo Levi’s Theorem we can derive that sequence $\left( \Gamma_{n}\right) $ converges almost everywhere to some integrable measurable function $\Gamma$ and $\int_{\left[ 0,\text{ }\infty\right) }\Gamma d\mu=\lim_{n\rightarrow \infty}\int_{\left[ 0,\text{ }\infty\right) }\Gamma_{n}d\mu\leq \lim_{n\rightarrow\infty}\operatorname{d}\left( \Delta_{n},\Phi_{2}\right) =0$, meaning that $\lim_{n\rightarrow\infty}\inf_{k\geq n}\Delta_{k}=\Phi_{2}$ almost everywhere. There are two cases to be clarified. First assume that $b_{1}<b_{2}$. Obviously, $\mu\left( \left( b_{1},~b_{2}\right) \right) >0$, so that there must be at least one point $x_{0}\in\left( b_{1}% ,~b_{2}\right) $ such that $\lim_{n\rightarrow\infty}\inf_{k\geq n}\Delta _{k}\left( x_{0}\right) =\Phi_{2}\left( x_{0}\right) $. But since $b_{1}<b_{2}$ the concave property implies that the graph of $\Phi_{2}% $ (resp. the graph of each function $\inf_{k\geq n}\Delta_{k}$) lies above (resp. below) the graph of the line of equation $y=x$ in the interval $\left( b_{1},~b_{2}\right) $. Consequently, $\lim_{n\rightarrow\infty}\inf_{k\geq n}\Delta_{k}\left( x_{0}\right) \leq x_{0}<\Phi_{2}\left( x_{0}\right) $. This, however, is absurd since $\lim_{n\rightarrow\infty}\inf_{k\geq n}% \Delta_{k}\left( x_{0}\right) =\Phi_{2}\left( x_{0}\right) $. Considering the second case when $b_{1}>b_{2}$ we can similarly get into a contradiction. Therefore, the statement is valid. \[lem4\]Let $b$ and $c\in\left( 0,\,\infty\right) $ be two arbitrary numbers. Then the following assertions are equivalent:*(1)* The equality $b=c$ holds.*(2)* The sets $\mathcal{Y}_{b}$ and $\mathcal{Y}_{c}$ are equal.*(3)* The equality $\operatorname*{dist}\left( \mathcal{Y}_{b},\mathcal{Y}_{c}\right) =0$ holds. We first note that the chain of implications (1) $\rightarrow$ (2) $\rightarrow$ (3) is obviously true. Thus we need only show the conditional (3) $\rightarrow$ (1). In fact, assume that $\operatorname*{dist}\left( \mathcal{Y}_{b},\mathcal{Y}_{c}\right) =0$ but $b\neq c$. Then $\rho\left( \mathcal{Y}_{b},\Delta\right) =0$ for all $\Delta\in\mathcal{Y}_{c}$. Nevertheless, this contradicts Lemma \[lem3\], since $b\neq c$. Therefore, the argument is valid. We can similarly prove that: \[lem5\]Let $b$ and $c\in\left( 0,\,\infty\right) $ be two arbitrary numbers. Then the following assertions are equivalent*:(1)* The equality $b=c$ holds.*(2)* The sets $\mathcal{A}% _{b}$ and $\mathcal{A}_{c}$ are equal.*(3)* The equality $\operatorname*{dist}\left( \mathcal{A}_{b},\mathcal{A}_{c}\right) =0$ holds. \[theo6\]Let $b$ and $c\in\left( 0,\,\infty\right) $ be two arbitrary numbers. Then the quantities $\operatorname*{dist}\left( \mathcal{A}% _{b},\mathcal{A}_{c}\right) $ and $\operatorname*{dist}\left( \mathcal{Y}_{b},\mathcal{Y}_{c}\right) $ define metrics on $\mathcal{Z}$ and $\mathcal{Z}^{\ast}$ respectively. Hence, the couples $\left( \mathcal{Z}% ,\operatorname*{dist}\right) $ and $\left( \mathcal{Z}^{\ast}% ,\operatorname*{dist}\right) $ are metric spaces. We need only show that $\operatorname*{dist}\left( \mathcal{Y}_{b}% ,\mathcal{Y}_{c}\right) $ is a metric on the set $\mathcal{Z}^{\ast}$, because the other case can be similarly proved. In fact, we first point out that the condition $\operatorname*{dist}\left( \mathcal{Y}_{b},\mathcal{Y}% _{c}\right) \geq0$ is obvious and, by Lemma \[lem4\] the equality holds if and only if $\mathcal{Y}_{b}=\mathcal{Y}_{c}$. We also note that the symmetry property trivially holds true. We are now left with the proof of the triangle inequality. In fact, let $\mathcal{Y}_{b_{j}}\in\mathcal{Z}^{\ast}$ and $\Phi_{j}\in\mathcal{Y}_{b_{j}}$ ($j\in\left\{ 1,~2,~3\right\} $) be arbitrary. Then by Proposition 5 (cf. [@AGB2005], page 15) we have that $\operatorname{d}\left( \Phi_{1},\Phi_{3}\right) \leq\operatorname{d}\left( \Phi_{1},\Phi_{2}\right) +\operatorname{d}\left( \Phi_{2},\Phi_{3}\right) $. Next, by taking the infimum over $\Phi_{3}\in\mathcal{Y}_{b_{3}}$ it follows that $$\rho\left( \Phi_{1},\mathcal{Y}_{b_{3}}\right) \leq\operatorname{d}\left( \Phi_{1},\Phi_{2}\right) +\rho\left( \Phi_{2},\mathcal{Y}_{b_{3}}\right) \leq\operatorname{d}\left( \Phi_{1},\Phi_{2}\right) +\operatorname*{dist}% \left( \mathcal{Y}_{b_{2}},\mathcal{Y}_{b_{3}}\right) ,$$ i.e. $\rho\left( \Phi_{1},\mathcal{Y}_{b_{3}}\right) \leq\operatorname{d}% \left( \Phi_{1},\Phi_{2}\right) +\operatorname*{dist}\left( \mathcal{Y}% _{b_{2}},\mathcal{Y}_{b_{3}}\right) $. Finally, taking the infimum over $\Phi_{2}\in\mathcal{Y}_{b_{2}}$ yields $\rho\left( \Phi_{1},\mathcal{Y}% _{b_{3}}\right) \leq\rho\left( \Phi_{1},\mathcal{Y}_{b_{2}}\right) +\operatorname*{dist}\left( \mathcal{Y}_{b_{2}},\mathcal{Y}_{b_{3}}\right) $, so that $$\operatorname*{dist}\left( \mathcal{Y}_{b_{1}},\mathcal{Y}_{b_{3}}\right) \leq\operatorname*{dist}\left( \mathcal{Y}_{b_{1}},\mathcal{Y}_{b_{2}% }\right) +\operatorname*{dist}\left( \mathcal{Y}_{b_{2}},\mathcal{Y}_{b_{3}% }\right) .$$ This was to be proven. By the law of trichotomy it is not hard to see that $\left( \mathcal{Z}% ,\preceq\right) $ and $\left( \mathcal{Z}^{\ast},\preceq\right) $ are lattices. Here too, the supremum and infimum binary operations on the lattices $\left( \mathcal{Z},\preceq\right) $ and $\left( \mathcal{Z}^{\ast}% ,\preceq\right) $ will be denoted by the usual symbols $\vee$ and $\wedge$ respectively. We also point out that $\left( \mathcal{Z},\preceq\right) $ and $\left( \mathcal{Z}^{\ast},\preceq\right) $ are infinite graphs. Between two vertices $\mathcal{A}_{b_{1}}$, $\mathcal{A}_{b_{2}}\in\mathcal{Z}$ we can define the edge in two different ways: one by $e=\operatorname*{dist}\left( \mathcal{A}_{b_{1}},\mathcal{A}_{b_{2}}\right) \in\left( 0,~\infty\right) $ and the other one by $\mathcal{A}_{e}\in\mathcal{Z}$ where $e=\operatorname*{dist}\left( \mathcal{A}_{b_{1}},\mathcal{A}_{b_{2}}\right) $. These two edges can apply for the vertices of $\mathcal{Z}^{\ast}$ as well. Dense subsets in $\mathcal{Y}_{b}^{\left( n\right) }$ ======================================================= \[theo7\]Let $b\in\left( 0,\,\infty\right) $ be an arbitrary number. Then $\mathcal{A}_{b}$ is a dense set in $\mathcal{Y}_{b}$. Fix arbitrarily any function $\Psi\in\mathcal{Y}_{b}$. Then there is some $\Phi\in\mathcal{Y}_{\mathrm{conc}}$ such that $\Psi=\frac{b\Phi}{\Phi\left( b\right) }$ (by Lemma \[lem1\]). Define $\Psi_{n}\left( x\right) =\frac{b\left( \Phi\left( x\right) \right) ^{1-1/(n+1)}}{\left( \Phi\left( b\right) \right) ^{1-1/(n+1)}}$, for all $x\in\left[ 0,\,\infty\right) $ and $n\in\mathbb{N}$. As we know from Theorem \[theo2\] (cf. [@AGB2005], page 6) function $\Phi^{1-1/(n+1)}\in\mathcal{A}$ for all $\Phi\in\mathcal{Y}_{\mathrm{conc}}$, $n\in\mathbb{N}$. Then $\left( \Psi _{n}\right) \subset\mathcal{A}$ (via Lemma \[lem1\], [@AGB2005], page 5). Hence, $\left( \Psi_{n}\right) \subset\mathcal{A}_{b}$, since $\Psi _{n}\left( b\right) =b$ for all $n\in\mathbb{N}$. We can easily show that $\left( \Psi_{n}\right) $ converges pointwise to $\Psi$. By Remark \[rem4\] it ensues that $\Psi\left( x\right) \leq h_{b}\left( x\right) $ and $\Psi_{n}\left( x\right) \leq h_{b}\left( x\right) $ for all $x\in\left[ 0,\,\infty\right) $ and $n\in\mathbb{N}$, where $h_{b}\left( x\right) =x+b$, $x\in\left[ 0,\,\infty\right) $. We know via Remark \[rem3\] that function $h_{b}$ is square integrable. Then by applying twice the Dominated Convergence Theorem one can verify that$$\lim_{n\rightarrow\infty}% %TCIMACRO{\dint _{\left[ 0,\,\infty\right) }}% %BeginExpansion {\displaystyle\int_{\left[ 0,\,\infty\right) }} %EndExpansion \Psi_{n}^{2}d\mu=% %TCIMACRO{\dint _{\left[ 0,\,\infty\right) }}% %BeginExpansion {\displaystyle\int_{\left[ 0,\,\infty\right) }} %EndExpansion \Psi^{2}d\mu\text{ \ and }\lim_{n\rightarrow\infty}% %TCIMACRO{\dint _{\left[ 0,\,\infty\right) }}% %BeginExpansion {\displaystyle\int_{\left[ 0,\,\infty\right) }} %EndExpansion \Psi_{n}\Psi d\mu=% %TCIMACRO{\dint _{\left[ 0,\,\infty\right) }}% %BeginExpansion {\displaystyle\int_{\left[ 0,\,\infty\right) }} %EndExpansion \Psi^{2}d\mu,$$ so that $\lim_{n\rightarrow\infty}\operatorname{d}\left( \Psi,\Psi _{n}\right) =0$, because $\Psi\left( x\right) \Psi_{n}\left( x\right) \leq\left( h_{b}\left( x\right) \right) ^{2}$ for all $x\in\left[ 0,\,\infty\right) $ and $n\in\mathbb{N}$ (by Remark \[rem4\]). This was to be proven. \[theo8\]Fix any pair of numbers $n\in\mathbb{N}\backslash\left\{ 1\right\} $ and $b\in\left( 0,~\infty\right) $. Then $\mathcal{A}% _{b}^{\left( n\right) }$ is dense in $\mathcal{Y}_{b}^{\left( n\right) }$. Pick arbitrarily some $\Delta\in\mathcal{Y}_{b}^{\left( n\right) }$. Since obviously $\mathcal{Y}_{b}^{CO\left( n\right) }$ is a proper subset of $\mathcal{Y}_{b}^{\left( n\right) }$, we will have two cases to take into consideration. First assume that $\Delta\in\mathcal{Y}_{b}^{CO\left( n\right) }$. This means that there can be found a counting number $k\geq n$ and a finite sequence $\Phi_{1},~\ldots,~\Phi_{k}\in\mathcal{Y}_{b}^{\left( 1\right) }=\mathcal{Y}_{b}$ such that $\Delta=\Phi_{1}\circ\ldots\circ \Phi_{k}$. Fix any integer $j\in\mathbb{N}$ and write $\Delta_{j}=\Psi _{j}\circ\Delta$, where $\Psi_{j}\left( x\right) =\left( b^{1/j}x\right) ^{j/(j+1)}$, $x\in\left[ 0,~\infty\right) $. Clearly, $\Psi_{j}% \in\mathcal{A}_{b}^{\left( 1\right) }$ for all $j\in\mathbb{N}$. Then applying Theorem \[theo2\] in [@AGB2005] and via the structure of set $\mathcal{A}_{b}^{CO\left( n\right) }$, we can deduce that $\Delta_{j}% \in\mathcal{A}_{b}^{CO\left( n\right) }$ for all $j\in\mathbb{N}$. It is not difficult to see that sequence $\left( \Delta_{j}\right) $ converge pointwise to $\Delta$. By Remark \[rem4\] we observe that $\Delta\leq h_{b}% $, $\Delta_{j}\leq h_{b}$ and hence, $\Delta\Delta_{j}\leq\left( h_{b}\right) ^{2}$ on $\left[ 0,~\infty\right) $. Then recalling twice the Dominated Convergence Theorem we can easily verify that $$\lim_{j\rightarrow\infty}\int_{\left[ 0,~\infty\right) }\left( \Delta _{j}\right) ^{2}d\mu=\int_{\left[ 0,~\infty\right) }\Delta^{2}d\mu =\lim_{j\rightarrow\infty}\int_{\left[ 0,~\infty\right) }\Delta\Delta _{j}d\mu.$$ Consequently, $\lim_{j\rightarrow\infty}\operatorname{d}\left( \Delta ,\Delta_{j}\right) =0$. In the second case we can suppose that $\Delta \in\mathcal{Y}_{b}^{\left( n\right) }\backslash\mathcal{Y}_{b}^{CO\left( n\right) }$. Then without loss of generality we may choose $\Phi_{1}% ,~\ldots,~\Phi_{k}\in\mathcal{Y}_{b}^{CO\left( n\right) }$, whose graphs are pairwise distinct, and some finite sequence $\left( t_{1},~\ldots ~t_{k}\right) \in H_{\left[ 0,~1\right] }$ with $\left( t_{1}% ,~\ldots~t_{k}\right) \subset\left( 0,~1\right) $ such that $\Delta =\sum_{i=1}^{k}t_{i}\Phi_{i}$. Consider $\Delta_{j}=\sum_{i=1}^{k}t_{i}\left( \Psi_{j}\circ\Phi_{i}\right) $, where $\Psi_{j}\left( x\right) =\left( b^{1/j}x\right) ^{j/(j+1)}$, $x\in\left[ 0,~\infty\right) $, $j\in \mathbb{N}$. Clearly, on the one hand we have that $\left( \Delta_{j}\right) \subset\mathcal{A}_{b}^{\left( n\right) }$ because $\left( \Psi_{j}% \circ\Phi_{i}\right) \subset\mathcal{A}_{b}^{CO\left( n\right) }$ for every fixed index $i\in\left\{ 1,~\ldots~k\right\} $ and on the other hand $\lim_{j\rightarrow\infty}\operatorname{d}\left( \Phi_{i},\Psi_{j}\circ \Phi_{i}\right) =0$, $i\in\left\{ 1,~\ldots~k\right\} $, because of the first part of this proof. Consequently, by the Minkowski inequality we can observe that $\lim_{j\rightarrow\infty}\operatorname{d}\left( \Delta ,\Delta_{j}\right) \leq\sum_{i=1}^{k}t_{i}\lim_{j\rightarrow\infty }\operatorname{d}\left( \Phi_{i},\Psi_{j}\circ\Phi_{i}\right) =0$. This completes the proof. Some criterium on the $L^{p}$-norm ================================== The result here below is worth being mentioned, which is an answer to the second open problem in [@AGB2005]. \[theo9\]Let $\Phi\in\mathcal{Y}_{\mathrm{conc}}$ be arbitrary. Then the following assertions are equivalent.*(1)* $\lim_{t\rightarrow \infty}\frac{\Phi\left( t\right) }{t}=\lim_{t\rightarrow\infty}% \varphi\left( t\right) \in\left( 0,\,\infty\right) $.*(2)* There is some constant $c\in\left[ 1,\,\infty\right) $ such that $c\Phi >\Phi_{\operatorname{id}}$ on $\left( 0,\,\infty\right) $.*(3)* There is some constant $c\in\left[ 1,\,\infty\right) $ and some strictly concave function $\Delta:\left[ 0,\,\infty\right) \rightarrow\left[ 0,\,\infty\right) $, differentiable on $\left( 0,\,\infty\right) $ and vanishing at the origin such that $c\Phi=\Phi_{\operatorname{id}}+\Delta$ on $\left[ 0,\,\infty\right) $. We first prove the conditional (1)$\rightarrow$(2). In fact, assume that $\lim_{t\rightarrow\infty}\frac{\Phi\left( t\right) }{t}\in\left( 0,\,\infty\right) $ but in the contrary for every counting number $k\in\mathbb{N}$ there is some $x_{k}\in\left( 0,\,\infty\right) $ for which $k\Phi\left( x_{k}\right) \leq x_{k}$. Obviously, $\limsup \limits_{k\rightarrow\infty}\frac{\Phi\left( x_{k}\right) }{x_{k}}\leq \lim_{k\rightarrow\infty}k^{-1}=0$ which is absurd since $\limsup \limits_{k\rightarrow\infty}\frac{\Phi\left( x_{k}\right) }{x_{k}}\in\left( 0,\,\infty\right) $ by the assumption. Next we show the implication (2)$\rightarrow$(3). In fact, assume that there is some constant $c\in\left[ 1,\,\infty\right) $ such that $c\Phi>\Phi_{\operatorname{id}}$ on $\left( 0,\,\infty\right) $ and write $\Delta:=c\Phi-\Phi_{\operatorname{id}}$. Clearly, $\Delta:\left[ 0,\,\infty\right) \rightarrow\left[ 0,\,\infty \right) $ is a function such that $\Delta\left( 0\right) =0$ and $\Delta$ is positive on $\left( 0,\,\infty\right) $. We also note that $\Delta$ is differentiable on $\left( 0,\,\infty\right) $. Writing $\delta$ for the derivative of $\Delta$, we can observe that $\delta=c\varphi-1$ on $\left( 0,\,\infty\right) $. To show that $\Delta$ is strictly concave it is enough if we prove that $$\left( y-x\right) \delta\left( y-0\right) <\Delta\left( y\right) -\Delta\left( x\right) <\left( y-x\right) \delta\left( x+0\right) =\left( y-x\right) \delta\left( x\right)$$ for all $x$, $y\in\left( 0,\,\infty\right) $ with $x<y$ (where, $\delta\left( t-0\right) $ respectively is the left derivative and $\delta\left( t+0\right) $ the right derivative of $\Delta$ at point $t$). In fact, fix arbitrarily two numbers $x$, $y\in\left( 0,\,\infty\right) $ such that $x<y$. But since $\Phi$ is strictly concave we have that$$\left( y-x\right) \varphi\left( y-0\right) <\Phi\left( y\right) -\Phi\left( x\right) <\left( y-x\right) \varphi\left( x+0\right) =\left( y-x\right) \varphi\left( x\right)$$ which easily leads to$$c\varphi\left( y-0\right) <\frac{c\Phi\left( y\right) -c\Phi\left( x\right) }{y-x}<c\varphi\left( x+0\right) =c\varphi\left( x\right) .$$ Hence,$$c\varphi\left( y-0\right) -1<\frac{c\Phi\left( y\right) -c\Phi\left( x\right) }{y-x}-1<c\varphi\left( x\right) -1,$$ i.e. $$\left( y-x\right) \delta\left( y-0\right) <\Delta\left( y\right) -\Delta\left( x\right) <\left( y-x\right) \delta\left( x\right) .$$ This ends the proof of the implication (2)$\rightarrow$(3). In the last step, we just point out that the conditional (3)$\rightarrow$(1) is obvious. Therefore, we can conclude on the validity of the argument. Denote $\widetilde{\mathcal{Y}_{\mathrm{conc}}}:=\left\{ \Phi\in \mathcal{Y}_{\mathrm{conc}}:\lim_{t\rightarrow\infty}\frac{\Phi\left( t\right) }{t}>0\right\} $. It is not difficult to check that $\widetilde {\mathcal{Y}_{\mathrm{conc}}}=\left\{ \Delta\in\mathcal{Y}_{\mathrm{conc}% }:c\Delta>\Phi_{\operatorname{id}}\text{ on }\left( 0,\,\infty\right) \text{ for some }c\in\left[ 1,\,\infty\right) \right\} $. Write $T_{\Delta }=\left\{ c\in\left[ 1,\,\infty\right) :c\Delta>\Phi_{\operatorname{id}% }\text{ on }\left( 0,\,\infty\right) \right\} $, $\Delta\in\widetilde {\mathcal{Y}_{\mathrm{conc}}}$. Some few words about set $\widetilde{\mathcal{Y}_{\mathrm{conc}}}$. \[rem9\]Let $\alpha\in\left( 0,\,\infty\right) $ be arbitrary. Then $\alpha\Delta\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}$ provided that $\Delta\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}$. Whenever $\Delta\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}$ we can choose a corresponding $c\in T_{\Delta}$ such that $c\Delta>\Phi_{\operatorname{id}}$ on $\left( 0,\,\infty\right) $. Now choose a constant $t_{0}\in\left( 1,\,\infty\right) $ such that $\alpha t_{0}\geq c$. Hence, $t_{0}\left( \alpha\Delta\right) \geq c\Delta>\Phi_{\operatorname{id}}$ on $\left( 0,\,\infty\right) $, i.e. $\alpha\Delta\in\widetilde{\mathcal{Y}% _{\mathrm{conc}}}$. \[rem10\]Every function $\Delta\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}$ can be written as the sum of a finite number of elements of $\widetilde {\mathcal{Y}_{\mathrm{conc}}}$. Conversely, the sum of a finite number of elements of $\widetilde{\mathcal{Y}_{\mathrm{conc}}}$ also belongs to $\widetilde{\mathcal{Y}_{\mathrm{conc}}}$. Next, we show that the quantities $\left\Vert f\right\Vert _{L^{p}}$ and $\sup_{\Phi\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}}\left( \Phi\left( 1\right) \right) ^{-1}\left\Vert \Phi\circ\left\vert f\right\vert \right\Vert _{L^{p}}$ are equivalent, in the sense that they are both either finite or infinite at the same time. This provides a kind of criterium for a measurable function to belong to $L^{p}$. \[theo10\]Let $f$ be any measurable function on an arbitrarily fixed measure space $\left( \Omega,\mathcal{F},\lambda\right) $ and $p\in\left[ 1,\infty\right) $ be any number. Then$$\left\Vert f\right\Vert _{L^{p}}\leq\sup_{\Phi\in\widetilde{\mathcal{Y}% _{\mathrm{conc}}}}\left( \Phi\left( 1\right) \right) ^{-1}\left\Vert \Phi\circ\left\vert f\right\vert \right\Vert _{L^{p}}\leq\left\Vert f\right\Vert _{L^{p}}+\lambda\left( \Omega\right) .$$ Pick any function $\Phi\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}$. Then$$\int_{\Omega}\left( \left( \Phi\left( 1\right) \right) ^{-1}\Phi \circ\left\vert f\right\vert \right) ^{p}d\lambda\leq\int_{\Omega}\left( \left\vert f\right\vert +1\right) ^{p}d\lambda$$ because $\Delta\leq\left( \Phi_{\operatorname{id}}+1\right) \Delta\left( 1\right) $ for all $\Delta\in\mathcal{Y}_{\mathrm{conc}}$. Consequently, via the Minkowski inequality, it follows that $\left( \Phi\left( 1\right) \right) ^{-1}\left\Vert \Phi\circ\left\vert f\right\vert \right\Vert _{L^{p}% }\leq\left\Vert f\right\Vert _{L^{p}}+\lambda\left( \Omega\right) $, which proves the inequality on the right hand-side of the above chain. To show the left side inequality fix any $\Delta\in\mathcal{Y}_{\mathrm{conc}}$ and write $\Delta_{n}=n^{-1}\Delta$, $n\in\mathbb{N}$. Clearly, $\left( \Delta _{n}\right) \subset\mathcal{Y}_{\mathrm{conc}}$. It is also evident that $\Phi_{\operatorname{id}}+\Delta_{n}\in\widetilde{\mathcal{Y}_{\mathrm{conc}}% }$, $n\in\mathbb{N}$. Then $$\begin{aligned} \sup_{\Phi\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}}\left( \Phi\left( 1\right) \right) ^{-1}\left\Vert \Phi\circ\left\vert f\right\vert \right\Vert _{L^{p}} & \geq\left( 1+n^{-1}\right) ^{-1}\left\Vert \left( \Phi_{\operatorname{id}}+\Delta_{n}\right) \circ\left\vert f\right\vert \right\Vert _{L^{p}}\\ & =\left( 1+n^{-1}\right) ^{-1}\left\Vert \left\vert f\right\vert +n^{-1}\left\vert f\right\vert \right\Vert _{L^{p}}\geq\left( 1+n^{-1}% \right) ^{-1}\left\Vert f\right\Vert _{L^{p}}.\end{aligned}$$ Passing to the limit yields $\sup_{\Phi\in\widetilde{\mathcal{Y}% _{\mathrm{conc}}}}\left( \Phi\left( 1\right) \right) ^{-1}\left\Vert \Phi\circ\left\vert f\right\vert \right\Vert _{L^{p}}\geq\left\Vert f\right\Vert _{L^{p}}$. Therefore, we have obtained a valid argument. \[theo11\]Let $\left( \Omega,\mathcal{F},\lambda\right) $ be any measure space and on it let $f$ be any measurable function. Then$$\lambda\left( \left\vert f\right\vert \geq\varepsilon\right) =\inf\left\{ \inf\left\{ \lambda\left( \Delta\circ\left\vert f\right\vert \geq\varepsilon c^{-1}\right) :c\in T_{\Delta}\right\} :\Delta\in\widetilde{\mathcal{Y}% _{\mathrm{conc}}}\right\}$$ for every number $\varepsilon\in\left[ 0,\,\infty\right) $. Throughout the proof $\varepsilon\in\left[ 0,\,\infty\right) $ will be any fixed number. We first note that the assertion is trivial when $\left( \left\vert f\right\vert =\infty\right) \neq\varnothing$. We shall then prove it when $\left( \left\vert f\right\vert <\infty\right) \neq\varnothing$. Pick some $\Delta\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}$ and $c\in T_{\Delta}$ such that $c\Delta>\Phi_{\operatorname{id}}$ on $\left( 0,\,\infty\right) $. It is not hard to see that $\left( \left\vert f\right\vert \geq\varepsilon\right) =\left( \Delta\circ\left\vert f\right\vert \geq\Delta\left( \varepsilon\right) \right) \subset\left( \Delta\circ\left\vert f\right\vert \geq\varepsilon c^{-1}\right) $ and thus$$\lambda\left( \left\vert f\right\vert \geq\varepsilon\right) =\lambda\left( \Delta\circ\left\vert f\right\vert \geq\Delta\left( \varepsilon\right) \right) \leq\lambda\left( \Delta\circ\left\vert f\right\vert \geq\varepsilon c^{-1}\right) .$$ Consequently, $$\lambda\left( \left\vert f\right\vert \geq\varepsilon\right) \leq \inf\left\{ \inf\left\{ \lambda\left( \Delta\circ\left\vert f\right\vert \geq\varepsilon c^{-1}\right) :c\in T_{\Delta}\right\} :\Delta\in \widetilde{\mathcal{Y}_{\mathrm{conc}}}\right\} .$$ To prove the converse statement, we need show that $$\lambda\left( \left\vert f\right\vert \geq\varepsilon\right) \geq \inf\left\{ \inf\left\{ \lambda\left( \Delta\circ\left\vert f\right\vert \geq\varepsilon c^{-1}\right) :c\in T_{\Delta}\right\} :\Delta\in \widetilde{\mathcal{Y}_{\mathrm{conc}}}\right\} .$$ In fact, for any $n\in\mathbb{N}$ set $\Delta_{n}=\Phi_{\operatorname{id}% }+n^{-1}\left( 1-e^{-\Phi_{\operatorname{id}}}\right) $. It is not difficult to see that $\Delta_{n}\in\mathcal{Y}_{\mathrm{conc}}$ and $\Delta_{n}% >\Phi_{\operatorname{id}}$ on $\left( 0,\,\infty\right) $, $n\in\mathbb{N}$. This means that $\left( \Delta_{n}\right) \subset\widetilde{\mathcal{Y}% _{\mathrm{conc}}}$ and moreover, $1\in T_{\Delta_{n}}$, $n\in\mathbb{N}$. Consequently, $$\begin{aligned} \inf\left\{ \inf\left\{ \lambda\left( \Delta\circ\left\vert f\right\vert \geq\varepsilon c^{-1}\right) :c\in T_{\Delta}\right\} :\Delta\in \widetilde{\mathcal{Y}_{\mathrm{conc}}}\right\} & \leq\lambda\left( \Delta_{n}\circ\left\vert f\right\vert \geq\varepsilon\right) \\ & =\lambda\left( \left\vert f\right\vert +n^{-1}\left( 1-e^{-\left\vert f\right\vert }\right) \geq\varepsilon\right) \text{.}%\end{aligned}$$ However, as $\left( \Delta_{n}\right) $ is a decreasing sequence it is obvious that $\left( \Delta_{n+1}\circ\left\vert f\right\vert \geq \varepsilon\right) \subset\left( \Delta_{n}\circ\left\vert f\right\vert \geq\varepsilon\right) $, $n\in\mathbb{N}$. Thus having passed to the limit we can observe that $$\inf\left\{ \inf\left\{ \lambda\left( \Delta\circ\left\vert f\right\vert \geq\varepsilon c^{-1}\right) :c\in T_{\Delta}\right\} :\Delta\in \widetilde{\mathcal{Y}_{\mathrm{conc}}}\right\} \leq\lambda\left( \left\vert f\right\vert \geq\varepsilon\right) .$$ Therefore, the proof is a valid argument. \[theo12\]Let $f\in L^{p}\left( \Omega,\mathcal{F},\lambda\right) $, $p\geq1$, where $\left( \Omega,\mathcal{F},\lambda\right) $ is any given measure space. Then $$\left\Vert f\right\Vert _{L^{p}}=\inf\left\{ \inf\left\{ c\left\Vert \Delta\circ\left\vert f\right\vert \right\Vert _{L^{p}}:c\in T_{\Delta }\right\} :\Delta\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}\right\} .$$ Pick arbitrarily some $\Delta\in\widetilde{\mathcal{Y}_{\mathrm{conc}}}$ and $c\in T_{\Delta}$ such that $c\Delta>\Phi_{\operatorname{id}}$ on $\left( 0,\,\infty\right) $. Clearly, $c\left\Vert \Delta\circ\left\vert f\right\vert \right\Vert _{L^{p}}\geq\left\Vert f\right\Vert _{L^{p}}$. We can then easily observe that $$\inf\left\{ \inf\left\{ c\left\Vert \Delta\circ\left\vert f\right\vert \right\Vert _{L^{p}}:c\in T_{\Delta}\right\} :\Delta\in\widetilde {\mathcal{Y}_{\mathrm{conc}}}\right\} \geq\left\Vert f\right\Vert _{L^{p}}.$$ To prove the converse of this inequality consider the sequence $\left( \Delta_{n}\right) \subset\widetilde{\mathcal{Y}_{\mathrm{conc}}}$, where $\Delta_{n}=\Phi_{\operatorname{id}}+n^{-1}\left( 1-e^{-\Phi _{\operatorname{id}}}\right) >\Phi_{\operatorname{id}}$ on $\left( 0,\,\infty\right) $, $n\in\mathbb{N}$. Then as $1\in T_{\Delta_{n}}$, $n\in\mathbb{N}$, we have$$\inf\left\{ \inf\left\{ c\left\Vert \Delta\circ\left\vert f\right\vert \right\Vert _{L^{p}}:c\in T_{\Delta}\right\} :\Delta\in\widetilde {\mathcal{Y}_{\mathrm{conc}}}\right\} \leq\left\Vert \Delta_{n}% \circ\left\vert f\right\vert \right\Vert _{L^{p}}.$$ Since $\left( \Delta_{n}\right) $ is a decreasing sequence it ensues that $\left( \Delta_{n}\circ\left\vert f\right\vert \right) $ is also a decreasing sequence which tends to $\left\vert f\right\vert $. As every member of sequence $\left( \Delta_{n}\circ\left\vert f\right\vert \right) $ is dominated by $\Delta_{1}\circ\left\vert f\right\vert \in L^{p}$, then by applying the Dominated Convergence Theorem it will entail that $$\inf\left\{ \inf\left\{ c\left\Vert \Delta\circ\left\vert f\right\vert \right\Vert _{L^{p}}:c\in T_{\Delta}\right\} :\Delta\in\widetilde {\mathcal{Y}_{\mathrm{conc}}}\right\} \leq\left\Vert f\right\Vert _{L^{p}}.$$ This completes the proof. \[cor3\]Suppose that $h:\mathbb{R\rightarrow R}$ is a continuous function. Then $\left\vert h\right\vert =\frac{1}{\sqrt[p]{\lambda\left( \Omega\right) }}\inf\left\{ \inf\left\{ \left( \Delta\circ\left\vert h\right\vert \right) c:c\in T_{\Delta}\right\} :\Delta\in\widetilde{\mathcal{Y}% _{\mathrm{conc}}}\right\} $. Fix any number $x\in\mathbb{R}$ and let $f\in L^{p}\left( \Omega ,\mathcal{F},\lambda\right) $ be the constant function defined by $f\equiv h\left( x\right) $ on $\Omega$. Then by applying Theorem \[theo10\] we can easily deduce the result. Given any number $k\in\mathbb{N}$ characterize all pairs of functions $\Phi$ and $\Delta\in\mathcal{Y}_{\mathrm{conc}}$ such that $\left\vert \left\{ x\in\left( 0,~\infty\right) :\Phi\left( x\right) =\Delta\left( x\right) \right\} \right\vert =k$. Characterize all pairs of functions $\Phi$ and $\Delta\in\mathcal{Y}% _{\mathrm{conc}}$ such that the sets $\left( 0,~\infty\right) $ and $\left\{ x\in\left( 0,~\infty\right) :\Phi\left( x\right) =\Delta\left( x\right) \right\} $ should be equinumerous. [99]{} <span style="font-variant:small-caps;">Agbeko, N. K.</span>: Concave function inequalities for sub-(super)martingales, Annales Univ. Sci. Budapest, Sectio Mathematica, **29** (1986), 9-17. <span style="font-variant:small-caps;">Agbeko, N. K.</span>: Necessary and sufficient condition for the maximal inequality of concave Young-functions, Annales Univ. Sci. Budapest, Sectio Mathematica, **32** (1989), 267-270. <span style="font-variant:small-caps;">Agbeko, N. K.</span>: Studies on concave Young-functions, Miskolc Math. Notes (**6**)2005, No. **1**, 3 - 18. (*Available online at: http://mat76.mat.uni-miskolc.hu/mnotes/files/6-1/*). <span style="font-variant:small-caps;">Burkholder, D. L.</span>: Distribution function inequalities for martingales, Annals of Probability, **1**(1973), 19 - 42. <span style="font-variant:small-caps;">Garcia-Cuerva, J. and Rubio De Francia, J. L.</span>: *Weighted Norm Inequalities and Related Topics*. North-Holland, Amsterdam, 1985. <span style="font-variant:small-caps;">Garsia, A. M.</span>: *Martingale inequalities*, *Seminar Notes on recent progress*, Benjamin, Reading, Massachussets, 1973. <span style="font-variant:small-caps;">Hamilton, A. G.</span>: *Numbers, sets and axioms: The apparatus of mathematics*, Cambridge University Press, 1982. <span style="font-variant:small-caps;">Kuratowski, K.</span>: *Topology*, vol. 1, Academic Press, New York, etc., 1966. <span style="font-variant:small-caps;">MacLane, S. and Birkhoff G.</span>: *Algèbre*, Tome 1, *Structures fondamentales*, Gauthier-Villars, 1971. <span style="font-variant:small-caps;">Mogyoródi, J.</span>: On a concave function inequality for martingales, Annales Univ. Sci. Bud. Sect. Math. **24**(1981), 255 - 271. <span style="font-variant:small-caps;">Reed, M. C.</span>: *Fundamental ideas of analysis*, John Wiley & Sons, New York ..., 1998. <span style="font-variant:small-caps;">Sinnamon, G.</span>: Embeddings of concave functions and duals of Lorentz spaces, Publ. Math. **46**(2002), 489-525. <span style="font-variant:small-caps;">Triebel, H.</span>: *Interpolation theory, function spaces, differential operators*, North-Holland, 1978.

--- abstract: | We report spectroscopic observations of the 2.63 day, detached, F-type main-sequence eclipsing binary . We use our observations together with existing $uvby$ photometric measurements to derive accurate absolute masses and radii for the stars good to better than 1.5%. We obtain masses of $M_1 = 1.269 \pm 0.017~M_{\sun}$ and $M_2 = 0.7542 \pm 0.0059~M_{\sun}$, radii of $R_1 = 1.477 \pm 0.012~R_{\sun}$ and $R_2 = 0.7232 \pm 0.0091~R_{\sun}$, and effective temperatures of $6770 \pm 150$ K and $5020 \pm 150$ K for the primary and secondary stars, respectively. Both components appear to have their rotations synchronized with the motion in the circular orbit. A comparison of the properties of the primary with current stellar evolution models gives good agreement for a metallicity of ${\rm [Fe/H]} = -0.17$, which is consistent with photometric estimates, and an age of about 2.2 Gyr. On the other hand, the K2 secondary is larger than predicted for its mass by about 4%. Similar discrepancies are known to exist for other cool stars, and are generally ascribed to stellar activity. The system is in fact an X-ray source, and we argue that the main site of the activity is the secondary star. Indirect estimates give a strength of about 1 kG for the surface magnetic field on that star. A previously known close visual companion to  is shown to be physically bound, making the system a hierarchical triple. author: - 'Jane C. Bright and Guillermo Torres,' title: 'Absolute dimensions of the F-type eclipsing binary V2154 Cygni' --- Introduction {#sec:introduction} ============  (also known as HD 203839, HIP 105584, BD+47 3386, and TYC 3594-1060-1; $V = 7.77$) is a 2.63 day eclipsing binary discovered by the [*Hipparcos*]{} team [@Perryman:1997], and found independently in 1996 by [@Martin:2003] in the course of a search for variable stars in the open cluster M39. Light curves in the $uvby$ Strömgren system were published by [@Rodriguez:2001], but the physical properties of the components were not derived by them because spectroscopy was lacking. The only spectroscopic work we are aware of are brief reports by [@Kurpinska:2000] listing preliminary values for the velocity amplitudes, and by [@Oblak:2004] giving preliminary masses and radii, though details of those analyses are unavailable. The very unequal depths of the eclipses ($\sim$0.3 mag for the primary and $\sim$0.05 mag for the secondary) suggest stars of rather different masses, making it an interesting object for followup because of the increased leverage for the comparison with stellar evolution models. This motivated us to carry out our own high-resolution spectroscopic observations of this star, which we report here.  is known from [*Tycho-2*]{} observations to have a close, 047 visual companion about two magnitudes fainter than the binary [$\Delta B_T = 2.18$ mag, $\Delta V_T = 2.15$ mag; @Fabricius:2000]. We show below that it is physically associated, making  a hierarchical triple system. While the primary of the eclipsing pair is an early F star, the secondary is a much smaller K star in the range where previous observations have shown discrepancies with models [see, e.g., @Torres:2013]. The measured radii of such stars are sometimes larger than predicted, and their temperatures cooler than expected, both presumably due to the effects of magnetic activity and/or spots [e.g., @Chabrier:2007; @Morales:2010].  therefore presents an opportunity to determine accurate physical properties of the stars in a system with a mass ratio significantly different from unity, and to investigate any discrepancies with theory in connection with measures of stellar activity. The layout of our paper is as follows. Our new spectroscopic observations are reported in Section \[sec:spectroscopy\], followed by a brief description in Section \[sec:photometry\] of the [@Rodriguez:2001] photometric measurements we incorporate into our analysis. The light curve fits are presented in Section \[sec:analysis\], along with consistency checks to support the accuracy of the results. With the spectroscopic and photometric parameters we then derive the physical properties of the system, given in Section \[sec:dimensions\], and compare them with current models of stellar structure and stellar evolution (Section \[sec:models\]). We discuss the results in the context of available activity measurements in Section \[sec:discussion\], and conclude with some final thoughts in Section \[sec:conclusions\]. Spectroscopic observations and analysis {#sec:spectroscopy} =======================================  was placed on our spectroscopic program in October of 2001, and observed through June of 2007 with two nearly identical echelle instruments [Digital Speedometer; @Latham:1992] on the 1.5m telescope at the Oak Ridge Observatory in the town of Harvard (MA), and on the 1.5m Tillinghast reflector at the Fred L. Whipple Observatory on Mount Hopkins (AZ). Both instruments (now decommissioned) used intensified photon-counting Reticon detectors providing spectral coverage in a single echelle order 45 Å wide centered on the b triplet at 5187 Å. The resolving power delivered by these spectrographs was $R \approx 35,\!000$, and the signal-to-noise ratios achieved for the 80 usable observations of  range from about 20 to 67 per resolution element of 8.5 . Wavelength solutions were carried out by means of exposures of a thorium-argon lamp taken before and after each science exposure, and reductions were performed with a custom pipeline. Observations of the evening and morning twilight sky were used to place the observations from the two instruments on the same velocity system and to monitor instrumental drifts [@Latham:1992]. Visual inspection of one-dimensional cross-correlation functions for each of our spectra indicated the presence of a star much fainter than the primary that we initially assumed was the secondary in . However, subsequent analysis with the two-dimensional cross-correlation algorithm TODCOR [@Zucker:1994] showed those faint lines to be stationary, while a third set of even weaker lines was noticed that moved in phase with the orbital period. This is therefore the secondary in the eclipsing pair, and the stationary lines correspond to the visual companion mentioned in the Introduction, as we show later, which falls within the 1 slit of the spectrograph. Consequently, for the final velocity measurements we used an extension of TODCOR to three dimensions [referred to here as TRICOR; @Zucker:1995] that uses three different templates, one for each star. In the following we refer to the binary components as stars 1 and 2, and to the tertiary as star 3. The templates were selected from a large library of synthetic spectra based on model atmospheres by R. L. Kurucz [see @Nordstrom:1994; @Latham:2002], computed for a range of temperatures ($T_{\rm eff}$), surface gravities ($\log g$), rotational broadenings ($v \sin i$, when seen in projection), and metallicities (\[m/H\]). We selected the optimum parameters for the templates as follows, adopting solar metallicity throughout. For the primary star we ran a grid of one-dimensional cross-correlations against synthetic spectra over a wide range of temperatures and $v \sin i$ values [see @Torres:2002], for a fixed $\log g$ of 4.0 that is sufficiently close to our final estimate presented later. The best match, as measured by the cross-correlation coefficient averaged over all exposures, was obtained for interpolated values of $T_{\rm eff} = 6770 \pm 150$ K and $v \sin i = 26 \pm 2$ . The secondary and tertiary stars are faint enough (by factors of 25 and 9, respectively; see below) that they do not affect these results significantly. For the secondary the optimal $v \sin i$ from grids of TRICOR correlations was $12 \pm 2$ . However, due to its faintness we were unable to establish its temperature from the spectra themselves, so we relied on results from the light curve analysis described later in Section \[sec:analysis\]. The central surface brightness ratio $J$ provides an accurate measure of the temperature ratio between stars 1 and 2. Using the primary temperature from above, the $J$ value for the $y$ band, and the visual flux calibration by [@Popper:1980], we obtained $T_{\rm eff} = 5020 \pm 150$ K. The surface gravity was adopted as $\log g = 4.5$, appropriate for a main-sequence star of this temperature. For the tertiary we again adopted $\log g = 4.5$, and grids of correlations with TRICOR for a range of temperatures indicated a preference for a value of 5500 K, to which we assign a conservative uncertainty of 200 K. Similar correlation grids varying $v \sin i$ indicated no measurable line broadening for the tertiary, so we adopted $v \sin i = 0$ kms, with an estimated upper limit of 2 . Radial velocities were then measured with TRICOR using values for the template parameters ($T_{\rm eff}$, $v \sin i$) in our library nearest to those given above: 6750 K and 25  for the primary, 5000 K and 12  for the secondary, and 5500 K and 0  for the tertiary. The light ratios we determined from our spectra are $L_2/L_1 = 0.036 \pm 0.004$ and $L_3/L_1 = 0.108 \pm 0.012$, corresponding to the mean wavelength of our observations (5187 Å). [lccccccc]{} 51874.5314 & 36.96 & 1.11 & $-$20.74 & 9.28 & 17.32 & 1.85 & 0.4586\ 52109.6581 & $-$41.58 & 0.67 & 128.03 & 5.55 & 19.35 & 1.11 & 0.8387\ 52123.6422 & 79.58 & 0.65 & $-$85.40 & 5.42 & 19.62 & 1.08 & 0.1546\ 52130.5621 & $-$52.47 & 0.75 & 135.05 & 6.22 & 18.83 & 1.24 & 0.7851\ 52151.5379 & $-$53.69 & 0.69 & 143.48 & 5.73 & 19.73 & 1.14 & 0.7588 Because our spectra are only 45 Å wide, systematic errors in the velocities can result from lines shifting in and out of this window as a function of orbital phase [see @Latham:1996]. To estimate this effect we followed a procedure similar to that of [@Torres:1997] and created artificial triple-lined spectra based on our adopted templates, which we then processed with TRICOR in the same way as the real spectra. A comparison of the input and output velocities showed a phase-dependent pattern with maximum shifts of about 0.2  for the primary, 6  for the secondary, and 1.2  for the tertiary. We applied these shifts as corrections to the individual raw velocities, and the final measurements including all corrections are listed in Table \[tab:rvs\], along with their uncertainties. The velocities of the third star appear constant within their uncertainties, and have a mean of $+19.31 \pm 0.13$  (weighted average). A similar correction for systematic errors was applied to the light ratios, and is already included in the values reported above. A weighted least-squares orbital fit to the primary and secondary velocities gives the elements and derived quantities presented in Table \[tab:specorbit\], where a circular orbit has been assumed. Tests allowing for eccentricity gave results consistent with zero, in agreement with similar experiments below based on the light curves. Initial solutions in which we included a possible systematic offset between the primary and secondary velocities, as may arise, e.g., from template mismatch, also gave a value consistent with zero. The observations and orbital fit are shown in Figure \[fig:rvs\]. The tertiary velocities, represented with triangles, are seen to be very close to the center-of-mass velocity, supporting the physical association. [lc]{}\     $P$ (days) & 2.6306359 $\pm$ 0.0000039\     $T_{\rm max}$ (HJD$-$2,400,000) & 52973.58847 $\pm$ 0.00091\     $\gamma$ () & +19.408 $\pm$ 0.076\     $K_1$ () & 72.699 $\pm$ 0.092\     $K_2$ () & 122.298 $\pm$ 0.723\     $e$ & 0.0 (fixed)\ \     $M_1\sin^3 i$ ($M_{\sun}$) & 1.268 $\pm$ 0.017\     $M_2\sin^3 i$ ($M_{\sun}$) & 0.7535 $\pm$ 0.0059\     $q\equiv M_2/M_1$ & 0.5944 $\pm$ 0.0036\     $a_1\sin i$ (10$^6$ km) & 2.6298 $\pm$ 0.0033\     $a_2\sin i$ (10$^6$ km) & 4.424 $\pm$ 0.026\     $a \sin i$ ($R_{\sun}$) & 10.140 $\pm$ 0.038\ \     $N_{\rm obs}$ & 80\     Time span (days) & 2381.4\     Time span (cycles) & 905.3\     $\sigma_1$ () & 0.69\     $\sigma_2$ () & 5.74\     $\sigma_3$ () & 1.21 Photometric observations {#sec:photometry} ======================== The light curves used for our analysis are those published by [@Rodriguez:2001][^1], and were obtained between July and November of 1998 with the 0.9m telescope at the Sierra Nevada Observatory (Spain). The 852 observations were made on 28 nights using $uvby$ filters, with HD204626 () as the comparison star and HD204977 () as the check star. The standard deviations of the difference in magnitude between the comparison and check stars, which may be taken as an indication of the precision of the observations, were 0.0085, 0.0035, 0.0032, and 0.0043 mag for $u$, $v$, $b$, and $y$, respectively. Light curve analysis {#sec:analysis} ==================== For the analysis of the light curves of this well-detached system we have adopted the Nelson-Davis-Etzel model [@Popper:1981; @Etzel:1981], as implemented in the JKTEBOP code[^2] [@Southworth:2013]. The free parameters of the fit are the period $P$ and reference epoch of primary minimum $T_{\rm min}$, the central surface brightness ratio $J \equiv J_2/J_1$, the sum of the relative radii $r_1+r_2$ normalized to the semimajor axis, the radius ratio $k \equiv r_2/r_1$, the inclination angle $i$, and a magnitude zero point $m_0$. Because of the presence of the third star in the aperture we also included the third light parameter $L_3$ (fractional brightness of star 3 divided by the total light, at phase 0.25 from primary eclipse). The mass ratio was held fixed at the spectroscopic value ($q = 0.5944$). Linear limb-darkening coefficients ($u_1$, $u_2$) were interpolated from the tables by [@Claret:2000] using the JKTLD code[^3] [@Southworth:2008], and gravity-darkening coefficients ($y_1$, $y_2$) were taken from the tabulations by [@Claret:2011] for the properties of the primary and secondary given earlier. Experiments with quadratic limb-darkening gave no improvement, so the linear law was used throughout. Initial fits that included the eccentricity as an additional free parameter indicated a value that was not significantly different from zero, consistent with the spectroscopic evidence, so the orbit was assumed to be circular. [lcccc]{} $P$ (days) & 2.630607 (+14/$-$19) & 2.6306290 (+61/$-$75) & 2.6306305 (+56/$-$59) & 2.6306316 (+74/$-$81)\ $T_{\rm min}$ (HJD$-$2,400,000) & 51048.61797 (+28/$-$22) & 51048.61815 (+12/$-$14) & 51048.61814 (+11/$-$14) & 51048.61808 (+15/$-$12)\ $r_1+r_2$ & 0.2169 (+26/$-$17) & 0.2172 (+12/$-$14) & 0.2167 (+13/$-$15) & 0.2163 (+13/$-$14)\ $k \equiv r_2/r_1$ & 0.492 (+17/$-$14) & 0.486 (+18/$-$9) & 0.492 (+15/$-$14) & 0.473 (+24/$-$4)\ $i$ (deg) & 88.76 (+32/$-$79) & 88.39 (+61/$-$33) & 88.57 (+43/$-$48) & 87.79 (+82/$-$6)\ $J$ & 0.120 (+11/$-$9) & 0.1267 (+84/$-$58) & 0.193 (+11/$-$9) & 0.246 (+16/$-$11)\ $L_3$ & 0.087 (+35/$-$59) & 0.054 (+51/$-$29) & 0.093 (+34/$-$47) & 0.028 (+87/$-$1)\ $m_0$ (mag) & 0.33069 (+45/$-$45) & 0.69957 (+40/$-$32) & 0.44492 (+37/$-$31) & 0.20558 (+44/$-$29)\ \ $r_1$ & 0.1454 (+24/$-$22) & 0.1462 (+13/$-$23) & 0.1453 (+20/$-$23) & 0.1469 (+11/$-$31)\ $r_2$ & 0.0715 (+16/$-$12) & 0.0710 (+15/$-$8) & 0.0714 (+12/$-$13) & 0.0695 (+21/$-$2)\ $L_2/L_1$ & 0.0264 (+21/$-$23) & 0.0280 (+22/$-$10) & 0.0435 (+22/$-$24) & 0.0513 (+60/$-$5)\ $\sigma$ (mmag) & 8.56 & 3.58 & 3.28 & 3.79\ \ $u_1$ & 0.722 & 0.748 & 0.696 & 0.615\ $u_2$ & 0.929 & 0.892 & 0.854 & 0.768\ $y_1$ & 0.393 & 0.354 & 0.305 & 0.260\ $y_2$ & 1.157 & 0.892 & 0.672 & 0.581 Separate solutions for each of the $uvby$ bands are presented in Table \[tab:LCfits1\]. As the errors provided by JKTEBOP are strictly internal and do not capture systematic components that may result, e.g., from red noise, the uncertainties given in the table were computed with the residual permutation (“prayer bead”) method, as follows. We shifted the residuals from the original fits by an arbitrary number of time indices (with wraparound), and added them back into the computed curves to create artificial data sets that preserve any time-correlated noise that might be present in the original data. We generated 500 such data sets for each of the passbands and fitted them with JKTEBOP. In each solution we simultaneously perturbed all of the quantities that were initially held fixed. We did this by adding Gaussian noise to the mass ratio corresponding to its measured error ($\sigma_q = 0.0036$), and Gaussian noise with $\sigma = 0.1$ to the limb-darkening and gravity-darkening coefficients. The standard deviations of the resulting distributions for each parameter were adopted as the uncertainties for the light curve elements. [lc]{} $r_1+r_2$ & 0.21696 $\pm$ 0.00087\ $k \equiv r_2/r_1$ & 0.4895 $\pm$ 0.0083\ $i$ (deg) & 88.55 $\pm$ 0.28\ $r_1$ & 0.1457 $\pm$ 0.0010\ $r_2$ & 0.07129 $\pm$ 0.00060\ $P$ (days) & 2.6306303 $\pm$ 0.0000038\ $T_{\rm min}$ (HJD$-$2,400,000) & 51048.618122 $\pm$ 0.000075 [cccc]{} $u$ & 0.119 (+12/$-$10) & 0.075 (+22/$-$13) & 0.0259 (+19/$-$16)\ $v$ & 0.1272 (+86/$-$82) & 0.069 (+22/$-$22) & 0.02851 (+75/$-$67)\ $b$ & 0.193 (+11/$-$11) & 0.086 (+22/$-$21) & 0.04311 (+81/$-$75)\ $y$ & 0.250 (+12/$-$14) & 0.101 (+21/$-$17) & 0.0560 (+13/$-$10) The results from the four passbands are fairly consistent within their uncertainties, with a few exceptions: (1) The ephemeris ($P$, $T_{\rm min}$) seems rather different for the $u$ band, which is the fit with the largest scatter. The fact that the $uvby$ measurements are simultaneous indicates this is almost certainly due to systematic errors affecting $u$ that are not uncommon. (2) The geometric parameters (most notably $k$ and $i$, and to a lesser extent $r_1+r_2$) seem systematically different for the $y$ band. Several features of that fit make us suspicious of these quantities, and of $L_3$ as well. In particular, $L_3$ is significantly lower than in the other bands, which runs counter to expectations given that the third star is cooler (redder) than the primary, and so its flux contribution ought to be larger in $y$, not smaller. Third light is always strongly (and positively) correlated with the inclination angle and with $k$ in this case, and indeed we see that both $i$ and $k$ are also low. Grids of JKTEBOP solutions over a range of fixed values of $k$ show that for all $k$ values the radius sum in the $y$ band is always considerably smaller than in the other three bands, which agree well among each other. Finally, we note that the $y$-band error bars for $k$, $i$, and $L_3$ are all highly asymmetric (always much larger in the direction toward the average of the $uvb$ results), which is not the case in the other bands. These features are symptomatic of strong degeneracies in $y$ that make the results highly prone to biases. We have therefore chosen not to rely on the geometric parameters from the $y$ band. Weighted averages of the photometric period and epoch (excluding the $u$ band) and of the geometric parameters (excluding the $y$ band) are given in Table \[tab:LCfits2\]. The photometric period agrees well with the spectroscopic one, within the errors. The final solutions for the wavelength-dependent quantities were carried out by holding the ephemeris and geometry fixed to these values, and the results are collected in Table \[tab:LCfits3\]. We illustrate these final fits in Figure \[fig:LC\], where the secondary eclipse is seen to be total. Consistency checks {#sec:consistency} ------------------ The spectroscopic light ratios reported in Section \[sec:spectroscopy\] ($L_2/L_1$ and $L_3/L_1$), which are independent of the light curve analysis above, offer an opportunity to test the accuracy of the light curve solutions. For the necessary flux transformation between the 5187 Å spectral window and the slightly redder Strömgren $y$ band (5470 Å) we used synthetic spectra from the PHOENIX library by [@Husser:2013], along with our adopted effective temperatures and surface gravities from Section \[sec:spectroscopy\], integrating the model fluxes over both passbands. An additional quantity that is needed to properly scale the spectral energy distributions is the radius ratio. As a sanity check we first used these spectra coupled with our measured radius ratio of $k = 0.4895$ to calculate the $y$-band light ratio between the primary and secondary, and obtained $L_2/L_1 = 0.055$, in good agreement with our light curve value. The flux ratio we then infer at 5187 Å based on the same parameters is 0.039, which is consistent with the spectroscopic measurement of $0.036 \pm 0.004$ (see Figure \[fig:bands\]). The scaling of the energy distributions of the tertiary and primary components requires knowledge of the radius ratio between those two stars, which our observations do not provide. We estimated it as follows. With our $y$-band light curve results from Table \[tab:LCfits3\] ($L_2/L_1$ and $L_3$) we calculated $L_3/L_1 = L_3 (1 + L_2/L_1)/(1 - L_3) = 0.119$. We then used the PHOENIX synthetic spectra and varied the radius ratio until we reproduced this value of $L_3/L_1$, which occurred for $R_3/R_1 = 0.56$. With the scaling set in this way, the predicted flux ratio at 5187 Å between the tertiary and primary is 0.100, which again agrees with the spectroscopically measured ratio of $0.108 \pm 0.012$, as illustrated in Figure \[fig:bands\]. These consistency checks between the spectroscopy and the photometry are an indication that the light curve fits are largely free from biases, and support the accuracy of the geometric elements used in the next section to derive the physical properties of the stars. Absolute dimensions {#sec:dimensions} =================== The absolute masses and radii of  are listed in Table \[tab:dimensions\]. The relative uncertainties are smaller than 1.5% for both components. The combined out-of-eclipse magnitudes of the system from [@Rodriguez:2001] and our fitted light ratios and third-light values enable us to deconvolve the light of the components. For the primary star we obtained the Strömgren indices $b-y = 0.243 \pm 0.035$, $m_1 = 0.139 \pm 0.063$, and $c_1 = 0.528 \pm 0.063$, along with $\beta = 2.691$. With these and the calibrations of [@Crawford:1975] we infer negligible reddening for the system (consistent with its small distance; see below), and an estimated photometric metallicity of ${\rm [Fe/H]} = -0.12$. Photometric estimates of the temperatures may be obtained from the $b-y$ index of the primary and the corresponding value for the secondary of $0.527 \pm 0.046$. The color/temperature calibration of [@Casagrande:2010] leads to values of $6840 \pm 200$ K and $5050 \pm 260$ K that are in good agreement with the spectroscopic values adopted in Section \[sec:spectroscopy\]. The deconvolved color of the third star ($b-y = 0.45 \pm 0.38$) is too uncertain to be useful, though the inferred temperature of $5500 \pm 870$ K again matches the value from Section \[sec:spectroscopy\]. The spectral types corresponding to the adopted temperatures are F2, K2, and G8 for the primary, secondary, and tertiary, respectively. Additional quantities listed in Table \[tab:dimensions\] include the luminosities, absolute magnitudes, and the distance ($90 \pm 9$ pc), which makes use of the bolometric corrections by [@Flower:1996]. The corresponding parallax, $11.2 \pm 1.1$ mas, is not far from the trigonometric values listed in the [*Hipparcos*]{} catalog ($\pi_{\rm HIP} = 11.77 \pm 0.59$ mas) and in the first data release of [*Gaia*]{} [$\pi_{\rm Gaia} = 13.35 \pm 0.82$ mas; @Brown:2016]. Our measured projected rotational velocities are also quite close to the expected synchronous values ($v_{\rm sync} \sin i$). As noted earlier, the third star was angularly resolved by the [ *Tycho-2*]{} experiment at a separation of 047 and a measured position angle of 59, at the mean epoch 1991.25. Subsequent astrometric measurements by a number of authors indicate a gradual decrease in the angular separation to 025 in 2010 [@Horch:2010], with no significant change in the position angle. This is inconsistent with being the result of a chance alignment with a background star, as the binary’s fairly large proper motion of 113 mas yr$^{-1}$ measured by [*Gaia*]{} would have carried the companion 2 away in that interval. The direction of motion would suggest a high inclined orbit, or possibly even an edge-on orientation. At our measured 90 pc distance the 047 separation implies a semimajor axis of roughly 42 au and an orbital period of $\sim$160 yr. [lcc]{} Mass ($M_{\sun}$)& $1.269 \pm 0.017$ & $0.7542 \pm 0.0059$\ Radius ($R_{\sun}$)& $1.477 \pm 0.012$ & $0.7232 \pm 0.0091$\ $\log g$ (cgs)& $4.2028 \pm 0.0089$ & $4.597 \pm 0.012$\ Temperature (K)& $6770 \pm 150$& $5020 \pm 150$\ $\log L/L_{\sun}$& $0.616 \pm 0.039$ & $-0.523 \pm 0.039$\ $BC_{\rm V}$ (mag)& $-0.02 \pm 0.10$& $-0.30 \pm 0.11$\ $M_{\rm bol}$ (mag)& $3.192 \pm 0.098$ & $6.041 \pm 0.097$\ $M_V$ (mag)& $3.17 \pm 0.14$ & $6.34 \pm 0.17$\ $m-M$ (mag)&\ Distance (pc)&\ Parallax (mas)&\ $v_{\rm sync} \sin i$ ()& $28.4 \pm 0.2$& $13.9 \pm 0.2$\ $v \sin i$ ()& $26 \pm 2$& $12 \pm 2$ Comparison with stellar evolution models {#sec:models} ======================================== The accurate properties for  are compared with predictions from current stellar evolution theory in Figure \[fig:mistlogg\]. The evolutionary tracks for the measured masses of the components were taken from the grid of MESA Isochrones and Stellar Tracks [MIST; @Choi:2016], which is based on the Modules for Experiments in Stellar Astrophysics package [MESA; @Paxton:2011; @Paxton:2013; @Paxton:2015]. The metallicity in the models was adjusted to ${\rm [Fe/H]} = -0.17$ to provide the best fit to the temperatures of the stars. This composition is not far from the photometric estimate reported earlier. The shaded areas in the figure indicate the uncertainty in the location of the tracks that comes from the errors in the measured masses. Solar-metallicity tracks are shown with dotted lines for reference. The age that best matches the radius of the primary is 2.2 Gyr (see below). An isochrone for this age is shown with a dashed line. The primary star is seen to be almost halfway through its main-sequence phase. At this relatively old age it is not surprising that we found the components’ rotation to be synchronized with the motion in a circular orbit, as the theoretically expected timescales for synchronization and orbit circularization are $\sim$1 Myr and $\sim$200 Myr, respectively [e.g., @Hilditch:2001]. The radii and temperatures are shown separately as a function of mass in Figure \[fig:mistmassradius\], in which the solid line represents the 2.2 Gyr isochrone for ${\rm [Fe/H]} = -0.17$ that reproduces the measured radius of the primary star at its measured mass. A solar metallicity isochrone for the same age is shown with the dashed line. The secondary star is seen to be larger than predicted for its mass by almost 4%, corresponding to a nearly $\sim$3$\sigma$ discrepancy. Similar deviations from theory are known to be present in other stars with convective envelopes, and are usually attributed to the effects of stellar activity [see, e.g. @Popper:1997; @Torres:2013]. The bottom panel of the figure shows that the temperatures of the two components are consistent with the theoretical values for their mass within the errors. This is somewhat unexpected for the secondary, as stellar activity typically causes both “radius inflation” and “temperature suppression”, though the latter effect is smaller and not as easy to detect. Aside from the brightness measurements and our spectroscopic estimates of $T_{\rm eff}$ and $v \sin i$ from Section \[sec:spectroscopy\], we have no direct information on the other fundamental physical properties of the tertiary. Based on our spectroscopically measured flux ratio of $L_3/L_1 = 0.108 \pm 0.012$ at 5187 Å and the above best-fit MIST isochrone, we infer $M_3 \approx 0.87~M_{\sun}$, $R_3 \approx 0.80~R_{\sun}$, and $T_{\rm eff} \approx 5490$ K. The temperature is consistent with that estimated directly from our spectra, and the radius ratio $R_3/R_1 \approx 0.54$ is not far from the value we found in a different way at the end of Section \[sec:analysis\]. Discussion {#sec:discussion} ========== The $\sim$4% discrepancy between the measured and predicted radius for the K2 secondary in  is in line with similar anomalies displayed by other late-type stars having significant levels of activity. While we do not detect any temperature suppression that often accompanies radius inflation, the fractional effect in $T_{\rm eff}$ seen in other cases is typically half that of radius inflation, or only about 100 K in this case, which is smaller than our formal uncertainty.  is an X-ray source listed in the ROSAT All-Sky Survey [@Voges:1999], and is also reported to have shown at least one X-ray flare during those observations [@Fuhrmeister:2003]. This is a clear indication of magnetic activity in the system, though in principle the source could be any of the three stars, or even all three. From the ROSAT count rate of $0.082 \pm 0.012$ counts s$^{-1}$ and the measured hardness ratio (${\rm HR1} = -0.22 \pm 0.14$) we infer an X-ray flux of $F_{\rm X} = 5.9 \times 10^{-13}$ erg cm$^{-2}$ s$^{-1}$, adopting the energy conversion factor recommended by [@Fleming:1995]. Using our distance estimate of 90 pc we then derive an X-ray luminosity of $L_{\rm X} = 5.7 \times 10^{29}$ erg s$^{-1}$. While fairly common in late-type objects (particularly if rotating rapidly), X-ray emission in stars much earlier than mid-F is generally not easy to explain because they lack sufficiently deep surface convective zones that are typically associated with magnetic activity generated by the dynamo effect. For this reason, X-rays in these stars are most often attributed to an unseen late-type companion [e.g., @Schroder:2007 and references therein], which can easily be hidden in the glare of the primary. Other mechanisms intrinsic to earlier-type stars are possible, such as shocks and instabilities in the radiatively driven winds, although these are not thought to be able to explain variability such as the X-ray flaring mentioned above [see, e.g. @Schmitt:2004; @Balona:2012]. We cannot completely rule out a priori that the primary in  is the main source of the X-rays, but its much thinner convective envelope makes this seem far less likely than an origin in a later-type star such as the secondary or tertiary. Indeed, the MIST models indicate that the mass of the convective envelope of the secondary is about 7.2% of its total mass, and that of the tertiary is 4.5% (the value for the Sun is 1.6%), whereas the fractional mass of the primary’s envelope is only $3 \times 10^{-4}$. The tertiary component in  is a possible source for the X-rays, if it were a rapidly rotating star. However, our spectroscopy suggests it is not a fast rotator: we measure $v \sin i < 2$  (Section \[sec:spectroscopy\]), although the projection factor is unknown so it is concievable the equatorial rotation is much faster. To estimate the true rotation period we used the age of the system (2.2 Gyr) along with the gyrochronology relations of [@Epstein:2014] and the estimated $B-V$ color of the star from the MIST isochrones, and inferred $P_{\rm rot} \approx 18$ days. If attributed entirely to the tertiary, the measured X-ray luminosity of  would be far in excess (by about an order of magnitude) of what is expected for a star of this mass and rotation period, according to studies of the relationship between stellar activity and rotation [e.g., @Pizzolato:2003]. This argues the X-rays are unlikely to originate mainly in the tertiary, although it is possible it has some small contribution. We are thus left with the secondary as the most probable site of the bulk of the X-ray emission in . With the bolometric luminosity given in Table \[tab:dimensions\] we compute $\log L_{\rm X}/L_{\rm bol} = -3.31$, a value that is close to the saturation level seen in very active stars. The study of [@Pizzolato:2003] indicates that this is in fact a typical value for a star of this mass with a rotation period of 2.63 days, supporting our conclusion that the secondary is the active star in the system. If that is the case, this provides a natural explanation for its inflated radius. Recent stellar evolution models that incorporate the effects of magnetic fields have had some success in explaining radius inflation in stars like the secondary [see, e.g., @Feiden:2012; @Feiden:2013]. To achieve this, those models introduce a tunable parameter that is the average strength of the surface magnetic field, $\langle Bf\rangle$, where $B$ is the photospheric magnetic field strength and $f$ the filling factor. Measurements of magnetic field strengths are very difficult to make in binary systems, let alone in triple-lined systems such as , but they are essential in order to validate the fits that these models provide. A rough estimate of $\langle Bf\rangle$ for the secondary may be obtained by taking advantage of a power-law relationship shown by [@Saar:2001] to exist between $\langle Bf\rangle$ and the Rossby number, $Ro \equiv P_{\rm rot}/\tau_c$, where $\tau_c$ is the convective turnover time. For consistency with the work of [@Saar:2001], we take $\tau_c$ from the theoretical calculations by [@Gilliland:1986], which give $\tau_c \approx 29$ days for a star with a temperature of 5020 K. The resulting Rossby number, $Ro \approx 0.091$, together with the relation by [@Saar:2001] then yields $\langle Bf\rangle \approx 1.1$ kG.[^4] An independent way of estimating the magnetic field strength makes use of the X-ray luminosity and the empirical relationship between that quantity and the total unsigned surface magnetic flux, $\Phi = 4\pi R^2 \langle Bf\rangle$. [@Pevtsov:2003] have shown in a study of magnetic field observations of the Sun and active stars that the relation holds over many orders of magnitude. With an updated version of that relation by [@Feiden:2013], and the measured radius of the secondary, we obtain $\langle Bf\rangle \approx 1.0$ kG, which is similar to our previous result. A magnetic field strength of this order is quite consistent with values measured in many other cool, active single stars [see, e.g., @Cranmer:2011; @Reiners:2012]. Our estimate of $\langle Bf\rangle \approx 1.0$ kG can serve as an input to stellar evolution calculations that model the effects of magnetic fields, and test their ability to match the measured size of the secondary.  is attended by a distant third star that is physically bound: we have shown that it has a similar radial velocity as the eclipsing pair, a brightness perfectly consistent with that expected for a star of its temperature at the same distance as the binary, and a motion on the plane of the sky that is incompatible with a background object but consistent with orbital motion in a highly inclined orbit around the binary (possibly even coplanar with it). The system is thus a hierarchical triple, which is not surprising given that [@Tokovinin:2006] have shown that up to 96% of all solar-type binaries with periods shorter than 3 days have third components. Conclusions {#sec:conclusions} =========== Our spectroscopic observations together with existing $uvby$ photometry have enabled us to derive accurate absolute masses and radii for the eclipsing components good to better than 1.5%, despite the faintness of the secondary (only 3.6% of the brightness of the primary).  thus joins the ranks of binary systems with the best determined properties [see @Torresetal:2010]. The highly unequal masses provide increased leverage for the comparison with stellar evolution models, and we find that the K2 secondary is about 4% larger than predicted for its mass, though its temperature appears normal. Thus, the star appears overluminous. The detection of the system as an X-ray source is evidence of activity, and we have argued that the source is the secondary component. This would provide at least a qualitative explanation for the radius anomaly, which is also seen in many other active stars with convective envelopes. We would expect the secondary to have significant spot coverage, but the star is much too faint compared to the primary for this to produce a visible effect on the light curves.  is a good test case for recent stellar evolution models that attempt to explain radius inflation in a more quantitative way by including the effects of magnetic fields. To this end, we have provided an estimate of the strength of the surface magnetic field on the secondary ($\sim$1 kG). Finally, we note that the study of this system would benefit from a detailed chemical analysis of the primary star based on high-resolution spectroscopy with broader wavelength coverage than the 45 Å afforded by the material at our disposal. This would remove the metallicity as a free parameter in the comparison with stellar evolution models, strengthening the results. [**Note added in proof:**]{} A high-resolution ($R \approx 44,000$) echelle spectrum of  with a signal-to-noise ratio of 220 in the b region was obtained recently at the Tillinghast reflector during the second quadrature (HJD 2,458,029.6, phase 0.73). It shows no sign of activity (e.g., H and K or H$\alpha$ emission) in the brighter primary, supporting our contention that this star is not particularly active. We are grateful to P. Berlind, M. Calkins, D. W. Latham, R. P. Stefanik, and J. Zajac for help in obtaining the spectroscopic observations of , and to R. J. Davis and J. Mink for maintaining the CfA echelle database over the years. We also thank J. Choi for assistance in calculating the extent of stellar envelopes, and the anonymous referee for helpful comments. We acknowledge support from the SAO Research Experience for Undergraduates (REU) program, which is funded by the National Science Foundation (NSF) REU and Department of Defense ASSURE programs under NSF grant AST-1659473, and by the Smithsonian Institution. G.T. acknowledges partial support for this work from NSF grant AST-1509375. This research has made use of the SIMBAD and VizieR databases, operated at CDS, Strasbourg, France, and of NASA’s Astrophysics Data System Abstract Service. Balona, L. A. 2012, , 423, 3420 Casagrande, L., Ram[í]{}rez, I., Mel[é]{}ndez, J., Bessell, M., & Asplund, M. 2010, , 512, A54 Chabrier, G., Gallardo, J., & Baraffe, I. 2007, , 472, L17 Choi, J., Dotter, A., Conroy, C., et al. 2016, , 823, 102 Claret, A. 2000, , 363, 1081 Claret, A., & Bloemen, S. 2011, , 529, A75 Cranmer, S. R., & Saar, S. H. 2011, , 741, 54 Crawford, D. L. 1975, , 80, 955 Epstein, C. R., & Pinsonneault, M. H. 2014, , 780, 159 Etzel, P. B. 1980, EBOP User’s Guide, 3rd. Ed., Dept. of Astron. Univ. California, Los Angeles Etzel, P. B. 1981, Photometric and Spectroscopic Binary Systems, Proc. NATO Adv. Study Inst., ed. E. B. Carling & Z. Kopal (Dordrecht: Reidel), p. 111 Fabricius, C., H[ø]{}g, E., Makarov, V. V., et al. 2002, , 384, 180 Fabricius, C., & Makarov, V. V. 2000, , 356, 141 Feiden, G. A., & Chaboyer, B. 2012, , 761, 30 Feiden, G. A., & Chaboyer, B. 2013, , 779, 183 Fleming, T. A., Molendi, S., Maccacaro, T., & Wolter, A. 1995, , 99, 701 Flower, P. J. 1996, , 469, 355 Fuhrmeister, B., & Schmitt, J. H. M. M. 2003, , 403, 247 Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2016, , 595, A2 Gilliland, R. L. 1986, , 300, 339 Hilditch, R. W. 2001, An Introduction to Close Binary Stars (Cambridge, UK: CUP), p. 153 Horch, E. P., Falta, D., Anderson, L. M., et al. 2010, , 139, 205 Husser, T.-O., Wende-von Berg, S., Dreizler, S., et al. 2013, , 553, A6 Kurpinska-Winiarska, M., & Oblak, E. 2000, The Formation of Binary Stars, IAU Symposium No. 200, ed. B. Reipurth and H. Zinnecker (Potsdam, Germany: AIP), 141 Latham, D. W. 1992, Complementary Approaches to Double and Multiple Star Research, IAU Coll. 135, ASP Conf. Ser. 32, ed. H. A. McAlister & W. I. Hartkopf (San Francisco, CA: ASP), 110 Latham, D. W., Nordström, B., Andersen, J., Torres, G., Stefanik, R. P., Thaller, M., & Bester, M. 1996, , 314, 864 Latham, D. W., Stefanik, R. P., Torres, G., Davis, R. J., Mazeh, T., Carney, B. W., Laird, J. B., & Morse, J. A. 2002, , 124, 1144 Martin, S. 2003, Interplay of Periodic, Cyclic and Stochastic Variability in Selected Areas of the H-R Diagram, ASP Conf. Ser., Vol. 292, ed. C. Sterkan (San Francisco, CA: ASP), 59 Morales, J. C., Gallardo, J., Ribas, I., et al. 2010, , 718, 502 Nordström, B., Latham, D. W., Morse, J. A. et al. 1994, , 287, 338 Oblak, E., Lastennet, E., Fernandes, J., Kurpinska-Winiarska, M., & Valls-Gabaud, D. 2004, Spectroscopically and Spatially Resolving the Components of the Close Binary Stars, ASP Conf. Ser., 318, ed. R. W. Hilditch, H. Hensberge and K. Pavlovski (San Francisco, CA: ASP), 175 Paxton, B., Bildsten, L., Dotter, A., et al. 2011, , 192, 3 Paxton, B., Cantiello, M., Arras, P., et al. 2013, , 208, 4 Paxton, B., Marchant, P., Schwab, J., et al. 2015, , 220, 15 Perryman, M. A. C., & ESA 1997, The Hipparcos and Tycho Catalogs (ESA SP-1200; Noordwijk: ESA) Pevtsov, A. A., Fisher, G. H., Acton, L. W., et al. 2003, , 598, 1387 Pizzolato, N., Maggio, A., Micela, G., Sciortino, S., & Ventura, P. 2003, , 397, 147 Popper, D. M. 1980, , 18, 115 Popper, D. M. 1997, , 114, 1195 Popper, D. M., & Etzel, P. B. 1981, , 86, 102 Pr[š]{}a, A., Harmanec, P., Torres, G., et al. 2016, , 152, 41 Reiners, A. 2012, Living Reviews in Solar Physics, 9, 1 Rodr[í]{}guez, E., Garc[í]{}a, J. M., Mart[í]{}n, S., & Claret, A. 2001, , 372, 588 Saar, S. H. 2001, 11th Cambridge Workshop on Cool Stars, Stellar Systems and the Sun, ASP Conf. Ser. 223, ed. R. J. García López, R. Rebolo, & M. R. Zapatero Osorio (San Francisco, CA: ASP), 292 Schmitt, J. H. M. M. 2004, Stars as Suns: Activity, Evolution and Planets, IAU Symp. 219, ed. A. K. Dupree and A. O. Benz (San Francisco, CA: ASP), 187 Schr[ö]{}der, C., & Schmitt, J. H. M. M. 2007, , 475, 677 Southworth, J. 2008, , 386, 1644 Southworth, J. 2013, , 557, A119 Tokovinin, A., Thomas, S., Sterzik, M., & Udry, S. 2006, , 450, 681 Torres, G. 2010, , 140, 1158 Torres, G. 2013, Astronomische Nachrichten, 334, 4 Torres, G., Andersen, J., & Gim[é]{}nez, A. 2010, , 18, 67 Torres, G., Stefanik, R. P., Andersen, J., et al. 1997, , 114, 2764 Torres, G., Neuh[ä]{}user, R., & Guenther, E. W. 2002, , 123, 1701 Voges, W., Aschenbach, B., Boller, T., et al. 1999, , 349, 389 Zucker, S., & Mazeh, T. 1994, , 420, 806 Zucker, S., Torres, G., & Mazeh, T. 1995, , 452, 863 [^1]: We note, incidentally, that the heliocentric Julian dates in the online electronic files should be corrected by subtracting exactly 790 days. [^2]: [ http://www.astro.keele.ac.uk/jkt/codes/jktebop.html]{} [^3]: <http://www.astro.keele.ac.uk/jkt/codes/jktld.html> [^4]: The same calculation applied to the tertiary star gives $\langle Bf\rangle \approx 70$ G, which is small compared to the secondary and supports the notion that it is not a very active star. The parameters for the primary star are outside of the range of validity of the [@Saar:2001] relation, but point to a magnetic field strength of only a few Gauss, again suggesting a very low activity level if the sustaining mechanism is the same as in late-type stars.

--- abstract: 'Avalanche control by explosion is a widely applied method to minimize the avalanche risk to infrastructure in snow-covered mountain areas. However, the mechanisms involved leading from an explosion to the release of an avalanche are not well understood. Here we test the hypothesis that weak layers fail due to the stress caused by propagating acoustic waves. The underlying mechanism is that the stress induced by the acoustic waves exceeds the strength of the snow layers. We compare field measurements to a numerical simulation of acoustic wave propagation in a porous material. The simulation consists of an acoustic domain for the air above the snowpack and a poroelastic domain for the dry snowpack. The two domains are connected by a wave field decomposition and open pore boundary conditions. Empirical relations are used to derive a porous model of the snowpack from density profiles of the field experiment. Biot’s equations are solved in the poroelastic domain to obtain simulated accelerations in the snowpack and a time dependent stress field. Locations of snow failure were identified by comparing the principal normal and shear stress fields to snow strength which is assumed to be a function of snow porosity. One air pressure measurement above the snowpack was used to calibrate the pressure amplitude of the source in the simulation. Additional field measurements of air pressure and acceleration measurements inside the snowpack were compared to individual field variables of the simulation. The acceleration of the air flowing inside the pore space of the snowpack was identified to have the highest correlation to the acceleration measurements in the snowpack.' address: - 'Department of Earth Sciences, Simon Fraser University, 8888 University Drive, BC V5A1S6 Burnaby, Canada' - 'WSL Institute for Snow and Avalanche Research SLF, Flüelastrasse 11, 7260 Davos Dorf, Switzerland' - 'Institute of Mechanical Systems, ETH Zurich, 8092 Zürich, Switzerland' author: - Rolf Sidler - Stephan Simioni - Jürg Dual - Jürg Schweizer bibliography: - '0-references.bib' title: Numerical simulation of wave propagation and snow failure from explosive loading --- snow ,acoustic wave propagation ,explosives ,avalanche control ,porous medium ,field experiments Introduction ============ During the last decades the number of people living and recreating in, or travelling through mountainous terrain has substantially increased. To ensure the reliability of infrastructure extensive engineering works such as supporting structures and snow sheds have been built to prevent damages due to large avalanches. Whereas these permanent protection measures are highly effective, they are also costly. Therefore, less expensive temporary preventive measures have become increasingly popular over the last decade. In particular, artificial avalanche release by explosion is among the key preventive measures. The aim is to trigger avalanches when their size is still small enough to not cause any damage and no people are exposed in the path of the avalanche [@mcclung:2006]. Releasing avalanches with explosives by hand or helicopter charging is, however, limited to locations or weather conditions allowing tolerably save access for avalanche control personnel. This limitation has been overcome by fixed avalanche control installations which trigger avalanches by the effect of explosions and allow remote operation even under most adverse weather conditions or during nighttime. The basic physical mechanisms that cause slab avalanches to release from explosives, and other causes, are well known and have been used to choose optimal locations of blast installations for years. What is lacking is a quantitative model incorporating the “known” physics associated with initiating failure of slab avalanches that can be used to examine the processes, improve understanding of the physical processes and make predictions that can be tested in the future. Historically, research on avalanche control has been focused on experimental evidence of waveforms, charge type and placement to support the work of avalanche control operations [@gubler:1976; @mellor:1973; @ueland:1993; @bones:2012; @binger:2015]. The most extensive measuring campaigns were performed by @gubler:1977. However, many of the more recent studies focused on small range effects [@bones:2012; @wooldridge:2012; @johnson:1993]. A more detailed review of the past research within snow and explosions is given by @simioni:2015. A model considering the porous character of snow based on Biot’s [-@biot:1962] equations has been proposed by @johnson:1982, but has rarely been applied to snow since [@albert:2013]. A mixed stress-energy failure criterion including simplified effects of explosive loading was developed by @cardu:2008. It is only recently that numerical tools are used to support a theoretical framework on the physical mechanisms that lead to the release of an avalanche. @miller:2011 considered the non-linear effects of an explosion and non-linear compaction of the snowpack for close ranges using the finite element method. Here we compare the measurements from field experiments on the wave propagation caused by an explosion to the results of a numerical simulation considering the porous character of snow. We tested the hypothesis that the stresses induced by the acoustic wave propagating through the snowpack locally exceed the snow strength and lead to failure. In the winter 2013-2014 we performed multiple field experiments with avalanche control explosives triggered at different elevations above the snow surface and measured the air pressure above the snowpack as well as acceleration at different depths within the snowpack and distances from the point of explosion [@simioni:2015]. In addition, we recorded weak layer failure with cameras. In the following we describe the numerical model that was used to perform the simulations. We focus on a specific experiment as a showcase for the test series, build a layered porous model for the prevailing snowpack, and evaluate the numerical results toward measured air pressure and acceleration. Finally, we compare locations where the stress in the numerical simulation exceeds the strength of the snowpack to the observed locations in the field. Methods ======= Field experiment ---------------- We chose the first experiment from a day with eight experiments on 27. February, 2015 as a showcase to compare with the numerical results. The geometry of the experiment is shown in Figure \[fig:layout\]. A 4.3 kg explosive charge was taped to a wood stick and placed 1 m above the snow surface. Three snow pits were excavated 12.3 m, 17.3 m, and 22.5 m horizontal distance from the point of the explosive charge. Microphones were placed 0.05 m above the undisturbed snow surface next to the snow pits. Three accelerometers were installed in cavities at 0.13 m, 0.48 m, and 0.83 m below the snow surface in each snow pit[@simioni:2015]. Special care was given to fit the diameters of the horizontal holes exactly with the diameters of the accelerometers to warrant the coupling of the sensors to the snowpack. Snowpack failure was recorded with compact cameras [@simioni:2015] ![\[fig:layout\] Longitudinal section of the measuring layout of the experiments from 27 February 2014 [@simioni:2015].](measuring_layout_paper_revP3){width="100.00000%"} The snowpack on the investigated day was 187 cm deep and consisted of a 45 cm thick layer of recently deposited snow (consisting of decomposing and fragmented precipitation particles) including two melt-freeze crusts above a well-consolidated base. The base was composed of layers of small rounded grains interspersed with several melt-freeze crusts and ice layers above hard layers of faceted crystals near the bottom of the snowpack. A potential weak layer was identified at a height of 85 cm from the ground. The snowpack was still dry but relatively warm with a minimum temperature of -1C. The point snow stability based on the snow profile was rated as good [@schweizer:2001]. An extended column test [@Simenhois:2009] indicated that the potential weak layer was very hard to trigger as it was buried below a 1 m thick well consolidated slab. The densities obtained by capacitive measurements in the three snow pits are shown in Figure \[fig:snow-profile\] [@denoth:1989; @eller:1996]. To localize weak layer failure during the experiments, compact cameras were installed in each snow pit and recorded the pit wall during the explosion. The single video stills allowed to visually identify weak layer failure due to movement of the snowpack overlaying the weak layer [@simioni:2015]. ![\[fig:snow-profile\] Density profiles measured with the capacitive Denoth probe in the three pits at distances of 12, 17 and 22 m from the point of triggering.](140227_densities){width="100.00000%"} Numerical model --------------- Seasonal snow is a highly porous material with air often taking up the larger part of the volume. @johnson:1982 showed that Biot’s [-@biot:1956] theory for wave propagation in porous materials can be successfully applied to snow. Acoustic wave propagation in such porous materials is characterized by the presence of a compressional and a shear wave in the ice skeleton and an additional second compressional wave that is propagating in the pore fluid that is also called the “slow” wave. Due to the high porosity and the proportions between material properties this second compressional wave mode is propagating in snow [@oura:1952; @ishida:1965] and can be recorded. This stands in contrast to other natural porous materials as, for example, sediments where the wave is diffusive and cannot be recorded. The energy dissipation mechanism in Biot’s theory is physically modeled by the viscosity of the pore fluid which is moved against the skeleton as acoustic waves propagate through the material. Biot’s model also accounts for the interaction between waves propagating in the porous frame and in the pore space of the material. To simulate acoustic wave propagation in snow we use a pseudo-spectral approach which is known to be accurate and efficient [@boyd:2001]. We use the algorithm of @sidler:2010a where the simulation consists of an upper acoustic domain that is connected to a poroelastic domain by a wave field decomposition to account for the boundary conditions at the interface [@gottlieb:1982; @carcione:1991b; @tessmer:1992]. For this study, the acoustic domain represents the air above the snowpack and the lower domain, where Biot’s [-@biot:1962] differential equations are solved, represents the snowpack. Interfaces of porous materials are not uniquely defined and have one degree of freedom that can be interpreted as the connection between the pore space [@deresiewicz:1963]. The connectivity of the pore space is expressed with the so called surface flow impedance that is zero for open pores and infinite for closed pore interfaces. For natural occurring materials the pore space is mostly connected and open pore boundary conditions apply. However, a coating on the surface, a deposit in the pore space or a mismatch of the pore throats between two porous materials or layers with different characteristics can lead to decreased connectivity. For the snow surface we assume open-pore type boundary conditions, where the air in the pore space is fully connected to the air above the snowpack and the pore spaces of adjacent layers are fully connected. The snowpack is considered two-dimensional in the model. The total size of the acoustic domain is 29 m in horizontal direction by 20 m in vertical direction. The poroelastic domain has the same length in horizontal direction, but is only 3.5 m in vertical direction. The acoustic domain of the model consists of 185 grid nodes in vertical direction and 725 grid nodes in horizontal direction. The poroelastic domain has the same number of grid nodes in the horizontal direction and 147 grid nodes in vertical direction. Due to the use of a Fourier operator the grid nodes are equally spaced at 0.04 m in horizontal direction. In vertical direction the spacing is irregular due to the Gauss-Lobatto points of the Chebyshev operator and a consequent grid stretching [@tessmer:1994; @peyret:2002]. Close to the interface the grid nodes are more densely spaced but almost regularly spaced at 0.04 m throughout most of the porous domain. The source was placed 4 m from the left boundary of the domain 1 m above the snow-air interface. A total of $5.6 \times 10^5$ time steps with a length of $2 \times 10^{-7}$ s were computed, which corresponds to a total length of 0.112 s for the entire simulation. A limitation in the presented simulation is that the non linear effects present in the vicinity of the explosion are not considered. These non-linear effects are believed to no longer be of relevance at the distances considered in this study. As the simulation also models the early part of the experiment some adjustments of the results of the simulation are necessary. The most significant adjustments and limitations of this simulation are: 1. To account for the supersonic wave velocity of the shock wave in the vicinity of the explosion we have adjusted the timing of the simulation to the measurements of the arrival of the direct air wave at the pressure receiver in the air closest to the explosion. 2. To account for the unknown pressure amplitude at the point of the explosion we have scaled the the entire simulation to the pressure of the direct air wave measured at the closest pressure sensor in the air. This is appropriate as the applied simulation is based on linear equations. The relative amplitude of field variables in the simulation does not depend on the amplitude of the source. Therefore it is possible to compute a simulation with a random source amplitude and scale the entire simulation with the actual pressure of the source or with a measurement of any of the field variables at any location in the simulation. 3. To account for the unknown source waveform and changes in the waveform due to non-linear effects we have chosen the Friedlander wavelet as the source waveform. This is the simplest form to express a blast wave [@friedlander:1946]. 4. To account for the effects of coupling between the acceleration of the pore fluid (air inside the snowpack) and the accelerometers we have used a simple scaling of the simulated fluid acceleration by a scaling factor. 5. The simulation does not make any adjustments at locations where snow failure is predicted. Snowpack model -------------- Ten properties of the porous material are required to solve Biot’s equations. The ice skeleton of the snow is defined by the bulk modulus of the frame material $K_s$, the bulk modulus of the matrix $K_m$, and the shear modulus $\mu_s$. The pore fluid is characterized by the fluid bulk modulus $K_f$ and the viscosity $\eta$ of the pore fluid. Additionally, the densities of the solid and fluid materials $\rho_s$ and $\rho_f$ have to be known. The geometry of the pore space is defined by the porosity $\phi$, the permeability $\kappa$, and the tortuosity $\mathcal{T}$. It is often not possible to measure all these properties at the required spatial resolution. However, the properties of snow are interrelated and [*a priori*]{} information, geometrical considerations, and empirical relationships can be used to estimate unknown snow properties from snow porosity or density [@sidler:2015]. The relations to obtain the porous properties of snow from its porosity are summarized in Table \[tab:snowmodel\]. Porous frame ----------------------- ----------------------------- ----------------------------------------------------------------------------------------------------- frame bulk modulus $K_{\rm s}$ (GPa) 10 matrix bulk modulus $K_{\rm m}$ (GPa) $K_{\rm S} (1-\phi)^\frac{30.85}{(7.76-\phi)}$ shear modulus $\mu_{\rm s}$ (GPa) $ \frac{3}{2} \frac{K_{\rm m} (1-2 \nu)}{1+\nu}$ density $\rho_s$ (kg/m$^3$) $(1-\phi) \cdot 916.7 $ permeability $\kappa$ (m$^2$) $0.2 \frac{\phi^3}{(\text{SSA})^2 (1-\phi)^2}$ tortuosity ${\cal T}$ $\frac{1}{2} \left( 1+\frac{1}{\phi} \right)$ Poisson’s ratio $\nu$ $\nu = 0.38 - 0.36 \, \phi$ specific surface area $\text{SSA}$ (m$^{2}$/kg) ${ \rm SSA} = -30.82 \frac{{\rm m^2}}{{\rm kg}} \ln(1-\phi) -17.93 \frac{{\rm m^2}}{{\rm kg}}$ Pore fluid density $\rho_{\rm f}$ (kg/m$^{3}$) 1.29 viscosity $\eta_{\rm}$ (Pa s) $1.7 \times 10^{-5}$ bulk modulus $K_{\rm f}$ (Pa) $1.4 \times 10^{5}$ : Material properties for a Biot-type porous snow model as a function of porosity $\phi$ [@sidler:2015]. \[tab:snowmodel\] The density profiles in the three snow pits show a spatially uniform distribution of the snowpack and we use a horizontally layered snowpack model for the simulation. Our density model is based on the density profile from the snow pit at 22 m horizontal distance from the explosion labeled X3 in Figure \[fig:snow-profile\] and is shown in Figure \[fig:porosity\]. A porosity model is obtained from this density model by assuming that the pore space is completely filled with air in dry snow. Based on the porosity model the remaining properties for the porous model are derived according to the relations shown in Table \[tab:snowmodel\]. ![\[fig:porosity\] Porosity distribution for the numerical model based on the snow profile in Hole 3.](porosity-model){width="100.00000%"} Source characterization ----------------------- The strong overpressure originating from an explosion leads to non-linear effects that are not covered by our numerical simulation. For such high pressures, the bulk modulus of the air is a function of the pressure amplitude and the temperature difference before and after the shock front [@cooper:1996]. The higher bulk modulus of air under higher pressure and temperature leads to shock waves propagating faster than sound waves. Moreover, as parts of the waveform with higher amplitude propagate faster than those with lower amplitude the wave front steepens up during propagation. When the shockwave is reflected at the interface between the air and the snowpack the incident and the reflected wave have positive interference and consequently a higher velocity in the vicinity of the interface that can lead to the formation of a so called “mach stem” [@mach:1878]. However, these effects are present only in the vicinity of the snowpack. The air pressure is supposed to decay due to non adiabatic effects and geometrical spreading. Therefore, at larger distances from the point of explosion, linear equations can be used to predict wave propagation. Using an elastic approach, it is not possible to characterize the amplitude and waveform from the energy content and the type of chemical reaction of the explosive. Instead, the waveform is estimated from pressure recordings to be of the form of a Friedlander wavelet, which is widely used for this purpose. This waveform is then scaled to fit the recorded pressure at the microphone that was placed at 12.3 m distance from the point of explosion. A similar approach was used by @albert:2013 who also used the recording of a receiver to characterize the waveform of the elastic equivalent source. The Friedlander wavelet and its frequency content used in the numerical simulation are shown in Figure \[fig:source-wvlt\]. a) ![\[fig:source-wvlt\] a) Friedlander wave form and b) frequency spectrum.](friedlander-waveform-500){width="100.00000%"} b) ![\[fig:source-wvlt\] a) Friedlander wave form and b) frequency spectrum.](friedlander-frequency-500){width="100.00000%"} \ Simulated snow failure locations -------------------------------- To evaluate the locations where the propagating wave field leads to a failure of the snowpack we compare the stress field of the numerical simulation to the compressional and shear strength of snow. In general, the nature of failure depends on the loading conditions. For brittle failure, @mellor:1975 suggested to use a fraction of the snow’s Young’s modulus to define the maximum stress snow may withstand before it starts to fail. Here we use a fraction of $1 \times 10^{-3}$ of the bulk modulus and a fraction of $0.5 \times 10^{-3}$ of the shear modulus to define the strength. The corresponding shear and compressional strength as a function of porosity based on these fractions and the relationships from Table \[tab:snowmodel\] are shown in Figure \[fig:strength\] and are compared to the compilation of snow strength measurements presented by @mellor:1975 and @shapiro:1997. a) ![\[fig:strength\] Maximum a) compressional and b) shear strength of snow. The black lines denote the strength to uniaxial and shear stress, respectively. For comparison, maximum strength of compilations from @mellor:1975 and @shapiro:1997 are shown.](axial-strength){width="100.00000%"} b) ![\[fig:strength\] Maximum a) compressional and b) shear strength of snow. The black lines denote the strength to uniaxial and shear stress, respectively. For comparison, maximum strength of compilations from @mellor:1975 and @shapiro:1997 are shown.](shear-strength){width="100.00000%"} \ The absolute values of the field variables were scaled to fit the pressure recording of the microphone at a horizontal distance of 12.3 m from the point of the explosion to evaluate the snow failure locations in the numerical simulation. As the underlying equations are linear the scaling of the simulation can be performed by a simple multiplication with a scaling factor. The factor itself depends on the amplitude of the source in the simulation and can be randomly chosen. The maximum principal normal and shear stresses for all grid nodes in the simulation are computed from snapshots of the stress tensor every 0.4 ms for the whole length of the simulation. The maximum principal normal stress $\sigma_{pn}$ and the principal shear stress $\tau_{p}$ can be obtained from the stress tensor $\sigma$ as $$\sigma_{pn} = \frac{\sigma_{xx} - \sigma_{yy}}{2} +\tau_{p},$$ and $$\tau_{p} = \sqrt{ \sigma_{xy}^2 +\left( \frac{\sigma_{xx} - \sigma_{yy}}{2} \right)^2 } ,$$ where the first indices $x$ and $y$ indicate a plane normal to the corresponding coordinate axis and the second indices denotes the direction in which the stress acts [@jaeger:2007]. The individual snapshots are then evaluated for locations where principal normal and shear stresses exceed the maximum strength of the snow model. These maximum strengths are computed from the porosity model. Based on the porosity model matrix the bulk and the shear moduli for all grid nodes are computed using the equations shown in Table \[tab:snowmodel\]. Locations where the simulated stresses have exceeded the computed strengths of the snowpack in one or more snapshots are considered locations of snow failure. Results ======= Air pressure ------------ The air pressure measurements above the snow surface and corresponding frequency spectra are shown in Figure \[fig:air-pressure\] and compared to the corresponding results of the numerical simulation. As the equations in the simulation are 2D a correction accounting for the differences between cylindrical and spherical spreading is applied for accuracy and completeness. In the 2D simulation the waves propagate cylindrically from the point of explosion and the amplitude decays proportional to $1/\sqrt{r}$, but in the field measurements the source is a point source which shows a spherical amplitude decay proportional to $1/r$ [@aki:1980]. The differences resulting from this processing step are rather small because of the relatively large distance from the source which leads to a small curvature of the waves. The differences are shown in Figure \[fig:air-pressure\]a). The simulated air pressure is shown in red while the air pressure corrected for spherical spreading is shown in blue. Omitting this step would not lead to any different conclusions for this study. The length of a typical signal is around 40 ms and the main frequency content is between 20Hz and 70Hz. The two receivers at larger distances from the point of the explosion show a stronger decrease of frequencies between $\sim$ 50Hz and 150Hz than the one closest to the point of explosion. Experiments with similar charge sizes and elevations of the point of explosion show very similar pressure wave forms and amplitudes. a) ![\[fig:air-pressure\] Time series of the pressure a) and the normalized amplitude spectrum b) for the simulated (color) and measured (black) air pressure 5 cm above the snowpack. The red (cylindrical) and blue (spherical) curves show the relatively small effect of the spreading correction necessary because of the 2D equations in the simulation. The velocity of the air pressure wave in the simulation is $V_{air} = 350$m/s.](mod-airpressure-fresh){width="100.00000%"} \ b) ![\[fig:air-pressure\] Time series of the pressure a) and the normalized amplitude spectrum b) for the simulated (color) and measured (black) air pressure 5 cm above the snowpack. The red (cylindrical) and blue (spherical) curves show the relatively small effect of the spreading correction necessary because of the 2D equations in the simulation. The velocity of the air pressure wave in the simulation is $V_{air} = 350$m/s.](mod-freq){width="100.00000%"} \ Acceleration in the snowpack ---------------------------- Simulated vertical acceleration of the ice skeleton 0.13 m, 0.48 m, and 0.83 m below the snow surface at a horizontal distance of 17.3 m from the explosion are compared to the measured acceleration at the same locations in Figure \[fig:17m-skeleton\]. The measured accelerations show a strong peak of acceleration at about 39 ms. This is approximately the same time as the air pressure wave arrives at the pressure sensor in the air above the snowpack. However, before the strong peak smaller peaks and negative acceleration are present in the field recordings. The simulated ice matrix acceleration shows small peaks of arriving wave fronts beginning approximately 15 ms after the triggering of the explosion. This earlier arrival corresponds to the fact that the acoustic wave speeds in the snowpack is higher than the wave speed in the air. Yet, the prominent recorded peak at 39 ms is considerably smaller in the simulated matrix acceleration. Subsequent arriving wave modes have a similar amplitude in the simulation and in the field measurements. The measured accelerations are compared to the pore fluid accelerations in Figures \[fig:12m-pore\], \[fig:17m-pore\], and \[fig:22m-pore\] for horizontal distances of 12.3 m, 17.3 m, and 22.5 m from the point of the explosion, respectively. Here, the prominent peak in the measured acceleration corresponds to the simulated air pressure wave propagating through the pore space of the snow pack. As the arrival times in the ice matrix responds well with the expected wave speeds in the snowpack we think that the response of the ice matrix is superimposed by the stronger response of the pore fluid. From the recording time of the modeled seismograms 11.7 ms is subtracted to account for the higher air wave velocities due to non-linear effects in the vicinity of the explosion. The numerically simulated accelerations shown in Figures \[fig:17m-skeleton\], \[fig:12m-pore\], \[fig:17m-pore\], and \[fig:22m-pore\] were scaled with a factor $10^{-2}$ (-40 dB). This scaling factor reproduces the coupling effect well for the receivers located 0.13 m and 0.48 m below the snow surface. For the receivers at a depth of 0.83 m below the snow surface the scaling factor is $10^{-1}$ (-20 dB). A part of this lower amplitude of the measured seismograms compared to the simulated particle acceleration can be explained with the higher inertia of the receiver compared to an air particle. In the simulation the stresses at the receiver locations act on an air particle with a much lower mass than that of a physical receiver. The resulting acceleration from this stress will therefore be higher than the measured acceleration. As stresses are applied to the receiver from both, the ice skeleton and the air in the pore space the response of the receiver will actually be a complex combination of the stress field, snow porosity, bulk modulus and density of the foam surrounding the receiver, and how well the foam is coupled to the ice skeleton. A method to estimate the coupling of a receiver to a visco-elastic ocean bottom, which represents a somewhat similar situation, has been shown by @sutton:1981. Vertical accelerations decrease rapidly with depth at all distances from the point of explosion. This effect can be explained with the strong attenuation for the second compressional wave. For the receiver 0.83 m below the snow surface it is not exactly clear whether the lower scaling factor is due to an increased coupling between snow and receiver or if the lower amplitude of the simulation is due to the snowpack properties in the model. It is save to say that predictions for wave propagating in the pore fluid are better higher in the snowpack as less layers are involved. However, the most plausible explanation is better coupling to the ice matrix due to the lower porosity as an overestimated attenuation would also lead to strong dispersion for the second compressional wave [@johnson:1982; @sidler:2015]. Such a dispersion can not be seen in the receivers at a depth of 0.83 m below the snow surface. a) ![\[fig:17m-skeleton\] Comparison of measured acceleration with modeled acceleration of the ice skeleton of the snow at a horizontal distance of 17.3m. The receivers were buried a) 0.13m, b) 0.48m, and c) 0.83 m blow the snow surface.](vy-17m-R6){width="100.00000%"} \ b) ![\[fig:17m-skeleton\] Comparison of measured acceleration with modeled acceleration of the ice skeleton of the snow at a horizontal distance of 17.3m. The receivers were buried a) 0.13m, b) 0.48m, and c) 0.83 m blow the snow surface.](vy-17m-R8){width="100.00000%"} \ c) ![\[fig:17m-skeleton\] Comparison of measured acceleration with modeled acceleration of the ice skeleton of the snow at a horizontal distance of 17.3m. The receivers were buried a) 0.13m, b) 0.48m, and c) 0.83 m blow the snow surface.](vy-17m-R10){width="100.00000%"} a) ![\[fig:12m-pore\] Measured acceleration compared to simulated acceleration of the pore fluid at a horizontal distance of 12.3m. The receivers were buried a) 0.13m, b) 0.48m, and c) 0.83 m blow the snow surface.](qy-12m-R0){width="100.00000%"} \ b) ![\[fig:12m-pore\] Measured acceleration compared to simulated acceleration of the pore fluid at a horizontal distance of 12.3m. The receivers were buried a) 0.13m, b) 0.48m, and c) 0.83 m blow the snow surface.](qy-12m-R2){width="100.00000%"} \ c) ![\[fig:12m-pore\] Measured acceleration compared to simulated acceleration of the pore fluid at a horizontal distance of 12.3m. The receivers were buried a) 0.13m, b) 0.48m, and c) 0.83 m blow the snow surface.](qy-12m-R4){width="100.00000%"} a) ![\[fig:17m-pore\] Same as Figure \[fig:12m-pore\], but for a horizontal distance of 17.3m.](qy-17m-R6){width="100.00000%"} \ b) ![\[fig:17m-pore\] Same as Figure \[fig:12m-pore\], but for a horizontal distance of 17.3m.](qy-17m-R8){width="100.00000%"} \ c) ![\[fig:17m-pore\] Same as Figure \[fig:12m-pore\], but for a horizontal distance of 17.3m.](qy-17m-R10){width="100.00000%"} a) ![\[fig:22m-pore\] Same as Figure \[fig:12m-pore\], but for a horizontal distance of 22.4m.](qy-22m-R12){width="100.00000%"} \ b) ![\[fig:22m-pore\] Same as Figure \[fig:12m-pore\], but for a horizontal distance of 22.4m.](qy-22m-R14){width="100.00000%"} \ c) ![\[fig:22m-pore\] Same as Figure \[fig:12m-pore\], but for a horizontal distance of 22.4m.](qy-22m-R16){width="100.00000%"} Generally it can be observed that the wave field in the snowpack is complex and can not well be described by identifying individual wave modes. This is due to the interaction of the propagating waves with the many interfaces present in the simulation. On every interface waves are reflected, transmitted and converted into all supported wave modes. The proportions of the waves that get reflected, transmitted and converted depends on the material properties of of the involved snow layers and the incidence angles. Moreover if the wavelengths are longer than the involved layers the reflection, transmission and conversion originates from a combination of the snow layers ‘sensed’ by the wave. In a direct numerical method where the wave field is evaluated on discrete grid points such as we use here these considerations are implicitly resolved by solving the underlying differential equations for the field variables. Here, these field variables are the horizontal and vertical matrix velocities, the relative horizontal and vertical pore fluid velocities, the horizontal and vertical stresses in the ice matrix, the shear stress in the ice matrix and the pore fluid pressure. As a consequence of the complex wave filed the shear and compressional stresses are difficult to observe in field measurements. In the simulation on the contrary all field variables can be evaluated individually and in combination with the field measurements it is possible to identify which field variables contribute to the recording. Snow failure {#sec:strength} ------------ On the investigated test day, eight experiments were performed. The charge sizes and charge elevation above the snow surface for the eight experiments are shown in Table \[tab:charges\]. \# of experiment Charge size (kg) Elevation above snow surface (m) ------------------ ------------------ ---------------------------------- 1 4.25 1.0 2 4.25 1.0 3 4.25 1.9 4 4.25 2.0 5 5.0 2.0 6 5.0 3.0 7 5.0 3.0 8 10.0 2.0 : Explosive charges for experiments on 27 February 2014. \[tab:charges\] Failure was observed mainly in the upper 30cm of the snowpack and at the weak layer at at a depth of about $\sim$1 m below the snow surface. Failures even deeper in the snowpack occurred with either very large charges or close to the point of the explosion. For some experiments, failure could not be observed at the close pit locations but only in the furthest pit. We assume that there was failure in the closer pits too, but could not be identified by analyzing the recorded video images. Figure \[fig:failloc\] shows the locations of the observed and simulated snow failure. Observed failed layers are indicated in the figure with black horizontal lines. The number at the right end of line indicates the number of the experiment in which the layer failed. In the field experiments the snow failure locations were only evaluated in the snow pits. If a snow pit showed layer failure at a specific depth it was assumed that this layer had also failed closer to the point of the explosion. The points of failure obtained from the numerical simulation are indicated by red dots. Each red dot represents a grid point where the principal stress has exceeded the computed strength of the snowpack in one or more snapshots of the simulation. One of the evaluated snapshots after 30 ms simulation time is shown in Figure \[fig:snapshots\]. In the upper acoustic domain, the air pressure wave can be seen as a red front at 14.5 m from the left boundary of the simulation. Principal shear stresses are shown in the lower poroelastic domain with blue and white color. Locations where the principal shear stress exceeded the shear strength of the snowpack during the past 30 ms of the simulation are indicated with dark red color in the lower domain. a) ![\[fig:failloc\] Simulated locations of snow failure (red dots) due to principal a) normal and b) shear stress compared to observed failure locations (black lines). The number behind the black line indicates the number of the experiment after which the failure was observed. ](failing-stress){width="100.00000%"} \ b) ![\[fig:failloc\] Simulated locations of snow failure (red dots) due to principal a) normal and b) shear stress compared to observed failure locations (black lines). The number behind the black line indicates the number of the experiment after which the failure was observed. ](failing-shear){width="100.00000%"} \ ![\[fig:snapshots\] Snapshot after 30 ms of the numerical simulation. The red star marks the point of the source. Blue and white color indicates principal shear stress in the snowpack. Locations where the principal shear stress of the propagating waves has exceeded the shear strength of the snowpack in the simulation are marked with red color.](snapshot_0070){width="100.00000%"} Discussion ========== Air pressure above the snow surface ----------------------------------- The air pressure measurements of the simulation fit the measurements of the experiments if the amplitude is corrected for the geometrical spreading of a point source. Such a correction is straightforward for the pressure recordings, where the wave travels in a relatively straight path through a homogeneous medium. Such a correction is more difficult for the recordings in the snowpack, where the waves are reflected and follow a complicated path between snow layers. Therefore, the measurements in the snowpack are not corrected for geometrical spreading and it can be assumed that the simulated wave field is slightly overestimating the amplitudes in the snowpack. The time delay for the arrival of the air wave in the simulation and the higher than expected velocity for the speed of sound are presumably due to non-linear effects in the propagation of pressure waves originating from explosions [@simioni:2015]. Close to an explosion the particle displacement is large enough that stresses are not linearly related to the strain as in Hook’s law. In air, large strains yield faster propagating waves than smaller strains and the waveform tends to acquire a steep front. The overpressure of the explosion propagates faster than the speed of sound in the form of a shockwave. The numerical simulation of the experiment does not take into account these non linear effects. However, at larger distances, where the particle displacement becomes smaller due to geometrical spreading, internal friction, and heat dissipation, the pressure wave caused by the explosion propagates in an elastic fashion. The change of waveform due to the nonlinear effects can be taken into account by adequately choosing the shape of the source waveform [e.g., @vandererden:2005]. For example, as we did here, by choosing a Friedlander source wavelet. The speed of sound, derived from the wave arrival time at the microphones and their distance, is around 350 m/s for the field experiment, which is higher than the expected 330 m/s for this temperature [@simioni:2015]. The theoretical shockwave speed $v_{shock}$ for the measured overpressure at 12.3 m from the explosion can be caculated as $$v_{shock} = \upsilon \sqrt{\frac{p_2 - p_1}{\upsilon_1 - \upsilon_2}},$$ where $p$ is the pressure and $\upsilon = 1/ \rho$ is the specific volume [@jaeger:2007 p. 363]. The subscripts 1 and 2 correspond to the regions in front of and behind the shockwave, respectively. For air, the ideal gas law can be used to obtain the density $\rho$ from the air pressure and temperature as $$\rho = \frac{p}{R_{spec} T},$$ where $T$ is the absolute temperature in Kelvin and $R_{spec} = 287.058$ J kg$^{-1}$K$^{-1}$ is the specific gas constant for dry air. The measured maximum air pressure at the microphone was $\sim$10 kPa in excess of the atmospheric pressure of $p_1 \cong 84$ kPa at the elevation of the study site. This pressure difference alone does not explain a shockwave that is propagating faster than the speed of sound. The observed velocity can be explained also with an increase of air temperature behind the wave front. To obtain the observed speed of $\sim$350 m/s the air temperature behind the shockwave has to be 9 $^\circ$C higher than in front of the shockwave. Acceleration in the snowpack ---------------------------- The measured acceleration corresponds much better with the simulated acceleration of the pore fluid than with the simulated acceleration of the ice matrix. This is a rather surprising result as intuitively one would expect that the accelerometers are coupled mainly to the ice skeleton of the snow. However, given that the porosity of the snow is usually higher than 0.5 the volume around the receiver mainly consists of air. It makes therefore sense to assume that the motion of the air around the receiver is represented in the recordings. The introduction of non physically based scaling factors is admittedly a flaw in the modeling process. However, the fact that a single scaling factor can reproduce the amplitude of most of the acceleration recordings so well is a strong indication that there is indeed some kind of coupling process involved between the snow and the accelerometers. The acceleration of the pore fluid in the simulation is almost an order of magnitude higher than the acceleration of the skeleton. Due to the open pore boundary conditions and the high porosity at the snow surface the air pressure from the blast wave is transmitted mostly to the pore space and only to a lesser extent to the ice skeleton of the snow. This wave propagating in the pore space of a porous material is sometimes also called the ’slow’ wave which is of high interest in hydrogeophysics and hydro carbon exploration as it directly connects hydrological properties of the porous materials to seismic wave propagation. Unlike in snow, the slow wave is a diffusive wave mode in these environments and cannot be directly measured. It is not clear why the simulated accelerations are higher than the measured accelerations. There are several mechanism that can potentially contribute to such a discrepancy. Complete coupling between the snow and the accelerometers may not be provided. Also, the coupling of the accelerometers is complicated due to the porous nature of the snowpack. The receiver is not only embedded in a solid material but in a combination of a fluid and a solid material. Due to the high porosity of the snow the receiver is actually mainly surrounded by air. The stiffer ice frame will, however, have a better coupling to the receiver than the pore fluid. Due to differences of the density between the air in the snowpack and the receiver the stresses applied to the receivers lead to accelerations that are different from the accelerations that would arise when the stresses would be applied to snow. Snow failure {#snow-failure} ------------ The simulated failure locations shown in Figure \[fig:failloc\] suggest that snow failure is caused in shear rather than compression or tension. This finding may be one of the reasons why explosives triggered above the snow surface are more effective in triggering avalanches than charges that are deployed within the snowpack. Explosive charges produce almost exclusively compressional waves. Yet, when the charge is placed above the snow surface the compressional waves are converted at the snow surface into all existing wave modes which are, in this case, the first and second compressional waves, the shear wave, a reflected wave in the air above the snowpack and a surface wave within the snowpack. The snow model features a relatively sharp increase in density at a depth of $\sim$0.5 m. Wave amplitudes and snow failure below this interface are considerably smaller (Figure \[fig:failloc\] and Fig. \[fig:snapshots\]) due to the large difference in impedance and the resulting reflectance of incident waves. The same observation is also true for the field measurements with the exception of a failure at 0.6 m and 1 m below the snow surface. Although the layer at 0.6 m only failed with a considerably larger charge and at 1 m depth a weak layer existed that might have favored crack propagation from a failure location closer to the point of explosion. This study is in many aspects complementary to the study of @miller:2011. While here we are missing the non-linear effects, the explicit source characterization, and the deformation of the snowpack in the immediate vicinity of the explosion we account for the porous nature of the snowpack and linear wave propagation at distances up to several tens of meters. The comparison of the simulated with the measured accelerations shown in this study reveal the importance of the porous nature of the snowpack. Most of the recorded signal coincides with the particle movement in the air and not with the ice skeleton of the snowpack. The porosity of the snowpack not only affects wave propagation inside the snowpack but also reflection and transmission of energy at the interface between the snowpack and the air above the snowpack. In a highly porous material as snow a large fraction of the transmitted energy is converted into the second compressional wave that is propagating in the air of the snowpack and is strongly attenuated. Neglecting the porous nature of the interface will consequently lead to overestimating of the energy propagating in the ice skeleton of the snow. Conclusions =========== We have compared field measurements of air pressure and accelerations within the snowpack caused by the detonation of an explosive charge over a horizontal snowpack to a numerical simulation of an acoustic source over a Biot-type porous material. The properties of snow as a porous material were derived from density profiles by using [*a priori*]{} information and empirical relationships. The interface between the air and the porous material in the simulation was considered to be of the open-pore type. The non-linear effects in the vicinity of the explosion were considered by using a Friedlander wave form for the source that was calibrated by the closest measurement of air pressure above the snowpack. The field measurements consisted of air pressure measurements 0.05 m above the snow surface and acceleration measurements in the snowpack. Accelerometers were installed at three snowpits to measure snowpack accelerations. In each snow pit the acceleration was measured at depths of 0.13 m, 0.48 m, and 0.83 m below the snow surface. Regularly spaced markers were placed on the pit walls in the three pits and were monitored with video cameras during the experiments. From the video images the failure locations were determined by visual inspection. The best fit of the amplitudes of the air pressure measurements was obtained when the air pressure in the simulation was corrected for the spherical spreading of a point source. The acceleration recordings in the snowpack fitted the modelled acceleration of the air moving inside the snowpack well. The simulated ice accelerations are missing the characteristic peak acceleration that is specific to the measured accelerations. This finding suggests that waves propagating the pore space, i.e. in the air, significantly contribute to wave propagation in snow due to an explosion. Such waves can be simulated by porous models only and are not considered in the standard elastic or viscoelastic seismological models. Snow failure locations in the simulation were evaluated by comparing the principal normal and shear stresses to snow strength. Snow strength was considered to be a fraction of the bulk or shear modulus of the snow and was derived from the porosity model of the simulation. Snapshots of the principal normal and shear stresses for every 0.4 ms of the simulation were then evaluated for locations were the stresses exceeded the local strength of the snow. The simulation results suggest that observed failures were mainly due to loading by shear stress. Acknowledgements {#acknowledgements .unnumbered} ================ Rolf Sidler was funded by a fellowship of the Swiss National Science Foundation. This study was partly funded by the Swiss Federal Office for the Environment FOEN.

--- abstract: 'We highlight differences in spectral types and intrinsic colors observed in pre-main sequence (pre-MS) stars. Spectral types of pre-MS stars are wavelength-dependent, with near-infrared spectra being 3-5 spectral sub-classes later than the spectral types determined from optical spectra. In addition, the intrinsic colors of young stars differ from that of main-sequence stars at a given spectral type. We caution observers to adopt optical spectral types over near-infrared types, since Hertzsprung-Russell (H-R) diagram positions derived from optical spectral types provide consistency between dynamical masses and theoretical evolutionary tracks. We also urge observers to deredden pre-MS stars with tabulations of intrinsic colors specifically constructed for young stars, since their unreddened colors differ from that of main sequence dwarfs. Otherwise, $V$-band extinctions as much as $\sim$0.6 mag erroneously higher than the true extinction may result, which would introduce systematic errors in the H-R diagram positions and thus bias the inferred ages.' bibliography: - 'pecaut.bib' title: Anomalous Spectral Types and Intrinsic Colors of Young Stars --- Introduction ============ Two of the most fundamental parameters of a star – the effective temperature () and luminosity – are based on simple, easy-to understand data such as the spectral type, extinction, and bolometric corrections. Determining these should be relatively error-free, right? Experience has taught observers that special care must be taken when characterizing pre-main sequence (pre-MS) stars, as these young stars require more attention than that of main-sequence dwarfs. When observers desire to characterize a star’s properties, they normally start with the spectral type. The spectral type of the star is determined by comparing characteristics of the spectrum with spectral standard stars. In doing this we can obtain an estimate of the temperature and a gross estimate of the surface gravity of the star. To quantify the extinction and reddening, the observer will compare the target star’s observed colors to tabulated intrinsic colors of stars of the same spectral type to determine a color excess and use a total-to-selective extinction ratio value to estimate the extinction. Once the extinction has been quantified, a distance can be estimated (if not known), by assuming an age and consulting a theoretical isochrone, or assuming the star is on the main-sequence and calculating a main-sequence distance. Alternatively, if a trigonometric or kinematic parallax is known, we may compute the luminosity of the star. Normally, the point of these calculations is to place the star on the Hertzsprung-Russell (H-R) diagram and compare it to theoretical evolutionary models to estimate an age and mass. Though this process seems fairly straightforward, there are many assumptions and systematic effects which can creep in and result in systematic errors in the fundamental parameters, such as temperature and luminosity (which are relatively model-free), and therefore the parameters derived from the evolutionary models. Assuming we are able to determine the spectral type with perfect fidelity, there are uncertainties in the tables which relate the spectral type, , intrinsic color and bolometric correction. There are also many different varieties of such tables, with slight variations in their intrinsic colors, underlying temperature scale, and bolometric corrections. Some tables even contain self-inconsistent values for the bolometric corrections and the bolometric luminosity of the Sun [@torres2010][^1]! For populations of young stars this presents many problems because systematic errors in the fundamental properties of young, pre-MS stars will propagate into systematic errors in masses and ages (see @soderblom2014 for a full discussion of ages of young stars). This can skew the inferred evolutionary lifetime of gas-rich disks. Since gas giant planets can only form when gas is present in the circumstellar disk, these ages are also used to constrain giant planet formation timescales. Problems are also present when individual stars are mischaracterized. If a young star hosts a directly-imaged substellar object, the mass of the substellar object is estimated by comparing the luminosity and assumed age of the object with evolutionary models. Systematic effects in assumed ages can then propagate to wrong assumptions about the model-derived masses (e.g., $\kappa$ And b; @hinkley2013 [@bonnefoy2014]), which may misdirect planet formation theories. Thus, systematic errors in individual parameters, such as spectral types and intrinsic colors, can propagate down and fundamentally limit our ability to test star and planet formation theories. Spectral Types ============== The spectral type of a target is one of the most useful measurements, since many other stellar properties are usually derived with some dependence on the spectral type. Young stars, like most stars, are typed by comparing their spectra with that of spectral standards, and this has historically been performed using optical spectra. However, many low-mass stars are brighter in the near-infrared (NIR) and so it seems completely reasonable to perform this same measurement with NIR spectra as well. Two interesting cases are that of TW Hya and V4046 Sgr. TW Hya, one of the most well-studied classical T-Tauri stars and a member of the youngest nearby moving group that bears its name (the TW Hydra Association), has typically been assigned a temperature type of K7 using optical spectra (K8IVe, @pecaut2013; K6Ve, @torres2006; K6e, @hoff1998; K7e, @delareza1989; K7 Ve, @herbig1978). However, @vacca2011 assigned a type of M2.5V using NIR spectra from SpeX, which implied a very young age of $\sim$3 Myr instead of the much older, more often quoted age of $\sim$ 10 Myr [@barrado2006]. This discrepancy is a source of great confusion – which spectral type should one adopt when characterizing young stars? V4046 Sgr, a young binary member of the $\beta$ Pictoris moving group harboring a gas-rich disk of its own, has also been typed in both the optical and NIR and the same effect is observed - the NIR spectral type is about 3-5 subtypes later than the optical spectral type [@kastner2014]. However, unlike TW Hya, V4046 Sgr has dynamical mass constraints from radial velocities and gas dynamics [@rosenfeld2012]. @kastner2014 have placed these two young stars on the H-R diagram assuming the optical spectral types in one case and the NIR spectral types in another case. Comparing these two sets of H-R diagram positions with theoretical evolutionary models, the NIR spectral types are inconsistent with the dynamical mass constraints and @kastner2014 thus urged caution against using the NIR spectral types on young stars. @stauffer2003 have studied this wavelength-dependent spectral type effect among the zero-age main sequence K dwarfs in the Pleiades [$\sim$135 Myr; @bell2014]. The @stauffer2003 study found that the spectral type of Pleiades K-type stars were systematically $\sim$1 subtype later in the red optical spectra than the blue optical spectra. Furthermore, they did not observe this spectral type anomaly in members of the older Praesepe cluster [$\sim$650-800 Myr; @gaspar2009; @bell2014; @brandt2015]. @stauffer2003 argued that spots were a major factor in this effect, and concluded that there must be more than one photospheric temperature present in the Pleiades K-dwarfs. Pre-main sequence stars are magnetically very active as well, and observations indicate large filling factors on their surfaces [@berdyugina2005], consistent with this effect. Intrinsic Colors ================ Many studies in the past two decades have pointed out that stellar intrinsic colors of young stars are different than that of typical main sequence dwarfs. @gullbring1998 had noted this for both classical and weak-lined T-Tauri stars in Taurus. @dario2010 have noted this in the Orion Nebula Cluster (ONC) and dereddened the very young ONC cluster members using a specially constructed spectral-type color sequence for young stars. However, their spectral type-color sequence was not published as part of their study. @luhman1999b and @luhman2010 [@luhman2010e] went much further and constructed a color-spectral type sequence for pre-MS late K- and M-type stars and brown dwarfs. Most recently, the @pecaut2013 study released a comprehensive tabulation of the intrinsic colors for pre-MS stars from F-type to late M-type with the most popular photometric bands – Johnson–Cousins $BVI_C$, 2MASS $JHK_S$ and the recently available WISE $W1$, $W2$, $W3$ and $W4$ bands. In addition, we fit the observed spectral energy distributions to Phoenix–NextGen synthetic spectra [@allard2012] to infer an effective temperature () and bolometric correction (BC) scale for pre-MS stars. This young spectral type–color––BC scale was constructed using all the known (as of July 2013) members of the nearest moving groups – the $\eta$ Cha Cluster, the TW Hydra Association (TWA), the $\beta$ Pictoris moving group and the Tucana–Horologium (Tuc-Hor) moving group. Two color-color plots for young stars are shown in Figure \[fig:color-color\]. We plot $V$–$K_S$ on the horizontal axis as a proxy for against $J$–$H$ (left) and $K_S$–$W1$ (right). The stars are predominantly clustered around the dwarf and giant sequence until the two diverge around $V$–$K_S$ of $\sim$4, where they lie between the dwarf and giant locus. This strongly suggests surface gravity is a major factor in the deviation of intrinsic colors from that of dwarfs, since we expect that pre-MS stars will have surface gravities somewhere between that of dwarfs and giants. An indication of the importance of accounting for the difference between pre-MS colors and dwarf colors is shown in Figure \[fig:spt\_jh\]. The $J$–$H$ colors of young stars are significantly redder than the dwarf colors at a given spectral type. If $J$–$H$ dwarf colors were used to estimate extinction for an unreddened M0 star, it would erroneously appear to have an $A_V \simeq 0.6$ mag of extinction! If dwarf colors are used to deredden a population of young stars, a systematic bias may be introduced, depending on the particular color used, which may cause their luminosities to be systematically overestimated and thus their ages would be systematically underestimated. A careful look at the temperature scale of pre-MS stars is warranted, since a star’s adopted spectral type is used to infer the . A frequently used temperature scale for M-type stars is the scale of @luhman2003, which is intermediate between dwarfs and giants. However, in the past few years two important developments have been made available: (1) with the release of the [*2MASS*]{} and [*WISE*]{} catalogs [@skrutskie2006; @cutri2012a], photometry is available which covers large sections of the star’s spectral energy distributions (SED) and (2) high-quality synthetic spectra are available from the Phoenix-NextGen group [@allard2012] which include updated molecular opacities and model low-temperature stars and brown dwarfs more successfully than ever possible previously. Thus it is possible to obtain tight constraints on the of an individual unreddened pre-MS member of one of the nearby, young moving groups through fitting the observed SED with synthetic models. @pecaut2013 fit Phoenix-NextGen BT-Settl models of @allard2012 to the observed SEDs of the members of $\eta$ Cha, TWA, $\beta$ Pic and Tuc-Hor to tie their updated spectral type intrinsic color tabulation to a -BC system. The results of this scale are shown in Figure \[fig:spt\_teff\]. Pre-MS stars are $\sim$200K systematically cooler than their main sequence counterparts at a given spectral type, for spectral types $\sim$G5–K5. Conclusions =========== When characterizing the basic observations and properties of young stars, such as spectral type, , reddening and extinction, care must be exercised to avoid systematic errors and biases. It is important that optical spectra be used to constrain the spectral type of the star, since these seem to best represent the effective temperature of the stars. Some evidence points to spots as a factor in wavelength-dependent spectral types, discussed in detail by @gullbring1998 and @stauffer2003. When considering or using tabulations of intrinsic colors, it is important to adopt intrinsic colors for pre-MS stars when the stars are still contracting to the main sequence. Comparisons with dwarf and giant colors as well as theoretical synthetic spectra of different surface gravity points to surface gravity as an important effect altering the intrinsic colors of pre-MS stars. If individual extinctions are mis-estimated using dwarf colors, stars placed on the H-R diagram may have their luminosities systematically over-estimated and the population may appear younger simply because observers are unable to account for reddening and extinction properly. This clearly inhibits our ability to accurately test theoretical evolutionary models and may systematically bias ages, which has consequences reaching down to mis-estimating age spreads in star-forming regions, underestimating the evolutionary timescales of disks and planet formation timescales, and systematic errors when inferring the initial mass function (IMF) in pre-MS stellar populations. [^1]: There are even a variety of values used for the bolometric magnitude of the Sun. Here we adopt $M_{bol,\odot}=4.7554\pm0.0004$, as advocated in @mamajek2012b.

--- abstract: 'We have measured radial velocities and metallicities of 16 RR Lyrae stars, from the QUEST survey, in the Sagittarius tidal stream at 50 kpc from the galactic center. The distribution of velocities is quite narrow ($\sigma=25$ km/s) indicating that the structure is coherent also in velocity space. The mean heliocentric velocity in this part of the stream is 32 km/s. The mean metallicity of the RR Lyrae stars is \[Fe/H\]$=-1.7$. Both results are consistent with previous studies of red giant stars in this part of the stream. The velocities also agree with a theoretical model of the disruption of the Sagittarius galaxy.' author: - 'A. Katherina Vivas' - Robert Zinn - Carme Gallart title: Velocities of RR Lyrae Stars in the Sagittarius Tidal Stream --- Introduction ============ Numerous observations have shown that the Sagittarius dwarf spheroidal galaxy (Sgr) is being disrupted by the tidal forces of the Milky Way. A long stream of its tidal debris has been observed multiple times in different parts of the sky. Many of these observations are described elsewhere in these proceedings. They include an all-sky view of M giant stars (Majewski et al. 2003), RR Lyrae stars (Vivas et al. 2001, Ivezic et al. 2000), A stars (Yanny et al. 2000), halo turnoff stars (Newberg et al. 2002) and main-sequence stars in color-magnitude diagrams (Mart[í]{}nez-Delgado et al. 2001). Each of these observations detects the stream as an over-density of the tracer above the halo background. Simulations of the disruption of satellite galaxies by the Milky Way show that tidal streams should be seen not only as over-densities but also as coherent structures in velocity space (e.g., Harding et al. 2001). The QUEST survey for RR Lyrae stars (Vivas et al. 2001, 2003, see also Zinn et al. in this volume) has observed part of the Sgr stream in a long, $2.3^\circ$-wide strip near the celestial equator. We present here a study of the radial velocities of a sub-sample of 16 RR Lyrae stars in the Sgr tidal stream. RR Lyrae stars stand out as one of the best tracers of the old halo stellar population because they are bright standard candles. Thus, they can provide excellent views of the stream in both the three-dimensional spatial distribution and the radial velocity distribution. The Data ======== The 16 RR Lyrae stars belong to the clump located at $\sim50$ kpc from the galactic center which has been related to the leading arm of the Sgr tidal stream. The QUEST survey found 84 stars in the Sgr stream, a factor of 10 higher than the background of halo stars. The spatial distribution of the clump indicates that is quite wide in right ascension, about $36^\circ$, from $13\fh 0$ to $15\fh 4$. We included in this study stars along all the stream in order to confirm its true size. All stars have mean magnitudes of $V \sim 19.2$. Because RR Lyrae stars are pulsating stars with periods of $\sim0.5$ days, exposure times of spectra should be kept short ($\la 30$ min) in order to avoid excessive broadening of the spectral lines by the changing pulsational velocity. Given the faintness of the stars in the clump, a large telescope was needed. Spectra of the 16 stars were taken with FORS2 at the VLT-Yepun in Paranal, Chile, during June-Aug 2002. We used grating 600B which gives a resolution of $\sim 6$Å, and covers a spectral range from 3400-6300Å. Exposure times varied between 20 and 30 minutes. For each star we obtained two spectra taken at random times on different nights. This allowed us to make measurements at two different phases during the pulsation cycle. A few radial velocity standards were also observed with the same instrumental setup. Radial Velocities ================= Radial velocities of RR Lyrae stars change during the pulsation cycle by up to $\sim 100$ km/s. Thus, it is important to know the exact phase at which the spectrum was taken in order to separate the systemic velocity of the star from the velocity due to the pulsations. We obtained phase information from the QUEST RR Lyrae catalog (Vivas et al 2003) which provides accurate ephemerides for all the stars in our sample. We determined the radial velocities by cross-correlation with each of the observed radial velocity standards. The error in a single measurement is estimated to be $\sim 20$ km/s. For each star we fitted a radial velocity curve template following the procedure described in Layden (1994). In a few cases, the spectra was taken near the phase of maximum brightness of the light curve. We did not use these observations since there is a strong discontinuity in the radial velocity curve at this point. The results are shown in Figure 1a. The histogram shows the distribution of the heliocentric radial velocities of the 16 stars. Taking out the two obvious outliers, the distribution is quite narrow, with a mean of 32 km/s and a standard deviation of only 25 km/s. The distribution does not resemble the one expected for a random sample of halo stars, which is the dashed, Gaussian curve in Figure 1a. For comparison we also show the distribution of velocities of four red giant stars from the Spaghetti survey (Dohm-Palmer, et al. 2001) which seem to be also associated with the Sgr stream in a region of the sky very close to ours. The presence of two outliers is not surprising. If the Sgr stream lies within a smooth distribution of halo stars following a $r^{-3}$ power-law, we expect 1-2 halo RR Lyrae stars in this volume of the sky. There is also good agreement between our observations and the predictions of the models of the disruption of Sgr. We compare with one of these models (Mart[í]{}nez-Delgado et al 2003) in Figure 1b. The red giants from the Spaghetti survey are also included. Metal Abundances ================ The metal abundances of the stars were measured using the modified $\Delta S$ technique described by Layden (1994), which is based on the equivalent widths of the Ca II K line and the Balmer lines. The error in a single measurement of \[Fe/H\] is 0.2 dex. The distribution of metallicities of the 16 stars of our sample is shown in Fig 2. Our results are in very good agreement with the 4 red giants from the Spaghetti survey. The mean metallicity of the RR Lyrae stars is \[Fe/H\]$= -1.7$. Notice however that the mean metallicity of the stream is significantly lower than in the core of the Sgr galaxy. This could be explained if Sgr once had a radial gradient in metallicity, an age-metallicity relation (since the RR Lyrae variables are exclusively very old stars) or a combination of both. Conclusions =========== We have measured VLT spectra for 16 RR Lyrae stars belonging to a part of the Sgr tidal stream, located at 50 kpc from the galactic center. The distribution of radial velocities is quite narrow, indicating that the Sgr clump is a coherent structure in velocity space. We do not find significant gradients of radial velocities or metallicities along the stream. This work is based on observations collected at the European Southern Observatory, Chile. The data were obtained as part of an ESO Service Mode run. This research project was partially supported by the National Science Foundation under grant AST-0098428. Dohm-Palmer, R. C. et al. 2001, , 555, 37 Harding, P. et al. 2001, , 122, 1397 Ivezic, Z. et al. 2000, , 120, 9631 Layden, A. C. 1994, , 108, 1016 Layden, A. C. & Sarajedini, A. 2000, , 119, 1760 Majewski, S. R. et al. 2003, , submitted Mart[í]{}nez-Delgado, D. et al. 2001, , 549, L199 Mart[í]{}nez-Delgado, D. et al. 2003, , submitted Newberg, H. J. et al. 2002, , 569, 245 Vivas, A. K. et al. 2001, , 554, L33 Vivas, A. K. et al. 2003, , submitted Yanny, B. et al. 2000, , 540, 825

--- abstract: 'We will describe radiative corrections to bremsstrahlung, focusing on applications to luminosity, fermion pair production, and radiative return at high-energy $e^+ e^-$ colliders. A precise calculation of the Bhabha luminosity process was essential at SLC and LEP, and will be equally important in ILC physics. We will review the exact results for two-photon radiative corrections to Bhabha scattering which led to the precision estimates for the BHLUMI MC. We will also compare the implementation of the virtual photon correction to bremsstrahlung for fermion pair production in the ${\cal KK}$ MC to similar exact expressions developed for other purposes, and discuss applications to radiative return in high energy $e^+ e^-$ colliders.' author: - 'S. Yost, S. Majhi, and B. F. L. Ward' title: | \ Virtual Corrections to Bremsstrahlung with Applications to Luminosity Processes and Radiative Return --- INTRODUCTION ============ In the 1990’s S. Jadach, B.F.L. Ward and S.A. Yost calculated the two real photon[@2real] and with M. Melles, the real plus virtual photon corrections[@real+virt] to the small angle Bhabha scattering process. These corrections were used to bring the theoretical uncertainty in the luminosity measurement, as calculated by the BHLUMI Monte Carlo (MC) program [@bhlumi], to within a 0.06% precision level for LEP1 parameters and 0.122% for LEP2 parameters[@precision]. A key component of the two-photon radiative corrections calculated for BHLUMI was the virtual photon contribution to hard photon bremsstrahlung. This radiative correction is also an important contribution to the fermion pair production process in $e^+ e^-$ annihilation implemented in the ${\cal KK}$ MC[@kkmc]. Comparisons of these results[@JMWY] to similar expressions obtained by other authors are reviewed. In particular, we focus attention on recent results for the virtual correction to hard photon radiation in radiative return applications[@KR; @phokhara; @epiphany2]. BHABHA LUMINOSITY PROCESS ========================= BHLUMI was developed into a high-precision tool for calculating the Bhabha luminosity process at SLC and LEP, and it can continue to be developed to meet the requirements of a future linear collider such as the ILC. A key advantage of the program is its exact treatment of the multi-photon emission phase space using a YFS-exponentiation procedure[@yfs], so that IR singularities are canceled exactly to all orders and the leading soft photon effects are exponentiated, leaving only well-behaved YFS residuals to be calculated exactly to the order needed. Table 1 shows a summary of the contributions to the theoretical uncertainty of the Bhabha luminosity process calculated by BHLUMI4.04, which includes the complete second order leading log (${\cal O}(\alpha^2 L^2)$) photonic radiative corrections[@precision]. The LEP1 CMS energy is taken to be the $Z$ mass, with an angular range between $1^\circ$ and $3^\circ$, while the LEP2 result is calculated at a CMS energy of 176 GeV and an angular range of $3^\circ - 6^\circ$. The portion of the error budget of interest here is the missing photonic ${\cal O}(\alpha^2 L)$ contribution, which is due to all two-photon radiative corrections at next-to-leading log (NLL) order. For ILC physics, it is desirable to reach $0.01\%$ precision. This is in reach for BHLUMI. Here, we will concentrate on the photonic contributions. The photonic part of the error estimate in Table 1 comes from comparing to an exact ${\cal O}(\alpha^2)$ calculation. There are three contributions: a double real emission term[@2real] which can reach 0.012%, a real plus virtual photon term[@real+virt] which is bounded by 0.02%, and a two-loop pure virtual correction[@precision; @BVNB] making up 0.014% in the LEP1 case. Adding these in quadrature gives the error quoted in Table 1. Since all of the exact ${\cal O}(\alpha^2)$ photonic radiative corrections are available, adding them to BHLUMI would remove almost all of the error quoted for these effects. The only remaining exact two photon contribution would then be “up-down interference.” The entire ${\cal O}(\alpha)$ up-down interference effect was 0.011% at $3^\circ$ and 0.099% at $9^\circ$[@updown], and the ${\cal O}(\alpha^2)$ contribution to up-down interference would be supressed by an additional factor on the order of ${\alpha\over\pi}L \approx 0.04.$ This contribution may be neglected for small-angle Bhabha scattering. However, a complete calculation of ${\cal O}(\alpha^2)$ effects, including the full up-down interference contribution, could prove useful if the ILC requires a wider angular acceptance for the luminosity monitor. Some new ${\cal O}(\alpha^2)$ computational tools and results on Bhabha scattering have appeared recently[@dixon; @penin; @italian; @riemann; @lorca]. Comparisons to these results will be useful in gauging the precision of different approaches to the higher order radiative corrections in Bhabha scattering. (6.0,2.7) (0,1.5) **Source of Uncertainty** **LEP1** **LEP2** ------------------------------------------ ---------- ---------- Missing Photonic ${\cal O}(\alpha^2 L)$ 0.027% 0.04% Missing Photonic ${\cal O}(\alpha^3L^3)$ 0.015% 0.03% Vacuum Polarization 0.04% 0.10% Light Pairs 0.03% 0.05% $Z$ Exchange 0.015% 0.0% [**Total**]{} 0.061% 0.122% (-0.25,0) (3.0,0.5)[{width="3.0in"}]{} (3.0,0) PAIR PRODUCTION AND RADIATIVE RETURN ==================================== Another important process at electron-positron colliders is fermion pair production. This process is calculated, for example, by the ${\cal KK}$ MC[@kkmc], and again, photonic radiative corrections are essential. In particular, we have presented explicit results real plus virtual photon emission from the initial or final state fermion line. In the case of initial state radiation, emitting a single hard photon permits the final fermion pair creation process to be investigated over a wide range of effective CM energies $s' = s(1-v)$, where $v$ is the energy fraction carried away by the hard photon. This is known as the “radiative return” method, and it can be used at a high energy collider to probe the $Z$ resonance over a range of energies, or at a lower energy collider to measure the pion form factor. Virtual photon emission is the most important radiative correction to radiative return. We have compared our results for this process in the context of the ${\cal KK}$ MC with several other results, including Ref. [@IN] (IN), which is fully differential, but lacking mass corrections, Ref. [@BVNB] (BVNB), which is differential only in $v$, but includes mass corrections, and Ref. [@KR] (KR), which is fully differential and includes mass corrections. The KR result was developed for calculating radiative return in the PHOKHARA MC, and is the newest available comparison. We have compared these results by using them to calculate the YFS residual $\overline\beta^{(2)}_1$, which includes the IR-finite part of single hard bremsstrahlung including virtual photon corrections. We have shown earlier that our result (JMWY) agrees with the IN and BVNB results analytically to NLL order ($({\alpha\over\pi})^2L$). We have recently shown similar analytical agreement for the KR result at NLL order[@ICHEP04; @compare-long]. The NLL result is in fact very compact, and represents the exact results to high accuracy over most of the range of hard photon energy fraction $v$. Without mass corrections, $$\overline\beta^{(2)}_{1\ \rm NLL} = L - 1 + 3\ln(1-r_1) + 2\ln r_2 \ln(1-r_1) - \ln^2(1-r_1) + 2\;{\rm Li}_2(r_1) + {r_1(1-r_1)\over 1 + (1-r_1)^2} + (r_1 \leftrightarrow r_2)$$ where $L = \ln(s/m_e^2)$, $r_i = 2p_i\cdot k/s$ with $p_i$ the incoming $e^{\pm}$ momenta and $k$ the hard photon momentum. This expression is taken as a baseline in comparing all of the exact expressions in a run of the ${\cal KK}$ MC shown in Fig. 1. Fig. 1 compares the NNLL contributions of the various exact expressions in a MC run with $10^8$ events at a CM energy of 200 GeV for a muon final state, using the YFS3ff generator (EEX3 option of ${\cal KK}$ MC). We find that the size of all of the NNLL contributions is less than $2\times 10^{-6}$ as a fraction of the Born cross section for $e^+ e^-\rightarrow \mu^+\mu^-$, for photon energy cut $v_{\rm max} < 0.75$. For $v_{\rm max} < 0.95$, all of the results except BVNB, which is not fully differential, agree to within $2.5\times 10^{-6}$ in units of the Born cross section. For the final data point, $v_{\rm max} = 0.975$, the KR and JMWY results differ by $3.5\times 10^{-5}$ without mass corrections, or $5\times 10^{-5}$ with mass corrections. Comparisons have also been run at 1 GeV CMS energy with comparable results[@epiphany2; @compare-long]. These comparisons show that we have a firm understanding of the precision tag for an important part of the order $\alpha^2$ corrections to fermion pair production in precision studies of the final LEP2 data analysis and future ILC physics. Work supported by Department of Energy contract DE-FG02-05ER41399. S. Yost thanks the organizers for an invitation to present this work at Loopfest IV. [99]{} S. Jadach, B.F.L. Ward and S.A. Yost, Phys. Rev. D47 (1993) 2682. S. Jadach, M. Melles, B.F.L. Ward and S.A. Yost, Phys. Lett. B377 (1996) 168. S. Jadach, E. Richter-Was, W. P[ł]{}aczek, B.F.L. Ward and Z. Was, Comp. Phys. Commun. 102 (1997) 229. B.F.L. Ward, S. Jadach, M. Melles and S.A. Yost, [Phys. Lett.]{} [B450]{} (1999) 262. D.R. Yennie, S.C. Frautschi, and H. Suura, [Ann. Phys.]{} [13]{} (1961) 379; K.T. Mahanthappa, [Phys. Rev.]{} [126]{} (1962) 329. F.A. Berends, W.L. Van Neerven and G.J.H. Burgers, Nucl. Phys.  B297 (1988) 429. M. Igarashi and N. Nakazawa, Nucl. Phys. [B288]{} (1987) 301. S. Jadach, M. Melles, B.F.L. Ward and S.A. Yost, [Phys. Rev.]{} [D65]{} (2002) 073030. C. Glosser, S. Jadach, B.F.L. Ward and S. Yost, Phys. Lett. B605 (2005) 123. J.H. Kühn and G. Rodrigo, Eur. Phys. J. [C25]{} (2002) 215. G. Rodrigo, H. Czyż, J.H. Kühn and M. Szopa, [Eur. Phys. J.]{} [C22]{} (2001) 81. S. Jadach, E. Richter-Was, B.F.L. Ward and Z. Was, Phys. Lett. B253 (1991) 469. S. Jadach, B.F.L. Ward and Z. Was, [Phys. Rev.]{} [D63]{} (2001) 113009; [Comput. Phys. Commun.]{} [130]{} (2000) 260. Z. Bern, L. Dixon and A. Ghinculov, Phys. Rev. D63 (2001) 053007. A.A. Penin, Phys. Rev. Lett. 95 (2005) 010408; Karlsruhe ITT preprint TTP-0513, to appear in Nucl. Phys. B (hep-ph/0508127). R. Bonciani and A. Ferroglia, Phys. Rev. D72 (2005) 056004. M. Czakon, J. Gluza and T. Riemann, Phys. Rev. D71 (2005) 073009; Nucl. Phys. (Proc. Suppl.) B135 (2004) 83. A. Lorca and T. Reimann, Nucl. Phys. (Proc. Suppl.) B135 (2004) 328; DESY report 04-226 (hep-ph/0412047); A. Lorca, contribution to ACAT 2005 (hep-ph/0509367). S.A. Yost, C. Glosser, S. Jadach and B.F.L. Ward, ICHEP 2004: Proceedings of the 32$^{\rm nd}$ International Conference on High Energy Physics, Beijing (World Scientific, Singapore, 2005) 478. S. Jadach, B.F.L. Ward and S.A. Yost, Baylor Univ. preprint BU-HEPP-05/03, to appear. S.A. Yost, S. Jadach and B.F.L. Ward, Acta Phys.  Polon. B36 (2005) 2379.

0[\_0]{} amssym.tex **** **Matts Roos and S. M. Harun-or-Rashid** Department of Physics, Division of High Energy Physics, University of Helsinki, Finland **ABSTRACT** The results of different analyses of the dynamical parameters of the Universe are converging towards agreement. Remaining disagreements reflect systematic errors coming either from the observations or from differences in the methods of analysis. Compiling the most precise parameter values with our estimates of such systematic errors added, we find the following best values: the baryonic density parameter $\Ombh =0.019\pm 0.02$, the density parameter of the matter component $\Omm =0.29\pm 0.06$, the density parameter of the cosmological constant $\Oml = 0.71\pm 0.07$, the spectral index of scalar fluctuations $n_s =1.02 \pm 0.08$, the equation of state of the cosmological constant $w_{\lambda} < -0.86$, and the deceleration parameter $q_0 = -0.56 \pm 0.04$. We do not modify the published best values of the Hubble parameter $H_0 = 0.73\pm 0.07$ and the total density parameter $\Om0\thinspace ^{+0.03}_{-0.02}$.\ INTRODUCTION ============ =5.0mm Our information on the dynamical parameters of the Universe describing the cosmic expansion comes from three different epochs. The earliest is the Big Bang nucleosynthesis which occurred a little over 2 minutes after the Big Bang, and which left its imprint in the abundances of the light elements affecting the baryonic density parameter $\Omega_b$. The discovery of anisotropic temperature fluctuations in the cosmic microwave background radiation at large angular scales (CMBR) by COBE-DMR [@smot], followed by small scale anisotropies measured in the balloon flights BOOMERANG [@dber] and MAXIMA [@ba-ha], by the radio telescopes Cosmic Background Imager (CBI) [@pe-ma], Very Small Array (VSA) [@scot] and Degree Angular Scale Interferometer (DASI) [@halv] testify about the conditions in the Universe at the time of last scattering, about 350000 years after Big Bang. The analyses of the CMBR power spectrum give information about every dynamical parameter, in particular $\Omega_0$ and its components $\Omega_b,\ \Omega_m$ and $\Omega_{\lambda}$, and the spectral index $n_s$. For an extensive review of CMBR detectors and results, see Bersanelli et al. [@bersa]. Very recently, also the expected fluctuations in the CMBR polarization anisotropies has been observed by DASI [@kova]. The third epoch is the time of matter structures: galaxy clusters, galaxies and stars. Our view is limited to the redshifts we can observe which correspond to times of a few Gyr after Big Bang. This determines the Hubble constant, successfully done by the Hubble Space Telescope (HST) [@free], and the difference $\Omega_{\lambda}-\Omega_m$ in the dramatic supernova Ia observations by the High-z Supernova Search Team [@ries] and the Supernova Cosmology Project [@perl]. The large scale structure (LSS) and its power spectrum has been studied in the SSRS2 and CfA2 galaxy surveys [@daco], in the Las Campanas Redshift Survey [@shec], in the Abell-ACO cluster survey [@retz], in the IRAS PSCz Survey [@saun] and in the 2dF Galaxy Redshift Survey [@peac],[@coll]. Various sets of CMBR data, supernova data and LSS data have been analyzed jointly. We shall only refer to global analyses of the now most recent CMBR power spectra and large scale distributions of galaxies. The list of other types of observations is really very long. To mention some, there have been observations on the gas fraction in X-ray clusters [@evrd], on X-ray cluster evolution [@ba-ek], on the cluster mass function and the Ly$\alpha$ forest [@wein], on gravitational lensing [@c-h-i], on the Sunyaev-Zel’dovich effect [@bi-ca], on classical double radio sources [@guer], on galaxy peculiar velocities [@zeha], on the evolution of galaxies and star creation versus the evolution of galaxy luminosity densities [@tota]. In this review we shall cover briefly recent observations and results for the dynamical parameters $H_0,\ \Omega_b,\ \Omega_m,\ \Omega_{\lambda},\ \Omega_0,\ n_s,\ w_\lambda$ and $q_0$. In Section 2 these parameters are defined in their theoretical context, in Section 3 we turn to the Hubble parameter, and in Section 4 to the baryonic density. The other parameters are discussed in Sections 5 and 6, which are organized according to observational method: supernovæ in Section 5, CMBR and LSS in Section 6. Section 7 summarizes our results.\ THEORY ====== The currently accepted paradigm describing our homogeneous and isotropic Universe is based on the Robertson–Walker metric $$\hbox{d}s^2=c^2\hbox{d}t^2-\hbox{d}l^2=c^2\hbox{d}t^2-R(t)^2\left({{\hbox{d}\sigma ^2}\over {1-k\sigma ^2}} +\sigma ^2\hbox{d}\theta^2+\sigma ^2\hbox{sin}^2\theta\ \hbox{d} \phi^2\right)\ \eqno(1)$$ and Einstein’s covariant formula for the law of gravitation, $$G_{\mu\nu}={{8\pi G}\over {c^4}}T_{\mu\nu}\ .\eqno(2)$$ In Eq. (1) d$s$ is the line element in four-dimensional spacetime, $t$ is the time, $R(t)$ is the cosmic scale, $\sigma$ is the comoving distance as measured by an observer who follows the expansion, $k$ is the curvature parameter, $c$ is the velocity of light, and $\theta,\ \phi$ are comoving angular coordinates. In Eq. (2) $G_{\mu\nu}$ is the Einstein tensor describing the curved geometry of spacetime, $T_{\mu\nu}$ is the energy-momentum tensor, and $G$ is Newton’s constant. From these equations one derives Friedmann’s equations which can be put into the form $${{\dot{R}^2 + kc^2}\over {R^2}}= {{8\pi G}\over {3}}(\rho_m+\rho_{\lambda})\ ,\eqno(3)$$ $${{2\ddot{R}}\over {R}}+{{\dot{R}^2 + kc^2}\over {R^2}}= -{{8\pi G}\over {c^2}}(p_m+p_\lambda)\ .\eqno(4)$$ Here $\rho$ are energy densities, the subscripts $m$ and $\lambda$ refer to matter and cosmological constant (or dark energy), respectively; $p_m$ and $p_{\lambda}$ are the corresponding pressures of matter and dark energy, respectively. Using the expression for the critical density today, $$\rho_c={3\over{8\pi G}}H_0^2\ ,\eqno(5)$$ where $H_0$ is the Hubble parameter at the present time, one can define density parameters for each energy component by $$\Omega=\rho/\rho_c\ .\eqno(6)$$ The total density parameter is $$\Omega_0=\Omega_m+\Omega_r+\Omega_{\lambda}\ .\eqno(7)$$ In what follows we shall ignore the very small radiation density parameter $\Omega_r$. The matter density parameter $\Omega_m$ can further be divided into a cold dark matter (CDM) component $\Omega_{CDM}$, a baryonic component $\Omega_b$ and a neutrino component $\Omega_{\nu}$. The pressure of matter is certainly very small, otherwise one would observe the galaxies having random motion similar to that of molecules in a gas under pressure. Thus one can set $p_m=0$ in Eq. (4) to a good approximation. If the expansion is adiabatic so that the pressure of dark energy can be written in the form $$p_\lambda=w_\lambda \rho_\lambda c^2\ ,\eqno(8)$$ and if dark energy and matter do not transform into one another, conservation of dark energy can be written $$\dot{\rho_\lambda}+3H\rho_\lambda(1 + w_\lambda)=0\ .\eqno(9)$$ One further parameter is the deceleration parameter $q_0$, defined by $$q=-{{R\ddot R}\over {\dot{R}^2}}=-{{\ddot R}\over {RH^2}}\ .\eqno(10)$$ Eliminating $\ddot R$ between Eqs. (4) and (10) one can see that $q_0$ is not an independent parameter. The curvature parameter $k$ in Eqs. (1), (3) and (4) describes the geometry of space: a spatially open universe is defined by $k=-1$, a closed universe by $k=+1$ and a flat universe by $k=0$. The curvature parameter is not an observable, but it is proportional to $\Omega_0-1$, so if $\Omega_0$ is observed to be 1, the Universe is spatially flat.\ THE HUBBLE PARAMETER ==================== From the definition of the Hubble parameter $H=\dot{R}/R$ one sees that it has the dimension of inverse time. Thus a characteristic time scale for the expansion of the Universe is the Hubble time $$\tau_H\equiv H_0^{-1}=9.78h^{-1}\times 10^{9} \hbox{yr}.\eqno(11)$$ Here $h$ is the commonly used dimensionless quantity $$h=H_0/(100\ \hbox{km\ s}^{-1}\ \hbox{Mpc}^{-1})\ .\eqno(12)$$ The Hubble parameter also determines the size scale of the observable Universe. In time $\tau_H$, radiation travelling with the speed of light has reached the Hubble radius $$r_H\equiv \tau_Hc = 3000 h^{-1}\hbox{Mpc}.\eqno(13)$$ Or, to put it differently, according to Hubble’s non–relativistic law, $$z = H_0{{r}\over {c}}\ ,\eqno(14)$$ objects at this distance would be expected to attain the speed of light which is an absolute limit in the theory of special relativity. However, in special relativity the redshift $z$ is infinite for objects at distance $r_H$ receding with the speed of light and thus unphysical. Therefore no information can reach us from farther away, all radiation is redshifted to infinite wavelengths and no particle emitted within the Universe can exceed this distance. Our present knowledge of $H_0$ comes from the Hubble Space Telescope (HST) Key Project [@free]. The goal of this project was to determine $H_0$ by a Cepheid calibration of a number of independent, secondary distance indicators, including Type Ia supernovae, the Tully-Fisher relation, the fundamental plane for elliptical galaxies, surface brightness fluctuations, and type II supernovae. Here we shall restrict the discussion to the best absolute determinations of $H_0$, which are those from supernovæ of type Ia. Visible bright supernova explosions are very brief events (one month) and very rare, historical records show that in our Galaxy they have occurred only every 300 years. The most recent one occurred in 1987 (code name SN1987A), not exactly in our Galaxy but in the nearby Large Magellanic Cloud (LMC). Since it now has become possible to observe supernovæ in very distant galaxies, one does not have to wait 300 years for the next one. The physical reason for this type of explosion (type SNII supernova) is the accumulation of Fe–group elements at the core of a massive red giant star of size 8–200 $M_{\odot}$ which already has burned its hydrogen, helium and other light elements. Another type of explosion (type SNIa supernova) occurs when a degenerate dwarf star of CNO composition enters a stage of rapid nuclear burning to Fe–group elements. The SNIa is the brightest and most homogeneous class of supernovæ with hydrogen-poor spectra, their peak brightness can serve as remarkably precise standard candles visible from very far. Additional information is provided by the colour, the spectrum, and an empirical correlation observed between the time scale of the sharply rising light curve and the peak luminosity, which is followed by a gradual decline. Although supernovæ are difficult to find, they can be used to determine $H_0$ out to great distances, 500 Mpc or $z\approx 0.1$, and the internal precision of the method is very high. At greater distances one can still find supernovæ, but Hubble’s linear law (14) is then no longer valid, the velocity starts to accelerate. Supernovæ of type II are fainter, and show a wider variation in luminosity. Thus they are not standard candles, but the time evolution of their expanding atmospheres provides an indirect distance indicator, useful out to some 200 Mpc. Two further methods to determine $H_0$ make use of correlations between different galaxy properties. Spiral galaxies rotate, and there the Tully-Fisher relation correlates total luminosity with maximum rotation velocity. This is currently the most commonly applied distance indicator, useful for measuring extragalactic distances out to about 150 Mpc. Elliptical galaxies do not rotate, they are found to occupy a “fundamental plane” in which an effective radius is tightly correlated with the surface brightness inside that radius and with the central velocity dispersion of the stars. In principle this method could be applied out to $z\approx 1$, but in practice stellar evolution effects and the non-linearity of Hubble’s law limit the method to $z\lesssim 0.1$, or about 400 Mpc. The resolution of individual stars within galaxies clearly depends on the distance to the galaxy. This method, called surface brightness fluctuations (SBF), is an indicator of relative distances to elliptical galaxies and some types of spirals. The internal precision of the method is very high, but it can be applied only out to about 70 Mpc. Observations from the HST combining all this methods [@free] and independent SNIa observations from observatories on the ground [@gibs] agree on a value $$H_0=73\pm 2\pm 7\ \hbox{km\ s}^{-1}\ \hbox{Mpc}^{-1}.\eqno(15)$$ Note that the second error in Eq. (15) which is systematical, is much bigger than the statistical error. This illustrates that there are many unknown effects which complicate the determination of $H_0$, and which in the past have made all determinations controversial. To give just one example, if there is dust on the sight line to a supernova, its light would be reddened and one would conclude that the recession velocity is higher than it in reality is. There are other methods such as weak lensing which do not suffer from this systematic error, but they have not yet reached a precision superior to that in Eq. (15).\ THE BARYONIC DENSITY ==================== The ratio of baryons to photons or the baryon abundance is defined as $$\eta\equiv{N_b\over N_{\gamma}}\simeq 2.75\times 10^{-8}\ \Ombh\,\eqno(16)$$ where $N_b$ is the number density of baryons and $N_{\gamma}=4.11 \times 10^8\ \hbox{m}^{-3}$ is the number density of photons. Thus the primordial abundances of baryonic matter in the standard Big Bang nucleosynthesis scenario (BBN) is proportional to $\Ombh$. Its value is obtained in direct measurements of the abundances of the light elements $^4$He, $^3$He, $^2$H or D, $^7$Li and indirectly from CMBR observations and galaxy cluster observations. If the observed abundances are indeed of cosmological origin, they must not significantly be affected by later stellar processes. The helium isotopes $^3$He and $^4$He cannot be destroyed easily but they are continuously produced in stellar interiors. Some recent helium is blown off from supernova progenitors, but that fraction can be corrected for by observing the total abundance in hydrogen clouds of different age, and extrapolating it to time zero. The remainder is then primordial helium emanating from BBN. On the other hand, the deuterium abundance can only decrease, it is easily burned to $^3$He in later stellar events. The case of $^7$Li is complicated because some fraction is due to later galactic cosmic ray spallation products. Among the light elements the $^4$He abundance is easiest to observe, but also least sensitive to $\Ombh$, its dependence is logarithmic, so that only very precise measurements are relevant. The best “laboratories” for measuring the $^4$He abundance are a class of low-luminosity dwarf galaxies called Blue Compact Dwarf (BCD) galaxies, which undergo an intense burst of star formation in a very compact region. The BCDs are among the most metal-deficient gas-rich galaxies known. Since their gas has not been processed during many generations of stars, it should approximate well the pristine primordial gas. Over the years the observations have yielded many conflicting results, but the data are now progressing towards a common value [@luri], in particular by the work of Yu. I. Izotov and his group. The analysis in their most recent paper [@izot], based on the two most metal-deficient BCDs known, gives the result $$\Omega_b(^4\hbox{He})\thinspace h^2= 0.017\pm 0.005\ \ \ \ (2\sigma\ \hbox{CL})\ ,\eqno(17)$$ where the error is statistical only. Usually one quotes the ratio $Y_p$ of mass in $^4$He to total mass in $^1$H and $^4$He, which in this case is 0.2452 with a systematic error in the positive direction estimated to be 2-4%. Because of the logarithmic dependence, this error translated to $\Ombh$ could be considerable, of the order of 100% . The $^3$He isotope can be seen in the Milky Way interstellar medium and its abundance is a strong constraint on $\Ombh$. The $^3$He abundance has been determined from 14 years of data by Balser et al. [@bals]. More interestingly, Bania et al. [@bani] combined Milky Way data with the helium abundance in stars [@char] to find $$\Omega_b(^3\hbox{He})\thinspace h^2 = 0.020^{+0.007}_{-0.003}\ \ \ (1\sigma\ \hbox{CL})\ .\eqno(18)$$ There are actually three different errors in their analysis, and their quadratic sum gives the total error. The first error is from the observed emission-line that includes the errors in the Gaussian fits to the observed line parameters. The second error is from the standard deviation of the observed continuum data and the third error is the percent uncertainty of all models that have been used in the analyses of reference [@bals]. For a constraint on $\Ombh$ from $^7$Li, Coc et al. [@coca] update the previous work of several groups. More importantly, they include NACRE data [@nacre] in their compilation, and the uncertainties are analysed in detail. There is some lack of information about the neutron-induced reaction in the NACRE compilation, but the main source of uncertainty for the lighter neutron-induced reaction (e.g. ${\rm ^1H(n,\gamma)^2H}$ and ${\rm ^3He(n,p)^3H}$) is the neutron lifetime (for the present value see the Review of Particle Physics [@grom]). However, there is no new information about the heavier neutron-induced reaction (e.g. ${\rm ^7Li}$) or for ${\rm ^3He(d,p)^3He}$, but in this compilation the Gaussian errors have been opted from the polynomial fit of Nollett & Burles [@noll]. We quote Coc et al. [@coca] for $$\Omega_b(^7\hbox{Li})\thinspace h^2= 0.015\pm 0.003\ \ \ \ (1\sigma\ \hbox{CL})\ .\eqno(19)$$ The strongest constraint on the baryonic density comes from the primordial deuterium abundance. Deuterium is observed as a Lyman-$\alpha$ feature in the absorption spectra of high-redshift quasars. A recent analysis [@burl] gives $$\Omega_b(^2\hbox{H})\thinspace h^2= 0.020\pm 0.001\ \ \ \ (1\sigma\ \hbox{CL})\ ,\eqno(20)$$ more precisely than any other determination. Some systematic uncertainties remain in the calculations arising from the reaction cross sections. Very recently Chiappini et al. [@chia] have redefined the production and destruction of ${\rm ^3He}$ in low and intermediate mass stars. They also propose a new model for the time evolution of deuterium in the Galaxy. Taken together, they conclude that $\Ombh \gtrsim 0.017$, in good agreement with the values in Eqs. (18) and (20). Let us now turn to the information from the cosmic microwave background radiation and from large scale structures. There are many analyses of joint CMBR data, in particular three large compilations. Percival et al. [@perc] combine the data from COBE-DMR [@smot] MAXIMA [@leea], BOOMERANG [@nett], DASI [@halv], VSA [@scot] and CBI [@pe-ma] with the 2dFGRS LSS data [@coll]. Wang et al.[@wang] combine the same $\hbox{CMBR}$ data (except VSA) with 20 earlier CMBR power spectra, take their LSS power spectra from the IRAS PSCz survey [@saun], and include constraints from Lyman $\alpha$ forest spectra [@crof] and from the Hubble parameter [@free] quoted in Eq. (15). Sievers et al. [@siev] also use the same CMBR data as Percival et al.[@perc] (except VSA), combine them with earlier LSS data, and use the HST Hubble parameter [@free] quoted in Eq. (15) and the supernova data referred to in Section 5 as supplementary constraints. All these analyses are maximum likelihood fits based on frequentist statistics, so the use of the Bayesian term “prior” for constraint is a misnomer. Assuming that the initial seed fluctuations were adiabatic, Gaussian, and well described by power law spectra, the values of a large number of parameters are obtained by fitting the observed power spectrum. Here we shall only discuss results on $\Ombh$ which is essentially measured by the relative magnitudes of the first and second acoustic peaks in the CMBR power spectrum, returning to this subject in more detail in Section 6. The data used in the three compilations are overlapping but not identical, and the central values show a spread over $\pm 0.0003$. This we treat as a systematic error to the straight unweighted average of the central values. Two compilations [@perc], [@wang] consider models with and without a tensor component. Since the fits are equally good in both cases we take their difference, $\pm 0.0008$, to constitute another systematic error. We shall use this averaging prescription also in Section 6 to obtain values of other parameters. All the analyses can then be summarized by the value $$\Omega_b(\hbox{CMBR})\thinspace h^2= 0.022\pm 0.002 \pm 0.001\ \ \ \ (1\sigma\ \hbox{CL})\ ,\eqno(21)$$ where the statistical error corresponds to references [@perc], [@wang]. ---------------------------------------------------------------------------------------------------------------------------------------------- [*Method*]{} $\eta$ $\Ombh$ [*Error*]{} [*References*]{} ------------------------ ------------------------------------- ----------------------------------- ------------------------ ------------------ ${\rm ^4He}$ abundance ${\rm 4.7\ ^{+1.0}_{-0.8} \times ${\rm 0.017\pm 0.005}$ $2\sigma$ stat. only [@izot] 10^{-10}}$ ${\rm ^3He}$ abundance ${\rm 5.4\ ^{+2.2}_{-1.2} \times ${\rm 0.020\ ^{+0.007}_{-0.003}}$ $1\sigma$ stat. only [@bani] 10^{-10}}$ ${\rm ^7Li}$ abundance ${\rm 5.0 \times 10^{-10}}$ ${\rm 0.015 \pm 0.003}$ $1\sigma$ stat. only [@coca] $^2$H abundance ${\rm 5.6 \pm 0.5 \times 10^{-10}}$ ${\rm 0.020 \pm 0.001}$ $1\sigma$ stat.+syst. [@burl] CMBR + 2dFGRS —— ${\rm 0.022 \pm 0.002 \pm 0.001}$ $1\sigma$ stat.+ syst. [@perc][@wang] ---------------------------------------------------------------------------------------------------------------------------------------------- : [The baryonic density parameter]{}[]{data-label="table1"} In Table 1 we summarize the results from Eqs. (17-21). From this table one can conclude that all determinations are consistent with the most precise one from deuterium [@burl]. A weighted mean using the quoted errors yields $0.0194\pm 0.0008$ which is dominated by deuterium. However, all light element abundance determinations generally suffer from the potential for systematic errors. As to CMBR, the statistical errors quoted in all compilations have been obtained by marginalizing, so they are certainly unrealistically small. We take a conservative approach and add a systematic error of $\pm 0.002$ linearly to each of the five data values before averaging. The weighted mean is then $$\Ombh = 0.019\pm 0.002\ ,\eqno(22)$$ in excellent agreement with all the uncorrected input values in Table 1. One further source of $\Omega_b$ information is galaxy clusters which are composed of baryonic and non-baryonic matter. The baryonic matter takes the forms of hot gas emitting X-rays, stellar mass observed in visual light, and perhaps invisible baryonic dark matter of unknown composition. Let us denote the respective fractions $f_{gas},\ f_{gal} $, and $ f_{bdm} $. Then $$f_{gas} + f_{gal} + f_{bdm} = \Upsilon{\Omega_b\over\Omega_m}\ ,\eqno(21)$$ where $\Upsilon$ describes the possible local enhancement or diminution of baryon matter density in a cluster compared to the universal baryon density. This relation could in principle be used to determine $\Omega_b$ when one knows $\Omega_m$ (or vice versa), since $f_{gas}$ and $f_{gal} $ can be measured, albeit with large scatter, while $ f_{bdm}$ can be assumed negligible. Cluster formation simulations give information on $\Upsilon$ [@eke],[@fre] to a precision of about 10%. However, the precision obtained for $\Ombh$ by adding several 10% errors in quadrature does not make this method competitive.\ SUPERNOVA Ia CONSTRAINTS ======================== In Section 3 we already mentioned briefly the physics of supernovæ. The SN Ia observations by the High-z Supernova Search Team (HSST) [@ries] and the Supernova Cosmology Project (SCP) [@perl] are well enough known not to require a detailed presentation here. The importance of these observations lies in that they determine approximately the linear combination $\Oml - \Omm$ which is orthogonal to $\Om0 = \Omm + \Oml$, see Figure 1. HSST use two quite distinct methods of light-curve fitting to determine the distance moduli of their 16 SNe Ia studied. Their luminosity distances are used to place constraints on six cosmological parameters: $h, \Omm, \Oml, q_0,$ and the dynamical age of the Universe, $t_0$. The MLCS method involves statistical methods at a more refined level than the empirical template model. The distance moduli are found from a $\chi^2$ analysis using an empirical model containing four free parameters. The MLCS method and the template method give moduli which differ by about $1\sigma$. Once the distance moduli are known, the parameters h, $\Omm,\ \Oml$ are determined by a maximum likelihood fit, and finally the Hubble parameter is integrated out. (The results are really independent of h.) One may perhaps be somewhat concerned about the assumption that each modulus is normally distributed. We have no reason to doubt that, but if the iterative $\chi^2$ analysis has yielded systematically skewed pdf’s, then the maximum likelihood fit will amplify the skewness. The authors state that “the dominant source of statistical uncertainty is the extinction measurement”. The main doubt raised about the SN Ia observations is the risk that (part of) the reddening of the SNe Ia could be caused by intervening dust rather than by the cosmological expansion, as we already noted after Eq. (15). Among the possible systematic errors investigated is also that associated with extinction. No systematic error is found to be important here, but for such a small sample of SNe Ia one can expect that the selection bias might be the largest problem. The authors do not express any view about which method should be considered more reliable, thus noting that “we must consider the difference between the cosmological constraints reached from the two fitting methods to be a systematic uncertainty”. We shall come back to this question later. Here we would like to point out that if one corrects for the unphysical region $\Omm < 0$ using the method of Feldman & Cousins [@feld], the best value and the confidence contours will be shifted slightly towards higher values of $\Om0$. This shift will be more important for the MLCS method than for the template method, because the former extends deeper into the unphysical $\Omm$ region. Let us now turn to SCP, which studied 42 SNe Ia. The MLCS method described above is basically repeated, but modified in many details for which we refer the reader to the source [@perl]. The distance moduli are again found from a $\chi^2$ analysis using an empirical model containing four free parameters, but this model is slightly different from the HSST treatment. The parameters $\Omm$ and $\Oml$ are then determined by a maximum likelihood fit to four parameters, of which the parameters $\mathcal{M}_B$ (an absolute magnitude) and $\alpha$ (the slope of the width-luminosity relation) are just ancillary variables which are integrated out (h does not enter at all). The likelihood contours in $(\Omega_m - \Omega_\lambda)$ plane of both supernovæ  projects (SCP and HSST) are shown in Figure-1. The authors then correct the resulting likelihood contours for the unphysical region $\Omm < 0$ using the method of Feldman & Cousins [@feld]. Since the number of SNe Ia is here so much larger than in HSST, the effects of selection and of possible systematic errors can be investigated more thoroughly. SCP quotes a total possible systematic uncertainty to $\Omm^{flat}$ and $\Oml^{flat}$ of 0.05. If we compare the observations along the line defining a flat Universe, SCP finds $\Oml - \Omm = 0.44 \pm 0.085 \pm 0.05$, whereas HSST finds $\Oml - \Omm = 0.36 \pm 0.10$ for the MLCS method and $\Oml - \Omm = 0.68 \pm 0.09$ for the template method. Treating this difference as a systematic error of size $\pm 0.16$ the combined SCP result is $0.52 \pm 0.10 \pm 0.16$. SCP and HSST then agree within their statistical errors – how well they agree cannot be established since they are not completely independent. We choose to quote a combined HSST and SCP value $$\Oml - \Omm = 0.5 \pm 0.1\ ,\eqno(22)$$ which excludes a flat de Sitter universe with $\Oml - \Omm = 1$ by $5\sigma$, and excludes a flat Einstein – de Sitter universe with $\Oml - \Omm = -1$ by $10\sigma$.\ CMBR AND LSS CONSTRAINTS ======================== The most important source of information on the cosmological parameters are the anisotropies observed in the CMBR temperature and polarization maps over the sky. The temperature angular power spectrum has been measured and analyzed since 1992 [@smot], whereas the polarization spectrum is very recent [@kova] and has not yet been analyzed to obtain values for the dynamical parameters. Given the temperature angular power spectrum, the polarization spectrum is predicted with essentially no free parameters. At the moment one can say that the temperature angular power spectrum supports the current model of the Universe as defined by the dynamical parameters obtained from the temperature angular power spectrum. Temperature fluctuations in the CMBR around a mean temperature in a direction $\alpha$ on the sky can be analyzed in terms of the autocorrelation function $C(\theta)$ which measures the average product of temperatures in two directions separated by an angle $\theta$, $$C(\theta)=\left\langle{\delta T\over T}(\alpha){\delta T\over T} (\alpha+\theta)\right\rangle\ .\eqno(23)$$ For small angles $(\theta)$ the temperature autocorrelation function can be expressed as a sum of Legendre polynomials $P_{\ell}(\theta)$ of order $\ell$, the wave number, with coefficients or powers $a_{\ell}^2$, $$C(\theta)={1\over 4\pi}\sum_{\ell=2}^{\infty}a_{\ell}^2(2\ell+1)P_{\ell}(\cos\theta)\ . \eqno(24)$$ All analyses start with the quadrupole mode $\ell=2$ because the $\ell=0$ monopole mode is just the mean temperature over the observed part of the sky, and the $\ell=1$ mode is the dipole anisotropy due to the motion of Earth relative to the CMBR. In the analysis the powers $a_{\ell}^2$ are adjusted to give a best fit of $C(\theta)$ to the observed temperature. The resulting distribution of $a_{\ell}^2$ values versus $\ell$ is the power spectrum of the fluctuations, see Figure 2. The higher the angular resolution, the more terms of high $\ell$ must be included. The exact form of the power spectrum is very dependent on assumptions about the matter content of the Universe. It can be parametrized by the vacuum density parameter $\Omega_k = 1 - \Om0$, the total density parameter $\Om0$ with its components $\Omm ,\ \Oml$, and the matter density parameter $\Omm$ withits components $\Omb ,\ \Omega_{CDM},\ \Omega_{\nu}$. Further parameters are the Hubble parameter [*h*]{}, the tilt of scalar fluctuations $n_s$, the CMBR quadrupole normalization for scalar fluctuations [*Q*]{}, the tilt of tensor fluctuations $n_t$, the CMB quadrupole normalization for tensor fluctuations [*r*]{}, and the optical depth parameter $\tau$. Among these parameters, really only about six have an influence on the fit. In Section 4 we already noted that the relative magnitudes of the first and second acoustic peaks are sensitive to $\Omb$. The position of the first acoustic peak in multipole $\ell$ - space is sensitive to $\Om0$, which makes the CMBR information complementary (and in $\Omm ,\ \Oml$ - space orthogonal) to the supernova information. A decrease in $\Om0$ corresponds to a decrease in curvature and a shift of the power spectrum towards high multipoles. An increase in $\Oml$ (in flat space) and a decrease in h (keeping $\Ombh$ fixed) both boost the peaks and change their location in $\ell$ - space. Let us now turn to the distribution of matter in the Universe which can, to some approximation, be described by the hydrodynamics of a viscous, non-static fluid. In such a medium there naturally appear random fluctuations around the mean density $\bar{\rho}(t)$, manifested by compressions in some regions and rarefactions in other regions. An ordinary fluid is dominated by the material pressure, but in the fluid of our Universe three effects are competing: radiation pressure, gravitational attraction and density dilution due to the Hubble flow. This makes the physics different from ordinary hydrodynamics, regions of overdensity are gravitationally amplified and may, if time permits, grow into large inhomogenities, depleting adjacent regions of underdensity. Two complementary techniques are available for theoretical modelling of galaxy formation and evolution: numerical simulations and semi-analytic modelling. The strategy in both cases is to calculate how density perturbations emerging from the Big Bang turn into visible galaxies. This requires following through a number of processes: the growth of dark matter halos by accretion and mergers, the dynamics of cooling gas, the transformation of cold gas into stars, the spectrophotometric evolution of the resulting stellar populations, the feedback from star formation and evolution on the properties of prestellar gas, and the build-up of large galaxies by mergers. As in the case of the CMBR, an arbitrary pattern of fluctuations can be mathematically described by an infinite sum of independent waves, each with its characteristic wavelength $\lambda$ or comoving wave number $k$ and its amplitude $\delta_k$. The sum can be formally expressed as a Fourier expansion for the density contrast at comoving spatial coordinate [**r**]{} and world time t, $$\delta(\hbox{\bf r},t)\propto \sum\delta_k(t)\hbox{e}^{i{\bf k}\cdot{\bf r}}\ ,\eqno(25)$$ where [**k**]{} is the wave vector. Analogously to Eq. (23) a density fluctuation can be expressed in terms of the dimensionless mass autocorrelation function $$\xi(r)=\langle\delta(\hbox{\bf r}_1)\delta(\hbox{\bf r+r}_1)\rangle\propto \sum\langle|\delta_k(t)|^2\rangle e^{i{\bf k}\cdot{\bf r}}\ .\eqno(26)$$ which measures the correlation between the density contrasts at two points [**r**]{} and [**r**]{}$_1$. The powers $|\delta_k|^2$ define the power spectrum of the rms mass fluctuations, $$P(k)=\langle|\delta_k(t)|^2\rangle\ .\eqno(27)$$ Thus the autocorrelation function $\xi(r)$ is the Fourier transform of the power spectrum. This is similar to the situation in the context of CMB anisotropies where the waves represented temperature fluctuations on the surface of the surrounding sky, and the powers $a_{\ell}^2$ were coefficients in the Legendre polynomial expansion Eq. (24). With the lack of more accurate knowledge of the power spectrum one assumes for simplicity that it is specified by a power law $$P(k)\propto k^{n_s}\ ,\eqno(28)$$ where $n_s$ is the spectral index of scalar fluctuations. Primordial gravitational fluctuations are expected to have an equal amplitude on all scales. Inflationary models also predict that the power spectrum of matter fluctuations is almost scale-invariant as the fluctuations cross the Hubble radius. This is the Harrison–Zel’dovich spectrum, for which $n_s=1$ ($n_s$ = 0 would correspond to white noise). Since fluctuations in the matter distribution has the same primordial cause as CMBR fluctuations, we can get some general information from CMBR. There, increasing $n_s$ will raise the angular spectrum at large values of $\ell$ with respect to low $\ell$. Support for $\ell\approx 1.0$ come from all the available analyses: combining the results of references [@perc], [@wang], [@siev] by the averaging prescription in Section 4, we find $$n_s = 1.02 \pm 0.06 \pm 0.05\ .\eqno(29)$$ Phenomenological models of density fluctuations can be specified by the amplitudes $\delta_k$ of the autocorrelation function $\xi(r)$. In particular, if the fluctuations are Gaussian, they are completely specified by the power spectrum $P(k)$. The models can then be compared to the real distribution of galaxies and galaxy clusters, and the phenomenological parameters determined. As we noted in Section 4, there are several joint compilations of CMBR power spectra and LSS power spectra of which we are interested in the three largest ones [@perc], [@wang], [@siev]. Combining their results for $\Omm$ by the averaging prescription in Section 4, we find $$\Omm = 0.29 \pm 0.05 \pm 0.04\ .\eqno(30)$$ If the Universe is spatially flat so that $\Om0=1$, this gives immediately the value $\Oml=0.71$ with slightly better precision than above. To check this assumption we can quote reference [@siev] from their Table 5 where they use all data, $$\Om0 = 1.00 \pm\ ^{0.03}_{0.02}\ .\eqno(31)$$ Note, however, that this result has been obtained by marginalizing over all other parameters, thus its small statistical errors are conditional on $n_s,\ \Omm, \ \Omb$ being anything, and we have no prescription for estimating a systematic error. A value for $\Oml$ can be found by adding $\Oml-\Omm$ in Eq. (22) to $\Omm$, thus $\Oml = 0.79 \pm 0.12$. A better route appears to be to combine Eqs. (30) and (31) to give $$\Oml = 0.71 \pm 0.07\ .\eqno(32)$$ Still a third route is to add $\Om0$ and $\Oml-\Omm$, or to subtract them, respectively. Then one obtains $$\Omm = 0.25 \pm 0.05\ ,\ \ \ \ \Oml = 0.75 \pm 0.05\ .$$ The routes making use of $\Oml-\Omm$ from Eq. (22) are, however, making multiple use of the supernova information, so we discard them. Before ending this Section, we can quote values also for $w_\lambda$ and $q_0$. The notation here implies that $w_\lambda$ is taken as the equation of state of a quintessence component, so that its value could be $w_\lambda > -1$. The equation of state of a cosmomological constant component is of course $w_\lambda = -1$. In a flat universe $w_\lambda$ is completely correlated to $\Oml$ and therefore also to $\Omm$. We choose to quote the analysis by Bean and Melchiorri [@bean] who combine CMBR power spectra from COBE-DMR [@smot], MAXIMA [@leea], BOOMERANG [@nett], DASI [@halv], the supernova data from HSST [@ries] and SCP [@perl], the HST Hubble constant [@free] quoted in Eq. (15), the baryonic density parameter $\Omb\ h^2 = 0.020 \pm 0.005$ and some LSS information from local cluster abundances. They then obtain likelihood contours in the $w_\lambda\ ,\Omm$ space from which they quote the $1\sigma$ bound $w_\lambda < -0.85$. If we permit ourselves to restrict their confidence range further by using our value $\Omm = 0.29 \pm 0.06$ from Eq. (30), the result is changed only slightly to $$w_\lambda < -0.86\ ,\ \ \ \ (1\sigma \ \hbox{CL}) \eqno(33).$$ Finally, the deceleration parameter is not an independent quantity, it can be calculated from $$q_0 = \hbox{${{1}\over {2}}$}\Omm - \Oml = \hbox{${{3}\over {2}}$}\Omm - \Om0 = -0.56 \pm 0.04\ .\eqno(34)$$ The error is so small because the $\Omm$ and the $\Oml$ errors are completely anticorrelated. Note that the negative value implies that the expansion of the Universe is accelerating. [*Parameters*]{} Values [*References*]{} -------------------- ------------------------------- ------------------ $H_0$ ${\rm 73 \pm 7}$ [@free] $\Ombh$ ${\rm 0.019 \pm 0.002}$ our compilation $\Omega_m$ ${\rm 0.29 \pm 0.06}$ our compilation $\Omega_{\lambda}$ ${\rm 0.71 \pm 0.07}$ our compilation $\Omega_0$ ${\rm 1.0\ ^{+0.03}_{-0.02}}$ [@siev] $n_s$ ${\rm 1.02 \pm 0.08}$ our compilation $w_{\lambda}$ [$<$ - 0.86]{} our compilation $q_0$ ${\rm -\ 0.56 \pm 0.04}$ our compilation : [Best values of the dynamical parameters. The errors include $1\sigma$ statistical errors and our estimates of systematic errors, except for $\Omega_0$ which is statistical only. The Hubble constant $H_0$ is given in units of ${\rm km \ s^{-1}Mpc^{-1}}$ ]{}[]{data-label="table2"} SUMMARY ======= Information on the dynamical parameters of the Universe are coming from the Big Bang nucleosynthesis, from the fluctuations in the temperature and polarization of the cosmic microwave background radiation, from the large scale structures of galaxies, from supernova observations and from many other cosmological effects that may not yet be of interesting precision. The results of different analyses are now converging towards agreement when in the past disagreements of the order of 100% have been known. In this review we have taken the attitude that remaining disagreements reflect systematic errors coming either from the observations or from differences in the methods of analysis. We have then compiled the most precise parameter values, combined them and added our estimates of such systematic errors. This we have done for the baryonic density parameter $\Ombh$, the density parameter of the matter component $\Omm$, the density parameter of the cosmological constant $\Oml$, the spectral index of scalar fluctuations $n_s$, the equation of state of the cosmological constant $w_{\lambda}$, and the deceleration parameter $q_0$. In addition we quote the best values of the Hubble parameter $H_0$ and the total density parameter $\Om0$ from other sources. In Table 2 we summarize our results. The conclusion is not new: that the Universe is spatially flat, that some 25% of gravitating matter is dark and unknown, and that some 70% of the total energy content is dark, possibly in the form of a cosmological constant.\ [***Acknowledgements:***]{} S. M. H. is indebted to the Magnus Ehrnrooth Foundation for support. [99]{} G. F. Smoot et al., 1992, ApJ 396, L1 P. de Bernardis et al., 2000, Nature 404, 955 A. Balbi et al., 2000, ApJ 545, L1; S. Hanany et al., 2000, ApJ 545, L5 T. J. Pearson et al., 2002, astro-ph/0205388; B. S. Mason et al., 2002, astro-ph/0205384 P. E. Scott et al., 2002, astro-ph/0205380 N. W. Halverson et al., 2002, ApJ 568, 38; C. Pryke et al., 2002, ApJ 568, 46 M. Bersanelli, D. Maino & A. Mennella, 2002, astro-ph/0209215, Submitted to “La Rivista del Nuovo Cimento”. J. Kovac et al., 2002, astro-ph/0209478 W. L. Freedman et al., 2001, ApJ 553, 47 A. G. Riess et al., 1998, AJ 116, 1009 S. Perlmutter et al., 1998, Nature 391, 51; 1999, ApJ 517, 565 L. N. da Costa et al., 1994, ApJ 424, L1 S. A. Shectman et al., 1996, ApJ 470, 172 J. Retzlaff et al., 1998, New Astron. A3, 631 W. Saunders et al., 2000, MNRAS 317, 55 J. A. Peacock et al., 2001, Nature 410, 169 M. Colless et al., 2001, MNRAS 328, 1039 A. E. Evrard, 1997, MNRAS 292, 289 N. A. Bahcall & X. Fan, 1998, ApJ 504, 1; V. R. Eke et al., 1998, MNRAS 298, 1145 D. H. Weinberg et al., 1999, ApJ 522, 563 M. Chiba & Y. Yoshii, 1999, ApJ 510, 42; P. Helbig, 2000, Proceedings IAU 201, astro-ph/0011031; M. Im, R. E. Griffiths & K. U. Ratnatunga, 1997, ApJ 475, 457 M. Birkinshaw 1999, Phys. Rept. 310, 97; J. E. Carlstrom, G. P. Holder and E. D. Reese, 2002, astro-ph/0208192 E. J. Guerra, R. A. Daly & L. Wan, 2000, ApJ 544, 659 I. Zehavi & A. Dekel, 1999, Nature 401, 252 T. Totani, 1997, IAU proceedings, Kyoto, Japan, astro-ph/9710312 B. K. Gibson & C. B. Brook, 2000, astro-ph/0011567; 2001, ASP Conference Services, 666 V. Luridiana, 2002, astro-ph/0209177; to appear in Proc. XXXVIIth Moriond Astrophys. Meeting “The Cosmological Model”, Les Arcs, France, March 16-23, 2002 T. X. Thuan & Y. I. Izotov, 2001, astro-ph/0112348; to appear in “Matter in the Universe”, ed P. Jetzer, K. Pretzl and R. von Steiger, Kluwer, Dordrecht (2002) D. S. Balser et al., 1999, ApJ 522, L73; 1999, ApJ 510, 759 T. M. Bania, R. T. Rood & D. S. Balser, 2002, Nature 415, 54 C. Charbonnel, 1995, ApJ 453, L41; C. Charbonnel et al., 1998, A & A 332, 204; 1998, A & A 336, 915; 1998, Space Sci. Rev. 84, 199 A. Coc et al., 2002, PR D65, 043510 C. Angulo et al., 1999, Nucl. Phys. A656, 3 D. E. Groom et al., 2000, Eur. Phys. J. C15, 1 K. M. Nollett & S. Burles, 2000, PR D61, 123505; 2001, ApJ 552, L1 S. Burles, K. M. Nollett & M. S. Turner, 2001, ApJ 552, L1 C. Chiappini, A. Renda & F. Matteucci, 2002, astro-ph/0209240 and A & A (accepted) W. J. Percival et al., 2002, astro-ph/0206256; 2001, MNRAS 327, 1297 A. T. Lee et al., 2001, ApJ 561, L1 C. B. Netterfield et al., 2002, ApJ 571, 604 X. Wang, M. Tegmark & M. Zaldarriaga, 2002, PR D65, 123001 A. C. Croft et al., 2002, ApJ (December 10 issue), astro-ph/0012324 J. L. Sievers et al., 2002, astro-ph/0205387 V. R. Eke, J. F. Navarro, & C. S. Frenk, 1998, ApJ, 503, 569 C. S. Frenk et al., 1999, ApJ 525, 554 G. J. Feldman & R. D. Cousins, 1998, PR D57, 3873 R. Bean & A. Melchiorri, 2002, PR D65, 041302

--- abstract: 'We present preliminary results of follow-up optical observations, both photometric and spectroscopic, of stellar X-ray sources, selected from the cross-correlation of ROSAT All-Sky Survey (RASS) and TYCHO catalogues. Spectra were acquired with the E[lodie]{} spectrograph at the 193-cm telescope of the Haute Provence Observatory (OHP) and with the REOSC echelle spectrograph at the 91-cm telescope of the Catania Astrophysical Observatory (OAC), while $UBV$ photometry was made at OAC with the same telescope. In this work, we report on the discovery of six late-type binaries, for which we have obtained good radial velocity curves and solved for their orbits. Thanks to the OHP and OAC spectra, we have also made a spectral classification of single-lined binaries and we could give first estimates of the spectral types of the double-lined binaries. Filled-in or pure emission H$\alpha$ profiles, indicative of moderate or high level of chromospheric activity, have been observed. We have also detected, in near all the systems, a photometric modulation ascribable to photospheric surface inhomogeneities which is correlated with the orbital period, suggesting a synchronization between rotational and orbital periods. For some systems has been also detected a variation of H$\alpha$ line intensity, with a possible phase-dependent behavior.' author: - 'A. Frasca, P. Guillout, E. Marilli, R. Freire Ferrero, K. Biazzo' title: 'Newly discovered active binaries in the RasTyc sample of stellar X-ray sources' --- Introduction {#sec:Intro} ============ The cross-correlation between the ROSAT All-Sky Survey ($\simeq$ 150000 sources) and the TYCHO mission ($\simeq$ 1000000 stars) catalogues has selected about 14000 stellar X-ray sources (RasTyc sample, [@Guillout99]). Although most of these soft X-ray sources are expected to be the youngest stars in the solar neighborhood, neither the contamination by older RS CVn systems nor the fraction of BY Dra binaries are actually known. This information is, however, of fundamental importance for studying the recent local star formation history and, for instance, for putting constrains on the scale height of the spatial distribution of nearby young stars around the galactic plane. We thus started a spectroscopic observation campaign aimed at a deep characterisation of a representative sub-sample of the RasTyc population. In addition to derive chromospheric activity levels (from H$\alpha$ emission) and rotational velocities (from Doppler broadening), high resolution spectroscopic observations allow to infer ages (by means of Lithium abundance) and to single out spectroscopic and active binaries. In this work we present some preliminary results of follow-up observations, both photometric and spectroscopic, of some RasTyc stars performed with the 193-cm telescope of OHP and the 91-cm telescope of the Catania Astrophysical Observatory (OAC). In particular, we analyse six new late-type binaries, for which we have obtained good radial velocity curves and orbital solutions. An accurate spectral classification for the single-lined binaries has been also performed and the projected rotational velocity $v\sin i$ has been measured for all stars. The chromospheric activity level and the lithium content have been also investigated using as diagnostics the H$\alpha$ emission and the Li[i]{}$\lambda\,6708$ line, respectively. [llcccccccr]{}\ RasTyc & Name & P$_{\rm orb}$ & $\gamma$ & $k$ (P/S) & $M\sin^3i$ & $v\sin i$ (P/S) & Sp. Type & $B-V$ & W$_{\rm LiI}$\ & & (days) & (kms$^{-1}$) & (kms$^{-1}$) & $M_{\odot}$ & (kms$^{-1}$) & & & (mÅ)\ \ 193137 & HD 183957 & 7.954 & $-$29.0 & 57.5/63.1 & 0.758/0.691 & 4.0/4.4 & K0-1V/K1-2V & 0.84 & $< 10$\ 215940 & OT Peg & 1.748 & $-$27.0 & 16.6/23.2 & 0.007/0.005 & 9.2/9.4 & K0V/K3-5V & 0.79 & 50\ 221428 & BD+334462 & 10.12 & $-$20.9 & 59.2/60.4 & 0.905/0.887 & 16.1/32.6 & G2 + K & 0.70 & 15:\ 040542 & DF Cam & 12.60 & $-$19.5 & 22.8 & SB1 & 35 & K2III & 1.14 & —\ 072133 & V340 Gem & 36.20 & +37.0 & 42.1 & SB1 & 40 & G8III & 0.83 & 70\ 102623 & BD+382140 & 15.47 & +47.4 & 31.3 & SB1 & 11.5 & K1IV & 1.03 & 40\ \ \ Observations and reduction {#sec:Obs} ========================== Spectroscopy ------------ Spectroscopic observations have been obtained at the [*Observatoire de Haute Provence*]{} (OHP) and at the [*M.G. Fracastoro*]{} station (Mt. Etna, 1750 m a.s.l.) of Catania Astrophysical Observatory (OAC). At OHP we observed in 2000 and 2001 with the E[lodie]{} echelle spectrograph connected to the 193-cm telescope. The 67 orders recorded by the CCD detector cover the 3906-6818 Å  wavelength range with a resolving power of about 42000 ([@Bar96]). The E[lodie]{} spectra were automatically reduced on-line during the observations and the cross-correlation with a reference mask was produced as well. The observations carried out at Catania Observatory have been performed in 2001 and 2002 with the REOSC echelle spectrograph at the 91-cm telescope. The spectrograph is fed by the telescope through an optical fiber (UV - NIR, $200\,\mu m$ core diameter) and is placed in a stable position in the room below the dome level. Spectra were recorded on a CCD camera equipped with a thinned back-illuminated SITe CCD of 1024$\times$1024 pixels (size 24$\times$24 $\mu$m). The échelle crossed configuration yields a resolution of about 14000, as deduced from the FWHM of the lines of the Th-Ar calibration lamp. The observations have been made in the red region. The detector allows us to record five orders in each frame, spanning from about 5860 to 6700 Å. The OAC data reduction was performed by using the [echelle]{} task of IRAF[^1] package following the standard steps: background subtraction, division by a flat field spectrum given by a halogen lamp, wavelength calibration using the emission lines of a Th-Ar lamp, and normalization to the continuum through a polynomial fit. Photometry ---------- The photometric observations have been carried out in 2001 and 2002 in the standard $UBV$ system also with the 91-cm telescope of OAC and a photon-counting refrigerated photometer equipped with an EMI 9789QA photomultiplier, cooled to $-15\degr$C. The dark noise of the detector, operated at this temperature, is about $1$ photon/sec. For each field of the RasTyc sources, we have chosen two or three stars with known $UVB$ magnitudes to be used as local standards for the determination of the photometric instrumental “zero points". Additionally, several standard stars, selected from the list of Landolt ([@Lan92]), were also observed during the run in order to determine the transformation coefficients to the Johnson standard system. A typical observation consisted of several integration cycles (from 1 to 3, depending on the star brightness) of 10, 5, 5 seconds, in the $U$, $B$ and $V$ filter, respectively. A 21$\arcsec$ diaphragm was used. The data were reduced by means of the photometric data reduction package PHOT designed for photoelectric photometry of Catania Observatory ([@LoPr93]). Seasonal mean extinction coefficient for Serra La Nave Observatory were adopted for the atmospheric extinction correction. Results ======= Radial velocity and photometry {#sec:RV} ------------------------------ The radial velocity (RV) measurements for the E[lodie]{} data have been performed onto the cross-correlation functions (CCFs) produced on-line during the data acquisition. Radial velocities for OAC spectra were obtained by cross-correlation of each echelle spectral order of the RasTyc spectra with that of bright radial velocity standard stars. For this purpose the IRAF task [fxcor]{}, that computes RVs by means of the cross-correlation technique, was used. The wavelength ranges for the cross-correlation were selected to exclude the H$\alpha$ and Na[I]{} D$_2$ lines, which are contaminated by chromospheric emission and have very broad wings. The spectral regions heavily affected by telluric lines (e.g. the O$_2$ lines in the $\lambda~6276-\lambda~6315$ region) were also excluded. The observed RV curves are displayed in Fig. \[fig:RV\], where, for SB2 systems, we used dots for the RVs of primary (more massive) components and open circles for those secondary (less massive) ones. We initially searched for eccentric orbits and found in any case very low eccentricity values (e.g. $e=0.010$ for HD 183957, $e=0.030$ for 221428). Thus, following the precepts of [-@Lucy71], we adopted $e=0$. The circular solutions are also represented in Fig. \[fig:RV\] with solid and dashed lines for the primary and secondary components, respectively. The orbital parameters of the systems, orbital period ($P_{\rm orb}$), barycentric velocity ($\gamma$), RV semi-amplitudes ($k$) and masses ($M\sin^3i$), are listed in Table \[tab:param\], where P and S refer to the primary and secondary components of the SB2 systems, respectively. With the only exception of HD 183957, for which any modulation is visible neither in OAC data nor in TYCHO $V_{\rm T}$ magnitudes, all sources show a photometric modulation well correlated with the orbital period, indicating a high degree of synchronization. The low amplitude of the light curve of 215940 and the very low values of $M\sin^3i$ imply a very low inclination of orbital/rotational axis. Spectral type and $v\sin i$ determination {#sec:Spty} ----------------------------------------- For SB1 systems observed with E[lodie]{} we have determined effective temperatures and gravity (i.e. spectral classification) by means of the TGMET code, available at OHP ([@Katz98]). We have also used ROTFIT, a code written by one of us ([@Frasca03]) in IDL (Interactive Data Language, RSI), which simultaneously find the spectral type and the $v\sin i$ of the target by searching, into a library of standard star spectra, for the standard spectrum which gives the best match of the target one, after the rotational broadening. As standard star library, we used a sub-sample of the stars of the TGMET list whose spectra were retrieved from the E[lodie]{} Archive ([@Prugniel01]). The ROTFIT code was also applied to the OAC spectra, using standard star spectra acquired with the same instrument. This was especially advantageous for DF Cam, for which we have no E[lodie]{} spectrum. For SB2 systems we made a preliminary classification on the basis of a visual inspection of E[lodie]{} and OAC spectra. However, we are developing a code for spectral type determination in double-lined binaries which will allow us to improve the spectral classification. We found at least two binaries composed by main sequence stars, while the remaining systems contain an evolved (giant or sub-giant) star. Measurements of $v\sin i$ were also made using the E[lodie]{} CCFs and the calibration relation between CCF width and $v\sin i$ proposed by [-@Queloz98]. The lower rotation rate ($v\sin i \simeq$4 kms$^{-1}$) has been detected for both components of HD 183957, which display also the lowest H$\alpha$ activity among the six sources. H$\alpha$ emission and Lithium content {#sec:Halpha} -------------------------------------- The H$\alpha$ line is an important indicator of chromospheric activity. Only the very active stars show always H$\alpha$ emission above the continuum, while in less active stars only a filled-in absorption line is observed. The detection of the chromospheric emission contribution filling in the line core is hampered in double-lined systems in which both spectra are simultaneously seen and shifted at different wavelengths, according to the orbital phase. Therefore a comparison with an “inactive” template built up with two stellar spectra that mimic the two components of the system in absence of activity is needed to emphasize the H$\alpha$ chromospheric emission. The inactive templates have been built up with rotationally broadened E[lodie]{} archive spectra (HD 10476, K1V for both components of HD 183957; $\gamma$ Cep, K1IV for 102623; $\delta$ Boo, G8III for 072133; HD 17382, K1V for 215940) or with OAC spectra of $\alpha$ Ari (K2III), for DF Cam, acquired during the observing campaigns. The two components of HD 183957 show only a small filling of their H$\alpha$ profiles (Fig. \[fig:Halpha2\]), while the other RasTyc stars display H$\alpha$ emission profiles with a variety of shapes, going from a simple symmetric emission profile (102623) to a double-peaked strong emission line (215940). It has been also observed a very broad, complex feature with a filled-in core and an emission blue wing (072133). A H$\alpha$ profile similar to that displayed by the latter star has been sometimes observed in some long-period RS CVn’s, like HK Lac (e.g. Catalano & Frasca 1994). RasTyc 072133 was classified as a semi-regular variable after Hipparcos, but it displays all the characteristics of a RS CVn SB1 binary. The E[lodie]{} spectra of 221428 in the H$\alpha$ region show that the secondary (less massive) component displays a H$\alpha$ line always in emission with a stronger intensity around phase 0$\fp$7. The OAC spectra of DF Cam always display a pure H$\alpha$ emission line, whose intensity varies with the orbital/rotational phase. Similarly to 072133, DF Cam, considered as a semi-regular variable after Hipparcos photometry, is very likely an active binary of the RS CVn or BY Dra class. The equivalent width of the lithium $\lambda$6708 line, $EW_{\rm Li}$, was measured on the E[lodie]{} spectra. For the three sources for which we were able to detect and measure $EW_{\rm Li}$, we deduced lithium abundance, $\log N(Li)$, in the range 1.3–1.8, according to [-@Pav96] NLTE calculations. We are grateful to the members of the staff of OHP and OAC observatories for their support and help with the observations. This research has made use of SIMBAD and VIZIER databases, operated at CDS, Strasbourg, France. Baranne A., Queloz D., Mayor M., et al., 1996, A&AS 119, 373 Catalano S. and Frasca A. 1994, A&A 287, 575 Frasca A., Alcalà J.M., Covino E., Catalano S., Marilli E. and Paladino R. 2003, A&A 405, 149 Guillout P., Schmitt J. H. M. M., Egret D., Voges W., Motch C. and Sterzik M. F. 1999, A&A 351, 1003 Katz D., Soubiran C., Cairel R., Adda M. and Cautain R. 1998, A&A 338, 151 Landolt, A. U. 1992, AJ, 104, 340 Lo Presti, C., & Marilli, E. 1993, PHOT. Photometrical Data Reduction Package. Internal report of Catania Astrophysical Observatory N. 2/1993 Lucy, L. B. and Sweeney, M. A., 1971, AJ 76, 544 Pavlenko Y.V. & Magazzù A. 1996, A&A 311, 961 Prugniel, P. and Soubiran, C. 2001, A&A 369, 1048 Queloz D., Allain S., Mermilliod J.-C., Bouvier J. and Mayor, M. 1998, A&A 335, 183 [^1]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.

--- address: | Theoretical Division, Los Alamos National Laboratory\ Los Alamos, New Mexico 87545, USA\ E-mail: nix@t2nix.lanl.gov [and]{} moller@moller.lanl.gov author: - and PETER MÖLLER title: 'MASSES AND DEFORMATIONS OF NEUTRON-RICH NUCLEI' --- =cmr8 1.5pt \#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{} \#1 \#2 \#3 \#4 \#1 \#2 \#3 \#4 psfig Introduction {#intro} ============ The accurate calculation of the ground-state mass and deformation of a nucleus far from stability, such as one of the neutron-rich nuclei considered in this conference, remains one of the most fundamental challenges of nuclear theory. Toward this goal, two major approaches—which also allow the simultaneous calculation of a wide variety of other nuclear properties—have been developed (along with numerous semi-empirical formulas for masses alone). At the most fundamental level, fully selfconsistent microscopic theories, starting with an underlying nucleon-nucleon interaction, have seen progress in both the nonrelativistic Hartree-Fock approximation and more recently the relativistic mean-field approximation. Although microscopic theories offer great promise for the future, their current accuracies are typically a few Mwhich is insufficient for most practical applications. At the next level of fundamentality, the macroscopic-microscopic method—where the smooth trends are obtained from a macroscopic model and the local fluctuations from a microscopic model—has been used in several recent global calculations that are useful for a broad range of applications. We will concentrate here on the 1992 version of the finite-range droplet model,$\,$[@MNMS; @MNK]with particular emphasis on its reliability for extrapolations to new regions of nuclei, but will also briefly discuss two other models of this type.$\,$[@APDT; @MS] Finite-Range Droplet Model {#frdms} ========================== In the finite-range droplet model, which takes its name from the macroscopic model that is used, the microscopic shell and pairing corrections are calculated from a realistic, diffuse-surface, folded-Yukawa single-particle potential by use of Strutinsky’s method.$\,$[@S]In 1992 we made a new adjustment of the constants of an improved version of this model to 28 fission-barrier heights and to 1654 nuclei with $N,Z \ge 8$ ranging from $^{16}$O to $^{263}$106 whose masses were known experimentally in 1989.$\,$[@A]The resulting microscopic enhancement to binding for even-even nuclei throughout the periodic system is shown in Fig. \[enhab\]. enhab fig1 201 [Calculated additional binding energy of even-even nuclei relative to the macroscopic energy of spherical nuclei, illustrating the crucial role of microscopic corrections.]{} This model has been used to calculate the ground-state mass, deformation, microscopic correction, odd-proton and odd-neutron spins and parities, proton and neutron pairing gaps, binding energy, one- and two-neutron separation energies, quantities related to $\beta$-delayed one- and two-neutron emission probabilities, $\beta$-decay energy release and half-life with respect to Gamow-Teller decay, one- and two-proton separation energies, and $\alpha$-decay energy release and half-life for 8979 nuclei with $N,Z \ge 8$ ranging from $^{16}$O to $^{339}136$ and extending from the proton drip line to the neutron drip line.$\,$[@MNMS; @MNK]These tabulated quantities are available electronically on the World Wide Web at the Uniform Resource Locator [http://t2.lanl.gov/publications/publications.html]{}. quad fig2 201 [Calculated quadrupole deformations of even-even nuclei, illustrating the transitions from spherical to deformed nuclei as one moves away from magic numbers.]{} Ground-State Deformations {#def} ========================= In our calculations, we specify a general nuclear shape in terms of deviations from a spheroidal shape by use of Nilsson’s $\epsilon$ parameterization.$\,$[@Ni]The ground-state shape is determined by initially minimizing the nuclear potential energy of deformation with respect to the two symmetric shape coordinates $\epsilon_2$ and $\epsilon_4$. During this minimization, we include a prescribed smooth dependence of the higher symmetric deformation $\epsilon_6$ on the two independent coordinates $\epsilon_2$ and $\epsilon_4$. This dependence is determined by minimizing the macroscopic potential energy of $^{240}$Pu with respect to $\epsilon_6$ for fixed values of $\epsilon_2$ and $\epsilon_4$. We then vary separately $\epsilon_6$ and the mass-asymmetric, or octupole, deformation $\epsilon_3$, with $\epsilon_2$ and $\epsilon_4$ held fixed at their previously determined values, to calculate any additional lowering in energy from these two degrees of freedom. For presentation purposes, it is sometimes more convenient to express the nuclear ground-state shape in terms of the $\beta$ parameterization, where the shape coordinates represent the coefficients in an expansion of the radius vector to the nuclear surface in a series of spherical harmonics. Figures \[quad\] and \[hex\] show our calculated quadrupole and hexadecapole deformations, respectively, in terms of $\beta_2$ and $\beta_4$, which are determined by transforming our calculated shapes from the $\epsilon$ parameterization. hex fig3 201 [Calculated hexadecapole deformations of even-even nuclei, illustrating the transitions from bulging to indented equatorial regions as one moves from smaller to larger magic numbers.]{} The inclusion of the $\epsilon_6$ and $\epsilon_3$ shape degrees of freedom is crucial for the isolation of such physical effects as the Coulomb redistribution energy, which arises from a central density depression.$\,$[@MNMS2]As illustrated in Fig. \[eps6\], an independent variation of the symmetric deformation $\epsilon_6$ is important for several regions of nuclei. For even-even nuclei, the maximum reduction in energy relative to that for a prescribed smooth $\epsilon_6$ dependence is 1.28 Mand occurs for $^{252}$Fm. As illustrated in Fig. \[eps3\], the mass-asymmetric deformation $\epsilon_3$ is important for nuclei in a few isolated regions. For even-even nuclei, the maximum reduction in energy relative to that for a symmetric shape is 1.29 Mand occurs for the neutron-rich nucleus $^{194}$Gd. For even-even nuclei close to the valley of $\beta$-stability, the maximum reduction in energy relative to that for a symmetric shape is 1.20 Mand occurs for $^{222}$Ra. eps6 fig4 198 [Calculated reduction in energy of even-even nuclei arising from an independent variation in $\epsilon_6$, relative to that for shapes with a prescribed smooth $\epsilon_6$ dependence. Note that the sign of the $\epsilon_6$ correction is reversed in this plot for clarity of display.]{} eps3 fig5 198 [Calculated reduction in energy of even-even nuclei arising from the inclusion of $\epsilon_3$ deformations, relative to that for symmetric shapes. Note that the sign of the $\epsilon_3$ correction is reversed in this plot for clarity of display.]{} Reliability for Extrapolations to New Regions of Nuclei {#extraps} ======================================================= For the original 1654 nuclei included in the adjustment, the theoretical error, determined by use of the maximum-likelihood method with no contributions from experimental errors,$\,$[@MNMS; @MNK]is 0.669 MAlthough some large systematic errors exist for light nuclei, they decrease significantly for heavier nuclei. Between 1989 and 1996, the masses of 371 additional nuclei heavier than $^{16}$O have been measured,$\,$[@AW]$^{\sen}\,$[@H]which provides an ideal opportunity to test the ability of mass models to extrapolate to new regions of nuclei whose masses were not included in the original adjustment. Figure \[frdm\] shows as a function of the number of neutrons from $\beta$-stability the individual deviations between these newly measured masses and those predicted by the 1992 finite-range droplet model. The new nuclei fall into three categories, with the first category corresponding to 273 nuclei lying on both sides of the valley of $\beta$-stability.$\,$[@AW]The second category corresponds to 91 proton-rich nuclei produced by fragmentation of $^{209}$Bi projectiles incident on a thick Be target in the experimental storage ring (ESR) at the Gesellschaft für Schwerionenforschung (GSI) in Darmstadt, Germany.$\,$[@K]The third category corresponds to seven proton-rich superheavy nuclei discovered in the separator for heavy-ion reaction products (SHIP) at GSI whose masses are estimated by adding the highest $\alpha$-decay energy release at each step in the decay chain to known masses.$\,$[@H]This procedure could seriously overestimate the experimental masses of some of the heavier nuclei because different energy releases have been observed in some cases.$\,$[@H]To account for this uncertainty, we have assigned a mass error of 0.5 Mfor each of these seven nuclei. Also, to account for errors of unknown origin, we have included an additional 0.076 Mcontribution$\,$[@N] to the mass errors for each of the 91 nuclei in the second category. The theoretical error of the 1992 finite-range droplet model \[FRDM (1992)\] for all of the 371 newly measured masses is 0.570 MThe reduction in error arises partly because most of the new nuclei are located in the heavy region, where the model is more accurate. Analogous deviations occur for version 1 of the 1992 extended-Thomas-Fermi Strutinsky-integral \[ETFSI-1 (1992)\] model of Aboussir, Pearson, Dutta, and Tondeur.$\,$[@APDT]In this model, the macroscopic energy is calculated for a Skyrme-like nucleon-nucleon interaction by use of an extended Thomas-Fermi approximation. The shell correction is calculated from single-particle levels corresponding to this same interaction by use of a Strutinsky-integral method, and the pairing correction is calculated for a $\delta$-function pairing interaction by use of the conventional BCS approximation. The constants of the model were determined by adjustments to the ground-state masses of 1492 nuclei with mass number $A \ge 36$, which excludes the troublesome region from $^{16}$O to mass number $A = 35$. The theoretical error corresponding to 1540 nuclei whose masses were known experimentally$\,$[@A] at the time of the original adjustment is 0.733 MThe theoretical error for 366 newly measured masses$\,$[@AW]$^{\sen}\,$[@H] for nuclei with $A \ge 36$ is 0.739 M Similar results hold for the 1994 Thomas-Fermi \[TF (1994)\] model of Myers and Swiatecki.$\,$[@MS]In this model, the macroscopic energy is calculated for a generalized Seyler-Blanchard nucleon-nucleon interaction by use of the original Thomas-Fermi approximation. For $N,Z \ge 30$ the shell and pairing corrections were taken from the 1992 finite-range droplet model, and for $N,Z \le 29$ a semi-empirical expression was used. The constants of the model were determined by adjustments to the ground-state masses of the same 1654 nuclei with $N,Z \ge 8$ ranging from $^{16}$O to $^{263}$106 whose masses were known experimentally in 1989 that were used in the 1992 finite-range droplet model. The theoretical error corresponding to these 1654 nuclei is 0.640 MThe reduced theoretical error relative to that in the 1992 finite-range droplet model arises primarily from the use of semi-empirical microscopic corrections in the extended troublesome region $N,Z \le 29$ rather than microscopic corrections calculated more fundamentally. The theoretical error for 371 newly measured masses$\,$[@AW]$^{\sen}\,$[@H] is 0.620 M frdm fig6 195 [Deviations between experimental and calculated masses for 371 new nuclei whose masses were not included in the 1992 adjustment of the finite-range droplet model.$\,$[@MNMS; @MNK]]{} As summarized in Table \[extrapt\], the theoretical error for the newly measured masses relative to that for the original masses to which the model constants were adjusted [*decreases*]{} by 15% for the FRDM (1992), increases by 1% for the ETFSI-1 (1992) model, and [*decreases*]{} by 3% for the TF (1994) model. These macroscopic-microscopic mass models can therefore be extrapolated to new regions of nuclei with differing amounts of confidence. [lcccccccc]{}\ & & & & & &\ \ Model & & ${N}_{\rm nuc}$ & Error & & ${N}_{\rm nuc}$ & Error & & Error\ & & & (M& & & (M& & ratio\ \ FRDM (1992) & & 1654 & 0.669 & & 371 & 0.570 & & 0.85\ ETFSI-1 (1992) & & 1540 & 0.733 & & 366 & 0.739 & & 1.01\ TF (1994) & & 1654 & 0.640 & & 371 & 0.620 & & 0.97\ Rock of Metastable Superheavy Nuclei {#rock} ==================================== rohic fig7 201 [Ten recently discovered superheavy nuclei,$\,$[@H+]$^{\sen}\,$[@O]superimposed on a theoretical calculation$\,$[@MNMS; @MNK] of the microscopic corrections to the ground-state masses of nuclei extending from the vicinity of lead to heavy and superheavy nuclei. The heaviest nucleus, whose location on the diagram is indicated by the flag, was produced through a gentle reaction between spherical $^{70}$Zn and $^{208}$Pb nuclei in which a single neutron was emitted.$\,$[@H+]]{} The heaviest nucleus known to man, $^{277}$112, was discovered$\,$[@H+] in February 1996 at the GSI by use of the gentle fusion reaction $^{70}$Zn + $^{208}$Pb $\rightarrow$ $^1$n + $^{277}$112. It is the latest in a series of about 10 recently discovered nuclei$\,$[@H+]$^{\sen}\,$[@O] lying on a rock of deformed metastable superheavy nuclei predicted to exist$\,$[@MNMS; @MNK; @MN]$^{\sen}\,$[@PS] near the deformed proton magic number at 110 and deformed neutron magic number at 162. These 10 superheavy nuclei are shown in Fig. \[rohic\] as tiny deformed three-dimensional objects. Most of the metastable superheavy nuclei that have been discovered live for only about a thousandth of a second, after which they generally decay by emitting a series of alpha particles. However, the decay products of the most recently discovered nucleus $^{277}$112 show for the first time that nuclei at the center of the predicted rock of stability live longer than 10 seconds. We have used the macroscopic-microscopic method recently to calculate the fusion barrier for several reactions leading to deformed superheavy nuclei.$\,$[@MNAHM]For the reaction $^{70}$Zn + $^{208}$Pb $\rightarrow$ $^1$n + $^{277}$112, the microscopic shell and pairing corrections associated primarily with the doubly magic $^{208}$Pb target nucleus lower the total potential energy at the touching configuration by about 12 Mrelative to the macroscopic energy. These shell and pairing corrections persist from the touching configuration inward to a position only slightly more deformed than the ground-state shape. The resulting maximum in the fusion barrier is about 2 Mlower than the center-of-mass energy that was used in the GSI experiment that produced $^{277}$112. One possibility to reach the island of spherical superheavy nuclei near $^{290}$110 that is predicted to lie beyond our present horizon involves the use of prolately deformed targets and projectiles that also possess large negative hexadecapole moments, which leads to large indented equatorial regions.$\,$[@IMNS] Summary and Conclusion {#sum} ====================== The FRDM (1992) and two other macroscopic-microscopic models have been used recently to calculate the ground-state masses and deformations of nuclei throughout our known chart and beyond, and the FRDM (1992) has also been used to simultaneously calculate a wide variety of other nuclear properties. These models are useful for extrapolating to new regions of nuclei whose masses were not included in the original adjustment. Macroscopic-microscopic models have also correctly predicted the existence and location of a rock of deformed metastable superheavy nuclei near $^{272}$110 that has recently been discovered. Nuclear ground-state masses and deformations will continue to provide an invaluable testing ground for nuclear many-body theories. The future challenge is for fully selfconsistent microscopic theories to predict these quantities with comparable or greater accuracy. Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported by the U. S. Department of Energy. References {#references .unnumbered} ========== [99]{} P. Möller, J. R. Nix, W. D. Myers, and W. J. Swiatecki, . P. Möller, J. R. Nix, and K. L. Kratz, . Y. Aboussir, J. M. Pearson, A. K. Dutta, and F. Tondeur, . W. D. Myers and W. J. Swiatecki, . V. M. Strutinsky, . G. Audi, Midstream Atomic Mass Evaluation, private communication (1989), with four revisions. S. G. Nilsson, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. [**29**]{}, 16 (1955). P. Möller, J. R. Nix, W. D. Myers, and W. J. Swiatecki, . G. Audi and A. H. Wapstra, . T. F. Kerscher, Ph. D. Thesis, Fakultät für Physik, Ludwig-Maximilians-Universität München (1996). S. Hofmann, . Yu. Norikov, private communication (1996). S. Hofmann et al., . S. Hofmann et al., . S. Hofmann et al., . Yu. A. Lazarev et al., [*Phys. Rev. Lett. *]{}[**73**]{}, 624 (1994). Yu. Ts. Oganessian, . P. Möller and J. R. Nix, . R. Bengtsson, P. Möller, J. R. Nix, and Jing-ye Zhang, Phys. Scr. [**29**]{}, 402 (1984). Z. Patyk and A. Sobiczewski, . P. Möller, J. R. Nix, P. Armbruster, S. Hofmann, and G. Münzenberg, , in press. A. Iwamoto, P. Möller, J. R. Nix, and H. Sagawa, .

--- address: - 'The Alan Turing Institute, 96 Euston Road, London NW1 2DB' - 'London School of Economics and Political Science, Houghton Street, London, WC2A 2AE' author: - Nikki Sonenberg - Edward Wheatcroft - Henry Wynn bibliography: - 'References.bib' title: Majorisation as a theory for uncertainty --- Introduction {#sec:intro} ============ Majorisation, also called rearrangement inequalities, yields a type of stochastic ordering in which two or more distributions are rearranged in decreasing order of their probability mass (discrete case) or probability density (continuous case) and then compared. When methods of majorisation are applied to probabilities and probability distributions, they provide a concept of the peakedness. This is independent of the ‘location’ of the probabilities, i.e., of the support of the distribution. This geometry-free property makes majorisation a good candidate as a foundation for the idea of uncertainty which is the focus of this paper. Majorisation is a partial, not total, ordering and implies that one or more of the class of order-preserving functions with respect to the ordering might be used as an entropy or ‘uncertainty metric’. Many well-known metrics fall into this category, one of which is Shannon entropy, widely used in information theory. Consider the question, ‘is uncertainty geometric?’. If we think our friend is in London, Birmingham or Edinburgh with probabilities $p_1,p_2, p_3$ respectively (where $p_1+p_2+p_3=1$), does it make any difference to our uncertainty, however we measure it, if the locations are changed to Reading, Manchester and Glasgow with the same probabilities, respectively? In fact, could we just permute the order of the first three cities? If our answer is no, i.e., that there is no difference, then we are in the realm of entropy and information. In the above cases, the Shannon entropy is $-\{p_1 \log(p_1) + p_2 \log(p_2) + p_3 \log(p_3)\}$ and we see that the subscript simply serves as a way to collect the probabilities, not to locate them in the geography of the UK. Another element of the majorisation approach is that it is, in a well-defined sense, dimension-free. In this paper, we show how this approach enables us to create, for a multivariate distribution, a univariate decreasing rearrangement (DR) by considering a decreasing threshold and ‘squashing’ all of the multivariate mass for which the density is above the threshold to a univariate mass adjacent to the origin. This creates the possibility of comparing multivariate distributions with different numbers of dimensions. We introduce a set of operations that can be applied to study uncertainty in a range of settings and illustrate these with examples. We see this work as a merging of methods used in applied mathematics and statistics with general methodology for the study of uncertainty. The methods discussed provide a foundation for the extension to Bayesian probabilities, a topic for further work. There is a large literature on majorisation. The classical results of Hardy, Littlewood and Polya [@Hardy1988] led to developments in a wide variety of fields. Marshall and Olkin’s [@Marshall2011] key volume on majorisation (later extended in [@Marshall2011]) built on these results. Applications in mathematical economics can be found in portfolio theory and income distributions built on classical work by Lorenz [@Lorenz1905] and Gini [@Gini1914] (see recent work by Arnold and Sarabia [@Arnold2018]). Majorisation has also been used in chemistry for mixing liquids and powders [@Klein1997] and in quantum information [@Partovi2011]. Statistical applications include experimental design [@Giovagnoli1987; @Pukelsheim1987], and in application to testing [@eaton1974monotonicity; @tong1988some]. Majorisation has been employed in the area of proper scoring rules by considering the partial and total ordering in the class of well-calibrated experts [@Degroot1986; @Degroot1985; @Degroot1988]. The common feature of these studies is the need to compare and quantify the degree of variation between distributions. We note that the theory of the decreasing rearrangement of functions, which we have used to a limited extent for probability densities, can be considered an area of functional analysis particularly in the area of inequalities of the kind which say that a rearrangement of a function increases or decreases some special functional [@lieb2001graduate].\ This paper is organised as follows. In the remainder of this section we introduce the concept of majorisation of probabilities and present related concepts and previous work. In Section \[sec:cont\_major\], we present results for the continuous case. In Section \[sec:multivariate\], we present the key idea of reducing multivariate distributions into a one dimensional decreasing rearrangement and illustrate this with examples. In Section \[sec:operations\], we collect together operations for the study of uncertainty and, in Section \[sec:algebra\], a lattice and an algebra for uncertainty. In Section \[sec:empirical\], we discuss empirical applications. Concluding remarks are given in Section \[sec:conclusion\]. Discrete majorisation and related work {#sec:discrete} -------------------------------------- We introduce majorisation for discrete distributions following Marshall *et al.* [@Marshall2011]. Consider two discrete distributions with $n$-vectors of probabilities $$p_1=(p^{(1)}_1,\ldots, p_n^{(1)}) \quad \text{ and } \quad p_2=(p^{(2)}_1,\ldots, p_n^{(2)}),$$ where $\sum_i p_i^{(1)}=\sum_i p_i^{(2)}=1$. Placing the probabilities in decreasing order: $$\tilde{p}^{(1)}_1 \geq \ldots \geq \tilde{p}_n^{(1)}\quad \text{ and } \quad \tilde{p}^{(2)}_1 \geq \ldots \geq \tilde{p}_n^{(2)},$$ it is then said that $p_2$ majorises $p_1$, written $p_1 \preceq p_2$ when, for all $n$, $$\sum_{i=1}^n \tilde{p}_i^{(1)} \leq \sum_{i=1}^n \tilde{p}_i^{(2)}.$$ This definition of majorisation is a partial ordering, that is, not all pairs of vectors are comparable. As argued by Partovi [@Partovi2011], this is not a shortcoming of majorisation, rather a consequence of its rigorous protocol for ordering uncertainty. Marshall *et al* [@Marshall2011] provide several equivalent conditions to $p_1\preceq p_2$. We consider three (A1-A3) of the best known in detail below.\ (A1) There is a doubly stochastic $n\times n$ matrix $P$, such that $$\begin{aligned} \label{equiv:doubly} p_1 = P p_2.\end{aligned}$$ This is a well known result by Hardy, Littlewood and Pólya [@Hardy1988]. The intuition of this result is that a probability vector which is a mixture of the permutations of another is more disordered. The relationship between a stochastic matrix $P$ and the stochastic transformation function in the refinement concept was presented by DeGroot [@Degroot1988].\ (A2) Schur [@Schur1923] demonstrated that, if (A1) holds for some stochastic matrix $P$, this leads to the following equivalent condition. For all continuous convex functions $h( \cdot )$, $$\begin{aligned} \label{condition3} \sum_{i=1}^n h(\tilde{p}_i^{(1)}) \leq \sum_{i=1}^n h(\tilde{p}_i^{(2)}),\end{aligned}$$ for all $n$. The sums in (\[condition3\]) are special cases of the more general Schur-convex functions on probability vectors. Details on the characteristics and properties of Schur-convex functions are provided by Marshall *et al.* [@Marshall2011]. In particular, entropy functions such as Shannon information, for which $h(y)=y\log(y)$, are Schur-convex. We also highlight a special case of the Tsallis information for which $$h(y)=\frac{y^{\gamma}-1}{\gamma}, \quad\gamma>0,$$ where, in the limit $\gamma\rightarrow 0$, Shannon information is obtained. The condition (A2) is equivalent to the majorisation ordering for distributions, and we consider it as a continuous extension to Equation (\[condition3\]) (see Section \[sec:cont\_major\] for details). The condition (A2) indicates that the ordering imposed by majorisation is stronger than the ordering by any single entropic measure and, in a sense, is equivalent to all such (entropic) measures taken collectively [@Partovi2011].\ (A3) Let $\pi(p) = (p_{\pi(1)}, \ldots, p_{\pi(n)})$ be the vector whose entries are a permutation $\pi$ of the entries of a probability vector $p$, with symmetric group $S$, then $$\begin{aligned} p_1 \in \mbox{conv}_{\pi \in S} (\{\pi(p_2)\}).\end{aligned}$$ That is to say, $p_1$ is in the convex hull of all permutations of entries of $p_2$. Majorisation is a special case of group-majorisation (G-majorisation) for the symmetric (permutation) group [@Giovagnoli1985]. The general type of groups for which the theory really works are reflection (Coxeter) groups, e.g. permutation and sign changes in $n$ dimensions (called the $B_n$ series). The main work on G-majorisation was by Eaton and Perlmann [@Eaton1977]. Rearrangements can be viewed as special instances of transportation plans, which move a given mass distribution to another distribution of the same total mass (see Buchard [@Burchard2009]). The recent use of transport mapping in UQ is to improve Markov chain Monte Carlo (MCMC) and sequential Monte Carlo (SMC) samplers [@Marzouk2016; @Parno2018]. Continuous majorisation {#sec:cont_major} ======================= Following Hardy *et al.* [@Hardy1988], in this section we describe continuous majorisation, extending on the discrete case presented in Section \[sec:discrete\]. Ryff [@Ryff1965] provided continuous analogues to discrete majorisation by replacing vectors and matrices with integrable functions and linear operators. \[drdefn\] Let $f(x)$ be a (univariate) pdf and define $m(y)=\mu\{z: f(z) \geq y\}$. The decreasing rearrangement of $f(x)$ is then $$\begin{aligned} \tilde{f}(z)=\mbox{sup}\{t: m(t) >z\},\; z >0.\end{aligned}$$ Let $\tilde{f}_1(z)$ and $\tilde{f}_2(z)$ be the DR of two pdfs $f_1(x)$ and $f_2(x)$, respectively and $\tilde{F}_1(z)$ and $\tilde{F}_2(z)$ their corresponding cdfs. We say that $f_2(x)$ majorises $f_1(x)$, written $f_1 \preceq f_2$, if and only if $$\tilde{F}_1(z) \leq \tilde{F}_2(z),\;\; \mbox{for all} \quad z > 0.$$ Similarly to the discrete case, we give three equivalent conditions for majorisation, $\preceq$:\ (B1) For some non-negative doubly stochastic kernel $P(x,t)$, $$\begin{aligned} f_1(x) = \int P(x,t) f_2(t) dt.\end{aligned}$$ (B2) For all continuous convex functions $h(\cdot)$, $$\begin{aligned} \int h(f_1(z)) dz \leq \int h(f_2(z))dz.\end{aligned}$$ (B3) Slice condition: $$\begin{aligned} \int(f_1(x)-c)_+ dx \leq \int(f_2(x)-c)_+dx, \quad c>0. \label{eq:slice}\end{aligned}$$ \[example|\_beta\] Consider the Beta$(a,b)$ distribution with $(a, b)=(3, 2)$ and pdf $p(z)=12(1-z)z^2$. We look for $z_1$ and $z_2$ (where $z_1<z_2$) such that $p(z_1)=p(z_2)=c$, that is the points at which $p(z)$ intersects the line $y=c$. The pdf and the horizontal line $y=c$ are both shown in Figure \[fig:DensityPlot\]. Setting $z=z_2-z_1$, this gives us the system of equations: $$\begin{aligned} \label{example_beta_system} \begin{cases} p(z_1)=12(1-z_1)z_1^2=y ,\\ p(z_2)=12(1-z_2)z_2^2=y,\\ z_2-z_1=z,\\ 0\leq z\leq 1. \end{cases}\end{aligned}$$ ![The identification of $z_{1}$ and $z_{2}$ for the probability density function of Beta(3,2).[]{data-label="fig:DensityPlot"}](DensityFunctionBeta.png){width="34.00000%"} The DR can be obtained from Equation (\[example\_beta\_system\]) by eliminating $z_1$ and $z_2$ and setting $\tilde{f} = y$. The elimination variety is $ 48z^6 - 96z^4 + 9y^2 + 48z^2 - 16y$. We obtain the explicit solution, $$\begin{aligned} \tilde{f}(z) = \left\{ \begin{array}{l} \frac{8}{9} + \frac{4}{9} (-27z^6 + 54z^4 - 27z^2 + 4)^{\frac{1}{2}}, \quad 0 \leq z \leq \frac{1}{\sqrt{3}} \\ \frac{8}{9} - \frac{4}{9} (-27z^6 + 54z^4 - 27z^2 + 4)^{\frac{1}{2}}, \quad \frac{1}{\sqrt{3}} \leq z \leq 1. \end{array} \right.\end{aligned}$$ This variety is shown in Figure \[fig:beta\] and the DR $\tilde{f}(z)$ is the section of the curve over $[0,1]$ decreasing from $\left(0, \frac{16}{9}\right)$ to $(1,0)$. The cdf of the DR of Beta$(2,3)$ is $\tilde{F}(z)=-4z^3 + 6z^2$. If $\tilde{F}(z)$ is the cdf corresponding to $\tilde{f}$ obtained by adjoining the equations $Y = F(z_2)-F(z_1)$, we obtain the second variety $4z^4 + 3Y^2 - 8Yz = 0,$ illustrated in Figure \[fig:beta\], and $\tilde{F}(z)$ is the upper portion of the curve from $(0,0)$ to $(1,1)$, namely $$\tilde{F}(z) = 2\left(\frac{2}{3} + \frac{\sqrt{-3z^2 + 4}}{3}\right)z.$$ It is hard to derive when $f_1(x) \leq f_2(x)$ for the general case in which $(a_1,b_1)$ and $(a_2,b_2)$ and $f_i(x) \sim \mbox{Beta}(a_i,b_i),\; i=1,2$. However, we can prove the following. Assume $a_1 , b_1 , a_2 , b_2 > 1$. If pdfs $f_1 (x)\sim \text{Beta}(a_1 , b_1 )$ and $f_2 (x)\sim\text{Beta}(a_2 , b_2 )$, have the same mode, then $\max_x f_1(x) \leq \max_x f_2(x)$ if and only if $X_1 \preceq X_2.$ We first prove that, under the same mode condition, $f_1(x)$ and $f_2(x)$ intersect at two distinct $x$-values at which the values of $f_1(x)$ and $f_2(x)$ are the same. Setting the modes equal, that is $$\frac{a_1-1}{a_1+b_1-2} = \frac{a_2-1}{a_2+b_2-2},$$ We find that, and assuming without loss of generality, that $a_2 > a_1$, we obtain $$\frac{f_1(x)}{f_2(x)} = \left\{x (1-x)^u \right\}^v C,$$ where $u= a_2-a_1,v = \frac{b_1-1}{b_2-1}$ and $C$ is a constant. If we set this equal to 1, we obtain two solutions given by $$x (1-x)^u = C^{-\frac{1}{v}}.$$ It is then straightforward to verify that the common value of $f_1(x)$ and $f_2(x)$ is the same at the two solutions. The proof of the theorem is completed by using the slice condition in Equation . Multivariate case: matching of uncertainty {#sec:multivariate} ========================================== The following construction shows how to induce a one dimensional DR from a multidimensional distribution. This is in fact the continuous multidimensional version of the following discrete version. Thus, suppose we have a 2-dimensional table with probabilities $p_{i,j}, i=1, \ldots, I; j= 1, \ldots, J$. Line up the probabilities in decreasing (non-increasing) order from $1$ to $n =IJ$ with support on $1, \ldots, n$, respectively. If we now count the number of $\{i,j\}$ such that the probability is greater than or equal to a constant $c$, it is the same as the original table as for the DR. \[multitouni\] A univariate decreasing rearrangement $\tilde{f}(z)$, compatible with $f(x)$, is, for all constants $c\geq 0$, $$\begin{aligned} \label{decreaseRearrangemult} \int_{\{x:f(x)\geq c \}}f(x)dx=\int_{\{z:\tilde{f}(z)\geq c\}}\tilde{f}(z)dz.\end{aligned}$$ Following [@Burchard2009]: as $$\begin{aligned} {\{x:f(x)\geq c \}} = {\{z:\tilde{f}(z)\geq c\}},\end{aligned}$$ then the volume of these sets are consistent. The notation on the lemma will be important to us. We shall use $f(x)$ for the multivariate pdf for a random variable $X = (X_1, \ldots, X_n)$ of dimension $n$, and $\tilde{f}(z)$ will be its one-dimensional pdf and we reserve $\tilde{F}(z)$ for the corresponding cdf, which will be the basis for the majorisation. The following lemma shows that the information/entropy for $X \sim f(x)$ and $Z \sim \tilde{f}(z)$ are the same. This is a crucial result and gives us confidence in the matching. Let $f(x)$ be a multidimensional pdf and $\tilde{f}(z)$ on $[0, \infty]$ its decreasing rearrangement. Then, given a convex function $\varphi(x)$, we have $$\int_S \varphi(f(x)) dx = \int_0^{\infty} \varphi(\tilde{f}(z)) dz,$$ where $S$ is the support of $f(x)$. (sketch) The proof consists of matching volume to length elements in $S$ and $[0,1)$. For $c>0$ and small $\delta c > 0$ we have $$\int_{x: f(x) \geq c, x \in S} f(x)dx - \int_{x: f(x) \geq c + \delta c, x \in S} f(x) dx = \int_{z: \tilde{f}(z) \geq c, z \in [0, \infty)} \tilde{f}(z) dz -\int_{z: \tilde{f}(z) \geq c +\delta c, z \in [0, \infty)} \tilde{f}(z) dz.$$ We can then write, approximately, $$u(c)A(c, \delta c)=u(c)L(c, \delta c),$$ where $A(c, \delta c)$ and $L(c, \delta c)$ are the corresponding increments in volume and length, respectively, as corresponding to the interval $[c,c+\delta c)$, that is $ f^{(-1)}([c,c+ \delta))$ and $ \tilde{f}^{(-1)}([c,c+ \delta))$, respectively. Cancelling $c$, we can equate $A(c, \delta c)$ and $L(c, \delta c)$, and this allows us to recapture and equate the integrals of any measurable function $u(\cdot)$: $$u(c)A(c,\delta c) = u(c) L(c, \delta c).$$ In particular, we can write $u(c) = \varphi(f(c)).$ In Examples \[multi\_norm\_ex\] and \[indep\_exp\_ex\], we demonstrate how to obtain a DR for the standard multivariate Normal distribution and the $n$-fold independent standard exponential distribution. In the examples below, we use the following idea to carry out computations. There may be cases in which, for a given $c$, the inverse set $f^{(-1)}(c)$ is described by some useful quantity $\delta$. Moreover $\delta$, expressed as a function of $x$, then becomes a random variable with a known (univariate) distribution. In that case, we can write Definition \[multitouni\] as $$\tilde{F}\big(\tilde{f}^{-1}(c) \big) =F_{\delta}\big(f_{{X}}^{-1}(c) \big)\nonumber \label{Fstuff},$$ then $$\begin{aligned} \label{eq:valid_DR} \tilde{f}(r) & = f_{\delta} \left( f_X^{(-1)}(\tilde{f}(r)) \right) \frac{\partial}{\partial r}\left(f_X^{(-1)}(\tilde{f}(r))\right).\end{aligned}$$ \[multi\_norm\_ex\] We provide a representation of Definition \[multitouni\] for a two-dimensional multivariate normal in Figure \[fig:DRM2\]. ![*Left panel:* Density plot of a two-dimensional standard multivariate normal. The dashed line and blue shaded region correspond to $f(x)=c$ and $\int_{\{x:f(x)\geq c \}}f(x)dx$ respectively. *Central panel:* A plot to demonstrate the connection between ${X}=(X_1, X_2)^T$ and $R^2$. The radius $r$ of a blue circle corresponds to $x=f^{-1}(c)$. *Right panel:* A DR $\tilde{f}(z)$ obtained for a multivariate normal. The blue shaded region corresponds to $\int_{\{z: \tilde{f}(z)\geq c\}}\tilde{f}(z) dz$.[]{data-label="fig:DRM2"}](MultivariateNorm1.png "fig:"){width="30.00000%"} ![*Left panel:* Density plot of a two-dimensional standard multivariate normal. The dashed line and blue shaded region correspond to $f(x)=c$ and $\int_{\{x:f(x)\geq c \}}f(x)dx$ respectively. *Central panel:* A plot to demonstrate the connection between ${X}=(X_1, X_2)^T$ and $R^2$. The radius $r$ of a blue circle corresponds to $x=f^{-1}(c)$. *Right panel:* A DR $\tilde{f}(z)$ obtained for a multivariate normal. The blue shaded region corresponds to $\int_{\{z: \tilde{f}(z)\geq c\}}\tilde{f}(z) dz$.[]{data-label="fig:DRM2"}](MultivariateNorm2.png "fig:"){width="30.00000%"} ![*Left panel:* Density plot of a two-dimensional standard multivariate normal. The dashed line and blue shaded region correspond to $f(x)=c$ and $\int_{\{x:f(x)\geq c \}}f(x)dx$ respectively. *Central panel:* A plot to demonstrate the connection between ${X}=(X_1, X_2)^T$ and $R^2$. The radius $r$ of a blue circle corresponds to $x=f^{-1}(c)$. *Right panel:* A DR $\tilde{f}(z)$ obtained for a multivariate normal. The blue shaded region corresponds to $\int_{\{z: \tilde{f}(z)\geq c\}}\tilde{f}(z) dz$.[]{data-label="fig:DRM2"}](MultiVariateNorm3.png "fig:"){width="30.00000%"} Let a real random vector ${X}=(X_1, \dots, X_n)^T$ be an $n$-variate standard normal distribution with density $$f_{{X}}(x_1, \dots, x_n)=\frac{1}{(2\pi)^{\frac{n}{2}}}\exp\Bigg\{-\frac{\sum_{i=1}^n x_i}{2} \Bigg\}.$$ We refer to a real random vector ${X}$ as a spherical Gaussian random vector with ${X}\sim\text{N}_n({0}, I_n)$, where ${0}$ is an $n$-vector of zeros and $I_n$ is an $n\times n$ identity matrix. Define the square of the radius of a spherical Gaussian random vector, that is, $$R^2 = \sum_{i=1}^nX_i^2.$$ To construct a DR, we slice the pdf of a multivariate normal at $f_{{X}}(x_1, \dots, x_n)=c$, then $$\begin{aligned} \label{eq:DR_derivation1} &c = \frac{1}{(2\pi)^{n/2}}\exp\Big\{-\frac{1}{2}\sum_{i=1}^n x_i^2 \Big\}. \end{aligned}$$ Given the relationship between the vector ${X}$, and a random variable $R^2$, define $r^2 =\sum_{i=1}^n x_i^2$, then $$\begin{aligned} r=\Big(-2\log\big((2\pi)^{n/2} c\big) \Big)^{1/2},\end{aligned}$$ where the volume of the $n$-dimensional Euclidean ball of radius $r$ is $$\label{eq:ball_volume} V_n(r)=\frac{\pi^{n/2}}{\Gamma\Big(\frac{n}{2}+1 \Big)}r^n.$$ Substituting Equation (\[eq:DR\_derivation1\]) into Equation (\[eq:ball\_volume\]), we obtain $$\begin{aligned} c=\frac{1}{(2\pi)^{n/2}}\exp\bigg\{-\frac{1}{2}\bigg(\frac{V_n(r)\Gamma(n/2+1)}{\pi^{n/2}} \bigg)^{2/n} \bigg\},\end{aligned}$$ noting the values of $c$ and $V_n(r)$ are dependent on each other. To generalise the expression above, we replace $c$ and $V_n(r)$ with $\tilde{f}(z)$ and $z$, respectively. The final form of the DR is $$\label{eq:DRM_mult_normal} \tilde{f}(z) = \frac{1}{(2\pi)^{n/2}}\exp\bigg\{-\frac{1}{2}\bigg(\frac{z}{V_n} \bigg)^{2/n} \bigg\},$$ where $V_n$ is the volume of the unit sphere in $\mathbb{R}^n$. We validate the form of the DR in Equation (\[eq:DRM\_mult\_normal\]) by employing the construction from Equation (\[eq:valid\_DR\]). Here, $R^2=\sum_{i=1}^n X_i^2$ follows a Chi-squared distribution with $n$ degrees of freedom, $R^2\sim \chi_n^2$, with pdf and cdf, $$\begin{aligned} f_{R^2}(y)=\frac{1}{2^{n/2}\Gamma(n/2)}y^{n/2-1}\exp\left\{-\frac{y}{2} \right\}, \qquad F_{R^2}(y)=\frac{1}{\Gamma(n/2)}\gamma\left(\frac{n}{2}, \frac{y}{2} \right),\end{aligned}$$ respectively. The result for the DR in Equation (\[eq:DRM\_mult\_normal\]) can be verified by employing Equation (\[eq:valid\_DR\]) with $\delta = R^2$ and directly substituting the required functions. \[indep\_exp\_ex\] Let the real random vector ${X}=(X_1, \dots, X_n)^T$ be an $n$-fold independent standard exponential distribution with density $$f_X(x_1, \dots, x_n)=\exp\left\{-\sum_{i=1}^n x_i \right\}.$$ As $f_X(x_1, \dots, x_n) = f_1(x_1) f_2(x_2) \cdots f_n(x_n)$, slicing the pdf at $c=f_{{X}}(x_1, \dots, x_n)$ yields $$\begin{aligned} \label{eq:DR_derivation2} &-\log(c)=\sum_{i=1}^n x_i.\end{aligned}$$ The volume of an $n$-dimensional simplex in which all $n$ variables are greater than $0$ but with sum less than $R$ is $$\label{eq:simplex_volume} V_n = \frac{R^n}{n!}.$$ Substituting Equation (\[eq:DR\_derivation2\]) into Equation (\[eq:simplex\_volume\]), we obtain $$\begin{aligned} c=\exp\left\{-(n!V_n)^{1/n} \right\},\end{aligned}$$ where the values of $c$ and $V_n$ depend on each other. To generalise this expression, we replace $c$ and $V_n$ with $\tilde{f}(z)$ and $z$, respectively. The DR can then be written as $$\label{eq:DRM_mult_exp} \tilde{f}(z)=\exp\left\{-(n!z)^{1/n} \right\}.$$ To verify the form of the DR in Equation (\[eq:DRM\_mult\_exp\]), we employ the construction in Equation (\[eq:valid\_DR\]). Define the random variable $R=\sum_{i=1}^n X_i$, such that $R\sim\text{Gamma}(n, 1)$, with pdf and cdf, $$\begin{aligned} f_R(y)=\frac{1}{\Gamma(n)}y^{n-1}e^{-y}, \qquad F_R(y)=\frac{1}{\Gamma(n)}\gamma(n, y).\end{aligned}$$ Similarly to Example \[multi\_norm\_ex\], we derive the DR, $\tilde{f}(z)$, for a Gamma-distributed random variable $R$, that is, $$\label{eq:DR_mult_exp} \tilde{f}(r)=f_R\big(f_{{X}}^{-1}(\tilde{f}(r))\big)\frac{\partial}{\partial r}(f_{{X}}^{-1}(\tilde{f}(r))).$$ The results for the DR in Equation (\[eq:DRM\_mult\_exp\]) can be verified by employing Equation (\[eq:valid\_DR\]) with the required functions. Some operations with $\preceq$ {#sec:operations} ============================== Inverse Mixing {#subsec:InverseMix} -------------- We present *inverse mixing* as a method for combining uncertainty given two distributions over two different populations. \[def:inversemixing\] Define the inverse mixture $$\tilde{f}_1\; [+] \;\tilde{f}_2= \left(\tilde{f}_1^{(-1)}(z)+\tilde{f}_2^{(-1)}(z) \right)^{(-1)},$$ and the $\alpha$-inverse mixture $$\tilde{f}_1\; [+]_{\alpha}\; \tilde{f}_2=\left(\tilde{f}_1\left(\frac{z}{1-\alpha}\right)^{(-1)} + \tilde{f}_2\left(\frac{z}{\alpha}\right)^{(-1)} \right)^{(-1)},$$ where $0 < \alpha < 1$ is the mixing parameter. For the case in which $\alpha= 1/2$, we claim that, in the discrete case, the *inverse mixture* corresponds to a combination of all the probabilities in both populations scaled by a factor $\alpha=\frac{1}{2}$, i.e., $\frac{1}{2}p_i\cup\frac{1}{2}q_i, i=1, \dots, 5$ and sorting them in a decreasing order. In the following examples, we demonstrate inverse mixing for the discrete and continuous cases. \[exampleworkplace\] Consider two distinct groups of people in a work place. Denote the probability of the $i$-th member of group one and the $i$-th member of group two obtaining a promotion to be $p_i$ and $q_i$, respectively. Let the probabilities $p_{1},...,p_{5}$ and $q_{1},...,q_{5}$ be those denoted in Table \[tab:Probabilities\_MF\], noting that $ p_1\geq p_2\geq\cdots\geq p_5$, $q_1\geq q_2\geq \cdots q_5$ and $\sum_{i}p_i=1$, $\sum_{i}q_i=1$, ----------------- ------- ------- ------- ------- ------- $i$ 1 2 3 4 5 \[0.5ex\] $p_i$ 0.577 0.192 0.128 0.064 0.038 $q_i$ 0.730 0.219 0.036 0.007 0.007 ----------------- ------- ------- ------- ------- ------- : Ordered probabilities for Example \[exampleworkplace\].[]{data-label="tab:Probabilities_MF"} To perform inverse mixing, we take the inverse of each pmf, combine them and sorting them into ascending order (demonstrated in panel (a) of Figure \[fig:inversetilt\]). The inverse is taken to obtain a pmf (panel (b), noting the change of scale on the $y$ axis). The result of direct mixing, obtained by summing ordered probabilities of two populations and scaling by a factor $\alpha$, i.e., $\frac{1}{2}(p_i+q_i), i=1, \dots, 5$, is shown in panel (c). Both mixtures provide information about the joint population, but the inverse mixture also preserves information about the individual subpopulations. ![(a) the addition of two inverse pmfs, (b) inverse mixture distribution with $\alpha=\frac{1}{2}$, (c) direct mixture distribution with $\alpha=\frac{1}{2}$.[]{data-label="fig:inversetilt"}](combine_inv.png){width="100.00000%"} We now consider inverse mixing for the continuous case. In the continuous case, we need to pay attention to the maximum values (modes) of the probability distributions. We demonstrate the importance of this condition in the following two examples. Given univariate and bivariate exponential distributions with the following form of decreasing rearrangements, $$\tilde{f}_1(z)=\exp\{-z\}, \quad \tilde{f}_2(z)=\exp\{-(2z)^{1/2}\},$$ we observe that $0<\tilde{f}_1(z), \tilde{f}_2(z)\leq 1$ and obtain the functional inverses, $$\tilde{f}_1^{(-1)}(z) = -\log(z), \quad \tilde{f}_2^{(-1)}(z)=\frac{1}{2}(\log(z))^2, \quad z\in(0, 1].$$ The left and central panels in Figure \[fig:inverse\_mixing\_con1\] show $\tilde{f}_1(z)$, $\tilde{f}_2(z)$, $\tilde{f}_1^{(-1)}(z)$ and $\tilde{f}_2^{(-1)}(z)$. The maximum value of these two probability functions occurs at the same point, $\tilde{f}_1(0)=\tilde{f}_2(0)=1$, so there is no kink in the inverse mixing of these two distributions. The inverse mixture of the two distributions is then $$f^{(1)}(z) = \left\{ \tilde{f}_1^{(-1)}\left(\frac{z}{1-\alpha}\right) + \tilde{f}_2^{(-1)}\left(\frac{z}{\alpha}\right) \right\}^{(-1)} = \left\{-\log\Big(\frac{z}{1-\alpha}\Big) +\frac{1}{2} \Big(-\log\Big(\frac{z}{\alpha}\Big) \Big)^2 \right\}^{(-1)} ,$$ for $0 \leq \alpha \leq 1$. The direct averaging of $f_1(x)$ and $f_2(x)$ gives: $$f^{(2)}(z) = \left\{ (1-\alpha)\tilde{f}_1^{(-1)}(z)+ \alpha \tilde{f}_2^{(-1)}(z)\right\} ^{(-1)}= \left\{(1-\alpha)(-\log(z))+\frac{\alpha}{2}(-\log(z))^2 \right\}^{(-1)}.$$ We specify $\alpha=1/2$ to obtain the following expression: $$f^{(1)}(z)=\Big\{-\log(2z)+\frac{1}{2}\big[\log(2z)\big]^2 \Big\}^{(-1)}.$$ Since the expression above is a quadratic in $\log(2z)$, we obtain the two solutions $f^{(1)}(z)=\frac{1}{2}\exp\{1+\sqrt{1+2z}\}$ and $f^{(1)}(z)=\frac{1}{2}\exp\{1-\sqrt{1+2z}\}$. Proceeding with the second solution, as the first solution does not integrate to 1, we obtain the mean, variance and Shannon entropy: $\frac{7}{2}, \frac{99}{4}$ and $\frac{3}{2}+\log(2)$, respectively. We perform similar calculations for direct mixing with $\alpha=1/2$. The pdf has the form, $$f^{(2)}(z)=\exp\left\{1-\sqrt{1+4z}\right\},$$ and, in this case, the values of the mean, variance and Shannon entropy are $\frac{7}{4}, \frac{99}{16}$ and $\frac{3}{2}$. Based on the pdfs, we have the following relationship for inverse mixing and direct averaging when $\alpha=1/2$, $$\label{eq:inverse_direct} f^{(2)}(z)=2f^{(1)}(2z),$$ illustrated in the right panel of Figure \[fig:inverse\_mixing\_con1\]. Here, $f^{(1)}(z)$ (red line) stretches the support of the distributions, and lowers the overall maximum, whereas $f^{(2)}(z)$ (blue line) preserves the maximum and shrinks the support, confirmed by Equation (\[eq:inverse\_direct\]). ![*Left panel:* plot of DR functions $\tilde{f}_1(z)$ (solid line) and $\tilde{f}_2(z)$ (dashed line). *Central panel:* plot of functional inverses of the DR functions, i.e. $\tilde{f}_1^{(-1)}(z)$ (solid line) and $\tilde{f}_2^{(-1)}(z)$ (dashed line). *Right panel:* plot of inverse mixing and direct averaging, $f^{(1)}(z)$ (red line) and $f^{(2)}(z)$ (blue line).[]{data-label="fig:inverse_mixing_con1"}](inverse_mixing_con1){width="100.00000%"} \[example:6\] We consider exponential distributions with means 1 and 2 and note that they are already DRs: $$\tilde{f}_1(z)=\exp\{-z\}, \quad\tilde{f}_2(z)=\frac{1}{2}\exp\{-z/2\}.$$ The left panel in Figure \[fig:inverse\_mixing\_con2final\] shows $\tilde{f}_1(z)$ and $\tilde{f}_2(z)$, depicted by solid and dotted lines respectively. We note that $0<\tilde{f}_1(z)\leq 1$ and $0<\tilde{f}_2(z)\leq\frac{1}{2}$, which indicates that there is different support for the functional inverses, i.e. $$\begin{aligned} &\tilde{f}_1^{(-1)}(z)=-\log(z), \quad z\in (0, 1],\\ &\tilde{f}_2^{(-1)}(z)=-2\log(2z), \quad z\in (0, 1/2].\end{aligned}$$ The inverse mixing of these two distributions with $\alpha=\frac{1}{2}$ is defined as $$f^{(1)}(z)=\tilde{f}_1\; [+]_{\frac{1}{2}}\; \tilde{f}_2=\left\{-\log(2z)-2\log(4z) \right\}^{(-1)}.$$ To avoid negative values of the expression inside the functional inverse, we propose the following modification: $$\label{eq:arg_inverse} \tilde{f}_1^{(-1)}(2z)+\tilde{f}_2^{(-1)}(2z) =\max\{0, -\log(2z)\}+\max\{0, -2\log(4z)\}.$$ In the central plot of Figure \[fig:inverse\_mixing\_con2final\], the dotted line corresponds to the function in Equation (\[eq:arg\_inverse\]). We note that the introduced modification results in a kink in $\tilde{f}_1^{(-1)}(2z)+\tilde{f}_2^{(-1)}(2z)$ at $z=0.25$. To obtain the inverse mixture, we are required to take another functional inverse by swapping the abscissa and ordinate. Therefore we observe a kink in $f^{(1)}(z)$ at $z=\log(2)$ in the right panel of Figure \[fig:inverse\_mixing\_con2final\] (blue line). We obtain the final form of the inverse mixing: $$\begin{aligned} f^{(1)}(z)&=\begin{cases} \frac{1}{2}\exp\{-z\}, &\mbox{if } 0<z<\log(2) \\ \frac{1}{2}\exp\{ \frac{-2\log(2)-z}{3} \}, &\mbox{if } z \geq \log(2), \end{cases}\end{aligned}$$ and we obtain the values of the mean, variance and Shannon entropy: $2.85$, $8.91$ and $1 + \frac{3\log(2)}{2}$. Similarly to example 5, we consider the direct averaging of these distributions with $\alpha=\frac{1}{2}$, i.e. $$f^{(2)}(z)=\big\{\frac{1}{2}\tilde{f}_1^{(-1)}(z)+\frac{1}{2}\tilde{f}_2^{(-1)}(z) \big\}^{(-1)}= \left\{-\frac{1}{2}\log(z)-\log(2z) \right\}^{(-1)}.$$ As with inverse mixing, to avoid negative values, we modify the argument of the functional inverse: $$\label{eq:arg_direct} \frac{1}{2}\tilde{f}_1^{(-1)}(z)+\frac{1}{2}\tilde{f}_2^{(-1)}(z)=\max\left\{ 0,-\frac{1}{2}\log(z)\right\}+\max\left\{0, -\log(2z) \right\}.$$ The solid line in the central plot of Figure \[fig:inverse\_mixing\_con2final\] corresponds to the function in Equation (\[eq:arg\_direct\]). We observe a kink in the function at $z=\frac{1}{2}$. As a result, we obtain a kink in $f^{(2)}(z)$ at $z=-\frac{1}{2}\log{\frac{1}{2}}$ in the right panel of Figure \[fig:inverse\_mixing\_con2final\] (red line). The final form of the direct averaging is $$\begin{aligned} f^{(2)}(z) &=\begin{cases} \exp\{-2z\}, &\mbox{if } 0<z<-\frac{1}{2}\log(\frac{1}{2}), \\ \exp\Big(\frac{-2z-2\log(2)}{3}\Big), &\mbox{if } z\geq -\frac{1}{2} \log(\frac{1}{2}), \end{cases}\end{aligned}$$ where the values of the mean, variance and Shannon entropy are $1.42$, $2.23$ and $1+\frac{\log(2)}{2}$. ![*Left panel:* plot showing DR functions $\tilde{f}_1(z)$ and $\tilde{f}_2(z)$. *Central panel:* plot showing the function $\tilde{f}_{12}^{(-1)}(z)$ which is employed in direct averaging and inverse mixing. *Right panel:* plot showing pdfs obtained from inverse mixing and direct averaging, $f^{(1)}(z)$ and $f^{(2)}(z)$.[]{data-label="fig:inverse_mixing_con2final"}](inverse_mixing_con2final){width="100.00000%"} From the representation of inverse mixing and direct averaging in Figure \[fig:inverse\_mixing\_con2final\], we can see that $f^{(1)}(z)$ stretches the support of the distribution, whilst $f^{(2)}(z)$ shrinks it. The maximum (mode) from the direct averaging is double the maximum of the inverse mixing. This example shows how distributions can be approximated using DR from higher dimensional distributions. Consider the pdf $$f(x)=3(1-x)^2,\quad x\in[0, 1],$$ which has the functional inverse $f^{(-1)}(x)=1-\sqrt{ x/3}$. We need to perform an expansion in $t=\log(x)$, substituting $x=\exp\{t\}$ in the test function, defined as $$h(t)=f^{(-1)}(\exp\{t\})=1-\sqrt{\exp\{t\}/3}.$$ We provide the first three terms of the Taylor series, $$h(t)\approx \Big(1-\frac{\sqrt{3}}{3}\Big)-\frac{\sqrt{3}}{6}t-\frac{\sqrt{3}}{24}t^2,$$ and obtain the approximation $$\hat{f}^{(-1)}(x) \approx \left(1-\frac{\sqrt{3}}{3}\right)-\frac{\sqrt{3}}{6}\log(x)-\frac{\sqrt{3}}{24}(\log(x))^2,$$ from which we obtain the functional inverse, $$\hat{f}(x)=\exp\left( -2+2\sqrt{2\sqrt{3}-1-2\sqrt{3}x}\right).$$ From the left panel in Figure \[fig:series\_plot\], we observe that the functional inverse (solid line) and its approximation (dotted line) are close to each other and, from the right panel, that $\hat{f}(x)$ (red line) is a good approximation to $f(x)$ (blue line). ![*Left panel:* plot of the functional inverse, $f^{(-1)}(x)$(solid line), and its approximation, $\hat{f}^{(-1)}(x)$ (dotted line). *Right panel:* plot of $f(x)$ (blue line) and its approximation $\hat{f}(x)$ (red line).[]{data-label="fig:series_plot"}](series_plot){width=".7\textwidth"} Independence, conditional distributions --------------------------------------- It is well known and axiomatic that Shannon information $S$ and entropy $(-S)$ are additive under independence: if $ X$ and $Y$ are independent random variables of the same dimension then, $$S(X+Y) = S(X) + S(Y).$$ It is a natural conjecture that entropy is a maximum in some sense when random variables are independent. This result holds for Shannon entropy. \[lem:lem4\_2\] Within the class of bivariate random variables $(X_1,X_2)$ with given marginal distributions $X_1 \sim f_1(x) $ and $X_2 \sim f_2(x)$, the maximum Shannon entropy is uniquely achieved when $X_1$ and $X_2$ are independent. Let $H( \cdot )$ be Shannon information. For random variables $X,Y$, we have the well-known expansion for the joint information: $$H(X,Y) = H(X) + \mbox{E}_X H(Y |X).$$ Also well known is the inequality, which follows from Jensen’s inequality, for the second term $$\mbox{E}_X H(Y |X) \geq H(Y),$$ with equality, and uniquely for Shannon information. The resulting additivity, $H(X,Y) = H(X) + H(Y)$, characterises Shannon information. We can argue by Lemma \[lem:lem4\_2\] that, within the class with fixed marginals, the independence case cannot be uniformly dominated within the ordering $\preceq$. This example shows that if we change the type of entropy, in this case to Tsallis, then the independent case may no longer achieve the minimum. Take $X_1,X_2 = 0,$ with probabilities $$p_{00} = \alpha \beta,\; p_{10} = (1-\alpha)\beta,\; p_{01} = \alpha (1-\beta),\; p_{11} = (1-\alpha)(1-\beta)$$ We can generate all distributions with the same margins with a perturbation $\epsilon$: $$p_{00} = \alpha \beta + \epsilon,\; p_{10} = (1-\alpha)\beta-\epsilon,\; p_{01} = \alpha (1-\beta)-\epsilon,\; p_{11} = (1-\alpha)(1-\beta)+\epsilon,$$ with the restriction that $|\epsilon| < \min(p_{00}, p_{10}, p_{01}, p_{11}).$ Taking the Tsallis entropy with $\gamma = 1$, we find the minimum when $$\epsilon = - \frac{(2\beta-1)(2\alpha-1)}{4},$$ which is zero if at least one of $\alpha$ or $\beta$ is $\frac{1}{2}$, which, interestingly, is a little less restricted than the case in which the distribution must be uniform i.e., all $p_{ij} = \frac{1}{4}$. Note that independence and inverse mixing are closely related. If $X_1 =\{0,1\}$ is a binary random variable with $\mbox{prob}\{X_1=1\} = \alpha$ and $X_2$ has $J$ levels, the one-dimensional DR of $(X_1,X_2)$ is the inverse mixture with mixing parameter $\alpha$. Thinking in terms of a $2 \times J$ table, we combine the top row of probabilities, the distribution of $X_2$ multiplied by $\alpha$, with the bottom row in which they are multiplied by $1-\alpha$. In the continuous case, if the DR pdfs of $X_1$ and $X_2$ are $\tilde{f}_1(z)$ and $\tilde{f}_2(z)$, respectively then, for the joint distribution, we can think of $\tilde{f}_1$ weighting $\tilde{f}_2$ (or vice versa) in a similar way. This leads to the formula which we write informally: $$\tilde{f}_{12} (z) = \int \tilde{f}_1 \left( \frac{z}{\tilde{f}_2^{-1}(x)} \right) dx = \int \tilde{f}_2 \left( \frac{z}{\tilde{f}_1^{-1}(x)} \right) dx.$$ Iterating this operation, we can cover independent random variables in several dimensions and recapture results, such as those in Section \[sec:cont\_major\]. As with Example \[example:6\], care has to be taken in handling supports and limits of integration. The following results enable us to propagate the ordering $\preceq$ via conditional distributions. Consider two pairs of joint distributed random variable $(X_1, X_3)$ and $(X_2,X_3)$ and suppose that the conditional distributions satisfy $X_1\; \vline \; X_3 \preceq X_2 \; \vline \; X_3$, for all values of the conditioning random variable, $X_3$. Then, for the joint distributions, $(X_1,X_3) \preceq (X_2,X_3)$. This follows since, with simple notation, the joint distributions are $f(x_1,x_3) = f(x_1| x_3) f(x_3)$ and $f(x_2,x_3) = f(x_2| x_3) f(x_3)$. By assumption, we have $\tilde{F}(x_1|x_3) \leq \tilde{F}(x_2|x_3)$, for all $x_3$. An inverse mixing with respect to $f(x_3)$, as mentioned above, completes the proof. This result can be understood easily in the discrete case using tables. It says that if there are two tables of probabilities with the same row margins then if every row of one table dominates the corresponding row of the other table then the whole table dominates the other whole table. Volume-contractive mappings --------------------------- We have seen that the area of the support is a key component of studying $\preceq$. For example, in the discrete case, if $n=0$ and $p= (p_1, p_2, p_3,0)$ are our probabilities with $p_1+p_2+p_3 = 1$, we can say we have support size 3. If we then split $p_3$ to form $q = (p_1, p_2, \frac{p_3}{2}, \frac{p_3}{2})$ then $q \preceq p$. In the continuous case, we refer to such an operation as dilation: locally, we have the same amount of density but stretch the support. This is a dilation in the continuous case via a special kind of transformation of the random variable, whose inverse we can call contractive. The result implies that the volume contractible mappings decrease uncertainty. A volume differential invertible mapping $ h: \mathbb R \rightarrow \mathbb R$, $y = h(x)$ will be called a volume-contractive mapping if the determinant of its Jacobian: $J = \vline \max{det} \left\{ \frac{\partial y_i}{\partial x_j}\right\} \vline$ satisfies $ 0 < J \leq 1$ for $x \in \mathbb R$. If $h( \cdot) $ is a volume contractible inverse mapping $\mathbb R \rightarrow \mathbb R$, then, for any random variable $X \sim f_X(x)$, it holds that: $$X \preceq Y = h(X).$$ We give a proof for the one dimensional case and, in addition, assume $f_X(x)$ and $f_Y(y)$ are invertible. Using the slice condition, we want to show that $$\mbox{prob} \{ f_X(X) \geq c\} \geq \mbox{prob} \{ f_Y(Y) \geq c\}.$$ Developing the left hand side, we see that $$\begin{aligned} \{ f_X (X) \geq C \} & \Leftrightarrow & \{X \geq f_X^{-1}(c) \} \\ & \Leftrightarrow & \{Y \geq h(f_X^{-1}(c)) \} \\ & \Leftrightarrow & \{X \geq f_X^{-1}(c) \}.\end{aligned}$$ Computing the density of $Y$ as $$f_Y(y) = |J^{-1}| f_X(h^{-1} (y)),$$ gives $$f_Y^{-1}(c) = h\left(f_X^{-1} ( J c)\right).$$ We thus need to establish whether $h(f_X^{-1} (c)) \geq h( f_X^{-1} ( J c)).$ We see the statement reduces to $c \geq J^{-1} c,$ which holds by assumption. Contractive flows in sensitivity analysis ----------------------------------------- A motivation for the previous subsection was to provide a method of analysis for systems within the general area of uncertainty quantification, one aspect of which is sensitivity analysis: the study of the propagation of variability through systems from input to output. We have seen above how a volume contractive mapping $Y=h(U)$ can decrease uncertainty. This subsection covers a closely related idea: how to show that, for two different inputs, the outputs may have more or less uncertainty. There are several areas of study in which it is hoped to decrease the variability of $Y$ via different types of intervention on the input $U$. Examples include the theory of antithetic variables in classical simulation, stochastic control, portfolio optimisation and robust design [@bates2002optimisation]. More generically, trying different random inputs $U$ is central to Monte Carlo simulation. We represent the intervention on $u$ as a deterministic transformation and we seek a way to reduce the variability of $Y$ by shifting $U$ in some way. Consider the composition: $$y_0 = G(u_0), \quad u_1 = h(u_0), \quad y_1 = G(u_1).$$ For some function intervention on $u$ given by $h(\cdot)$, we express schematically $$\begin{array}{ccc} y_0 & \leftarrow & u_0 \\ \rotatebox{-90}{$\dashrightarrow$} & & \rotatebox{-90}{$\rightarrow$} \\ y_1 & \leftarrow & u_1 \end{array}$$ Here, we are interested in what kind of interventions will result in volume contractive mappings from $y_0$ to $y_1$, induced by the intervention $h(u)$ in the dashed arrow in the diagram. When this holds, we can say that there is less uncertainty about the stochastic output $Y_1$ than about $Y_0$, for any input $U_0$. A note of caution is that the induced function $y_0 \rightarrow y_1$ needs to be properly defined, in which case we can say that $h(u)$ is compatible. It is somewhat easier in explanation when the $u$-space and $y$-space have the same dimension. In addition, as we now see, local developments are easier. It is convenient to express $h(u)$ as a [*flow*]{} of the form: $$\begin{aligned} h(u,\epsilon) = h(u_0) + \epsilon \xi(u_0) +\mbox{O}(\epsilon^2).\end{aligned}$$ In one dimension, $$\begin{aligned} y_1 = G(u_0 + \epsilon \xi(u_0) ) + \mbox{O}(\epsilon^2),\end{aligned}$$ then $$\begin{aligned} \frac{d y_1}{d y_0} & = G'(u_0 + \epsilon \xi (u_0)) \frac{d u_0}{dy_0} + \mbox{O}(\epsilon^2),\\ & = 1 + \epsilon\frac{\xi(u_0) G'' (u_0) + G'(u_0)\xi'(u_0)}{G'(u_0)} + \mbox{O}(\epsilon^2).\end{aligned}$$ Thus, if $\xi(u) = c > 0 $, that is a constant, then $\frac{d y_1}{d y_0} < 1$ (locally) if and only if $$\begin{aligned} \frac{G''(u_0)}{G'(u_0)} = \frac{d}{du_0} \log G'(u_0) < 1.\end{aligned}$$ For the multivariate linear case, we assume that the dimension of the $u$- and $y$- space are the same, namely $n$, for $n \times n$ matrices $A,B$ write: $$\begin{aligned} y_0 = A u_0, \quad u_1 = u_0 + \epsilon B u_0 + \mbox{O}(\epsilon^2),\quad y_1 = Au_1,\end{aligned}$$ so that $$\begin{aligned} y_1 = y_0 + \epsilon A B A^{-1}y_0 + \mbox{O}(\epsilon^2).\end{aligned}$$ Then locally we want $$\begin{aligned} |\mbox{det} (I + \epsilon A B A^{-1})| < 1,\end{aligned}$$ for small $\epsilon$. If $\{\lambda_i\}$ are the eigenvalues of $ABA^{-1}$ then the condition is $$\begin{aligned} | \prod_{i=1}^n (1+ \epsilon \lambda_i)| < 1.\end{aligned}$$ There is a particular problem when the input space and output space have different dimensions. Thus let the $y$-space above be one dimensional and the $u$-space $n$-dimensional. We can write for an $n$-vector $a$ $$y_0 = a^Tu_0, \quad u_1 = u_0 + \epsilon B u_0 + \mbox{O}(\epsilon^2)\quad y_1 = a^Tu_1,$$ so that $$y_1 = (a^T + \epsilon a^T B)u_0 + \mbox{O}(\epsilon^2).$$ Then, when we consider the random input version, and with the additional condition that the components of the random $U_0$ are independent, we have that $Y_0 \prec Y_1 $, when $B^T a \leq 0$ (componentwise). Geometrically, this says that the vector $a$ must lie in the dual cone generated by the columns of $B$. Consider the problem when the input space and output space have different dimensions. We study the quadratic case, whose importance stems from the fact that, under suitable smoothness conditions, all function are locally quadratic. Let $A$ be an $n \times n$ symmetric positive definite matrix and consider the quadratic form $$y = u^TAu.$$ The function $y$ has a minimum at the origin. Starting at the point $u_0$, the natural shift which would contract the output is the direction of steepest descent which is given by the negative of the gradient at $u_0$, namely $$\frac{\partial y}{\partial u}\; \vline_{_{\;u=u_0}} = 2Au.$$ The flow in that direction would be expressed by $$u_1 = u_0 - \epsilon Au_0 +\mbox{O}(\epsilon^2),\quad y_0=u_0^TAu_0,$$ which gives $$\begin{aligned} y_1 & = & u_1^T A u_1 + O(\epsilon^2),\\ & = & y_0 - 2\epsilon u_0^T A^2 u_0 + \mbox{O} (\epsilon^2), \\ & = & u_0^T (A - 2\epsilon A^2) u_0 + \mbox{O}(\epsilon^2). \end{aligned}$$ Taking the spectral decomposition of $A: A= \sum \lambda_i z_i z_i^T$, where the $\{\lambda_i\}$ are non-negative eigenvalues and the $\{z_i\}$ are standard unit length eigenvectors of $n\times 1$ dimension, we have $$y_1 = \sum_{i=1}^n (\lambda_i - 2\epsilon \lambda_i^2) (u_0^Tz_i)^2 + O(\epsilon^2).$$ If we take the input $U_0$ to be a vector of iid standard $N(0,1)$ random variables, then the variables $\{Z_i = (U_0^Tz_i)^{2}\}$ are independent $\chi^2$ random variables with 1 degree of freedom. Then it is straightforward to show that $$\sum_{i=1}^n \lambda_i Z_i \prec \sum_{i=1}^n (\lambda_i - 2\epsilon \lambda_i^2)\; Z_i,$$ so that approximately $Y_0 \prec Y_1$ where $Y_0 = U_0^T A U_0$, $Y_1 = U_1^T A U_1$ and $U_1 = U_0 - \epsilon AU_0 +\mbox{O}(\epsilon^2)$. Algebra for uncertainty {#sec:algebra} ======================= Let us recall some of the notation we have used. The majorisation ordering $\preceq$ is defined by: $$X_1 \preceq X_2 \Leftrightarrow \tilde{F}_1(z) \leq \tilde{F}_2(z)\; \mbox{for all}\; x>0,$$ where $\tilde{F}$ means the (one dimensional) cdf for the decreasing rearrangement density $\tilde{f}$. We may write, equivalently, $\tilde{F}_1(z) \preceq \tilde{F}_2(z).$ Recall also that we can compare distributions in different dimensions via the construction given in Section \[sec:multivariate\]. We have the general concept of inverse mixing $ \tilde{h}_1\; [+] \; \tilde{h}_2$, as in Definition \[def:inversemixing\], applied to invertible functions $ h_1(x)$ and $h_2(x)$, whether or not they are densities. Denote by $\tilde{F}_1 \otimes \tilde{F}_2$ the associated cdf of the density, $$\begin{aligned} \label{eq:x} \frac{1}{2} ( \tilde{f}_1(z) \; [+] \; \tilde{f}_2(z)) = \left(\tilde{h}_1(2z)^{(-1)} + \tilde{h}_2(2z)^{(-1)} \right)^{(-1)}. \end{aligned}$$ The first type of algebra is based on the following definition. \[def:lattice\] For any two DR cdfs $\tilde{F}_1$ and $\tilde{F}_2 (z)$, we define $$\tilde{F}_1(z) \vee \tilde{F}_2 (z) = \max(\tilde{F}_1(z), \tilde{F}_2(z)),$$ $$\tilde{F}_1(z) \wedge \tilde{F}_2 (z) = \min(\tilde{F}_1(z), \tilde{F}_2(z)),$$ which themselves are cdfs. The partial ordering $\preceq$ under the meet and join $\vee$ and $\wedge$ defines a lattice which we refer to as the [*uncertainty lattice*]{}. It is satisfying that the ‘meet’ and ‘join’ which are defined once $\preceq$ is established can be manifested by the max and min of Definition \[def:lattice\]. We stress that, because we can embed a multidimensional distribution with density $f(x)$ into the one dimensional DR with pdf $\tilde{f}(z)$ and cdf $\tilde{F}(z)$, we can claim that the lattice is universal. The inverse mixing cdf $\otimes$ can be combined with $\vee$ (or $\wedge$). To make the notation more appropriate, we will replace $\vee$ by $\oplus$ and then $\oplus$ and $\otimes$ yield a so-called max-plus (also called tropical) algebra [@maclagan2015introduction]. For this to be valid, we need distributivity. \[dist\] We have $$\tilde{F}_3 \otimes ( \tilde{F}_1 \oplus \tilde{F}_2) = (\tilde{F}_3 \otimes \tilde{F}_1) \oplus (\tilde{F}_3 \otimes \tilde{F}_2),$$ where $\tilde{F}_1,\tilde{F}_2$ and $\tilde{F}_3$ are all cdfs for DR where $\tilde{F}_1(z) \oplus \tilde{F}_2 (z) = \max(\tilde{F}_1(z),\tilde{F}_2(z))$ and $\otimes$ is given in Equation (\[eq:x\]). This follows from the useful expression of inverse mixing for cdfs, namely: $$\tilde{F}_1(z) \otimes \tilde{F}_2(z) = \left( \tilde{F}_1(2z)^{(-1)} + \tilde{F}_2(2z)^{(-1)} \right)^{(-1)}.$$ Swapping min for max, we obtain $$\min\left(\tilde{F}_3(2z)^{(-1)} + \tilde{F}_1(2z)^{(-1)}, \tilde{F}_3(2z)^{(-1)} + \tilde{F}_2(2z)^{(-1)} \right) =$$ $$\tilde{F}_3(2z)^{(-1)} + \min \left(\tilde{F}_1(2z)^{(-1)}, \tilde{F}_2(2z)^{(-1)}\right),$$ which is the key step and common to maxplus (tropical) proofs. We can now define a (semi) ring which we shall call the [*uncertainty ring*]{}. We first note that the $\otimes$ unit element will be $0$ and the $\oplus$ unit element will be $-\infty$. We immediately have the issue that, to define the ring, we need to remove the fact that $\tilde{f}$ is a density and $\tilde{F}$ is a non-negative function with $\tilde{F}(z) \rightarrow 1$ as $z \rightarrow \infty$. The polynomials which comprise the ring require powers and monomials.\ Consider the pdf arising from $\tilde{F}_1 \otimes \tilde{F}_1$. This is: $$\begin{aligned} \tilde{f}(z) & = & \left( \tilde{f}(2z)^{(-1)} + \tilde{f}(2z)^{(-1)} \right)^{(-1)}, \\ & = & \frac{1}{2} \tilde{f}\left(\frac{z}{2} \right), \end{aligned}$$ which is the pdf for the scaled random variable $Y= 2 X$ where $X \sim \tilde{f}(z)$. The pdf for $Y$ is $\tilde{F}\left(\frac{z}{2}\right)$. In general the $k$-th $\otimes$ power is $$\otimes^n \tilde{F} (z) = \tilde{F}\left( \frac{z}{n} \right).$$ The intuition of this expression is that increasing powers represent increasing dilation and form a decreasing chain with respect to our $\preceq$ ordering. A monomial with respect to $\otimes$ takes the form $$\prod_{i=1}^{m} \otimes^{\alpha_i} \tilde{F}_i(z).$$ Adjoining the base field $\mathbb R$, and appealing to Lemma \[dist\], we can define a ring of tropical polynomials [@glicksberg1959convolution]. The uncertainty ring is the toric semi ring of non-decreasing, twice-differentiable functions on $[0, \infty)$ on $\otimes$ and $\oplus$ and with $\oplus$ identity as $-\infty$. To obtain proper decreasing densities we need to impose the additional condition that $\tilde{f}(z)$ is decreasing, $\tilde{F}(0) = 0$ and $\tilde{F}(z) \rightarrow 1$ as $z \rightarrow \infty$. Assuming that we have proper pdfs, we summarise the operations that we have: 1. Scalar multiplication $\tilde{F} \rightarrow \beta\tilde{F}$, $ \beta \in \mathbb R$. 2. Inverse mixing $\tilde{F}_1 \otimes \tilde{F}_2$. 3. Maximum and minimum of $\tilde{F}_1, \tilde{F}_2$, denoted $\vee$ and $\wedge$, respectively. Noting that $\vee$ is written $\oplus$, when discussing the ring. We can also define a min-plus algebra and may use $\oplus$. 4. Powers and monomials. 5. Convolution $\tilde{F}_1 * \tilde{F}_2$. This refers to the DR pdf of the sum of independent random variables $X_1 \sim F_1$ and $X_2 \sim F_2$. Further natural developments using ring concepts such as ideals are the subject of further work. In fact, convolutions themselves form a semi-group [@feller2008introduction], but we do not delve into the relationship between our ring and that semi-group. It is instructive to work over the binary field so that we do not have to use full scales from $\mathbb R$, but only $\{0,1\}$. This also has the advantage that, in every polynomial, we have proper pdfs and cdfs. An analogy is Boolean algebra. In Example \[sec6:example\_exp\] we illustrate these operations and the complexity obtained from a single distribution. \[sec6:example\_exp\] Exponential with unit mean. Let $X_1 \sim \exp\{-x\}, $ with $x>0$. We have, $$\begin{aligned} \tilde{F}_1 & = & 1-e^{-z}, \\ \tilde{F}_2 = \tilde{F}_1 * \tilde{F}_1 & = & 1 - (1+\sqrt{2z}) e^{-\sqrt{2z}}, \\ \tilde{F}_3 = \otimes^2 \tilde{F}_1 & = & \frac{z}{2} \left( 1- e^{-z} \right), \\ \tilde{F}_4 = \otimes^2 \tilde{F}_2 & = & \frac{1}{2} \left( z- e^{-\sqrt{2z}} \right) \\ \tilde{F}_5 = \tilde{F}_1 \otimes \tilde{F}_2 & = & \frac{z}{2} - \left( \frac{z}{2} + \frac{z^{\frac{3}{2}}}{\sqrt{2}} \right) e^{-\sqrt{2z}}.\end{aligned}$$ For the uncertainty lattice in Definition \[def:lattice\], we have $$\tilde{F}_4 \preceq \tilde{F}_5 \preceq \tilde{F}_2 \preceq \tilde{F}_1, \quad \text{and} \quad \tilde{F}_5 \preceq \tilde{F}_3 \preceq \tilde{F}_1,$$ and neither $\tilde{F}_2$ or $\tilde{F}_3$ dominate the other. Under $\vee$ and $\wedge$ we can include $ \tilde{F}_3 \vee \tilde{F}_2$ and $\tilde{F}_3 \wedge \tilde{F}_2,$ and show that $$\tilde{F}_4 \preceq \tilde{F}_3 \wedge \tilde{F}_2 \preceq \tilde{F}_3 \vee \tilde{F}_2 \preceq \tilde{F}_1.$$ Algebra and entropies --------------------- Given the equivalence of the majorisation in Section \[sec:cont\_major\], we study how the above structures affect the manipulation of uncertainty measured by a single metric, $$H(f(x)) = \int h(f(x)) dx,$$ for some convex function $h( \cdot)$. Since $$H(f(x)) = \int_{0}^{\infty} h(\tilde{f}(z)) dz,$$ we can, without loss of generality, consider $H$ as a functional of the DR with the advantage that the operations $\otimes,\oplus$ can be applied. The following, then, is a collection of operations and results, which we can claim as a toolbox for handling uncertainty, which can be applied to any uncertainty ($-H$), in this class. We omit the proofs. 1. $\tilde{f}_1 \preceq \tilde{f_2} \Rightarrow H(\tilde{f}_1) \leq H(\tilde{f}_2).$ 2. $H\left(\tilde{f}_1 \otimes \tilde{f}_2 \right) = H\left(\tilde{\frac{1}{2}f}_1\right) + H\left(\frac{1}{2}\tilde{f}_2\right).$ 3. $H\left(\tilde{f}_1 \oplus \tilde{f}_2 \right) \geq \max \left\{H\left(\tilde{f}_1\right), H\left(\tilde{f}_2\right) \right\}.$ 4. $H\left(\tilde{f}_1 \otimes (\tilde{f}_2 + \tilde{f}_3) \right) \geq H\left(\frac{1}{2}\tilde{f}_1\right) + \max\left\{H\left(\frac{1}{2}\tilde{f}_2\right), H\left(\frac{1}{2}\tilde{f}_3\right)\right\}.$ Uncertainty toolbox: future scenarios ------------------------------------- We present two practical situations in which the results in Section \[sec:algebra\] are employed to handle the uncertainty in practical situations, by combining them in two different ways. Suppose there is initially to be $i=1,2$ races with a different number of horses $n_i$ ranked in order of their probability of winning (in the mind of a punter or bookmaker): $$p_{(i,1)} \geq \cdots \geq p_{(i,n_i)},$$ such that $\sum_{j=1}^{n_i}p_{i,j}=1, i=1,2$. If the two sets of horses should be combined into a single race, the issue then is how to combine the probabilities. The simplest approach is to divide each probability by two and combine all the probabilities, ranking them in the process, i.e, $\tilde{F}_1 \otimes \tilde{F}_2$, that is inverse mixing with $\alpha = \frac{1}{2}$. One can imagine some effect which may lead to having $\alpha \neq \frac{1}{2}$ such as the track being wet and not suited to one set of horses. Consider the case in which we have a single race (say race 1) with two punters who rank the horses in the same order, but with different probabilities, i.e., $p_{(1,1)}\geq \ldots \geq p_{(1,n_1)}$ and $q_{(1,1)} \geq \ldots \geq \ldots q_{(1,n_1)}$, respectively. To define a joint betting strategy, a set of odds combining each of their own odds could take different approaches: an optimistic, more certain, approach with $\max(\tilde{F}_1(z),\tilde{F}_2(z))$ or a more pessimistic, uncertain, approach with $\min(\tilde{F}_1(z),\tilde{F}_2(z))$. We note that the argument in the last paragraph is predicated on the two punters’ initial rank order being the same. If not, there is a danger that the same horse may appear twice in the min or max ordering. The min or max may then refer to a kind of hypothetical race. Nonetheless, we suggest that they are useful notionally. The same issue arises if one considers an average of the two actual probabilities, direct mixing, $\frac{1}{2}(p_{(1,1)} +q_{(1,1)} )\geq \ldots \geq\frac{1}{2}(p_{(1,n_1)}+q_{(1, n_1)}),$ which corresponds to taking $\frac{1}{2}(\tilde{F}_1(z) + \tilde{F}_2(z))$. Now suppose there are two sets of horses of size $n_1$ and $n_2$ and two punters. We see then that the toolbox provides various ways to combine uncertainties within rows and columns. With obvious notation, we have punter 1 punter 2 -------- ------------------- ------------------ race 1 $\tilde{F}_{11}$ $\tilde{F}_{21}$ race 2 $ \tilde{F}_{12}$ $\tilde{F}_{22}$ \[commiteeexample\] Consider a scenario with two committees, each with the same number of members. Each committee covers a different area of oversight, for example different technologies to solve a particular problem, say ‘electricity’ and ‘gas’. We assume that, within each committee, there is a common agreement about a range of possible future scenarios in the committee’s area. We define two future scenario types: 1. an active situation: to choose between the two technologies on some grounds, in which the two committees’ assessments would not be combined. 2. a passive situation: where only the probability of a particular technology being used can be assessed, which may be a consequence of unforeseen events. In the latter case, the inverse mixing is a way to combine the uncertainties into an overall assessment. The actual future would be one gas-based alternative or one electricity alternative, the “horse” that won the combined race. The difficulties that may arise, as pointed out already, are that the private initial order of members of the [*same*]{} committee may not be the same as one familiar in subjects such as choice theory and rater assessment. Note that independence between committees tends to increase uncertainty, and, in our development, the unidimensional DR is equivalent to inverse mixing. This shows, heuristically, that, when two committees act independently, there is more, rather than less, uncertainty. Empirical decreasing rearrangements {#sec:empirical} =================================== We extend the applications of DR to analyse a data set collected from an experiment in a large number of trials or from a product of computer simulations. We present two approaches for deriving the empirical DR and its associated cdf from a data set. In Section \[subsec:Discrete\_Climate\], we assess the uncertainties associated with climate projections, in which we transform continuous variables into discrete ones by grouping values, which is easy to perform in two dimensions. In Section \[subsec:Cont\_Heat\], algorithms are presented to obtain approximations for $\tilde{f}(z)$ and $\tilde{F}(z)$. This allows us to perform majorisation over data sets in higher dimensions, which we demonstrate analysing the risk associated with individual infrastructure decisions in an energy system. Discrete Majorisation - climate projections {#subsec:Discrete_Climate} ------------------------------------------- The UKCP18 climate projections [@UKCP18] consider four different scenarios of greenhouse gas concentrations in the atmosphere, called Representative Concentration Pathways (RCP). We consider variables over the 12km gridboxes that cover the UK. UKCP18 uses ensemble methods in which the model is run multiple times with slightly differing initial conditions and parameter values to account for observational and parametric uncertainty. We consider two variables: 1. Increase in mean air temperature at a height of 1.5 metres, 2. Percentage increase in precipitation, where each variable is relative to the baseline period of 1981-2010. The projections correspond to mean daily values over the period from 2050 to 2079. The data, illustrated in Figure \[fig:climate\_scatter\], is discretised by dividing each variable into ranges and counting the number of ensemble members that fall into each category in the two dimensions. The temperature anomaly is divided into five categories, whilst the increase in precipitation is divided into four categories, therefore an ensemble member falls into one of 20 categories. ![Scatterplot showing how ensemble members are categorised according to the value each variable. Each point represents an ensemble member and each colour represents a different RCP.[]{data-label="fig:climate_scatter"}](climate_scatter.png) We present the joint distribution for two contrasting scenarios for RCP2.6 and RCP8.5, of temperature anomaly and percentage change in precipitation is shown in Table \[table:clim\_dist\]. The ordered probabilities for each RCP are shown in Figure \[fig:climate\_example\_pdf\_cdf1\]. We use this to obtain empirical cdfs of DR together with the maximum and minimum of the cdfs depicted in the left and central panel plots in Figure \[fig:climate\_example\_pdf\_cdf2\]. We observe that the empirical cdf for RCP2.6 lies above that of RCP8.5. The uncertainty is therefore related to the level of assumed greenhouse gas concentration in the atmosphere. In particular, RCP2.6 carries the lowest level of uncertainty among the considered scenarios since its cdf corresponds to $\tilde{F}_1(z) \lor\tilde{F}_2(z)\lor\tilde{F}_3(z)\lor\tilde{F}_4(z)$, where the subscript indicates the scenario. In contrast, RCP8.5 carries the most uncertainty, since its cdf corresponds to $\tilde{F}_1(z) \land\tilde{F}_2(z)\land\tilde{F}_3(z)\land\tilde{F}_4(z)$. [|ll|&gt;cc&gt;cc&gt;cc&gt;cc&gt;cc|]{} & &\ & & & & & &\ & [**$<$0%**]{} & 0.01 & 0.00 & 0.18 & 0.02 & 0.17 & 0.09 & 0.01 & 0.10 & 0.00 & 0.04\ & [**0-5%**]{} & 0.01 & 0.00 & 0.22 & 0.03 & 0.28 & 0.16 & 0.04 & 0.20 & 0.00 & 0.08\ & [**5-10%**]{} & 0.00 & 0.00 & 0.02 & 0.01 & 0.05 & 0.07 & 0.01 & 0.11 & 0.00 & 0.06\ & [**$>$10%**]{} & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.01 & 0.00 & 0.02 & 0.00 & 0.02\ ![Ordered probabilities under each RCP.[]{data-label="fig:climate_example_pdf_cdf1"}](climate_example_pdf_cdf1.pdf) ![*Left panel*: Empirical cdf for the DR for each scenario. *Central panel*: the representation of $\max(\tilde{F}_1(z), \tilde{F}_2(z), \tilde{F}_3(z), \tilde{F}_4(z))$ and $\min(\tilde{F}_1(z), \tilde{F}_2(z), \tilde{F}_3(z), \tilde{F}_4(z))$. *Right panel*: cdf of inverse mixing with equal weights specified for each scenario.[]{data-label="fig:climate_example_pdf_cdf2"}](climate_example_pdf_cdf2.pdf){width="100.00000%"} We can apply inverse mixing, described in Section \[subsec:InverseMix\], to combine the uncertainties in the four scenarios. For the discrete case, we specify an equal weighting for each pdf, and we illustrate cdf of the inverse mixing in the right panel in Figure \[fig:climate\_example\_pdf\_cdf2\]. Majorisation in higher dimensions {#subsec:Cont_Heat} --------------------------------- We denote a data set $x_{ij}, i=1, \dots, m$ and $j=1, \dots, n$, where $m$ and $n$ correspond to the total number of data points and dimensions respectively. To obtain the DR, we require the density function values to construct the distribution function $m(y)$ (see Section \[sec:cont\_major\]). We assume that the observed data are a sample from a population with unknown pdf $f_X(x_1, \dots, x_n)$, from which we estimate the pdf $\hat{f}_X(x_1, \dots, x_n)$. In our examples, we employ kernel density estimation to obtain $\hat{f}_X(x_1, \dots, x_n)$ using the `ks` package in `R`, which automatically selects the bandwidth parameters [@ks2020]. Algorithm 1 describes the approach adopted in this paper to obtain empirical DRs $\tilde{f}_{\hat{f}}(z)$. We note from Definition \[drdefn\] that obtaining the DR is a two-stage process. At the first stage, we are required to obtain a measure (distribution) function $m(y)$. For instance, in 1D, a distribution function $m(y)$ returns the length of the intervals on the $x$-axis at which $f(x)\geq y$. At the second stage, we employ the measure function to derive the DR. Based on data $x_{ij}\in R, i=1, \dots, m$ and $j=1, \dots, n$, fit a pdf $\hat{f}_X(x_1, \dots, x_n)$ using kernel density estimation Produce a uniform and/or space-filling set $S$ of size $N$ across the input space $R$, with “sites” $s\in S$ Plot the estimated measure function values, $m_{\hat{f}}(y)$ against $y$ Swap the abscissa with ordinate, so that $\tilde{f}_{\hat{f}}(z)$ and $z$ correspond to $y$ and $m_{\hat{f}}(y)$. As part of this algorithm, we employ Monte Carlo integration [@Fok1989] to estimate the volume of domain $S_y$ to derive the measure function $m_{\hat{f}}(y)$. In particular, [@Fok1989] proposed specifying another domain $R$ (a hypercube or a hyperplane) of known volume $\text{Vol}(R)$, such that $ S_y\in R. $ The ratio of two volumes, $p=\text{Vol}(S_y)/\text{Vol}(R)$, and the volume $\text{Vol}(S_y)$ are estimated by $$\hat{p}=N_y/N\quad \text{and}\quad \hat{\text{Vol}}(S_y)=\hat{p}\text{Vol}(R).$$ We employ Algorithm 1 on a bivariate data set. We start by generating a random sample of size $m=200$, i.e. $x_{ij}, i=1, \dots, 200$ and $j=1, 2$, from a standard bivariate normal distribution, since we have the closed form expression for DR of a standard 2-dimensional normally distributed random variable $X$: $$\label{eq:DR2D} \tilde{f}(z)=\frac{1}{2\pi}\exp\Big\{-\frac{1}{2}\frac{z}{\pi} \Big\}.$$ To perform the algorithm, we produce a uniform sample of points of size $N=2500$ across the domain $R=[-5, 5]\times [-5, 5]$ of $\text{Vol}(R)=10^2$. Figure \[fig:EmpDRFinal1\] demonstrates the implementation of Algorithm 1 as well as comparing $\tilde{f}_{\hat{f}(z)}$ to $\tilde{f}(z)$. In the left panel, we depict the estimated values of the distribution function $m_{\hat{f}}(y)$ against $y$. Note that the smoothness of the estimated distribution function depends on $M$, i.e. we expect to obtain a smooth representation of the distribution function with a large value of $M$ (cutoffs in density). We observe that the empirical DR (red dashed line) is overlapping with the DR in equation (\[eq:DR2D\]) (blue solid line) in the right panel. ![*Left panel*: A plot of the estimated measure function $m_{\hat{f}}(y)$ against $y$. *Right panel*: A DR plot where the blue solid line and the red dashed line correspond to $\tilde{f}(z)$ and $\tilde{f}_{\hat{f}}(z)$.[]{data-label="fig:EmpDRFinal1"}](EmpDRFinal.pdf){width=".8\textwidth"} We present Algorithm 2 for obtaining an empirical cdf of the DR, an approximation to $\tilde{F}(z)$, denoted as $\tilde{F}_{\hat{f}}(z)$. Specify an equally spaced vector $\boldsymbol{z}^*=(z_1^*, z_2^*, \dots, z_l^*)$ Fit a linear interpolator (spline) through $\{z_i, \tilde{f}_{\hat{f}}(z_i)\}_{i=1}^M$ (these values were derived in Algorithm 1) to obtain values of $\tilde{f}_{\hat{f}}(z_i^*), i=1, \dots, l$ Plot $\tilde{F}_{\hat{f}}(z^*)$ against $z^*$. The weighting of computed probabilities by $$\frac{1}{\sum_{k=1}^{n-1}P(z_k<z<z_{k+1})},$$ in Algorithm 2 comes from the assumption that $z$ is upper bounded and we can only compute probabilities at the values specified in $\boldsymbol{z}^*$. Therefore, we expect $\sum_{k=1}^{n-1}P(z_k<z<z_{k+1})=1$. However, we tend to observe this sum to be slightly less than one due to errors introduced by numerical integration. Our proposed weighting is similar to normalisation performed as part of the construction of a histogram. We proceed to demonstrate the implementation of Algorithm 2 in Figure \[fig:CDFDR2D\] on the bivariate data set used previously. We have a closed form expression for $\tilde{F}(z)$ of a standard 2-dimensional normally distributed random variable $X$: $$\label{eq:CDF_DR2D} \tilde{F}(z)=1-\exp\Big\{-\frac{z}{2\pi} \Big\}.$$ ![*Left panel*: binned probability representation of the empirical DR, $\tilde{f}_{\hat{f}}(z)$, obtained as part of Algorithm 2. *Right panel*: empirical cdf of the DR, $\tilde{F}_{\hat{f}}(z)$ obtained from Algorithm 2. The blue solid line and red dashed line correspond to $\tilde{F}(z)$ and $\tilde{F}_{\hat{f}}(z)$ respectively.[]{data-label="fig:CDFDR2D"}](CDFModifiedScheme2D.pdf){width=".75\textwidth"} Based on the right panel in Figure \[fig:CDFDR2D\], we conclude that by employing Algorithm 2, the empirical cdf $\tilde{F}_{\hat{f}}(z)$ is an accurate representation of $\tilde{F}(z)$. ![Empirical cdf of DRs obtained for random samples from the standard normal distribution with $n=1, \dots, 4$ (dimension).[]{data-label="fig:EmpiricalCDF"}](EmpiricalCDF.png){width="50.00000%"} We used Algorithms 1 and 2 to construct $\tilde{F}_{\hat{f}}(z)$ in Figure \[fig:EmpiricalCDF\] for a random sample from the multivariate standard normal with varying dimensions. We observe that the empirical cdf of the DR based on a sample from the univariate normal distribution majorises the remaining empirical cdfs. This is an expected result and confirms our expectation that uncertainty increases as we increase the number of dimensions. The accuracy of the representation of the DR is sensitive to the number of data points in the sample, $m$, while the smoothness of DR depends on $M$. ### District heating example {#subsubsec:DHE} We compare the uncertainty associated with three potential design options for supplying heat in a model based on a system in Brunswick, Germany [@REUSEHEAT]. District heating networks allow heat from a centralised source to be distributed to buildings through a network of insulated pipes [@werner2013district]. This allows for a range of potential sources of heat to be connected and, in recent years, the idea of ‘reusing’ excess heat produced by nearby sources such as factories has gained traction as part of efforts to decarbonise the energy sector. This is done by using excess heat to heat water which is then distributed through the network. Traditionally, high temperature sources such as from heavy industry have been used, there has been increased interest in recent years in low temperature sources such as data centres, metro systems and sewage which require electric heat pumps to ‘upgrade’ the temperature before being suitable for use in the system. The Brunswick case is a demonstrator on the EU funded ReUseHeat project [@REUSEHEAT] that aims to demonstrate the use of low temperature sources of heat for use in district heating networks. The city’s existing district heating network is powered by a Combined Heat and Power (CHP) plant, which uses natural gas as a fuel and outputs both heat for use in the network and electricity. The network in the newly constructed area of interest will be connected to the CHP and, in addition, there is an option to use excess heat from a data centre to provide at least some of the heat to the district. There are therefore three potential design options to be considered and these are shown in Table \[tab:Design\]. The question of interest is then which design option should be chosen, taking into consideration both cost and carbon emissions. Although not used in actual decision making for the city or in the ReUseHeat project itself, a simple model was described in [@Volodina2020] which outputs both Net Present Cost (NPC) in € and $\text{CO}_2$-equivalent emissions (in metric tonnes). We use these simulations to demonstrate majorisation as a tool for decision making. Local and global sensitivity analysis was performed for each design option by varying a number of inputs to the model, resulting in a total of 81 simulations. In addition, three scenarios were considered and these are shown in Table \[tab:Scenarios\], further details of which are given in [@Volodina2020]. In Figure \[fig:ScatterPlot\], we illustrate the outputs of the model under sensitivity analysis by plotting emissions against NPC for each of the three design options and under each scenario. The highest level of emissions comes from design option 1 due to the use of natural gas whilst design option 3 is shown to be the most environmentally friendly. However, there is an inverse relationship between carbon emissions and cost with design option 3 being the most expensive. ![Net Present Costs against carbon emissions for all three design options under the three scenarios.[]{data-label="fig:ScatterPlot"}](ScatterPlot.png){width="70.00000%"} We demonstrate majorisation and DR in the context of the model outputs under the three scenarios. In particular, we obtain $\tilde{f}_{\hat{f}}(z)$ and $\tilde{F}_{\hat{f}}(z)$ based on the distribution of points under the sensitivity analysis for each design option and under each scenario by employing Algorithms 1 and 2. To apply equal importance to both outputs, we scale the data on $[0, 1]$ and generate a uniform set $S$ across $[0, 1]\times [0, 1]$ of size $N=2500$. To produce a smooth representation of the DR and its cdf, we use a value of $M=5000$, which corresponds to the number of cutoffs in the density values. ----------------- ------------------------------- **Design type** **Description** Design option 1 Combined Heat and Power (CHP) Design option 2 CHP and Heat Pump Design option 3 Heat Pump ----------------- ------------------------------- : Description of design options in the District Heating study [@Volodina2020]. \[tab:Design\] -------------- ---------------------- --------------------- ------------------------------------------ **Scenario** **Emission Penalty** **Consumer demand** **Commodity prices** Green 100€/metric tonne -1% annual change $\uparrow$ gas, $\downarrow$ electricity Neutral 40€/metric tonne small fluctuations small fluctuations Market no penalty +1% annual change $\downarrow$ gas, $\uparrow$ electricity -------------- ---------------------- --------------------- ------------------------------------------ : Description of scenarios in the District Heating study [@Volodina2020]. \[tab:Scenarios\] ![Empirical cdf for decreasing rearrangements $\tilde{F}_{\hat{f}}(z)$ for (i) all three design options plotted together for each individual scenario (*first row*), (ii) all three scenarios plotted together for each individual design option (*second row*).[]{data-label="fig:CDF_HeatExample"}](ScenarioPlotHeat.pdf "fig:"){width="100.00000%"} ![Empirical cdf for decreasing rearrangements $\tilde{F}_{\hat{f}}(z)$ for (i) all three design options plotted together for each individual scenario (*first row*), (ii) all three scenarios plotted together for each individual design option (*second row*).[]{data-label="fig:CDF_HeatExample"}](DesignPlotHeat.pdf "fig:"){width="100.00000%"} Plots of the empirical cdfs of the DR $\tilde{F}_{\hat{f}}(z)$ are shown in Figure \[fig:CDF\_HeatExample\]. In the first row, the cdfs obtained for all three design options are considered together under individual scenarios. A feature here is that, under the green and neutral scenarios, the empirical cdf for design option 3 lies above that for design option 2, which lies above that for design option 1 whilst, under the market scenario, the ordering of the empirical cdfs changes and the empirical cdf for design option 1 lies above that for design option 2. We conclude that under all three scenarios, the (unknown) distribution function associated with design option 3 majorises the cdfs for both design options 1 and 2. We therefore consider that, for the outputs of interest, design option 3 is less uncertain than the alternatives. The second row of plots shows empirical cdfs for each scenario plotted together under individual design options. This gives a slightly different view, allowing us to compare how the uncertainty under each design option varies under the different scenarios. For example, under design option 1, there is clear ordering of the cdfs which implies that the Market scenario is less uncertain than the Neutral scenario which, in turn, is less uncertain than the Green scenario. We now demonstrate the uncertainty tools from Section \[sec:algebra\] in order to combine the uncertainty under different scenarios and produce orderings of design options. In particular, under each design option, we find the minimum of the empirical cdfs associated with individual scenarios to obtain an approximation to $\tilde{F}_1(z)\lor\tilde{F}_2(z)\lor\tilde{F}_3(z)$. This is shown in the left panel of Figure \[fig:maxminDR\] and can be considered to represent an ‘optimistic’ approach. We find that design option 3 majorises the other design options. ![*Left panel*: representation of $\max(\tilde{F}_1(z), \tilde{F}_2(z), \tilde{F}_3(z))$. *Right panel*: representation of $\min(\tilde{F}_1(z), \tilde{F}_2(z), \tilde{F}_3(z))$.[]{data-label="fig:maxminDR"}](maxminDR.pdf){width=".7\textwidth"} We also produce an approximation to $\tilde{F}_1(z)\land\tilde{F}_2(z)\land\tilde{F}_3(z)$, which corresponds to a ‘pessimistic’ approach. The results are shown in the right panel of Figure \[fig:maxminDR\] in which we obtain the maximum of the empirical cdfs associated with individual scenarios. In this case, we observe a clear ordering between design options: design option 3 majorises design option 2, which majorises design option 1. Under both the pessimistic and optimistic outlooks, we therefore conclude that design option 3 is less uncertain than the two alternatives. ![cdfs from inverse mixing with different weightings on each scenario: *Left panel*: equal weights on each scenario. *Central panel*: $\alpha_{G}=0.7$, $\alpha_{N}=0.2$ and $\alpha_{M}=0.1$. *Right panel*: $\alpha_{G}=0.05$, $\alpha_{N}=0.05$ and $\alpha_{M}=0.9$[]{data-label="fig:HeatInverseMixing"}](HeatInverseMixing.pdf){width="100.00000%"} We employ inverse mixing to combine the uncertainty associated with all three scenarios under individual design options. This is done by estimating probabilities from the empirical DR, by Algorithm 2 and ordering the probabilities. We consider the defined weights in inverse mixing as probabilities of the occurrence of each scenario. Let $\alpha_{G}, \alpha_{N}$ and $\alpha_{M}$ be the weights applied to the Green, Neutral and Market scenarios, respectively. We consider three different cases for the weights: (i) equal weights, (ii) $\alpha_G=0.7$, $\alpha_M=0.15$ and $\alpha_N=0.15$ and (iii) $\alpha_G=0.05$, $\alpha_M=0.9$ and $\alpha_N=0.05$. The results of the inverse mixing in the three cases are shown in the left, middle, and right panels of Figure \[fig:HeatInverseMixing\], respectively. In cases (i) and (ii), there are clear orderings in which the cdf of design option 3 lies above the cdf of design option 2 which lies above that of design option 1. In case (iii), however, there is no ordering between the empirical cdfs. In particular, the cdfs for design options 1 and 2 cross. However, the pdf associated with design option 3 majorises the pdfs for both design options 1 and 2 and we conclude that design option 3 is the least risky option in all three cases. It is important to note that, whilst the above results provide useful guidance for comparing uncertainty, the uncertainty is only one aspect of such decisions and one would want to take into account the actual costs and carbon emissions (rather than just their variability) in each case. However, here we have demonstrated majorisation to be an intuitive approach to comparing uncertainty and ultimately aiding informed decisions in such settings. Concluding remarks {#sec:conclusion} ================== The concept of uncertainty is the subject of much discussion, particularly at the technical interface between scientific modelling and statistics. It is our contention that uncertainty is close in spirit to entropy, but that restriction to a limited definition of entropy can be lifted. The fact that there are different types points to the existence of a wider framework which may then widen the scope of uncertainty. The idea presented is that a candidate for a wider framework is a stochastic ordering under for which most, if not all, types of entropy are order preserving. Here, we suggest majorisation, which only compares the rank order of probability mass, continuous or discrete. We have shown that any two distributions can be compared, and consider this to be a principal contribution of the paper. We demonstrated this approach to assess the uncertainty in applications of climate projections and energy systems.

--- abstract: 'Given a directed graph $D=(V,A)$ we define its intersection graph $I(D)=(A,E)$ to be the graph having $A$ as a node-set and two nodes of $I(D)$ are adjacent if their corresponding arcs share a common node that is the tail of at least one of these arcs. We call these graphs facility location graphs since they arise from the classical uncapacitated facility location problem. In this paper we show that facility location graphs are hard to recognize and they are easy to recognize when the underlying graph is triangle-free. We also determine the complexity of the vertex coloring, the stable set and the facility location problems on that class.' author: - Mourad Baïou - Laurent Beaudou - Zhentao Li - Vincent Limouzy bibliography: - 'paper-FLG.bib' title: On a class of intersection graphs --- Introduction ============ In this paper we study the following class of intersection graphs. Given a directed graph $D=(V,A)$, we denote by $I(D)=(A,E)$ the [*intersection graph of $D$*]{} defined as follows: - the node-set of $I(D)$ is the arc-set of $D$, - two nodes $a=(u,v)$ and $b=(w,t)$ of $I(D)$ are adjacent if one of the following holds: $u=w$ or $v=w$ or $t=u$ or $(u,v)=(t,w)$ (see Figure \[adjac\]). \(a) at (0,0); (b) at ($ (a) + (2,0) $); (d) at ($ (a) + (6,0) $); (e) at ($ (d) + (0,-1) $); (c) at ($ (d) + (0,1) $); (g) at ($ (d) + (2,0) $); (h) at ($ (g) + (0,-1) $); (f) at ($ (g) + (0,1) $); (i) at ($ (g) + (3,.5) $); (j) at ($ (i) + (-.75,-1) $); (k) at ($ (i) + (.75,-1) $); (l) at ($ (i) + (2,.5) $); (m) at ($ (l) + (0,-2) $); (n) at ($ (c) + (1,1) $); (o) at ($ (i) + (0,1.5) $); (p) at ($ (l) + (0,1) $); (q) at ($ (a) + (1,-3) $); (r) at ($ (i) + (0,-3.5) $); (a1) at ($ (a) + (2,0.5) $); (b1) at ($ (a1) + (-.75,-1) $); (c1) at ($ (a1) + (+.75,-1) $); (l1) at ($ (g) + (6,0) $); (m1) at ($ (l1) + (-2,0) $); (a1) circle (1.5pt); (b1) circle (1.5pt); (c1) circle (1.5pt); \(c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (f) circle (1.5pt); (g) circle (1.5pt); (h) circle (1.5pt); (l) circle (1.5pt); (m) circle (1.5pt); \(e) – (d) node\[midway, left\] [$a$]{}; (d) – (c) node\[midway, left\] [$b$]{}; \(f) – (g) node\[midway, left\] [$b$]{}; (g) – (h) node\[midway, left\] [$a$]{}; (a1) – (b1) node\[midway, left\] [$a$]{}; (a1) – (c1) node\[midway, right\] [$b$]{}; \(l) to\[bend right\] (m); (l1) node\[left\] [$a$]{}; (m1) node\[right\] [$b$]{}; (m) to\[bend right\] (l) ; We focus on two aspects: the recognition of these intersection graphs and some combinatorial optimization problem in this class. De Simone and Mannino [@Simone] considered the recognition problem and provided a characterization of these graphs based on the structure of the (directed) neighborhood of a vertex. Unfortunately this characterization does not yield a polynomial time recognition algorithm. Intersection graphs we consider arise from the [*uncapacitated facility location problem*]{} (UFLP) defined as follows. We are given a directed graph $D=(V,A)$, costs $f(v)$ of opening a facility at node $v$ and cost $c(u,v)$ of assigning $v$ to $u$ (for each $(u,v) \in A$). We wish to select a subset of facilities to open and an assignment of each remaining nodes to a selected facility so as to minimize the cost of opening the selected facilities plus the cost of arcs used for assignment. This problem can be formulated as a linear integer program as follows. $$\mbox{min } \sum_{(u,v)\in A}c(u,v)x(u,v)+\sum_{v\in V}f(v)y(v)$$ $$\left\{ \begin{aligned} \sum_{(u,v)\in A}x(u,v) + y(u) = 1 \quad & \forall u\in V,\\ x(u,v)\leq y(v) \quad & \forall (u,v)\in A,\\ x(u,v)\geq 0 \quad & \forall (u,v)\in A, \\ y(v)\geq 0 \quad & \forall v\in V,\\ x(u,v)\in \{0,1\} \quad & \forall (u,v)\in A,\\ y(v)\in \{0,1\} \quad & \forall v\in V. \end{aligned} \right.$$ If we remove the variables $y(v)$ for all $v$ from the formulation above, we get $$\mbox{min } \sum_{(u,v)\in A}(c(u,v)-f(u))x(u,v)+\sum_{v\in V}f(v)$$ $$\left\{ \begin{aligned} \sum_{(u,v)\in A}x(u,v) \leq 1 \quad & \forall u\in V,\\ x(u,v)+\sum_{(v,w)\in A}x(v,w) \leq 1 \quad & \forall (u,v)\in A,\\ x(u,v)\geq 0 \quad & \forall (u,v)\in A,\\ x(u,v)\in \{0,1\} \quad & \forall (u,v)\in A. \end{aligned} \right.$$ This is exactly the maximal clique formulation of the [*maximum stable set problem*]{} associated with $I(D)$, where the weight of each node $(u,v)$ of $I(D)$ is $f(u)-c(u,v)$. This correspondence is well known in the literature (see in [@AVS; @CorTh; @Simone]). We may consider several combinatorial optimization problems on directed graph that may be reduce to the maximum stable set problem on an undirected graph. For example in [@Chv-Eben], Chvátal and Ebenegger reduce the max cut problem in a directed graph $D=(V,A)$ to the maximum stable set problem in the following intersection graph called the [ *line graph of a directed graph*]{}: we assign a node to each arc $a\in A$ and two nodes are adjacent if the head of one (corresponding) arc is the tail of the other. They prove that recognizing such graphs is <span style="font-variant:small-caps;">np</span>-complete. Balas [@Balas] considered the asymmetric assignment problem. He defined an intersection graph of a directed graph $D$ where nodes are arcs of $D$ and two nodes are adjacent if the two corresponding arcs have the same tail, the same head or the same extremities without being parallel. Balas uses this correspondence to develop new facets for the asymmetric assignment polytope. We may generalize the notion of line graphs to directed graphs in many ways. The simplest involves deciding 1. if arcs that share a head are adjacent, 2. if arcs that share a tail are adjacent, and 3. if two arcs are adjacent if the head of one arc is the tail of the other. It is not too difficult to show the recognition problem is easy if we choose non-adjacency for (3). Choosing non-adjacency for (3) means that we could separate all vertices $v$ of a digraph $D$ into two vertices (one for all arcs entering that vertex and one for all arcs leaving it) and the line graph of the resulting digraph $D'$ is the same as the line graph of $D$. Furthermore, the line graph of $D'$ is the line graph of its underlying digraph. So all classes obtained by choosing non-adjacency for (3) are easy to recognize as it simply involves recognizing if a line graph is bipartite [@beineke_1970] (where some sides of the bipartition are possibly forced to have degree 1 from out choice of (1) and (2)). So suppose arcs of type (3) are adjacent. Choosing adjacency for (1) and (2) gives the line graphs of the underlying undirected graph, and these are easy to recognize [@beineke_1970]. Choosing non-adjacency for both (1) and (2) leads to the line graphs defined by Chvátal and Ebenegger and it is <span style="font-variant:small-caps;">np</span>-complete to recognize them [@Chv-Eben]. And picking exactly one of (1) and (2) to be adjacent and non-adjacency for the other leads to the same class of graphs (as we can simply reverse all arcs of a digraph before taking its line graph) and we wish to determine the complexity of recognizing this very last class. Finally, note that since the stable set problem in our class is equivalent to the facility location problem, one may use all the material developed for facility location problem to solve the stable set problem in these graphs. It is well known that in practice the facility location problem may be solved efficiently via several approaches: polyhedra, approximation algorithms and heuristics. This paper is organized as follows. Section \[def\] contains some basic definition and notations. Other definitions and notations will be given when needed. In section \[RLFG\], we show that facility location graphs are hard to recognize and in Section \[RTFLG\] we show that the subclass of triangle-free facility location graphs are recognizable in polynomial time. Section \[RP\] is devoted to some combinatorial optimization problem in facility location graphs. In particlular we show that the maximum stable set problem remains <span style="font-variant:small-caps;">np</span>-complete in triangle-free facility location graphs but the vertex coloring problem is solvable in polynomial time in this class. We also discuss the facility location problem and show it is <span style="font-variant:small-caps;">np</span>-complete in some restricted class of graphs. We provide concluding remarks in Section \[conc\]. Definitions and notations {#def} ========================= Let $G$ be an undirected graph, we say that $G$ is a [*facility location*]{} (FL) graph if there exists a directed graph $D$ such that $G=I(D)$. Any FL graph will be denoted by $I(D)$, this notation helps to indicate the directed graph $D$ from which our FL graph may be obtained. Such a graph $D$ is called the [*preimage*]{} of $G$. Let $D=(V,A)$ be a directed graph. Given an arc $a=(u,v)\in A$, the node $u$ is called the [*tail*]{} of $a$ and $v$ is called the [ *head*]{} of $a$. Sometimes we use the notation $t(a)$ (respectively $h(a)$) for the tail (respectively the head) of $a$. A [*sink*]{} is a node which is a tail of no arc in $A$. A [*branch*]{} in a directed graph is an arc $(u,v)$ where $v$ is a sink and is the head of only $(u,v)$. A [*cycle*]{} $C$ in $D$ is a cycle in the underlying undirected graph of $D$. I.e., an ordered sequence of arcs $a_1, a_2, \dots,a_p,a_{p+1}$, where $a_i$ and $a_{i+1}$ are incident, for $i=1,\dots,p$, with $a_{p+1}=a_1$. If $a_1$ and $a_p$ are not incident, then this sequence is called a [*path*]{}. We denote by $A(C)$ the arcs of $C$ and by $V(C)$ its nodes, that are the endnodes of the arcs in $A(C)$. The nodes of $V(C)$ may be partitioned into three sets (1) $\dot{C}$, the nodes that are the tail of two arcs in $A(C)$, (2) $\hat{C}$, the nodes that are the head of two arcs in $A(C)$ and (3) $\tilde{C}$, the nodes that are the tail of one arc and the head of the other arc in $A(C)$. When $\hat{C}$ (or $\dot{C}$) is empty, the cycle $C$ is the classical [*directed*]{} cycle. Similarly the nodes of a path $P$, except its extremities, may be partitioned into three sets $\dot{P}$, $\hat{P}$ and $\tilde{P}$. When $\hat{P}=\dot{P}=\emptyset$, $P$ is called a [*directed path*]{}. We define a cycle $C$ in an undirected graph as a sequence of ordered nodes, instead of ordered edges, this is useful when we study the correspondence between $I(D)$ and $D$. The nodes of $C$ are denoted by $A(C)$ and its edges by $E(C)$. A [*path*]{} is defined similarly. Define $x_1,\dots,x_n$ to be $n$ Boolean variables. Define a [*literal*]{} $\lambda_i$ be either a Boolean variable $x_i$ or its negation $\bar{x_i}$. A [*clause*]{} $C$, is a disjunction of literals $\lambda_i$, that is $C=\lambda_{i_1}\vee\dots\vee\lambda_{i_k}$. $F=C_1\wedge\dots\wedge C_m$ is a conjunction of $m$ clauses. In the [*satisfiablity problem*]{} <span style="font-variant:small-caps;">sat</span>, we want to decide if there exists values $x_i$ such that an input conjunction of disjunctions (of literals) $F$ evaluates to [*true*]{}. If such values exist, we say $F$ is [*satisfiable*]{}. If each clause is a disjunction of at most three literals then the problem is called 3-satisfiability. The problem 3-<span style="font-variant:small-caps;">sat</span> has been shown <span style="font-variant:small-caps;">np</span>-complete by Karp [@karp]. An undirected graph $G$ is triangle-free if it does not contain a clique of size 3. A [*wheel*]{} $W_n$ is a graph obtained from a cycle $C_n$by adding a vertex adjacent to all vertices of the cycle. Recognizing facility location graphs is <span style="font-variant:small-caps;">np</span>-complete {#RLFG} ================================================================================================= The main result of this section is the following: \[NP-complete\] Recognizing facility location graphs is <span style="font-variant:small-caps;">np</span>-complete. The proof of this theorem is given in subsection \[proof-NP-complete\]. We first give a sketch of this proof and some useful lemmas before providing the detailed proof. Proof sketch ------------ We will reduce the problem 3-<span style="font-variant:small-caps;">sat</span> to the recognition of FL graphs. We assume we are given an instance of the problem 3-<span style="font-variant:small-caps;">sat</span>. That is, we have $n$ Boolean variables $x_1,\dots,x_n$ and a Boolean formula $F=C_1\wedge\dots\wedge C_m$, where each clause $C_j=\lambda_{j_1}\vee\lambda_{j_2}\vee\lambda_{j_3}$, for $j=1\dots,m$. From $F$ we construct an undirected graph $G_F$ and we show that $F$ is satisfiable if and only if $G_F$ is a facility location graph. We build $G_F$ using gadgets for variables and clauses. Values for variables are stored, replicated and negated through the “branches” of the variable gadgets. These branches are then connected to the clauses gadgets of clauses that contain these variables (and their negation). More precisely, the construction of $G_F$ follows three steps: (1) for each variable $x_i$, we construct a graph called $\textsc{Gad}^1_i$ (<span style="font-variant:small-caps;">Gad</span> stands for gadget), (2) for each clause $C_j$, another gadget called $\textsc{Gad}^2_j$ is constructed and (3) we connect the graphs $\textsc{Gad}^1_i$ and $\textsc{Gad}^2_j$ to produce $G_F$. Each graph $\textsc{Gad}^1_i$ contains $2m$ branches where each branch express the fact that the variable $x_i$ (or $\bar{x}_i$) is present in the clause $C_j$, $j=1,\dots,m$. Each graph $\textsc{Gad}^2_j$ contains exactly three branches where each branch expresses the litterals of this clause $\lambda_{j_1}$, $\lambda_{j_2}$ and $\lambda_{j_3}$. The three following subsections are devoted to the construction of the graphs $\textsc{Gad}^1_i$, $\textsc{Gad}^2_j$ and $G_F$. The construction of the graphs $\textsc{Gad}^1_i$ ------------------------------------------------- \[wheel-obs\] There are 15 directed graphs whose intersection graph is the wheel $W_5$. We list 5 of them that will be useful for our reduction. From these graphs, we obtain all the remaining directed graphs by identifying the head of the pending arc with the tail of one of the arcs entering this pending arc. \(a) at (-2,1.5); (b) at ($ (a) + (126:1.5) $); (c) at ($ (a) + (-162:1.5) $); (d) at ($ (a) + (-90:1.5) $); (e) at ($ (a) + (-18:1.5) $); (f) at ($ (a) + (54:1.5) $); \(b) – (c) – (d) – (e) – (f) – (b); (a) – (b); (a) – (c); (a) – (d); (a) – (e); (a) – (f); \(a) circle (1.5pt); (b) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (f) circle (1.5pt); \(a) node\[above\] [$a$]{}; (b) node\[left\] [$b$]{}; (c) node\[left\] [$c$]{}; (d) node\[below\] [$d$]{}; (e) node\[right\] [$e$]{}; (f) node\[right\] [$f$]{}; \(x) at (3,2.25); (y) at (2,3); (z) at (1,2.25); (t) at (2,1.5); (u) at (2,0); (v) at (2,-0.5); \(y) – (z) node\[midway,above left\][$d$]{}; (y) – (x) node\[midway,above right\][$c$]{}; (z) – (t) node\[midway,below left\][$e$]{}; (x) – (t) node\[midway,below right\][$b$]{}; (t) – (y) node\[midway,right\][$a$]{}; (t) – (u) node\[midway,right\][$f$]{}; (v) node\[below\] [$I_5$]{}; \(x) circle (1.5pt); (y) circle (1.5pt); (z) circle (1.5pt); (t) circle (1.5pt); (u) circle (1.5pt); \(x) at (0,2.25); (y) at (-1,3); (z) at (-2,2.25); (t) at (-1,1.5); (u) at (-1,0); (v) at (-1,-0.5); \(y) – (z) node\[midway,above left\][$c$]{}; (y) – (x) node\[midway,above right\][$b$]{}; (z) – (t) node\[midway,below left\][$d$]{}; (x) – (t) node\[midway,below right\][$f$]{}; (t) – (y) node\[midway,right\][$a$]{}; (t) – (u) node\[midway,right\][$e$]{}; (v) node\[below\] [$I_4$]{}; \(x) circle (1.5pt); (y) circle (1.5pt); (z) circle (1.5pt); (t) circle (1.5pt); (u) circle (1.5pt); \(x) at (-3,2.25); (y) at (-4,3); (z) at (-5,2.25); (t) at (-4,1.5); (u) at (-4,0); (v) at (-4,-0.5); \(y) – (z) node\[midway,above left\][$b$]{}; (y) – (x) node\[midway,above right\][$f$]{}; (z) – (t) node\[midway,below left\][$c$]{}; (x) – (t) node\[midway,below right\][$e$]{}; (t) – (y) node\[midway,right\][$a$]{}; (t) – (u) node\[midway,right\][$d$]{}; (v) node\[below\] [$I_3$]{}; \(x) circle (1.5pt); (y) circle (1.5pt); (z) circle (1.5pt); (t) circle (1.5pt); (u) circle (1.5pt); \(x) at (-6,2.25); (y) at (-7,3); (z) at (-8,2.25); (t) at (-7,1.5); (u) at (-7,0); (v) at (-7,-0.5); \(y) – (z) node\[midway,above left\][$f$]{}; (y) – (x) node\[midway,above right\][$e$]{}; (z) – (t) node\[midway,below left\][$b$]{}; (x) – (t) node\[midway,below right\][$d$]{}; (t) – (y) node\[midway,right\][$a$]{}; (t) – (u) node\[midway,right\][$c$]{}; (v) node\[below\] [$I_2$]{}; \(x) circle (1.5pt); (y) circle (1.5pt); (z) circle (1.5pt); (t) circle (1.5pt); (u) circle (1.5pt); \(x) at (-9,2.25); (y) at (-10,3); (z) at (-11,2.25); (t) at (-10,1.5); (u) at (-10,0); (v) at (-10,-0.5); \(y) – (z) node\[midway,above left\][$e$]{}; (y) – (x) node\[midway,above right\][$d$]{}; (z) – (t) node\[midway,below left\][$f$]{}; (x) – (t) node\[midway,below right\][$c$]{}; (t) – (y) node\[midway,right\][$a$]{}; (t) – (u) node\[midway,right\][$b$]{}; (v) node\[below\] [$I_1$]{}; \(x) circle (1.5pt); (y) circle (1.5pt); (z) circle (1.5pt); (t) circle (1.5pt); (u) circle (1.5pt); Consider the two undirected graphs $I$ and $I'$ of Figure \[fig:demi\_inv\]. These two graphs are isomorphic, they are obtained from the wheel $W_5$ by adding four nodes. This restricts the number of possible preimage for the wheel to only 2. When we join the nodes $j$ and $j'$ we obtain the desired graph called <span style="font-variant:small-caps;">Inv</span> which is again the intersection graph of exactly two directed graphs. The graph <span style="font-variant:small-caps;">Inv</span> will be useful for the construction of $\textsc{Gad}^2_j$. \(a) at (-2,1.5); (b) at ($ (a) + (126:1.5) $); (c) at ($ (a) + (-162:1.5) $); (d) at ($ (a) + (-90:1.5) $); (e) at ($ (a) + (-18:1.5) $); (f) at ($ (a) + (54:1.5) $); (g) at ($ (b) + (90:1.5) $); (h) at ($ (f) + (90:1.5) $); (i) at ($ (d) + (-2,0) $); (j) at ($ (d) + (2,0) $); (x) at ($(d) + (0,-1)$); \(b) – (c) – (d) – (e) – (f) – (b); (c) – (i) – (d) – (c); (d) – (j) – (e) – (d); (a) – (b); (a) – (c); (a) – (d); (a) – (e); (a) – (f); (b) – (g); (f) – (h); (x) node\[below\] [$I$]{}; \(a) circle (1.5pt); (b) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (f) circle (1.5pt); (g) circle (1.5pt); (h) circle (1.5pt); (i) circle (1.5pt); (j) circle (1.5pt); \(a) node\[above\] [$a$]{}; (b) node\[left\] [$b$]{}; (c) node\[left\] [$c$]{}; (d) node\[below\] [$d$]{}; (e) node\[right\] [$e$]{}; (f) node\[right\] [$f$]{}; (g) node\[above\] [$g$]{}; (h) node\[above\] [$h$]{}; (i) node\[left\] [$i$]{}; (j) node\[right\] [$j$]{}; \(a) at (4,1.5); (b) at ($ (a) + (126:1.5) $); (c) at ($ (a) + (-162:1.5) $); (d) at ($ (a) + (-90:1.5) $); (e) at ($ (a) + (-18:1.5) $); (f) at ($ (a) + (54:1.5) $); (g) at ($ (b) + (90:1.5) $); (h) at ($ (f) + (90:1.5) $); (i) at ($ (d) + (-2,0) $); (j) at ($ (d) + (2,0) $); (x) at ($(d) + (0,-1)$); \(b) – (c) – (d) – (e) – (f) – (b); (c) – (i) – (d) – (c); (d) – (j) – (e) – (d); (a) – (b); (a) – (c); (a) – (d); (a) – (e); (a) – (f); (b) – (g); (f) – (h); (x) node\[below\] [$I'$]{}; \(a) circle (1.5pt); (b) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (f) circle (1.5pt); (g) circle (1.5pt); (h) circle (1.5pt); (i) circle (1.5pt); (j) circle (1.5pt); \(a) node\[above\] [$a'$]{}; (b) node\[left\] [$b'$]{}; (c) node\[left\] [$c'$]{}; (d) node\[below\] [$d'$]{}; (e) node\[right\] [$e'$]{}; (f) node\[right\] [$f'$]{}; (g) node\[above\] [$g'$]{}; (h) node\[above\] [$h'$]{}; (i) node\[left\] [$j'$]{}; (j) node\[right\] [$i'$]{}; Recall that $W_5$ may be the intersection of 15 directed graphs as described by Remark \[wheel-obs\]. Let us discuss the possible directed graphs having $I$ as their intersection graph. The subgraph of $I$ induced by the nodes $a,b,c,d,e,f$ is a wheel $W_5$ and thus is the intersection of 15 directed graphs obtained from those of Figure \[fig:w5isflg\] as described by Remark \[wheel-obs\]. In $I$ there are two pendent nodes $g$ and $h$ adjacent to two neighbors $b$ and $f$ in $I$. This implies that the adjacency between $b$ and $f$ cannot be of the form as represented in the graphs $I_2$, $I_3$ and $I_4$ of Figure \[fig:w5isflg\]. Hence we have only two possible directed graphs $I_1$ or $I_5$, since in neither of these two graphs we may identify the head of the pending arc with the tail of one of the arcs entering this pending arc. Moreover, there is one way of adding the arcs $i$ and $j$ in each case. Consequently there are only two graphs $D_1$ and $D_2$ such that $I(D_1)=I(D_2)=I$. These graphs are represented in Figure \[fig:demi\_inv1\]. It is obvious that there are also two directed graphs whose intersection graph is $I'$. For convenience these two graphs will be shown in Figure \[fig:demi\_inv2\]. \(x) at (4,2.25); (y) at (3,3); (z) at (2,2.25); (t) at (3,1.5); (u) at (3,0); (v) at (3,-1.5); (p) at (3,4.5); (q) at (0.5,2.25); (w) at (5.5,2.25); (k) at (2,-1); \(y) – (z) node\[midway,left, above\][$e$]{}; (y) – (x) node\[midway,right, above\][$d$]{}; (z) – (t) node\[midway,left, below\][$f$]{}; (x) – (t) node\[midway,right, below\][$c$]{}; (t) – (y) node\[midway,right\][$a$]{}; (t) – (u) node\[midway,right\][$b$]{}; (p) – (y) node\[midway,right\][$j$]{}; (x) – (w) node\[midway, below\][$i$]{}; (q) – (z) node\[midway,above\][$h$]{}; (u) – (v) node\[midway,right\][$g$]{}; (k) node\[below\] [$D_1$]{}; \(x) circle (1.5pt); (y) circle (1.5pt); (z) circle (1.5pt); (t) circle (1.5pt); (u) circle (1.5pt); (p) circle (1.5pt); (q) circle (1.5pt); (v) circle (1.5pt); (w) circle (1.5pt); \(x) at (10,2.25); (y) at (9,3); (z) at (8,2.25); (t) at (9,1.5); (u) at (9,0); (v) at (9,-1.5); (p) at (9,4.5); (q) at (6.5,2.25); (w) at (11.5,2.25); (k) at (8,-1); \(y) – (z) node\[midway,left, above\][$d$]{}; (y) – (x) node\[midway,right, above\][$c$]{}; (z) – (t) node\[midway,left, below\][$e$]{}; (x) – (t) node\[midway,right, below\][$b$]{}; (t) – (y) node\[midway,right\][$a$]{}; (t) – (u) node\[midway,right\][$f$]{}; (p) – (y) node\[midway,right\][$i$]{}; (w) – (x) node\[midway, below\][$g$]{}; (z) – (q) node\[midway,above\][$j$]{}; (u) – (v) node\[midway,right\][$h$]{}; (k) node\[below\] [$D_2$]{}; \(x) circle (1.5pt); (y) circle (1.5pt); (z) circle (1.5pt); (t) circle (1.5pt); (u) circle (1.5pt); (p) circle (1.5pt); (q) circle (1.5pt); (v) circle (1.5pt); (w) circle (1.5pt); \(x) at (4,2.25); (y) at (3,3); (z) at (2,2.25); (t) at (3,1.5); (u) at (3,0); (v) at (3,-1.5); (p) at (3,4.5); (q) at (0.5,2.25); (w) at (5.5,2.25); (k) at (2,-1); \(y) – (z) node\[midway,left, above\][$e'$]{}; (y) – (x) node\[midway,right, above\][$d'$]{}; (z) – (t) node\[midway,left, below\][$f'$]{}; (x) – (t) node\[midway,right, below\][$c'$]{}; (t) – (y) node\[midway,right\][$a'$]{}; (t) – (u) node\[midway,right\][$b'$]{}; (p) – (y) node\[midway,right\][$i'$]{}; (x) – (w) node\[midway, below\][$j'$]{}; (q) – (z) node\[midway,above\][$h'$]{}; (u) – (v) node\[midway,right\][$g'$]{}; (k) node\[below\] [$D'_1$]{}; \(x) circle (1.5pt); (y) circle (1.5pt); (z) circle (1.5pt); (t) circle (1.5pt); (u) circle (1.5pt); (p) circle (1.5pt); (q) circle (1.5pt); (v) circle (1.5pt); (w) circle (1.5pt); \(x) at (10,2.25); (y) at (9,3); (z) at (8,2.25); (t) at (9,1.5); (u) at (9,0); (v) at (9,-1.5); (p) at (9,4.5); (q) at (6.5,2.25); (w) at (11.5,2.25); (k) at (8,-1); \(y) – (z) node\[midway,left, above\][$d'$]{}; (y) – (x) node\[midway,right, above\][$c'$]{}; (z) – (t) node\[midway,left, below\][$e'$]{}; (x) – (t) node\[midway,right, below\][$b'$]{}; (t) – (y) node\[midway,right\][$a'$]{}; (t) – (u) node\[midway,right\][$f'$]{}; (p) – (y) node\[midway,right\][$j'$]{}; (w) – (x) node\[midway, below\][$g'$]{}; (z) – (q) node\[midway,above\][$i'$]{}; (u) – (v) node\[midway,right\][$h'$]{}; (k) node\[below\] [$D'_2$]{}; \(x) circle (1.5pt); (y) circle (1.5pt); (z) circle (1.5pt); (t) circle (1.5pt); (u) circle (1.5pt); (p) circle (1.5pt); (q) circle (1.5pt); (v) circle (1.5pt); (w) circle (1.5pt); Let us call <span style="font-variant:small-caps;">Inv</span> (<span style="font-variant:small-caps;">Inv</span> stands for inverter) the graph obtained from $I$ and $I'$ by identifying the nodes $j$ and $j'$. We use $j$ for the name of the resulting node, see Figure \[fig:inv\]. \(a) at (-2,1.5); (b) at ($ (a) + (126:1.5) $); (c) at ($ (a) + (-162:1.5) $); (d) at ($ (a) + (-90:1.5) $); (e) at ($ (a) + (-18:1.5) $); (f) at ($ (a) + (54:1.5) $); (g) at ($ (b) + (90:1.5) $); (h) at ($ (f) + (90:1.5) $); (i) at ($ (d) + (-2,0) $); (j) at ($ (d) + (2,0) $); (a’) at (2,1.5); (b’) at ($ (a') + (126:1.5) $); (c’) at ($ (a') + (-162:1.5) $); (d’) at ($ (a') + (-90:1.5) $); (e’) at ($ (a') + (-18:1.5) $); (f’) at ($ (a') + (54:1.5) $); (g’) at ($ (b') + (90:1.5) $); (h’) at ($ (f') + (90:1.5) $); (i’) at ($ (d') + (2,0) $); \(a) circle (1.5pt); (b) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (f) circle (1.5pt); (g) circle (1.5pt); (h) circle (1.5pt); (i) circle (1.5pt); (j) circle (1.5pt); (a’) circle (1.5pt); (b’) circle (1.5pt); (c’) circle (1.5pt); (d’) circle (1.5pt); (e’) circle (1.5pt); (f’) circle (1.5pt); (g’) circle (1.5pt); (h’) circle (1.5pt); (i’) circle (1.5pt); \(b) – (c) – (d) – (e) – (f) – (b); (c) – (i) – (d) – (c); (d) – (j) – (e) – (d); (b’) – (c’) – (d’) – (e’) – (f’) – (b’); (c’) – (j) – (d’) – (c’); (d’) – (i’) – (e’) – (d’); \(a) – (b); (a) – (c); (a) – (d); (a) – (e); (a) – (f); (b) – (g); (f) – (h); (a’) – (b’); (a’) – (c’); (a’) – (d’); (a’) – (e’); (a’) – (f’); (b’) – (g’); (f’) – (h’); \(a) node\[above\] [$a$]{}; (b) node\[left\] [$b$]{}; (c) node\[left\] [$c$]{}; (d) node\[below\] [$d$]{}; (e) node\[right\] [$e$]{}; (f) node\[right\] [$f$]{}; (g) node\[above\] [$g$]{}; (h) node\[above\] [$h$]{}; (i) node\[left\] [$i$]{}; (j) node\[above\] [$j$]{}; (a’) node\[above\] [$a'$]{}; (b’) node\[left\] [$b'$]{}; (c’) node\[above left\] [$c'$]{}; (d’) node\[below\] [$d'$]{}; (e’) node\[right\] [$e'$]{}; (f’) node\[right\] [$f'$]{}; (g’) node\[above\] [$g'$]{}; (h’) node\[above\] [$h'$]{}; (i’) node\[right\] [$i'$]{}; (inv) at (8,2) [$~~~$]{}; (gg) at (6,2); (gg’) at (10,2); (gg) circle (1.5pt); (gg’) circle (1.5pt); (inv.east) circle (1.5pt); (inv.west) circle (1.5pt); (gg) node\[above\] [$g$]{}; (gg’) node\[above\] [$g'$]{}; (inv.east) node\[above right\] [$b'$]{}; (inv.west) node\[above left\] [$b$]{}; (gg) – (inv.west); (inv.east) – (gg’); From the discussion above there are only two directed graphs that for convenience we call $\overleftrightarrow{\textsc{Inv}}$ and $\overline{\textsc{Inv}}$, such that $\textsc{Inv}=I(\overleftrightarrow{\textsc{Inv}})=I(\overline{\textsc{Inv}})$. The graph $\overleftrightarrow{\textsc{Inv}}$ (respectively $\overline{\textsc{Inv}}$) is obtained from $D_1$ and $D'_1$ (respectively $D_2$ and $D'_2$) by identifying $j$ and $j'$. Notice that the other possibilities of identifying $j$ and $j'$ will not lead to the graph $\textsc{Inv}$. Now we are ready to build graphs $\textsc{Gad}^1_i$ that correspond to the variables $x_i$, for each $i$ in $\{1,\dots,n\}$. For each variable $x_i$ we construct $m$ copies of the graph $I$, where the nodes $a,\ldots,j$ of each copy are renamed, respectively, $a^i_1,\ldots,j^i_1$ up to $a^i_m,\ldots,j^i_m$. The graph $\textsc{Gad}^1_i$ is obtained by identifying the node $j^i_l$ with $i^i_{l+1}$ and we call $i^i_{l+1}$ the resulting node, for $l=1,\dots,m-1$. Also we rename the node $j^i_m$ by $i^i_{m+1}$, see Figure \[fig:gadget1\]. \(a) at (-2,1.5); (b) at ($ (a) + (126:1.5) $); (c) at ($ (a) + (-162:1.5) $); (d) at ($ (a) + (-90:1.5) $); (e) at ($ (a) + (-18:1.5) $); (f) at ($ (a) + (54:1.5) $); (g) at ($ (b) + (90:1.5) $); (h) at ($ (f) + (90:1.5) $); (i) at ($ (d) + (-2,0) $); (j) at ($ (d) + (2,0) $); (a’) at (2,1.5); (b’) at ($ (a') + (126:1.5) $); (c’) at ($ (a') + (-162:1.5) $); (d’) at ($ (a') + (-90:1.5) $); (e’) at ($ (a') + (-18:1.5) $); (f’) at ($ (a') + (54:1.5) $); (g’) at ($ (b') + (90:1.5) $); (h’) at ($ (f') + (90:1.5) $); (i’) at ($ (d') + (2,0) $); (am) at (10,1.5); (bm) at ($ (am) + (126:1.5) $); (cm) at ($ (am) + (-162:1.5) $); (dm) at ($ (am) + (-90:1.5) $); (em) at ($ (am) + (-18:1.5) $); (fm) at ($ (am) + (54:1.5) $); (gm) at ($ (bm) + (90:1.5) $); (hm) at ($ (fm) + (90:1.5) $); (im) at ($ (dm) + (-2,0) $); (im1) at ($ (dm) + (2,0) $); \(a) circle (1.5pt); (b) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (f) circle (1.5pt); (g) circle (1.5pt); (h) circle (1.5pt); (i) circle (1.5pt); (j) circle (1.5pt); (a’) circle (1.5pt); (b’) circle (1.5pt); (c’) circle (1.5pt); (d’) circle (1.5pt); (e’) circle (1.5pt); (f’) circle (1.5pt); (g’) circle (1.5pt); (h’) circle (1.5pt); (i’) circle (1.5pt); (am) circle (1.5pt); (bm) circle (1.5pt); (cm) circle (1.5pt); (dm) circle (1.5pt); (em) circle (1.5pt); (fm) circle (1.5pt); (gm) circle (1.5pt); (hm) circle (1.5pt); (im) circle (1.5pt); (im1) circle (1.5pt); \(b) – (c) – (d) – (e) – (f) – (b); (c) – (i) – (d) – (c); (d) – (j) – (e) – (d); (b’) – (c’) – (d’) – (e’) – (f’) – (b’); (c’) – (j) – (d’) – (c’); (d’) – (i’) – (e’) – (d’); (bm) – (cm) – (dm) – (em) – (fm) – (bm); (cm) – (im) – (dm) – (cm); (dm) – (im1) – (em) – (dm); \(a) – (b); (a) – (c); (a) – (d); (a) – (e); (a) – (f); (b) – (g); (f) – (h); (am) – (bm); (am) – (cm); (am) – (dm); (am) – (em); (am) – (fm); (bm) – (gm); (fm) – (hm); (a’) – (b’); (a’) – (c’); (a’) – (d’); (a’) – (e’); (a’) – (f’); (b’) – (g’); (f’) – (h’); (i’) – (im); ($ (a) + (25:0.4) $) node [$a_1^i$]{}; (b) node\[left\] [$b_1^i$]{}; (c) node\[left\] [$c_1^i$]{}; (d) node\[below\] [$d_1^i$]{}; (e) node\[right\] [$e_1^i$]{}; (f) node\[right\] [$f_1^i$]{}; (g) node\[above\] [$g_1^i$]{}; (h) node\[above\] [$h_1^i$]{}; (i) node\[left\] [$i_1^i$]{}; (j) node\[below\] [$i_2^i$]{}; ($ (a') + (25:0.4) $) node [$a_2^i$]{}; (b’) node\[left\] [$b_2^i$]{}; (c’) node\[above left\] [$c_2^i$]{}; (d’) node\[below\] [$d_2^i$]{}; (e’) node\[right\] [$e_2^i$]{}; (f’) node\[right\] [$f_2^i$]{}; (g’) node\[above\] [$g_2^i$]{}; (h’) node\[above\] [$h_2^i$]{}; (i’) node\[below\] [$i_3^i$]{}; ($ (am) + (20:0.5) $) node [$a_m^i$]{}; (bm) node\[left\] [$b_m^i$]{}; (cm) node\[above left\] [$c_m^i$]{}; (dm) node\[below\] [$d_m^i$]{}; (em) node\[right\] [$e_m^i$]{}; (fm) node\[right\] [$f_m^i$]{}; (gm) node\[above\] [$g_m^i$]{}; (hm) node\[above\] [$h_m^i$]{}; (im) node\[below\] [$i_m^i$]{}; (im1) node\[right\] [$i_{m+1}^i$]{}; The discussion above implies the following lemma, see Figures \[fig:inv\_depic1\] and \[fig:inv\_depic2\]. \[GAD1\] For each directed graph $D$ with $I(D)=\textsc{Gad}^1_i$ exactly one of the following two assumptions holds: - $h(b^i_j)=t(g^i_j)$ and $t(f^i_j)=h(h^i_j)$ for each $j=1,\dots,m$, - $t(b^i_j)=h(g^i_j)$ and $h(f^i_j)=t(h^i_j)$ for each $j=1,\dots,m$. The construction of the graphs $\textsc{Gad}^2_j$ ------------------------------------------------- We will use the graph $\textsc{Inv}$ of the previous subsection to construct $\textsc{Gad}^2_j$. We have three triangles $\Delta_1=\{r_j,a_j,f_j\}$, $\Delta_2=\{s_j,b_j,c_j\}$ and $\Delta_3=\{t_j,e_j,d_j\}$ with the addition of a branch pending from each node of these triangles, that is we add the edges $r_jr'_j$, $a_jf'_j$; $s_js'_j$, $b_jb'_j$; $t_jt'_j$, $e_je'_j$, $d_jd'_j$. These triangles are connected using their branches. We choose one triangle say $\Delta_1$ and we connect it to $\Delta_2$ and $\Delta_3$ via two graphs identical to $\textsc{Inv}$ using two of its branches $a_ja'_j$ and $f_jf'_j$. The triangles $\Delta_2$ and $\Delta_3$ are connected by identifying the branches $c_jc'_j$ and $d_jd'_j$ (the nodes $c'_j$ and $d'_j$ are removed), see Figure \[fig:gadget2\]. (r’) at (0,0); (r) at ($ (r') + (0,-2) $); (a) at ($ (r) + (-120:2) $); (f) at ($ (r) + (-60:2) $); (g) at ($ (a) + (-120:3) $) [$~~~$]{}; (h) at ($ (f) + (-60:3) $) [$~~~$]{}; (b) at ($ (a) + (-120:6) $); (e) at ($ (f) + (-60:6) $); (s) at ($ (b) + (-120:2) $); (c) at ($ (b) + (-60:2) $); (d) at ($ (e) + (-120:2) $); (t) at ($ (e) + (-60:2) $); (s’) at ($ (s) + (-120:2) $); (t’) at ($ (t) + (-60:2) $); (cen) at (12,-7); (n) at ($ (cen) + (90:2) $); (sw) at ($ (cen) + (-150:2) $); (se) at ($ (cen) + (-30:2) $); (n’) at ($ (cen) + (90:4) $); (sw’) at ($ (cen) + (-150:4) $); (se’) at ($ (cen) + (-30:4) $); \(r) – (r’); (t) – (t’); (s) – (s’); (r) – (a) – (f) – (r); (s) – (b) – (c) – (s); (t) – (d) – (e) – (t); (d) – (c); (g.east) – (a); (g.west) – (b); (h.east) – (e); (h.west) – (f); \(a) circle (1.5pt); (b) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (f) circle (1.5pt); (r) circle (1.5pt); (r’) circle (1.5pt); (s) circle (1.5pt); (s’) circle (1.5pt); (t) circle (1.5pt); (t’) circle (1.5pt); (g.east) circle (1.5pt); (h.east) circle (1.5pt); (g.west) circle (1.5pt); (h.west) circle (1.5pt); (g.east) node\[above\] [$a'$]{}; (g.west) node\[left\] [$b'$]{}; (h.east) node\[right\] [$e'$]{}; (h.west) node\[above\] [$f'$]{}; (a) node\[above left\] [$a$]{}; (b) node\[above left\] [$b$]{}; (c) node\[above right\] [$c$]{}; (d) node\[above left\] [$d$]{}; (e) node\[above right\] [$e$]{}; (f) node\[above right\] [$f$]{}; (r) node\[above right\] [$r$]{}; (r’) node\[above\] [$r'$]{}; (s) node\[above left\] [$s$]{}; (s’) node\[below left\] [$s'$]{}; (t) node\[right\] [$t$]{}; (t’) node\[below right\] [$t'$]{}; Before establishing the main lemma of this section let us notice the following remark. \[Delta\] Call $\Delta$ the undirected graph defined by a triangle with three branches pending from each of its three nodes. There are only three possible directed graphs, $D_1$, $D_2$, $D_3$ such that $\Delta=I(D_1)=I(D_2)=I(D_3)$, as shown in Figure \[fig:k3s\] \(a) at (0,2); (b) at ($ (a) + (-1.5,-2) $); (c) at ($ (a) + (0,-2) $); (d) at ($ (a) + (1.5,-2) $); (e) at (4,1.5); (f) at ($ (e) + (0,1.5) $); (g) at ($ (e) + (-0.75,-1.5) $); (h) at ($ (e) + (0.75,-1.5) $); (k) at (8,2); (i) at ($ (k) + (-1.3,-2) $); (j) at ($ (k) + (1.3,-2) $); (m) at (12,2); (l) at ($ (m) + (0,2) $); (n) at ($ (m) + (0,-2) $); (o) at ($ (b) + (0,-2) $); (p) at ($ (c) + (0,-2) $); (q) at ($ (d) + (0,-2) $); (r) at ($ (g) + (0,-2) $); (s) at ($ (h) + (0,-2) $); (t) at ($ (f) + (0,2) $); (u) at ($ (i) + (-120:2) $); (w) at ($ (j) + (-60:2) $); (v) at ($ (k) + (90:2) $); (p1) at ($ (c) + (0,-2.5) $); (s1) at ($ (h) + (-0.5,-2.5) $); (s1) at ($ (h) + (-0.5,-2.5) $); (v1) at (8,-2.5); \(a) circle (1.5pt); (b) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (f) circle (1.5pt); (g) circle (1.5pt); (h) circle (1.5pt); (i) circle (1.5pt); (j) circle (1.5pt); (k) circle (1.5pt); (o) circle (1.5pt); (p) circle (1.5pt); (q) circle (1.5pt); (r) circle (1.5pt); (s) circle (1.5pt); (t) circle (1.5pt); (u) circle (1.5pt); (v) circle (1.5pt); (w) circle (1.5pt); \(a) – (b); (a) – (c); (a) – (d); \(b) – (o); (c) – (p); (d) – (q); (p1) node\[below\] [$D_1$]{}; \(f) – (e); (e) – (h); (e) – (g); \(t) – (f); (g) – (r); (h) – (s); (s1) node\[below\] [$D_2$]{}; \(i) – (j); (j) – (k); (k) – (i); \(u) – (i); (w) – (j); (v) – (k); (v1) node\[below\] [$D_3$]{}; \[GAD2\] Let $D$ be a directed graph such that $I(D)=\textsc{Gad}^2_j$. Then the two following assumptions hold: - The arcs $r'_j$, $s'_j$ and $t'_j$ cannot all enter the arcs $r_j$, $s_j$ and $t_j$, respectively, - Any other configuration for theses three adjacencies is possible. \(i) Assume that in $D$ all the arcs $r'_j$, $s'_j$ and $t'_j$ enter the arcs $r_j$, $s_j$ and $t_j$, respectively. That is $h(r'_j)=t(r_j)$, $h(s'_j)=t(s_j)$ and $h(t'_j)=t(t_j)$. Therefore, from Remark \[Delta\], none of the triangles $\Delta_1$, $\Delta_2$ and $\Delta_3$ has the configuration $D_1$ of Figure \[fig:k3s\]. We may also check that $\Delta_2$ and $\Delta_3$ cannot have the same configuration, that is cannot be both of the form $D_2$ or $D_3$ of Figure \[fig:k3s\]. Hence assume that $\Delta_2$ has the form of $D_2$ and $\Delta_3$ has the form of $D_3$. Since the arc $e_j'$ must enter the arc $e_j$ it follows from Lemma \[GAD1\] that $$\begin{aligned} \label{1r} f'_j \mbox{ must enter the arc } f_j. \end{aligned}$$ Since $c_j$ enters the arc $d_j$ and using the fact that $s'_j$ enters $s_j$ we conclude that $b_j$ must enter the arc $b'_j$. Now using again Lemma \[GAD1\] we obtain $$\begin{aligned} \label{2r} a_j \mbox{ must enter the arc } a'_j. \end{aligned}$$ Now combining the two facts and we have that $\Delta_1$ must be of the form $D_2$ of Figure \[fig:k3s\] and that the arc $r_j$ must enter $r'_j$, which is not possible. \(ii) The proof of this assumption is presented in the appendix. It lists all the possible configurations of $D$ in Figure \[fig:preimages\_gad2\] The construction of the graph $G_F$ ----------------------------------- Let $F=C_1\wedge\dots\wedge C_m$, where each clause $C_j=\lambda_{j_1}\vee\lambda_{j_2}\vee\lambda_{j_3}$, for $j$ in $\{1,\dots,m\}$. Each $\lambda_{j_k}$ correspond to the variable $x_{j_k}$ or its negation $\bar{x}_{j_k}$. From $F$ we construct an undirected graph $G_F$ as follows. Let $\textsc{Gad}^1_i$ be the undirected graph associated with each Boolean variable $x_i$, $i=1,\dots,n$. And let $\textsc{Gad}^2_j$ be the undirected graph associated with each clause $C_j$, $j=1,\dots,m$. In the construction of $G_F$, there is no connection between the graphs $\textsc{Gad}^1_i$ themselves and between the graphs $\textsc{Gad}^2_j$ themselves. The only connections are between $\textsc{Gad}^1_i$ and $\textsc{Gad}^2_j$ where the variable $x_i$ or $\bar{x}_i$ appears in the clause $C_j$. Moreover these connections are made through their branches. Specifically, for each clause $C_j=\lambda_{j_1}\vee\lambda_{j_2}\vee\lambda_{j_3}$ we do the following: - if $\lambda_{j_1} = x_{j_1}$, we identify vertex $r_j$ with vertex $g_j^{j_1}$ and vertex $r'_j$ with vertex $b_j^{j_1}$, - if $\lambda_{j_1} = \bar{x}_{j_1}$, we identify vertex $r_j$ with vertex $h_j^{j_1}$ and vertex $r'_j$ with vertex $f_j^{j_1}$, - if $\lambda_{j_2} = x_{j_2}$, we identify vertex $s_j$ with vertex $g_j^{j_2}$ and vertex $s'_j$ with vertex $b_j^{j_2}$, - if $\lambda_{j_2} = \bar{x}_{j_2}$, we identify vertex $s_j$ with vertex $h_j^{j_2}$ and vertex $s'_j$ with vertex $f_j^{j_2}$, - if $\lambda_{j_3} = x_{j_3}$, we identify vertex $t_j$ with vertex $g_j^{j_3}$ and vertex $t'_j$ with vertex $b_j^{j_3}$, - if $\lambda_{j_3} = \bar{x}_{j_3}$, we identify vertex $t_j$ with vertex $h_j^{j_3}$ and vertex $t'_j$ with vertex $f_j^{j_3}$. Proof of Theorem \[NP-complete\] {#proof-NP-complete} -------------------------------- Since the problem 3-<span style="font-variant:small-caps;">sat</span> is NP-complete, it is sufficient to prove that the Boolean formula $F$, as defined in the previous subsection, is true if and only if the graph $G_F$ is a facility location graph. Assume that the graph $G_F$ is a facility location graph and let $D$ be a directed graph such that $I(D)=G_F$. Define an assignment of the Boolean variables $x_i$, $i=1\dots,n$ as follows: $$x_i=\left\lbrace \begin{array}{ll} 1 & \mbox{ if the arc }g^i_1 \mbox{ enters the arc } b^i_1 \mbox{ in } D,\\ 0 & \mbox{otherwise} \end{array} \right.$$ Notice that from Lemma \[GAD1\] whenever the arc $g^i_1$ enters the arc $b^i_1$, then $g^i_j$ enters the arc $b^i_j$ for each $j=1,\dots,m$. Let $C_j$ be any clause of $F$. From Lemma \[GAD2\] (i), we must have that $r_j$ enters $r'_j$, or $s_j$ enters $s'_j$ or that $t_j$ enters $t'_j$ in any directed graph whose intersection graph is $\textsc{Gad}^2_j$. We may assume that $r_j$ enters $r'_j$. By the definition of $G_F$ the branch $r_jr'_j$ is identified with $g^i_jb^i_j$ when $x_i$ is present in $C_j$ and in this case $x_i=1$ and so $C_j$=1. Otherwise the branch $r_jr'_j$ is identified with $h^i_jf^i_j$ when $\bar{x}_i$ is present in $C_j$. So the arc $h^i_j$ enters the arc $f^i_j$ and from Lemma \[GAD1\] we have that the arc $b^i_j$ enters the arc $g^i_j$ and by definition we have $x_i=0$, which implies that $C_j=1$. Now assume that there is an assignment of the variables $x_i$, $i=1,\dots,n$ satisfying $F$. Let us construct a directed graph $D$ such that $G_F=I(D)$. For each graph $\textsc{Gad}^1_i$ we build a directed graph such that each arc $g^i_j$ enters the arc $b^i_j$ when $x_i=1$ and each arc $b^i_j$ enters $g^i_j$ when $x_i=0$. This is possible from Lemma \[GAD1\]. Now given a clause $C_j=\lambda_{j_1}\vee\lambda_{j_2}\vee\lambda_{j_3}$, from Lemma \[GAD2\] the graph $D$ cannot exist only when the assumption (i) of Lemma \[GAD2\] is not satisfied. But one can check that this may happen only when $\lambda_{j_1}=\lambda_{j_2}=\lambda_{j_3}=0$, which is not possible. Recognizing triangle-free facility location graphs {#RTFLG} ================================================== Notice that the maximum cliques on the graph $G_F$ built by our reduction in the section have size 3. Hence it is natural to ask if recognizing triangle-free facility location graphs remains difficult. In this section we show that this recognition may be done in polynomial time. In subsection \[struc\] we examine the structure of general FL graphs. In subsection \[app-tri-free\], we restrict ourselves to triangle-free graphs and we give the main result of this section. Some structural properties {#struc} -------------------------- In [@BBpg], Baïou and Barahona gave a characterization of preimages of cycles. [@BBpg] \[cycle\] Given a directed graph $D=(V,A)$, a subset of arcs $C\subseteq A$ induce a chordless cycle of size at least four in $I(D)$, if and only if $C$ may be partitioned into two subsets $C'$ and $C''$, such that $C'$ is a cycle in $D$ and there is a 1-to-1 correspondence between the nodes in $\hat{C'}$ and the arcs in $C''$, where each arc $(v,\bar{v})$ of $C''$ correspond to a node $v\in\hat{C'}$ where (i) $\bar{v}\in V\setminus V(C')$ or (ii) $\bar{v}\in \tilde{C'}$ with $\bar{v}$ is one of the two neighbors of $v$ in $C'$, see Figure \[cycles\] for an example. \(a) at (0,2); (b) at ($ (a) + (2,-2) $); (c) at ($ (b) + (0,-2) $); (d) at ($ (c) + (-2,-2) $); (e) at ($ (d) + (-2,2) $); (f) at ($ (e) + (0,2) $); (g) at ($(d)+(0.5,1.5)$); \(a) circle (1.5pt); (b) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (f) circle (1.5pt); \(a) – (b) node\[midway,right\][$a_1$]{}; (b) – (c) node\[midway,right\][$a_2$]{}; (c) – (d) node\[midway,right\][$a_3$]{}; \(e) – (d) node\[midway,left\][$a_4$]{}; (f) – (e) node\[midway,left\][$a_5$]{}; (a) – (f) node\[midway,left\][$a_6$]{}; (d) to\[bend left=45\] (c); (g) node\[above\][$b$]{}; Notice that we may have $C''=\emptyset$. In this case $C$ is a directed cycle in $D$. Notice the following remark. \[pendent\] Let $G$ be an undirected graph. Let $e=uv$ an edge of $G$ where $u$ has degree one and $v$ has degree two. Then $G$ is a FL graph if and only if $G-u$ is also a FL graph. \[sink\] If $G$ is a FL graph, then there exists a digraph $D$ such that $G=I(D)$ and every sink node in $D$ has exactly one entering arc. Since $G$ is a FL graph, there exists a directed graph $D'$ such that $G=I(D')$. If $u$ is a sink node with the entering arcs $(u_1,u),\dots,(u_k,u)$, $k\geq 2$. We may split the node $u$ and so each arc $(u_i,u)$ is replaced by $(u_i,u_i')$, where each node $u_i'$ is a sink with only one entering arc. If $D$ is the resulting directed graph we have that $G=I(D)$. \[degre2\] Given an undirected graph $G$. If there is an edge $e=bc$ in $G$ such that $b$ and $c$ both have degree two and no common neighbour, then the following statements are equivalent: 1. $G$ is a FL graph, 2. $G - e$ is a FL graph. Let us call $a$ the other neighbour of $b$ and $d$ the other neighbour of $c$ (see Figure \[trans\]). [*(i) $\Rightarrow$ (ii)*]{}. Assume that $G$ is a FL graph and hence $G=I(D)$. Assume that the arcs $a$ and $c$ have no common vertex in $D$, in this case the arc $b$ must share exactly one endnode with $a$ and one endnode with $c$. Replace the arc $b=(r,s)$ by $b=(r',s)$ (respectively $b=(r,s')$) when $r$ (respectively $s$) is an endnode of $c$. The nodes $r'$ and $s'$ are new nodes. Call the resulting graph $D'$. We have $G - e=I(D')$. Now assume that $a$ and $c$ have a common endnode. Let $a=(u,v)$ and $c=(w,t)$. Since $a$ and $c$ are not adjacent we must have $v=t$ and $u\not=w$. Let $b=(r,s)$. If $r\not=v$, then as in the previous case we replace the arc $b=(r,s)$ by $b=(r',s)$ (respectively $b=(r,s')$) when $r$ (respectively $s$) is an endnode of $c$ and we obtain a new graph $D'$ with $G - e=I(D')$. If $r=v$, replace the arc $c=(w,t)$ by $c=(w,t')$, and if in addition we have $s=w$ we replace the arc $b=(r,s)$ by $b=(r,s')$, where $s'$ and $t'$ are two new nodes. It is easy to check that we obtain a new graph $D'$ with $G-e=I(D')$. [*(ii) $\Rightarrow$ (i).*]{} Now assume that $G - e=I(D')$. Let $G''$ be the graph obtained from $G-e$ by adding a node $b''$ and the edge $b''c$. To avoid confusion, we rename the node $b$ by $b'$, see Figure \[trans\]. From Remark \[pendent\], we know there exists a directed graph $D''$ such that $G''=I(D'')$. \(a) at (0,2); (b) at ($ (a) + (0,3) $); (c) at ($ (b) + (3,0) $); (d) at ($ (a) + (3,0) $); (e) at ($ (a) + (10,0) $); (f) at ($ (e) + (0,3) $); (g) at ($ (f) + (3,0) $); (h) at ($ (g) + (0,-1.5) $); (i) at ($ (h) + (0,-1.5) $); (j) at (-2,1.5); (k) at (5,1.5); (l) at ($ (e) + (-2,-0.5) $); (m) at ($ (i) + (+2,-0.5) $); \(a) circle (1.5pt); (b) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (f) circle (1.5pt); (g) circle (1.5pt); (h) circle (1.5pt); (i) circle (1.5pt); \(j) to \[bend left\] (k); (l) to \[bend left\] (m); (a) – (b); (b) – (c); (c) – (d); \(e) – (f); (g) – (h); (h) – (i); \(a) node\[left\] [$a$]{}; (b) node\[above\] [$b$]{}; (c) node\[above\] [$c$]{}; (d) node\[right\] [$d$]{}; (e) node\[left\] [$a$]{}; (f) node\[above\] [$b'$]{}; (g) node\[above\] [$b''$]{}; (h) node\[right\] [$c$]{}; (i) node\[right\] [$d$]{}; By Lemma \[sink\] we may pick $D''$ such that every sink has a most one entering arc. Let $b'=(r,s)$ and $b''=(t,u)$. The nodes $b'$ and $b''$ have no node in common. Indeed, if $b'$ and $b''$ have a common node it must be that $s=u$, and since in $G''$ the nodes $b'$ and $b''$ are not adjacent. And $s$ must be a sink since $b'$ and $b''$ have no common neighbour. It follows that $s$ is a sink having at least two entering nodes, which is a contradiction. Moreover, we may assume that $a\not=(s,r)$ and $c\not=(u,t)$. If $a=(s,r)$. Since $a$ is the unique neighbor of $b'=(r,s)$, we may replace the arc $b'$ by $b=(r,s')$ with $s'$ a new node. The same arguments hold when $c=(u,t)$. Therefore, the connections between the arcs $b'$ and $a$ and between $b''$ and $c$ are of three types as shown in the Figure \[connection\] below. \(a) at (0,2); (b) at ($ (a) + (0,-3) $); (c) at ($ (b) + (3,0) $); (d) at ($ (a) + (7,0) $); (e) at ($ (d) + (0,-3) $); (f) at ($ (d) + (3,0) $); (g) at ($ (f) + (4,0) $); (h) at ($ (g) + (3,0) $); (i) at ($ (h) + (0,-3) $); (j) at ($ (b) + (1.5,-1.5) $); (k) at ($ (e) + (1.5,-1.5) $); (l) at ($ (g) + (1.5,-4.5) $); \(a) circle (1.5pt); (b) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (f) circle (1.5pt); (g) circle (1.5pt); (h) circle (1.5pt); (i) circle (1.5pt); \(b) circle(0.25); (c) circle(0.25); (d) circle(0.25); (f) circle(0.25); (g) circle(0.25); (a) – (b) node\[midway,right\][$a(c)$]{}; (b) – (c) node\[midway,below\][$b'(b'')$]{}; (d) – (e) node\[midway,right\][$a(c)$]{}; (d) – (f) node\[midway,above\][$b'(b'')$]{}; (g) – (h) node\[midway,above\][$b'(b'')$]{}; (h) – (i) node\[midway,right\][$a(c)$]{}; (j) node\[below\] [(I)]{}; (k) node\[below\] [(II)]{}; (l) node\[below\] [(III)]{}; In some cases a simple identification of the arcs $b'$ and $b''$ give rise to a graph $D$ with $I(D)=G$. Identifying the arcs $b'=(r,s)$ and $b''=(t,u)$ means that we remove them and we shrink $r$ with $t$ and $s$ with $u$, where $r'$ and $s'$ are the resulting nodes, respectively. Finally we put $b=(r',s')$. The cases where such an operation may be done are summarized in the Table \[tab:comp\] below. -- ------- ----- ------ ------- (I) (II) (III) (I) (II) (III) -- ------- ----- ------ ------- : Compatibility between types[]{data-label="tab:comp"} If a connection of type (II) occurs, say between $a$ and $b'$, we may replace the arc $a=(r,v)$ by $a=(s,v)$. The head of $a$ is unchanged but its tail now coincide with the head of $b'$. This transformation will not alter $G''$, that is $G''$ is again the intersection graph of the resulting directed graph. Moreover the connection between $a$ and $b'$ is now of type (III). Therefore the only remaining case to study is when the connection between $a$ and $b'$ and between $c$ and $b''$ are both of type (III). Recall that $c$ is adjacent to only $b''$ and $d$ in $G''$. Let $d=(v,w)$. If the head of the arc $c$ is $v$ in $D''$, we set $c=(t,v)$. Again $G''$ is the intersection graph of this new directed graph and the connection of $c$ and $b''$ is of type (II). Now assume that $w=u$, in this case we may set $b''$ to be the arc going from the head of $c$ to $t$ and we obtain a connection of type (I) between, $b''$ and $c$. Notice that Lemma \[degre2\] is not true when $b$ and $c$ are two adjacent nodes of degree two with a common neighbor $a$, that is $a,b,c$ is a triangle. In fact, the graph shown in Figure \[2tri\] (a) is not a FL graph, but the graph obtained by removing the edge $e=bc$ is a FL graph. And the graph shown in Figure \[2tri\](b) is a FL graph, but if we remove the edge $e=bc$ the resulting graph is not anymore a FL graph. But we may easily check that when $a$ has only one neighbor different from $b$ and $c$, then Lemma \[degre2\] holds. \(a) at (0,2); (b) at ($ (a) + (0,-2) $); (c) at ($ (b) + (2,1) $); (d) at ($ (c) + (0,-2) $); (e) at ($ (c) + (2,1) $); (f) at ($ (e) + (0,-2) $); (g) at ($ (e) + (4,1) $); (h) at ($ (g) + (0,-2) $); (j) at ($ (g) + (2,-1) $); (k) at ($ (j) + (0,-2) $); (l) at ($ (k) + (0,-2) $); (i) at ($ (k) + (-2,-1) $); (m) at ($ (j) + (2,-1) $); (n) at ($ (j) + (4,0) $); (o) at ($ (n) + (0,-2) $); (p) at ($ (d) + (0,-1.5)$); (q) at ($ (l) + (1,-0.5) $); \(a) circle (1.5pt); (b) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (f) circle (1.5pt); (g) circle (1.5pt); (h) circle (1.5pt); (i) circle (1.5pt); (j) circle (1.5pt); (k) circle (1.5pt); (l) circle (1.5pt); (m) circle (1.5pt); (n) circle (1.5pt); (o) circle (1.5pt); \(a) – (b); (a) – (c); (b) – (c); (c) – (d); (c) – (e); (e) – (f); (c) – (f); (j) – (g); (j) – (h); (j) – (k); (k) – (l); (k) – (i); (i) – (l); (k) – (m); (j) – (m); (m) – (n); (m) – (o); (n) – (o); (c) node\[above\][$a$]{}; (e) node\[above\][$b$]{}; (f) node\[below\][$c$]{}; (m) node\[above\][$a$]{}; (n) node\[above\][$b$]{}; (o) node\[below\][$c$]{}; (p) node\[below\][(a)]{}; (q) node\[below\][(b)]{}; Application to triangle-free facility location graphs {#app-tri-free} ----------------------------------------------------- \[fork\] Let $G$ be a connected triangle-free graph that does not contain two adjacent nodes having both degree two. If $G$ is a FL graph, that is there exists a directed graph $D=(V,A)$ with $G=I(D)$, then for each vertex $v\in V$ we have (i) $v$ is the tail of at most one arc or (ii) $v$ is the tail of exactly two arcs $(v,v_1)$ and $(v,v_2)$ and there is no arc leaving $v_1$ and $v_2$. Since $G$ is triangle-free, $v$ is the tail of at most two arcs. Therefore we may assume it is the tail of exactly two arcs $(v,v_1)$ and $(v,v_2)$. Notice that there is no arc entering $v$, otherwise $G$ contains a triangle. Assume that for each node $v_i$, the arc $(v_i,v'_i)$ exists. Notice that there are no other arcs leaving $v_1$ or $v_2$. Call $b=(v,v_1)$ and $c=(v,v_2)$. We have that $b$ and $c$ are two adjacent nodes in $G$ having degree two. \[triangle-free\] Let $G$ be a triangle-free graph. Define $G'$ to be the graph obtained from $G$ by removing each edge $e=bc$ where $b$ and $c$ have degree two. Then $G$ is a FL graph if and only if $G'$ admits at most one cycle. From Lemma \[degre2\], we may assume that $G$ does not contain an edge $e=bc$ where $b$ and $c$ have degree two. [*Necessity.*]{} Let $G$ be a triangle-free facility location graph, that is there exist a directed graph $D=(V,A)$, with $G=I(D)$. Also assume that there are no two adjacent nodes of degree two. Suppose that there is a connected component of $G$ containing two cycles $C_1$ and $C_2$. We may assume that both $C_1$ and $C_2$ are chordless cycles. By Lemma \[cycle\], $C_1$ may be partitioned into $C'_1$ and $C''_1$ and $C_2$ may be partitioned into $C'_2$ and $C''_2$. We have that $C'_1$ and $C'_2$ are two cycles in $D$ and $C''_1$ and $C''_2$ are as defined in Lemma \[cycle\]. From Lemma \[fork\], we have $\dot{C'_1}=\dot{C'_2}=\emptyset$. That is both $C'_1$ and $C'_2$ are directed cycles and that $C''_1=C''_2=\emptyset$. Notice that $C_1$ and $C_2$ have no node in common, otherwise $C'_1$ and $C'_2$ must share at least one arc, and since they are both directed cycles, we must create a triangle in $G$. Since $C_1$ and $C_2$ belong to the same connected component we must have a path $P$ in $G$ connecting a vertex $u$ of $C_1$ and a vertex $v$ of $C_2$. Let $P=(v_1=(u_1,u'_1), v_2=(u_2,u'_2),\dots,v_{p-1}=(u_{p-1},u'_{p-1}), v_p=(u_p,u'_p))$, with $u=v_1$ and $v=v_p$. Then $u'_1=u'_2$ and $u'_{p-1}=u'_p$. In this case, $P$ cannot be a directed path in $G$. It follows, that there exists at least one node $u_k$ that contradicts Lemma \[fork\]. [*Sufficiency.*]{} Consider a connected component of $G$. Suppose that it consists of a tree. Let us construct a directed graph $D$ with $G=I(D)$. Pick any node $r$ as a root. Let $r=(u_0,v_0)$. Let $r_1,\dots,r_k$ be the children of $r$ in $G$, we set $r_i=(v_i,u_0)$ for $i=1,\dots,k$. Now each node $r_i$ play the role of $r$ and we repeat this step. This procedure ends with a directed graph $D$ such that $G=I(D)$. Suppose that there is a cycle $C$. This cycle must be chordless. Let $C'$ be a directed cycle where each arc in $C'$ correspond to a node in $C$. The rest of this component consist of disjoint trees each intersect $C$ in one node. If this node is chosen to be the root of the tree, then the procedure above may be applied to get a directed graph $D$ such that $G=I(D)$. As a consequence we obtain the following result. Given an undirected triangle-free graph $G=(V,E)$, we may decide whether or not $G$ is a facility location graph in $O(|E|)$. In $O(|E|)$ we may remove all the edges $e=bc$ with both $b$ and $c$ of degree two. Then we apply a breadth-first search in $O(|E|)$. If a node is encountered more than twice or there are two nodes that were encountered twice, then there are two cycles. Otherwise $G$ is a facility location graph. Consequences and related problems {#RP} ================================= The vertex coloring problem ---------------------------- A [*vertex color*]{} of a graph is an assignment of colors to the nodes of the graph such that no two adjacent nodes receive the same color. The minimum number needed for a such coloring is called the [*chromatic number*]{} and denoted by $\chi(G)$. It is well known that finding $\chi(G)$ is <span style="font-variant:small-caps;">np</span>-complete for triangle-free graphs. A direct consequence of the previous section shows that $\chi(G)\leq 3$ when $G$ is a triangle-free facility location graph. Let $G=(V,E)$ be a triangle-free facility location problem and $G'$ the graph defined in Theorem \[triangle-free\]. It follows that each connected component of $G'$ contains at most one cycle. If there is an odd cycle then $\chi(G')=3$, otherwise $\chi(G')=2$. Let us extend the coloring of $G'$ to $G$. The reconstructing of $G$ from $G'$ implies that at each step we add an edge $e=bc$ between two pendent nodes of $G'$. We keep calling the graph obtained at each step $G'$, until the last step that produce $G$. Let $b'$ and $c'$ be, respectively, the unique neighbors of $b$ and $c$ in $G'$. If $b$ and $c$ have not the same color then we add $e=bc$ without altering the existing coloring. Assume that $b$ and $c$ have the same color. If $b'$ and $c'$ have different colors, then we may assign the color of $c'$ to $b$, or the color of $b'$ to $c$. Finally if $b'$ and $c'$ have the same color, then we pick arbitrarily $b$ or $c$ and we assign him a third available color. From the discussion above, we have the following: \[coloration\] If $G$ is a triangle-free facility location graph, then $\chi(G)\leq 3$. Moreover, $\chi(G)$ may be computed in $O(|E|)$. A natural question arises: whether or not coloring facility locations graphs is polynomial. Unfortunately the answer is no. This will be shown using a reduction from the edge coloring problem. Given an undirected graph $G=(V,E)$ an [*edge color*]{} of $G$ is a coloring of the edges that gives two different colors for each pair of incident edges. Coloring facility locations graphs is <span style="font-variant:small-caps;">np</span>-complete. Given a undirected graph $G=(V,E)$ and positive integer $k$, the problem of deciding wether or not we can color the edges of $G$ with $k$ colors has been proved to be <span style="font-variant:small-caps;">np</span>-complete by Holyer in [@Hoy]. We will reduce this problem to the problem of whether or not we can color the nodes of a facility location graph with $k$ colors. Precisely, from $G=(V,E)$ we build a directed graph $D$ such that the edges of $G$ may be colored with $k$ colors if and only if the nodes of $I(D)$ may be colored with $k$ colors. Let $E=\{e_1,\dots,e_m\}$. We build $D$ as follows: for each edge $e_i=uv\in E$ we add a node $v_i$ and the arcs $(u,v_i)$ and $(v,v_i)$. For each node $v_i$ we add $k-1$ arcs $(v_i,v_{i_1}),\dots,(v_i,v_{i_{k-1}})$, where $v_{i_1},\dots,v_{i_{k-1}}$ are new nodes. Assume that $G$ admits an edge coloring with $k$ colors. Assign the color of each edge $e_i=uv$ to the nodes $(u,v_i)$ and $(v,v_i)$ of $I(D)$ and color the nodes $(v_i,v_{i_1}),\dots,(v_i,v_{i_{k-1}})$ with the other $k-1$ colors. This is a vertex coloring of $I(D)$ with $k$ colors. Now assume $I(D)$ admits a vertex coloring with $k$ colors. Since each node $(v_i,v_{i_1}),\dots,(v_i,v_{i_{k-1}})$ of $I(D)$ must receive a different colors (because they form a clique in $I(D)$) we have that both nodes $(u,v_i)$ and $(v,v_i)$ must have the same color, then assign this color ro the edge $e_i=uv$ of $G$. The resulting coloring is an edge coloring of $G$ with $k$ colors or less. The stable set problem ---------------------- Given an undirected graph $G=(V,E)$, a subset of nodes $S\subseteq V$ of an undirected graph is called a [*stable set*]{} if there is no edge between any two nodes of $S$. The [*maximum stable set problem*]{} is to find a stable set of maximum size. This size is usually called the [*stability number*]{} and denoted by $\alpha(G)$. If we associate a weight $w(v)$ to each vertex $v\in V$, then the [ *maximum weighted stable set problem*]{} if to find a stable set $S$ with $\sum_{v\in S}w(v)$ maximum. The maximum stable set problem is <span style="font-variant:small-caps;">np</span>-complete for triangle-free graph. One may show this result using the following transformation due to Poljak [@Poljak]. Given any undirected graph $G=(V,E)$ replace any edge $e=uv$ in $E$ by a path $uu',u'u'',u''v$. The resulting graph $\textsc{Sub}_G$ is triangle-free and $\alpha(\textsc{Sub}_G)=\alpha(G)+|E|$. This shows that the maximum stable set problem is <span style="font-variant:small-caps;">np</span>-complete in triangle-free graphs. Using Theorem \[triangle-free\] we have that $\textsc{Sub}_G$ is also a facility location graph, since the removal of the edges $u'u''$ yields a graph where each connected component is a star. As a consequence we obtain the following result, \[tfg-complete\] The maximum stable set problem is <span style="font-variant:small-caps;">np</span>-complete in triangle-free facility location graphs. Since from Theorem \[coloration\] one may color the vertices of any triangle-free facility location graph with 3 colors in $O(|E|)$, this immediately implies a 3-approximation algorithm for the maximum stable set problem. This remains true for the maximum weighted stable set problem. In fact, let $V_1$, $V_2$, $V_3$ be a partition of $V$ where each subset $V_i$ is stable. Let $V'_i\subseteq V_i$, be the nodes of $V_i$ having only positive weights, for $i=1,\dots,3$. Let $w(V'_1)=\mbox{max }\{w(V'_2), w(V'_3)\}$ and $S^*$ the stable set of maximum weight. We have $$w(S^*)\leq w(V'_1)+w(V'_2)+w(V'_3)\leq 3w(V'_1).$$ The facility location problem ----------------------------- Recall that the uncapacitated facility location problem (UFLP) associated with a directed graph $D$ is equivalent to the maximum weighted stable set problem with respect to $I(D)$. Therefore, from Theorem \[tfg-complete\] we have the following corollary. The uncapacitated facility location problem associated with directed graph $D$ is <span style="font-variant:small-caps;">np</span>-complete even when $D$ does not contain the four graphs of Figure \[fig:triangle\] as subgraphs. \(a) at (0,2); (b) at ($ (a) + (-1.5,-2) $); (c) at ($ (a) + (0,-2) $); (d) at ($ (a) + (1.5,-2) $); (e) at (4,1.5); (f) at ($ (e) + (0,1.5) $); (g) at ($ (e) + (-0.75,-1.5) $); (h) at ($ (e) + (0.75,-1.5) $); (k) at (8,2); (i) at ($ (k) + (-1.3,-2) $); (j) at ($ (k) + (1.3,-2) $); (m) at (12,2); (l) at ($ (m) + (0,2) $); (p1) at ($ (c) + (0,-1.5) $); (s1) at ($ (h) + (-0.5,-1.5) $); (v1) at (8,-1.5); (o) at (12,1.5); (p) at ($ (o) + (0,1.5) $); (q) at ($ (o) + (0,-1.5) $); (o1) at ($ (v1) + (4,0) $); \(a) circle (1.5pt); (b) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (f) circle (1.5pt); (g) circle (1.5pt); (h) circle (1.5pt); (i) circle (1.5pt); (j) circle (1.5pt); (k) circle (1.5pt); (o) circle (1.5pt); (p) circle (1.5pt); (q) circle (1.5pt); \(a) – (b); (a) – (c); (a) – (d); (p1) node\[below\] [$T_1$]{}; \(f) – (e); (e) – (h); (e) – (g); (s1) node\[below\] [$T_2$]{}; \(i) – (j); (j) – (k); (k) – (i); (v1) node\[below\] [$T_3$]{}; \(o) to\[bend left=45\] (p); (p) to\[bend left=45\] (o); (o) – (q); (o1) node\[below\] [$T_4$]{}; In the following we will show that the UFLP remains <span style="font-variant:small-caps;">np</span>-complete even for a more restricted class of graphs. An undirected graph $G=(V,E)$ is called [*cubic*]{} is the degree of each vertex is 3. A [*bridge*]{} is an edge such that its deletion increase the number of connected components. A [*bridgeless*]{} graph is a graph with no bridge. We have the following well know result. [@GJS] \[GJ\] The maximum stable set problem in cubic graphs is <span style="font-variant:small-caps;">np</span>-complete. In [@GJS] it has been shown that the [*minimum vertex cover*]{} problem is <span style="font-variant:small-caps;">np</span>-complete, here we look for a subset of nodes with minimum cardinality, such that each edge has at least one endnode in this set. Notice that if $S$ is a minimum vertex cover, then $\bar{S}=V\setminus S$ is a maximum stable set. Then both problems minimum vertex cover and maximum stable set are equivalent in the same graph without any transformation. We also notice that the proof in [@GJS] use a reduction of 3-<span style="font-variant:small-caps;">sat</span> to the minimum vertex cover problem. The graph constructed from a 3-<span style="font-variant:small-caps;">sat</span> instance is bridgeless and each node has at most degree 3. Moreover, each node with degree 2 has two non-adjacent nodes of degree 3. Thus we can remove this nodes and connect its two neighbors. It is easy to check that if one can solve the minimum vertex cover problem in this new graph, then one may solve it in the original graph too. From this discussion and Theorem [@GJS] we have the following corollary. \[coro-GJS\] The maximum stable set problem in a bridgeless cubic graph is <span style="font-variant:small-caps;">np</span>-complete. In addition to the forbidden subgraphs $T_1$, $T_2$, $T_3$ and $T_4$ we also add the subgraphs $F_1$ and $F_2$ of Figure \[fig:3in\], and the UFLP remains <span style="font-variant:small-caps;">np</span>-complete. \(a) at (0,2); (b) at ($ (a) + (0,1.5) $); (c) at ($ (a) + (1,1.5) $); (d) at ($ (a) + (-1,1.5) $); (e) at ($ (a) + (0,-1.5) $); (a2) at ($ (a) + (4,0) $); (c2) at ($ (c) + (4,0) $); (d2) at ($ (d) + (4,0) $); (e2) at ($ (e) + (4,0) $); (s1) at ($ (e) + (0,-0.5) $); (s2) at ($ (e2) + (0,-0.5) $); \(a) circle (1.5pt); (b) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (a2) circle (1.5pt); (c2) circle (1.5pt); (d2) circle (1.5pt); (e2) circle (1.5pt); \(b) – (a); (c) – (a); (d) – (a); (a) – (e); (c2) – (a2); (d2) – (a2); (a2) to\[bend left=45\] (e2); (e2) to\[bend left=45\] (a2); (s1) node\[below\] [$F_1$]{}; (s2) node\[below\] [$F_2$]{}; \[UFLP-NP\] The uncapacitated facility location problem is <span style="font-variant:small-caps;">np</span>-complete for graphs that do not contain any of $T_1$, $T_2$, $T_3$, $T_4$, $F_1$ and $F_2$ as a subgraph. Let $G=(V,E)$ be an undirected bridgeless cubic graph. From $G$ define the subdivision of it, $\textsc{Sub}_G$, as in the previous subsection, that is each edge $e=uv\in E$ is replaced by path of size three. Now we construct a directed graph $D$ containing none of the graphs $T_1$, $T_2$, $T_3$, $T_4$, $F_1$ and $F_2$ as a subgraph and such that $I(D)=\textsc{Sub}_G$. Thus from Corollary \[coro-GJS\] the maximum weighted stable set problem is <span style="font-variant:small-caps;">np</span>-complete in bridgeless cubic graphs, and by equivalence we have that UFLP is also <span style="font-variant:small-caps;">np</span>-complete in graphs satisfying the theorem’s hypothesis. Now let us give the construction of $D$. From Petersen’s theorem [@pet], the graph $G$ contains a perfect matching $M$. Let $G'$ be the graph obtained by removing $M$. Each component of $G'$ is a chordless cycle. Let $C=v_0,v_1,\dots,v_p$ be one of these cycles. In $\textsc{Sub}_G$ this cycle corresponds to a cycle $C'=v_0,v_1,v_2,\dots,v_{3p},v_{3p+1},v_{3p+2}$. Let us construct a directed graph $D$ with $I(D)=\textsc{Sub}_G$. Each cycle $C'$ of $\textsc{Sub}_G$ may be defined in $D$ by the directed cycle where the arc $v_i$ enters the arc $v_{i+1}$ for each $i=0,\dots,3p+1$, and the arc $v_{3p+2}$ enters the arc $v_0$ (an arc $a$ enters an arc $b$ means that the head of $a$ coincide with the tail of $b$). To complete the definition of $D$ we need to consider all the edges of $M$ and their subdivisions. Let $e=uv\in M$ and $u_1,u_2,u_3,u_4$ the corresponding path in $\textsc{Sub}_G$. Complete the construction of $D$ by creating for every such edge $e$ two arcs $u_2$ and $u_3$ having the same tail where $u_2$ enters the arc $u_1$ and $u_3$ enters the arc $u_4$. This transformation is depicted in Figure \[illust\]. By construction we have that $I(D)=\textsc{Sub}_G$ and that each node is the head of at most two arcs, hence $F_1$ is not present in $D$ and it is easy to see that with construction $F_2$ cannot occur. Also there are no $T_1$, $T_2$, $T_3$ and $T_4$ in $D$ since $\textsc{Sub}_G$ is triangle-free, see Figure \[illust\]. Concluding remarks {#conc} ================== In this paper we studied the class of facility location graphs. These graphs come from the classical and well studied uncapacitated facility location problem. We have shown that the recognition problem of these graphs is <span style="font-variant:small-caps;">np</span>-complete in general and polynomially solvable in free-triangle graphs. As a consequence, we observed that the stable set problem still <span style="font-variant:small-caps;">np</span>-complete on a more restricted class than triangle-free graphs and that three colors suffice to color the vertex set of a triangle-free facility location graph. We also studied the complexity of two problems (1) the vertex coloring problem in facility location graphs and (2) the uncapacitated facility location problem in graphs that do not contain as a subgraph the graphs $T_1$, $T_2$, $T_3$ , $T_4$, $F_1$ and $F_2$. Let us discuss a natural attempt for restricting more this class of graphs. We know from [@BBdo1; @Stauffer] that if the graph $F_3$ of Figure \[Y\] is forbidden, then UFLP is polynomially solvable. Now consider a graph without any of the subgraphs $T_1,\dots,T_4$, $F_1$ and $F_2$ and containing the subgraph $F_3$. There is at least an arc leaving the node $u$ of $F_3$, otherwise by definition any feasible solution of UFLP must contain $u$ and in that case $u$ may be splitted into several copies depending on the number of arcs entering it. The arc leaving $u$ must have a head that do not belong to $F_3$, which lead to the graph $F_4$ of Figure \[Y\]. Now if we consider a directed graph $D$ with no $F_4$ and since we do not have $F_1$ and $F_2$, the intersection graph $I(D)$ is claw-free and hence the maximum stable set problem is polynomially solvable [@Minty; @Sbihi; @Faenza]. Equivalently, the UFLP is polynomially solvable if, in addition to the hypothesis of Theorem \[UFLP-NP\], we forbid also the subgraph $F_4$. \(a) at (0,2); (c) at ($ (a) + (1,1.5) $); (d) at ($ (a) + (-1,1.5) $); (e) at ($ (a) + (0,-1.5) $); (a1) at ($ (a) + (4,1.5) $); (c1) at ($ (c) + (4,1.5) $); (d1) at ($ (d) + (4,1.5) $); (e1) at ($ (e) + (4,1.5) $); (f) at ($ (e1) + (0,-1.5) $); (s1) at ($ (e) + (0,-0.5) $); (s2) at ($ (f) + (0,-0.5) $); \(a) circle (1.5pt); (c) circle (1.5pt); (d) circle (1.5pt); (e) circle (1.5pt); (a1) circle (1.5pt); (c1) circle (1.5pt); (d1) circle (1.5pt); (e1) circle (1.5pt); (f) circle (1.5pt); \(c) – (a); (d) – (a); (a) – (e); (c1) – (a1); (d1) – (a1); (a1) – (e1); (e1) – (f); \(e) node\[right\] [$u$]{}; (e1) node\[right\] [$u$]{}; (s1) node\[below\] [$F_3$]{}; (s2) node\[below\] [$F_4$]{}; Acknowledgements {#acknowledgements .unnumbered} ================ The authors wish to thank Reza Naserasr for fruitful discussions. Appendix {#appendix .unnumbered} ======== Figures to explicitely prove Lemma \[GAD2\].

--- abstract: 'We prove several results on the lifespan, regularity, and uniqueness of solutions of the Cauchy problem for the homogeneous complex and real equations (HCMA/HRMA) under various a priori regularity conditions. We use methods of characteristics in both the real and complex settings to bound the lifespan of solutions with prescribed regularity. In the complex domain, we characterize the $C^3$ lifespan of the HCMA in terms of analytic continuation of Hamiltonian mechanics and intersection of complex time characteristics. We use a conservation law type argument to prove uniqueness of solutions of the Cauchy problem for the HCMA. We then prove that the Cauchy problem is ill-posed in $C^3$, in the sense that there exists a dense set of $C^3$ Cauchy data for which there exists no $C^3$ solution even for a short time. In the real domain we show that the HRMA is equivalent to a Hamilton–Jacobi equation, and use the equivalence to prove that any differentiable weak solution is smooth, so that the differentiable lifespan equals the convex lifespan determined in our previous articles. We further show that the only obstruction to $C^1$ solvability is the invertibility of the associated Moser maps. Thus, a smooth solution of the Cauchy problem for HRMA exists for a positive but generally finite time and cannot be continued even as a weak $C^1$ solution afterwards. Finally, we introduce the notion of a “leafwise subsolution" for the HCMA that generalizes that of a solution, and many of our aforementioned results are proved for this more general object.' address: - 'Department of Mathematics, Stanford University, Stanford, CA 94305, USA' - 'Department of Mathematics, Northwestern University, Evanston, IL 60208, USA' author: - 'Yanir A. Rubinstein' - Steve Zelditch title: 'The Cauchy problem for the homogeneous Monge–Ampère equation, III. Lifespan' --- Introduction ============ This article is the third in a series [@RZ1; @RZ2] whose aim is to study existence, uniqueness, and regularity of solutions of the initial value problem (IVP) for geodesics in the space $$\label{HoEq} \textstyle\calH_\o = \{\vp\in C^{\infty}(M) \,:\, \omega_\vp:= \omega+\i{\partial{\bar\partial}}\vp>0\}$$ of [Kähler ]{}metrics on a compact [[K]{}\^]{}manifold $(M, \omega)$ in the class of $\omega$, where $\calH_\o$ is equipped with the metric [@M; @S; @D1] $$\gM(\zeta,\eta)_{\vp}:= \int_M \zeta\eta\, {\omega_{\vp}^m},\quad \vp \in {\mathcal{H}}_{\o},\quad \zeta,\eta \in T_{\vp} {\mathcal{H}}_{\o}\isom C^\infty(M).$$ This initial value problem is a special case of the Cauchy problem for the homogeneous complex/real Monge–Ampère equation (HCMA/HRMA). The IVP is long believed to be ill-posed, and a motivating problem is to prove that this is indeed the case, to determine which initial data give rise to solutions, especially those of relevance in geometry (‘geodesic rays’), to construct the solutions, and to determine the lifespan $T_\span$ of solutions for general initial data. In this article, we prove a number of results on the lifespan, regularity, and uniqueness of solutions of the Cauchy problem for the HCMA and the HRMA equations under various a priori regularity conditions. The results are based on a study of the ‘characteristics’ of the HCMA/HRMA equations, or more precisely on the relations between solutions of these equations and Hamiltonian mechanics, and to solutions of related Hamilton–Jacobi equations. First, we characterize the $C^3$ lifespan of the HCMA and prove uniqueness of classical solutions. We then introduce the notion of a leafwise subsolution of the HCMA that generalizes the notion of a solution, and derive obstructions to its existence. This can be considered as a method of ‘complex characteristics’. Combining these results we establish that the IVP for the HCMA is locally ill-posed in $C^3$. This puts a restriction on Cauchy data, and addresses questions about the Cauchy problem raised by the work of Mabuchi, Semmes, and Donaldson [@M p. 238],[@S],[@D1 p. 27]. We then study the notion of a leafwise subsolution for the HRMA, and prove its uniqueness. This allows us to characterize the Legendre transform subsolution of the prequels [@RZ1; @RZ2], and determine the $C^1$ lifespan of the HRMA. A key ingredient here is an apparently new connection between HRMA and Hamilton–Jacobi equations. Obstructions to solvability, uniqueness, and the smooth lifespan of the HCMA ---------------------------------------------------------------------------- We begin in the complex domain, where Semmes and Donaldson [@S; @D1] gave a formal solution of the IVP in terms of holomorphic characteristics. Namely, the Cauchy data $(\omega_{\vp_0}, \dot{\vp}_0)$ of the IVP determines a Hamiltonian flow $\exp t X_{\dot{\vp}_0}^{\omega_{\vp_0}}$. If the orbits $\exp t X_{\dot{\vp}_0}^{\omega_{\vp_0}} z$ of the flow admit analytic continuations in time up to imaginary time $T$, one obtains a family of maps $$\label {MAPS} f_\tau(z) = \exp -\sqrt{-1} \tau X_{\dot{\vp}_0}^{\omega_{\vp_0}} z: S_T \times M \to M,$$ where $$S_T=[0,T]\times{\mathbb{R}},$$ with $\tau=s+\i t\in S_T$, $s\in[0,T]$ and $t\in{\mathbb{R}}$. The formal solution $\vp_s$ is then given by the formula, $$\label {FORMAL} (f_s^{-1})^\star \omega_{\vp_0} - \omega_{\vp_0} = \i {\partial{\bar\partial}}\vp_s, \quad s\in[0,T].$$ There are several obstructions to solving the IVP in this manner, which must vanish if there exists a $C^3$ solution. The most obvious one is that the Hamilton orbits need to possess analytic continuations to a strip $S_T$. This analytic extension problem for orbits should already be an ill-posed problem, and we say that the Cauchy data is “$T$-good" if the extension exists and $f_s$ is smooth (see Definitions \[TGoodDef\]–\[TGoodDef\]). This is a Cauchy problem for a holomorphic map into a nonlinear space, and we do not study it directly here; but in §\[1.2\] we describe some results on obstructions to closely related linear Cauchy problems. In several settings, such as torus-invariant Cauchy data on toric varieties, the Hamilton orbits for smooth Cauchy data do possess analytic continuations (see Proposition \[ToricLifespanProp\] below). As the following theorem shows, the only additional obstruction to solving the HCMA smoothly is that the space-time complex Hamilton orbits may intersect. To state the result precisely, let $(M,J,\o)$ be a compact closed connected [[K]{}\^]{}manifold of complex dimension $n$. The IVP for geodesics is equivalent to the following Cauchy problem for the HCMA $$\label{HCMARayEq} \begin{aligned} (\pi_2^\star\omega + \i{\partial{\bar\partial}}\vp)^{n+1} = 0, \quad (\pi_2^\star\omega + \i{\partial{\bar\partial}}\vp)^{n} \ne 0, \;\; &\mbox{on} \; S_{T} \times M, \cr \vp(0,t,\,\cdot\,) = \vp_0(\,\cdot\,), \quad \partial_s\vp(0,t,\,\cdot\,) = \dot\vp_0(\,\cdot\,), \;\; &\mbox{on} \; \{0\}\times{\mathbb{R}}\times M. \end{aligned}$$ where $\pi_2: S_{T} \times M \to M$ is the projection, and where $\vp$ is is required to be $\pi_2^\star\o$-plurisubharmonic (psh) on $S_T\times M$. The rest of the notions in the following theorem are defined in §\[HamFlowsHCMASection\]. \[HCMACauchyCthreeThm\] [(Smooth lifespan and uniqueness)]{} Let $(M,\omega_{\vp_0})$ be a compact [[K]{}\^]{}manifold. The Cauchy problem (\[HCMARayEq\]) with $\omega_{\vp_0}\in C^1$ and $\dot\vp_0\in C^3(M)$ has a solution in $C^3(S_T \times M)\cap PSH(S_T \times M,\pi_2^\star\o)$ if and only if the Cauchy data is $T$-good and the maps $f_s$ defined by are $C^1$ and admit a $C^1$ inverse for each $s\in[0,T]$. The solution is unique in $C^3(S_T \times M)\cap PSH(S_T\times M,\pi_2^\star\o)$. This result is important in clarifying the nature of the obstructions to solving the HCMA. The existence proof follows by a rather straightforward combination of the Semmes–Donaldson arguments [@S; @D1]. The uniqueness proof, somewhat surprisingly, does not readily adapt from the ${{\mathbb C}}^n$ setting studied by Bedford–Burns [@BB]. Unlike in their setting, the proof is not local in nature, and requires a global conservation law type argument. The key difference is that the stripwise equations vary from leaf to leaf, and one has to prove an a priori estimate that ensures that the stripwise elliptic problems are not degenerating. The uniqueness proof is also completely different from the corresponding proof for the Dirichlet problem, where the maximum principle is available. Henceforth, we describe breakdown in time of solutions in terms of lifespans. Let the $C^{k,\alpha}$ lifespan $T^{k,\alpha}_\span$ (respectively, lifespan $T_\span$) of the Cauchy problem (\[HCMARayEq\]) be the supremum over all $T\ge 0$ such that (\[HCMARayEq\]) admits a solution in $C^{k,\alpha}(S_T \times M)\cap PSH(S_T\times M,\pi_2^\star\o)$ (respectively, in $PSH(S_T\times M,\pi_2^\star\o))$. We thus have the following characterization of the smooth lifespan of the HCMA. The same result holds also for the $C^3$ lifespan $T^3_\span$. \[HCMACauchyLifespanCor\] The smooth lifespan $T^\infty_\span$ of the Cauchy problem (\[HCMARayEq\]) with smooth initial data is the supremum over $T\ge 0$ such that the Cauchy problem is $T$-good and the maps $f_s$ defined by (\[FsMapEq\]) are smoothly invertible for each $s\in[0,T]$. \[1.2\] Leafwise subsolutions for the HCMA and ill-posedness ------------------------------------------------------------ In the apparent absence of weak solutions beyond the convex lifespan, motivated by the detailled results of the prequel [@RZ2] on the Legendre subsolution in the special case of the HRMA, we are led to introduce a notion of a [*leafwise subsolution*]{}, that should be an “optimal" subsolution in some situations. \[OptimalSubsolutionDef\] Assume that the Cauchy problem (\[HCMARayEq\]) is $T$-good. We call a $\pi_2^\star\o$-psh function $\vp$ on $S_T\times M$ a [$T$-leafwise subsolution]{} of the HCMA (\[HCMARayEq\]) if it satisfies the initial conditions of the Cauchy problem (\[HCMARayEq\]), if $(\pi_2^\star\o+\i{\partial{\bar\partial}}\vp)^n\ne0$, and if for each $z\in M$, we have $$\label {LeafwiseDefEq} \gamma_z^\star(\pi_2^\star\o+\i{\partial{\bar\partial}}\vp)=0,$$ where $\gamma_z(\tau)=(\tau, f_{\tau}(z))$, and $f_\tau(z):= \exp-\i \tau X_{\dot\vp_0}^{\omega_{\vp_0}}.z$. The proof of Theorem \[HCMACauchyCthreeThm\] shows that a $C^3$ solution on $[0,T]\times M$ is a $T$-leafwise subsolution, but “leafwise subsolutions" are more general: a subsolution of HCMA may solve along leaves without solving the HCMA globally since the invertibility condition on $f_s$ in Theorem \[HCMACauchyCthreeThm\] may fail, e.g., when the leaves intersect. One of our main results is that the problem of existence of a leafwise subsolution for the Cauchy problem for the HCMA is already locally ill-posed in time. In particular, this implies the ill-posedness in $C^3$ of the Cauchy problem for the HCMA itself. \[GENERIC\] [(Local ill-posedness)]{} For each ${\vp_0} \in {\mathcal{H}}_{\omega}$ there exists a dense set of $\dot\vp_0\in C^3(M)$ for which $T_\span^\infty=T^3_\span=0$, i.e., the IVP admits no $C^3$ solution for any $T>0$. The proof is given in Section \[LeafwiseSection\]. The Cauchy problem for the HCMA is a multi-dimensional, nonlinear generalization of the Cauchy problem for the Laplace equation $\Delta u = 0$ on each strip $S_T$, which is one of the classic ill-posed problems of Hadamard [@H; @L; @P]. One might think that Theorem \[GENERIC\] could be obtained directly from the well-known ill-posedness of the Cauchy problem for the Laplace equation on a strip. However, this is not the case: the leafwise equations are inhomogeneous and depend on the solution of the HCMA itself. Second, and perhaps more basic, is that the strip on which the problem is posed depends on the solution of the HCMA. The standard argument of Hadamard (for the classical Laplace equation) of perturbing the Cauchy data so as to lie outside the range of the Dirichlet-to-Neumann operator therefore cannot be applied directly as it would also perturb the leaves themselves! The actual proof does employ the Dirichlet-to-Neumann operators along each leaf but also uses a geometric perturbation argument. First, we analyze the obstructions for a leafwise subsolution in detail, and show that, for each $z \in M$, the pull-back of a leafwise subsolution under $\gamma_z$ (the map of restriction to the leaf through $z$ defined in Definition \[OptimalSubsolutionDef\]) satisfies a certain real-analyticity condition on the initial boundary $\{0\}\times{\mathbb{R}}$ of $S_T$; more precisely, a certain function of the Cauchy data is real analytic on ${{\mathbb R}}$ and possesses an analytic continuation to a [*two-sided*]{} strip $[-T,T]\times {\mathbb{R}}$ of width precisely $T$. We refer to Proposition \[LeafwiseProp\] for the precise statement. Second, we combine Theorem \[HCMACauchyCthreeThm\] and Proposition \[LeafwiseProp\] with a geometric perturbation argument and basic properties of the Hilbert transform to derive a real-analyticity condition, independent of $T$. Theorem \[GENERIC\] is thus based on the analytic continuation obstruction of Theorem \[HCMACauchyCthreeThm\]. In the remainder of the paper we concentrate on the second obstruction, i.e., the invertibility of the Moser maps $f_s$ appearing in Theorem \[HCMACauchyCthreeThm\]. It is present even in the simplest case of toric Kähler manifolds. As we will see, even when the strip-wise Cauchy problems can all be solved, there does not generally exist a global in time solution of the HCMA. Complementary results for the HRMA ---------------------------------- The local ill-posedness result, Theorem \[GENERIC\], does not apply to the study of the HRMA. The Cauchy problem for the HRMA arises precisely when the Cauchy data is torus-invariant, which is, of course, non-generic in the space of all possible Cauchy data. And in fact, the Cauchy problem for the HRMA has a positive smooth lifespan [@RZ2]. Moreover, as we observe in Proposition \[ToricLifespanProp\], there is no obstruction to analytically continuing orbits. In the remainder of the article our goal is thus to derive results for the HRMA that are somewhat of a complementary nature to those for the HCMA described above. First, we would like to understand how our characterization of the smooth (or $C^3$) lifespan specializes to the setting of the HRMA. Second, we would like to understand lifespan of solutions with less regularity than that described in Theorem \[HCMACauchyCthreeThm\], that is less than $C^3$. Theorem \[HCMACauchyCthreeThm\] clarifies the breakdown of classical solutions of the Cauchy problem for the HCMA already in the toric case. When the Cauchy data is $(S^1)^n$-invariant, the equation reduces to the HRMA $$\label{HRMARayEq} \begin{aligned} \!\!\h{\rm MA}\, \psi = 0, \;\,\mskip2mu &\mbox{on} \; [0,T] \times {\mathbb{R}}^n, \quad\; \psi(0,\,\cdot\,) = \psi_0(\,\cdot\,), \quad\; \partial_s\psi(0,\,\cdot\,) = \dot\psi_0(\,\cdot\,), \;\; \mbox{on} \; {\mathbb{R}}^n, \end{aligned}$$ that describes geodesics in the space of toric [[K]{}\^]{}metrics, where $\h{MA}$ denotes the real operator that associates a Borel measure to a convex function and equals $\det\nabla^2 f\, dx^1\w\cdots\w dx^{n+1}$ on $C^2$ functions (see [@RZ2 §2.2] and §\[ConeLifespanSection\]). Here $\psi_0$ is a smooth strictly convex function (moreover, with strictly positive Hessian) on ${\mathbb{R}}^n$ that corresponds to a torus-invariant [[K]{}\^]{}metric, i.e., $\o_{\vp_0}=\i{\partial{\bar\partial}}\psi_0$ over the open orbit and $\dot\vp_0$ is a smooth torus-invariant function on $M$, considered as a smooth bounded function $\dot\psi_0$ on ${\mathbb{R}}^n$. Thus, we view ${\mathbb{R}}^n$ as the real slice of $M$ (minus its divisor at infinity). We also refer to the real slices of the leaves of the foliation as leaves. Also, $\overline{\Im\nabla\psi_0}=P$ is a compact convex polytope in ${\mathbb{R}}^n$. Translating the definition of $\pi_2^*\omega_0$-psh solutions to the HCMA to the real setting yields a corresponding class for the HRMA. Before defining the class we recall the definition of the operator. Let $M({\mathbb{R}}^{n+1})$ denote the space of differential forms of degree $n+1$ on ${\mathbb{R}}^{n+1}$ whose coefficients are Borel measures (i.e., currents of degree $n+1$ and order 0). [(See [@RT Proposition 3.1])]{} Define by $$\h{\rm MA} f:= d\frac{\partial f}{\partial x^1}\w \cdots\w d\frac{\partial f}{\partial x^{n+1}},$$ an operator $\h{\rm MA}: C^2({\mathbb{R}}^{n+1})\ra M({\mathbb{R}}^{n+1})$. Then $\h{\rm MA}$ has a unique extension to a continuous operator on the cone of convex functions. A result of Alexandrov shows that for any convex function $f$, the measure $\calM\calA\, f$, defined by $ (\calM\calA\, f)(E):=\h{\rm Lebesgue measure of\ } \partial f(E), $ where $\del f$ denotes the subdifferential mapping of $f$ (see [@RZ2 §2.1]), is a Borel measure ([@RT Section 2]). Furthermore, according to Rauch–Taylor $\h{\rm MA} f = \calM\calA\, f$ for every convex function $f$ on ${\mathbb{R}}^{n+1}$ [@RT Proposition 3.4]. \[AlexandrovDef\] A convex function $\rho$ on $[0,T]\times{\mathbb{R}}^n$ is an Alexandrov weak solution of the HRMA $\h{\rm MA}\, \rho =0$, if the image of the subdifferential mapping $\del\rho:[0,T]\times{\mathbb{R}}^n\ra{\mathbb{R}}^{n+1}$ is a set of Lebesgue measure zero. We now define our class of “admissible solutions" to the HRMA to be Alexandrov weak solutions with the property that the image under the spatial sub-differential of the solution is a fixed polytope for all times. The assumption means that the solutions are potentials of non-degenerate [[K]{}\^]{}metrics for each $s$ that stay in the same [[K]{}\^]{}class. It seems that only these solutions are relevant to toric Kähler geometry. \[AdmissibleDef\] An admissible subsolution to is a convex function on $[0,T]\times{\mathbb{R}}^n$ satisfying (i) $\psi(0,\,\cdot\,) = \psi_0(\,\cdot\,)$ and $\partial_s\psi(0,\,\cdot\,) = \dot\psi_0(\,\cdot\,)$ on ${\mathbb{R}}^n$, and (ii) $\psi(s):{\mathbb{R}}^n\ra{\mathbb{R}}$ is strictly convex, and $\overline{\Im\,\del\psi(s)}=P$ for each $s\in[0,T]$. An admissible solution in addition is a weak solution of $\h{\rm MA}\, \psi = 0$ in the sense of Alexandrov. Note that this definition assumes $\psi_0$ and $\dot\psi_0$ to satisfy the regularity, growth, and convexity assumptions of the previous paragraphs. In the previous article, we showed that the Legendre transform method for solving the HRMA breaks down at the [*convex lifespan* ]{} $$\label{ToricTspanEq} T_{\span}^\cvx := \sup\,\{\,s>0: \psi^\star_0-s\dot \psi_0\circ(\nabla\psi_0)^{-1} \hbox{\rm\ is convex}\},$$ where $\psi_0^\star$ denotes the Legendre transform of $\psi_0$ [@RZ2 Theorem 1]. The next proposition shows that there is no obstruction for the Hamilton orbits to admit analytic extensions to strips nor for the maps to be smooth, and that the only obstruction to smooth solvability is the invertibility of these maps, that we refer to as Moser maps (see Definition \[MoserMapDef\]). \[ToricLifespanProp\] Let $(M,J,\o_{\vp_0})$ be a toric [[K]{}\^]{}manifold, and let $\dot \vp_0\in C^3(M)$ be torus-invariant. Then,(i) The Cauchy problem (\[HCMARayEq\]) for $(\o_{\vp_0},\dot\vp_0)$ is $T$-good for every $T>0$. (ii) The maps $f_s(z)=\exp-\i sX_{\dot\vp_0}^{\o_{\vp_0}}.z$ (\[MAPS\]) are invertible if and only if $\,s\in[0,T^\cvx_\span)$. This result, together with Theorem \[HCMACauchyCthreeThm\], determines the smooth lifespan for toric geodesics, as well as characterizes all smooth toric geodesic rays. \[ToricGeodesicRayCor\] [(Characterization of smooth toric geodesics)]{} (i) The smooth lifespan of the Cauchy problem with smooth Cauchy data coincides with the convex lifespan , $T^\infty_\span=T_\span^\cvx$. (ii) Smooth geodesic rays in the space of toric metrics are in one-to-one correspondence with admissible solutions of the Cauchy problem (\[HRMARayEq\]) with $\psi_0\in C^\infty({\mathbb{R}}^n)$, $\nabla^2\psi_0>0$, $\overline{\Im\nabla\psi_0}=P$, $\dot\psi_0\in C^\infty\cap L^\infty({\mathbb{R}}^n)$, and $\dot \psi_0\circ(\nabla\psi_0)^{-1}$ a concave function on $P$. Next, we show that in the case of the HRMA the leafwise obstruction vanishes and characterizes the Legendre transform subsolution among all subsolutions of the Cauchy problem. \[OptimalSubsolutionToricProp\] (i) The Legendre transform potential, given by $$\label{OptimalSubsolutionToricEq} \psi_L(s,x) := (\psi^\star_0-s \dot\vp_0\circ(\nabla\psi_0)^{-1})^\star(z),\quad x\in {\mathbb{R}}^n, \; s\in{\mathbb{R}}_+,$$ is the unique admissible leafwise subsolution to the HRMA (\[HRMARayEq\]) for all $T>0$. (ii) The corresponding unique admissible leafwise subsolution to the HCMA (\[HCMARayEq\]) is given by $$\label {ToricHCMALeafwiseSubSolEq} \vp_L(s+\i t,e^{x+\i\th}):=\psi_L(s,x)-\psi_0(x).$$ Observe that the uniqueness result in (i) holds under much weaker regularity than that needed in Theorem \[HCMACauchyCthreeThm\]. However, the possibility remains that a solution could persist beyond $\Tspancvx$, but not be given by the Legendre transform method. But by following the lead of Theorem \[HCMACauchyCthreeThm\] in the case of the HRMA, we show that there cannot exist any $C^1$ weak solution in the Alexandrov sense beyond $T^{\cvx}_{\span}$. The result is a regularity statement. \[ConeLifeSpanThm\] [($C^1$ lifespan of HRMA)]{} Any admissible $C^1$ weak solution to the Cauchy problem (\[HRMARayEq\]) with Cauchy data $\psi_0\in C^\infty({\mathbb{R}}^n)$, $\nabla^2\psi_0>0$, $\overline{\Im\nabla\psi_0}=P$, $\dot\psi_0\in C^\infty\cap L^\infty({\mathbb{R}}^n)$ is smooth. Thus, $T^1_\span=T^\cvx_\span$. This generalizes a classical theorem of Pogorelov on the developability of flat (in a suitable sense) $C^1$ surfaces in ${\mathbb{R}}^3$. In the language of geodesics in the infinite dimensional symmetric space ${\mathcal{H}}_\o$ [@M; @S; @D1], it shows that the exponential map fails to be globally defined even when $C^1$ weak solutions are allowed. It is interesting to observe that Pogorelov’s result for $n=1$ involves a quite intricate proof [@Po; @Sa]. In higher dimensions, this result has been known previously under the rather stronger assumption of $C^2$ regularity or more, i.e., for classical solutions [@HN; @Fo1; @Fo2; @U]. The proof of Theorem \[ConeLifeSpanThm\] uses the following characterization of the HRMA in terms of a Hamilton–Jacobi equation: \[HJThm\] [(HRMA and Hamilton–Jacobi)]{} $\eta\in C^1([0,T\times{\mathbb{R}}^n)$ is an admissible weak solution of the HRMA if and only if it is a classical solution of the Hamilton–Jacobi equation $$\label{HJEq} F(\nabla \eta)=0, \qquad \eta(0,\,\cdot\,)=\psi_0,$$ where $F(\sigma,\xi)=\sigma-\dot \psi_0\circ (\nabla\psi_0)^{-1}(\xi)$, where $\sigma\in{\mathbb{R}},\xi\in {\mathbb{R}}^n$. Theorem \[HJThm\] reduces the HRMA to a first-order equation for which a well-known theory for solutions exists—based on the method of characteristics. The Hamilton–Jacobi equation is a ‘conservation law’ for the HRMA. It may be viewed as combining the conservation law $\dot{\varphi}_s \circ f_s = \dot{\varphi}_0$ of Proposition \[FTPHIDOT\] (see for notation) with the explicit formula for $f_s^{-1}$ in ; see also . This makes rigorous as well as generalizes to weak solutions the folklore idea [@HN; @Fo1; @Fo2; @U] that classical solutions to HRMA—despite being of second-order—can be obtained by integrating along ‘characteristics’ just like a first-order equation, indeed they are affine along lines determined by the Cauchy data. Theorem \[ConeLifeSpanThm\] follows from Proposition \[OptimalSubsolutionToricProp\] and Theorem \[HJThm\]. Except for one step (Proposition \[ConeadmissibleIsOptimalProp\]), the proof is short and we give it here: Given the results of [@RZ2], the main new step of the proof of Theorem \[ConeLifeSpanThm\] is the following generalization to weak $C^1$ admissible solutions of HRMA  of the fact (see Section \[HamFlowsHCMASection\]) that every $C^3$ $\pi_2^\star\o$-psh solution of the HCMA  is a leafwise subsolution. \[ConeadmissibleIsOptimalProp\] Let $(M,J,\o_{\vp_0})$ be a toric [[K]{}\^]{}manifold, and let $\dot \vp_0\in C^\infty(M)$ be torus-invariant. Assume that the corresponding Cauchy problem for the HCMA (\[HCMARayEq\]) is $T$-good. Then any $C^1$ $\pi_2^\star\o$-psh solution of the HCMA (\[HCMARayEq\]) up to time $T$ is the unique $T$-leafwise subsolution. The proof of Proposition \[ConeadmissibleIsOptimalProp\] is based on Theorem \[HJThm\] and uniqueness of $C^1$ solutions of the Hamilton–Jacobi equation. We now complete the proof Theorem \[ConeLifeSpanThm\], assuming Proposition \[ConeadmissibleIsOptimalProp\]. This is possible since the $T$-good assumption is satisfied in the toric setting. The proof is simple and is given in Lemma \[AnalyticContinuationHamOrbitsToricLemma\]. By Proposition \[OptimalSubsolutionToricProp\] there exists a unique leafwise subsolution $\vp_L$ of the toric HCMA (see ), induced by the Legendre transform potential . By Proposition \[ConeadmissibleIsOptimalProp\] any $\pi_2^\star\o$-psh $C^1$ solution of on a toric variety must coincide with $\vp_L$. However, $\vp_L\not\in C^1$ for $T>T_\span^\cvx$ [@RZ2 Proposition 1]. Hence, there exists no admissible $C^1$ weak solution of the IVP for $T>T_\span^\infty$, concluding the proof of Theorem \[ConeLifeSpanThm\]. For sufficiently regular $\eta, $ Theorem 1.13 can be proved in a symplectic geometric way by observing that the Lagrangian submanifold $\Lambda_{\nu}: = \mbox{graph} (d \eta)$ of $T^*({{\mathbb R}}\times {{\mathbb R}}^n)$ lies in a level set of the Hamiltonian $F$. When $\Lambda_{\eta}$ is sufficiently smooth, it must then be invariant under the Hamilton flow of $F$. When $\Lambda_{\eta}$ is Lipschitz, for instance, we can use flat forms and chains to prove the latter statement, and obtain: \[COR\] Let $\eta\in C^{1,1}([0,T]\times{\mathbb{R}}^n)$ be an admissible weak solution of the HRMA. Then the Lipschitz Lagrangian submanifold $\Lambda_{\eta}:=\h{graph}\,(d\eta)\subset T^\star{\mathbb{R}}^{n+1}$ is foliated by straight line segments along each of which $\nabla\eta$ is constant. We postpone the details of this symplectic approach to the HRMA and the proof of this proposition to a sequel [@RZ3], where we also pursue a complex analogue for the HCMA. Organization ------------ The charactrization of the smooth lifespan and uniqueness of classical solutions (Theorem \[HCMACauchyCthreeThm\]) is proved in Section \[HamFlowsHCMASection\]. The ill-posedness of the leafwise problem, Theorem \[GENERIC\], is proved in Section \[LeafwiseSection\]. Proposition \[ToricLifespanProp\] concerning the obstructions to solvability and the characterization of the smooth lifespan in the toric setting is proved in Section \[LifespanHRMASection\]. The characterization of the Legendre potential as the unique leafwise subsolution is proved in Section \[OptimalSubsolutionHRMASection\]. The characterization of the $C^1$ lifespan for the HRMA is given in Section \[ConeLifespanSection\], where we also prove the equivalence between HRMA and a Hamilton–Jacobi equation. Smooth lifespan of the HCMA: Proof of Theorem \[HCMACauchyCthreeThm\] {#HamFlowsHCMASection} ===================================================================== Before proving Theorem \[HCMACauchyCthreeThm\], we need to introduce some terminology and background related to the ill-posedness of the Cauchy problem. \[THamAnalyticDef\] We say that the Cauchy problem (\[HCMARayEq\]) with smooth initial data $(M, \omega_{\vp_0},\dot\vp_0)$ is $T$-Hamiltonian analytic if for every $z \in M$ the orbit of $z$ under the Hamiltonian flow of $\dot\vp_0$ with respect to $\omega_{\vp_0}$ admits a holomorphic extension to the strip $S_T$. Here, by a holomorphic extension of a map $\gamma:{\mathbb{R}}\ra M$ to $S_T$ we mean a holomorphic map $\tilde\gamma:S_T\ra M$ such that $\tilde\gamma(0,t)=\gamma(t)$. Such an extension is unique when it exists (this can be seen either by the Cauchy-Riemann equations or by the Monodromy Theorem). When it exists for $z\in M$ we denote it by $\exp-\i\tau X_{\dot\vp_0}^{\omega_{\vp_0}}.z, \,\tau\in S_T$ the holomorphic strip extending the Hamiltonian orbit $\exp tX_{\dot\vp_0}^{\omega_{\vp_0}}.z$. \[MoserMapDef\] The Moser maps are defined by $$\label{FsMapEq} f_\tau(z):= \exp-\i \tau X_{\dot\vp_0}^{\omega_{\vp_0}}.z, \quad \tau=s+\i t\in S_T.$$ Thus, by definition, the Moser maps are the analytic continuation to complex time of the Hamiltonian flow of $\dot\vp_0$ with respect to the symplectic structure $(M,\o_{\vp_0})$. This terminology will be justified by the fact that for solutions of the HCMA these maps act as Moser maps in the usual sense of symplectic geometry, see below. \[TGoodDef\] We say that the Cauchy problem (\[HCMARayEq\]) is $T$-good if it is $T$-Hamiltonian analytic and if the Moser map $f_{\tau}$ is a differentiable map of $M$ for each $\tau\in S_T$. HCMA and invertibility of the Moser maps {#HCMAInvertibilityMoserSubsection} ---------------------------------------- We now begin the proof of Theorem \[HCMACauchyCthreeThm\]. In this subsection we show one direction, namely, that a $C^3$ solution of the HCMA gives rise to smoothly invertible Moser maps in the sense of Definition \[MoserMapDef\]. The proof can be extracted from the arguments of [@S; @D1]. For the sake of completeness, we present the rather simple argument. A $C^3$ function $\vp$ on $S_T\times M$ satisfies HCMA if and only if the form $\pi_2^\star\omega+\i{\partial{\bar\partial}}\vp$ has a non-trivial kernel. Since this form is of type $(1,1)$ and, according to (\[HCMARayEq\]), nondegenerate on $M$-slices it follows that the kernel defines a one-dimensional integrable complex distribution on $S_T\times M$. It also follows that its leaves are holomorphic copies of $S_T$ inside $S_T\times M$, and the leaf passing through $(0,z)$, for each $z\in M$, may be parametrized in the form $$\label {GAMMADEF} \{(\tau,\Gamma_z(\tau))\,:\, \tau\in S_T\}\subset S_T\times M,\;\;\; \text{with}\;\Gamma_z: S_T \to M.$$ For each $\tau\in S_T$ define the map $f_\tau: M \to M$ by $$\label {ftau} f_\tau(z):=\Gamma_z(\tau).$$ By the transversality condition and the fact that the leaves do not intersect each other (follows from uniqueness for ODEs with $C^1$ coefficients—here we used the $C^3$ assumption for the second time) it follows that $f_\tau$ is a $C^1$ diffeomorphism. It remains to prove that the maps $f_\tau$ are Moser maps in the sense of Definition \[MoserMapDef\]. Since the strips are constructed by integrating the vector field $\frac{\del}{\del\tau}+\frac{d f_\tau}{d\tau}$ in $S_T\times M$, this vector field lies in the kernel of $\pi_2^\star\omega+\i{\partial{\bar\partial}}\vp$. Therefore, $$\label {FtauPullbackEq} f_\tau^\star\o_{\vp_\tau}=\o_{\vp_0}.$$ Now, since $\vp_0,\dot\vp_0$ are invariant under the ${\mathbb{R}}$-action $(\tau,z)\mapsto (\tau+\i c,z), c\in{\mathbb{R}}$, uniqueness of smooth solutions implies that so is $\vp_\tau$. The proof of uniqueness is postponed to Lemma \[ExistenceGivenTGoodInvertibleLemma\] below, however its proof does not rely on the rest of this subsection. By abuse of notation we write $\vp_s=\vp_\tau$ when no confusion arises, where $\tau=s+\i t$. Next, $$\label {FlowLeavesEqs} \frac{df_\tau}{dt}=X_{\dot\vp_s}^{\o_{\vp_s}}\circ f_\tau=-J\nabla_{g_{\vp_s}}{\dot\vp_s}\circ f_\tau,\qquad \frac{df_\tau}{ds}=-\nabla_{g_{\vp_s}}{\dot\vp_s}\circ f_\tau,\quad f_0=\id,$$ since $\frac{\del}{\del\tau}-\nabla^{1,0}_{g_{\vp_s}}{\dot\vp_s}\in\ker (\pi_2^\star\omega+\i{\partial{\bar\partial}}\vp)\big|_{(\tau,\Gamma_z(\tau))}$, indeed $$\iota_\frac{\del}{\del\tau}(\pi_2^\star\omega+\i{\partial{\bar\partial}}\vp)= \i{\bar\partial}\frac{\del\vp}{\del \tau}=\i {\bar\partial}\dot\vp_s,$$ and $$\iota_{\nabla_{g_{\vp_s}}{\dot\vp_s}}(\pi_2^\star\omega+\i{\partial{\bar\partial}}\vp) = \iota_{\nabla_{g_{\vp_s}}{\dot\vp_s}}\o_{\vp_s} = d^c\dot\vp_s=\i({\bar\partial}-\del)\dot\vp_s,$$ and we use the convention $\frac{\del}{\del \tau}=\frac12\frac{\del}{\del s}-\frac\i2\frac{\del}{\del t}$ and $Y^{1,0}=\frac12Y-\frac\i2JY$. It then follows from (\[FtauPullbackEq\]) that $$\label {MoserDiffDecompositionEq} f_{s+\i t} = h_{s+\i t}\circ f_{s},$$ with $h_{s+\i t}$ a $C^1$ symplectomorphism of $(M,\omega_{\vp_s})$. Also, from and $$\label{h} h_{s+\i t}=\exp tX_{\dot\vp_s}^{\o_{\vp_s}}.$$ We conclude therefore from and that the maps $f_\tau$ defined by satisfy , i.e., for each $z\in M$, induce analytic continuation to the strip of the Hamiltonian orbit $\exp tX_{\dot\vp_0}^{\o_{\vp_0}}.z$. Hence we have shown both that the Cauchy data is $T$-good and that the Moser maps of Definition \[MoserMapDef\] are $C^1$ and admit $C^1$ inverses for each $s\in[0,T]$. This completes the proof of the first half of Theorem \[HCMACauchyCthreeThm\]. We conclude this subsection with some further properties of the Moser maps. In view of , the Moser maps which are relevant to the solution of HCMA are the ones with $t = 0$, and their definition only requires analytic continuation of the Hamiltonian flow of $X_{\dot\vp_0}^{\o_{\vp_0}}$ to a rectangle $[0, T]\times(-\epsilon, \epsilon)$. However, such an analytic continuation necessarily induces one to the strip $S_T$. \[STRIP\] Suppose that $h_{\sqrt{-1} t}= \exp tX_{\dot\vp_0}^{\omega_{\vp_0}} z$ admits an analytic continuation to $[0, T]\times(-\epsilon, \epsilon)$. Then $h_{\sqrt{-1}t }$ admits an analytic continuation to $S_T$. Indeed, by , $h_{s + \sqrt{-1}t}(z)$ is the orbit of a Hamiltonian flow for fixed $s$ and varying $t\in {\mathbb{R}}$. Hence it may be holomorphically extended by the group law $$\exp (t_1 + t_2) X_{\dot\vp_s}^{\o_{\vp_s} }(z) = \exp t_1 X_{\dot\vp_s}^{\o_{\vp_s}} (\exp t_2X_{\dot\vp_s}^{\o_{\vp_s}}z).$$ Therefore, by , one may define $f_{s+\i t}$ for all $s+\i t\in S_T$. \[NoGroupLawRemark\] [In comparison to this group law for fixed $s$, $f_{s + \sqrt{-1} t}$ does not satisfy a group law in the complex parameter $s + \sqrt{-1} t$ and thus we cannot conclude that the flow has an analytic continuation to a half-plane by the same argument. This may be seen from the fact that $X_{\dot{\vp}_s}^{\omega_{\vp_s}}$ does not Lie-commute with its image under $J$. Indeed, commutativity fails even for generic Cauchy data in the case of toric varieties—see Remark \[ToricNoCommutativityRemark\]. ]{} Existence of classical solutions for the HCMA {#AnalyticContinuationHCMASubsection} --------------------------------------------- In this subsection we continue the proof of Theorem \[HCMACauchyCthreeThm\], and establish the existence of a classical solution to the HCMA under our assumptions. We now assume that the Cauchy problem for $(\o_{\vp_0},\dot\vp_0)$ is $T$-good and solve the HCMA under the additional assumption of invertibility. \[ExistenceGivenTGoodInvertibleLemma\] Let $\o_{\vp_0}\in C^1$ and $\dot\vp_0\in C^3$. Assume that the Cauchy problem for $(\o_{\vp_0},\dot\vp_0)$ is $T$-good, and that for each $\tau\in S_T$ the map $f_\tau$ given by (\[FsMapEq\]) is smoothly invertible. Then the HCMA (\[HCMARayEq\]) admits a $C^3$ $\pi_2^\star\o$-psh solution. Define a $C^3$ function on $S_T\times M$ by $$\label{ExplicitGreenFunctionSolEq} \begin{aligned} \vp(s+\i t,z) := &-\i\partial_{\omega_{\vp_0}}^\star {\bar\partial}_{\omega_{\vp_0}}^\star G_{{\omega_{\vp_0}}}^2 \big( (f_\tau^{-1})^\star \omega_{\vp_0} - \omega_{\vp_0} \big)(z) \cr & +\vp_0(z) +\frac sV\int_M\dot\vp_0\o_{\vp_0}^n, \end{aligned}$$ where $G_{{\omega_{\vp_0}}}$ denotes Green’s function for the Laplacian $\Delta_{{\omega_{\vp_0}}}=-{\bar\partial}\circ{\bar\partial}^\star-{\bar\partial}^\star\circ{\bar\partial}$ acting on forms. The operator $\i\partial_{\omega_{\vp_0}}^\star {\bar\partial}_{\omega_{\vp_0}}^\star G_{{\omega_{\vp_0}}}^2$ is a pseudo-differential operator of order $-2$ with smooth coefficients. By our assumptions it then follows that $\vp$ is $C^3$. We claim that $\vp$ solves the HCMA (\[HCMARayEq\]). First, observe that since $f_{\i t}(z)=\exp tX_{\dot\vp_0}^{\o_{\vp_0}}.z$ is a symplectomorphism the forumula (\[ExplicitGreenFunctionSolEq\]) implies that $\vp(\i t,z)=\vp(0,z)=\vp_0(z)$. Next, $$\begin{aligned} \frac{\del\vp(\i t,z)}{\del s} = &-\i\partial_{\omega_{\vp_0}}^\star {\bar\partial}_{\omega_{\vp_0}}^\star G_{{\omega_{\vp_0}}}^2 \big( \calL_{-\frac{df_{\i t}}{ds}}\omega_{\vp_0} \big)(f_{\i t}(z)) \cr &+\frac1V\int_M\dot\vp_0\o_{\vp_0}^n. \end{aligned}$$ Since $$\frac{df_{\i t}}{ds}=JX_{\dot\vp_0}^{\o_{\vp_0}}(f_{\i t}(z))=-\nabla_{g_{\vp_0}}\dot\vp_0(f_{\i t}(z)),$$ we have $$\calL_{-\frac{df_{\i t}}{ds}}\omega_{\vp_0}=\i{\partial{\bar\partial}}\dot\vp_0,$$ and the ${\partial{\bar\partial}}$-lemma [@GH p. 149] implies that $\frac{\del\vp(\i t,z)}{\del s}=\dot\vp_0(z)$. Finally, applying the ${\partial{\bar\partial}}$-lemma again implies that holds where $\vp_\tau:=\vp(\tau,\,\cdot\,)$, for all $\tau\in S_T$. Since $f_\tau$ is a diffeomorphism and moreover a smooth homotopy to the identity map it follows that $\o_{\vp_\tau}$ is a [[K]{}\^]{}metric for each $\tau\in S_T$. In particular, $(\pi_2^\star\o_{\vp_0}+\i{\partial{\bar\partial}}\vp)^n\ne0$. Differentiating (\[FtauPullbackEq\] ) we find that $\frac{\del}{\del \tau}+\frac{df_\tau}{d\tau}$ is a holomorphic vector field in the kernel of $\pi_2^\star\o_{\vp_0}+\i{\partial{\bar\partial}}\vp$. It follows that $(\pi_2^\star\o_{\vp_0}+\i{\partial{\bar\partial}}\vp)^{n+1}=0$ on $S_T\times M$, as required. This concludes the proof of existence. Uniqueness of classical solutions for the HCMA {#UniquenessHCMASubsection} ---------------------------------------------- In this subsection we complete the proof of Theorem \[HCMACauchyCthreeThm\], and establish the uniqueness of classical solutions to the HCMA under our assumptions. Before giving the proof let us emphasize some of the subtleties involved. First, the uniqueness we establish is essentially equivalent to showing that any solution must be ${\mathbb{R}}$-invariant when the Cauchy data is ${\mathbb{R}}$-invariant. A subtle point is that the HCMA is only equivalent to the geodesic equation under the assumption of ${\mathbb{R}}$-invariance, which is implicit in the arguments of Semmes and Donaldson. In general, the HCMA is equivalent to the more complicated WZW equation. Thus, the uniqueness proof cannot a priori use the identities we established in §\[HCMAInvertibilityMoserSubsection\] for $C^3$ ${\mathbb{R}}$-invariant solutions. We need to derive these identities in the proof, and we do so by first establishing short-time uniqueness and then extending this to a global statement. Thus, if we only wanted to prove uniqueness of ${\mathbb{R}}$-invariant solutions, the proof would simplify considerably. Alternatively, one could have defined the class of admissible subsolutions to be ${\mathbb{R}}$-invariant $\pi_2^\star \o$-psh functions. It follows from Lemma \[UniquenessGivenTGoodInvertibleLemma\] below that such a restriction would be redundant. Second, the proof does not follow directly from the arguments of Bedford–Kalka [@BK] and Bedford–Burns [@BB Proposition 1.1], where uniqueness is proved for a simpler situation, namely for the equation $(\i{\partial{\bar\partial}}u)^m=0$ on ${{\mathbb C}}^m$. Parts of the proof are local in nature, essentially the Cauchy–Kowalevskaya theorem on each strip, and thus adapt to our setting. However, the relative [[K]{}\^]{}potential $\pi_2^\star\o$ makes the situation more complicated since the leafwise equations are now not the fixed Laplace equation on $S_T$ but rather an inhomogeneous Poisson equation that varies from strip to strip, and one has to make sure that this equation does not degenerate. Thus, we need to invoke a global conservation law type argument that is special for our HCMA . \[UniquenessGivenTGoodInvertibleLemma\] Let $\o_{\vp_0}\in C^1$ and $\dot\vp_0\in C^3$. Assume that the Cauchy problem for $(\o_{\vp_0},\dot\vp_0)$ is $T$-good, and that for each $\tau\in S_T$ the map $f_\tau$ given by (\[FsMapEq\]) is smoothly invertible. Then any $C^3$ $\pi_2^\star\o$-psh solution of the HCMA (\[HCMARayEq\]) is unique, and in particular ${\mathbb{R}}$-invariant. Assume that $\vp,\rho\in C^3$ are both $\pi_2^\star\o$-psh solutions of (\[HCMARayEq\]). Then the equation and the equality of the Cauchy data implies that all the second derivatives of $\vp$ and $\rho$, possibly with the exception of the second $s$ derivative, agree on the hypersurface $\Sigma:=\{0\}\times{\mathbb{R}}\times M$. Now the form $\pi_2^\star\omega+\i{\partial{\bar\partial}}\Phi$ restricts to a positive form on $\Sigma$ ensuring that $g_\vp$ is non-degenerate (i.e., $\Sigma$ is non-characteristic). Also, the Monge–Ampère equation on the initial hypersurface can be rewritten as $$\label{SemmesGeodEq} \ddot \vp|_{s=0}=\frac12|\nabla\dot\vp|_{g_\vp}^2|_{s=0};$$ this was shown by Semmes [@S] for all $s$, assuming $\vp$ is an ${\mathbb{R}}$-invariant solution, but holds by his argument at $\{s=0\}$ without that assumption since $\del_t\vp,\del^2_t\vp,\del_s\del_t\vp$, and $\del_t\del_z\vp$ vanish on $\{0\}\times{\mathbb{R}}\times M$ as the initial data is ${\mathbb{R}}$-invariant. Note that expresses the second $s$ derivative of a solution in terms of the other second derivatives, all restricted to $\Sigma$. Since we know $\vp_0$ is a [[K]{}\^]{}potential, it follows that $\vp$ and $\rho$ agree to second order on $\Sigma$. Thus, $\ker(\pi_2^\star\o+\i{\partial{\bar\partial}}\vp)|_\Sigma=\ker(\pi_2^\star\o+\i{\partial{\bar\partial}}\rho)|_\Sigma$ along the hypresurface. Thus, by the uniqueness of solutions of first order ODEs with $C^1$ coefficients, the leaves of the foliation by strips defined by each of the solutions $\vp,\rho$ must coincide. Thus the maps defined by and for $\vp$ and $\rho$ are identical, and we denote them simply by $\Gamma_z(\tau)=f_\tau(z)$. By the construction of the foliation, on each leaf the [[K]{}\^]{}form $\pi_2^\star\o+\i{\partial{\bar\partial}}\vp$ satisfies . We claim that always holds for $s=0$. Recall, that we proved for all $s\in[0,T]$, but only under the assumption of ${\mathbb{R}}$-invariance of the solution. To prove this claim, note first $$\iota_\frac{\del}{\del\tau}(\pi_2^\star\omega+\i{\partial{\bar\partial}}\vp)\Big|_{s=0}= \i{\bar\partial}\frac{\del\vp}{\del \tau}\Big|_{s=0}=\i {\bar\partial}\dot\vp_0, \quad \h{when $s=0$},$$ since $\dot\vp_0$ is ${\mathbb{R}}$-invariant. Similarly, since $\vp_0$ is ${\mathbb{R}}$-invariant, $$\iota_{\nabla_{g_{\vp_0}}{\dot\vp_0}}(\pi_2^\star\omega+\i{\partial{\bar\partial}}\vp)|_{s=0} = \iota_{\nabla_{g_{\vp_0}}{\dot\vp_0}}\o_{\vp_0} = d^c\dot\vp_0=\i({\bar\partial}-\del)\dot\vp_0.$$ Thus, $\frac{\del}{\del\tau}-\nabla^{1,0}_{g_{\vp_0}}{\dot\vp_0}\in\ker (\pi_2^\star\omega+\i{\partial{\bar\partial}}\vp)\big|_{(\i t,\Gamma_z(\i t))}$, Therefore, since also $\frac{\del}{\del\tau}+\frac{df_\tau}{d\tau} \in\ker (\pi_2^\star\omega+\i{\partial{\bar\partial}}\vp)\big|_{(\i t,\Gamma_z(\i t))}$, we conclude that $$\label {FlowLeavesTimeZeroEqs} \frac{df_\tau}{dt}\Big|_{s=0}=X_{\dot\vp_0}^{\o_{\vp_0}}\circ f_{\i t}=-J\nabla_{g_{\vp_0}}{\dot\vp_0}\circ f_{\i t},\qquad \frac{df_\tau}{ds}\Big|_{s=0}=-\nabla_{g_{\vp_0}}{\dot\vp_0}\circ f_{\i t},$$ as claimed. Let $\Gamma_z$ be as in and and suppose that $\Gamma_z(S_T)\not=\{z\}$, i.e., that the leaf passing through $z$ is not trivial. For each $z\in M$, put $e_z:=\gamma_z^\star \varphi\,$, $\tilde e_z:=\gamma_z^\star \rho$, and let $\omega_z:=\gamma_z^\star\pi_2^\star\omega=\Gamma_z^\star\omega$. First, note that $\omega_z$ is strictly positive $(1,1)$-form on $S_T$. Indeed, write $\omega_z=\i a_z d\tau\wedge d\bar\tau=2a_z ds\wedge dt$. Then by , $$\label{AFunctionTimeZeroDefEq} \begin{aligned} a_z(\i t) &= -\i \omega\Big(d\Gamma_z|_{\tau=\i t}\Big(\frac{\del}{\del\tau}\Big), d\Gamma_z|_{\tau=\i t}\Big(\frac{\del}{\del\bar \tau}\Big)\Big) \Big|_{\Gamma_z(\i t)} \cr &=-\i \omega\Big( \frac{\del f_\tau(z)}{\del\tau}\Big|_{\tau=\i t}, \frac{\del f_\tau(z)}{\del\bar \tau}\Big|_{\tau=\i t} \Big) \Big|_{f_{\i t}(z)} \cr &=-\frac{\i}4 \omega(-\nabla_{g_{\vp_0}}\dot\vp_0+\i J\nabla_{g_{\vp_0}}\dot\vp_0, -\nabla_{g_{\vp_0}}\dot\vp_0-\i J\nabla_{g_{\vp_0}}\dot\vp_0)|_{f_{\i t}(z)} \cr &= \frac12\omega(\nabla_{g_{\vp_0}}\dot\vp_0, J\nabla_{g_{\vp_0}}\dot\vp_0)|_{f_{\i t}(z)} = \frac12|\nabla_{g_{\vp_0}}\dot\vp_0|^2_{g}({f_{\i t}(z)})\ge0. \end{aligned}$$ Since $g_{\vp_0}$ and $g$ are (strictly positive) metrics, $a_z$ vanishes at some $\i t\in \{0\}\times{\mathbb{R}}\subset S_T$ if and only if $d\dot\vp_0(f_{\i t}(z))=d\dot\vp_0(z)=0$ (by $f_{\i t}=\exp tX_{\dot\vp_0}^{\o_{\vp_0}}$ so in particular $f_{\i t}^\star \dot\vp_0=\dot\vp_0$). Thus, if $a_z(\i t)=0$ for some $t$, then $a_z(\i t)$ for all $t\in {\mathbb{R}}$. Now, for fixed $z\in M$ and $t\in{\mathbb{R}}$, equation is an ODE in $s$ for $f_{s+\i t}(z)$. If $a_z(\i t)=0$, then its initial condition is $f_{\i t}(z)=z$ and the initial derivative is zero. Thus, in this case $f_\tau(z)=z$ for all $\tau\in S_T$, and the leaf through $z$ is trivial, i.e., $\Gamma_z(S_T)=\{z\}$. Since we assumed at the beginning of this paragraph that the leaf through $z$ was non-trivial, we thus conclude that $a_z|_{s=0}>0$, and by continuity also $C>a_z|_{s\in[0,3\eps]}>0$, for some $C,\eps>0$. Denote the Laplacian associated to $\o_z$ by $\Delta_z$. Then for each $z$ with a non-trivial leaf, the leafwise problem restricted to $S_{2\eps}\times M$ is equivalent to the Cauchy problem, $$\label{CauchyLaplaceLeaveLocalEq} \begin{aligned} 1 + \Delta_z \a_z & = 0, \quad \h{\ on \ } S_{2\eps}, \cr \a_z(\i t) & = \vp_0(\Gamma_z(\i t)) \quad \h{\ on \ } \{0\}\times{\mathbb{R}}, \cr\dis \frac{\del\a_z}{\del s}(\i t) & = \dot\vp_0 (\Gamma_z(\i t) - d\vp_0(\nabla_{g_{\vp_0}}\dot\vp_0)(\Gamma_z(\i t)), \quad \h{\ on \ } \{0\}\times{\mathbb{R}}. \end{aligned}$$ The last equation follows from . Thus, $e_z$ and $\tilde e_z$ solve . Hence, since $\Delta_z = a_z^{-1} \Delta_0$, $\zeta_z:=e_z-\tilde e_z$ solves the Cauchy problem for $\Delta_0\zeta_z=0$ on $S_{2\eps}$ with zero initial data, where $\Delta_0$ denotes the Euclidean Laplacian on $S_{2\eps}$. It is well-known that bounded solutions to the Cauchy problem on [*bounded*]{} domains for this classical Euclidean equation are unique (cf., e.g., [@L p. 19]). However, we could not find a reference that treats our particular situation, namely the non-compact strip as in the following Lemma. \[LaplaceUniquenessLemma\] Let $u\in C^2\cap L^\infty(S_T)$ be a solution of $\Delta_0 u=0$ on $S_T$, with $u|_{s=0}=a\in C^2({\mathbb{R}})$, and $\del u/\del s|_{s=0}=b\in C^2({\mathbb{R}})$. Then $u$ is unique. Since the equation is linear it suffices to consider the case of zero Cauchy data $a=b=0$, and prove any solution must then vanish. Also, it suffices to consider the case $T=\pi$, since if $u$ is a non-trivial solution of $\Delta_0u=0$ on $S_T$ with $a=b=0$ then $v(s,t):=u(\frac T\pi s,\frac T\pi t)$ solves the same equation on $S_\pi$. Let $P$ denote the Dirichlet Poisson kernel of the strip $S_\pi$, $$\label{PoissonStripEq} P(s, t) = \frac{\sin s}{\cosh t - \cos s}.$$ According to a theorem of Widder [@W1 Theorem 4], any harmonic function bounded below on the strip $S_\pi$ can be expressed as $$\label{PoissonStEq} u(s,t)-\inf u=[Ae^t+Be^{-t}]\sin s+\frac1{2\pi}\int_{\mathbb{R}}P(s,a-t)d\alpha(a) +\frac1{2\pi}\int_{\mathbb{R}}P(\pi-s,a-t)d\beta(a),$$ for some constants $A,B\ge0$, and some (measurable) nondecreasing functions $\alpha,\beta:{\mathbb{R}}\ra{\mathbb{R}}$. Moreover, the integrals converge in the interior of $S_\pi$. Evaluating at $s=0$ gives, by the continuity of $u$ $$-\inf u =\frac1{2\pi}\lim_{s\ra0^+}\int_{\mathbb{R}}P(s,a-t)d\alpha(a).$$ Thus $-\inf u\, dt=d\alpha(t)$. Plugging this back into , thus $$\label{PoissonStSecondEq} u(s,t)=[Ae^t+Be^{-t}]\sin s+\frac1{2\pi}\int_{\mathbb{R}}P(\pi-s,a-t)d\beta(a).$$ Therefore, $$0=\frac{\del u}{\del s}(0,t)=Ae^t+Be^{-t} + \frac1{2\pi}\int_{\mathbb{R}}\frac{d\beta(a)}{\cosh(a-t)+1}$$ Since each of the terms is nonnegative they all vanish. Hence, $A=B=0$, and $\frac1{2\pi}\int_{\mathbb{R}}\frac{d\beta(a)}{\cosh(a-t)+1}=0,$ and therefore $d\beta=0$. Plugging back into , we conclude that $u=0$, as desired. It follows that $e_z=\tilde e_z$, whenever $\Gamma_z(S_T)\not=\{z\}$. On the other hand, if $\Gamma_z(S_T)=\{z\}$ then $\vp(\tau,z)=\rho(\tau,z)$ by using , and that $\vp(\i t,z)=\rho(\i t,z)$. Since the foliation foliates all of $S_{2\eps}\times M$, it follows that $\vp=\rho$ on that set. Thus, we have short-time uniqueness for $C^3$ solutions of the HCMA . In particular, it follows that both $\vp$ and $\rho$ are ${\mathbb{R}}$-invariant for $s\in[0,\eps]$. Also, – hold since again they were derived assuming only ${\mathbb{R}}$-invariance. Thus, extends to a strip: $$\label{SemmesGeodGlobalEq} \ddot \vp=\frac12|\nabla\dot\vp|_{g_\vp}^2,$$ on $S_{\eps}\times M$. Consequently [@S; @D1], $$\label {ConservationLawHCMAEq} \dot\vp_s\circ f_s=\dot\vp_0.$$ Indeed, this holds when $s=0$, and differentiating in $s$ and using , , , and we obtain it must holds for all $s\in[0,\eps]$, where we used that $\dot\vp_s$ is constant along its Hamilton orbits (the factor of $1/2$ in can be traced to our normalizations and corresponds to switching between the Hermitian and the Riemannian metrics associated to $\o_{\vp}$, cf. [@R §2.1.4.1,§2.2.3]). Finally, we can now also apply which was valid for any ${\mathbb{R}}$-invariant solution, and compute $$\label{AFunctionDefEq} \begin{aligned} a_z(s+\i t) &= -\i \omega\Big(d\Gamma_z\Big(\frac{\del}{\del\tau}\Big), d\Gamma_z\Big(\frac{\del}{\del\bar \tau}\Big)\Big) \Big|_{\Gamma_z(\tau)} \cr &=-\i \omega\Big( \frac{\del f_\tau(z)}{\del\tau}, \frac{\del f_\tau(z)}{\del\bar \tau} \Big) \Big|_{f_\tau(z)} \cr &=- \frac\i4\omega(-\nabla_{g_{\vp_s}}\dot\vp_s+\!\i J\nabla_{g_{\vp_s}}\dot\vp_s, -\nabla_{g_{\vp_s}}\dot\vp_s-\!\i J\nabla_{g_{\vp_s}}\dot\vp_s)|_{f_{\tau}(z)} \cr &= \frac12\omega(\nabla_{g_{\vp_s}}\dot\vp_s, J\nabla_{g_{\vp_s}}\dot\vp_s)|_{f_{\tau}(z)} = \frac12|\nabla_{g_{\vp_s}}\dot\vp_s|^2_{g}({f_{\tau}(z)}). \end{aligned}$$ Therefore, by , , and compactness it follows that if $0<a|_{s=0}<$ then there exists constants $c,C>0$ determined by $z$ and the Cauchy data such that the a priori estimate $c<a_z(\tau)<C$ holds for each $\tau\in S_T$ for which a solution exists. Thus, we can now repeat the argument for the Cauchy problem with ${\mathbb{R}}$-invariant initial data given by $\vp|_{\{\eps\}\times{\mathbb{R}}}=\rho|_{\{\eps\}\times{\mathbb{R}}}$ and $\dot\vp|_{\{\eps\}\times{\mathbb{R}}}=\dot\rho|_{\{\eps\}\times{\mathbb{R}}}$, and conclude that in fact must hold on $S_T$. Thus $\rho=\vp$. This concludes the proof of Lemma \[UniquenessGivenTGoodInvertibleLemma\]. Theorem \[HCMACauchyCthreeThm\] now follows by combining §\[HCMAInvertibilityMoserSubsection\], and Lemmas \[ExistenceGivenTGoodInvertibleLemma\] and \[UniquenessGivenTGoodInvertibleLemma\]. As can be seen from the proof, $\vp$ is a smooth solution of the IVP if and only the Moser maps $f_s$ are smoothly invertible and the ‘conservation law’ holds. Of course, this is a weaker statement than Lemma \[ExistenceGivenTGoodInvertibleLemma\]. Nevertheless we record it here. Let $\o_{\vp_0}\in C^1$ and $\dot\vp_0\in C^3$. Assume that the Cauchy problem for $(\o_{\vp_0},\dot\vp_0)$ is $T$-good, and that for each $\tau\in S_T$ the map $f_\tau$ given by (\[FsMapEq\]) is smoothly invertible. Then and are equivalent. We already saw that implies . For the converse, note that under the assumptions, it follows from Lemma \[ExistenceGivenTGoodInvertibleLemma\] that there exists a solution, and that the Moser maps determined by the Cauchy data satisfy –; thus differentiating immediately gives . In the setting of the HRMA, we will interpret in terms of a Hamilton–Jacobi equation (Theorem \[HJThm\]) and show that this ‘conservation law’ persists also for certain weak solutions (Proposition \[FTPHIDOT\]). Ill-posedness of leafwise Cauchy problems {#LeafwiseSection} ========================================== The goal of this section is to prove Theorem \[GENERIC\], showing that the Cauchy problem for the HCMA is not even locally well-posed. As the proof of Theorem \[HCMACauchyCthreeThm\] shows, the leaves of the Monge–Ampère foliation are obtained as the analytic continuation of the Hamiltonian flow of $(\omega_{\vp_0},\dot\vp_0)$. The Monge–Ampère distribution picks out as the $M$-component the Hamiltonian vector field associated to $(\omega_{\vp_0},\dot\vp_0)$ and not an arbitrary multiple of it precisely because the $S_T$-component of the distribution is $\partial/\partial\tau$. In other words, the leaves (strips) of the foliation are graphs (of maps $S_T\ra S_T\times M$) of (complex) time-parametrized Hamiltonian flow of $(\omega_{\vp_0},\dot\vp_0)$. As we will show, this puts a serious restriction on the Cauchy data. So far, we have operated under the assumption that we have $T$-good Cauchy data (Definitions \[THamAnalyticDef\] and \[TGoodDef\]). Yet the analytic continuation of each Hamiltonian orbit should be an ill-posed problem. The closely related problem of solving the leafwise Cauchy problem for the equation should also ill-posed, and the goal of this section is to give a proof of this latter ill-posedness. The latter problem seems simpler than the former since it is a linear problem for a function on a strip rather than a Cauchy problem for a holomorphic map into a nonlinear space. Hence we concentrate on the leafwise problem here. However, it is natural also to linearize the nonlinear problem (cf. [@D2]) and prove ill-posedness for the existence of $T$-Hamiltonian analytic data. We pursue this approach in a sequel. As above, we suppose that we are given $(\omega_{\vp_0}, \dot{\vp}_0)$ for which the orbit $\exp t X_{\dot{\vp}_0}^{\omega_{\vp_0}}$ admits an analytic continuation to the strip $S_T$. Let $\gamma_z$ be as in Definition \[OptimalSubsolutionDef\]. Then $\a_z = \gamma_z^\star \varphi$ satisfies . Since $\Gamma_z^\star \omega$ has a global potential $\Phi_z$ on $S_T$, we may also write the equation in terms of the Euclidean Laplacian $\Delta_0$ as $$\label {DELTAZERO} \Delta_0 \chi = 0, \;\;\; \text{\rm where}\;\;\chi = (\Phi_z + \a_z).$$ However, $\Phi_z$ is not unique since the addition of any harmonic function on the strip gives another potential. In the case where the image of the complex Hamiltonian orbit $\Gamma_z$ lies in an open set $U \subset M$ in which $\omega$ has a potential $\Phi_0$, we have $\Phi_z = \Gamma_z^\star \Phi_0$. In general, the closure of the image of $\Gamma_z(\i t)$ lies in the level set $\{\dot{\vp}_0 = \dot{\vp}_0(z)\}.$ We will see that toric varieties always satisfy these conditions. However, simple examples (e.g., elliptic curves) show that there need not exist a potential for $\omega$ defined in a neighborhood of the orbit. The following lemma shows that one may find a reasonable replacement for that, on each leaf separately. The growth estimate we derive here is not optimal, but suffices for our purposes. \[PolyGrowthLemma\] Let $\vp$ be a smooth solution to the HCMA . There exists a global [[K]{}\^]{}potential $\Phi_z$ for $\Gamma_z^\star\o$ on $S_T$ with polynomial growth at infinity. The claim would be obvious if there exists a potential for $\omega$ on a neighborhood of the image of $\Gamma_z(S_T)$, but as mentioned earlier such a potential need not exist. Instead, we will find suitable [[K]{}\^]{}potentials along each leaf. As before, denote $\o_z=\gamma_z^\star \pi_2^\star \omega = 2a_z ds \wedge dt.$ As shown in the proof of Lemma \[ExistenceGivenTGoodInvertibleLemma\], $a_z>0$ if and only if the leaf through $z$ is non-trivial, i.e., $\Gamma_z(S_T)\not=\{z\}$, which we assume throughout this section. Thus, by compactness, there exist some constants $c,C>0$ (depending on $z$) such that $0 <c < a_z(s, t) < C$ on $S_T$. For convenience, in this section we omit the subscript and denote $a\equiv a_z$. We wish to find $\Phi_z\in C^\infty(S_T)$ of polynomial growth so that $\Delta_0 \Phi_z = a$, i.e., $\i{\partial{\bar\partial}}\Phi_z = \gamma_z^\star \pi_2^\star \omega$. Throughout this proof $\del=\del_\tau$. We rewrite the Poisson equation above as $$\label{PE} {\bar\partial}(\i \partial \Phi_z) = -2a ds \wedge dt$$ and use existence theorems for the inhomogeneous ${\bar\partial}$-equation on the strip. Introduce the subharmonic weight $\psi = \log (1 + |\tau|^2)$ and observe that $$a ds \wedge dt \in L^2_{(1,1)}(S_T, \psi )$$ where $L^2_{(1,1)}(S_T, \psi)$ is the space of $(1,1)$ forms $a(s, t) ds \wedge dt$ so that $$\int_{S_T} e^{- \psi} |a|^2 ds \wedge dt < \infty.$$ By Hörmander’s weighted $L^2$ existence theorem for the ${\bar\partial}$-equation [@Ho1 Theorem 4.4.2], there exists $u \in L^2_{(0, 1)}(S_T)$ such that ${\bar\partial}u = -2a ds \wedge dt$ and $$\int_{S_T} |u|^2 (1 + |\tau|^2)^{-3} d s \wedge dt \leq 4\int_{S_T} |a|^2 e^{- \psi} ds \wedge dt.$$ Applying the same theorem to ${\bar\partial}\Phi_z= \bar{u}$ with $\psi = 3 \log (1 + |\tau|^2)$, we then obtain a solution $\Phi_z$ of $\i {\partial{\bar\partial}}\Phi = 2a ds \wedge dt$ satisfying $$\label{5} \int_{S_T} |\Phi_z|^2 (1 + |\tau|^2)^{-5} ds dt < \infty.$$ We now show that this $L^2$ estimate implies the polynomial growth of $\Phi_z$. Note that $\partial_s \Phi_z$ and $\partial_t \Phi_z$ satisfy a Poisson equation on $S_T$ satisfying the same estimates. Indeed, by (and the assumption of existence of a smooth solution) under $\Gamma_z$ these vector fields push-forward to the Hamilton vector fields for $\dot{\vp}_s$, respectively $J$ of these fields. Hence the Lie derivative with respect to these fields of $\omega$ are bounded and we can use them as the right hand side in place of $a ds \wedge dt$ above and repeat the argument to get the estimate for these derivatives and for repeated mixed derivatives. By the Sobolev inequality $\sup_{S_T} f^2 \leq C \int_{S_T} |(1 - \Delta_0) f|^2 ds dt $ for a strip, we have $$\sup_{S_T} \Phi_z^2 (1 + |\tau|^2)^{-5} \leq C \int_{S_T} |(1 - \Delta_0)(\Phi_z (1 + |\tau|^2)^{-5/2}) |^2 ds \wedge dt.$$ It is straightforward to check that the integral is finite: this follows from the weighted $L^2$ estimates for $\Phi_z$ and $\Delta_0\Phi_z$, and the fact that derivatives of $(1 + |\tau|^2)^{-r} $ for $r > 0 $ decay more rapidly with each derivative. It follows that $$\label{EST} |\Phi_z| \leq C (1 + |\tau|^2)^{5/2} \;\; \text{\rm on}\;\; S_T.$$ In the proof of Theorem \[GENERIC\] we will be able to specialize to a situation where $\Phi_z$ is actually of the form $\Gamma_z^\star\Phi_0$. However, Lemma \[PolyGrowthLemma\] is needed to derive the general obstruction in Proposition \[LeafwiseProp\] below that holds for all $z\in M$. The obstruction to solvability of , and hence to the existence of a leafwise subsolution (and in particular to the existence of a $C^3$ solution of the HCMA), is summarized in the following propostion. \[LeafwiseProp\] Let $\vp$ be a $C^3$ solution to the HCMA , and $z\in M$. Let $D = \frac{1}{\i} \frac{d}{dt}$ on ${{\mathbb R}}$, and set $$\begin{aligned} q_z(t) &: = \frac{\partial \a_z}{\partial s}(\i t) = \dot\vp_0 (\Gamma_z(\i t) - d\vp_0(\nabla_{g_{\vp_0}}\dot\vp_0)(\Gamma_z(\i t)), \cr p_z(t) &: = -\del_s\Phi_z(0,t) - D\coth TD (\Phi_z+\gamma_z^\star\vp_0)(0,t), \end{aligned}$$ where $\Phi_z$ is given by Lemma \[PolyGrowthLemma\]. Then, $$\widehat{(q_z-p_z)}(\xi)=o(e^{-T|\xi|}).$$ Thus, $q_z-p_z$ admits an analytic continuation to the interior of $S_T\cup \overline{S_T} =[-T,T]\times{\mathbb{R}}$. Henceforth we denote by $PW_T({{\mathbb R}})$ the Paley–Wiener space $$\label{PW} PW_T({{\mathbb R}}):= \{f \in L^2({{\mathbb R}})\,: \, |\hat{f}(\xi) | =o(e^{-T|\xi|}) \}.$$ Our convention for the Fourier transform is $$\calF(f)(\xi)\equiv\hat f(\xi):=\int_{\mathbb{R}}e^{-\i t\xi}f(t)dt.$$ It is well-known that if $f \in PW_T({{\mathbb R}})$, then $f$ is the restriction to ${{\mathbb R}}$ of a holomorphic function on any two-sided strip $S_b\cup\overline{S_b}=[-b,b]\times{\mathbb{R}}$ with $b < T$ [@St p. 121]. Despite the lack of uniqueness of $\Phi_z$ it seems simpler to work with the equation rather than $\Delta_z \a = -1$ since the Euclidean equation is simpler and it too has real analytic coefficients. We then wish to represent the solution $\chi_z$ as a Poisson integral in terms of its boundary values on $\partial S_{T}$. We first assume $T = \pi$. We recall the following theorem of Widder [@W1 Theorem 3]: If $u(s, t)$ is (i) continuous on $S_{\pi}$ and harmonic on its interior; (ii) satisfies the bounds $u(0, t) e^{- |t|} \in L^1({{\mathbb R}}),\, u(\pi, t) e^{- |t|} \in L^1({{\mathbb R}})$ and $\int_0^{\pi} |u(s, t)| ds = o(e^{|t|}), $ then $$\label{u} u(s, t) = \frac{1}{2 \pi} \int_{{{\mathbb R}}} P(s, a - t) u(0,a) da + \frac{1}{2 \pi} \int_{{{\mathbb R}}} P(\pi - s, a- t) u(\pi,a)da,$$ where $P$ is defined by . The assumptions for Widder’s theorem are satisfied when $$u=\chi=\a_z+\Phi_z,$$ with $\Phi_z$ the potential constructed in Lemma \[PolyGrowthLemma\]. Indeed, then $\chi$ has polynomial growth at infinity on $S_T$ ($\a_z$ itself is a bounded continuous function). Consequently, is valid when $u = \chi$. We next consider the implications of this equation for $q_z$. As in [@W2], it simplifies the notation to put $$Q(s, t) = \frac{1}{4} \frac {\cos \frac{\pi}{2} s} {\cosh \frac{\pi}{2} t + \sin \frac{\pi s}{2}} = \frac14 P\Big(\frac{\pi}{2} s + \frac{\pi}{2}, \frac{\pi}{2} t\Big)$$ on the strip $s \in (-1,1), t \in {{\mathbb R}}$. One has [@W2 (5)] $$Q(s, t) = \frac{1}{2\pi} \int_{{{\mathbb R}}} e^{-\i t a}\;\; \frac{\sinh(1-s)a}{\sinh 2a} da, \;\; s \in (-1, 1).$$ Then, $$\label{UstEq} \begin{aligned} u(s, t) & = \int_{{{\mathbb R}}} Q(s,a-t) u(-1,a)da + \int_{{{\mathbb R}}} Q(-s,a-t) u(1,a)da \cr & = \int_{{{\mathbb R}}} e^{\i t a} \frac{\sinh (1 - s) a}{\sinh 2 a} \hat u(-1,a)da + \int_{{{\mathbb R}}} e^{\i t a} \frac{\sinh (1 + s) a}{\sinh 2 a} \hat u(1,a)da. \end{aligned}$$ Note that this formula holds even when $u(\pm 1,\,\cdot\,)$ is of polynomial growth. Then $\hat u(\pm 1,\,\cdot\,)$ is a temperate distribution while $\frac{\sinh (1 \mp s) a}{\sinh 2 a}$ is a Schwartz function for $s\in(-1,1)$, and so the second equality holds by the definition of the Fourier transform of a temperate distribution [@Ho2 Definition 7.1.9]. By a change of variable, for the strip $(s,t)\in[0,T]\times{\mathbb{R}}$ and for $\Delta_0\chi=0$ with boundary values $\chi(0,\,\cdot\,)$ and $\chi(T,\,\cdot\,)$ we obtain, $$\chi(s, t) = \int_{{{\mathbb R}}} e^{\i t a} \frac{\sinh (T - s)a}{\sinh T a} \hat \chi(0,a)da + \int_{{{\mathbb R}}} e^{\i t a} \frac{\sinh sa}{\sinh T a} \hat \chi(T,a)da.$$ Thus, $$\begin{aligned} \del_s\chi(0,t) = - \int_{{{\mathbb R}}} e^{\i t a} a\coth Ta \hat \chi(0,a) da + \int_{{{\mathbb R}}} e^{\i t a} \frac{ a}{\sinh T a} \hat \chi(T,a) da. \end{aligned}$$ Note that differentiation at the boundary is allowed since we can consider as a distributional equation in $t$ with parameter $s$, and so one can pair with any Schwartz function of $t$ and then differentiate in $s$. Thus, $$\begin{aligned} q_z(t) & = -\del_s\Phi_z(0,t) - D\coth TD \chi(0,\,\cdot\,) + \frac{D}{\sinh TD}\chi(T,\,\cdot\,) \cr & =:p_z(t) + \frac{D}{\sinh TD}\chi(T,\,\cdot\,), \end{aligned}$$ where $D = \frac{1}{\i} \frac{d}{dt}$ on ${{\mathbb R}}$. Inverting, $q_z$ must lie in the domain of the operator $\calA_{T,z}:\calS'\ra \calS'$ given by $$\calA_{T,z}:u\mapsto \frac{\sinh TD}{D}(u-p_z) = \int_{{\mathbb{R}}}e^{ita}\frac{\sinh Ta}{a}\widehat{(u-p_z)}(a)da .$$ Here $\calS'$ denotes temperate distributions on ${\mathbb{R}}$. By our earlier estimates $\chi(T,\,\cdot\,)$ is continuous and of at most polynomial growth, hence belongs to $\calS'$. Thus it follows that one has $\widehat{(q_z-p_z)}(\xi)=o(e^{-T|\xi|})$, since these are the Fourier coefficients of $\chi(T,\,\cdot\,)$. In particular $q_z-p_z\in L^2({\mathbb{R}})$, so $q_z-p_z\in PW_T({\mathbb{R}})$. By a Paley–Wiener type theorem [@St p. 121] it follows that $q_z-p_z$ admits an analytic continuation $$\label {LCALPHI} (q_z-p_z) (s + \i t):= \int_{{{\mathbb R}}} e^{(s +\i t) a} \frac{ a}{\sinh T a} \hat \chi(T,a) da$$ to the interior of a two-sided strip of width $2T$. As mentioned in the Introduction, the Paley–Wiener condition on $q_z$ may be viewed as characterizing the range of a Dirichlet-to-Neumann map. These leafwise Dirichlet-to-Neumann maps are induced by the global Dirichlet-to-Neumann map for the HCMA, defined by $${\mathcal{N}}^T(\vp_0, \vp_T) = \dot{\vp}_0$$ from the endpoint $\vp_T$ at time $T$ of the geodesic arc from $\vp_0$ to $\vp_T$ to the initial velocity $\dot{\vp}_0$ of the geodesic. Lifespan of generic Cauchy data ------------------------------- We now complete the proof of Theorem \[GENERIC\]. Assume that the Cauchy problem for with Cauchy data $(\vp_0,\dot\vp_0)$ admits a $C^3$ solution $\vp$. For simplicity, we take the reference [[K]{}\^]{}metric to be $\omega_{\vp_0}$ and then the initial relative [[K]{}\^]{}potential becomes zero. Then reduces to $$\label{CauchyLaplaceLeaveLocalEqa} \begin{aligned} 1 + \Delta_z \a_z & = 0, \quad \h{\ on \ } S_T, \cr \a_z(\i t) & = 0 \quad \h{\ on \ } \{0\}\times{\mathbb{R}}, \cr\dis \frac{\del\a_z}{\del s}(\i t) & = \dot\vp_0 (z) \quad \h{\ on \ } \h{\ on \ } \{0\}\times{\mathbb{R}}. \end{aligned}$$ The last line follows since $\dot \vp_0$ is constant along its Hamiltonian flow orbits. It follows that $q_z$ is a constant, and therefore Proposition \[LeafwiseProp\] implies that $$\label{ZerothPWConditionEq} p_z = (- \partial_s - A_T) \Phi_z |_{s = 0}\in PW_T({\mathbb{R}})$$ where $$A_T: = D \coth T D.$$ Note that $A_T$ is an approximation (with respect to $T$) to the Dirichlet-to-Neumann operator for the half-plane, which is the operator $$|D| f(t) = \int_{{{\mathbb R}}} e^{\i t a} |a| \hat{f}(a) da.$$ We further observe that, at least for some $z$, $\Phi_z$ is the pullback of a [Kähler ]{}potential defined in a neighborhood of $\Gamma_z(S_T)$. Indeed, let $z_0$ be a non-degenerate maximum point of $\dot{\vp}_0$ (one always exists for a generic $\dot\vp_0$, which as far as proving Theorem \[GENERIC\] we may assume is the case), so that the orbit of $z_0$ is $\{z_0\}$ and find a potential in a neighborhood of $z_0$. If $z$ is sufficiently close to $z_0$ then the orbit of $z$ under the Hamilton flow of $\dot{\vp}_0$ is non-trivial and is contained in the level set $\{\dot{\vp}_0 = \dot{\vp}_0(z)\}\subset M$, which is close to $\{z_0\}$ (by the Morse theorem). Moreover, this Hamilton orbit is contractible in $M$ to $z_0$. By – the slices $\Gamma_z(\{s\}\times{\mathbb{R}})$ are all homotopic to the this initial Hamilton orbit. Hence, $\Gamma_z(S_T)$ itself is contractible to $z_0$. Since $\o$ has a local [[K]{}\^]{}potential on any contractible neighborhood of $z_0$ in $M$, the conclusion follows. Assume from now on that $$\label {GlobalPotentialAssumpEq} \Phi_z = \Gamma_z^\star \Phi_0,$$ where $\Phi_0$ is a smooth function defined on some neighborhood of $\Gamma_z(S_T)$ in $M$. For simplicity of notation, put $$H:= \dot{\vp}_0, \quad X_H:=X_{\dot\vp_0}^{\o_{\vp_0}}.$$ Let $\nabla$ denote the gradient with respect to the associated metric $g_{\vp_0}$. Then $$\partial_s \Phi_z |_{s = 0} = \Gamma_z^\star J X_H \Phi_0|_{s=0}.$$ By and , the conclusion of Proposition \[LeafwiseProp\] can be rewritten as $$\label {SecondPWConditionEq} \Gamma_z^\star J X_H \Phi_0 + A_T \Gamma_z^\star \Phi_0 = \Gamma_z^\star d\Phi_0(\nabla H) + A_T \Gamma_z^\star \Phi_0 \in PW_T({\mathbb{R}}).$$ We now study this equation under particular deformations of the Cauchy data. We denote by $${\mathcal{C}}_{T,z} = \{(\vp_0, \dot\vp_0,z) \in C^3(M) \times C^3(M)\times M: (\vp_0,\dot\vp_0) \h{\ is $T$-good and\ } \eqref{ZerothPWConditionEq} \h{\ holds}\}.$$ We claim that for $z$ near a maximum point (as above), the complement of ${\mathcal{C}}_{T, z}$ in $C^3(M) \times C^3(M)\times M$ is dense. By assumption $ (0, \dot{\vp}_0,z)\in\calC_{T,z} $. We fix such a $z$ for the rest of the argument. We may, and do, choose $z$ so that in addition it is a regular point for $H$. Our first goal is to find a perturbation of $\dot\vp_0$ with the property that the orbit $\Gamma_z(\i {\mathbb{R}})$ is unchanged. Let $h$ be a $C^3$ function on $M$, and set for each $\eps\ge 0$, $$\begin{aligned} H_{\epsilon} &:= H + \epsilon (H - H(z)) h, \cr V_z &:= \{w\in M\,:\, H_\eps(w)= H_\eps(z))\}. \end{aligned}$$ Note that $V_z$ indeed is independent of $\eps\ge0$; by assumption $z\in M$ is a regular point for $H$ so that $V_z$ is a (real) hypersurface. Also, $$\begin{aligned} X_{H_{\epsilon} } := X^{\omega_{\vp_0}}_{H_{\epsilon} } = (1 + \epsilon h) X_H \quad \h{along $V_z$}. \end{aligned}$$ Denote $\hat\Gamma_z(t):=\Gamma_z(\i t)$. Then define $$\hat{\Gamma}_z^{\epsilon} ( t):= \exp t X_{H_{\epsilon}} (z) =\hat{\Gamma}_z( g_\eps(t)),$$ since $\Gamma_z(\i {\mathbb{R}})\subset V_z$, where $g_{\epsilon}: {{\mathbb R}}\to {{\mathbb R}}$ is a diffeomorphism defined by $$\label{gepsprimeEq} g_\eps'(t):= \frac{d}{dt} g_{\epsilon}(t) = 1 + \epsilon h\circ \hat{\Gamma}_z( t), \quad g_\eps(0)=0.$$ Thus, $$\label{gepsEq} g_{\epsilon}(t) = t + \eps\int_0^t h\circ\hat\Gamma_z(a) da.$$ To derive a contradiction, we assume that there exists some $\eps_0>0$ for which $\{(0,H_\eps,z)\}_{\eps\in[0,\eps_0]}\subset\calC_{T,z}$. In particular, by , for sufficiently small $\eps\ge0$, $$\label{EP} (\hat\Gamma_z^{\epsilon})^\star d \Phi_0(\nabla H_{\epsilon}) + A_T (\hat\Gamma_z^{\epsilon})^\star \Phi_0 \in PW_T({{\mathbb R}}).$$ Thus, $$\label{EP'} \frac{d}{d \epsilon}\Big|_{\epsilon = 0} \Big( (\hat\Gamma_z^{\epsilon})^\star d \Phi_0(\nabla H_{\epsilon}) + A_T (\hat\Gamma_z^{\epsilon})^\star \Phi_0 \Big) \in PW_T({{\mathbb R}}).$$ Note that $$d \Phi_0(\nabla H_{\epsilon})|_{V_z} = (1+\eps h) d\Phi_0 (\nabla H)|_{V_z}.$$ Also $$\label {EpsVariationAzEq} \frac{d}{d \epsilon}\Big|_{\epsilon = 0} (\hat\Gamma_z^{\epsilon})^\star \Phi_0 = h_z \frac{d}{dt} \hat\Gamma_z^\star \Phi_0,$$ where by $$\label{hzEq} h_z(t)= \int_0^t h\circ\hat \Gamma_z(a) da.$$ Set also, $$\tilde h_z=\hat\Gamma_z^\star h.$$ Then $$\label{hztildeEq} h_z'=\tilde h_z.$$ First, $$\label{FIRST} \begin{aligned} \frac{d}{d \epsilon}\Big|_{\epsilon = 0} (\hat\Gamma_z^{\epsilon})^\star d \Phi_0(\nabla H_{\epsilon}) = h_z \big(\hat\Gamma_z^\star d\Phi_0 (\nabla H))\big)' + \tilde h_z \hat\Gamma_z^\star d\Phi_0 (\nabla H). \end{aligned}$$ Second, $$\label{EP'2} \frac{d}{d \epsilon}\Big|_{\epsilon = 0} A_T (\hat\Gamma_z^{\epsilon})^\star \Phi_0 = A_T (h_z \frac{d}{dt} \hat\Gamma_z^\star \Phi_0).$$ Combining , and , we get that $$\label{VAR} h_z \big(\hat\Gamma_z^\star d\Phi_0 (\nabla H)) \big)' + \tilde h_z \hat\Gamma_z^\star d\Phi_0 (\nabla H) + A_T (h_z \frac{d}{dt} \hat\Gamma_z^\star \Phi_0) \in PW_T({{\mathbb R}}),$$ for all nonnegative functions $h \in C^{\infty}(M)$ for which $(0,H_\eps,z)\in\calC_{T,z}$. In this formula, $$\label{azbzEq} a_z:= \hat\Gamma_z^* \Phi_0, \;\;\;\;\; b_z:=\hat\Gamma_z^* d\Phi_0(\nabla H)$$ are two fixed functions on ${\mathbb{R}}$ satisfying (by ) $$\label{BASIC} b_z+ A_Ta_z \in PW_T({\mathbb{R}}).$$ By , rewrite equation as $$\label{VARB} h_z b_z'+ \tilde h_z b_z +A_T(h_z a_z') = (h_z b_z)'+A_T(h_z a_z') \in PW_T({\mathbb{R}}).$$ Our goal is now to show that is impossible for a dense set of $h$. We denote by ${{{\operatorname{Hilb}}}}:\calS'\ra \calS'$ the Hilbert transform, defined by $$({{{\operatorname{Hilb}}}}\, f)(t)=-\int_{\mathbb{R}}\i\,\sign\,(\xi) e^{\i t\xi}\hat f(\xi) d\xi.$$ \[HilbertLemma\] Let $h_z$ be given by . Then $ \i h_z b_z+h_za_z'= 2h_z\Gamma_z^\star {\bar\partial}\Phi_0(X_H) $ admits a holomorphic extension to $S_T$. By , $$\i\xi\,\widehat{h_zb_z}+\xi\coth T\xi\,\widehat{h_za_z'}=o(e^{-T|\xi|}).$$ Now, $\coth T\xi-\sign(\xi)=O(e^{-2T|\xi})$. Since $a_z'\in L^\infty({\mathbb{R}})$ and $h_z/t\in L^\infty({\mathbb{R}})$, this implies $$\i\xi\,\widehat{h_zb_z}+\xi\sign(\xi)\,\widehat{h_za_z'}=o(e^{-T|\xi|}),$$ or $\widehat{h_zb_z}-\i\sign(\xi)\,\widehat{h_za_z'}=o(e^{-T|\xi|})$, i.e., $h_zb_z+{{{\operatorname{Hilb}}}}(h_za_z')\in PW_T$. Since by definition ${{{\operatorname{Hilb}}}}$ maps $PW_T$ to itself, we also have ${{{\operatorname{Hilb}}}}(h_zb_z)-h_za_z'\in PW_T$. Multiply the former equation by $\i$ and add it to the latter to obtain $$(I-\i{{{\operatorname{Hilb}}}})(\i h_z b_z-h_za_z')\in PW_T,$$ and by conjugation $(I+\i{{{\operatorname{Hilb}}}})(\i h_z b_z+h_za_z')\in PW_T$. Since $I+\i{{{\operatorname{Hilb}}}}$ is twice the orthogonal projection operator onto the positive frequency space, it follows that $$\calF(\i h_z b_z+h_za_z')(\xi)=o(e^{-T\xi}), \quad \h{for all $\xi>0$}.$$ By the proof of [@St Theorem 3.1] it follows that $\i h_z b_z+h_za_z'$ admits a holomorphic extension to the one-sided strip $S_T$. The lemma now follows from and , and $d\Phi_0(X_H+\i JX_H)=2 d\Phi_0(X^{0,1}_H)=2{\bar\partial}\Phi_0(X_H)$. \[hzACLemma\] There exist $\alpha<\beta\in{\mathbb{R}}$ and $\eps\in(0,T]$ all independent of $T$ (but depending on $z$) such that $h_z$, given by , admits a holomorphic extension to the two-sided rectangle $[-\eps,\eps]\times [\alpha,\beta]\subset S_T\cup \overline{S_T}$. In particular, $\hat\Gamma_z^\star h$ is real-analytic on $[\alpha,\beta]$. First, observe that is true for the constant function $h\equiv 1$ on $M$: then $(0,(1+\eps)\dot\vp_0,z)\in\calC_{T',z}$ for all $\eps\in[0,\eps_0]$, for some $T'<T$. In fact, there exists then a solution to the HCMA for some $T'<T$ by reparametrizing $\vp$ in the $s$ variable, and $T'\ra T$ as $\eps_0\ra0$. This proves the claim. Lemma \[HilbertLemma\] implies that $h_z(\i b_z+a_z')$, respectively $h_z(-\i b_z+a_z')$, admits a holomorphic extension to $S_T$, respectively, $\overline{S_T}$. When $h\equiv1$, then $\tilde h_z\equiv 1$ and $h_z(t)=t$, on ${\mathbb{R}}$. Thus, by the previous paragraph, these estimates hold for $h_z=t$. Hence, $\i b_z+a_z'$, respectively $-\i b_z+a_z'$, admits a holomorphic extension to $S_T$, respectively, $\overline{S_T}$. By dividing, and since $h_z$ is real, it follows by the Schwarz reflection principle that $h_z$ admits a holomorphic extension to some rectangle $[-\eps,\eps]\times[\alpha,\beta]\subset S_T$ whenever $\i b_z+a_z'$ does not vanish on $[\alpha,\beta]$ (here we also used the fact that zeros of holomorphic functions cannot have an accumulation point). In particular, $h_z$ is real analytic on ${\mathbb{R}}\setminus W_z$, where $$W_z:=\{t\in{\mathbb{R}}\,:\,a_z'=\Gamma_z^\star X_H\Phi_0(\i t)= d\Phi_0(X_H)\circ \Gamma_z(\i t)=0\}.$$ The proof is complete if ${\mathbb{R}}{\setminus}W_z$ contains an open interval. Since $W_z$ is closed it thus suffices to rule out the case where $W_z={\mathbb{R}}$, i.e., $a_z'=0$, and hence $a_z=0$, on ${\mathbb{R}}$. If this holds for every point in a neigborhood of $z$ then $\Phi_0$ must be a function of $H$ on some neighborhood of $z$ in $M$. Clearly, this is a non-generic property and perturbing either $H$ or adding to $\Phi_0$ the real part of a generic local holomorphic function (this does not require changing $\o_{\vp_0}$) will destroy this property. Thus, $h_z$ must be real-analytic at least on some open interval on ${\mathbb{R}}$, and by and differentiation so is $\hat\Gamma_z^\star h$. Now, let $[\alpha,\beta]\subset {\mathbb{R}}\setminus W_z$ be any nonempty interval, and note that $W_z$ is independent of $T$. We now complete the proof of Theorem \[GENERIC\]. By taking $\beta-\alpha$ sufficiently small we may assume that $\hat\Gamma_z:[\alpha,\beta]\ra M$ is an embedded curve. Consider the map $R_z:C^3(M)\ra C^3([\alpha,\beta])$ defined by $R_zf:=\hat\Gamma_z^\star f|_{[\alpha,\beta]}$. Observe that $R_z$ is a bounded surjective linear operator. Hence, it defines an open map. Let $B$ be any open ball in $C^3(M)$ containing the zero function $0$. If for some $T>0$, $\{(\vp_0,\dot\vp_0+f,z)\,:\, f\in B\}\subset \calC_{T,z}$ then Lemma \[hzACLemma\] implies that $R_z(B)$ is contained in the subset of real-analytic functions in $C^3([\alpha,\beta])$, with $[\alpha,\beta]$ independent of $T>0$. However, the latter is not an open subset in $C^3([\alpha,\beta])$. This concludes the proof of Theorem \[GENERIC\]. The smooth lifespan of the HRMA {#LifespanHRMASection} =============================== In this section we restrict to toric manifolds and prove Proposition \[ToricLifespanProp\] concerning the analytic continuation of orbits of Hamiltonian orbits and the invertibility of the associated Moser maps. The first part, concerning the infinite analytic continuation of the Hamiltonian flow defined by the Cauchy data, is proved in Lemma \[AnalyticContinuationHamOrbitsToricLemma\]. The second part, concerning the invertibility of the Moser maps, is proved in Lemma \[InvertibilityMoserMapsToricLemma\]. Some background on toric [[K]{}\^]{}manifolds {#ToricBackgroundSubsection} --------------------------------------------- We briefly recall some background facts on toric [[K]{}\^]{}manifolds. For more detailled background we refer to [@R; @RZ1] and references therein. A symplectic toric manifold is a compact closed [[K]{}\^]{}manifold $(M,\o)$ whose automorphism group contains a complex torus $({{\mathbb C}}^\star)^n$ whose action on a generic point is isomorphic to $({{\mathbb C}}^\star)^n$, and for which the real torus $(S^1)^n\subset ({{\mathbb C}}^\star)^n$ acts in a Hamiltonian fashion by isometries. We will work with coordinates on the open dense orbit of the complex torus given by $z_j=e^{x_j/2+\i\th_j}, j=1,\ldots,n$, with $(x,\th)=(x_1,\ldots,x_n,\th_1,\ldots,\th_n)\in{{\mathbb R}}^n\times (S^1)^n$. Let $M_\open\isom({{\mathbb C}}^\star)^n$ be the open orbit of the complex torus in $M$ and write $$\label{OpenOrbitPotentialEq} \o|_{M_\open}=\i{\partial{\bar\partial}}\psi_\o.$$ We call $\psi_\o$ the open-orbit [[K]{}\^]{}potential of $\o$. The real torus $(S^1)^n\subset ({{\mathbb C}}^\star)^n$ acts in a Hamiltonian fashion with respect to $\o$. The image of the moment map $\nabla\psi_\o$ is a convex Delzant polytope $P\subset{\mathbb{R}}^n$ and depends only on $[\o]$ (note that $\o$ only determines $P$ up to translation; we fix a strictly convex $\psi_\o$ satisfying (\[OpenOrbitPotentialEq\]) to fix $P$). We further assume that this is a lattice polytope. Being a lattice Delzant polytope means that: (i) at each vertex meet exactly $n$ edges, (ii) each edge is contained in the set of points $\{p+tu_{p,j}\,:\, t\ge0\}$ with $p\in{{\mathbb Z}}^n$ a vertex, $u_{p,j}\in{{\mathbb Z}}^n$ and $$\label{UpjEdgesDef} \h{span}\{u_{p,1},\ldots,u_{p,n}\}={{\mathbb Z}}^n.$$ Equivalently, there exist outward pointing normal vectors $\{v_j\}_{j=1}^d\subset{{\mathbb Z}}^n$ that are primitive (i.e., their components have no common factor) to the $d$ facets in $\partial P$ and $P$ may be written as $$\label{PdefEq} P=\{y\in{\mathbb{R}}^n\,:\, l_j(y):=\langle y,v_j\rangle-\lambda_j\le0,\quad j=1,\ldots,d\},$$ with $\lambda_j=\langle p,v_j\rangle\in{{\mathbb Z}}$ with $p$ any vertex on the $j$-th facet, and $y$ the coordinate on ${\mathbb{R}}^n$. Given a toric metric $\o_\vp$ its corresponding open-orbit [[K]{}\^]{}potential $\psi$ is a strictly convex function on ${\mathbb{R}}^n$ in logarithmic coordinates. Therefore its gradient $\nabla\psi$ is one-to-one onto $P=\overline{\Im\nabla\psi}$. Its Legendre dual $u:=\psi^\star$, called the symplectic potential, is a strictly convex function on $P$. Recall the following formulas that will be used throughout $$\label {GradientLegendreTransformEq} (\nabla\psi)^{-1}(y)=\nabla u(y),$$ $$\label{HessianLegendreTransformEq} (\nabla^2 \psi)^{-1}|_{(\nabla\psi)^{-1}(y)}=\nabla^2 u|_y,$$ and if $\eta(s)$ is a one-parameter family of [[K]{}\^]{}potentials and $u(s):=\eta(s)^\star$ the corresponding symplectic potentials then $$\label{VariationPotentialEq} \dot\eta(s) =-\dot u(s)\circ \nabla\eta(s).$$ The proofs of these identities, assuming at least $C^2$ regularity, can be found in [@R pp. 84–87]. Complexifying Hamiltonian flows on toric manifolds {#ComplexifyingHamFlowsSubsection} -------------------------------------------------- First we establish the following result regarding the existence of analytic continuations for the Hamiltonian orbits. It shows that on a toric manifold any smooth Cauchy data is good, and moreover gives an explicit expression for the associated Moser maps. \[AnalyticContinuationHamOrbitsToricLemma\] Let $(M,J,\o)$ be a toric [[K]{}\^]{}manifold. Given a toric [[K]{}\^]{}potential $\vp_0$ let $\psi_0$ be a smooth strictly convex function on ${\mathbb{R}}^n$ such that over the open orbit $\o_{\vp_0}=\i{\partial{\bar\partial}}\psi_0$, and let $\dot\vp_0$ be a smooth torus-invariant function on $M$. For every $z\in M_\open$, the orbit of the Hamiltonian vector field $X_{\dot\vp_0}^{\o_{\vp_0}}$ admits an analytic continuation to the strip $S_\infty$. Moreover, it is given explicitly by $$f_{\tau}(z) = \exp-\i\tau X_{\dot\vp_0}^{\o_{\vp_0}}: z\mapsto z-\tau(\nabla^2\psi_0)^{-1}\nabla_{x}\dot\vp_0,\quad \tau\in S_\infty.$$ This expression remains valid on the divisor at infinity if we restrict to the orbit coordinates $\tilde x$ on a slice containing $z$. Here (and in similar expressions below) by $(\nabla^2\psi_0)^{-1}\nabla_{x}\dot\vp_0$ we mean the usual matrix multiplication of the matrix $(\nabla^2\psi_0)^{-1}(x)$ and the vector $\nabla_{x}\dot\vp_0(x)$. The moment coordinates $y$ on the polytope $P$ and the angular coordinates on the regular orbits are action-angle coordinates for the $(S^1)^n$ Hamiltonian action on $(M,\o_{\vp_0})$, in other words $$\label{SymplecticFormMomentCoordsEq} (\nabla\psi_0)_\star\o_{\vp_0}=\sum_{j=1}^n dy_j\w d\th_j,\quad \h{\ over\ \ $(P\setminus\del P)\times(S^1)^n$}.$$ The Hamiltonian vector field of $\dot\vp_0$ is given in these coordinates by $$\label{PushForwardHamVectorFieldEq} (\nabla\psi_0)_\star X^{\o_{\vp_0}}_{\dot\vp_0} = -\sum_{j=1}^n\frac{\del\dot\vp_0}{\del y_j}\big((\nabla\psi_0)^{-1}(y)\big)\frac{\del}{\del\th_j}, \quad y\in P\setminus\del P.$$ Therefore the Hamiltonian flow of $X^{\o_{\vp_0}}_{\dot\vp_0}$ is given, in terms of the moment coordinates, by $$\label{HamFlowToricPullbackEq} \nabla\psi_0\circ\exp t X^{\o_{\vp_0}}_{\dot\vp_0}\circ(\nabla\psi_0)^{-1}.(y,\th) = (y,\th-t\nabla_y\dot\vp_0\circ(\nabla\psi_0)^{-1}), \quad \h{\ over\ \ $(P\setminus\del P)\times(S^1)^n$},$$ and in terms of the coordinates on $M_\open$ by $$\exp t X^{\o_{\vp_0}}_{\dot\vp_0}.(x,\th) = (x,\th-t(\nabla^2\psi_0)^{-1}\nabla_x\dot\vp_0).$$ It therefore admits a holomorphic extension to a map $\exp \i\tau X^{\o_{\vp_0}}_{\dot\vp_0}, \tau=s+\i t$, given in these coordinates by (using (\[GradientLegendreTransformEq\])-(\[HessianLegendreTransformEq\])) $$\label{HamFlowOpenOrbitEq} \begin{aligned} \exp -\i\tau X^{\o_{\vp_0}}_{\dot\vp_0}.(x,\th) & = (x-s(\nabla^2\psi_0)^{-1}\nabla_x\dot\vp_0\,,\;\th-t(\nabla^2\psi_0)^{-1}\nabla_x\dot\vp_0), \cr & \qquad\qquad\qquad \qquad\qquad\qquad \qquad\qquad s\in{\mathbb{R}}_+,t\in{\mathbb{R}}. \end{aligned}$$ For each $z\in M_\open$, this is a holomorphic map of $S_\infty$ into $M_\open\subset M$ since in terms of the complex coordinates $z_j:=x_j+\i\th_j$ it is given by an affine map $$\tau\mapsto z-\tau(\nabla^2\psi_0)^{-1}\nabla_{x}\dot\vp_0,\quad \tau\in S_\infty.$$ It remains to consider orbits of points $z\in M\setminus M_\open$ (for these points Equation (\[HamFlowOpenOrbitEq\]) is not valid), and this essentially amounts to some toric bookkeeping. Let $F\subset\partial P$ be a codimension $k$ face of $P$ cut out by the equations (see (\[PdefEq\])) $$\label{FFaceDefEq} F:=\{y\in\del P\,:\,\langle y, v_{j_i}\rangle=\lambda_{j_i},\quad i=1,\ldots,k\},$$ and assume that $z$ corresponds to a point in the interior of $F$. More precisely, assume that for a sequence of points $\{z_i\}\subset M_\open$ converging to $z$ the points $\nabla\psi_0(z_i)$ converge to a point in the interior of $F$. On points in $M$ that correspond to points in $F\setminus \del F$ the stabilizer of the $(S^1)^n$-action action is $k$-dimensional. In other words, when restricted to $F\setminus \del F$, the vector fields $\frac{\del}{\del\th_1},\ldots,\frac{\del}{\del\th_n}$ span an $(n-k)$-dimensional distribution. Without loss of generality we may assume that in (\[FFaceDefEq\]) we have $\{j_i,\ldots,j_k\}=\{1,\ldots,k\}$ (otherwise rename the labels). Let $p\in\del F$ be a vertex and let $u_{p,1},\ldots,u_{p,n}$ be the vector defining the edges emanating from $p$, as in (\[UpjEdgesDef\]). Without loss of generality assume the vectors $u_{p,1},\ldots,u_{p,n-k}$ span $F$. On $F\setminus\del F$ the vectors $\{v_1,\ldots,v_k,u_{p,1},\ldots,u_{p,n-k}\}$ span ${\mathbb{R}}^n$ and one may find $n-k$ unit vectors $\tilde u_{p,1},\ldots,\tilde u_{p,n-k}$ such that $\{v_1,\ldots,v_k,\tilde u_{p,1},\ldots,\tilde u_{p,n-k}\}$ form an orthonormal basis. Let $U$ denote the orthogonal matrix obtained from these $n$ column vectors. Let $\tilde y:=yU$ and $\tilde\th:=\th U$. Then in these coordinates (\[SymplecticFormMomentCoordsEq\]) becomes $$\label{SymplecticFormAdaptedMomentCoordsEq} (\nabla\psi_0)_\star\o_{\vp_0}=\sum_{j=1}^n d\tilde y_j\w d\tilde\th_j,\quad \h{\ over\ \ $(P\setminus\del P)\times(S^1)^n$}.$$ The advantage of this formula is that it specializes to the following formula when restricted to $F\setminus \del F$: $$\label{SymplecticFormAdaptedMomentCoordsRestrictionEq} \o_{\vp_0}|_{(F\setminus \del F)\times(S^1)^{n-k}} = \sum_{j=k+1}^{n} d\tilde y_j\w d\tilde\th_j.$$ Hence, in these coordinates the Hamiltonian flow of $\dot\vp_0$ is given by $$(\tilde y,\tilde\th) \mapsto (\tilde y,\tilde\th_{1},\ldots,\tilde\th_k, \tilde\th_{k+1}+t\nabla_{\tilde y_{k+1}}\dot u_0, \ldots, \tilde\th_{n}+t\nabla_{\tilde y_{n}}\dot u_0).$$ In order to describe the complexification of this map in $M$, we use local holomorphic slice-orbit coordinates $(z',z'')\in{{\mathbb C}}^k\times({{\mathbb C}}^\star)^{n-k}$ (see, e.g., [@SoZ1]) that can be described as follows. The stabilizer of $({{\mathbb C}}^\star)^n$ at $z$ is $({{\mathbb C}}^\star)^k$. The tangent space $T_zM$ decomposes to the tangent space to the orbit of $z$, $T_z(({{\mathbb C}}^\star)^{n-k}.z)$, and its normal $(T_z(({{\mathbb C}}^\star)^{n-k}.z))^{\perp}$. Intersecting each of these spaces with the unit ball in $T_zM$ we therefore obtain local holomorphic coordinates $(z',z'')\in{{\mathbb C}}^k\times({{\mathbb C}}^\star)^{n-k}$. The coordinates $z'$ are called the slice coordinates, while the $z''$ are called the orbit coordinates. We may write $z''_j=\tilde x_j/2+\i\tilde\th_j, \, j=k+1,\ldots,n$, with $\tilde x_j=\nabla u_0(\tilde y_j)$. On $F\setminus \del F$ the matrix $\nabla_{\tilde y}u$ is of rank $n-k$ with the bottom $(n-k)\times(n-k)$ block an invertible matrix. The same reasoning as before now shows that we have a formula analogous to (\[HamFlowOpenOrbitEq\]) where we replace $(\nabla^2_x\psi)^{-1}$ by that block of $\nabla^2_{\tilde y}u$, and $\nabla_x\dot\vp_0$ by $(\nabla_{\tilde x_{k+1}}\dot\vp_0,\ldots,\nabla_{\tilde x_{n}}\dot\vp_0)$. Once again we see that the resulting maps extend to the strip $S_\infty$, and this concludes the proof of the Lemma. \[ToricNoCommutativityRemark\] [As the Lemma shows, the Hamiltonian orbits admit an analytic continuation to the whole upper half plane. In relation to Remark \[NoGroupLawRemark\], we point out that nevertheless the Moser maps do not generically obey a group law in the holomorphic variable $\tau$. To see this, change variables to the action-angle variables $(y, \theta)$. Since $y=y(x)$, $X^{\o_{\vp_0}}_H = -\sum_j \frac{\partial H}{\partial I_j} \frac{\partial}{\partial \theta_j}$ and $J X^{\o_{\vp_0}}_H = -\sum_j \frac{\partial H}{\partial I_j} \frac{\partial}{\partial x_j}$ (with a slight abuse of notation as compared to ). Then $$[X^{\o_{\vp_0}}_H, J X^{\o_{\vp_0}}_H ] = - \sum_{j, k} \frac{\partial H}{\partial I_k} \frac{\partial^2 H}{\partial x_k \partial I_j} \frac{\partial}{\partial \theta_j},$$ vanishing only if the matrix $\begin{pmatrix} \frac{\partial^2 H}{\partial x_k \partial I_j} \end{pmatrix}$ has a kernel, which is generically false. ]{} Moser flows on toric manifolds {#MoserFlowsSubsection} ------------------------------ Having derived an explicit expression for the analytic continuations of the Hamiltonian orbits for all imaginary time, we now turn to investigate the invertibility of the resulting Moser maps. \[InvertibilityMoserMapsToricLemma\] Let $(M,J,\o)$ be a toric [[K]{}\^]{}manifold. Given a toric [[K]{}\^]{}potential $\vp_0$ let $\psi_0$ be a smooth strictly convex function on ${\mathbb{R}}^n$ such that over the open orbit $\o_{\vp_0}=\i{\partial{\bar\partial}}\psi_0$, and let $\dot\vp_0$ be a smooth torus-invariant function on $M$. The Moser maps $f_s(z)=\exp-\i sX_{\dot\vp_0}^{\o_{\vp_0}}.z$ defined by Lemma \[AnalyticContinuationHamOrbitsToricLemma\] are smoothly invertible if and only if $$\label{ToricTspanSecondEq} s< T_\span^\cvx:= \sup\,\{\,a>0: \psi^\star_0-a\dot \vp_0\circ(\nabla\psi_0)^{-1} \hbox{\rm\ is convex}\}.$$ Note that the formula for $T^\cvx_\span$ is well-defined independently of the choice of the open-orbit [[K]{}\^]{}potential $\psi_0$ for $\o_{\vp_0}$. From the proof of Lemma \[AnalyticContinuationHamOrbitsToricLemma\] (cf.) we have the following formula for the Moser maps, restricted to the open orbit, $$\label{MoserMapOpenOrbitEq} f_s(z)=z-s(\nabla^2\psi_0)^{-1}\nabla_{x}\dot\vp_0,\quad \tau\in S_\infty, z\in M_\open,$$ or in terms of the moment coordinates $$\label{MoserMapOpenOrbitMomentCoordEq} f_s(\nabla u_0(y)) = \nabla_y u_0(y) + s\nabla_{y}\dot u_0,\quad s\in {\mathbb{R}}_+, y\in P\setminus\partial P.$$ Since $\nabla_y=\nabla^2_yu_0.\nabla_x$, applying the gradient with respect to $y$ to this equation we obtain $$\nabla^2 u_0(y). \nabla_x f_s(\nabla u_0(y)) = \nabla^2_y(u_0+s\dot u_0).$$ Since $\nabla^2 u_0$ is invertible for $y\in P\setminus\del P$, it follows that the gradient of $f_s$ is invertible at $z\in M_\open$ if and only if $u_0+s\dot u_0$ is strictly convex on $P\setminus\del P$. The analysis for $z\in M\setminus M_\open$ is similar, following the technicalities outlined in the proof of Lemma \[AnalyticContinuationHamOrbitsToricLemma\]. Since by definition $u_0=\psi^\star_0$ and using (\[GradientLegendreTransformEq\]) we obtain (\[ToricTspanSecondEq\]). This concludes the proof of Proposition \[ToricLifespanProp\]. Leafwise subsolutions for HRMA {#OptimalSubsolutionHRMASection} ============================== The toric setting is special in that first the Moser maps exist for all $s\ge0$, and second that $\o$ admits a [[K]{}\^]{}potential on the whole open orbit $M_\open$. As in the discussion below , the Cauchy problem takes the following form: $$\label {CauchyLaplaceLeaveToricEq} \begin{aligned} \Delta\chi_z & = 0,\quad \h{\ on \ } S_\infty, \cr \chi_z(\i t) & = \psi_0\circ f_{\i t}(z),\quad \h{\ on \ } \del S_\infty, \cr\dis \frac{\del\chi_z}{\del s}(\i t) & = \dot\vp_0\circ f_{\i t}(z) - \nabla_{g_{\vp_0}}\dot\vp_0(\psi_0)\circ f_{\i t}(z), \quad \h{\ on \ } \del S_\infty. \end{aligned}$$ We now turn to proving that the HRMA (\[HRMARayEq\]) admits a unique leafwise subsolution. First, we record some useful formulas for the Moser maps on a toric manifold. They follow from the proof of Lemma \[AnalyticContinuationHamOrbitsToricLemma\] by substituting $y=\nabla\psi_0$ in (\[MoserMapOpenOrbitMomentCoordEq\]) and using (\[GradientLegendreTransformEq\]). \[MoserPathFormulaToricCor\] Let $\psi_s$ be a smooth solution of the HRMA (\[HRMARayEq\]), and let $f_s$ denote the associated Moser diffeomorphisms given by Lemma \[AnalyticContinuationHamOrbitsToricLemma\]. Then on the open-orbit, $$\label{SecondFsToricMomentEq} f^{-1}_s = (\nabla\psi_0)^{-1}\circ\nabla\psi_s = \nabla u_0\circ(\nabla u_s)^{-1},\quad s\in[0,T^\cvx_\span),$$ and if we let $u_s(y)=u_0(y)+s\dot u_0(y)$, then $$\label{SecondAllTimeFsToricMomentEq} f_s = \nabla u_s\circ(\nabla u_0)^{-1} ,\quad \h{\ all \ } s\ge0.$$ These expressions remain valid globally on $M$ if we use the Euclidean gradient in the orbit coordinates $\tilde x$ along each slice. Observe that and (\[SecondAllTimeFsToricMomentEq\]) are in agreement with Proposition \[ToricLifespanProp\] (i),(ii), respectively. Next, we show that each of the Cauchy problems admits a unique global smooth solution. In the toric setting the harmonic extension to the generalized leaves of the foliation is especially simple since the initial conditions are constant on the boundary of the strip. In particular, the harmonic functions must be linear along the leaves of the foliation. \[ToricHarmonicAlongLeavesProp\] For every $z\in M_\open$ the Cauchy problem for the Laplace equation admits a unique smooth solution, given by $$\label{LinearSolutionAlongLeavesHRMAEq} \chi_z(\tau):= \langle \nabla\psi_0(z),\nabla(u_0+s\dot u_0)\circ\nabla\psi_0(z)\rangle - (u_0+s\dot u_0)\circ\nabla\psi_0(z).$$ Note first that uniqueness holds for the Cauchy problem for the Laplace equation on a half-plane (this can be obtained from a suitable generalization of Lemma \[LaplaceUniquenessLemma\] to the case $T=\infty$). We claim that a solution to is given by (\[LinearSolutionAlongLeavesHRMAEq\]). First, $\chi_z$ is linear in $s$ and independent of $t$, hence harmonic. Moreover, by , $$\chi_z(\i t) = \langle \nabla\psi_0(z),\nabla u_0\circ\nabla\psi_0(z)\rangle - u_0\circ\nabla\psi_0(z) = u_0^\star(z)=\psi_0(z),$$ and by and , $$\begin{aligned} \frac{\del\chi_z}{\del s}(\i t) & = \langle \nabla\psi_0(z),\nabla\dot u_0\circ\nabla\psi_0(z)\rangle - \dot u_0\circ\nabla\psi_0(z) \cr & = \langle\nabla\psi_0(z),-\nabla^2\psi_0.\nabla\dot\vp_0\rangle +\dot\vp_0(z) \cr & = -g_{\vp_0}(\nabla\psi_0(z),\nabla\dot\vp_0) +\dot\vp_0(z) \cr & = -\nabla_{g_{\vp_0}}\dot\vp_0(\psi_0)(z)+\dot\vp_0(z). \end{aligned}$$ Finally, observe that in (\[CauchyLaplaceLeaveToricEq\]) one may eliminate $f_{\i t}$ since the data $(\psi_0,\dot\vp_0)$ is $(S^1)^n$-invariant. Thus, $\chi_z$ satisfies the initial conditions. [To see how this Lemma fits in with Proposition \[LeafwiseProp\], note that $\Phi_z=\gamma_z^\star(\psi_0-\vp_0)$. Thus, $p_z(t)=-\del_s\Phi_z=d(\psi_0+\vp_0)(\nabla_{g_{\vp_0}}\dot\vp_0)(\Gamma_z(\i t))$, and $q_z-p_z=\dot\vp_0(\Gamma_z(\i t))-d\psi_0(\nabla_{g_{\vp_0}}\dot\vp_0)(\Gamma_z(\i t))$ is a constant (depending on $z$), and so naturally admits an analytic continuation to a whole half-plane. ]{} We now turn to proving that the Cauchy problem admits a unique leafwise subsolution, equal precisely to the Legendre transform potential $\psi_L$ given by . Note that we use interchangeably $z$ and $x=\log|z|^2$, as $\psi_L$ is independent of $\theta$. To show that (\[OptimalSubsolutionToricEq\]) defines a leafwise subsolution on the open orbit it suffices to show that for every $z\in M_\open$ the function $F_z^\star \psi_L$ solves the Cauchy problem (\[CauchyLaplaceLeaveToricEq\]). Now, $$\psi_L(s,z)=u_s^\star(z)=\sup_{y\in P}[\langle y,x\rangle - u_s(y)],$$ with the supremum achieved in at least one point $y$ that is contained in the set $(\nabla u_s)^{-1}(z)$. It then follows from (\[SecondAllTimeFsToricMomentEq\]) and that $F_z^\star \psi_L=u_s^\star\circ f_\tau(z)=\chi_z(s+\i t)$, and thus by Lemma \[ToricHarmonicAlongLeavesProp\] $\psi_L$ defines a leafwise subsolution. This proves the existence part of Proposition \[OptimalSubsolutionToricProp\] (i). To prove the existence part of Proposition \[OptimalSubsolutionToricProp\] (ii), it suffices to note that every leafwise subsolution for the HRMA (\[HRMARayEq\]) on the open orbit gives rise to a global leafwise subsolution for the HCMA (\[HCMARayEq\]) by letting $$\vp(s+\i t,z)=\psi_L(s,z)-\psi_0(z).$$ This can be seen as follows. Note first that according to Lemma \[AnalyticContinuationHamOrbitsToricLemma\] the maps $F_z$ are smooth. Second, note that according to our description of the Moser maps in orbit coordinates, it follows that $f_\tau$ preserves the interior of each codimension $k$ toric subvariety of the divisor at infinity $D$. And so, given $z\in D=M\setminus M_\open$, the condition $F_z^\star(\pi_2^\star\o+\i{\partial{\bar\partial}}\vp)$ is equivalent to a Cauchy problem for the Laplace equation, where we now let $N$ be the open toric variety obtained as the interior of the codimension $k$ toric subvariety containing $z$. This Cauchy problem then admits a unique smooth global solution, by working in orbit coordinates, as in Lemma \[AnalyticContinuationHamOrbitsToricLemma\]. And since, as already noted, $F_z$ preserves $N$, the harmonicity of $F_z^\star(\psi_N+\vp)$ implies that $F_z^\star(\pi_2^\star\o+\i{\partial{\bar\partial}}\vp)=0$ on $N$, where here $\psi_N$ is a local [[K]{}\^]{}potential for $\o$ on $N$. Finally, we prove the uniqueness of the leafwise subsolution just constructed. Let $\eta(\tau,z)$ be another leafwise subsolution. It will suffice to prove that $\eta=\psi_L$ on the product of $S_\infty$ and the open-orbit $M_\open$. Observe that by (\[SecondAllTimeFsToricMomentEq\]),(\[LinearSolutionAlongLeavesHRMAEq\]), and the fact that $\nabla\psi_0:{\mathbb{R}}^n\ra P\setminus\del P$ is an isomorphism we have $$\eta(\tau,\nabla u_s(y))=\langle y,\nabla u_s(y)\rangle-u_s(y).$$ for every $y\in P\setminus \del P$ and $\tau\in S_\infty$. Since $\psi_L$ satisfies the same equation and by [@RZ2 Lemma 7.1] $\Im\nabla u_s|_{P\setminus\partial P}=\Im\nabla u_0|_{P\setminus\partial P}={\mathbb{R}}^n$ it follows that $\eta=\psi_L$. [As a by-product, Propositions \[ToricLifespanProp\] and \[OptimalSubsolutionToricProp\] give an alternative and conceptual proof that the Legendre transform solves the homogeneous real Monge–Ampère equation. Of course, these results show considerably more since they give information for all time, where the Legendre duality breaks down to some extent. As studied in detail in [@RZ2], the leafwise subsolution $\psi$ measures precisely the extent to which the Legendre duality breaks down. ]{} HRMA and the Hamilton–Jacobi equation {#ConeLifespanSection} ====================================== We now turn to showing that there exists no admissible $C^1$ weak solution of the IVP for $T>T_\span^\infty$ and establishing the relation between the HRMA and the Hamilton–Jacobi equation. By a weak solution we mean a solution in the sense of Alexandrov. The first step is the observation that any $C^1$ weak solution of HRMA is a classical solution of a Hamilton–Jacobi equation. Some steps resemble the arguments of Proposition 13.1 in [@RZ2 §13]. Recall that the initial Neumann data $\dot\psi_0$ of the HRMA is a bounded function on ${\mathbb{R}}^n$ obtained by restricting the global Neumann data $\dot\vp_0$ on the toric manifold to the open-orbit. Given the Cauchy data $ (\psi_0, \dot\psi_0)$ of , we set $$\dot u_0:=-\dot\psi_0\circ(\nabla\psi_0)^{-1}.$$ \[GMapLemma\] Let $\eta$ be a $C^1$ admissible solution for the HRMA (\[HRMARayEq\]). Define the set-valued map, $$G:s\in{\mathbb{R}}_+\mapsto \Im\,\nabla\eta(\{s\}\times{\mathbb{R}}^n)\subset{\mathbb{R}}^{n+1}.$$ Then $G(s)=G(0)=\{(-\dot u_0(y),y)\,:\, y\in P{\setminus}\del P\}$, for each $s\in[0,T)$. Since $\eta$ is admissible, $\nabla_x\eta({\mathbb{R}}^n)=P{\setminus}\del P$. Thus, $\nabla\eta(\{s\}\times{\mathbb{R}}^n)\subset {\mathbb{R}}\times (P{\setminus}\del P)$. Note that $G(0)$ is the graph of $-\dot u_0$ over $P{\setminus}\del P$. We now prove that $G(s)\subset G(0)$. The idea is that $s \to G(s)$ is a continuous set-valued map. If $G(s) $ is not contained in $G(0)$, it would sweep out a set of positive Lebesgue measure in ${\mathbb{R}}\times (P{\setminus}\del P)\subset{\mathbb{R}}^{n+1}$ as $s$ varies, contrary to the assumption that $\eta$ is a weak solution. Since $\eta(s)$ is $C^1$ and strictly convex, its gradient map $\nabla_x \eta (s): \times {{\mathbb R}}^n \to P\backslash \partial P$ is a homeomorphism to its image, i.e. has a $C^0$ single valued inverse, and for each $x_0 \in P{\setminus}\del P$, $(\nabla_x\eta(s))^{-1}:P{\setminus}\del P\ra{\mathbb{R}}^n$ maps an open neighborhood of $\nabla_x\eta(s, x_0)\in P$ to an open neighborhood $U$ of $x_0$ in ${\mathbb{R}}^n$. To clarify the picture, consider the diagram: $$\label{DIAGRAM} \begin{array}{ccccc} \{s\} \times {{\mathbb R}}^n & \mapright{\nabla \eta}{75} & G(s) = \{\dot{\eta}(s,x), \nabla_x \eta(s, x))\} \subset {{\mathbb R}}\times ( P{\setminus}\del P) \\ & & \\ \mapdown{\pi}& & \mapdown{\pi}\\ & & \\ {{\mathbb R}}^n & \mapleft{(\nabla_x \eta(s))^{-1}}{75} & P{\setminus}\del P \end{array}$$ Here, $\pi$ is the natural projection. Since $\nabla_x \eta(s, x): {{\mathbb R}}^n \to P{\setminus}\del P$ is a homeomorphism, also $\pi: G(s) \to P{\setminus}\del P$ is a homeomorphism. Thus, $G(s)$ is a graph over $P{\setminus}\del P$. Now suppose that there exists $z=(\dot\eta(s,x_0),\nabla_x\eta(s,x_0))\in G(s)\setminus G(0)$, i.e. $\dot\eta(s,x_0)\not=-\dot u_0\circ\nabla_x\eta(s,x_0)$. Then $\pi^{-1}(U) \subset G(s)$ is a graph passing through $z\not\in G(0)$. Since $G(s)$ is a continuous set-valued mapping and $\pi: G(0) \to U$ is a different graph than $\pi: G(s) \to U$, the intermediate graphs $\pi: G(\sigma) \to U$ for $\sigma \in [0, s]$ must fill out the region in between the graphs and create a set of positive Lebesgue measure. To be more precise, put $S:=\sup\{\sigma\,:\, G(\sigma) \h{ contains } (-\dot u_0\circ\nabla_x\eta(s,x),\nabla_x\eta(s,x))\}\le s$. Again by continuity, ${\mathbb{R}}\times U\cap \Big(\cup_{\sigma\in[S,s]}G(\sigma) \Big)$ must contain a set of positive Lebesgue measure in ${\mathbb{R}}^{n+1}$. This is impossible, though, by Definition \[AlexandrovDef\]. Thus, we have shown that $G(s)\subset G(0)$ But since the projection of $G(s)$ onto the ${\mathbb{R}}^n$ factor equals $P{\setminus}\del P$ for each $s$, and $G(0)$ is a graph over $P{\setminus}\del P$, the containment just proved implies the equality $G(s)=G(0)$. Thus, by Lemma \[GMapLemma\] and the differentiability assumption, for each $(s,x)\in[0,T]\times {\mathbb{R}}^n$ there exists a unique $y\in P\setminus\del P$ such that $$\Big(\frac{\del\eta}{\del s}(s,x),\nabla_x\eta(s,x)\Big)=(-\dot u_0(y),y),$$ or, in other words, $$\label{DotEtaEq} \frac{\del\eta}{\del s}(s,x)= -\dot u_0\circ \nabla_x\eta(s,x),$$ which concludes the proof of one direction of Theorem \[HJThm\]. For the converse, suppose that $\eta\in C^1([0,T]\times{\mathbb{R}}^n)$ is a solution of the Hamilton–Jacobi equation . Then $\Im\nabla\eta\subset G(0)$, and since $G(0)$ has zero Lebesgue measure in ${\mathbb{R}}^{n+1}$, $\eta$ is a weak solution of the HRMA. [The proof can be generalized to handle admissible solutions that are only partially $C^1$ regular in the sense of [@RZ2 §10]. ]{} Let $\psi_L$ denote the leafwise subsolution of the HRMA (\[HRMARayEq\]) given by Proposition \[OptimalSubsolutionToricProp\] and let $\eta$ be a $C^1$ admissible solution of (\[HRMARayEq\]) (see Definition \[AdmissibleDef\]). Both $\psi$ and $\eta$ are convex functions on $[0,T]\times{\mathbb{R}}^n$. By Theorem \[HJThm\] both $\psi$ and $\eta$ are solutions of the Hamilton–Jacobi equation . The method of characteristics implies that $C^1$ solutions of are unique as long as the characteristics of the equation do not intersect each other. The equation for the projected characteristic curves ${\bf x}(s)$ is (see, e.g., [@Evans Chapter 3]) $$\dot {\bf x}(s)=\big(1,\nabla_\xi\dot u_0({\bf p}_\xi(s))\big), \qquad {\bf x}(0)=(0,x_0),$$ while $z(s)$, the solution at ${\bf x}(s)$, satisfies $$\dot z(s)=\big(1,\nabla_\xi\dot u_0({\bf p}_\xi(s))\big)\cdot\big(p_\sigma(s),{\bf p}_{\xi}(s)\big), \qquad z(0)=\psi_0(x_0),$$ and ${\bf p}(s)=(p_\sigma(s),{\bf p}_{\xi}(s))$, the gradient of the solution at ${\bf x}(s)$, satisfies $$\dot {\bf p}(s)=0,\qquad {\bf p}(0)=(\dot\psi_0(x_0),\nabla\psi_0(x_0)).$$ Therefore, ${\bf x}(s)=\big(s,x_0+s\nabla \dot u_0(\nabla\psi_0(x_0))\big)$. Thus, the projected characteristic do not intersect as long as the map $(s,x)\mapsto (s,x+s\nabla \dot u_0(\nabla\psi_0(x)))$ is invertible, or equivalently as long as $$x\mapsto \nabla u_0\circ\nabla\psi_0(x)+s\nabla\dot u_0\circ\nabla\psi_0(x)$$ is invertible on ${\mathbb{R}}^n$; this is precisely as long as $\nabla u_0+s\nabla\dot u_0$ is invertible on $P{\setminus}\del P$, or as long as $u_0+s\dot u_0$ is strictly convex, i.e., precisely for $s<\Tspancvx$. Thus $\eta=\psi_L$ for $s\le\Tspancvx$. In fact, the equation for ${\bf x}(s)$ shows that the characteristics for the Hamilton–Jacobi equation precisely coincide with the leaves of the HRMA foliation. Moreover, the equation for $z(s)$ shows that $$\begin{aligned} z({\bf x}(s)) &=\psi_0(x_0)+s\dot\psi_0(x_0)+s\langle \nabla\dot u_0\circ\nabla\psi_0(x_0), \nabla\psi_0(x_0)\rangle \cr &=-u_0\circ(\nabla\psi_0(x_0)) +\langle \nabla\dot u_0\circ\nabla\psi_0(x_0), \nabla\psi_0(x_0)\rangle \cr &\quad +s\dot\psi_0(x_0)+s\langle \nabla\dot u_0\circ\nabla\psi_0(x_0), \nabla\psi_0(x_0)\rangle \cr &=-u_0\circ(\nabla\psi_0(x_0)) +\langle \nabla u_0\circ\nabla\psi_0(x_0), \nabla\psi_0(x_0)\rangle \cr &\quad -s\dot u_0(\nabla\psi_0(x_0))+s\langle \nabla\dot u_0\circ\nabla\psi_0(x_0), \nabla\psi_0(x_0)\rangle \cr &=-(u_0+s\dot u_0)\circ(\nabla\psi_0(x_0)) +\langle \nabla(u_0+s\dot u_0)\circ\nabla\psi_0(x_0), \nabla\psi_0(x_0)\rangle , \cr \end{aligned}$$ while from the equation for ${\bf x}(x)$ we have $${\bf x}(s) = (\nabla u_0+s\nabla \dot u_0)\circ\nabla\psi_0(x_0).$$ Altogether, letting $u_s:=u_0+s\dot u_0$, we have $$z(\nabla u_s(y))=-u_s(y)+\langle \nabla u_s(y),y\rangle,$$ or in other words, $z(s,x)=u_s^\star(x)=\psi_L(s,x)$. Note that in the proof above we show in essence that any $C^1$ solution of the HRMA is given by the Hopf–Lax formula [@Hopf; @Evans]. Finally, we relate the orbits of the Moser map, the Hamiltonian orbits, and the characteristics in ${{\mathbb R}}^{n+1}$ of the HRMA. The following generalizes to weak $C^1$ solutions of the HRMA the well-known ‘conservation law’ of smooth solutions of the HCMA. \[FTPHIDOT\] Let $\eta$ be a $C^1$ weak solution of the HRMA , and let $\vp=\eta-\psi_0$, considered as a function $M$. Also, let $f_s$ be the Moser maps $f_s(z)=\exp-\i sX_{\dot\vp_0}^{\o_{\vp_0}}.z$ defined in and Proposition \[ToricLifespanProp\]. Then $$\dot{\vp}_s\circ f_s = \dot{\vp}_0.$$ Further, the $f_s$-orbits $(s, f_s(x))$ are the leaves of the real Monge–Ampère foliation, namely the projected characteristics of the Hamilton–Jacobi equation . By combining , and Propositions \[ToricLifespanProp\] and \[ConeadmissibleIsOptimalProp\], one sees that this equation is equivalent to the Hamilton–Jacobi equation in Theorem \[HJThm\]. To prove the last statement we note that the leaves of the Monge–Ampère foliation are orbits of the complexified Hamiltonian action $\exp t X_{\dot{\vp}_0}^{\omega_{\vp_0}}$. The real orbits lie on the orbits of the Hamiltonian $(S^1)^n$-action and the real slice of this torus orbit is a point. Hence the real slice is the imaginary time orbit, i.e., the orbit of $f_{s}$. [**Acknowledgments.**]{} This material is based upon work supported in part by a NSF Postdoctoral Research Fellowship and grants DMS-0603850, 0904252. [HHHH]{} A.D. Alexandrov, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected to it, Leningrad State Univ. Ann., Math. Ser. 6 (1939), 3–35. E. Bedford, D. Burns, Holomorphic mapping of annuli in ${{\mathbb C}}^n$ and the associated extremal function, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4) 6 (1979), 381–414. E. Bedford and M. Kalka, Foliations and complex Monge–Ampère equations, Comm. Pure Appl. Math. 30 (1977), 543–571. S.K. Donaldson, Symmetric spaces, Kähler geometry and Hamiltonian dynamics, Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, 196, Amer. Math. Soc., Providence, RI, 1999, 13–33. , Holomorphic discs and the complex Monge–Ampère equation, J. Symp. Geom. 1 (2002), 171–196. L.C. Evans, Partial differential equations, Second Ed., AMS, 2011. R.L. Foote, Differential geometry of real Monge–Ampère foliations, Math. Z. 194 (1987), 331–350. , A geometric solution to the Cauchy problem for the homogeneous Monge–Ampère equation, Proc. Workshops in Pure Math. 11 (1991), 31–39, The Korean Academic Council. P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley-Interscience, 1978. J. Hadamard, Sur les problèmes aux derivées partielles et leur signification physique, Princeton University Bulletin 13 (1902), 49–52. P. Hartman, L. Nirenberg, On spherical image maps whose Jacobians do not change sign, Amer. J. Math. 81 (1959), 901–920. E. Hopf, Generalized solutions of non-linear equations of first order, J. Math. Mech. 14 (1965), 951–973. L. Hörmander, An introduction to complex analysis in several variables, Third Ed., North-Holland, 1990. , The analysis of linear partial differential operators I, Second Ed., Springer, 1990. M.M. Lavrentiev, Some improperly posed problems of mathematical physics, Springer, 1967. T. Mabuchi, Some symplectic geometry on compact Kähler manifolds I, Osaka J. Math. 24 (1987), 227–252. L.E. Payne, Improperly posed problems in partial differential equations, Regional Conference Series in Applied Mathematics, SIAM, 1975. A.V. Pogorelov, Extrinsic geometry of convex surfaces, American Mathematical Society, 1973. J. Rauch, B.A. Taylor, The Dirichlet problem for the multidimensional Monge–Ampère equation, Rocky Mountain J. Math. 7 (1977), 345–364. Y.A. Rubinstein, Geometric quantization and dynamical constructions on the space of Kähler metrics, Ph.D. Thesis, Massachusetts Institute of Technology, 2008. Y.A. Rubinstein, S. Zelditch, The Cauchy problem for the homogeneous Monge–Ampère equation, I. Toeplitz quantization, J. Differential Geom. 90 (2012), 303–327. , The Cauchy problem for the homogeneous Monge–Ampère equation, II. Legendre transform, Adv. Math. 228 (2011), 2989–3025. , The Cauchy problem for the homogeneous Monge–Ampère equation, IV, preprint, in preparation. I.Kh. Sabitov, Isometric immersions and embeddings of locally Euclidean metrics, Cambridge Scientific Publishers, 2008. S. Semmes, Complex Monge–Ampère and symplectic manifolds, Amer. J. Math. 114 (1992), 495–550. J. Song, S. Zelditch, Bergman metrics and geodesics in the space of Kähler metrics on toric varieties, Anal. PDE 3 (2010), 295–358. E.M. Stein, R. Shakarchi, Complex analysis, Princeton University Press, 2003. V. Ushakov, The explicit general solution of the trivial Monge–Ampère equation, Comm. Math. Helv. 75 (2000), 125–133. D.V. Widder, Functions harmonic in a strip, Proc. Amer. Math. Soc. 12 (1961), 67–72. , Fourier cosine transforms whose real parts are non-negative in a strip, Proc. Amer. Math. Soc. 16 (1965), 1246–1252.

--- author: - | Mangesh Gupte[^1]\ Google Inc\ <mangesh@cs.rutgers.edu> - | Darja Krushevskaja\ Rutgers University\ <darja@cs.rutgers.edu> - | S. Muthukrishnan\ Rutgers University\ <muthu@cs.rutgers.edu> bibliography: - 'algorithmica.bib' title: Analyses of Cardinal Auctions --- [^1]: This work was done while the author was graduate student at Rutgers University.

--- abstract: 'We establish exact relations between the winding of “energy” (eigenvalue of Hamiltonian) on the complex plane as momentum traverses the Brillouin zone with periodic boundary condition, and the presence of “skin modes” with open boundary condition in non-hermitian systems. We show that the nonzero winding with respect to any complex reference energy leads to the presence of skin modes, and vice versa. We also show that both the nonzero winding and the presence of skin modes share the common physical origin that is the non-vanishing current through the system.' author: - Kai Zhang - Zhesen Yang - Chen Fang bibliography: - 'NonHermitianBib.bib' title: 'Correspondence between winding numbers and skin modes in non-hermitian systems' --- [^1] [^2] \[s:intro\]Introduction ======================= Some systems that are coupled to energy or particle sources or drains, or driven by external fields can be effectively modeled Hamiltonians having non-hermitian terms[@Persson2000; @Volya2003; @Rotter2009; @Choi2010; @Diehl2011; @Reiter2012]. For example, one may add a diagonal imaginary part in a band Hamiltonian for electrons to represent the effect of finite quasiparticle lifetime[@Shen2018; @Zhou2018; @Papaj2019; @Kozii2017]. One may also introduce an imaginary part to the dielectric constant in Maxwell equations to represent metallic conductivity in a photonic crystal[@Longhi2009; @Zloshchastiev2016; @Sounas2017; @El-Ganainy2018; @Bliokh2019]. As non-hermitian operators in general have complex eigenvalues, the eigenfunctions of Schrödinger equations are no longer static, but decay or increase exponentially in amplitude with time[@Brody2013; @Gong2018]. A topic in recent condensed-matter research is the study of topological properties in band structures, which are generally given by the wave functions, *not* the energy, of all occupied bands (or more generally, a group of bands capped from above and below by finite energy gaps)[@Hasan2010; @Qi2011; @Bernevig2013; @Chiu2016; @Armitage2018]. In non-hermitian systems, however, one immediately identifies a different type of topological numbers in bands, given by the phase winding of the “energy” (eigenvalue of Hamiltonian), *not* the wave functions, in the Brillouin zone (BZ)[@Shen2018a]. This *winding number*, together with several closely related winding numbers if other symmetries are present, give topological classification that is richer than that of their hermitian counterparts[@Lee2016; @Leykam2017; @Gong2018; @Yin2018; @Xiong2018; @Ghatak2019]. Besides winding in energy in complex plane, another unique phenomenon recently proposed in non-hermitian systems is the presence of “skin modes” in open systems[@Yao2018; @Yao2018a; @Kunst2018; @Kunst2019; @Lee2019]. A typical spectrum of open hermitian system consists of a large number of bulk states, and, if at all, a small number of edge states, and as the system increases in size $L$, the numbers of the bulk and of the edge states increase as $L^d$ and $L^{d-n}$ respectively, where $d$ is the dimension and $0<n\le{d}$. However, in certain non-hermitian systems, a finite fraction, if not all, of eigenstates are concentrated on one of the edges. These skin modes decay exponentially away from the edges just like edge states, but their number scales as the volume ($L^d$), rather than the area, of the system. In this Letter, we show an exact relation between the new quantum number, *i. e.*, the winding number of energy with periodic boundary, and the existence of skin modes with open boundary, for any one-band model in one dimension. To do this, we first extend the one-band Hamiltonian with finite-range hopping $H(k)$ to a holomorphic function $H(z)=P_{n+m}(z)/z^m$ ($n,m>0$), where $P_{n+m}(z)$ is an $(n+m)$-polynomial, and the Brillouin zone maps to unit circle $|z|=1$ (or $z=e^{ik}$). The image of the unit circle under $H(z)$ is the spectrum of the system with periodic boundary, and generally, it forms a loop on the complex plane, $\mathcal{L}_{\mathrm{BZ}}\in\mathbb{C}$. Then we show that as long as $\mathcal{L}_{\mathrm{BZ}}$ has finite interior, or roughly speaking encloses finite area, skin modes appear as eigenstates with open boundary condition; but when $\mathcal{L}_{\mathrm{BZ}}$ collapses into a curve having no interior on the complex plane, the skin modes disappear. In other words, skin modes with open boundary appear if and only if there be $E_b\in\mathbb{C}$ with respect to which $\mathcal{L}_{\mathrm{BZ}}$ has nonzero winding. Finally, we show that the winding of the periodic boundary spectrum, and hence the presence of skin modes with open boundary, are related to the total direct current of the system. We prove that if the current vanishes for all possible state distribution functions $n(H,H^\ast)$, the winding and the skin modes also vanish, and vice versa. The relations we establish among nonzero winding, presence of skin modes and non-vanishing current are summarized in Fig. \[fig:1\]. These results are extended to higher dimensions and more bands. ![\[fig:1\]The reciprocal relations among the three phenomena unique to non-hermitian systems: the non-vanishing direct current, nonzero winding number of energy and the presence of skin modes. The validity of any one is the sufficient and necessary condition for the validity of the other two.](Fig1){width="1\linewidth"} Hamiltonian as holomorphic function =================================== We start with an arbitrary one-band tight-binding Hamiltonian in one dimension, only requiring that hoppings between $i$ and $j$-sites only exist within a finite range $-m\le{}i-j\le{}n$. $$H=\sum_{i,j}t_{i-j}|i\rangle\langle{j}|=\sum_{k\in\mathrm{BZ}}H(k)|k\rangle\langle{k}|,$$ where $H(k)=\sum_{r=-m,\dots,n}t_r(e^{ik})^r$ is the Fourier transformed $t_{r}$ ($t_0$ being understood as the onsite potential). For periodic boundary condition, we have $0\le{k}<2\pi$, and $e^{ik}$ moves along the unit circle on the complex plane. For future purposes, we define $z:=e^{ik}$, and consider $z$ as a general point on the complex plane. Therefore for each Hamiltonian $H(k)$, we now have a holomorphic function $$H(z)=t_{-m}z^{-m}+\dots+t_nz^n=\frac{P_{m+n}(z)}{z^m},$$ where $P_{m+n}(z)$ is a polynomial of order $m+n$. $H(z)$ has one composite pole at $z=0$, the order of which is $m$, and has $m+n$ zeros, *i. e.*, the zeros of the $(m+n)$-polynomial. Along any oriented loop $\mathcal{C}$ and any given reference point $E_b\in\mathbb{C}$, one can define the winding number of $H(z)$ $$\label{eq:winding} w_{\mathcal{C},E_b}:=\frac{1}{2\pi}\oint_{\mathcal{C}}\frac{d}{dz}\arg[H(z)-E_b]dz.$$ Specially, for $\mathcal{C}=\mathrm{BZ}$, $w_{\mathcal{C},E_b}$ is the winding of the phase of $H(z)-E_b$ along BZ, considered as a new topological number unique to non-hermitian systems[@Lee2016; @Leykam2017; @Shen2018a; @Gong2018; @Yin2018; @Xiong2018; @Ghatak2019]. Complex analysis relates the winding number of any complex function $f(z)$ to the total number of zeros and poles enclosed in $\mathcal{C}$, that is, $$\label{eq:4} w_{\mathcal{C},E_b}=N_{zeros}-N_{poles},$$ where $N_{zeros,poles}$ is the counting of zeros (poles) weighted by respective orders. See Fig. \[fig:2\](a,b) for the pole, the zeros and the winding of $\mathcal{L}_{\mathrm{BZ}}$ for a specific Hamiltonian. In fact, we always have $N_{poles}=m$, so that the winding number is determined by the number of zeros of $P_{m+n}(z)-z^mE_b$ that lie within the unit circle. As we will see later, the advantage of extending the Hamiltonian into a holomorphic function lies in exactly this relation between the winding numbers and the zeros. Generalized Brillouin zone ========================== In Ref.[@Yao2018], it is shown that energy spectrum of one-band non-hermitian systems with open boundary may deviate drastically from that with periodic boundary, due to the presence of skin modes[@Xiong2018; @Ghatak2019]. Furthermore, in Ref.[@Yao2018; @Yokomizo2019], the authors introduce a new concept of generalized Brillouin zone to signify the difference between periodic and open boundary: instead of evaluating $H(z)$ along BZ, the open-boundary energy spectrum is recovered as one evaluates $H(z)$ on another closed loop called GBZ as $L$ goes to infinity. The GBZ is determined by the equation $$\label{eq:GBZ} \mathrm{GBZ}:=\{z||H^{-1}_m(H(z))|=|H^{-1}_{m+1}(H(z))|\},$$ where $H_i^{-1}(E)$’s satisfying $|H^{-1}_i(E)|\le|{H}^{-1}_{i+1}(E)|$ are the $m+n$ branches of the inverse function of $H(z)$. (In Ref.[@Yokomizo2019], $m=n$ is assumed, and we extend the results to $m\neq{n}$ cases in appendix \[AppendixA\].) We emphasize that using GBZ, one can compute the open boundary spectrum of systems of large or infinite size by solving some algebraic equations such as Eq.(\[eq:GBZ\]), a process we sketch using the following steps. To begin with, one finds the inverse functions of $H(z)$, and orders them in ascending amplitude, thus obtaining $H^{-1}_i(E)$, where $i=1,...,m+n$ because the $P_{m+n}(z)-Ez^m$ is an order $m+n$-polynomial of $z$. Then, as there are two variables $({\operatorname{Re}}(E),{\operatorname{Im}}(E))$ in Eq.(\[eq:GBZ\]), by codimension counting its solution on the complex plane forms one or several close loops, which are nothing but the open boundary energy spectrum. Finally, one substitutes these solutions back into $H^{-1}_m(E)$. It is noted that if we are only interested in the spectrum, we may stop at the second last step, but we need GBZ in order to articulate some of our key results. With GBZ thus defined, we state our central result (for proof see appendix \[AppendixB\]): GBZ is a single loop in complex plane that encloses the origin and exactly $m$ zeros of $P_{m+n}(z)-Ez^m$ for arbitrary $E\in \mathbb{C}$. This seemingly technical result has following consequences. First, this means within GBZ the total number of zeros and poles (weighted by respective orders) cancel, so that the winding of $H(z)-E$ vanishes. Next, the arbitrariness of $E$ ensures that GBZ is invariant under a shift of energy origin in the complex plane $H(z)\rightarrow{H}_z-E_b$. Combining these two points, we see that the image of GBZ under $H(z)$ on the complex plane, denoted by $\mathcal{L}_{\mathrm{GBZ}}$, has zero winding with respect to any $E_b\in\mathbb{C}$, or symbolically, $$\label{eq:w=0} w_{\mathrm{GBZ},E_b}=0.$$ Therefore, we finally see that the open-boundary spectrum of $H(z)$ cannot be a circle or eclipse like the periodic-boundary counterpart, and it cannot even be form a loop enclosing any finite area, because in that case one can choose $E_b$ inside that area so that the winding of $\mathcal{L}_{GBZ}$ with respect to $E_b$ is nonzero. The only possibility is that $\mathcal{L}_{\mathrm{GBZ}}$ *collapses* into a curve as shown in Fig. \[fig:2\](d). In this specific example ($m=n=1$ and see caption for parameters), we plot the GBZ in Fig. \[fig:2\](c) and $\mathcal{L}_{\mathrm{GBZ}}$ in Fig. \[fig:2\](d) as $z$ moves counterclockwise along GBZ. We see that while GBZ is more or less a circle, its image $\mathcal{L}_{\mathrm{GBZ}}$ keeps “back-stepping” itself: except for a few turning and branching points, any point in $\mathcal{L}_{\mathrm{GBZ}}$ has two or an even number of pre-images in the GBZ, so that the end result looks like more connected segments of curves than a closed loop. ![\[fig:2\]We show the BZ (a) with periodic-boundary spectrum (b), and GBZ (c) with open boundary spectrum (d) for the model $H(z)=(2i z^2+(3+i) z+1)/z$, and $E_0=H(z=a)=H(z=b)=3$ is the reference energy with respect to which winding is defined. We remark that the orientation of GBZ in (c) is arbitrarily chosen.](Fig2){width="1\linewidth"} Skin modes and nonzero winding numbers ====================================== GBZ not only gives the open boundary spectrum, but also yields information on the eigenstates with open boundary[@Yao2018; @Yokomizo2019]. In fact, each point $z\in{\mathrm{GBZ}}$ represents an eigenstate, the wave function of which is in the form $\langle{s}|\psi(z)\rangle\propto{|z|^s}$, where $s=1,\dots,L$ labels the sites. When $|z|>1$ ($|z|<1$), the wave function is concentrated near $(s=1)$-edge ($(s=L)$-edge) and exponentially decays with distance from the edge \[see Fig. \[fig:3\](a3, b3, c3) for examples\]. Therefore, any part of GBZ that lies within (without) the unit circle corresponds to a set of skin modes. In extreme cases, when the entire GBZ is inside (outside) the unit circle, all eigenstates are skin modes on the left (right) side of the chain. In short, any deviation of GBZ from BZ signifies the existence of skin modes. For a given $H(z)$, if $w_{\mathrm{BZ},E_b}\neq0$, then from Eq.(\[eq:w=0\]) we have $w_{\mathrm{GBZ},E_b}=0$, hence GBZ must deviate from the unit circle, that is, skin modes must exist with open boundary. Let us now try to prove the inverse statement: if GBZ and BZ differ from each other, then one can always find an $E_b\in\mathbb{C}$ such that $w_{\mathrm{BZ},E_b}\neq0$. GBZ and BZ may differ from each other in three typical ways: (i) as in Fig. \[fig:3\](a1), GBZ contains the unit circle, and we define $U$ as the region inside GBZ but outside BZ (colored in red); (ii) as in Fig. \[fig:3\](b1), GBZ is contained in the unit circle, and we define $V$ as the region outside GBZ but inside BZ (colored in blue); (iii) as in Fig. \[fig:3\](c1), one part of GBZ is outside and another part inside the unit circle. For case-(i), pick $z_0\in{U}$ and $E_0=H(z_0)$. $z_0$ is then a zero of $H(z_0)-E_0$, and from Eq.(\[eq:w=0\]), we know there are exactly $m$ zeros inside GBZ, so inside BZ there are at most $m-1$ zeros, and from Eq.(\[eq:4\]) we have $w_{\mathrm{BZ},E_0}<-1\neq0$ \[see example in Fig. \[fig:3\](a2)\]. For case-(ii), pick $z'_0\in{V}$ and $E'_0=H(z'_0)$, then use similar arguments to see $w_{\mathrm{BZ},E'_0}>1\neq0$ \[see example in Fig. \[fig:3\](b2)\]. We postpone the proof for case-(iii) to appendix \[AppendixC\], but mention here that for $z_0\in{U}$ and $z'_0\in{V}$, the periodic-boundary spectrum $\mathcal{L}_{\mathrm{BZ}}$, taking the shape of a fish \[see Fig. \[fig:3\](c2)\], has opposite windings with respect to $E_0$ and $E'_0$. {width="1\linewidth"} Winding numbers, skin modes and direct current ============================================== From the above results, we see that if and only if $\mathcal{L}_\mathrm{BZ}$ does not enclose any $E_b\in\mathbb{C}$, then the skin modes do not exist. When this is the case, $\mathcal{L}_{\mathrm{BZ}}$ always “back-steps” itself just like $\mathcal{L}_{\mathrm{GBZ}}$, or more precisely, along $\mathcal{L}_{\mathrm{BZ}}$, for any small segment $\delta{H}$ centered at some $E$, there must be another segment $-\delta{H}$ centered at exactly the same $E$. What is the physical meaning of this condition? We show that this is equivalent to the absence of total direct current with periodic boundary. To define the current, we assume that the particles have some charge (taken to be unity), so the total direct current can be derived as $J=\sum_kn_kv_k=\sum_kn_k{H'}(k)dk$, where $n_k$ is some distribution function. Now we make a general physical assumption that $n_k$ only depends on the “energy” of the state, that is $n_k=n(H(k),H^\ast(k))$, but does not depend on $k$ explicitly. (Here $n$ depends on both the real and the imaginary parts of $H(k)$, so is not necessarily holomorphic.) For example, the Bose distribution $n_k=(e^{Re[H(k)]/k_BT}-1)^{-1}$ satisfies such a condition. When the curve $\mathcal{L}_{\mathrm{\mathrm{BZ}}}$ has no interior, we have $$J=\int_0^{2\pi}n(H,H^\ast)\frac{dH(k)}{dk}dk=\oint_{\mathcal{L}_{\mathrm{BZ}}}n(H,H^\ast)dH=0,$$ that is, the total direct current vanishes. In appendix \[AppendixD\], we prove the inverse statement that if there is any $E_b\in\mathbb{C}$ with respect to which $H(z)$ has nonzero winding, then one can always find some $n(H,H^\ast)\neq 0$ such that $J\neq 0$. This equivalence is intuitively understood: if a direct current is driven through a ring, then as one cuts open the ring, the charge starts concentrating on one end of the open chain, being pushed by the driving voltage. Discussion ========== So far we have established the reciprocal relations shown in Fig. \[fig:1\] for one-band model in one dimension. Some of the results may be extended to the cases of more bands and/or higher dimensions. For example, in $d$-dimension, one should consider a multi-variable holomorphic function $H(z_1,z_2,\dots,z_d): \mathbb{C}^d\rightarrow\mathbb{C}$, where $z_j:=e^{ik_j}$, and the general spectrum of $H(z_1,\dots,z_d)$ is in general a continuum on the complex plane. Are there skin modes when we have open boundary along $0<l\le{d}$ directions, but periodic boundary along the other $d-l$ directions? We have two conjectures for two extreme cases: (i) if $l=d$, that is, if all directions are open, skin modes vanish if and only if each component direct current vanishes for arbitrary $n(H,H^\ast)$; and (ii) if $l=1$, that is, if only one direction is open, the skin modes vanish if and only if the entire spectrum of $H(z_1,\dots,z_d)$ collapse into a curve having no interior. The “only if” part of (i) and the “if” part of (ii) are trivial, but the other parts seem not quite so. Extension of the relation between the direct current and the winding numbers in periodic boundary to multiple-band systems is straightforward. Now $H_{ab}(z)$ becomes a matrix function of $z:=e^{ik}$, where $a,b=1,\dots,n$ label the orbitals. The direct current in this case becomes $J=\sum_{i=1,\dots,n}J_i$, where $$J_i=\int_0^{2\pi}n(E_i,E^\ast_i)\frac{dE_i(k)}{dk}dk=\oint_{\mathcal{L}_{i,\mathrm{BZ}}}n(E_i,E^\ast_i)dE_i.$$ While $J_i=0$ implies $J=0$, $J=0$ does *not* necessitate $J_i=0$ for each $i$. In fact, one part of the trajectory of $E_i(k)$ may be back-stepped by another part of the trajectory of $E_{j\neq{i}}(k)$ so that their contribution to $J$ cancel out. Therefore, $J=0$ is equivalent to the collapse of the spectrum, not of each individual band, but of all bands, into a curve that has no interior. In more precise terms, $J=0$ for arbitrary $n(E,E^\ast)$ if and only if for any $E_b\in\mathbb{C}$ and $E_b\notin\mathcal{L}_{i,\mathrm{BZ}}$, the total winding number of all bands with respect to $E_b$ vanishes, or symbolically $$\int_0^{2\pi}\frac{d\log\det[H(z)-E_bI_{n\times{n}}]}{dk}=0.$$ When there are additional conserved charges in the Hamiltonian, for example some spin component, we can simply replace the total current $J$ with the component current for each conserved charge $J_c$. At this point, we do not know exactly how the nonzero direct current or the winding numbers are related to the skin modes in multi-band systems, but from physical intuition, we conjecture that $J\neq0$ implies skin modes with open boundary, and vice versa. Derivation of Eq.(5) {#AppendixA} ==================== In this section, we first calculate the generalized Brillouin zone of a heuristic single-band model, and further give the general formal proof of Eq.(5) in the main text. Finally, we generalize the discussion to the two-band model with chiral symmetry. Model ----- Consider a single-band model with the following real space Hamiltonian $$\label{a_model} \hat{H}= \sum_{i=1}^L t_{-1} \hat{c}_{i+1}^{\dagger} \hat{c}_{i} + t_1 \hat{c}_{i}^{\dagger} \hat{c}_{i+1} + t_2 \hat{c}_{i}^{\dagger} \hat{c}_{i+2} + t_3 \hat{c}_{i}^{\dagger} \hat{c}_{i+3},$$ and the corresponding eigenequation is $$\label{a_matrix} H\Psi=E\Psi,\qquad H=\left(\begin{array}{llllllll} {0} & {t_1} & {t_2} & t_3 & 0 & {\cdots} & {0} & {0} \\ {t_{-1}} & {0} & {t_1} & t_2 & t_3 & {\cdots} & {0} & {0} \\ {0} & {t_{-1}} & {0} & t_1 & t_2 & {\ldots} & {0} & {0} \\ 0 & {0} & {t_{-1}} & {0} & t_1 & {\ldots} & {0} & {0} \\ 0 & 0 & {0} & {t_{-1}} & {0} & {\ldots} & {0} & {0} \\ {\vdots} & {\vdots} & {\vdots} & {\vdots} & {\vdots} & {\ddots} & {\vdots} & {\vdots} \\ 0 & 0 & 0 & {0} & {0} & {\ldots} & {0} & {t_1} \\ {0} & {0} & {0} & 0 & 0 & {\ldots} & {t_{-1}} & {0}\end{array}\right), \qquad \Psi=\left(\begin{array}{c}{\psi_{1}} \\ {\psi_{2}} \\ {\psi_{3}} \\ {\psi_{4}} \\ {\psi_{5}} \\ {\vdots} \\ {\psi_{L-1}} \\ {\psi_{L}}\end{array}\right).$$ Before we proceed to solve Eq. (\[a\_matrix\]), we first review the procedure of exact solution. Recall the one-dimensional infinite square well problem in quantum mechanics. Although the translational symmetry is broken at the boundary of the well, we also need to solve the (translational invariance) Schrödinger equation $(\hat{p}^2/2m+V_0)|\phi\rangle=E|\phi\rangle$. For any given $E$, there exist two linear independent plane wave solutions $|k\rangle$ and $|-k\rangle$. If their linear superposition $|\phi\rangle=c_1|k\rangle+c_2|-k\rangle$ satisfy the boundary condition, we say $|\phi\rangle$ is the eigenstate of the Hamiltonian with the corresponding eigenvalue $E$. Back to Eq. (\[a\_matrix\]), we first notice that the eigenequation can be separated to the [*bulk equation*]{} $$t_{-1}\psi_s - E\psi_{s+1}+t_1 \psi_{s+2}+t_2\psi_{s+3}+t_3\psi_{s+4}=0, \label{Bequation}, \quad s=1,2,...,L-4,$$ and [*boundary equation*]{} $$\label{split} \begin{split} -E \psi_1 + t_1 \psi_2 + t_2 \psi_3 + t_3 \psi_4 &= 0, \\ t_{-1}\psi_{L-3}-E \psi_{L-2}+t_1 \psi_{L-1}+t_2 \psi_L &= 0, \\ t_{-1}\psi_{L-2}-E\psi_{L-1}+t_1 \psi_{L}&=0, \\ t_1 \psi_{L-1}-E \psi_{L}&=0. \end{split}$$ Here the [*bulk equation*]{} corresponds to the (translational invariance) Schrödinger equation, and the [*boundary equation*]{} refers to boundary condition in the one-dimensional infinite square well problem. Since the [*bulk equation*]{} has discrete translational symmetry, for a given $E$, it has four linear independent eigenfunctions, which can be written as $$\Psi_i(E)=(z_i,z_i^2,...,z_i^{L-1},z_i^{L}),\quad i=1,2,3,4,$$ where $z_i$ satisfy the following characteristic polynomial equation for given $E$ $$\label{CEquation} f(z_i,E) :=H(z_i)-E= t_{-1}/z_i + t_1 z_i + t_2 z_i^2 + t_3 z_i^3 - E =0,$$ The solution of Eq. (\[a\_matrix\]) can be written as the linear superposition of $\Psi_i(E)$ satisfying the [*boundary equation*]{}. To be more precise, $$\begin{aligned} \Psi(E)&=c_1 \Psi_1(E)+c_2 \Psi_2(E)+c_3 \Psi_3(E)+c_4 \Psi_4(E)\\ &=(\psi_1,\psi_2,...,\psi_{L-1},\psi_L)^t, \end{aligned}$$ where $$\label{Gsolution} \psi_n=\sum_{i=1}^4 c_i z_i^n =c_1z_1^n+c_2z_2^n+c_3z_3^n+c_4z_4^n,\quad n=1,...,L,$$ and the solution of Eq. (\[CEquation\]) are ordered as follows $$|z_1| \leq |z_2| \leq |z_3|\leq |z_4|. \label{Order}$$ Substituting Eq. (\[Gsolution\]) to the [*boundary equation*]{} Eq. (\[split\]), one can obtain the following matrix equation after an appropriate transformations $$H_B \begin{pmatrix} c_1 \\ c_2 \\ c_3 \\ c_4 \end{pmatrix}= \begin{pmatrix} A(z_1) & A(z_2) & A(z_3) & A(z_4) \\ B_1(z_1)z_1^L&B_1(z_2)z_2^L &B_1(z_3)z_3^L&B_1(z_4)z_4^L\\ B_2(z_1)z_1^L&B_2(z_2)z_2^L &B_2(z_3)z_3^L&B_2(z_4)z_4^L\\ B_3(z_1)z_1^L&B_3(z_2)z_2^L &B_3(z_3)z_3^L&B_3(z_4)z_4^L\\ \end{pmatrix} \begin{pmatrix} c_1 \\ c_2 \\ c_3 \\ c_4 \end{pmatrix}=0, \label{Mform}$$ where $$\begin{aligned} A(z_{i})&=-E z_i+t_1 z_i^2+t_2 z_i^3+t_3 z_i^4,\\ B_1(z_j)&=t_2+t_1/z_j-E/z_j^2+t_{-1}/z_j^3,\\ B_2(z_j)&=t_1-E/z_j+t_{-1}/z_j^2,\\ B_3(z_j)&=t_{-1}/z_j - E. \end{aligned}$$ The non-trivial solution $(c_1,c_2,c_3,c_4)$ requires $$\det[H_B] = 0. \label{Det}$$ It is clear that the determinant of $H_B$ contains $4!=24$ terms, and each term is a product of elements from different rows and columns of the matrix. Hence Eq. (\[Det\]) can be further expressed as $$\begin{aligned} &\sum_{i\neq j \neq k \neq l=1}^4 A(z_i)B_1(z_j)B_2(z_k)B_3(z_l)\times(z_j z_k z_l)^L \\ &=F_1(z,E)\times(z_2z_3z_4)^L+F_2(z,E)\times(z_1z_3z_4)^L+F_3(z,E)\times(z_1z_2z_4)^L+F_4(z,E)\times(z_1z_2z_3)^L\\ &=0,\\ \label{Detform} \end{aligned}$$ where the coefficient $F_i(z,E)$ is a function of $z=(z_1,z_2,z_3,z_4)$ and $E$. Since both the degrees of $A(z_i)$ and $B_{1/2/3}(z_i)$ are finite and independent of $L$, the leading term of Eq. (\[Detform\]) can be ordered by $$(z_2z_3z_4)^L\geq(z_1z_3z_4)^L\geq (z_1z_2z_4)^L\geq(z_1z_2z_3)^L$$ according to Eq. (\[Order\]) in the thermodynamic limit. Hence in the $L\rightarrow \infty$ limit, if $|z_1(E)|<|z_2(E)|$, the only leading term of Eq. (\[Detform\]) is $F_1(z,E)\times(z_2z_3z_4)^L$, which requires $F_1(z,E)=0$. Since $F_1(z,E)$ is a function of $A(z,E)$ and $B_{1/2/3}(z,E)$, the order of $E$ in $F_1(z,E)$ is independent of $L$. This means there only exist finite solutions of $E$ in $F_1(z,E)=0$. As a result, they can not form a continuous spectrum. On the other hand, if $$|z_1(E)|=|z_2(E)|, \label{GBZcondition}$$ there exist two leading terms in the $L\rightarrow \infty$ limit, which implies $$\det[H_B]=0\rightarrow F_1(z,E)\times(z_2z_3z_4)^L+F_2(z,E)\times(z_1z_3z_4)^L=0.$$ In this case $$\frac{F_1(z,E)}{F_2(z,E)}=-\left(\frac{z_1}{z_2}\right)^L.$$ This means the order of $E$ depends on $L$, which will form a continuous band in the thermodynamic limit. The set of $z$ satisfying Eq. (\[GBZcondition\]) is called generalized Brillouin zone (GBZ). From the above derivation, the GBZ condition $|z_1(E)|=|z_2(E)|$ is related to the bulk Hamiltonian $$H(z)=t_{-1}/z + t_1 z + t_2 z^2 + t_3 z^3. \label{Hamz}$$ To be more precise, the order of the pole in $H(z)$ determines the form of boundary matrix $H_B$ in Eq. (\[Mform\]), and finally determines the condition for the continuous band Eq. (\[GBZcondition\]). In the thermodynamic limit, the spectrum as the image of GBZ on $H(z)$ is labeled by $\mathcal{L}_{GBZ}$, and needs to satisfy $$|H_m^{-1}(\mathcal{L}_{GBZ})|=|H_{m+1}^{-1}(\mathcal{L}_{GBZ})|.$$ where $H^{-1}(E)$ is the inverse function of $H(z)$. Here we give numerical calculations (Fig. (\[figa1\])) to support the above conclusions. Next we generalize the above procedure to a general single band Hamiltonian. ![ (a) presents eigenvalues of Hamiltonian Eq. (\[a\_model\]) with the number of sites $L=100$ and all parameters $t_{i=-1,1,2,3}$ equal to 1. Each eigenvalue corresponds to four solutions of Eq. (\[Hamz\]), which are ordered by absolute value and marked in red, orange, darker blue and darker green colors respectively. Then (b) shows that the first two solutions form the GBZ (gray continuous loop). []{data-label="figa1"}](FigA1.pdf){height="5cm" width="11cm"} General case ------------ Consider the following general single band real space Hamiltonian, $$\hat{H}=\sum_{i,j=1}^L t_{i-j}\hat{c}_j^{\dagger}\hat{c}_{i},$$ where the hopping parameters only exist in a finite range $-m \leq i - j \leq n$ and $L$ is the number of sites. For each site, the largest hopping range to left is $n$ and to right is $m$. The system reduces to Hermitian when $t_{i-j}$ equals to $t_{j-i}^*$. Similar to the derivation in the above section, this Hamiltonian can be divided into two parts: the bulk and the boundary. The bulk, ranging from $(n+1)_{th}$ site to $(L-m)_{th}$ site, maintains translational symmetry, while the boundary, including the remaining parts of two ends, has no longer the translation invariance. Starting from the eigenequation $$H \Psi = E \Psi \label{Eequa}$$ we can solve the eigenequation from two parts, the bulk and boundary equations. *Bulk equation:* The bulk equation is $$t_{-m} \psi_s + t_{-m+1} \psi_{s+1} + \dots + (t_0 - E)\psi_{s+m} + t_1 \psi_{s+m+1}+\dots + t_n \psi_{s+m+n} = 0, \quad s=1,...,L-(m+n), \label{GBequation}$$ where $\psi_s$ denotes the $s_{th}$ component of wavefunction. For the sake of simplicity, the value of $t_0$ is usually taken as zero since it does not affect the eigenfunction. For a given $E$, the bulk equation has $m+n$ linear independent eigenfunctions, which can be written as $$\Psi_i(E)=(z_i,z_i^2,...,z_i^{L-1},z_i^{L}),\quad i=1,...,m+n,$$ where $z_i$ satisfy the following characteristic polynomial equation for given $E$ $$f(z_i,E) :=H(z_i)-E= \sum_{j=-m}^{n} t_j z_i^{j} - E =0. \label{CEquation1}$$ According to the linear superposition principle, the following wavefunction is also the eigenfunction of Eq. (\[Eequa\]) $$\begin{aligned} \Psi(E)&=\sum_{i=1}^{m+n}c_i\Psi_i(E)\\ &=(\psi_1,\psi_2, \dots ,\psi_s, \dots , \psi_{L-1},\psi_L)^t, \end{aligned}$$ where $$\psi_s=\sum_{i=1}^{m+n} c_i z_i^s, \quad s=1,...,L, \label{GSolution}$$ and the solutions of Eq. (\[CEquation1\]) are ordered as follows $$|z_1| \leq |z_2| \leq... \leq |z_{m+n-1}|\leq |z_{m+n}|. \label{Order}$$ *Boundary equation:* Consider the $m+n$ boundary equations and substitute the solution Eq. (\[GSolution\]) into it, then one can obtain $m$ constraint equations about $\{\psi_1,\psi_2, \dots, \psi_m\}$ and $n$ limited equations about $\{\psi_{L-n+1},\psi_{L-n+2}, \dots, \psi_L\}$, where $L$ is the total number of the lattice sites. Open boundary means that there are no components of wave function beyond the two ends of the chain, namely, $\psi_{i<1}=0$ and $\psi_{i>L}=0$. In order to express the boundary equation, we first define $$T_s(\psi):=t_{-m} \psi_s + t_{-m+1} \psi_{s+1} + \dots + (t_0 - E)\psi_{s+m} + t_1 \psi_{s+m+1}+\dots + t_n \psi_{s+m+n}.$$ Then the $m+n$ boundary equations can be expressed as $$\begin{split} &T_s(\psi)=0, \qquad s=-m+1, -m+2, \dots, 0; \\ &T_s(\psi)=0, \qquad s=L-m-n+1, L-m-n+2, \dots, L-m, \end{split}$$ with the open boundary condition $$\psi_{i<1}=\psi_{i>L}=0 \label{OBcondition}$$ In fact, the form of boundary equations ensures two things. There is always at least one term, of $m+n$ terms in Eq. (\[GBequation\]), removed according to open boundary Eq. (\[OBcondition\]). And we need to make sure that the term with coefficient $E$ does not disappear. For example, $T_{-m+1}(\psi)= - E\psi_{1} + t_1 \psi_{2}+\dots + t_n \psi_{n+1} = 0 $ due to $\psi_{i<1}=0$ and $T_{L-n}(\psi)=t_{-m}\psi_{L-m} +t_{-m+1}\psi_{L-m+1} + \dots -E \psi_L =0$ due to $\psi_{i>L}=0$. According to Eq. (\[GSolution\]), these $m+n$ boundary equations can be written as the following form: $$H_b \begin{pmatrix} c_1 \\ c_2 \\ \vdots \\ c_m \\ c_{m+1} \\ \vdots \\ c_{m+n} \end{pmatrix}= \begin{pmatrix} A_1(z_1) & A_1(z_2) & \dots & A_1(z_{m+n}) \\ A_2(z_1) & A_2(z_2) & \dots & A_2(z_{m+n}) \\ \vdots & \vdots & \vdots & \vdots \\ A_m(z_1) & A_m(z_2) & \dots & A_m(z_{m+n}) \\ B_1(z_1)z_1^L & B_1(z_2)z_2^L & \dots & B_1(z_{m+n})z_{m+n}^L \\ \vdots & \vdots & \vdots & \vdots \\ B_n(z_1)z_1^L & B_n(z_2)z_2^L & \dots & B_n(z_{m+n})z_{m+n}^L \end{pmatrix} \begin{pmatrix} c_1 \\ c_2 \\ \vdots \\ c_m \\ c_{m+1} \\ \vdots \\ c_{m+n} \end{pmatrix}=0, \label{GMfrom}$$ where $z_i$ represents $i_{th}$ roots of Eq. (\[CEquation1\]), $A_i(z_j)$ and $B_i(z_j)$ are polynomials about $z_j$ with finite order. The matrix elements $A_i(z_j)$ can be obtained by substituting all the terms $\psi_r$ of $T_{-m+i}(\psi)$ into $z_j^r$, likely, $B_i(z_j)$ is obtained by substituting all the terms $\psi_r$ of $T_{L-m-n+i}(\psi)$ into $z_j^{r-L}$. For example, $A_1(z_j) = -E z_j + t_1z_j^2 + \dots + t_n z_j^{n+1}$ and $B_n(z_j) = t_{-m}z_j^{-m} + t_{-m+1} z_j^{-m+1} + \dots - E$. Notice that $m,n \ll L$ and $L$ representing the size of lattice tends to infinity in the thermodynamic limit. Nontrivial solutions of Eq. (\[GMfrom\]) further require that $$\det[H_b]= 0. \label{GDet}$$ The determinant of $H_b$, a $(m+n)*(m+n)$ square matrix, is the summation of $S_{m+n}=(m+n)!$ terms, and each term is the product of matrix elements belonging to different rows and columns. Therefore, each term contains the product of $n$ different roots $z_j^L$, and Eq. (\[GDet\]) can be expressed as: $$\begin{aligned} &\det[H_b]\\ &=F_{1}(z,E)\times(z_{m+1}z_{m+2}...z_{m+n-1}z_{m+n})^L+F_{2}(z,E)\times(z_{m}z_{m+2}...z_{m+n-1}z_{m+n})^L+...\\ &=0,\\ \label{Detform1} \end{aligned}$$ where $F_{1}$ and $F_{2}(z,E)$ are the coefficients of $(z_{m+1}z_{m+2}...z_{m+n-1}z_{m+n})^L$ and $(z_{m}z_{m+2}...z_{m+n-1}z_{m+n})^L$ respectively, and the subscript $i$ of $F_i(z,E)$ increases as the magnitude of $(z_i\dots z_j)^L$ gradually decreases. In this way, $F_1(z,E)$ always corresponds to the term with the largest magnitude, namely, $(z_{m+1}z_{m+2}...z_{m+n-1}z_{m+n})^L$, and so on. Based on the similar reasons with the previous part, the continuous band requires $$|z_m(E)|=|z_{m+1}(E)|,$$ such that the leading order of Eq. (\[Detform1\]) is equivalent to $$\left(\frac{z_{m+1}}{z_m}\right)^L=-\frac{F_{1}(z,E)}{F_{2}(z,E)}. \label{GConstraint}$$ In fact, Eq. (\[GConstraint\]) gives the constrain condition on $E$ because $z$ is the function of $E$. The exact solutions of $|z_m(E)|=|z_{m+1}(E)|$ give the spectrum in the thermodynamic limit and corresponding $z$ forms the generalized Brillouin zone. The right side of Eq. (\[GConstraint\]) includes the detailed information about open boundary while the left side does not. It means that boundary conditions have little effect on the properties of the bulk if $N$ tends to infinity, but we need pay special attention to the fact that the above statements are based on the open boundary, it is no longer true for nonlocal boundary terms, such as periodic boundary. Two-band chiral model --------------------- The above discussions can be naturally extended to multi-band cases, especially two-band Hamiltonian with chiral symmetry. Generically, the bulk Hamiltonian is written as $$H(z) = \begin{pmatrix} 0 & \frac{P^{(1)}_{m_1+n_1}(z)}{z^{m_1}} \\ \frac{P^{(2)}_{m_2+n_2}(z)}{z^{m_2}} & 0 \end{pmatrix}. \label{ChiralH}$$ The off-diagonal terms of the matrix are two different functions that are holomorphic at the entire complex plane except the origin. Next we show that the commonly used two-band tight-binding model can always mapped to the single-band model due to the presence of chiral symmetry. According to Eq. (\[ChiralH\]), one can immediately write down the characteristic equation, $$\label{a_twoce} F(z,E) = \frac{P_{m+n}(z)-E^2 z^{m}}{z^{m}} = 0,$$ where $m=m_1+m_2$ and $n=n_1+n_2$, and this equation establishes the mapping between $z$ and $E$. Additionally, chiral symmetry ensures that the eigenvalues $(E,-E)$ always appear in pairs. Therefore, it comes to the conclusion that one can always take the base energy as $E^{2}$ in single-band model, which corresponds to the $\pm E$ in two-band chiral Hamiltonian, and they have the same characteristic equation that determines the generalized Brillouin zone. Then we can get the same conclusion as the single-band model $$\label{a_twogbz} |z_{m}(E)|=|z_{m+1}(E)|,$$ where $m$ represents the multiplicity of the pole of characteristic equation Eq. (\[a\_twoce\]), and this conclusion is confirmed numerically in Fig. (\[figa4\]). GBZ encircles the same number of zeros and poles {#AppendixB} ================================================ In this section, we prove that GBZ is a single loop in complex plane that encloses the origin and exactly $m$ zeros of $H(z)-E_b=0$ for arbitrary $E_b\in\mathbb{C}$, and give numerical supports for this statement. Here we focus on the cases where GBZ is a simple closed loop and characteristic equation (Hamiltonian in single-band cases) has the form $(P_{m+n}(z)-E_b z^m)/z^m $. Assume that there is a simple closed loop $C_{z}$ in complex plane with $({\operatorname{Re}}[z], {\operatorname{Im}}[z])$ as the coordinate system, which can be mapped into the set $\mathcal{L}_{C_z}$ living in $({\operatorname{Re}}(E), {\operatorname{Im}}(E))$ space (complex energy plane) by $H(z)$. If choose the point $E$ in complex energy plane, then one can find corresponding zeros of $H(z)-E=0$ in the $({\operatorname{Re}}(z), {\operatorname{Im}}(z))$ space. We first propose the following lemma before coming to the final proof. ***Lemma:*** If there exists a smooth path $E_s$ (in the complex plane) that is controlled by parameter $s$ and has no intersections with the set $\mathcal{L}_{C_z}$, then any two points connected through this path $E_s$ have the same number of zeros of $H(z)-E_s=0$ within the area surrounded by $ C_z $. This lemma will be frequently used, and we have proved it rigorously. ![ An illustration to the proof. The $z_s^{A/B}$ curves represent the loci of $z$ roots(zeros) of $ P_{m+n}(z) - z^m E_s$. As parameter $s$ changes from $0$ to $1$, the energy changes from $ E_{\infty} $ to arbitrary selected base energy $ E_b $, and the corresponding zeros move continuously from $ z_0^{A/B} $ to $ z_1^{A/B} $. There are two different cases. In one case, the locus of $ E_s $ has no intersections with the set $\mathcal{L}_{GBZ}$, hence the corresponding zeros will not enter or escape from GBZ so that the total number of zeros is invariant during the process. In another case, there are intersections between $ E_s $ and the set $\mathcal{L}_{GBZ}$, and the same number of zeros will passed through and entered into GBZ, thus ensuring that the total number of zeros within GBZ is unchanged. []{data-label="figa2"}](FigA2.pdf){height="8cm" width="10cm"} ***proof:*** The zeros of Hamiltonian are determined by $P_{m+n}(z)-z^m E_b=0$. Defining two analytic functions of $z$ as $$\begin{split} g_1(z)=P_{m+n}(z)-z^m E_1 \\ g_2(z)=P_{m+n}(z)-z^m E_2 \end{split}$$ where $E_2$ is located in the neighborhood of $E_1$ and expressed as $E_2=E_1+\delta$ with $|\delta|$ tends to zero. Let both $E_1$ and $E_2$ do not belong to the set $ \mathcal{L}_{C_z}$. There always exist a small enough $\delta$ to ensure $z_c \in C_z$ satisfies the following inequation $$|g_1(z_c)| > |g_2(z_c)-g_1(z_c)|=|\delta z_c^m|$$ Then it follows from Rouche’s theorem that $g_1(z)$ possess the same number of zeros as that of $g_2(z)$ in the region enclosed by $C_{z}$. By constantly using the trick of neighboring $\delta$, one can obtain a smooth path $E_s$ that connects $ E_{\infty} $ with $ E_b $. There are two cases, one is that $E_s$ does not intersect with the set $ \mathcal{L}_{C_z}$ such that $g_1(z)=P_{m+n}(z)-z^m E_{\infty}=0$ and $g_2(z)=P_{m+n}(z)-z^m E_{b}=0$ have the same number of zeros inside the region surrounded by $C_z$. This case just corresponds to case $1$ illustrated in Fig. (\[figa2\]), in which $C_z$ is taken as $GBZ$. In another case, the path $E_s$ has interactions with $\mathcal{L}_{C_z}$, just like case $2$ in Fig. (\[figa2\])($C_z$ is taken as $GBZ$), one can always find at least one path where zeros inside and outside GBZ exchange with each other and the total number of zeros inside GBZ is invariant. For a given base energy $E_b$, the holomorphic function defined on the complex plane except the origin, $H(z)-E_b$, possess $m+n$ zeros expressed as $z=z_{i=1,2\dots,m+n}$ and $m_{th}$ order pole at the origin of complex. The form of complex function is that $$t_{-m}+t_{-m+1} z+\dots + t_{-1} z^{m-1}+ t_1 z^{m+1}+\dots + t_{n} z^{m+n} = E$$ has asymptotic behavior that $m$ roots verge to zero and the remaining roots tend to infinity as $E$ tends to infinity. Here notice that the number $m$ of zeros located in the region bounded by $C_{z}$ is the same as multiplicity of pole if taking $E$ as $E_{\infty}$. Now we have two conclusion: (i:) Two points $E_{s=0}$ and $E_{s=1}$, connected by smooth curve $E_s$, have the same number of zeros within simple closed loop $C_z$; (ii:) The number of zeros of $P_{m+n}(z)-E_{\infty}=0$ is the same as the multiplicity of the pole. Combining this two points and taking $E_{s=0}$ as $E_{\infty}$, the conclusion is obtained that for arbitrary base energy $E_b$ connected to $E_\infty$ via the smooth path $E_s$, the number of zeros is the same as the multiplicity of the pole inside $C_{z}$. Here we take $C_z$ as $GBZ$, then, for all $E \in \mathbb{C} \setminus \mathcal{L}_{GBZ}$, there always exist at least one path connecting $E_{\infty}$ with $E$ such that $P_{m+n}(z)-z^m E$ has $m$ zeros inside GBZ as $E_\infty$ does. Thus the winding number of energy spectrum with open boundary is calculated $$w_{GBZ,E_b}= \frac{1}{2\pi i} \oint_{C_z} \frac{H'(z)}{H(z)-E_b} dz = N_{zeros}-N_{poles} = 0$$ where $N_{zeros}$ denotes the number of zero points and $N_{poles}$ indicates the multiplicity of the pole. ![ (a) represents the energy spectrum of non-Hermitian system determined by Hamiltonian Eq. (\[b\_ham1\]) under open boundary condition with parameters $ \{t_3,t_2,t_1,t_{-1},t_{-2}\}=\{1/2, 1, 1/2, 1/2, 2 \}$ and the number of site $L=110$, and the green arrows denote the direction of energy band as $z$ clockwise along GBZ depicted in (c); (b) shows the energy spectrum with periodic boundary and the color gradually turns from black to red as k change from $0$ to $2\pi$. Given three base energy $E_1=0$ in red color, $E_2=1.2 I$ in purple color and $E_3=2+I$ in blue color, and their corresponding zeros are colored with the same color on the $z$ plane; In Fig.(c), gray unit circle and yellow loop indicate GBZ of the system with periodic and open boundary respectively, and the cross notation denotes the pole with multiplicity two. Here only the first two, with smaller absolute values, of the five zeros are depicted; (d) presents the distribution of five roots of Eq. (\[b\_ham1\]) as $E$ varying uniformly from $-10-10I$ to $10+10I$. The red and pink denote the first two roots, and blue, purple and black regions denote last three roots respectively, which also shows that GBZ(yellow curve) is exactly the boundary between them. []{data-label="figa3"}](FigA3.pdf){height="8cm" width="10cm"} Example 1 --------- Here we take a special single-band model to further confirm numerically the main conclusions, and the bulk Hamiltonian is shown as: $$\label{b_ham1} H(z)= \sum_{i=-2}^3 t_i z^i,$$ where $z$ is complex variable and $t_i$ denotes the hopping parameter. Then transform Eq.(\[b\_ham1\]) into the form, $(t_3 z^5 + t_2 z^4 + t_1 z^3 - E_b z^2 + t_{-1} z + t_{-2})/z^2=0$. We fix the other coefficients, then let $E_b$ arbitrarily take values in the complex plane except for the energy spectrum $\mathcal{L}_{GBZ}$, and always get five zeros, two of which are inside GBZ and the rest are outside it, which is illustrated in Fig.(\[figa3\]). It further comes to a conclusion that if and only if GBZ is the boundary of the region composed of the first two solutions ordered by absolute value, the winding regarding to any choices of base energy $E_b$ vanish. In Fig.(\[figa3\]), We have chosen three representative base energy $E_b$, which are marked in different colors, and the winding with respect to base energy $E_1, E_2, E_3$ equal to $-2,-1,0$ respectively under periodic boundary conditions, as well as all of which equal to zero with open boundary. It is worth noting that even if the spectrum surrounds the $E_1$(red point) in Fig. \[figa3\](a), the winding of the energy is still zero. The reason is that as z moves clockwise along GBZ, its image $\mathcal{L_{GBZ}}$ keeps “back-stepping” itself and has no interior. These following significant conclusions have been confirmed numerically: (i:) Once we have determined the multiplicity m of the pole of $H(z)$, GBZ always contains m zeros accordingly, regardless of how the base energy takes values on the complex plane except for $\mathcal{L}_{GBZ}$. (ii:) If $z$ directionally circles BZ, the corresponding $\mathcal{L}_{BZ}$ will surround a finite area, while if $z$ moves along GBZ, $\mathcal{L}_{GBZ}$ always wraps around itself and contains zero area, which refers to the collapse from $\mathcal{L}_{BZ}$ to $\mathcal{L}_{GBZ}$. (iii:) GBZ is precisely the boundary between the first $m$ zeros and the last $n$ zeros. ![ (a) illustrates the eigenvalues of non-Hermitian system determined by Hamiltonian Eq.(\[S20\]) under open boundary condition with parameters $ \{t_1,t_2,t_3,w_1,w_2,w_3\}=\{1,4,1,1,1,\frac{1}{2}\}$ and size $L=75$, and the red point denotes two-degenerate edge state protected by chiral symmetry. The arrows represents the evolution orientation of eigenvalues as $z$ anticlockwise along the GBZ; (b) shows the GBZ in orange color and BZ in dashed gray color, and the roots $z_i$, obtained by substituting eigenvalues into $H^{-1}(E)$, are ordered in ascending amplitude and marked in different colors. []{data-label="figa4"}](FigA4.pdf){height="7cm" width="12cm"} Example 2 --------- Consider a two-band Hamiltonian with respect to chiral symmetry in real-space representation, which is shown as: $$H = \sum_{i=1}^L t_1 a_i^{\dagger}b_i + t_2 a_{i+1}^{\dagger}b_i + t_3 a_i^{\dagger}b_{i+1} + w_1 b^{\dagger}_{i+1}a_i+w_2 b_i^{\dagger}a_{i+2} + w_3 b_i^{\dagger}a_{i+3}$$ with the number of unit cell $L$ and two degrees per unit cell. Then the corresponding bulk Hamiltonian is written as: $$H(z)= \begin{pmatrix} 0 & \frac{t_2+t_1 z +t_3 z^2}{z} \\ \frac{w_1+w_2 z^3+w_3 z^4}{z} & 0 \end{pmatrix}, \label{S20}$$ in which the complex variable $z$ is expressed as $e^{ik}$ under periodic boundary conditions. The characteristic equation is written as $F(z,E)=\det[H(z)-E I_{2\times 2}]=\frac{P_{m+n}(z)-E^2 z^m}{z^m}=0$, with $m=2$ and $n=4$. According to the arguments in appendix \[AppendixA\], the GBZ is constructed by $|z_m(E)|=|z_{m+1}(E)|$, which has been confirmed numerically in Fig.(\[figa4\]). Form Fig.(\[figa4\]), one can extract the following informations: (i:) GBZ is formed by $z_2$ and $z_3$, which are roots of characteristic equation with regards to eigenvalues of $H$, and this ensures that GBZ contains the same number $m$ of zeros and poles. (ii:) $E$ and $-E$ always correspond to the same roots due to chiral symmetry, hence the two-part connected bulk spectrum always circulates around itself and encloses zero area as $z$ varies along the GBZ, and their behavior is symmetric about the origin. So one can always map two-band chiral model to a single-band Hamiltonian. The Proof for case-(iii) {#AppendixC} ======================== Here we give a rigorous proof for the case (iii) in main text. In this case, one part of GBZ outside and another part inside the unit circle(BZ), then we define the region inside GBZ but outside BZ as $U$, the region inside BZ but outside GBZ as $V$, and the region inside both GBZ and BZ as $W$. Before the proof, we must display two facts, one is that GBZ must enclose the origin, and another is that GBZ formed by $H^{-1}_m({\mathcal{L}_{GBZ})}$ and $H^{-1}_{m+1}({\mathcal{L}_{GBZ})}$ always encircles the first $m$ zeros that have ordered by absolute value. Then we first prove that for any choice of base energy $E_b$, the roots of $H(z)-E_b=0$ can not appear in $U$ and $V$ regions at the same time. A obvious fact is that the magnitude of $z^{\prime}_0$ inside $V$ is always less than that of $z_0$ inside $U$ region. Assuming that there exist at least two zeros $z_0$ and $z^{\prime}_0$ correspond to the same base energy $E_b$, then it will happen that for the $E_b$, the root $z^{\prime}_0$ with smaller absolute value does not belongs to GBZ but the root $z_0$ with larger absolute value belongs to. This distinctly contradicts the facts we have displayed. Hence the assumption is not true, that is to say, there are no two roots at most correspond to the same $E_b$, i.e., the roots of $H(z)-E_b=0$ can not appear in $U$ and $V$ regions at the same time for any $E_b$. Furthermore, it comes to the conclusion that the other roots of base energy corresponding to $z^{\prime}_0$ may only appear in region $W$ but not in region $U$. The next steps are the same as in case (i) and (ii). Pick $z_0 \in U$ and $E_0=H(z_0)$, then $z_0$ is a zero of $H(z)-E_0$, likely pick $z^{\prime}_0 \in V$ and $E_0^{\prime}=(z^{\prime}_0)$, then $z^{\prime}_0$ is a zero of $H(z)-E_0$. Here we notice that $E_0 \neq E^{\prime}_0$ that has been proved above. With the fact that GBZ encloses $m$ zeros, therefore in case (iii) there are always $E_0$ and $E_1$ such that $w_{BZ,E_0}<-1$ and $w_{BZ,E^{\prime}_0}>1$, and $E_0$ is not equal to $E^{\prime}_0$. Here the case (iii) has been rigorously proved. Prove the inverse statement of Eq.(7) {#AppendixD} ===================================== In this section we prove the statement that if there is any $E_b \in \mathcal{C}$ with respect to which $H(z)$ has nonzero winding, then one can always find some $n(H,H^{\ast})\neq 0$ such that $J \neq 0$. The natural extension of the definition of total current from Hermitian systems is that $$J=\int_0^{2\pi} n(H,H^{\ast})\frac{dH(k)}{dk}dk=\oint_{\mathcal{L}_{BZ}}n(H,H^*)dH. \label{Current1}$$ If the interior area of $\mathcal{L}_{BZ}$ is nonzero, it means that $\mathcal{L}_{BZ}$ is composed of one or several close loops. One can always find the base energy $E_b$ surrounded by one closed loop $\mathcal{L}^{\prime}_{BZ} \subseteq \mathcal{L}_{BZ}$. Here we denote the interior area of $\mathcal{L}^{\prime}_{BZ}$ as $S(\mathcal{L}^{\prime}_{BZ})$, and we have $$S(\mathcal{L}^{\prime}_{BZ}) \neq 0$$ Hence we can always define the distribution function $n(H,H^*)$ as follows $$\left\{ \begin{array}{lr} n(H,H^{\ast})=\frac{1}{2i}(H-H^{\ast}), &H \in \mathcal{L}^{\prime}_{BZ} \\ n(H,H^{\ast})=0, &H \in \mathcal{L}_{BZ} \setminus \mathcal{L}^{\prime}_{BZ} \end{array} \right.$$ such that $$J=\oint_{\mathcal{L}_{BZ}}n(H,H^*)dH = \oint_{\mathcal{L}^{\prime}_{BZ}}{\operatorname{Im}}(H) dH,$$ obviously, the imaginary part of which is zero. Then the total current becomes $$J=\oint_{\mathcal{L}^{\prime}_{BZ}}{\operatorname{Im}}(H) d{\operatorname{Re}}(H) =S(\mathcal{L}^{\prime}_{BZ}) \neq 0.$$ The finite area enclosed by $\mathcal{L}_{BZ}$ ensures the existence of the current under periodic boundary conditions, and which collapse into skin modes with periodic boundary. Here the inverse statement of Eq.(7) has been proven. [^1]: These two authors contributed equally [^2]: These two authors contributed equally

--- abstract: 'We study nonequilibrium properties of an electronic Mach-Zehnder interferometer built from integer quantum Hall edge states at filling fraction $\nu{=}1$. For a model in which electrons interact only when they are inside the interferometer, we calculate exactly the visibility and phase of Aharonov-Bohm fringes at finite source-drain bias. When interactions are strong, we show that a lobe structure develops in visibility as a function of bias, while the phase of fringes is independent of bias, except near zeros of visibility. Both features match the results of recent experiments \[Neder *et al.* Phys. Rev. Lett. **96**, 016804 (2006)\].' author: - 'D. L. Kovrizhin$^{1,2}$ and J. T. Chalker$^{1}$' title: 'Exactly Solved Model for an Electronic Mach-Zehnder Interferometer' --- Questions about phase coherence in interacting quantum systems out of equilibrium are of fundamental and wide-ranging importance. Despite great progress over the past decade, many aspects of nonequilibrium problems remain poorly understood. One recent example of this situation is the unexpected behaviour observed in state-of-the-art experiments on electronic Mach-Zehnder interferometers (MZIs) [@heiblum2; @roulleau07; @bieri08] driven out of equilibrium by an applied bias voltage. In these experiments the visibility of Aharonov-Bohm (AB) fringes in the conductance shows a lobe–like structure as a function of bias, while the phase of oscillations is independent of bias even with different interferometer arm lengths, except at zeros of the visibility where it jumps by $\pi $. These observations have attracted a lot of attention. It was immediately appreciated [@heiblum2] that they lie outside a single-particle description. Moreover, since integer quantum Hall edge states scale to non-interacting chiral Fermi gases at low energy, the [*finite-range*]{} of electron-electron interactions seems to be crucial. The effort to understand interaction effects in MZIs at integer filling is therefore linked with work on non-linear effects in non-chiral Luttinger liquids [@glazman], as well as to interferometry of fractional quantum Hall quasiparticles [@fractional]. The most obvious consequence anticipated from interactions is dephasing. This may arise from external noise [@marquardt04] or internally [@buttiker01; @chalker07], but in both cases is expected to suppress AB fringe visibility smoothly with increasing bias, in contrast to observations. It has been found, however, that zeros in visibility can arise if the edge channels that form the interferometer arms are coupled to another channel: such an extra channel may be a feature of sample design [@sukhorukov07], and is present intrinsically at $\nu{=}2$ [@sukhorukov08]. Although those results are encouraging, they do not seem sufficiently universal to explain all current experiments. In this context, two recent papers [@neder08; @sim08] that obtain visibility oscillations from calculations of interaction effects at $\nu{=}1$ represent an interesting advance. These papers contain illuminating physical insights, and similar phenomena have been shown to exist in another context [@marquardt08], but approximations used in [@neder08; @sim08] are not standard ones and their reliability is hard to judge. In this Letter we present an exact calculation for a simplified model of an interferometer. It reproduces the main signatures observed experimentally [@heiblum2; @roulleau07; @bieri08] and shows that the lobe pattern is a many-body effect, which would not appear in any approximation that treats single particles moving in a static mean-field potential. The model is illustrated in the inset to Fig. \[fig2\]. As in previous studies, two quantum Hall edge channels, both with the same propagation direction, are coupled at two quantum point contacts (QPCs). The simplifying feature of the model is that electrons interact only when they are *inside* the interferometer. This allows us to combine a description of the contacts using fermion operators with a treatment of interactions using bosonization. Within the MZI we take interactions only between two electrons on the same arm and with fixed strength independent of distance, although it would be feasible to relax these restrictions. We consider an initial state in which Fermi seas in the two channels are filled to different chemical potentials, to represent the bias voltage, and evolve this state forward in time using the Schödinger equation. At long times the system reaches a stationary regime. In this regime we calculate current and differential conductance as a function of chemical potential difference and enclosed AB flux. Our main results are presented in Figs. \[fig2\] and \[fig3\], and discussed following an outline of their derivation; details will be presented elsewhere [@kovrizhin09prbmz]. The solution we describe is significant more broadly as a rare example of a solved non-equilibrium scattering problem. One earlier instance is that of tunneling between fractional quantum Hall edge states [@fendley], while another is the interacting resonant level model, treated recently by a form of Bethe Ansatz [@andrei], and using boundary field theory [@boulat]. The remarkable structure observed experimentally [@heiblum2; @roulleau07; @bieri08] makes the MZI particularly interesting in this context. The Hamiltonian $\hat{H} = \hat{H}_{kin} + \hat{H}_{int} + \hat{H}_{tun}$ for the model has three contributions, representing respectively: kinetic energy, interactions, and tunneling at contacts. We formulate $\hat{H}$ initially for edges of length $L$ with periodic boundary conditions, then take the limit $L\to\infty$. Then $$\hat{H}_{kin}=-i\hbar v_{F}\sum_{\eta =1,2}\int_{-L/2}^{L/2}\hat{\psi}_{\eta }^{+}(x)\partial _{x}\hat{\psi}_{\eta }(x)dx, \label{H_kin}$$where $v_{F}$ is the Fermi-velocity and $\eta =1,\,2$ is the channel index. The Fermi field operators can be written as $\hat{\psi}% _{\eta }(x)=L^{-1/2}\sum_{k}\hat{c}_{k\eta }e^{ikx}$, with $k=2\pi n_{k}/L$ and $n_{k}$ integer, and $\{\hat{c}_{k\eta },\hat{c}_{q\eta ^{\prime }}^{+}\}=\delta _{kq}\delta _{\eta \eta ^{\prime }}$. Interactions are described by $$\hat{H}_{int}=\frac{1}{2}\sum_{\eta =1,2}\int_{-L/2}^{L/2}U_{\eta }(x,x^{\prime })\hat{\rho}_{\eta }(x)\hat{\rho}_{\eta }(x^{\prime })dxdx^{\prime }\,, \label{H_int}$$where $\hat{\rho}_{\eta }\left( x\right) =\hat{\psi}_{\eta }^{+}(x)\hat{\psi} _{\eta }(x)$ is the electron density operator. In our model $U_{\eta }(x,x^{\prime })=0$ for $x,x^{\prime }\notin (0,d_{\eta })$. Finally, the QPCs are represented by $$\hat{H}_{tun}=v_{a}e^{i\alpha }\hat{\psi}_{1}^{+}(0)\hat{\psi}% _{2}(0)+v_{b}e^{i\beta }\hat{\psi}_{1}^{+}(d_{1})\hat{\psi}_{2}(d_{2})+% \mathrm{h.c.} \label{H_tun}$$The AB-phase appears here as $\varphi _{AB}\equiv \beta - \alpha $. The total current $I$ from channel 1 to 2 has contributions $I_a$ and $I_b$ arising from each QPC, which can be written in terms of expectation values of operators acting at points infinitesimally before the QPC. Each contribution can be separated into a term that is not sensitive to coherence between the edges, and another that is sensitive. We define $t_{a,b}=\sin \theta _{a,b}$ and $r_{a,b}=\cos \theta _{a,b}$ with $\theta _{a,b}=v_{a,b}/\hbar v_{F}$, and denote expectation values by $\langle \ldots \rangle$. A straightforward calculation yields for QPC $b$ the expressions $I_b = I_b^{(1)} + I_b^{(2)}$, with $$\begin{aligned} I_{b}^{(1)}&=&ev_{F}t_{b}^{2}\langle \hat{\rho}_{1}(d_1) -\hat{\rho} _{2}(d_2)\rangle \nonumber\\ %\begin{equation} I_{b}^{(2)}&=&ev_{F}t_{b}r_{b}[ie^{i\beta }\langle \hat{G}_{12} \rangle +\mathrm{h.c.}]\;, \nonumber %\end{equation}%\end{aligned}$$ where $\hat{G}_{12}=\hat{\psi}_{1}^{+}(d_{1})\hat{\psi} _{2}(d_{2}) $. Terms in $I_a$ are obtained from these for $I_b$ by replacing $d_1$ and $d_2$ with $0$, and $v_b$ with $v_a$. Since there is no coherence between channels before QPC $a$, $I_a^{(2)}=0$ and the term responsible for AB oscillations in current is $I_b^{(2)}$. The bias voltage is $V=(\mu_1-\mu_2)/e$ and the differential conductance is $\mathcal{G}=e {\rm d}I/{\rm d}\mu _{1}$ (with $\mu_2$ fixed). $\mathcal{G}$ oscillates with $\varphi _{AB}$, having maximum and minimum values $\mathcal{G}_{\max }$ and $\mathcal{G}_{\min }$, and [*AB fringe visibility*]{} is defined as $(\mathcal{G}_{\max }-\mathcal{G}% _{\min })/(\mathcal{G}_{\max }+\mathcal{G}_{\min })$. The central task is therefore to calculate the correlator $\langle \hat{G}_{12} \rangle$, and our approach is as follows. (i) We work in the interaction representation, evolving operators with $\hat{H}_0 = \hat{H}_{kin} + \hat{H}_{int}$ and treating $\hat{H}_{tun}$ as the ‘interaction’. Then $\hat{\psi}_\eta\left( x,t\right) =e^{i\hat{H}_{0}t/\hbar}\hat{\psi}_\eta \left( x\right) e^{-i\hat{H}_{0}t/\hbar}$ (note that we distinguish operators in the Schrödinger and interaction representations by the absence or presence of a time argument). The wavefunction of the system, denoted at $t{=}0$ by $|Fs\rangle$, evolves with the S-matrix $\hat{S}\left( t\right) =\mathrm{T}\exp \{-(i/\hbar)\int^{t}_0\hat{H} _{tun}\left( t^{\prime }\right) dt^{\prime }\}$, where $\mathrm{T}$ indicates time ordering. (ii) Time evolution of operators is calculated using bosonization to diagonalise $\hat{H}_0$. (iii) Results are written in terms of operators in the Schrödinger picture, with boson operators re-expressed using fermion ones. This yields an expression for $\hat{G}_{12}$ suitable for straightforward numerical evaluation. We next outline these three steps. Step (i): Evaluation of $\hat{S}(t)$ hinges on our restriction of interactions to the interior of the MZI. Specifically, separating $\hat{H}_{tun}$ into parts $\hat{H}_{tun}^a$ and $\hat{H}_{tun}^b$ due to each QPC, we find from step (ii) that $[\hat{H}_{tun}^a(t_1),\hat{H}_{tun}^b(t_2)]=0$ and $[\hat{G}_{12}(t_1),\hat{H}^b(t_2)]=0$, provided $t_1 \geq t_2$. The first commutator allows us to factorise the S-matrix as $\hat{S}(t) = \hat{S}^b(t) \hat{S}^a(t)$, where $\hat{S}^a(t)$ is calculated using $\hat{H}_{tun}^a$ and $\hat{S}^b(t)$ using $\hat{H}_{tun}^b$. The second ensures that $[\hat{S}^b(t)]^+\hat{G}_{12}(t)\hat{S}^b(t) = \hat{G}_{12}(t)$, so that an explicit form for $\hat{S}^b(t)$ is not required in the calculation. Since QCP $a$ acts before interactions, $\hat{S}^a(t)$ is easy to evaluate: we have $[\hat{H}_{tun}^a(t_1),\hat{H}_{tun}^a(t_2)]=0$ for any $t_1,t_2 \geq 0$ and so may omit time ordering. In particular, we will need to compute the action of $\hat{S}^a(t)$ on fermionic operators. It is a rotation in the space of channels and can be written $\tilde{\hat{\psi}}_{\eta }(x)=[\hat{S}^a(t)]^{+}\hat{\psi}_{\eta^{\prime} }(x)\hat{S}^a(t)$. For $0<x< v_F t$ we find $$\begin{aligned} \tilde{\hat{\psi}}_{\alpha }(x)&=&\sum_{\beta }{\cal S}_{\alpha \beta }^{a}\hat{\psi}% _{\beta }(x),\nonumber\\ {\cal S}^{a}&=&\left( \begin{array}{cc} r_{a} & -it_{a}e^{i\alpha } \\ -it_{a}e^{-i\alpha } & r_{a}% \end{array}% \right)\;. \label{Sa}\end{aligned}$$ Step (ii): We compute time evolution under $\hat{H}_0$ using bosonization [@vonDelft]. Fermion operators are written in the form $$\hat{\psi}_{\eta }(x)=(2\pi a)^{-1/2}\hat{F}_{\eta }e^{i\frac{2\pi }{L}\hat{N }_{\eta }x}e^{-i\hat{\phi}_{\eta }(x)}, \label{boson_id}$$ where $\hat{F}_{\eta }$ are Klein factors with commutation relations $ \{\hat{F}_{\eta },\hat{F}_{\eta ^{\prime }}^{+}\}=2\delta _{\eta \eta ^{\prime }}$ and bosonic fields are defined as $$\hat{\phi}_{\eta }\left( x\right) =-\sum_{q>0}\left( 2\pi /qL\right) ^{1/2}(e^{iqx}\hat{b}_{q\eta }+\mathrm{h.c.})e^{-qa/2}, \label{phi_x}$$with $a$ an infinitesimal regulator. Plasmon creation operators obey bosonic commutation relations $[\hat{b}_{q\eta },\hat{b}_{k\eta ^{\prime }}^{+}]=\delta _{qk}\delta _{\eta \eta ^{\prime }}$ and are expressed for $q>0$ in terms of fermions as $$\hat{b}_{q\eta }^{+}=i\left( 2\pi /qL\right) ^{1/2}\sum_{k=-\infty}^{\infty}\hat{c}_{k+q\eta }^{+}\hat{c}_{k\eta } \;. \label{b_op}$$Since $\hat{H}_0$ does not couple channels, we restrict attention to a single channel and omit channel labels until we reach step (iii). The kinetic energy $\hat{H}_{kin}$ for a single edge has the bosonized form $$\hat{H}_{kin} =\frac{\hbar v_{F}}{2}\int_{-L/2}^{L/2}\frac{dx}{2\pi }% (\partial _{x}\hat{\phi}\left( x\right) )^{2}+\frac{2\pi }{L}\frac{\hbar v_{F}}{2}\hat{N}(\hat{N}+1)$$ where $\hat{N}\equiv \sum_{k}\hat{c}_{k}^{+}\hat{c}_{k}$ is the particle number operator. Similarly, $\hat{H}_{int}$ is quadratic when written using the bosonic representation of the density operators, $\hat{\rho}\left( x\right) =-\frac{1 }{2\pi }\partial _{x}\hat{\phi}\left( x\right) +\hat{N}/L$. The time dependence of $\hat{\phi}\left( x,t\right)$ can be found by solving the equation of motion. Since our choice of non-uniform interactions leads to a coupling between the plasmon and number operators, we make the separation $\hat{\phi}\left( x,t\right)= \hat{\phi}^{(0)}\left( x,t\right)+\hat{\phi}^{(1)}\left( x,t\right)$, where $\hat{\phi}^{(0)}\left( x,t\right) \propto \hat{N}/L$ and $\hat{\phi}^{(1)}\left( x,t\right)$ is independent of $\hat{N}$, satisfying $$2 \pi \hbar (\partial_t + v_F \partial_x) \hat{\phi}^{(1)}\left( x,t\right) = -\int U(x,y) \partial_y \hat{\phi}^{(1)}\left( y,t\right) {\rm d}y\,.$$ The solution can be written in the form $$\hat{\phi}^{(1)}\left( x,t\right) = \int_{-L/2}^{L/2} K(x,y;t)[ \hat{\phi}\left( y\right) -\hat{\phi}^{(0)}(y)]{\rm d}y\,,\nonumber$$ where the Green function $K(x,y;t)$ can be constructed in the usual way from the eigenfunctions of the time-independent equation, $$2 \pi \hbar v_F(\partial_x - ip) f_p(x) = -\int U(x,y) \partial_y f_p(y) {\rm d}y\,.\nonumber$$ We now specialise to interactions that are constant within the interferometer: $U(x,x^{\prime })=g$ for $x,x^{\prime }\in (0,d)$ and $U(x,x^{\prime })=0$ otherwise. This form of the potential is the one treated approximately in [@neder08]. It is characterised by the dimensionless coupling constant $\gamma =gd/2\pi \hbar v_{F}$. We find in the limit $L \to \infty$ $$f_{p}\left( x\right) = \left\{ \begin{array}{ll} e^{ipx} & x \leq0 \\ r_{p}+s_{p}e^{ipx} & 0<x<d \\ e^{ipx-i\delta _{p}} & x \geq d% \end{array}% \right. \;.$$The coefficients $s_{p}=(1+t_{p})^{-1}$ and $r_{p}=t_{p}s_{p}$, with $t_{p}=(i\gamma/ p d)(1 - e^{ipd})$, are obtained from matching $f_{p}(x)$ at $x=0,d.$ The phase shifts of plasmons $\delta _{p}$ due to the interactions are given by $e^{-i\delta _{p}}=(1+t_{p}^{\ast })/(1+t_{p})$. Similarly, we find $\hat{\phi }^{(0)}\left( x\right) =2\pi \bar{\gamma}\hat{N}x/L$ for $x\in (0,d),$ where $\bar{\gamma}=\gamma (1+\gamma )^{-1}$. In this way we find an expression for $K(x,y;t)$. Setting $x=d$, it simplifies at long times to $$K(d,y;t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} {\rm d} p\, e^{i(p[d-y-v_Ft] - \delta_p)}\,.$$ Using this and Eqns. (\[phi\_x\]) and (\[b\_op\]), we write $\hat{\phi}^{(1)}\left( x,t\right)$ as a bilinear in the fermion operators $\hat{c}^+_k$ and $\hat{c}_k$. Step (iii): We employ this result to construct an expression for $\hat{G}_{12}(t)$ in terms of fermion operators in the Schrödinger representation. To this end, we start from Eq. (\[boson\_id\]) in the interaction representation at time $t$ and substitute for $\hat{\phi}_\eta(d_\eta,t)$ as described. We also eliminate the combination ${\cal F} \equiv (2\pi a)^{-1/2}\hat{F}_{\eta }e^{i\frac{2\pi }{L}\hat{N }_{\eta }d_\eta}$ by inverting the bosonization identity, Eq. (\[boson\_id\]), writing $${\cal F}(t) = e^{i\hat{H}_{kin}t/\hbar} {\cal F} e^{-i\hat{H}_{kin}t/\hbar} = \hat{\psi}_{\eta }(z)e^{i\hat{\phi}_{\eta }(z)}\nonumber$$ for $z_\eta=d_\eta - v_Ft$. Finally, we substitute for $\hat{b}_{q \eta}$ and $\hat{b}_{q \eta}^+$ in $\hat{\phi}_\eta(x)$ using Eq. (\[b\_op\]). The result (omitting an unimportant, constant phase) is the operator identity $$\hat{\psi}_\eta(d_\eta,t)=e^{-i\hat{Q}_\eta}\hat{\psi}_\eta(z_\eta)\;. \label{psi_dt}$$Here $\hat{Q}_\eta = \int_{-\infty }^{\infty }Q_\eta(x-z_\eta)\hat{\rho}_\eta(x) dx$, where the kernel $Q_\eta(x)=L^{-1}\sum_{q=-\infty }^{\infty }\tilde{Q}_\eta(q)e^{iqx}$ has for our choice of interaction the Fourier transform $$\tilde{Q}_\eta(q)=2\pi \gamma d_\eta j_{0}^{2}(qd_\eta/2)(1+ \gamma e^{-iqd_\eta/2}j_{0}(qd_\eta/2))^{-1}, \label{kernel}$$in which $j_{0}(x)=x^{-1}\sin x$. In this way we arrive at the expression $$\langle G_{12}(t) \rangle = e^{i\bar{\Phi}}\langle Fs|[\hat{S}^a(t)]^{+}\hat{\psi} _{1}^{+}(z_1)e^{i\hat{R}}\hat{\psi}_{2}(z_{2})\hat{S}^a(t)\left\vert Fs\right\rangle\;.\nonumber$$ Here $\bar{\Phi}$ is an initial phase that is independent of voltage, and $\hat{R}=\hat{Q}_1 - \hat{Q}_2$. The action of $\hat{S}^a(t)^{+}$ and $\hat{S}^a(t)$ on the operators they enclose is given by Eq. (\[Sa\]), and evaluation of $\langle\hat{G}_{12}(t)\rangle$ reduces to the calculation of correlators of the form $C_{\mu\eta}=\langle Fs|\hat{c}_{\mu}% ^{+}\exp({i\sum_{\alpha \beta}\mathrm{M}_{\alpha \beta}\hat{c}_{\alpha}^{+}\hat{c}_{\beta}})c_{\eta}|Fs\rangle $, where the indices specify both channel and momentum, and the matrix $\mathrm{M}$ is obtained from $[\hat{S}^{a}(t)]^{+}\hat{R}\hat{S}^{a}(t)$. One can show that $C_{\mu\eta}=\mathrm{{D}}_{\eta\mu}^{-1}\det \mathrm{{D}}$ with $\mathrm{{D}}$ constructed from the matrix elements of $\exp(i\mathrm{M})$ between the single-particle states that are occupied in the Slater determinant $|Fs\rangle$. We calculate $C_{\mu\eta}$ numerically, achieving convergence of the results when keeping up to $10^{3}$ basis states and $400$ particles in each channel. The physical interpretation of the solution we have presented is as follows. Each electron passing QPC $b$ at time $t$ has an accumulated phase from its interactions with other electrons. The phase is a collective effect and is represented by the operator $\hat{Q}_\eta$ in Eq. (\[psi\_dt\]). Contributions from interactions with particles at a distance $x$ from the one at QPC $b$ have a weight determined by the kernel $Q_\eta(x)$, illustrated in the inset to Fig. \[fig3\]. This weight is largest near $x=0$, showing that interactions with nearby electrons are most important. Moreover, since $Q_\eta(x) = 0$ for $x<-d_\eta$, a given electron is uninfluenced by the ones behind, that enter the interferometer after it exits. The precise form of the kernel reflects the full many-body physics of the problem: a similar kernel appears in Eq. (11) of Ref. [@neder08], but with a simpler form because of the approximations employed there. A consequence of the phase $\hat{Q}_\eta$ is that many-particle interference influences the MZI conductance. As an illustration, consider the quantum amplitudes for two particles to pass through the interferometer on all possible paths connecting given initial and final states. Paths for which both particles propagate on the same arm of the interferometer have an interaction contribution to their phase that varies with their separation and is absent if the two particles propagate on different arms. Destructive interference between paths with different interaction phases generates the observed lobe structure. We now turn to our results. The parameters in the model are: the dimensionless interaction strength $\gamma$, the transmission probabilities $t_a^2$ and $t_b^2$, the ratio $d_2/d_1$ of arm lengths, and the dimensionless bias voltage $eV\sqrt{d_1 d_2}/2\pi \hbar v_F$. We consider $1\leq 2\pi \gamma \leq 10$, $1\leq d_2/d_1 \leq 1.2$ and first discuss behaviour with $t_a^2 = t_b^2 = 1/2$. The dependence of visibility of AB fringes on bias voltage and interaction strength is presented in Fig. \[fig2\], taking equal arm lengths and transmission probabilities of $1/2$ at both QPCs. The key features of all three curves in this figure match those of the experiment (see Figs. 2 and 3 of [@heiblum2]): with increasing bias there is a sequence of lobes in the visibility, which have decreasing amplitude and are separated by zeros. The phase of AB fringes is also influenced by interactions. Results are displayed in Fig \[fig3\]. For an MZI with different arm lengths (as in this figure), the fringe phase without interactions varies linearly with bias, because the Fermi wavevector $k_F$ is linear in bias and the phase difference between particles traversing the two arms is $k_F(d_2 - d_1)$. With increasing interaction strength the phase dependence on bias develops into a series of smooth steps, each of height $\pi$. The risers of these steps coincide with minima of the visibility. Strikingly, with strong interactions phase steps at minima of the visibility persist for $d_1=d_2$, even though in this case phase would be independent of bias without interactions. The stepwise phase variation we find at large interaction strength also matches observations (see Fig. 2 of [@heiblum2]). Behaviour is insensitive to the transmission probability $t_b^2$ at QPC $b$, apart from the overall scale for visibility. Departures from $t_a^2=1/2$, however, eliminate the exact zeros in visibility, leaving only sharp minima. A difference in arm lengths has a similar though much weaker effect. The width in bias voltage of the central visibility lobe defines an energy scale. In our model this scale is of order $g$ at large $\gamma$. Taking $v_F = 2.5\times 10^{4} {\rm ms^{-1}}$, $d=10\mu {\rm m}$ and the permittivity $\epsilon =12.5$ of GaAs, we estimate from the capacitance of an edge channel $g\sim 10{\rm \mu eV}$. This is similar to the experimentally observed value of about $14\ \mathrm{\mu eV}$ [@heiblum2]. Our calculations rely on a simplified form for interactions, but we believe our choice is quite reasonable. Our central approximation is to neglect interactions between an electron inside the MZI and one outside. In practice, such interactions will anyway be screened by the metal gates that define the QPCs. We also neglect interactions between a pair of electrons that are both outside the MZI. This is unimportant: before electrons reach the MZI, such interactions do not cause scattering because of Pauli blocking, while after electrons pass through the MZI, these interactions cannot affect the current. Within the MZI we represent interactions by a capacitative charging energy. Such a choice is standard in the theory of quantum dots and has been applied previously to interferometers [@buttiker01; @neder08]. In summary, we have calculated the visibility of Aharonov-Bohm fringes in the differential conductance of an electronic MZI out of equilibrium, taking exact account of interactions between electrons. From our calculations we obtain a lobe pattern in the dependence of visibility on bias, and jumps in the phase of fringes at zeros of the visibility, as observed experimentally [@heiblum2; @roulleau07; @bieri08]. We thank F. H. L. Essler for fruitful discussions and acknowledge support from EPSRC grants EP/D066379/1 and EP/D050952/1. [99]{} I. Neder [*et al.*]{}, Phys. Rev. Lett. **96**, 016804, (2006). P. Roulleau [*et al.*]{}, Phys. Rev. B **76**, 161309(R) (2007). E. Bieri [*et al.*]{}, Phys. Rev. B [**79**]{}, 245324 (2009). A. Imambekov and L. I Glazman, Science [ **323**]{}, 228 (2009) C. de C. Chamon [*et al*]{}, Phys. Rev. B **55**, 2331 (1997); F. E. Camino, W. Zhou, and V. J. Goldman, [*ibid.*]{} [**72**]{}, 075342 (2005); K. T. Law, D. E. Feldman, and Y. Gefen, [*ibid.*]{} **74**, 045319 (2006); D. E. Feldman and A. Kitaev, Phys. Rev. Lett. **97**, 186803 (2006); V. V. Ponomarenko and D. V. Averin, [*ibid.*]{} **99**, 066803 (2007). F. Marquardt and C. Bruder, Phys. Rev. Lett. **92**, 056805 (2004). G. Seelig and M. Büttiker, Phys. Rev. B ** 64**, 245313 (2001). J. T. Chalker, Y. Gefen, and M. Y. Veillette, Phys. Rev. B **76** 085320 (2007). E. V. Sukhorukov and V. V. Cheianov, Phys. Rev. Lett. **99**, 156801 (2007). I. P. Levkivskyi, E. V. Sukhorukov, Phys. Rev. B **78**, 045322 (2008). I. Neder and E. Ginossar, Phys. Rev. Lett. **100**, 196806 (2008). Seok-Chan Youn, Hyun-Woo Lee, and H.-S. Sim, Phys. Rev. Lett. **100** 196807 (2008). B. Abel and F. Marquardt, Phys. Rev. B [**78**]{}, 201302 (2008). D.L. Kovrizhin and J. T. Chalker, in preparation. P. Fendley, A. Ludwig, and H. Saleur, Phys. Rev. Lett. [**75**]{}, 2196 (1995). P. Mehta and N. Andrei, Phys. Rev. Lett. [**96**]{}, 216802 (2006). E. Boulat, H. Saleur, and P. Smitteckert, Phys. Rev. Lett. [**101**]{}, 140601 (2008). See: J. von Delft and H. Schoeller, Annalen Phys. **7**, 225 (1998); T. Giamarchi, [*Quantum Physics in One Dimension*]{} (Oxford Univ. Press, Oxford, 2004).

--- abstract: 'We develop a new thermodynamic formalism to investigate the transient behaviour of maps on the real line which are skew-periodic ${\mathbb{Z}}$-extensions of expanding interval maps. Our main focus lies in the dimensional analysis of the recurrent and transient sets as well as in determining the whole dimension spectrum with respect to $\alpha$-escaping sets. Our results provide a one-dimensional model for the phenomenon of a dimension gap occurring for limit sets of Kleinian groups. In particular, we show that a dimension gap occurs if and only if we have non-zero drift and we are able to precisely quantify its width as an application of our new formalism.' address: - 'Faculty of Mathematics, University of Vienna, Oskar Morgensternplatz 1, 1090 Vienna, Austria' - 'Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602 Japan' - 'FB03 – Mathematik und Informatik, Universität Bremen, 28359 Bremen, Germany' author: - Maik Gröger - Johannes Jaerisch - Marc Kesseböhmer title: Thermodynamic formalism for transient dynamics on the real line --- [^1] Introduction ============ The main motivation of this article is the connection between transient phenomena of dynamical systems and its manifestation in dimensional quantities. Since transience can impose major obstructions to an ergodic-theoretic description of (fractal-)geometric features its further understanding is vital and has attracted a lot of attention. For instance, it has a strong tradition in complex dynamics with landmark results like the ones obtained for complex quadratic polynomials in [@MR1626737] or [@MR2373353]. A closely related and paralleling line of research established corresponding results for families of transcendental functions, see for example [@MR2302520; @MR2465667; @MR2197375; @MR871679]. In both cases, a particular striking effect revealing the preponderance of transience is the occurrence of a so-called dimension gap. In fact, the origin of this phenomenon goes back to the rich field of geometric group theory which we explain in more detail further below. In the framework of thermodynamic formalism, transient effects in topological Markov chains have been seminally studied by Sarig [@MR1738951; @MR1818392]. Directly related to this are fractal-geometric applications of thermodynamic formalism for infinite conformal graph directed Markov systems which have been systematically worked out by Mauldin and Urbanski in [@MR2003772]. In there, strong mixing conditions were introduced to guarantee that the recurrent behaviour governs the system. One main goal of this paper is to set up a new thermodynamic formalism in the absence of such strong mixing conditions to provide a systematic approach to the geometric phenomenon of a dimension gap. More precisely, we introduce the concept of fibre-induced pressure which allows us to express the occurrence and the width of a dimension gap for skew-periodic ${\mathbb{Z}}$-extensions of expanding interval maps exclusively in terms of this newly developed pressure. Furthermore, we obtain effective analytic relations between the fibre-induced pressure and the classical pressure of the base transformation allowing us to determine the crucial dimensional quantities in a number of examples explicitly. Let us now illustrate the phenomenon of a dimension gap in the setting of actions of non-elementary Kleinian groups $G$ on the hyperbolic space $\mathbb{H}^{n}$. By a general result of Bishop and Jones [@MR1484767] we know that the Hausdorff dimension of both the radial limit set $\Lambda_{r}\left(G\right)$ and the uniformly radial limit set $\Lambda_{ur}\left(G\right)$ of $G$ are equal to the Poincaré exponent of $G$ given by $$\delta_{G}\coloneqq\inf\left\{ s\geq0:\sum_{g\in G}{\mathrm{e}}^{-s\cdot d_{H}\left(0,g0\right)}<\infty\right\} ,\label{eq:definition Poincare exponent group}$$ where $d_{H}$ denotes the hyperbolic distance on $\mathbb{H}^{n}$. Recall that $\Lambda_{r}\left(G\right)$ and $\Lambda_{ur}\left(G\right)$ represent recurrent dynamics of the geodesic flow on $\mathbb{H}^{n}/G$. Clearly, for a normal subgroup $N<G$ we have that $\delta_{G}\geq\delta_{N}$ and moreover, $$\dim_{H}\left(\Lambda_{r}\left(G\right)\right)=\delta_{G}>\delta_{N}=\dim_{H}\left(\Lambda_{r}\left(N\right)\right)\iff G/N\;\text{is non-amenable,}$$ with $\dim_{H}(\,\cdot\,)$ the Hausdorff dimension of the corresponding set. This was first proved by Brooks for certain Kleinian groups fulfilling $\delta_{G}>(n-1)/2$ in [@MR783536] and later generalised to a wider class of groups without this restriction by Stadlbauer [@Stadlbauer11]. Note that by a result of Falk and Stratmann $\delta_{N}\ge\delta_{G}/2$, see [@MR2097162]. If $G$ is additionally geometrically finite, then the strict inequality $\delta_{N}>\delta_{G}/2$ holds by a result of Roblin [@MR2166367] (see also [@MR3299281]). Furthermore, if $\Lambda\left(G\right)$ denotes the limit set of the Kleinian group $G$, then $\delta_{G}=\dim_{H}\left(\Lambda\left(G\right)\right)$ and, since $\Lambda\left(N\right)=\Lambda\left(G\right)$, this implies the following criterion for the occurrence of a *dimension gap*: $$\dim_{H}\left(\Lambda_{r}\left(N\right)\right)=\dim_{H}\left(\Lambda_{ur}\left(N\right)\right)<\dim_{H}\left(\Lambda\left(N\right)\right)\iff G/N\;\text{is non-amenable.}$$ In other words, a certain amount of transient behaviour causes a dimension gap from the dimension of the full limit set compared to the restriction of the limit set to certain recurrent parts. It is remarkable that for Kleinian groups the presence of a dimension gap depends only on the group-theoretic property of ** amenability. Accordingly, a natural example for the occurrence of a dimension gap is given by a Schottky group $G=N\rtimes\mathbb{F}_{2}$ where $\mathbb{F}_{2}$ denotes the free group with two generators. Nevertheless, only little is known in the literature concerning the concrete size of this dimension gap. The occurrence of a dimension gap is closely related to the decay of certain return probabilities. In fact, Kesten [@MR0112053; @MR0109367] has shown for symmetric random walks on countable groups that exponential decay of return probabilities is equivalent to non-amenability. However, for amenable groups exponential decay can also be caused by non-symmetric random walks. To be more precise, for groups admitting a recurrent random walk (e.g. ${\mathbb{Z}}$) it is shown in [@MR3436756] that exponential decay of return probabilities is equivalent to a lack of certain symmetry condition on the thermodynamic potential related to the random walk (see also Remark \[rem:characterisation of recurrence\] for further details). We are aiming at investigating these closely linked phenomena for a class of maps on ${\mathbb{R}}$ which can be considered as models of ${\mathbb{Z}}$-extensions of Kleinian groups. In fact, if the Kleinian group $G=N\rtimes{\mathbb{Z}}$ is a Schottky group, then the elements in $\Lambda_{r}\left(N\right)$ can be characterised as the limits of $G$-orbits for which the ${\mathbb{Z}}$-coordinate returns infinitely often to some point in $\mathbb{Z}$ (compare this with our definition of a recurrent set, see Section \[subsec:Recurrent-and-transient\]). Since ${\mathbb{Z}}$ is amenable, these limit points have full Hausdorff dimension by Brooks’ amenability criterion. We will see later (end of Section \[sec:Examples\]) that this property also follows from the fact that the ${\mathbb{Z}}$-coordinate has zero drift with respect to a canonical invariant measure obtained from the Patterson-Sullivan construction. Our models witness drift behaviour and we show that indeed non-zero drift is equivalent to the occurrence of a dimension gap, see Theorem \[thm:-dimension gap\]. It is therefore also very natural to consider subsets of the transient dynamics with fixed drift in more detail. This motivates the definitions of various escaping sets in our one-dimensional models, see Section \[subsec:Escaping-sets\]. The related dimension spectra will allow us to determine the size of the dimension gap explicitly. Similar results will be shown in the forthcoming paper [@JKG20] on ${\mathbb{Z}}$-extensions with reflective boundaries allowing us to illuminate earlier results in [@MR1438267; @MR2959300; @MR3610938] which studied a family suggested by van Strien modelling induced maps of Fibonacci unimodal maps. Let us point out that drift arguments where also prominent in the proofs of [@MR2959300]. Our leading motivating example for which we obtain dimensional results on the transient behaviour stems from the family of (a-)symmetric random walks, see Example \[exa:Classical Random Walk\]. More precisely, let ${F}$ be an expanding interval map with finitely many $C^{1+\epsilon}$ full branches, say $\left.{F}\right|_{I_{i}}:I_{i}\rightarrow[0,1]$ with disjoint intervals $I_{i}\subset[0,1]$ with non-empty interior, $i\in I\coloneqq\{1,\dots,m\}$, $m\geq2$. Set $h_{i}\coloneqq H_{i}^{-1}:\left[0,1\right]\to\overline{I_{i}}$ for the continuous continuation $H_{i}$ of $\left.{F}\right|_{I_{i}}$ to $\overline{I_{i}}$ and define the corresponding coding map $\pi:\Sigma\coloneqq I^{{\mathbb{N}}}\rightarrow[0,1]$ by $\pi\left(\omega_{1},\omega_{2},\ldots\right)\coloneqq x$ for $\bigcap_{n\in{\mathbb{N}}}h_{\omega_{1}}\circ\cdots\circ h_{\omega_{n}}\left(\left[0,1\right]\right)=\left\{ x\right\} $. The *repeller* of $F$ is then $\pi\left(\Sigma\right)\subset[0,1]$ and we set ${\mathcal{R}}\coloneqq\pi\left(\Sigma\right)\setminus\mathcal{D}+{\mathbb{Z}}$, where we subtract the countable set $\mathcal{D}$ to avoid technical problems stemming from possible discontinuities at the boundaries of the $I_{i}$’s, see (\[eq:D\]) for the precise definition of $\mathcal{D}$. We assume that the *step length function* $\Psi:[0,1]\rightarrow{\mathbb{Z}}$ is constant on each of the intervals $I_{i}$ and consider the *$\Psi$-lift* of ${F}$ given by $$\begin{aligned} {F}_{\,\Psi}:{\mathcal{R}}& \rightarrow{\mathcal{R}}:\quad x\mapsto\sum_{k\in{\mathbb{Z}}}\left(k+F(x-k)+\Psi(x-k)\right){\mathbbm{1}}_{[0,1]}(x-k).\end{aligned}$$ Since this function is periodic up to a certain skewness induced by $\Psi$, we refer to this map as a *skew-periodic interval map*. In order to study the various escaping sets of ${F}_{\,\Psi}$ later it will be necessary to consider ${\mathbb{R}}$-extensions rather than ${\mathbb{Z}}$-extensions. This is the reason why in our abstract set-up we will consider *skew product dynamical systems* defined via $$\sigma\rtimes f:\Sigma\times{\mathbb{R}}\rightarrow\Sigma\times{\mathbb{R}},\quad(\sigma\rtimes f)(\omega,x)\coloneqq(\sigma(\omega),x+f(\omega)),\label{eq:definition skew product dynamical system}$$ for some function $f:\Sigma\to{\mathbb{R}}$. We want to stress that after neglecting a countable set ${F}_{\,\Psi}$ is topological conjugate to the ${\mathbb{Z}}$-extension $\sigma\rtimes\left(\Psi\circ\pi\right):\Sigma\times{\mathbb{Z}}\to\Sigma\times{\mathbb{Z}}$, see Lemma \[fac:factor\]. \[exa:Classical Random Walk\] We model a classical one-step random walk via a skew-periodic interval map ${F}_{\,\Psi}$. For this fix $c_{1},c_{2}\in\left(0,1\right)$ with $c_{1}+c_{2}\leq1$ and consider the map $$F:x\mapsto\begin{cases} c_{1}^{-1}x & \text{for }x\in[0,c_{1}]\\ c_{2}^{-1}x & \text{for }x\in\left(1-c_{2},1\right] \end{cases},$$ with code space $\Sigma:=\left\{ 1,2\right\} ^{{\mathbb{N}}}$ and set $\Psi\coloneqq-{\mathbbm{1}}_{[0,c_{1}]}+{\mathbbm{1}}_{\left(1-c_{2},1\right]}$ (see Figure \[fig:Classical-Random-Walk\]). In this setting we will also refer to $F_{\,\Psi}:{\mathcal{R}}\to{\mathcal{R}}$ as the *random walk model*. For any $\left(p_{1},p_{2}\right)$-Bernoulli measure $\mu$ on $\Sigma$, we can model the classical (a-)symmetric random walk on ${\mathbb{Z}}$ with transition probability $p_{1}$ to go one step left and probability $p_{2}$ one step right. The random walk starting in $0\in\mathbb{{\mathbb{Z}}}$ would then be given by the stochastic process $\left(\left\lfloor F_{\,\Psi}^{n}\right\rfloor \right)_{n\in{\mathbb{N}}}$ with respect to the probability measure $\mu\circ\pi^{-1}$ on the repeller $\pi\left(\Sigma\right)$. Note that $\pi\left(\Sigma\right)$ is a Cantor set with Hausdorff dimension $\delta<1$ if and only if $c_{1}+c_{2}<1$ where $\delta$ is the unique number $s$ with $c_{1}^{s}+c_{2}^{s}=1$. Otherwise, $\pi\left(\Sigma\right)$ is the unit interval and hence, ${\mathcal{R}}={\mathbb{R}}\setminus(\mathcal{D}+{\mathbb{Z}})$. (-3.9,0.) – (3.9,0.); in [-3,-2,-1,1,2,3]{} (0pt,2pt) – (0pt,-2pt) node\[below\] [$\x$]{}; (0.,-3.9) – (0.,3.9); in [-3,-2,-1,1,2,3]{} (2pt,0pt) – (-2pt,0pt) node\[left\] [$\y$]{}; (0pt,-10pt) node\[right\] [$0$]{}; (-3.9,-3.9) rectangle (3.9,3.9); plot(,[2.50\*()+3.5]{}); plot(,[5/3\*()+7/3]{}); plot(,[2.5\*()+2]{}); plot(,[5/3\*()+5/3]{}); plot(,[2.50\*()+0.5]{}); plot(,[5/3\*()+3/3]{}); plot(,[2.5\*()-1]{}); plot(,[5/3\*()+1/3]{}); plot(,[2.50\*()-2.5]{}); plot(,[5/3\*()-1/3]{}); plot(,[2.5\*()-4]{}); plot(,[5/3\*()-3/3]{}); Crucial for our analysis will be the *fibre-induced pressure* $\mathcal{P}$ and its close connection to the so-called *$\alpha$-Poincaré exponent* $\delta_{\alpha}$ defined in the next section. Indeed, the new pressure $\mathcal{P}$ is a natural generalisation of the notion of Gurevich pressure (cf. [@MR1738951] and Remark \[rem:original-gurevich\]) and is necessary to perform our analysis for general escaping rates, see Section \[subsec:Escaping-sets\]. In particular, Gurevich pressure is defined for topological Markov chains whereas our new notion is defined for more general ${\mathbb{R}}$-extensions. Further, we will see that our new quantity can also be deduced from the *classical pressure* $\mathfrak{P}$. In fact, we will show in Theorem \[thm:fibre-induced pressure via base pressure\] below that $$\mathcal{P}\left(f,\psi\right)=\inf_{s\in{\mathbb{R}}}\mathfrak{P}\left(s\psi+f\right),$$ for $f,\psi:\Sigma\rightarrow{\mathbb{R}}$ Hölder continuous and $\psi$ satisfying natural conditions fulfilled in our setting. Generalising the concept of the classical Poincaré exponent, the $\alpha$-Poincaré exponent will reveal a natural connection to the analysis of limit sets of Kleinian groups as well as the dimension theory of Birkhoff averages, see the remark after Theorem \[thm: Multifractal Decomposition\] and Remark \[rem:delta\_=00005Calpha as Birkhoff average\]. Main results ------------ We define the *geometric potential* $\varphi:\Sigma\to(-\infty,0)$ in a Hölder continuous way such that $\varphi\left(\omega\right)\coloneqq-\log\left|{F}'\left(\pi\left(\omega\right)\right)\right|$ except possibly on a finite set. Denote by $\delta\text{\ensuremath{>0}}$ the unique $s$ such that $\mathfrak{P}\left(s\varphi\right)=0$. Here, $\mathfrak{P}\left(f\right)$ refers to the classical topological pressure defined for any continuous function $f:\Sigma\to{\mathbb{R}},$ see end of Section \[sec:Preliminaries-and-basic\]. Then by *Bowen’s formula* we have for the repeller $\pi(\Sigma)$ of ${F}$ that $$\delta=\dim_{H}(\pi(\Sigma)).\label{eq:Hausdorff dim Bowen formula}$$ Further, we can associate to $\Psi$ a *symbolic step length function* $\psi:\Sigma\to{\mathbb{Z}}$ which is constant on one-cylinder sets such that $\psi=\Psi\circ\pi$ except possibly on a finite set. Let us introduce the *$\alpha$-Poincaré exponent* $$\delta_{\alpha}:=\inf\left\{ s\ge0\mid\sum_{{\omega\in I^{\star}},\,{\left|S_{\omega}\left(\psi-\alpha\right)\right|\leq K}}{\mathrm{e}}^{s\cdot S_{\omega}\varphi}<\infty\right\} ,\label{eq:definition Poincare exponent}$$ where $I^{\star}$ denotes the set of all finite words over the *alphabet* $I$ and $K>0$ is sufficiently large (the term $S_{\omega}f$, $\omega\in I^{\star}$ is defined in (\[eq:birkhoff sum and max sum\]) for any $f:\Sigma\to\mathbb{R}$). For this exponent we observe $$\alpha\in{\mathbb{R}}\setminus\left[\underline{\psi},\overline{\psi}\right]\implies\delta_{\alpha}=0\label{eq:delta_alpha=00003D0}$$ and for $\alpha\in\text{\ensuremath{\big[}}\underline{\psi},\overline{\psi}\big]$ the critical exponent $\delta_{\alpha}$ can be expressed as the unique zero of the fibre-induced pressure for suitable potentials and is positive on $(\underline{\psi},\overline{\psi})$ (see Theorem \[thm:critical exponent via base pressures\] as well as (\[eq:Psiupperlower\]) for the definition of $\underline{\psi}$ and $\overline{\psi}$). We will also see that the map $\alpha\mapsto\delta_{\alpha}$ is real-analytic on $\big(\underline{\psi},\overline{\psi}\big)$ and unimodal but not necessarily concave (cf. Section \[sec:Multifractal-decomposition\] and Section \[subsec:First-examples\] for examples). ### Recurrent and transient sets and dimension gap\[subsec:Recurrent-and-transient\] For ${F}_{\,\Psi}$ we define the *recurrent set* $$\mathbf{R}:=\left\{ x\in{\mathcal{R}}\mid\exists K\in{\mathbb{R}}\:\text{such that }\left|{F}_{\,\Psi}^{n}(x)\right|\leq K\mbox{ for infinitely many }n\in{\mathbb{N}}\right\}$$ and the *uniform recurrent set* $$\mathbf{R}_{u}:=\left\{ x\in{\mathcal{R}}\mid\exists K\in{\mathbb{R}}\;\forall n\in{\mathbb{N}}\;\left|{F}_{\,\Psi}^{n}(x)\right|\leq K\right\} .$$ The *positive (resp. negative) transient set* is given by $$\mathbf{T}_{1}^{\pm}\coloneqq\left\{ x\in{\mathcal{R}}\mid\lim_{n\to\infty}{F}_{\,\Psi}^{n}(x)=\pm\infty\right\} .$$ We also consider the supersets $$\mathbf{T}_{2}^{\pm}\coloneqq\left\{ x\in{\mathcal{R}}\mid\limsup_{n\to\infty}\mp{F}_{\,\Psi}^{n}(x)<\infty\right\} ,\,\,\ \mathbf{T}_{3}^{\pm}\left(r\right)\coloneqq\left\{ x\in{\mathcal{R}}\mid\forall n\geq0\;\pm{F}_{\,\Psi}^{n}(x)>r\right\} ,$$ for some $r\in{\mathbb{R}}$. Observe that $$\mathbf{T}_{1}^{\pm}\subseteq\mathbf{T}_{2}^{\pm}=\bigcup_{k\in{\mathbb{Z}}}\mathbf{T}_{3}^{\pm}\left(k\right).\label{eq:T2UnionT3}$$ The following theorems will be a consequence of our general multifractal decomposition for escaping sets presented in the next subsection. The corresponding proofs can be found in Section \[sec:Multifractal-decomposition\]. The first theorem is the analogue of the result of Bishop and Jones in our setting. Last, recall that for the Hölder continuous function $\delta\varphi:\Sigma\to{\mathbb{R}}$ there exists a unique Gibbs measure $\mu_{\delta\varphi}$, see end of Section \[sec:Preliminaries-and-basic\]. \[thm:Bishop and Jones\]Let ${F}_{\,\Psi}:{\mathcal{R}}\rightarrow{\mathcal{R}}$ be the $\Psi$-lift of an expanding interval map ${F}$. Then we have for the recurrent and uniformly recurrent sets $$\dim_{H}(\mathbf{R})=\dim_{H}(\mathbf{R}_{u})=\delta_{0}.$$ We say the system ${F}_{\,\Psi}$ has a *dimension gap* if the Hausdorff dimension $\delta_{0}$ of the (uniformly) recurrent set is strictly less than the Hausdorff dimension $\delta$ of ${\mathcal{R}}$. In fact, this is the case if and only if the system has a *drift* $\mu_{\delta\varphi}\left(\psi\right)\neq0$, see Theorem \[thm:-dimension gap\] below. Furthermore, we are able to provide direct methods to determine $\delta_{0}$, see Theorem \[thm:critical exponent via base pressures\]. This allows us to easily calculate $\delta_{0}$ for our examples and to precisely quantify the dimension gap, see for instance Figure \[fig:delta\_0Graph\]. \[thm:-Transient-sets\]For the transient sets the following implications hold: - $\mu_{\delta\varphi}\left(\psi\right)\geq0$ implies $\dim_{H}\left(\mathbf{T}^{-}\right)=\delta_{0}$ and $\dim_{H}\left(\mathbf{T}^{+}\right)=\delta$, - $\mu_{\delta\varphi}\left(\psi\right)\leq0$ implies $\dim_{H}\left(\mathbf{T}^{+}\right)=\delta_{0}$ and $\dim_{H}\left(\mathbf{T}^{-}\right)=\delta$, where $\mathbf{T}^{\pm}$ may be chosen to be one of the sets $\mathbf{T}_{1}^{\pm}$, $\mathbf{T}_{2}^{\pm}$ or $\mathbf{T}_{3}^{\pm}\left(r\right)$ for any $r\in{\mathbb{R}}$. ### \[subsec:Escaping-sets\]Escaping sets For $\alpha\in{\mathbb{R}}$ let us now define the $\alpha$*-escaping set* for ${F}_{\,\Psi}$ by $$\begin{aligned} \mathbf{E}(\alpha) & \coloneqq\left\{ x\in{\mathcal{R}}\mid\exists K>0\;\left|{F}_{\,\Psi}^{n}(x)-n\alpha\right|\leq K\;\text{for infinitely many }n\in{\mathbb{N}}\right\} ,\end{aligned}$$ and the *uniformly* $\alpha$*-escaping set* for ${F}_{\,\Psi}$ by $$\begin{aligned} \mathbf{E}_{u}(\alpha): & =\left\{ x\in{\mathcal{R}}\mid\exists K>0\;\forall n\in{\mathbb{N}}\;\left|{F}_{\,\Psi}^{n}(x)-n\alpha\right|\leq K\right\} .\end{aligned}$$ As stated above, the following result proves our statement on the occurrence of a dimension gap. \[thm:-dimension gap\] We have $\delta_{\alpha}=\delta$ if and only if $\mu_{\delta\varphi}(\psi)=\alpha$. In particular, a dimension gap occurs if and only if $\mu_{\delta\varphi}(\psi)\neq0$. \[Multifractal decomposition with respect to $\alpha$-escaping\] \[thm: Multifractal Decomposition\] For $\alpha\in{\mathbb{R}}$ we have $$\dim_{H}(\mathbf{E}(\alpha))=\dim_{H}(\mathbf{E}_{u}(\alpha))=\delta_{\alpha}.$$ We note that $\delta_{\alpha}$ also allows for a multifractal spectral interpretation of certain Birkhoff averages (cf. Remark \[rem:delta\_=00005Calpha as Birkhoff average\]). However, the results of Theorem \[thm: Multifractal Decomposition\] need more subtle ideas adopted from the analysis of Kleinian groups and we believe that this connection is also of independent interest. ### First examples and further consequences\[subsec:First-examples\] The previous theorems applied to Example \[exa:Classical Random Walk\] with fixed $c_{1,}c_{2}\in(0,1)$ lead to the following observations. We determine the graph of $\alpha\mapsto\dim_{H}\left(\mathbf{E}\left(\alpha\right)\right)=\dim_{H}(\mathbf{E}_{u}(\alpha))=\delta_{\alpha}(c_{1},c_{2})$ for the random walk model, see Figure \[fig:asymmetric-Random-Walk-Spectrum\]. \[ domain=-1:1, samples=197,line width=1.2pt, \] [(-ln(4.0)+(1.0+(x))\*ln(1.0+(x))+(1.0-(x))\*ln(1.0-(x)))/((1.0+(x))\*ln(0.5)+(1.0-(x))\*ln(1.0-0.5))]{}; \[ domain=-1:1, samples=197,line width=1.2pt, dashed \] [(-ln(4.0)+(1.0+(x))\*ln(1.0+(x))+(1.0-(x))\*ln(1.0-(x)))/((1.0+(x))\*ln(0.3)+(1.0-(x))\*ln(1.0-0.3))]{}; \[ domain=-1:1, samples=197,line width=1.2pt, dotted \] [(-ln(4.0)+(1.0+(x))\*ln(1.00000+(x))+(1.0-(x))\*ln(1.0000-(x)))/((1.0+(x))\*ln(0.95)+(1.0-(x))\*ln(1.0-0.95))]{}; Moreover, the Hausdorff dimension of the (uniformly) recurrent set for the random walk model given by Theorem \[thm:Bishop and Jones\] is $$\delta_{0}\left(c_{1},c_{2}\right)=\frac{\log4}{\log(1/c_{1})+\log(1/c_{2})},$$ see Figure \[fig:delta\_0Graph\] for a one-parameter plot of $c\mapsto\delta_{0}\left(c,1-c\right)$. \[ domain=0:1, samples=270,line width=1.2pt, \] [-ln(4.0)/ ln(x\*(1-x)+0.00001)]{}; \[ domain=0:0.13, samples=5,line width=1.2pt, \] (1,x); (0,x); As an extension of Example \[exa:Classical Random Walk\] we will consider asymmetric step widths and interval maps with more than two branches. The corresponding calculations and precise formulas are postponed to Section \[sec:Examples\]. We end the introduction by providing an application of our results to the above mentioned ${\mathbb{Z}}$-extensions of Kleinian groups. More precisely, we partially recover a result of [@rees_1981] and provide a direct alternative proof, see end of Section \[sec:Examples\]. Recall that a group $G$ is of *divergence type* if the series in (\[eq:definition Poincare exponent group\]) is infinite for the critical exponent $\delta_{G}$. \[thm:Kleingroup divergence type\]If $G=N\rtimes{\mathbb{Z}}$ is a Schottky group, then $\delta_{N}=\delta_{G}$ and $N$ is of divergence type. Preliminaries and basic definitions\[sec:Preliminaries-and-basic\] ================================================================== Let $I$ be a finite set, $I^{\star}\coloneqq\bigcup_{k=1}^{\infty}I^{k}\cup\left\{ {{\varnothing}}\right\} $ the set of all finite words over $I$ containing the *empty word* ${{\varnothing}}$ and set $\Sigma\coloneqq I^{{\mathbb{N}}}$. For $\omega\in I^{\star}$ we denote by $|\omega|$ the unique $k\in{\mathbb{N}}$ such that $\omega\in I^{k},$ and we refer to $|\omega|$ as the *length* of $\omega$. Note that ${{\varnothing}}$ is the unique word of length zero. For $\omega=\left(\omega_{1},\ldots,\omega_{n}\right)\in I^{n}$ and $1\leq k\leq n$, or $\omega\in\Sigma$ and $k\in{\mathbb{N}}$ resp., we set $\omega|_{k}\coloneqq\left(\omega_{1},\ldots,\omega_{k}\right)$. For $\omega=\left(\omega_{1},\ldots,\omega_{n}\right)\in I^{n}$ and $\nu=\left(\nu_{1},\ldots,\nu_{k}\right)\in I^{k}$, or $v\in\Sigma$ resp., we define the concatenation $\omega\nu\coloneqq\left(\omega_{1},\ldots,\omega_{n},\nu_{1},\ldots,\nu_{k}\right)$, or $\omega\nu\coloneqq\left(\omega_{1},\ldots,\omega_{n},\nu_{1},\nu_{2},\ldots\right)$. We denote by $\sigma:\Sigma\rightarrow\Sigma$ the (left) shift map given by $\sigma(\omega)_{i}:=\omega_{i+1}$ for every $i\in{\mathbb{N}}$. We endow $\Sigma$ with the metric $d(\omega,\tau):=\exp(-\max\left\{ k\ge0\mid\omega_{1}=\tau_{1},\dots,\omega_{k}=\tau_{k}\right\} )$. In this way we obtain $\left(\Sigma,\sigma\right)$ a continuous dynamical system over the compact metric space $(\Sigma,d)$. Recall the definition of a skew-periodic interval map ${F}_{\,\Psi}$ with $\Psi:[0,1]\to{\mathbb{Z}}$ and of a skew product dynamical system ** $\left(\Sigma\times{\mathbb{R}},\sigma\rtimes f\right)$ with $f:\Sigma\to{\mathbb{R}}$, see (\[eq:definition skew product dynamical system\]).Let us now give the precise definition of the auxiliary countable set, where we set $h_{{\varnothing}}\coloneqq id|_{\left[0,1\right]}$, $$\mathcal{D}\coloneqq\bigcup_{\left(\omega_{1},\ldots,\omega_{n}\right)\in I^{\star}}h_{\omega_{1}}\circ\cdots\circ h_{\omega_{n}}\left(\left\{ 0,1\right\} \right).\label{eq:D}$$ Note that $\pi^{-1}(\mathcal{D})$ is the set of all sequences in $\Sigma$ which are eventually constant. \[fac:factor\]${F}_{\,\Psi}$ and $\left.\sigma\rtimes\left(\Psi\circ\pi\right)\right|_{\Sigma\backslash\pi^{-1}(\mathcal{D})\times{\mathbb{Z}}}$ are topological conjugate. Let us define the map ${\widetilde}{\pi}:\Sigma\times{\mathbb{Z}}\to{\mathbb{R}}$ by ${\widetilde}{\pi}\left(\omega,\ell\right)\coloneqq\pi\left(\omega\right)+\ell$ and restrict its domain to$\widetilde{\pi}^{-1}\left(\mathcal{R}\right)$ (which equals $\Sigma\backslash\pi^{-1}(\mathcal{D})\times{\mathbb{Z}}$). Then for $\left(\omega,\ell\right)\in\widetilde{\pi}^{-1}\left(\mathcal{R}\right)$ we have $${F}_{\,\Psi}({\widetilde}{\pi}\left(\omega,\ell\right))=\sum_{k\in{\mathbb{Z}}}\left(k+F(\pi\left(\omega\right)+\ell-k)+\Psi(\pi\left(\omega\right)+\ell-k)\right){\mathbbm{1}}_{[0,1]}(\pi\left(\omega\right)+\ell-k).$$ Since $F\circ\pi=\pi\circ\sigma$ on $\Sigma\setminus\pi^{-1}\left(\mathcal{D}\right)$, we have $${F}_{\,\Psi}({\widetilde}{\pi}\left(\omega,\ell\right))={F}(\pi\left(\omega\right))+\ell+\Psi(\pi\left(\omega\right))={\widetilde}{\pi}(\sigma\rtimes\left(\Psi\circ\pi\right)\left(\omega,\ell\right)).\qedhere$$ For $n\in{\mathbb{N}}$ and $\omega\in I^{\star}$, we set $$S_{n}f\coloneqq\sum_{k=0}^{n-1}f\circ\sigma^{k},\;S_{0}f\coloneqq0\quad\textnormal{as well as}\quad S_{\omega}f\coloneqq\sup_{x\in\left[\omega\right]}S_{|\omega|}f\left(x\right),\label{eq:birkhoff sum and max sum}$$ where $\left[\omega\right]\coloneqq\left\{ x\in\Sigma:x|_{\left|\omega\right|}=\omega\right\} $ denotes the *cylinder set* in $\Sigma$ over $\omega$. We say $f:\Sigma\rightarrow{\mathbb{R}}$ is *Hölder continuous* if there exists $\alpha>0$ such that $$\sup_{\omega,\tau\in\Sigma}\left|f(\omega)-f(\tau)\right|/d(\omega,\tau)^{\alpha}<\infty.$$ Note that by the Hölder continuity there exists a constant $D_{f}$ such that for all $n\in{\mathbb{N}}$ and $\omega\in I^{n}$ we have $$\sup_{x,y\in\left[\omega\right]}\left|S_{n}f\left(x\right)-S_{n}f\left(y\right)\right|\leq D_{f}.\label{eq:boundedDistortion}$$ We recall an important result from [@MR0419727] adopted to our situation. \[lem:nodrift implies recurrence\] Suppose $\mu$ is a $\sigma$-invariant ergodic Borel probability measure and $f:\Sigma\rightarrow{\mathbb{R}}$ is $\mu$-integrable. We have $$\mu(f)=0\qquad\iff\qquad\liminf_{\ell\to\infty}\left|S_{\ell}f\right|=0\quad\mu\text{-a.e.}$$ If $\mu$ is non-atomic, we apply the result from [@MR0419727]. For this we consider the natural extension of $\left(\Sigma,\sigma\right)$ which is given by the two-sided shift $I^{{\mathbb{Z}}}$ making $\sigma:I^{{\mathbb{Z}}}\to I^{{\mathbb{Z}}}$ a bi-measurable map with respect to the unique $\sigma$-invariant measure $\widetilde{\mu}$ on $I^{{\mathbb{Z}}}$ such that $\mu=\widetilde{\mu}\circ h^{-1}$. Here, $h$ denotes the canonical projection from $I^{{\mathbb{Z}}}$ to $\Sigma$. Then we can transfer the corresponding result from [@MR0419727] for $\widetilde{f}\coloneqq f\circ h$ back to our situation. If $\mu$ has atoms, then by $\sigma$-invariance and ergodicity we in fact have $\mu=n^{-1}\sum_{k=0}^{n-1}\delta_{\sigma^{k}x}$, where $x\in\Sigma$ is some periodic point with period $n$. With this at hand the equivalence is obvious. The following fact puts the previous lemma into the context of infinite ergodic theory. Let $\mu$ be a $\sigma$-invariant Borel probability measure and assume that $f:\Sigma\rightarrow{\mathbb{R}}$ is $\mu$-integrable. Then the $(\sigma\rtimes f)$-invariant measure $\mu\times\lambda$ (with $\lambda$ the Lebesgue measure on ${\mathbb{R}}$) is conservative if and only if $\mu$-a.e. $\liminf_{\ell\to\infty}\left|S_{\ell}f\right|=0$. Next, let us recall the classical thermodynamical formalism for full shifts. On the compact subspace $J^{{\mathbb{N}}}\subset\Sigma$ the *classical topological pressure* $\mathfrak{P}(f,J)$ of the continuous *potential* $f:\Sigma\rightarrow{\mathbb{R}}$ is given by $$\ \mathfrak{P}(f,J)\coloneqq\lim_{n\rightarrow\infty}\frac{1}{n}\log\sum_{\omega\in J^{n}}\exp\left(S_{\omega}f\right)\qquad\text{and }\qquad\mathfrak{P}(f)\coloneqq\mathfrak{P}(f,I),$$ in here the limit always exists (see for example [@MR648108]). If $f$ is Hölder continuous and if $J\subset I$ is not a singleton, then there exists a unique $\sigma$-invariant Borel probability measure $\mu_{f,J}$ on $J^{{\mathbb{N}}}$– called the *Gibbs measure* (for $f$ restricted to $J^{{\mathbb{N}}}$) – fulfilling the *Gibbs property*, that is, there exists $c\geq1$ such that for all $n\in{\mathbb{N}}$, $\omega\in J^{n}$ and $x\in[\omega]\cap J^{{\mathbb{N}}}$ we have $$c^{-1}\leq\frac{\mu_{f,J}\left(\left[\omega\right]\right)}{\exp(S_{n}f(x)-n\mathfrak{P}(f,J))}\leq c.\label{eq:Gibbs}$$ If $J=\left\{ i\right\} $, then $\mu_{f,J}$ denotes the Dirac measure $\delta_{x}$ on the constant sequence $x=\left(i,i,\ldots\right)$. Moreover, for $J=I$ we set $\mu_{f}\coloneqq\mu_{f,I}$. Fibre-induced Pressure and recurrence properties\[sec:Fibre-induced-Pressure\] ============================================================================== In this section we will always assume that $\psi,f:\Sigma\rightarrow{\mathbb{R}}$ are Hölder continuous functions and let $$\underline{\psi}\coloneqq\inf_{\omega\in\Sigma}\liminf_{n\rightarrow\infty}S_{n}\psi\left(\omega\right)/n\quad\text{and}\quad\overline{\psi}\coloneqq\sup_{\omega\in\Sigma}\limsup_{n\rightarrow\infty}S_{n}\psi\left(\omega\right)/n.\label{eq:Psiupperlower}$$ This gives rise to a skew product dynamical system $\sigma\rtimes\psi:\Sigma\times{\mathbb{R}}\to\Sigma\times{\mathbb{R}}$ and we aim at defining the new notion of fibre-induced pressure for functions $f\circ\pi_{1}:\Sigma\times{\mathbb{R}}\to{\mathbb{R}}$ depending only on the first coordinate where $\pi_{1}:\Sigma\times{\mathbb{R}}\rightarrow\Sigma$. Recall that $D_{\text{\ensuremath{\psi}}}$ denotes the distortion constant of $\psi$ defined in (\[eq:boundedDistortion\]). For $K>0$ and $n\in{\mathbb{N}}$ we define $$\mathcal{C}_{n}(K)\coloneqq\mathcal{C}_{n}\left(\psi,K\right)\coloneqq\left\{ \omega\in I^{n}\mid\left|S_{\omega}\psi\right|\leq K\right\} ,\;\mathcal{C}\left(K\right)\coloneqq\mathcal{C}\left(\psi,K\right)\coloneqq\bigcup_{n\in{\mathbb{N}}}\mathcal{C}_{n}\left(K\right),$$ $$\zeta_{n}(f,\psi,K)\coloneqq\sum_{\omega\in\mathcal{C}_{n}(K)}{\mathrm{e}}^{S_{\omega}f}$$ as well as $$\mathcal{P}(f,\psi,K)\coloneqq\limsup_{n\rightarrow\infty}\frac{1}{n}\log\zeta_{n}(f,\psi,K)\quad\text{and }\;\mathcal{P}(f,\psi)\coloneqq\lim_{K\to\infty}\mathcal{P}(f,\psi,K).$$ We will call $\mathcal{P}(f,\psi)$ the *fibre-induced pressure* of $f$ with respect to $\psi.$ \[Fibre-induced vs. classical pressure\] \[thm:fibre-induced pressure via base pressure\]Let $\psi,f:\Sigma\rightarrow{\mathbb{R}}$ be Hölder continuous functions and fix $K>D_{\text{\ensuremath{\psi}}}$. If $0\in\big(\underline{\psi},\overline{\psi}\big)$, then there exists a unique number $t(f)\in{\mathbb{R}}$ with $\int\psi\,\,d\mu_{t(f)\psi+f}=0$. For this number we have $$\mathcal{P}(f,\psi,K)=\mathfrak{P}(t(f)\psi+f)=\min_{s\in{\mathbb{R}}}\mathfrak{P}(s\psi+f).\label{eq:InducedPressureViaMinClassicalPressure}$$ Further, if we assume that $\psi$ is constant on one-cylinder sets, allowing also for $0\notin\big(\underline{\psi},\overline{\psi}\big)$, we have $$\mathcal{P}(f,\psi,K)=\inf_{s\in{\mathbb{R}}}\mathfrak{P}(s\psi+f)$$ and the value of the fibre-induced pressure will be finite if and only if $0\in\big[\underline{\psi},\overline{\psi}\big]$. For $0\in\{\underline{\psi},\overline{\psi}\}$ and setting $I_{0}\coloneqq\left\{ i\in I\colon\psi\left(i,\ldots\right)=0\right\} $ we have in this situation $$\mathcal{P}(f,\psi,K)=\mathfrak{P}(f,I_{0}).$$ In any case if $0\in\big(\underline{\psi},\overline{\psi}\big)$ or $\psi$ is constant on one-cylinder sets and $0\in\big[\underline{\psi},\overline{\psi}\big]$, then we have $$\sum_{\omega\in\mathcal{C}(K)}{\mathrm{e}}^{S_{\omega}f-|\omega|\mathcal{P}(f,\psi,K)}=\infty.$$ We first consider the case $0\in\big(\underline{\psi},\overline{\psi}\big)$. It is well known that the function $H:t\mapsto\mathfrak{P}(t\psi+f)$ is real-analytic and convex ([@MR0234697]). By our assumption on $\psi$, we have that $\lim_{t\rightarrow\pm\infty}H(t)=\infty$. It follows that $H$ is not affine and hence, $H$ is strictly convex. We conclude that there is a unique $t(f)\in{\mathbb{R}}$ such that $H'(t(f))=\int\psi d\mu_{t(f)\psi+f}=0$ and $H(t(f))$ must be the unique minimum of $H$. We have $\mu_{t(f)\psi+f}$-a.e. that $\liminf_{\ell\to\infty}\left|S_{\ell}\psi\right|=0$, by Lemma \[lem:nodrift implies recurrence\], and as a consequence of the Borel-Cantelli Lemma we have $\sum_{\ell\ge1}\mu_{t(f)\psi+f}\left\{ \left|S_{\ell}\psi\right|\leq\varepsilon\right\} =\infty$ for every $\varepsilon>0$. This fact combined with the Gibbs property (\[eq:Gibbs\]) implies for $\epsilon\coloneqq K-D_{\text{\ensuremath{\psi}}}>0$ and $c\geq1$ from (\[eq:Gibbs\]), $$\begin{aligned} \infty=\sum_{\ell\ge1}\mu_{t(f)\psi+f}\left\{ \left|S_{\ell}\psi\right|\leq\varepsilon\right\} & \leq c\sum_{\omega\in\mathcal{C}(K)}{\mathrm{e}}^{S_{\omega}\left(f+t\left(f\right)\psi\right)-|\omega|\mathfrak{P}(t(f)\psi+f)}\\ & \leq c{\mathrm{e}}^{|t(f)|K}\sum_{\omega\in\mathcal{C}(K)}{\mathrm{e}}^{S_{\omega}f-|\omega|\mathfrak{P}(t(f)\psi+f)}.\end{aligned}$$ From this it is easy to see that $\mathcal{P}(f,\psi,K)-\mathfrak{P}(t(f)\psi+f)\ge0$. Combining this with the obvious estimate $\mathcal{P}(f,\psi,K)\le\mathfrak{P}(s\psi+f)$, for any $s\in{\mathbb{R}}$, completes the proof for the case $0\in\big(\underline{\psi},\overline{\psi}\big)$. As seen above in any case we have $\mathcal{P}(f,\psi,K)\le\inf_{s\in{\mathbb{R}}}\mathfrak{P}(s\psi+f)$. Hence, for the boundary cases $0\in\big\{\underline{\psi},\overline{\psi}\big\}$ we are left to verify that $\mathcal{P}(f,\psi,K)\geq\inf_{s\in{\mathbb{R}}}\mathfrak{P}(s\psi+f)$. Let us only consider the case $\overline{\psi}=0$ and $\psi$ is constant on one-cylinders. Then $\inf_{s\in{\mathbb{R}}}\mathfrak{P}(s\psi+f)=\lim_{s\to+\infty}\mathfrak{P}(s\psi+f)$. Let us now consider the ergodic invariant Gibbs measure, respectively atomic measure, $\mu_{f,I_{0}}$. Since $\mu_{f,I_{0}}\left(\psi\right)=0$ and by applying Lemma \[lem:nodrift implies recurrence\], we obtain for $c\geq1$ from (\[eq:Gibbs\]), $$\begin{aligned} \infty=\sum_{\ell\ge1}\mu_{f,I_{0}}\left\{ \left|S_{\ell}\psi\right|\leq K\right\} & =\adjustlimits\sum_{\ell\ge1}\sum_{\omega\in\mathcal{C}_{\ell}(K)\cap I_{0}^{\ell}}\mu_{f,I_{0}}\left(\left[\omega\right]\right)\\ & \leq c\adjustlimits\sum_{\ell\ge1}\sum_{\omega\in\mathcal{C}_{\ell}(K)\cap I_{0}^{\ell}}{\mathrm{e}}^{S_{\omega}f-|\omega|\mathfrak{P}(f,I_{0})}\\ & \leq c\adjustlimits\sum_{\ell\ge1}\sum_{\omega\in\mathcal{C}_{\ell}(K)}{\mathrm{e}}^{S_{\omega}f-|\omega|\mathfrak{P}(f,I_{0})}.\end{aligned}$$ From this it is easy to see that $\mathcal{P}(f,\psi,K)-\mathfrak{P}(f,I_{0})\ge0$. To see that $\inf_{s\in{\mathbb{R}}}\mathfrak{P}(s\psi+f)=\mathfrak{P}(f,I_{0})$ we first note that obviously $\mathfrak{P}(s\psi+f)\geq\mathfrak{P}(f,I_{0})$ for any $s\in{\mathbb{R}}$. By the sub-additivity of the classical pressure, we have for any $n\in{\mathbb{N}}$ and $s>0$ that $$\mathfrak{P}(s\psi+f)\leq\frac{1}{n}\log\sum_{\omega\in I^{n}}{\mathrm{e}}^{S_{\omega}\left(s\psi+f\right)}\to\frac{1}{n}\log\sum_{\omega\in I_{0}^{n}}{\mathrm{e}}^{S_{\omega}f}\;\text{for }s\to\infty.$$ Hence, $$\inf_{s\in{\mathbb{R}}}\mathfrak{P}(s\psi+f)\leq\frac{1}{n}\log\sum_{\omega\in I_{0}^{n}}{\mathrm{e}}^{S_{\omega}f}\to\mathfrak{P}(f,I_{0})\quad\text{for }n\to\infty.$$ In particular, we have $\mathcal{P}(f,\psi,K)>-\infty$ . Finally, if $0\notin\big[\underline{\psi},\overline{\psi}\big]$, then $\inf_{s\in{\mathbb{R}}}\mathfrak{P}(s\psi+f)=-\infty$ and by the obvious estimate $\mathcal{P}(f,\psi,K)=-\infty=\inf_{s\in{\mathbb{R}}}\mathfrak{P}(s\psi+f)$. If $0\in\big(\underline{\psi},\overline{\psi}\big)$ or if $\psi$ is constant on one-cylinder sets, then the fibre-induced pressure is given by $$\mathcal{P}(f,\psi)=\mathcal{P}(f,\psi,K),$$ for $K>D_{\text{\ensuremath{\psi}}}$. We can now characterise when the fibre-induced pressure and classical pressure coincide. \[cor:full pressure if no drift\] Let $f:\Sigma\rightarrow{\mathbb{R}}$ and $\psi:\Sigma\rightarrow{\mathbb{R}}$ be Hölder continuous functions. Then we have $$\mathcal{P}(f,\psi)=\mathfrak{P}(f)\quad\text{if and only if}\quad\mu_{f}(\psi)=0.$$ First, assume $\mu_{f}(\psi)=0$. With $H$ defined as in the proof of Theorem \[thm:fibre-induced pressure via base pressure\], we have $H'\left(0\right)=0$. We distinguish two cases. If $H$ is strictly convex, then $0\in\big(\underline{\psi},\overline{\psi}\big)$ and we have $t(f)=0$. Thus, by Theorem \[thm:fibre-induced pressure via base pressure\], $\mathcal{P}(f,\psi)=\mathfrak{P}(f)$. If $H$ is affine, then $\psi$ is cohomologous to zero. Hence, also in this case, we have $\mathcal{P}(f,\psi)=\mathfrak{P}(f)$. Now, suppose $\mu_{f}(\psi)\neq0$. Again by the proof of Theorem \[thm:fibre-induced pressure via base pressure\], $\mathcal{P}(f,\psi)\le\inf_{s\in{\mathbb{R}}}\mathfrak{P}(s\psi+f)$. Since $\frac{\partial}{\partial t}\mathfrak{P}(t\psi+f)_{|t=0}=\mu_{f}(\psi)\neq0$, we conclude $\mathcal{P}(f,\psi)<\mathfrak{P}(f)$. For later use we need the following auxiliary statement. \[lem:strictly decreasing\]If $f<0$ and $0\in[\underline{\psi},\overline{\psi}]$, then $s\mapsto\mathcal{P}(sf,\psi)$ is finite, continuous, strictly decreasing and we have $$\lim_{s\to\pm\infty}\mathcal{P}(sf,\psi)=\mp\infty.$$ By definition, we have for $K>D_{\text{\ensuremath{\psi}}}$ and $s<t$, $$\mathcal{P}(tf,\psi,K)=\limsup_{n\rightarrow\infty}\frac{1}{n}\log\sum_{\omega\in\mathcal{C}_{n}\left(K\right)}{\mathrm{e}}^{\left((t-s)+s\right)S_{\omega}f}\leq(t-s)\max f+\mathcal{P}(sf,\psi,K).$$ The claim follows because $\mathcal{P}(sf,\psi,K)\in{\mathbb{R}}$, $s\in{\mathbb{R}}$, which follows from the arguments given in the proof of Theorem \[thm:fibre-induced pressure via base pressure\]. \[rem:Real-Analytic\] For $f,\psi$ Hölder continuous functions, due to the correspondence between the fibre-induced and the classical pressure as stated in Theorem \[thm:fibre-induced pressure via base pressure\], we obtain that $\left(s,a\right)\mapsto\mathcal{P}\left(sf,\psi-a\right)$ is real-analytic with respect to $s\in{\mathbb{R}}$ and $a\in(\underline{\psi},\overline{\psi})$. We end this section introducing the new notion of $\psi$-recurrent potentials. As already mentioned in the introduction this clarifies the connection of our setting to the ideas developed in [@MR1738951] and [@MR3436756]. Let $f,\psi:\Sigma\rightarrow{\mathbb{R}}$ be Hölder continuous. We say that $f$ is a *$\psi$-recurrent potential* if for all $K>D_{\psi}$ $$\sum_{\omega\in\mathcal{C}\left(K\right)}{\mathrm{e}}^{S_{\omega}f-|\omega|\mathcal{P}\left(f,\psi\right)}=\infty.$$ The following corollary is an immediate consequence of Theorem \[thm:fibre-induced pressure via base pressure\]. \[cor:REcurrent\] Let $\psi:\Sigma\rightarrow{\mathbb{R}}$ be a Hölder continuous function with $\underline{\psi}<0<\overline{\psi}$ or $\psi$ is constant on one-cylinders. Then any Hölder continuous $f:\Sigma\rightarrow{\mathbb{R}}$ is a $\psi$-recurrent potential. \[rem:original-gurevich\] We would like to point out that our definition of recurrent potentials is analogous to the corresponding definition for topological Markov chains ([@MR1738951]). For this suppose that $\psi:\Sigma\rightarrow{\mathbb{R}}$ is constant on $1$-cylinders such that $\underline{\psi}=\min\psi<0<\max\psi=\overline{\psi}$. Then we have $D_{\psi}=0$ and $\mathcal{P}(f,\psi)=\mathcal{P}(f,\psi,K)$ for every $K>0$. Moreover, $G:=\left\{ S_{\omega}\psi\mid\omega\in I^{\star}\right\} $ is a countable semi-group and if $G$ is a discrete subset of ${\mathbb{R}}$, by Lemma \[lem:D\] below, $G$ is a countable group. Further, $\sigma\rtimes\psi:\Sigma\times G\rightarrow\Sigma\times G$ is a transitive topological Markov chain with alphabet $I\times G$. Now, Sarig’s definition in [@MR1738951] of the Gurevich pressure of $f\circ\pi_{1}$ with respect to this Markov chain coincides with our fibre-induced pressure and Sarig’s definition of a recurrent potential also coincides with ours (see also [@MR2551790; @MR2861747; @MR3922537]). For the closely related concept of induced pressure for Markov shifts we refer to [@MR3190211]. \[lem:D\]Assume that $\psi:\Sigma\rightarrow{\mathbb{R}}$ is constant on one-cylinder sets and $0\in\big(\underline{\psi},\overline{\psi}\big)$. Then for any $N>0$ and for every $\epsilon>0$ there is a finite set $\Lambda\subset I^{\star}$ such that for any $\omega\in\mathcal{C}\left(N\right)$ there exists $\nu\in\Lambda$ such that $\omega\nu\in\mathcal{C}\left(\epsilon\right)$. If $0\in\big(\underline{\psi},\overline{\psi}\big)$, the map $t\mapsto\mathfrak{P}\left(t\psi\right)$ is real-analytic, strictly convex and we have $\lim_{t\to\pm\infty}\mathfrak{P}\left(t\psi\right)=\infty$. Hence, it has a unique minimum in $t_{0}\in{\mathbb{R}}$. Then for the unique Gibbs measure $\mu_{t_{0}\psi}$ with respect to the potential $t_{0}\psi$ we get $$\mu_{t_{0}\psi}\left(\psi\right)=\frac{d}{dt}\mathfrak{P}\left(t\psi\right)|_{t=t_{0}}=0$$ and this Gibbs measure is also non-atomic and ergodic with respect to $\sigma$. By Lemma \[lem:nodrift implies recurrence\], we obtain $$\liminf_{\ell\to\infty}\left|S_{\ell}\psi\right|=0\qquad\mu_{t_{0}\psi}\text{-a.e.}\label{eq:recurrence-1}$$ For $k=1,\ldots,m\coloneqq\left\lceil 2N/\epsilon\right\rceil $ we set $$I_{k}\coloneqq\left(\left(k-1\right)\epsilon/2,k\epsilon/2\right]\quad\text{and}\quad I_{-k}\coloneqq\left[-k\epsilon/2,\left(-k+1\right)\epsilon/2\right).$$ We can ignore all $k\in\left\{ \pm1,\ldots,\pm m\right\} $ such that $S_{\omega}\psi\notin I_{k}$ for all $\omega\in I^{\star}$. For the remaining $k$ we find $\omega_{k}\in I^{\star}$ such that $S_{\omega_{k}}\psi\in I_{k}$. Since every non-empty cylinder $[\omega]$ with $\omega\in I^{\star}$ carriespositive $\mu_{t_{0}\psi}$-measure, using (\[eq:recurrence-1\]) we find $x\in\left[\omega_{k}\right]$ and $n\geq\left|\omega_{k}\right|$ such that $\left|S_{n}\psi\left(x\right)\right|<\epsilon/2$. With $\nu_{k}\coloneqq\left(x_{\left|\omega_{k}\right|+1},\ldots,x_{n}\right)$ we have $\left|S_{\omega_{k}\nu_{k}}\psi\right|<\epsilon/2$ applying (\[eq:boundedDistortion\]). Hence using (\[eq:boundedDistortion\]) once more, for an arbitrary $\omega\in I^{\star}$ with $S_{\omega}\psi\in I_{k}$, we have for all $x\in\left[\omega\nu_{k}\right]$ $$\begin{aligned} \left|S_{\left|\omega\nu_{k}\right|}\psi\left(x\right)\right| & =\left|S_{\left|\omega\nu_{k}\right|}\psi\left(x\right)-S_{\left|\omega_{k}\nu_{k}\right|}\psi\left(\omega_{k}\sigma^{\left|\omega\right|}\left(x\right)\right)+S_{\left|\omega_{k}\nu_{k}\right|}\psi\left(\omega_{k}\sigma^{\left|\omega\right|}\left(x\right)\right)\right|\\ & \leq\left|S_{\left|\omega\nu_{k}\right|}\psi\left(x\right)-S_{\left|\omega_{k}\nu_{k}\right|}\psi\left(\omega_{k}\sigma^{\left|\omega\right|}\left(x\right)\right)\right|+\left|S_{\left|\omega_{k}\nu_{k}\right|}\psi\left(\omega_{k}\sigma^{\left|\omega\right|}\left(x\right)\right)\right|,\\ & \leq\left|S_{\left|\omega\right|}\psi\left(x\right)-S_{\left|\omega_{k}\right|}\psi\left(\omega_{k}\sigma^{\left|\omega\right|}\left(x\right)\right)\right|+\epsilon/2\\ & \leq\epsilon/2+\epsilon/2=\epsilon\end{aligned}$$ proving the desired statement. \[rem:characterisation of recurrence\]For a subgroup $G<{\mathbb{Z}}$ and $\psi:\Sigma\rightarrow G$ it is shown in [@MR3436756] that $f\circ\pi_{1}$ is recurrent (according to Sarig, see [@MR1738951]) if and only if there is a character $c:G\rightarrow(0,\infty)$ such that $\mu_{f-\log c\circ\psi}\times\lambda_{G}$ is the conservative equilibrium measure of $f\circ\pi_{1}$, where $\lambda_{G}$ denotes the counting measure on $G$ (for the definition of an equilibrium measure in this setting, see [@MR1818392]). If $\psi$ is constant on one-cylinders and $0\in\big(\underline{\psi},\overline{\psi}\big)$, then $f-\log c\circ\psi$ is of the form $f+t(f)\psi$ as stated in Theorem \[thm:fibre-induced pressure via base pressure\]. Further, we can derive (\[eq:InducedPressureViaMinClassicalPressure\]) from [@MR3436756 Theorem 1(1)] where we note again that the fibre-induced and Gurevich pressure coincide in this setting, see the previous remark. Moreover, by [@MR3436756 Proposition 1.4] we have that $\mathcal{P}(f,\psi)=\mathfrak{P}(f)$ if and only if $f$ is *symmetric on average*, i.e. $$\sup_{m\in{\mathbb{Z}}}\limsup_{n\rightarrow\infty}\frac{\sum_{\left|\omega\right|\le n,S_{\omega}\psi=m}{\mathrm{e}}^{S_{\omega}f-\left|\omega\right|\mathcal{P}\left(f,\psi\right)}}{\sum_{\left|\omega\right|\le n,S_{\omega}\psi=-m}{\mathrm{e}}^{S_{\omega}f-\left|\omega\right|\mathcal{P}\left(f,\psi\right)}}<\infty.$$ By Corollary \[cor:full pressure if no drift\] this condition is in fact equivalent to $\mu_{f}(\psi)=0$. Multifractal decomposition with respect to $\alpha$-escaping sets\[sec:Multifractal-decomposition\] =================================================================================================== In this section $\psi:\Sigma\to{\mathbb{Z}}$ denotes the symbolic step length function which is constant on one-cylinder sets and such that $\psi=\Psi\circ\pi$ except possibly on a finite set. Note that under these assumption on $\psi$ we have $D_{\psi}=0$. For a given drift parameter $\alpha\in\big[\underline{\psi},\overline{\psi}\big]$ let us set $$\psi_{\alpha}:=\psi-\alpha.$$ Note that we have $$\begin{aligned} \mathbf{E}(\alpha) & \cap\left[0,1\right]=\pi\left\{ \omega\in\Sigma\mid\exists K>0\;\left|S_{n}\psi_{\alpha}(\omega)\right|\leq K\;\text{for infinitely many }n\in{\mathbb{N}}\right\} \setminus\mathcal{D}\end{aligned}$$ and $$\begin{aligned} \mathbf{E}_{u}(\alpha)\cap\left[0,1\right] & =\pi\left\{ \omega\in\Sigma\mid\exists K>0\;\forall n\in{\mathbb{N}}\;\left|S_{n}\psi_{\alpha}(\omega)\right|\leq K\right\} \setminus\mathcal{D},\end{aligned}$$ for the countable set $\mathcal{D}$, see (\[eq:D\]). Recall the definition of the $\alpha$-Poincaré exponent $\delta_{\alpha}$ with $K>{D_{\psi}}$, see (\[eq:definition Poincare exponent\]). Next we give the proof of the implication (\[eq:delta\_alpha=00003D0\]) stated in the introduction. \[Proof of implication (\[eq:delta\_alpha=00003D0\])\] For $\alpha\in{\mathbb{R}}\setminus\big[\underline{\psi},\overline{\psi}\big]$ we have that $S_{n}\psi_{\alpha}$ diverges uniformly to either $+\infty$ or $-\infty$ and hence, being a finite sum, $\sum_{{\omega\in I^{\star}},\,{\left|S_{\omega}\left(\psi-\alpha\right)\right|\leq K}}{\mathrm{e}}^{s\cdot S_{\omega}\varphi}$ is finite for all $s\geq0$. The following assertion will be crucial for determining the multifractal decomposition with respect to our escaping sets, and to derive explicit formulas for dimension gaps. \[thm:critical exponent via base pressures\] We have the following three different characterisations of the $\alpha$-Poincaré exponent $\delta_{\alpha}$: 1. \[enu:-delta\_Alpha\_Pressure\_new\] For $\alpha\in\big[\underline{\psi},\overline{\psi}\big]$ we have that $\delta_{\alpha}$ is uniquely determined by $${\displaystyle \mathcal{P}\left(\delta_{\alpha}\varphi,\psi_{\alpha}\right)=0.}$$ 2. \[enu:There-exist-a(ii)\]For $\alpha\in\big(\underline{\psi},\overline{\psi}\big)$ we have that $\delta_{\alpha}$ is determined by the unique solution $\left(\delta_{\alpha},q_{\alpha}\right)\in{\mathbb{R}}^{2}$ of the equations $${\displaystyle \mathfrak{P}\left(\delta_{\alpha}\varphi+q_{\alpha}\psi_{\alpha}\right)=0\quad\text{and}\quad\frac{\partial}{\partial q}\mathfrak{P}(\delta_{\alpha}\varphi+q\psi_{\alpha})|_{q=q_{\alpha}}=0}.$$ In particular, we have $\int\psi\,d\mu_{\delta_{\alpha}\varphi+q_{\alpha}\psi_{\alpha}}=\alpha$. 3. \[enu:delta\_alpha via MF-Formalism\]Let $s:{\mathbb{R}}\times{\mathbb{R}}\to{\mathbb{R}}$ be the implicitly defined function given by $\mathfrak{P}\left(s\left(q,a\right)\varphi+q\psi+a{\mathbbm{1}}\right)=0$. Then $s$ is convex and for the *Legendre transform* of $s$ defined by $$\widehat{s}\left(\alpha_{1},\alpha_{2}\right)\coloneqq\sup_{(q,a)\in{\mathbb{R}}^{2}}q\alpha_{1}+a\alpha_{2}-s\left(q,a\right)$$ we have for $\alpha\in\big(\underline{\psi},\overline{\psi}\big)$ $$\delta_{\alpha}=-\inf_{r\in{\mathbb{R}}}\widehat{s}\left(\alpha r,r\right)=\inf_{r\in{\mathbb{R}}}s\left(r,-\alpha r\right).$$ In particular, we have that on the line $g$ through the origin with slope $-\alpha$ in the $\left(q,a\right)$-plane there exists exactly one point $\left(r,-\alpha r\right)$ such that the gradient of $s$ in this point is perpendicular to $g$. The height of $s$ in $\left(r,-\alpha r\right)$ is $\delta_{\alpha}$ and the plane which is tangential to the graph of $s$ in $\left(r,-\alpha r,s\left(r,-\alpha r\right)\right)$ intersects the $s$-axis in $-\delta_{\alpha}$ (cf. Figure \[fig:Graph-of-1\]). ![\[fig:Graph-of-1\]The contour plot for the random walk model with $c_{1}=0.1$, $c_{2}=0.5$ of $s\left(q,a\right)$ which is implicitly defined by $\mathfrak{P}\left(s\left(q,a\right)\varphi+q\psi+a{\mathbbm{1}}\right)=0$. The fourth contour line is determined by $s\left(q,a(q)\right)=1$, i.e. corresponds to height $\delta$ (defined in (\[eq:Hausdorff dim Bowen formula\])). The line $g$ goes through the origin with slope $-\alpha$.](ContourPlot){width="50.00000%"} \(i) To prove that the $\alpha$-Poincaré exponent coincides with the zero of the fibre-induced pressure first note that for all $\alpha\in\big[\underline{\psi},\overline{\psi}\big]$ we have that there is a unique zero ${\widetilde}{\delta}_{\alpha}\in{\mathbb{R}}$ of the function $s\mapsto\mathcal{P}\left(s\varphi,\psi_{\alpha}\right)$, by Lemma \[lem:strictly decreasing\]. By definition, we have $\mathcal{P}\left(s\varphi,\psi_{\alpha}\right)=\limsup_{n\to\infty}n^{-1}\log\sum_{{\omega\in\mathcal{C}_{n}\left(\psi_{\alpha},K\right)}}{\mathrm{e}}^{sS_{\omega}\varphi}<0$ for $s>{\widetilde}{\delta}_{\alpha}$ and $K>0$. Hence, $$\sum_{{\omega\in I^{\star}},\,{\left|S_{\omega}\left(\psi-\alpha\right)\right|\leq K}}{\mathrm{e}}^{sS_{\omega}\varphi}=\sum_{n\in{\mathbb{N}}}\sum_{{\omega\in\mathcal{C}_{n}\left(\psi_{\alpha},K\right)}}{\mathrm{e}}^{sS_{\omega}\varphi}=\sum_{n\in{\mathbb{N}}}\exp\left(n\cdot\frac{1}{n}\log\sum_{{\omega\in\mathcal{C}_{n}\left(\psi_{\alpha},K\right)}}{\mathrm{e}}^{sS_{\omega}\varphi}\right)<\infty$$ implying $s\geq\delta_{\alpha}$, and by the arbitrariness of $s>{\widetilde}{\delta}_{\alpha}$ we have ${\widetilde}{\delta}_{\alpha}\geq\delta_{\alpha}.$ For the reverse inequality assume $s<{\widetilde}{\delta}_{\alpha}$ . Then $\mathcal{P}\left(s\varphi,\psi_{\alpha}\right)>0$. Again by definition of the pressure, there exists a sequence $n_{j}\nearrow\infty$ such that $n_{j}^{-1}\log\sum_{{\omega\in\mathcal{C}_{n_{j}}\left(\psi_{\alpha},K\right)}}{\mathrm{e}}^{sS_{\omega}\varphi}>0$ for all $j\in{\mathbb{N}}$. Hence, $$\sum_{{\omega\in I^{\star}},\,{\left|S_{\omega}\left(\psi-\alpha\right)\right|\leq K}}{\mathrm{e}}^{sS_{\omega}\varphi}=\sum_{n\in{\mathbb{N}}}\sum_{{\omega\in\mathcal{C}_{n}\left(\psi_{\alpha},K\right)}}{\mathrm{e}}^{sS_{\omega}\varphi}\geq\sum_{j\in{\mathbb{N}}}\exp\left(0\right)=\infty.$$ This gives $s\leq\delta_{\alpha}$. Since this holds for every $s<{\widetilde}{\delta}_{\alpha}$, we conclude ${\widetilde}{\delta}_{\alpha}\leq\delta_{\alpha}$. \(ii) By the Hölder continuity of $\varphi$ and $\psi$ we have that the function $p:(s,t)\mapsto\text{\ensuremath{\mathfrak{P}}}\left(s\varphi+t\psi_{\alpha}\right)$ is convex and real-analytic with respect to both coordinates. For fixed $q\in{\mathbb{R}}$ the function $s\mapsto p(s,q)$ is strictly monotonically decreasing from $+\infty$ to $-\infty$, hence there is a unique number $s\left(q\right)$ such that $p\left(s\left(q\right),q\right)=0$. Also the function $s:q\mapsto s\left(q\right)$ is real-analytic and convex by the Implicit Function Theorem and is tending to infinity for $q\to\pm\infty$. Hence, there is a unique number $q_{\alpha}\in{\mathbb{R}}$ minimising $s(q)$, or equivalently, such that $s'\left(q_{\alpha}\right)=0=-\frac{\partial}{\partial q}p\left(s(q_{\alpha}),q_{\alpha}\right)=\int\psi_{\alpha}\,d\mu_{s(q_{\alpha})\varphi+q_{\alpha}\psi_{\alpha}}$. To see that $\delta_{\alpha}$ in fact coincides with $s\left(q_{\alpha}\right)$ we combine this observation, our first alternative characterisation of $\delta_{\alpha}$ and Theorem \[thm:fibre-induced pressure via base pressure\] (with $f=s\left(q_{\alpha}\right)\varphi$) to get $$\mathcal{P}(s\left(q_{\alpha}\right)\varphi,\psi_{\alpha})=\mathfrak{P}\left(s\left(q_{\alpha}\right)\varphi+q_{\alpha}\psi_{\alpha}\right)=0.$$ Obviously, by the definition of $\psi_{\alpha}$ we also find $\int\psi\,d\mu_{s(q_{\alpha})\varphi+q_{\alpha}\psi_{\alpha}}=\alpha$. \(iii) For the last alternative characterisation we observe that the contour lines of $s$ of height $d$ can be given explicitly by $$a_{d}\left(q\right)\coloneqq-\mathfrak{P}\left(d\varphi+q\psi\right)\quad\text{with }a_{d}'\left(q\right)=-\int\psi\,d\mu_{d\varphi+q\psi}.$$ Given $\alpha\in\big(\underline{\psi},\overline{\psi}\big)$, we find that for $d=\delta_{\alpha}$ and $q=q_{\alpha}$ we have by (ii) that the line through the origin with slope $-\alpha$ is tangential to the contour line $a_{\delta_{\alpha}}$ of height $\delta_{\alpha}$ in the point $\left(q_{\alpha},-\alpha q_{\alpha}\right)$ (see Figure \[fig:Graph-of-1\]). This shows that $$\delta_{\alpha}=\inf_{r\in{\mathbb{R}}}s\left(r,-\alpha r\right).$$ Now, the gradient of $s$ in $\left(q_{\alpha},-\alpha q_{\alpha}\right)$ is $\left(\alpha t_{\alpha},t_{\alpha}\right)$ with $t_{\alpha}\coloneqq1/\int\varphi\,d\mu_{\delta_{\alpha}\varphi+q_{\alpha}\psi_{\alpha}}$ and hence, for all $t\in{\mathbb{R}}$ $$s\left(q_{\alpha},-\alpha q_{\alpha}\right)\geq-\widehat{s}\left(\alpha t,t\right),$$ with equality if $t=t_{\alpha}$ (see e.g. [@rockafellar1970 Theorem 23.5]). This proves $\delta_{\alpha}=\sup_{t\in{\mathbb{R}}}-\widehat{s}\left(\alpha t,t\right)$. \[rem:delta\_=00005Calpha as Birkhoff average\]As a consequence of Remark \[rem:Real-Analytic\], the fact that $\varphi<0$, and the Implicit Function Theorem, we have that the mapping $\alpha\mapsto\delta_{\alpha}$ is real-analytic on $\big(\underline{\psi},\overline{\psi}\big)$. Also note that the characterisations in Theorem \[thm:critical exponent via base pressures\] in particular show that we have the following multifractal spectral identity $$\delta_{\alpha}=\dim_{H}\left\{ \pi\left(\omega\right)\colon\omega\in\Sigma,\,\lim_{n\rightarrow\infty}\frac{S_{n}(\psi\circ\pi)\left(\omega\right)}{n}=\alpha\right\} .$$ See also [@MR1858487; @MR1837214; @MR1916371; @MR3286501; @JordanRams2017] and Remark \[rem:Multifractal\_Alternative\_Proof\]. We are now in the position to prove Theorems \[thm:-dimension gap\] and \[thm: Multifractal Decomposition\] stated in the introduction. \[Proof of Theorem \[thm:-dimension gap\]\]By definition of $\delta$ and Theorem \[thm:critical exponent via base pressures\], we have $\delta_{\alpha}=\delta>0$ if and only if $\mathfrak{P}(\delta\varphi)=0=\mathcal{P}\left(\delta\varphi,\psi_{\alpha}\right)$. Hence by Corollary \[cor:full pressure if no drift\], this holds if and only if $\mu_{\delta\varphi}(\psi_{\alpha})=0$ which is equivalent to $\mu_{\delta\varphi}\left(\psi\right)=\alpha$. Before we proceed with the proof of Theorem \[thm: Multifractal Decomposition\], let us fix some further properties of $\alpha\mapsto\delta_{\alpha}$ in the following proposition. \[prop:Properties of delta\_alpha\]If $\underline{\psi}<\overline{\psi}$, then the function $\alpha\mapsto\delta_{\alpha}$ given by the $\alpha$-Poincaré exponent is strictly increasing on $\left(\underline{\psi},\alpha_{\text{max}}\right)$ and strictly decreasing on $\left(\alpha_{\text{max}},\overline{\psi}\right)$ and hence has a unique maximum in $\alpha_{\text{max }}\coloneqq\mu_{\delta\varphi}\left(\psi\right)$ with value $\delta$. Further, we have $$\frac{d}{d\alpha}\delta_{\alpha}=\frac{q_{\alpha}}{\mu_{\delta_{\alpha}\varphi+q_{\alpha}\psi_{\alpha}}\left(\varphi\right)}$$ and in particular, $$\lim_{\alpha\to\overline{\psi}}\frac{d}{d\alpha}\delta_{\alpha}=-\infty\quad\text{and}\quad\lim_{\alpha\to\underline{\psi}}\frac{d}{d\alpha}\delta_{\alpha}=+\infty.$$ First, for $\alpha\in\big(\underline{\psi},\overline{\psi}\big)$, we check that with $\mu=\mu_{\delta_{\alpha}\varphi+q_{\alpha}\psi_{\alpha}}$, $$\frac{\partial}{\partial\alpha}\left(\mathfrak{P}(\delta_{\alpha}\varphi+q_{\alpha}\psi_{\alpha})\right)=\frac{d}{d\alpha}\delta_{\alpha}\int\varphi d\mu+\frac{d}{d\alpha}q_{\alpha}\int\psi d\mu-\frac{d}{d\alpha}q_{\alpha}\alpha-q_{\alpha}.$$ By Theorem \[thm:critical exponent via base pressures\], we conclude that $\frac{\partial}{\partial\alpha}\left(\mathfrak{P}(\delta_{\alpha}\varphi+q_{\alpha}\psi_{\alpha})\right)=0$. Hence, $q_{\alpha}=\frac{d}{d\alpha}\delta_{\alpha}\int\varphi d\mu$. We are left to prove the monotonicity. We find that the contour lines of $s$ given by the function $q\mapsto a_{c}\left(q\right)$ implicitly via $s\left(q,a_{c}(q)\right)=c$ intersect the $a$-axis in $-\mathfrak{P}\left(c\varphi\right)$ for all $c\in{\mathbb{R}}$. Since $a_{\delta}\left(0\right)=0$, $a_{\delta}'\left(0\right)=-\alpha_{\text{max}}$ and $-\mathfrak{P}\left(c\varphi\right)<0$ for $c<\delta$, we find – using the characterisation of $\delta_{\alpha}$ from Theorem \[thm:critical exponent via base pressures\] (iii) – that $q_{\alpha}<0$ for $\alpha\in\left(\underline{\psi},\alpha_{\text{max}}\right)$ and $q_{\alpha}>0$ for $\alpha\in\left(\alpha_{\text{max}},\overline{\psi}\right)$. \[Proof of Theorem \[thm: Multifractal Decomposition\]\] For the upper bound we use a straight forward covering argument and the definition of the $\alpha$-Poincaré exponent. The lower bound uses the fibre-induced pressure and a concrete construction of a Frostman measure inspired by [@MR1484767]. Let $$L_{n}(\alpha,k):=\bigcup_{\omega\in\mathcal{C}_{n}\left(\psi_{\alpha},k\right)}\pi\left[\omega\right],$$ for $k\in{\mathbb{N}}$ and set $$L(\alpha,k)\coloneqq\limsup_{n\rightarrow\infty}L_{n}(\alpha,k)=\bigcap_{n\in{\mathbb{N}}}\bigcup_{m\geq n}L_{m}(\alpha,k).$$ Clearly, $\mathbf{E}\left(\alpha\right)\subset\bigcup_{k\in{\mathbb{N}}}L(\alpha,k)$. By the stability of the Hausdorff dimension it suffices to prove that $\dim_{H}\left(L(\alpha,k)\right)\le\delta_{\alpha}$ for every $k\in{\mathbb{N}}$. For every $s>\delta_{\alpha}$, we have by the definition of the Poincaré exponent for any $N\in{\mathbb{N}}$ $$\sum_{n\geq N}\sum_{\omega\in\mathcal{C}_{n}\left(\psi_{\alpha},k\right)}\left|\pi\left[\omega\right]\right|^{s}\leq\sum_{n\geq N}\sum_{\omega\in\mathcal{C}_{n}\left(\psi_{\alpha},k\right)}{\mathrm{e}}^{S_{\omega}s\varphi}\leq\sum_{\omega\in\mathcal{C}\left(\psi_{\alpha},k\right)}{\mathrm{e}}^{S_{\omega}s\varphi}<\infty.$$ This shows that the $s$-dimensional Hausdorff measure of $L\left(\alpha,k\right)$ is finite and hence $\dim_{H}(L(\alpha,k))\le s$ for every $s>\delta_{\alpha}$ giving the upper bound. For the lower bound we start with the case $\alpha\in\big\{\underline{\psi},\overline{\psi}\big\}$. For $I_{0}\coloneqq\left\{ i\in I\colon\psi_{\alpha}\left(i,\ldots\right)=0\right\} $ we clearly have $\pi\left(I_{0}^{{\mathbb{N}}}\right)\subset\mathbf{E}_{u}\left(\alpha\right)$. By Theorem \[thm:fibre-induced pressure via base pressure\], we have $0=\mathcal{P}\left(\delta_{\alpha}\varphi,\psi_{\alpha}\right)=\mathfrak{P}(\delta_{\alpha}\varphi,I_{0})$ and hence by Bowen’s formula $\dim_{H}\left(\pi\left(I_{0}^{{\mathbb{N}}}\right)\right)=\delta_{\alpha}$ providing the lower bound for this case. For $\alpha\in\big(\underline{\psi},\overline{\psi}\big)$ we consideronly the case $\delta_{\alpha}>0$. It then suffices to construct for every positive $s<\delta_{\alpha}$ a subset $C\subset\mathbf{E}_{u}\left(\alpha\right)$ such that $\dim_{H}C>s$. Fix such positive $s<\delta_{\alpha}$ and let $\epsilon>0$. Then $\mathcal{P}\left(s\varphi,\psi_{\alpha}\right)>0$ by Theorem \[thm:critical exponent via base pressures\] (i). Hence, for $\epsilon>0$, $$\limsup_{n\to\infty}\sum_{\omega\in\mathcal{C}_{n}\left(\psi_{\alpha},\epsilon\right)}{\mathrm{e}}^{sS_{\omega}\varphi}=\infty.$$ Hence, for arbitrarily large $M>0$ we find $\ell\in{\mathbb{N}}$ with $\sum_{\omega\in\mathcal{C}_{\ell}\left(\psi_{\alpha},\epsilon\right)}{\mathrm{e}}^{sS_{\omega}\varphi}>M$. Set $\Gamma:=\mathcal{C}_{\ell}\left(\psi_{\alpha},\epsilon\right)$. Further, let $\Lambda\subset I^{\star}$ denote the finite set of words of length less than $r\in{\mathbb{N}}$ witnessing the conditions stated in Lemma \[lem:D\] for $N=2\epsilon$. Set $\Gamma_{1}\coloneqq\Gamma$. Then for every $\omega\in\Gamma_{1}$ we have $\left|S_{\omega}\psi_{\alpha}\right|\leq\epsilon$. Suppose we have defined $\Gamma_{n}$ such that $\left|S_{\omega}\psi_{\alpha}\right|\leq\epsilon$ for all $\omega\in\Gamma_{n}$. Then we define inductively $\Gamma_{n+1}$ as follows: Since for $\omega\in\Gamma_{n}$ we have $\left|S_{\omega}\psi_{\alpha}\right|\leq\epsilon$ it follows that $\left|S_{\omega\nu}\psi_{\alpha}\right|\leq2\epsilon$ for every $\nu\in\Gamma$. Then we find an element $\tau_{\omega\nu}\in\Lambda$ such that $\left|S_{\omega\nu\tau_{\omega\nu}}\psi_{\alpha}\right|\leq\epsilon$ and we set $$\Gamma_{n+1}\coloneqq\left\{ \omega\nu\tau_{\omega\nu}:\omega\in\Gamma_{n},\nu\in\Gamma\right\} .$$ Next, using Hölder continuity of $\varphi$ and bounded distortion, we find suitable constants $c_{i}$, $i=1,\ldots,5$, such that for all $\omega\in\Gamma_{n}$, $n\in{\mathbb{N}}$, $$\begin{aligned} \sum_{\nu\tau_{\omega\nu}:\omega\nu\tau_{\omega\nu}\in\Gamma_{n+1}}\left|\pi\left[\omega\nu\tau_{\omega\nu}\right]\right|^{s} & \geq c_{1}\sum_{\nu\tau_{\omega\nu}:\omega\nu\tau_{\omega\nu}\in\Gamma_{n+1}}{\mathrm{e}}^{sS_{\omega\nu\tau_{\omega\nu}}\varphi}\geq c_{2}{\mathrm{e}}^{sS_{\omega}\varphi}\sum_{\nu\tau_{\omega\nu}:\omega\nu\tau_{\omega\nu}\in\Gamma_{n+1}}{\mathrm{e}}^{sS_{\nu\tau_{\omega\nu}}\varphi}\nonumber \\ \nonumber \\ & \geq c_{3}{\mathrm{e}}^{sS_{\omega}\varphi}\sum_{\gamma\in\Gamma}{\mathrm{e}}^{sS_{\gamma}\varphi}\geq c_{4}M{\mathrm{e}}^{sS_{\omega}\varphi}\geq c_{5}M\left|\pi\left[\omega\right]\right|^{s}\geq\left|\pi\left[\omega\right]\right|^{s},\label{eq:GeometricInequality}\end{aligned}$$ where the last inequality holds for $\ell$, and hence for $M$, chosen sufficiently large. In the next step we define the Cantor set $$C\coloneqq\limsup_{n\rightarrow\infty}\bigcup_{\omega\in\Gamma_{n}}\pi\left[\omega\right]$$ and a probability measure $\mu$ supported on $C$ defined by its marginals as follows: $\mu\left(\left[0,1\right]\right)=1$ and for $\omega\in\Gamma_{n}$ and $\omega\nu\tau_{\omega\nu}\in\Gamma_{n+1}$ $$\mu\left(\pi\left[\omega\nu\tau_{\omega\nu}\right]\right)\coloneqq\frac{\left|\pi\left[\omega\nu\tau_{\omega\nu}\right]\right|^{s}}{\sum_{\widetilde{\omega}\in\Gamma_{n+1},\left[\widetilde{\omega}\right]\subset\left[\omega\right]}\left|\pi\left[\widetilde{\omega}\right]\right|^{s}}\mu\left(\pi\left[\omega\right]\right).$$ Note that the existence of $\mu$ is guaranteed by Kolmogorov’s Consistency Theorem. Inductively, using the definition of $\mu$ in tandem with inequality (\[eq:GeometricInequality\]) we verify that for all $n\in{\mathbb{N}}$ and $\omega\in\Gamma_{n}$ we have $$\mu\left(\pi\left[\omega\right]\right)\leq\left|\pi\left[\omega\right]\right|^{s}.$$ For each interval $L\subset\left[0,1\right]$ we let $$\begin{gathered} \Gamma\left(L\right)\coloneqq\bigg\{\omega\in\Gamma_{n}\colon n\in{\mathbb{N}},\;\pi\left[\omega\right]\cap L\neq{\varnothing},\;\left|\pi\left[\omega\right]\right|\text{\ensuremath{\leq}}\left|L\right|\text{ and }\widetilde{\omega}\in\Gamma_{n-1}:\left[\widetilde{\omega}\right]\supset\left[\omega\right]\Rightarrow\left|\pi\left[\widetilde{\omega}\right]\right|>\left|L\right|\bigg\}.\end{gathered}$$ Since $$\mu\left(L\right)\leq\sum_{\omega\in\Gamma\left(L\right)}\mu\left(\pi\left[\omega\right]\right)\leq\sum_{\omega\in\Gamma\left(L\right)}\left|\pi\left[\omega\right]\right|^{s}\leq\operatorname*{card}\left(\Gamma\left(L\right)\right)\left|L\right|^{s}$$ and $\operatorname*{card}\left(\Gamma\left(L\right)\right)\leq2\cdot\operatorname*{card}\left(I\right)^{\ell+r}$, the Mass Distribution Principle gives $$\dim_{H}\left(C\right)\geq s.$$ Since $s<\delta_{\alpha}$ was arbitrary,and $\mu\left(C\right)=\mu\left(C\setminus\mathcal{D}\right)$ the desired lower bound is established. \[rem:Multifractal\_Alternative\_Proof\] Let $\mu_{\alpha}:=\mu_{\delta_{\alpha}\varphi+q\psi_{\alpha}}\circ\pi^{-1}$. Since $\mu_{\delta_{\alpha}\varphi+q\psi_{\alpha}}(\psi_{\alpha})=0$, Lemma \[lem:nodrift implies recurrence\] implies that $\mu_{\alpha}\left(\mathbf{E}(\alpha)\right)=1$. By Young’s formula we have $\dim_{H}(\mu_{\alpha})=\delta_{\alpha}$, which then gives an alternative proof for the lower bound of the Hausdorff dimension of $\mathbf{E}\left(\alpha\right)$. \[Proof of Theorem \[thm:Bishop and Jones\]\] First suppose that $0\in\big[\underline{\psi},\overline{\psi}\big]$. Then this corollary is a special case of Theorem \[thm: Multifractal Decomposition\] with $\alpha=0$ by noting that $\mathbf{R}=\mathbf{E}\left(0\right)$ and $\mathbf{R}_{u}=\mathbf{E}_{u}\left(0\right)$. If $0\notin\big[\underline{\psi},\overline{\psi}\big]$, then $\mathbf{R}=\mathbf{E}\left(0\right)=\mathbf{R}_{u}=\mathbf{E}_{u}\left(0\right)={\varnothing}$ and $\delta_{0}=0$. \[Proof of Theorem \[thm:-Transient-sets\]\]We only consider the first case, the second case follows in exactly the same manner. For this we assume $\alpha=\mu_{\delta\varphi}\left(\psi\right)\ge0$. Since $\mathbf{E}_{u}\left(\alpha'\right)\subset\mathbf{T}_{1}^{+}\subset\mathbf{T}_{2}^{+}$ for all $\alpha'>0$, we obtain $$\begin{aligned} \delta & =\dim_{H}\left(\mathbf{E}_{u}\left(\alpha\right)\right)=\sup_{\alpha'>\alpha}\dim_{H}\left(\mathbf{E}_{u}\left(\alpha'\right)\right)\leq\dim_{H}\left(\mathbf{T}_{1}^{+}\right)\leq\dim_{H}\left(\mathbf{T}_{2}^{+}\right)\leq\delta.\end{aligned}$$ For the set $\mathbf{T}_{3}^{+}\left(r\right)$, $r\in{\mathbb{R}}$ note that $$\bigcup_{\ell\geq0}{F}_{\Psi}^{-\ell}\left(\mathbf{T}_{3}^{+}\left(r\right)\right)\supset\mathbf{T}_{1}^{+}.$$ Since for each $\ell\in{\mathbb{N}}$ the set $F_{\Psi}^{-\ell}\left(\mathbf{T}_{3}^{+}\left(r\right)\right)$ is a countable union of bi-Lipschitz images of $\mathbf{T}_{3}^{+}\left(r\right)$, $$\begin{aligned} \delta & =\dim_{H}\left(\mathbf{T}_{1}^{+}\right)\leq\dim_{H}\left(\bigcup_{\ell\geq0}{F}_{\Psi}^{-\ell}\left(\mathbf{T}_{3}^{+}\left(r\right)\right)\right)\\ & =\sup_{\ell\geq0}\dim_{H}\left({F}_{\Psi}^{-\ell}\left(\mathbf{T}_{3}^{+}\left(r\right)\right)\right)=\dim_{H}\left(\mathbf{T}_{3}^{+}\left(r\right)\right)\leq\delta.\end{aligned}$$ Similarly, we have $\mathbf{E}\left(\alpha'\right)\subset\mathbf{T}_{1}^{-}\subset\mathbf{T}_{2}^{-}$ for all $\alpha'<0$ and hence, by Theorem \[thm: Multifractal Decomposition\], giving the lower bound $$\dim_{H}\left(\mathbf{E}\left(0\right)\right)=\delta_{0}=\sup_{\alpha'<0}\delta_{\alpha'}=\sup_{\alpha'<0}\dim_{H}\left(\mathbf{E}\left(\alpha'\right)\right)\leq\dim_{H}\left(\mathbf{T}_{1}^{-}\right)\leq\dim_{H}\left(\mathbf{T}_{2}^{-}\right).$$ And for $\mathbf{T}_{3}^{-}\left(r\right),$ $r\in{\mathbb{R}}$ similarly as above $$\dim_{H}\left(\mathbf{E}\left(0\right)\right)\leq\dim_{H}\left(\mathbf{T}_{1}^{-}\right)\leq\sup_{\ell\geq0}\dim_{H}\left({F}_{\Psi}^{-\ell}\left(\mathbf{T}_{3}^{-}\left(r\right)\right)\right)=\dim_{H}\left(\mathbf{T}_{3}^{-}\left(r\right)\right).$$ For the upper bound we assume without loss of generality that $\delta_{0}<\delta$ (otherwise nothing is to be shown) and fix $0<\varepsilon<\delta-\delta_{0}$. By Proposition \[prop:Properties of delta\_alpha\], we have $q_{0}\le0$. Hence, we obtain the following bound for $n\in{\mathbb{Z}}$, $$\begin{aligned} \sum_{\substack{\left|\omega\right|>N\\ S_{\omega}\psi<-n } }\left|\pi\left[\omega\right]\right|^{\delta_{0}+\epsilon}\le & \phantom{{{\mathrm{e}}^{-q_{0}n}}}\sum_{\substack{\left|\omega\right|>N\\ S_{\omega}\psi<-n } }{\mathrm{e}}^{(\delta_{0}+\epsilon)S_{\omega}\varphi+q_{0}S_{\omega}\psi}{\mathrm{e}}^{-q_{0}S_{\omega}\psi}\\ \le & {\mathrm{e}}^{-q_{0}n}\sum_{\substack{\left|\omega\right|>N\\ S_{\omega}\psi<-n } }{\mathrm{e}}^{(\delta_{0}+\epsilon)S_{\omega}\varphi+q_{0}S_{\omega}\psi_{0}}\\ \leq & {\mathrm{e}}^{-q_{0}n}\sum_{\substack{\left|\omega\right|>N} }{\mathrm{e}}^{(\delta_{0}+\epsilon)S_{\omega}\varphi+q_{0}S_{\omega}\psi_{0}}<\infty,\end{aligned}$$ which shows $\dim_{H}\left(\mathbf{T}_{3}^{-}\left(r\right)\cap\left[n-r-1,n-r\right]\right)\leq\delta_{0}$, for all $n\in{\mathbb{Z}}$. As a consequence of the countable stability of the Hausdorff dimension we obtain $\dim_{H}\left(\mathbf{T}_{3}^{-}\left(r\right)\right)\leq\delta_{0}$. In particular, for $r\in{\mathbb{Z}}$ we use (\[eq:T2UnionT3\]) and the countable stability of the Hausdorff dimension once more to finish the proof. The last upper bound in the above proof could also been seen via the general multifractal formalism for limiting behaviour of $\left(S_{n}\psi/S_{n}\varphi\right)$ provided e.g. in [@MR2719683; @MR2672614] by observing the inclusion, for $r\in{\mathbb{R}}$ and $n\in{\mathbb{Z}}$, $$\mathbf{T}_{3}^{-}\left(r\right)\cap\left[n,n+1\right]\subset\pi\left\{ \omega\in\Sigma\colon\liminf_{\ell\rightarrow\infty}\frac{S_{\ell}\psi}{S_{\ell}\varphi}\leq0\right\} +n.$$ Let us consider the implicitly defined function $s:{\mathbb{R}}\to{\mathbb{R}}$ by $\mathfrak{P}\left(s(q)\varphi+q\psi\right)=0$ as in the proof of Theorem \[thm:critical exponent via base pressures\]. By the general multifractal formalism, for $\alpha=\mu_{\delta\varphi}\left(\psi\right)\ge0$, we have $$\dim_{H}\left(\pi\left\{ \omega\in\Sigma\colon\liminf_{\ell\rightarrow\infty}\frac{S_{\ell}\psi}{S_{\ell}\varphi}\le0\right\} \right)=\inf_{q\in{\mathbb{R}}}s\left(q\right).$$ This is to say that we have to find $s_{0}$ and $q_{0}$ with $\mathfrak{P}\left(s_{0}\varphi+q_{0}\psi\right)=0$ and $s'\left(q_{0}\right)=\frac{\partial}{\partial q}\mathfrak{P}\left(s_{0}\varphi+q\psi\right)|_{q=q_{0}}=0$. By Theorem \[thm:critical exponent via base pressures\], this shows $\inf_{q\in{\mathbb{R}}}s\left(q\right)=\delta_{0}$. The upper bound then follows from the countable stability of the Hausdorff dimension. Examples and an application to Kleinian groups\[sec:Examples\] ============================================================== Let us consider an interval map $F$ with two expansive branches with slopes $1/c_{1}$ and $1/c_{2}$, respectively, where $c_{1},c_{2}\in(0,1)$ with $c_{1}+c_{2}\leq1$ (see for instance Example \[exa:Classical Random Walk\] in the introduction). Then the corresponding geometric potential is given by $\varphi\left(\omega\right)\coloneqq\log\left(c_{\omega_{1}}\right)$ for $\omega=\left(\omega_{1},\omega_{2},\ldots\right)$. Moreover, note that $\delta$ solves the Moran-Hutchinson formula $c_{1}^{\delta}+c_{2}^{\delta}=1.$ We determine the dimension spectrum $\alpha\mapsto\delta_{\alpha}$ of the escaping sets for ${F}_{\Psi}$ for different parameters $c_{1},c_{2}$ and step length functions $\Psi$. In the following we also make the convention that $0\cdot\log0=0$. First, we consider arbitrary $c_{1},c_{2}\in(0,1)$ and pick a symmetric step length function $\Psi$ such that $\psi\left(\omega\right)=\left(-1\right)^{\omega_{1}}$. Then we have to solve the two equations for $\alpha\in(-1,1)$: $$\begin{aligned} 1 & =\exp\left(\mathfrak{P}\left(s\varphi+q\psi_{\alpha}\right)\right)={\mathrm{e}}^{-q\alpha}\left(c_{1}^{s}{\mathrm{e}}^{-q}+c_{2}^{s}{\mathrm{e}}{}^{q}\right)\eqqcolon z_{\alpha}(s,q)\end{aligned}$$ and $$0=\frac{\partial z_{\alpha}}{\partial q}(s,q)=-\alpha+{\mathrm{e}}^{-q\alpha}\left(-c_{1}^{s}{\mathrm{e}}^{-q}+c_{2}^{s}{\mathrm{e}}{}^{q}\right).$$ Using Theorem \[thm:critical exponent via base pressures\] (i)&(ii), this gives for $\alpha\in[-1,1]$ $$\delta_{\alpha}=\delta_{\alpha}(c_{1},c_{2})=-\frac{\left(\frac{1+\alpha}{2}\right)\log\left(\frac{1+\alpha}{2}\right)+\left(\frac{1-\alpha}{2}\right)\log\left(\frac{1-\alpha}{2}\right)}{\left(\frac{1+\alpha}{2}\right)\log(1/c_{1})+\left(\frac{1-\alpha}{2}\right)\log(1/c_{2})}.$$ Note that this expression is in fact the quotient of the measure-theoretic entropy over the Lyapunov exponent of $F$ with respect to the $\left(\frac{1+\alpha}{2},\frac{1-\alpha}{2}\right)$-Bernoulli measure. Moreover, $$\lim_{\alpha\searrow-1}\frac{d}{d\alpha}\delta_{\alpha}=\infty\quad\text{and}\quad\lim_{\alpha\nearrow1}\frac{d}{d\alpha}\delta_{\alpha}=-\infty,$$ see Figure \[fig:asymmetric-Random-Walk-Spectrum\] for the graph of $\alpha\mapsto\delta_{\alpha}$. In order to determine the Hausdorff dimension of the (uniformly) recurrent set we need to determine $\delta_{0}$ depending on the parameters $\left(c_{1},c_{2}\right)$. We obtain $$\delta_{0}\left(c_{1},c_{2}\right)=\frac{\log4}{\log(1/c_{1})+\log(1/c_{2})},$$ see Figure \[fig:delta\_0Graph\] for a one-parameter plot of $c\mapsto\delta_{0}\left(c,1-c\right)$. Finally, for $\alpha=0$, using Theorem \[thm:fibre-induced pressure via base pressure\] we obtain for the fibre-induced pressure $$\mathcal{P}\left(t\varphi,\psi\right)=\log2+t\cdot\left(\log c_{1}+\log c_{2}\right)/2.$$ \[exa:asymmetric setp\]Here, we set $c_{1}=c_{2}=c$ and consider $\Psi$ such that $\psi\left(\omega\right)=m_{1}(2-\omega_{1})+m_{2}(\omega_{1}-1)$ with $m_{1}<m_{2}$ (a-)symmetric. A similar calculation as in the first example leads us to the dimension spectrum $$\delta_{\alpha}=\delta_{\alpha}(c,m_{1},m_{2})=-\frac{\left(\frac{\alpha-m_{1}}{m_{2}-m_{1}}\right)\log\left(\frac{\alpha-m_{1}}{m_{2}-m_{1}}\right)+\left(\frac{m_{2}-\alpha}{m_{2}-m_{1}}\right)\log\left(\frac{m_{2}-\alpha}{m_{2}-m_{1}}\right)}{\log(1/c)},$$ for $\alpha\in[m_{1},m_{2}]$. Again, note that this expression is the quotient of the measure-theoretic entropy over the Lyapunov exponent of $F$ with respect to the $\left(\frac{\alpha-m_{1}}{m_{2}-m_{1}},\frac{m_{2}-\alpha}{m_{2}-m_{1}}\right)$-Bernoulli measure. \[exa:more intervals\]For the last example we change the map $F$. Assume that $F$ has $g_{1}+g_{2}$ intervals where $g_{1}$, $g_{2}\geq1$ such that on each interval the slope $1/c\geq g_{1}+g_{2}$. Further, we choose $\Psi$ in such a way that on $g_{1}$ intervals it is $-1$ and on the others it is $+1$. An analogous calculation as in the previous examples gives for $\alpha\in[-1,1]$ the following dimension spectrum $$\delta_{\alpha}=\delta_{\alpha}(c,g_{1},g_{2})=-\frac{g_{1}\cdot\left(\frac{1-\alpha}{2g_{1}}\right)\log\left(\frac{1-\alpha}{2g_{1}}\right)+g_{2}\cdot\left(\frac{1+\alpha}{2g_{2}}\right)\log\left(\frac{1+\alpha}{2g_{2}}\right)}{\log(1/c)}.$$ Once more, observe that this expression is a quotient of the measure-theoretic entropy over the Lyapunov exponent of $F$ for a corresponding Bernoulli measure. For a particular choice consider $g_{1}=1$, $g_{2}=2$ and $c=1/3$, see also Figure \[fig:more examples\]. We have $\dim_{H}\left(\mathbf{E}\left(1\right)\right)=\log2/\log3$ and this means that the Hausdorff dimension of $\mathbf{E}\left(1\right)$ coincides with the dimension of the $1/3$-Cantor set. \[ domain=-1:2, samples=270,line width=1.2pt, \] [(ln(3)/ln(2)-((x+1)\*ln(x+1)+(2-x)\*ln(2-x))/(3\*ln(2)))]{}; \[ domain=0:1, samples=197,line width=1.2pt, dashed \] [(-(x\*ln(x)+(1-x)\*ln(1-x))/(ln(2)))]{}; \[ domain=-1:1, samples=197,line width=1.2pt, \] [(-((1-x)\*ln((1-x)/2)+(1+x)\*ln((1+x)/4))/(2\*ln(3)))]{}; \[ domain=-1:1, samples=197,line width=1.2pt, dashed \] [(-((1-x)\*ln((1-x)/16)+(1+x)\*ln((1+x)/4))/(2\*ln(10)))]{}; Finally, we provide an application of our results partially recovering a result of [@rees_1981] about extensions of Kleinian groups. Recall that if $G=\left\langle g_{1},\dots,g_{k}\right\rangle $ is a Schottky group, the limit set of $G$ can be represented by the subshift of finite type $\Sigma_{G}:=\left\{ \omega=(\omega_{1},\omega_{2}\dots)\in I^{{\mathbb{N}}}\mid\omega_{i}\neq-\omega_{i+1},\,\,i\in{\mathbb{N}}\right\} $ with alphabet $I=\left\{ \pm1,\dots\pm k\right\} $. Then $\pi$ denotes the natural coding map of the limit set with respect to $\Sigma_{G}$. Further, the $\sigma$-invariant $\delta_{G}$-dimensional Patterson-Sullivan measure of $G$ is given by the $\sigma$-invariant Gibbs measure on $\Sigma_{G}$ with respect to the Hölder continuous geometric potential $\delta_{G}\cdot\varphi:\Sigma_{G}\rightarrow{\mathbb{R}}$ associated with $G$ ([@Bowen1979]). Concerning the general theory of the Patterson-Sullivan measure and its $\sigma$-invariant version, see [@MR0450547] and [@MR556586], respectively. Let $G=\left\langle g_{1},\dots,g_{k}\right\rangle $ be a Schottky group. Let $\mu$ be the $\sigma$-invariant version of the $\delta_{G}$-dimensional Patterson-Sullivan measure on the subshift of finite type $\Sigma_{G}$ with alphabet $I=\left\{ \pm1,\dots\pm k\right\} $. Then we have $\mu\left(\left[i\right]\right)=\mu\left(\left[-i\right]\right)$ for all $i\in I$. Denote by $m$ the $\delta_{G}$-dimensional Patterson-Sullivan measure. Recall that $|\xi-\eta|^{-2\delta_{G}}dm(\xi)\times dm(\eta)$ defines a $G$-invariant measure on the geodesics on $\mathbb{H}^{n}\backslash G$ represented by $\Lambda(G)\times\Lambda(G)$. By disintegration of this measure, we obtain the $\sigma$-invariant version $\mu$ on $\Sigma_{G}$ which satisfies for all $i\in I$, $$\mu([i])=\int_{\pi([i])}\sum_{j\neq i}\int_{\pi([j])}\left|\xi-\eta\right|^{-2\delta_{G}}dm(\eta)dm(\xi).$$ Since $$g_{i}\left\{ \pi[i]\times\left(\bigcup_{j\neq i}\pi[j]\right)\right\} =\left(\bigcup_{j\neq-i}\pi[j]\right)\times\pi[-i],$$ the $G$-invariance of $|\xi-\eta|^{-2\delta_{G}}dm(\xi)\times dm(\eta)$ implies $$\begin{aligned} \mu([i]) & =\int_{\pi([i])}\sum_{j\neq i}\int_{\pi([j])}\left|\xi-\eta\right|^{-2\delta_{G}}dm(\eta)dm(\xi)\\ & =\int_{\pi([-i])}\sum_{j\neq-i}\int_{\pi([j])}\left|\xi-\eta\right|^{-2\delta_{G}}dm(\eta)dm(\xi)=\mu([-i]).\end{aligned}$$ This shows the assertion. Now, let $G$ be a Schottky group and $N<G$ a normal subgroup such that $G/N=\left\langle g\right\rangle \cong{\mathbb{Z}}$ and without loss of generality $g\in\left\{ g_{1},\ldots,g_{k}\right\} $. Then, using Kronecker delta notation, we set $\psi:\Sigma_{G}\rightarrow{\mathbb{Z}}$, $\psi(\omega):=\delta_{g_{\omega_{1}},g}-\delta_{g_{\omega_{1}},g^{-1}}$. Relating the hyperbolic distance to $\varphi$ as in [@MR2041265] as well as replacing the fullshift $\Sigma$ with $\Sigma_{G}$ in the definition of Poincaré exponents, we deduce that $\delta=\delta_{G}$ and $\delta_{0}=\delta_{N}$. Further, note that $0\in\left(\underline{\psi},\overline{\psi}\right)=(-1,1)$ and that the first and the last assertion of Theorem \[thm:fibre-induced pressure via base pressure\] and Corollary \[cor:full pressure if no drift\] remain valid also for mixing subshifts of finite type. Hence, combining the previous proposition with Corollary \[cor:full pressure if no drift\] and Theorem \[thm:fibre-induced pressure via base pressure\] applied to $\delta_{G}\varphi$, gives Corollary \[thm:Kleingroup divergence type\]. [10]{} G. Atkinson. Recurrence of co-cycles and random walks. , 13(3):486–488, 1976. A. Avila and M. Lyubich. Hausdorff dimension and conformal measures of [F]{}eigenbaum [J]{}ulia sets. , 21(2):305–363, 2008. L. Barreira and B. Saussol. Variational principles and mixed multifractal spectra. , 353(10):3919–3944, 2001. C. J. Bishop and P. W. Jones. Hausdorff dimension and [K]{}leinian groups. , 179(1):1–39, 1997. R. Bowen. Hausdorff dimension of quasi-circles. , 50:11–25, 1979. R. Brooks. The bottom of the spectrum of a [R]{}iemannian covering. , 357:101–114, 1985. H. Bruin and M. Todd. Transience and thermodynamic formalism for infinitely branched interval maps. , 86(1):171–194, 2012. V. Cyr. Countable [M]{}arkov shifts with transient potentials. , 103(6):923–949, 2011. V. Cyr and O. Sarig. Spectral gap and transience for [R]{}uelle operators on countable [M]{}arkov shifts. , 292(3):637–666, 2009. K. Falk and B. O. Stratmann. Remarks on [H]{}ausdorff dimensions for transient limit sets of [K]{}leinian groups. , 56(4):571–582, 2004. A.-H. Fan, T. Jordan, L. Liao, and M. Rams. Multifractal analysis for expanding interval maps with infinitely many branches. , 367(3):1847–1870, 2015. D.-J. Feng, K.-S. Lau, and J. Wu. Ergodic limits on the conformal repellers. , 169(1):58–91, 2002. M. Gr[ö]{}ger, J. Jaerisch, and M. Kesseb[[ö]{}]{}hmer. Escaping sets for transient dynamics with reflective boundary. , 2019. G. Iommi, T. Jordan, and M. Todd. Transience and multifractal analysis. , 34(2):407–421, 2017. J. Jaerisch. A lower bound for the exponent of convergence of normal subgroups of [K]{}leinian groups. , 25(1):298–305, 2015. J. Jaerisch. Recurrence and pressure for group extensions. , 36(1):108–126, 2016. J. Jaerisch and M. Kesseb[ö]{}hmer. The arithmetic-geometric scaling spectrum for continued fractions. , 48(2):335–360, 2010. J. Jaerisch and M. Kesseb[ö]{}hmer. Regularity of multifractal spectra of conformal iterated function systems. , 363(1):313–330, 2011. J. Jaerisch, M. Kesseb[ö]{}hmer, and S. Lamei. Induced topological pressure for countable state [M]{}arkov shifts. , 14(2):1350016, 31, 2014. T. Jordan and M. Rams. Birkhoff spectrum for piecewise monotone interval maps. , pages 1–22, 2017. B. Karpińska and M. Urbański. How points escape to infinity under exponential maps. , 73(1):141–156, 2006. M. Kesseböhmer and B. O. Stratmann. A multifractal formalism for growth rates and applications to geometrically finite [K]{}leinian groups. , 24(1):141–170, 2004. H. Kesten. Full [B]{}anach mean values on countable groups. , 7:146–156, 1959. H. Kesten. Symmetric random walks on groups. , 92:336–354, 1959. R. D. Mauldin and M. Urba[[ń]{}]{}ski. , volume 148 of [*Cambridge Tracts in Mathematics*]{}. Cambridge University Press, Cambridge, 2003. Geometry and dynamics of limit sets. C. McMullen. Area and [H]{}ausdorff dimension of [J]{}ulia sets of entire functions. , 300(1):329–342, 1987. S. J. Patterson. The limit set of a [F]{}uchsian group. , 136(3-4):241–273, 1976. Y. Pesin and H. Weiss. The multifractal analysis of [B]{}irkhoff averages and large deviations. In [*Global analysis of dynamical systems*]{}, pages 419–431. Inst. Phys., Bristol, 2001. M. Rees. . , 1(1):107–133, 1981. L. Rempe. Hyperbolic dimension and radial [J]{}ulia sets of transcendental functions. , 137(4):1411–1420, 2009. T. Roblin. Un théorème de [F]{}atou pour les densités conformes avec applications aux revêtements galoisiens en courbure négative. , 147:333–357, 2005. R. T. Rockafellar. . Princeton landmarks in mathematics and physics. Princeton University Press, 1970. D. Ruelle. Statistical mechanics of a one-dimensional lattice gas. , 9:267–278, 1968. O. M. Sarig. Thermodynamic formalism for countable [M]{}arkov shifts. , 19(6):1565–1593, 1999. O. M. Sarig. Thermodynamic formalism for null recurrent potentials. , 121:285–311, 2001. K. Schmidt. . Macmillan Company of India, Ltd., Delhi, 1977. Macmillan Lectures in Mathematics, Vol. 1. M. Shishikura. The [H]{}ausdorff dimension of the boundary of the [M]{}andelbrot set and [J]{}ulia sets. , 147(2):225–267, 1998. O. Shwartz. Thermodynamic formalism for transient potential functions. , 366(2):737–779, 2019. M. Stadlbauer. An extension of [K]{}esten’s criterion for amenability to topological [M]{}arkov chains. , 235:450–468, 2013. B. Stratmann and R. Vogt. Fractal dimensions for dissipative sets. , 10(2):565–577, 1997. D. Sullivan. The density at infinity of a discrete group of hyperbolic motions. , 50:171–202, 1979. M. Urbański and A. Zdunik. Geometry and ergodic theory of non-hyperbolic exponential maps. , 359(8):3973–3997, 2007. P. Walters. , volume 79 of [*Graduate Texts in Mathematics*]{}. Springer-Verlag, New York, 1982. [^1]: MG was supported by the DFG grants JA 1721/2-1 and GR 4899/1-1, JJ was supported by JSPS KAKENHI 17K14203 and MK acknowledges support by the DFG grant KE 1440/3-1. This project was also part of the activities of the Scientific Network Skew product dynamics and multifractal analysis (DFG grant OE 538/3-1).

--- abstract: | In this work we study arrangements of $k$-dimensional subspaces $V_1,\ldots,V_n \subset {\mathbb{C}}^\ell$. Our main result shows that, if every pair $V_{a},V_b$ of subspaces is contained in a dependent triple (a triple $V_{a},V_b,V_c$ contained in a $2k$-dimensional space), then the entire arrangement must be contained in a subspace whose dimension depends only on $k$ (and not on $n$). The theorem holds under the assumption that $V_a \cap V_b = \{0\}$ for every pair (otherwise it is false). This generalizes the Sylvester-Gallai theorem (or Kelly’s theorem for complex numbers), which proves the $k=1$ case. Our proof also handles arrangements in which we have many pairs (instead of all) appearing in dependent triples, generalizing the quantitative results of Barak et. al. [@BDWY-pnas]. One of the main ingredients in the proof is a strengthening of a Theorem of Barthe [@Bar98] (from the $k=1$ to $k>1$ case) proving the existence of a linear map that makes the angles between pairs of subspaces large on average. Such a mapping can be found, unless there is an obstruction in the form of a low dimensional subspace intersecting many of the spaces in the arrangement (in which case one can use a different argument to prove the main theorem). author: - 'Zeev Dvir[^1]' - 'Guangda Hu[^2]' bibliography: - 'hidimSG.bib' title: '**Sylvester-Gallai for Arrangements of Subspaces**' --- Introduction ============ The Sylvester-Gallai (SG) theorem states that for $n$ points ${\boldsymbol{v}}_1,{\boldsymbol{v}}_2,\ldots,{\boldsymbol{v}}_n \in {\mathbb{R}}^\ell$, if for every pair ${\boldsymbol{v}}_i,{\boldsymbol{v}}_j$ there is a third point ${\boldsymbol{v}}_k$ on the line passing through ${\boldsymbol{v}}_i,{\boldsymbol{v}}_j$, then all points must lie on a single line. This was first posed by Sylvester [@Syl93], and was solved by Melchior [@Mel40]. It was also conjectured independently by Erd[ö]{}s [@Erd43] and proved shortly after by Gallai. We refer the reader to the survey [@BM90] for more information about the history and various generalizations of this theorem. The complex version of this theorem was proved by Kelly [@Kel86] (see also [@EPS06; @DSW12] for alternative proofs) and states that if ${\boldsymbol{v}}_1,{\boldsymbol{v}}_2,\ldots,{\boldsymbol{v}}_n \in {\mathbb{C}}^\ell$ and for every pair ${\boldsymbol{v}}_i,{\boldsymbol{v}}_j$ there is a third ${\boldsymbol{v}}_k$ on the same complex line, then all points are contained in some complex plane (over the complex numbers, there are planar examples and so this theorem is tight). In [@DSW12] (based on earlier work in [@BDWY-pnas]), the following quantitative variant of the SG theorem was proved. For a set $S \subset {\mathbb{C}}^\ell$ we denote by $\dim(S)$ the smallest $d$ such that $S$ is contained in a $d$-dimensional subspace of ${\mathbb{C}}^\ell$. \[thm:osg\] Given $n$ points ${\boldsymbol{v}}_1,{\boldsymbol{v}}_2,\ldots,{\boldsymbol{v}}_n \in {\mathbb{C}}^\ell$, if for every $i\in[n]$ there exists at least $\delta n$ values of $j\in[n]\setminus\{i\}$ such that the line through ${\boldsymbol{v}}_i$ and ${\boldsymbol{v}}_j$ contains a third point ${\boldsymbol{v}}_k$, then $\dim\{{\boldsymbol{v}}_1,{\boldsymbol{v}}_2,\ldots,{\boldsymbol{v}}_n\}\leq 10/\delta$. (The dependence on $\delta$ is asymptotically tight). From here on, we will work with homogeneous subspaces (passing through zero) instead of affine subspaces (lines/planes etc). The difference is not crucial to our results and the affine version can always be derived by intersecting with a generic hyperplane. In this setting, the above theorem will be stated for a set of one-dimensional subspaces, each spanned by some ${\boldsymbol{v}}_i$ (and no two ${\boldsymbol{v}}_i$’s being a multiple of each other) and collinearity of ${\boldsymbol{v}}_i,{\boldsymbol{v}}_j,{\boldsymbol{v}}_k$ is replaced with the three vectors being linearly dependent (i.e., contained in a 2-dimensional subspace). One natural high dimensional variant of the SG theorem, studied in [@Han65; @BDWY-pnas], replaces 3-wise dependencies with $t$-wise dependencies (e.g, every triple is in some coplanar four-tuple). In this work, we raise another natural high-dimensional variant in which the [*points*]{} themselves are replaced with $k$-dimensional subspaces. We consider such arrangements with many 3-wise dependencies (defined appropriately) and attempt to prove that the entire arrangement lies in some low dimensional space. We will consider arrangements $V_1,\ldots,V_n \subset {\mathbb{C}}^\ell$ in which each $V_i$ is $k$-dimensional and with each pair satisfying $V_{i_1} \cap V_{i_2} = \{{\boldsymbol{0}}\}$. A dependency can then be defined as a triple $V_{i_1},V_{i_2},V_{i_3}$ of $k$-dimensional subspaces that are contained in a single $2k$-dimensional subspace. The pair-wise zero intersections guarantee that every pair of subspaces defines a unique $2k$-dimensional space (their span) and so, this definition of dependency behaves in a similar way to collinearity. For example, we have that if $V_{i_1},V_{i_2},V_{i_3}$ are dependent and $V_{i_2},V_{i_3},V_{i_4}$ are dependent then also $V_{i_1},V_{i_2},V_{i_4}$ are dependent. This would not hold if we allowed some pairs to have non zero intersections. In fact, if we allow non-zero intersection then we can construct an arrangement of two dimensional spaces with many dependent triples and with dimension as large as $\sqrt{n}$ (see below). We now state our main theorem, generalizing Theorem \[thm:osg\] (with slightly worse parameters) to the case $k>1$. We use the standard $V + U$ notation to denote the subspace spanned by all vectors in $V \cup U$. We use big ‘O’ notation to hide absolute constants. \[thm:sg\] Let $V_1,V_2,\ldots,V_n\subset {\mathbb{C}}^\ell$ be $k$-dimensional subspaces such that $V_{i}\cap V_{i'}=\{{\boldsymbol{0}}\}$ for all $i\neq i'\in[n]$. Suppose that, for every $i_1\in[n]$ there exists at least $\delta n$ values of $i_2\in[n]\setminus\{i_1\}$ such that $V_{i_1}+V_{i_2}$ contains some $V_{i_3}$ with $i_3 \not\in \{i_1,i_2\}$. Then $$\dim(V_1+V_2+\cdots+V_n)= O(k^4/\delta^2).$$ The condition $V_i \cap V_{i'} = \{{\boldsymbol{0}}\}$ is needed due to the following example. Set $k=2$ and $n=\ell(\ell-1)/2$ and let $\{{\boldsymbol{e}}_1,{\boldsymbol{e}}_2,\ldots,{\boldsymbol{e}}_\ell\}$ be the standard basis of ${\mathbb{R}}^\ell$. Define the $n$ spaces to be $V_{ij} = \operatorname{span}\{{\boldsymbol{e}}_i,{\boldsymbol{e}}_j\}$ with $1\leq i<j\leq\ell$. Now, for each $(i,j) \neq (i',j')$ the sum $V_{ij} + V_{i'j'}$ will contain a third space (since the size of $\{i,j,i',j'\}$ is at least three). However, this arrangement has dimension $\ell > \sqrt{n}$. The bound $O(k^4/\delta^2)$ is probably not tight and we conjecture that it could be improved to $O(k/\delta)$, possibly with a modification of our proof. One can always construct an arrangement with dimension $2k/\delta$ by partitioning the subspaces into $1/\delta$ groups, each contained in a single $2k$ dimensional space. ##### Overview of the proof: A preliminary observation is that it suffices to prove the theorem over ${\mathbb{R}}$. This is because an arrangement of $k$-dimensional Complex subspaces can be translated into an arrangement of $2k$-dimensional Real subspaces (this is proved at the end of Section \[sec:adsystem\]). Hence, we will now focus on Real arrangements. The proof of the theorem is considerably simpler when the arrangement of subspaces $V_1,\ldots,V_n$ satisfies an extra ‘robustness’ condition, namely that every two spaces have an angle bounded away from zero. More formally, if for every two unit vectors ${\boldsymbol{v}}_1 \in V_{i_1}$ and ${\boldsymbol{v}}_2 \in V_{i_2}$ we have $|{\langle {\boldsymbol{v}}_1,{\boldsymbol{v}}_2 \rangle}| \leq1-\tau$ for some absolute constant $\tau>0$. This condition implies that, when we have a dependency of the form $V_{i_3} \subset V_{i_1} + V_{i_2}$, every unit vector in $V_{i_3}$ can be obtained as a linear combination [*with bounded coefficients*]{} (in absolute value) of unit vectors from $V_{i_1},V_{i_2}$. Fixing an orthogonal basis for each subspace and using the conditions of the theorem, we are able to construct many local linear dependencies between the basis elements. We then show (using the bound on the coefficients in the linear combinations) that the space of linear dependencies between all basis vectors, considered as a subspace of ${\mathbb{R}}^{kn}$, contains the rows of an $nk \times nk$ matrix that has large entries on the diagonal and small entries off the diagonal. Since matrices of this form have high rank (by a simple spectral argument), we conclude that the original set of basis vectors must have small dimension. To handle the general case, we show that, unless some low dimensional subspace $W$ intersects many of the spaces $V_i$ in the arrangement, we can find a change of basis that makes the angles between the spaces large on average (in which case, the previous argument works). This gives us the overall strategy of the proof: If such a $W$ exists, we project $W$ to zero and continue by induction. The loss in the overall dimension is bounded by the dimension of $W$, which can be chosen to be small enough. Otherwise (if such $W$ does not exist) we apply the change of basis and use it to bound the dimension. The change of basis is found by generalizing a theorem of Barthe [@Bar98] (see [@DSW14] for a more accessible treatment) from the one dimensional case (arrangement of points) to higher dimension. We state this result here since we believe it could be of independent interest. To state the theorem we must first introduce the following, somewhat technical, definition. \[def:adbasic\] Given a list of vector spaces $\mathcal{V}=(V_1,V_2,\ldots,V_n)$ ($V_i\subseteq{\mathbb{R}}^\ell$), a set $H\subseteq [n]$ is called a [*$\mathcal{V}$-admissible basis set*]{} if $$\dim(\sum_{i\in H}V_i)=\sum_{i\in H}\dim(V_i)=\dim(\sum_{i\in[n]}V_i),$$ i.e. if every space with index in $H$ has intersection $\{{\boldsymbol{0}}\}$ with the span of the other spaces with indices in $H$, and the spaces with indices in $H$ span the entire space $\sum_{i\in[n]}V_i$. A [*$\mathcal{V}$-admissible basis vector*]{} is any indicator vector ${\boldsymbol{1}}_H$ of some $\mathcal{V}$-admissible basis set $H$ (where the $i$-th entry of ${\boldsymbol{1}}_H$ equals $1$ if $i\in H$ and $0$ otherwise). The following theorem is proved in Section \[sec:barthe\]. \[thm:barthe\] Given a list of vector spaces $\mathcal{V}=(V_1,V_2,\ldots,V_n)$ ($V_i\subseteq{\mathbb{R}}^\ell$) with $V_1+V_2+\cdots+V_n={\mathbb{R}}^\ell$ and a vector ${\boldsymbol{p}}\in{\mathbb{R}}^n$ in the convex hull of all $\mathcal{V}$-admissible basis vectors. Then for any $\varepsilon>0$, there exists an invertible linear map $M:{\mathbb{R}}^\ell\mapsto{\mathbb{R}}^\ell$ such that $$\Big\|\sum_{i=1}^np_i\operatorname{Proj}_{M(V_i)}-I_{\ell\times\ell}\Big\|\leq\varepsilon,$$ where $\|\cdot\|$ is the spectral norm and $\operatorname{Proj}_{M(V_i)}$ is the orthogonal projection matrix onto $M(V_i)$. The connection to the explanation given in the proof overview is as follows: If there is no subspace $W$ of low dimension that intersects many of the spaces $V_1,\ldots,V_n$ then, one can show that there exists a vector ${\boldsymbol{p}}$ in the convex hull of all $\mathcal{V}$-admissible basis vectors such that the entries of ${\boldsymbol{p}}$ are not too small. This is enough to show that the average angle between pairs of spaces is large since otherwise one can derive a contradiction to the inequality which says that the sum of orthogonal projections of any unit vector must be relatively small. The proof of the one dimensional case in [@Bar98] (which does not have an $\epsilon$ error) proceeds by defining a strictly convex function $f(t_1,\ldots,t_m)$ on ${\mathbb{R}}^m$ and shows that the function is bounded. This means that there must exist a point in which all partial derivatives of $f$ vanish. Solving the resulting equations gives an invertible matrix that defines the required change of basis. We follow a similar strategy, defining an appropriate bounded function $f(t_1,\ldots,t_m,R_1,\ldots,R_n)$ in more variables, where the extra variables $R_1,\ldots,R_n$ represent the action of the orthogonal group $\mathbf{O}(k)$ on each of the spaces. However, in our case, we cannot show that $f$ is strictly convex and so a maximum might not exist. However, we are still able to show that there exists a point in which all partial derivatives are very small (smaller than any $\epsilon>0$), which is sufficient for our purposes. ##### Connection to Locally Correctable Codes. A $q$-query Locally Correctable Code (LCC) over a field ${\mathbb{F}}$ is a $d$-dimensional subspace $C \subset {\mathbb{F}}^n$ that allows for ‘local correction’ of codewords (elements of $C$) in the following sense. Let ${\boldsymbol{y}} \in C$ and suppose we have query access to ${\boldsymbol{y}}'$ such that ${\boldsymbol{y}}_i = {\boldsymbol{y}}'_i$ for at least $(1-\delta)n$ indices $i \in [n]$ (think of ${\boldsymbol{y}}'$ as a noisy version of ${\boldsymbol{y}}$). Then, for every $i$, we can probabilistically pick $q$ positions in ${\boldsymbol{y}}'$ and, from their (possibly incorrect values), recover the correct value of ${\boldsymbol{y}}_i$ with high probability (over the choice of queries). LCC’s play an important role in theoretical computer science (mostly over finite fields but recently also over the Reals, see [@Dvir-rigidity]) and are still poorly understood. In particular, when $q$ is constant greater than 2, there are exponential gaps between the dimension of explicit constructions and the proven upper bounds. In [@BDWY11] it was observed that $q$-LCCs are essentially equivalent to configurations of points with many local dependencies[^3]. A variant of Theorem \[thm:osg\] shows for example that the maximal dimension of a $2$-LCC in ${\mathbb{R}}^n$ has dimension bounded by $(1/\delta)^{O(1)}$. Our results can be interpreted in this framework as dimension upper bounds for $2$-query LCC’s in which each coordinate is replaced by a ‘block’ of $k$ coordinate. Our results then show that, even under this relaxation, the dimension still cannot increase with $n$. The case of $3$-query LCC’s over the Reals is still wide open (some modest progress was made recently in [@DSW14]) and we hope that the methods developed in this work could lead to further progress on this tough problem. ##### Organization. In Section \[sec:adsystem\], we define the notion of $(\alpha,\delta)$-systems (which generalizes the SG condition) and reduce our $k$-dimensional Sylvester-Gallai theorem to a more general theorem, Theorem \[thm:main\], on the dimension of $(\alpha,\delta)$-systems (this part also includes the reduction from Complex to Real arrangements). Then, in Section \[sec:barthe\], we prove the generalization of Barthe’s theorem (Theorem \[thm:barthe\]). Finally, in section \[sec:main\], we prove the our main result regarding $(\alpha,\delta)$-systems. Reduction to (alpha,delta)-systems {#sec:adsystem} ================================== The notion of an $(\alpha,\delta)$-system is used to ‘organize’ the dependent triples in the arrangement in a more convenient form so that each space is in many triples and every pair of spaces is together only in a few dependent triples. We also allow dependent [*pairs*]{} as those might arise when we apply a linear map on the arrangement. \[def:adsystem\] Given a list of vector spaces $\mathcal{V}=(V_1,V_2,\ldots,V_n)$ ($V_i\subseteq{\mathbb{R}}^\ell$), we call a list of sets $\mathcal{S}=(S_1,S_2,\ldots,S_w)$ an [*$(\alpha,\delta)$-system*]{} of $\mathcal{V}$ ($\alpha\in{\mathbb{Z}}^+$, $\delta>0$) if 1. Every $S_j$ is a subset of $[n]$ of size either $3$ or $2$. 2. If $S_j$ contains $3$ elements $i_1$, $i_2$ and $i_3$, then $V_{i_1}\subseteq V_{i_2}+V_{i_3}$, $V_{i_2}\subseteq V_{i_1}+V_{i_3}$ and $V_{i_3}\subseteq V_{i_1}+V_{i_2}$. If $S_j$ contains $2$ elements $i_1$ and $i_2$, then $V_{i_1}=V_{i_2}$. 3. Every $i\in[n]$ is contained in at least $\delta n$ sets of $\mathcal{S}$. 4. Every pair $\{i_1,i_2\}$ ($i_1\neq i_2\in[n]$) appears together in at most $\alpha$ sets of $\mathcal{S}$. Note that we allow $\delta>1$ in an $(\alpha,\delta)$-systems. This is different from the statement of the Sylvester-Gallai theorem where $\delta\in[0,1]$. We have the following simple observations. \[lem:awd\] Let $\mathcal{S}=(S_1,S_2,\ldots,S_w)$ be an $(\alpha,\delta)$-system of some vector space list $\mathcal{V}$. Then $\delta n^2/3\leq w\leq\alpha n^2/2$ and $\delta/\alpha\leq3/2$. We consider the sum $\sum_{j\in[w]}|S_j|$. By the definition of $(\alpha,\delta)$-system, $$n\cdot\delta n\leq\sum_{j\in[w]}|S_j|\leq3w\quad\Longrightarrow\quad\delta n^2/3\leq w.$$ Then we consider the number of pairs $\sum_{j\in[w]}\binom{|S_j|}{2}$, we can see $$w\leq\sum_{j\in[w]}\binom{|S_j|}{2}\leq\alpha\binom{n}{2}\leq\alpha n^2/2.$$ It follows that $\delta/\alpha\leq3/2$. \[lem:removebad\] Let $\mathcal{V}=(V_1,V_2,\ldots,V_n)$ ($V_i\subseteq{\mathbb{R}}^\ell$) be a list of vector spaces and $\mathcal{S}=(S_1,S_2,\ldots,S_w)$ be a list of sets. If $w\geq\delta n^2$ and $\mathcal{S}$ satisfies the first, second and fourth requirements in Definition \[def:adsystem\], then there exists a sublist $\mathcal{V}'$ of $\mathcal{V}$ and a sublist $\mathcal{S}'$ of $\mathcal{S}$ such that $|\mathcal{V}'|\geq\delta n/(2\alpha)$ and $\mathcal{S}'$ is an $(\alpha,\delta/2)$-system of $\mathcal{V}'$. We iteratively remove all $V_i$’s that appear in less than $\delta n/2$ sets, and the sets they appear in. There are $n$ $V_i$’s in total, so we can remove at most $n\cdot\delta n/2$ sets. When the procedure ends, we still have at least $\delta n^2-\delta n^2/2\geq\delta n^2/2$ sets. So we do not remove all of $V_1,V_2,\ldots,V_n$. For a remaining $V_i$, since it appears in at least $\delta n/2$ sets, we must still have at least $\delta n/(2\alpha)$ vector spaces left. Let $\mathcal{V}'$ be the list of these spaces and $\mathcal{S}'$ be the list of the remaining sets. We can see that $\mathcal{S}'$ is an $(\alpha,\delta/2)$-system of $\mathcal{V}'$. \[lem:linearmap\] Let $\mathcal{V}=(V_1,V_2,\ldots,V_n)$ ($V_i\subseteq{\mathbb{R}}^\ell$) be a list of vector spaces with an $(\alpha,\delta)$-system $\mathcal{S}=(S_1,S_2,\ldots,S_w)$. Then for any linear map $P:{\mathbb{R}}^\ell\mapsto{\mathbb{R}}^\ell$, $\mathcal{S}$ is also an $(\alpha,\delta)$-system of $\mathcal{V}'=(V_1',V_2',\ldots,V_n')$, where $V_i'=P(V_i)$. This is trivial since, if $V_{i_1}\subseteq V_{i_2}+V_{i_3}$, then $$V_{i_1}'=P(V_{i_1})\subseteq P(V_{i_2}+V_{i_3})=P(V_{i_2})+P(V_{i_3})=V_{i_2}'+V_{i_3}'.\qedhere$$ \[lem:removezero\] Let $\mathcal{V}=(V_1,V_2,\ldots,V_n)$ ($V_i\subseteq{\mathbb{R}}^\ell$) be a list of vector spaces with an $(\alpha,\delta)$-system $\mathcal{S}=(S_1,S_2,\ldots,S_w)$. Suppose we remove all zero ($\{{\boldsymbol{0}}\}$) spaces in $\mathcal{V}$ in the following way: 1. Let $n'$ be the number of nonzero (not $\{{\boldsymbol{0}}\}$) vector spaces in $\mathcal{V}$, and $\phi$ be a one-to-one mapping from the indices of nonzero spaces to $[n']$. We define $V_1'=V_{\phi^{-1}(1)}$, $V_2'=V_{\phi^{-1}(2)}$,…, $V_{n'}'=V_{\phi^{-1}(n')}$ to be all the nonzero spaces. 2. For each $S_j$ ($j\in[w]$), we define $S_j'=\{\phi(i):i\in S_j,V_i\neq\{{\boldsymbol{0}}\}\}$. 3. We remove the $S_j$’s that are empty. Let $\mathcal{S}'$ be the list of the remaining sets in $S_1',S_2',\ldots,S_w'$. Then $\mathcal{S}'$ is an $(\alpha,\delta')$-system of $\mathcal{V}'=(V_1',V_2',\ldots,V_{n'}')$, where $\delta'=\delta n/n'$. We first consider an $S_j$ containing 3 elements $i_1$, $i_2$ and $i_3$. If none of $V_{i_1}$, $V_{i_2}$, $V_{i_2}$ is $\{{\boldsymbol{0}}\}$, we have $S_j'=\{\phi(i_1),\phi(i_2),\phi(i_3)\}$ and it satisfies the second requirement. If exactly one of them is $\{{\boldsymbol{0}}\}$, say $V_{i_3}=\{{\boldsymbol{0}}\}$, we can see $S_j'=\{\phi(i_1),\phi(i_2)\}$ and $V_{\phi(i_1)}'=V_{\phi(i_2)}'$ by $V_{i_1}\subseteq V_{i_2}+\{{\boldsymbol{0}}\}$, $V_{i_2}\subseteq V_{i_1}+\{{\boldsymbol{0}}\}$. If exactly two of them are $\{{\boldsymbol{0}}\}$, say $V_{i_2}=V_{i_3}=\{{\boldsymbol{0}}\}$, then $V_{i_1}\subseteq V_{i_2}+V_{i_3}$ must also be $\{{\boldsymbol{0}}\}$, contradiction. If all $V_{i_1}$, $V_{i_2}$, $V_{i_3}$ are $\{{\boldsymbol{0}}\}$, $S_j'=\emptyset$ and it is removed. We then consider an $S_j$ containing 2 elements $i_1$ and $i_2$. If neither of $V_{i_1}$, $V_{i_2}$ is $\{{\boldsymbol{0}}\}$, we have $S_j'=\{\phi(i_1),\phi(i_2)\}$ and $V_{\phi(i_1)}'=V_{\phi(i_2)}'$. If one of them is $\{{\boldsymbol{0}}\}$, the other must be $\{{\boldsymbol{0}}\}$ by $V_{i_1}=V_{i_2}$, and $S_j'$ is removed. In summary, the first two requirements of the definition of $(\alpha,\delta)$-system are satisfied. We can also see that each $i\in[n']$ is contained in at least $\delta'n'=\delta n$ sets and each pair $\{i_1,i_2\}$ with $i_1\neq i_2\in[n']$ is contained in at most $\alpha$ sets, because we have only removed the sets containing only indices of zero spaces. Therefore the third and fourth requirements are also satisfied. Combining the above two lemmas, we have the following corollary. \[cor:remove\] Let $\mathcal{V}=(V_1,V_2,\ldots,V_n)$ ($V_i\subseteq{\mathbb{R}}^\ell$) be a list of vector spaces with an $(\alpha,\delta)$-system, and $P:{\mathbb{R}}^\ell\mapsto{\mathbb{R}}^\ell$ be any linear map. Define $\mathcal{V}'=(V_1',V_2',\ldots,V_{n'}')$ to be the list of nonzero spaces in $P(V_1),P(V_2),\ldots,P(V_n)$. Then $\mathcal{V}'$ has an $(\alpha,\delta')$-system, where $\delta'=\delta n/n'$. Theorem \[thm:sg\], will be derived from the following, more general statement, saying that the dimension $d$ is small if there is a $(\alpha,\delta)$-system. A vector space $V\subseteq{\mathbb{R}}^\ell$ is [*$k$-bounded*]{} if $\dim V\leq k$. \[thm:main\] Let $\mathcal{V}=(V_1,V_2,\ldots,V_n)$ ($V_i\subseteq{\mathbb{R}}^\ell$) be a list of $k$-bounded vector spaces with an $(\alpha,\delta)$-system and $d=\dim(V_1+V_2+\cdots+V_n)$, then $d=O(\alpha^2 k^4/\delta^2)$. We can easily reduce the high dimensional Sylvester-Gallai problem in ${\mathbb{C}}^\ell$ (Theorem \[thm:sg\]) to the setting of Theorem \[thm:main\] in ${\mathbb{R}}^\ell$ as shown below. Let $B_j=\{{\boldsymbol{v}}_{j1},{\boldsymbol{v}}_{j2},\ldots,{\boldsymbol{v}}_{jk}\}$ be a basis of $V_j$. Define $$V_j'=\operatorname{span}\big\{\operatorname{Re}({\boldsymbol{v}}_{j1}),\operatorname{Re}({\boldsymbol{v}}_{j2}),\ldots,\operatorname{Re}({\boldsymbol{v}}_{jk}),\operatorname{Im}({\boldsymbol{v}}_{j1}),\operatorname{Im}({\boldsymbol{v}}_{j2}),\ldots,\operatorname{Im}({\boldsymbol{v}}_{jk})\big\}\quad\forall j\in[n].$$ \[clm:complex\] $V_j'=\{\operatorname{Re}({\boldsymbol{v}}):{\boldsymbol{v}}\in V_j\}$ for every $j\in[n]$. For every ${\boldsymbol{v}}'\in V_j'$, there exist $\lambda_1,\lambda_2,\ldots,\lambda_k,\mu_1,\mu_2,\ldots,\mu_k\in{\mathbb{R}}$ such that $$\begin{split} {\boldsymbol{v}}'&=\sum_{s=1}^k\Big(\lambda_s\operatorname{Re}({\boldsymbol{v}}_{js})+\mu_s\operatorname{Im}({\boldsymbol{v}}_{js})\Big)=\sum_{s=1}^k\Big(\lambda_s\operatorname{Re}({\boldsymbol{v}}_{js})+\mu_s\operatorname{Re}(-i{\boldsymbol{v}}_{js})\Big) \\ & =\operatorname{Re}\left(\sum_{s=1}^k(\lambda_s-i\mu_s){\boldsymbol{v}}_{js}\right). \end{split}$$ Since $\lambda_1,\lambda_2,\ldots,\lambda_k,\mu_1,\mu_2,\ldots,\mu_k$ can take all values in ${\mathbb{R}}$, we can see the claim is proved. \[clm:triple\] Given a set $A$ with $r\geq3$ elements, we can construct a family of $r^2-r$ triples of elements in $A$ with following properties: 1) Every triple contains three distinct element; 2) Every element of $A$ appears in exactly $3(r-1)$ triples; 3) Every pair of two distinct elements in $A$ is contained together in at most $6$ triples. We call a $2k$-dimensional subspace $U \subset {\mathbb{C}}^\ell$ [*special*]{} if it contains at least three of $V_1,V_2,\ldots,V_n$. We define the [*size*]{} of a special space as the number of spaces among $V_1,V_2,\ldots,V_n$ contained in it. For a special space with size $r$, we add the $r^2-r$ triples of indices of the spaces in it with the properties in Claim \[clm:triple\]. Let $\mathcal{S}$ be the family of all these triples. We claim that $\mathcal{S}$ is a $(6,3\delta)$-system of $\mathcal{V}=(V_1',V_2',\ldots,V_n')$. For every triple $\{j_1,j_2,j_3\}\in\mathcal{S}$, we can see that $V_{j_1},V_{j_2},V_{j_3}$ are contained in the same $2k$-dimensional special space. And by $V_{j_1}\cap V_{j_2}=\{{\boldsymbol{0}}\}$, the space must be $V_{j_1}+V_{j_2}$ and hence $V_{j_3}\subseteq V_{j_1}+V_{j_2}$. By Claim \[clm:complex\], $$V_{j_3}'=\{\operatorname{Re}({\boldsymbol{v}}):{\boldsymbol{v}}\in V_{j_3}\}\subseteq\{\operatorname{Re}({\boldsymbol{u}})+\operatorname{Re}({\boldsymbol{w}}):{\boldsymbol{u}}\in V_{j_1},{\boldsymbol{w}}\in V_{j_2}\}=V_{j_1}'+V_{j_2}'.$$ Similarly, $V_{j_1}'\subseteq V_{j_2}'+V_{j_3}'$ and $V_{j_2}'\subseteq V_{j_1}'+V_{j_3}'$. One can see every pair in $[n]$ appears in at most $6$ triples because the corresponding two spaces are contained in at most one special space, and the pair appears at most $6$ times in the triples constructed from this special space. For every $j\in[n]$, there are at least $\delta n$ values of $j'\in[n]\setminus\{j\}$ such that there is a special space containing $V_j$ and $V_{j'}$. This implies that the number of triples that $j$ appears in is $$\sum_{\text{special space }U\atop V_j\subseteq U}3\big(\operatorname{size}(U)-1\big)=3\sum_{\text{special space }U\atop V_j\subseteq U}\big|\{j'\neq j:V_{j'}\subseteq U\}\big|\geq3\delta n.$$ Therefore $\mathcal{S}$ is a $(6,3\delta)$-system of $\mathcal{V}$. By Theorem \[thm:main\], $$\dim(V_1'+V_2'+\cdots+V_n')=O(6^2(2k)^4/(3\delta)^2)=O(k^4/\delta^2).$$ Note that $$\begin{split} V_1+V_2+\cdots+V_n & \subseteq \operatorname{span}\big\{\operatorname{Re}({\boldsymbol{v}}_{js}),\operatorname{Im}({\boldsymbol{v}}_{js})\big\}_{j\in[n],s\in[k]}\quad(\text{span with complex coefficients}), \\ V_1'+V_2'+\cdots+V_n' & = \operatorname{span}\big\{\operatorname{Re}({\boldsymbol{v}}_{js}),\operatorname{Im}({\boldsymbol{v}}_{js})\big\}_{j\in[n],s\in[k]}\quad(\text{span with real coefficients}). \end{split}$$ We thus have $\dim(V_1+V_2+\cdots+V_n)\leq\dim(V_1'+V_2'+\cdots+V_n')=O(k^4/\delta^2)$. A generalization of Barthe’s Theorem {#sec:barthe} ==================================== We prove Theorem \[thm:barthe\] in the following 3 subsections. In the fourth and last subsection, we state a convenient variant of the theorem (Theorem \[thm:barthec\]) that will be used later in the proof of our main result. The idea of the proof is similar to [@Bar98] (see also [@DSW14 Section 5]), which considers the maximum point of a function, and using the fact that all derivatives are $0$ the result is proved. Here we consider a similar function $f$ defined in Section \[sec:barthefunc\]. However, since our problem is more complicated, it is unclear whether we can find a maximum point at which all derivatives are $0$. Instead we will show that there is a point with very small derivatives in Section \[sec:optimumpt\], which is sufficient for our proof of the theorem in Section \[sec:barthepf\]. The function and basic properties {#sec:barthefunc} --------------------------------- Let $k_1,k_2,\ldots,k_n$ be the dimensions of $V_1,V_2,\ldots,V_n$ respectively and $m=k_1+k_2+\cdots+k_n$. Throughout our proof, we use pairs $(i,j)$ with $i\in[n]$, $j\in[k_i]$ to denote the element of $[m]$ of position $\sum_{i'<i}k_{i'} + j$. We define a vector ${\boldsymbol{\gamma}}\in{\mathbb{R}}^m$ as $$\gamma_{ij}=p_i\qquad\forall i\in[n],j\in[k_i].$$ For every $i\in[n]$, we fix $\{{\boldsymbol{v}}_{i1},{\boldsymbol{v}}_{i2},\ldots,{\boldsymbol{v}}_{ik_i}\}$ to be some basis of $V_i$ (not necessarily orthonormal). A set $I\subseteq[m]$ is called a [*good basis set*]{} if $$I=\bigcup_{i\in H}\big\{(i,1),(i,2),\ldots,(i,k_i)\big\}$$ for some $\mathcal{V}$-admissible basis set $H$. We can see that for any good basis set $I$, the set $\{{\boldsymbol{v}}_{ij}:(i,j)\in I\}$ is a basis of ${\mathbb{R}}^\ell$. For a list of vectors ${\boldsymbol{a}}_1,{\boldsymbol{a}}_2,\ldots,{\boldsymbol{a}}_q$ ($q\in{\mathbb{Z}}^+$), we use $[{\boldsymbol{a}}_1,{\boldsymbol{a}}_2,\ldots,{\boldsymbol{a}}_q]$ to denote the matrix consisting of columns ${\boldsymbol{a}}_1,{\boldsymbol{a}}_2,\ldots,{\boldsymbol{a}}_q$. Let $\mathbf{O}(s)$ be the group of $s\times s$ orthogonal matrices. The function $f:{\mathbb{R}}^m\times \mathbf{O}(k_1)\times \mathbf{O}(k_2)\times\cdots\times \mathbf{O}(k_n)\mapsto{\mathbb{R}}$ is defined as $$f({\boldsymbol{t}},R_1,\ldots,R_n)=\langle{\boldsymbol{\gamma}},{\boldsymbol{t}}\rangle-\ln\det\left(\sum_{i\in[n],j\in[k_i]}e^{t_{ij}}{\boldsymbol{x}}_{ij}{\boldsymbol{x}}_{ij}^T\right),$$ where, for every $i\in[n]$, the vectors ${\boldsymbol{x}}_{ij}$ are given by $$[{\boldsymbol{x}}_{i1},\ldots,{\boldsymbol{x}}_{ik_i}]=[{\boldsymbol{v}}_{i1},\ldots,{\boldsymbol{v}}_{ik_i}]R_i.$$ We note that here for every $i\in[n]$, $j\in[k_i]$, ${\boldsymbol{x}}_{ij}$ is a function of $R_i$ and $\{{\boldsymbol{x}}_{i1},\ldots,{\boldsymbol{x}}_{ik_i}\}$ is another basis of $V_i$. The next lemma shows that the function $f$ is bounded over its domain. The proof is similar to Proposition 3 in [@Bar98]. For completeness, we include the proof here. \[lem:bounded\] There is a constant $C\in{\mathbb{R}}$ such that $f({\boldsymbol{t}},R_1,\ldots,R_n)\leq C$ for all ${\boldsymbol{t}}\in{\mathbb{R}}^m$ and $R_i\in \mathbf{O}(k_i)$ ($i\in[n]$). In this proof, we use $\mathcal{F}=\binom{[m]}{\ell}$ to denote the family of all $\ell$-subsets of $[m]$. For a set $I\subseteq[m]$, let ${\boldsymbol{1}}_I\in\{0,1\}^m$ be the indicator vector of $I$, i.e. the $i$-th entry is $1$ iff $i\in I$. By the definition of the vector ${\boldsymbol{\gamma}}$, we can pick $\mu_I\in[0,1]$, $\sum_{I\in\mathcal{F}}\mu_I=1$ so that $${\boldsymbol{\gamma}}=\sum_{I\in\mathcal{F}}\mu_I{\boldsymbol{1}}_I,$$ and $\mu_I\neq0$ only when $I$ is a good basis set. In the proof, we will use the Cauchy-Binet formula which states that for a $\ell\times m$ matrix $A$ and an $m\times\ell$ matrix $B$, $$\label{eqn:cauchybinet} \det(AB)=\sum_{I\in\mathcal{F}}\det(A_I)\det(B_I),$$ where $A_I$ denotes the $\ell\times\ell$ matrix that consists of the subset of $A$’s columns with indices in $I$, and $B_I$ denotes the $\ell\times\ell$ matrix that consists of the subset of $B$’s rows with indices in $I$. We use $t_I$ to denote the sum of the entries in ${\boldsymbol{t}}$ with indices in $I$, and $L_I$ be the $\ell\times\ell$ submatrix of $[{\boldsymbol{x}}_{11},\ldots,\ldots,{\boldsymbol{x}}_{nk_n}]$ containing only the columns with indices in $I$. We then have $$\label{eqn:gammat} \langle{\boldsymbol{\gamma}},{\boldsymbol{t}}\rangle=\Big\langle\sum_I\mu_I{\boldsymbol{1}}_I,{\boldsymbol{t}}\Big\rangle=\sum_I\mu_It_I.$$ Using equations (\[eqn:cauchybinet\]) and (\[eqn:gammat\]), $$\begin{split} \det\left(\sum_{i\in[n],j\in[k_i]}e^{t_{ij}}{\boldsymbol{x}}_{ij}{\boldsymbol{x}}_{ij}^T\right) & = \det\left(\Big[{\boldsymbol{x}}_{11},\ldots,{\boldsymbol{x}}_{nk_n}\Big]\left[\begin{array}{c}e^{t_{11}}{\boldsymbol{x}}_{11}^T \\ \vdots \\ e^{t_{nk_n}}{\boldsymbol{x}}_{nk_n}^T\end{array}\right]\right) \\ & = \sum_{I\in\mathcal{F}} e^{t_I}\det(L_I)\cdot\det(L_I^T)\hspace{2.01cm}\text{(By~(\ref{eqn:cauchybinet}))} \\ & = \sum_{I\in\mathcal{F}:\mu_I\neq0}\mu_I\left(\frac{e^{t_I}}{\mu_I}\right)\det(L_I)^2+\sum_{I\in\mathcal{F}:\mu_I=0}e^{t_I}\det(L_I)^2 \\ & \geq \prod_{I\in\mathcal{F}:\mu_I\neq0}\left(\frac{e^{t_I}\det(L_I)^2}{\mu_I}\right)^{\mu_I}+0\hspace{0.99cm}\text{(AM-GM inequality)}\\ & = e^{\langle{\boldsymbol{\gamma}},{\boldsymbol{t}}\rangle}\cdot\prod_{I\in\mathcal{F}:\mu_I\neq0}\left(\frac{\det(L_I)^2}{\mu_I}\right)^{\mu_I}\hspace{1cm}\text{(By~(\ref{eqn:gammat}))}. \end{split}$$ Take the logarithm of both sides, $$f({\boldsymbol{t}},R_1,\ldots,R_n)=\langle{\boldsymbol{\gamma}},{\boldsymbol{t}}\rangle-\ln\det\left(\sum_{i\in[n],j\in[k_i]}e^{t_{ij}}{\boldsymbol{x}}_{ij}{\boldsymbol{x}}_{ij}^T\right)\leq\sum_{I:\mu_I\neq0}\mu_I\ln\left(\frac{\mu_I}{\det(L_I)^2}\right).$$ The right side is a function of the orthogonal matrices $R_1,R_2,\ldots,R_n$ because $L_I$ is a function of them. We use $\widetilde{f}(R_1,R_2,\ldots,R_n)$ to denote the right side of the above inequality. For $\mu_I\neq0$, $I$ must be a good basis set. Hence $\det(L_I)\neq0$ no matter what the orthogonal matrices $R_1,R_2,\ldots,R_n$ are, and $\widetilde{f}$ is a well-defined continuous function. Since $\widetilde{f}$ is defined on the compact set $\mathbf{O}(k_1)\times \mathbf{O}(k_2)\times\cdots\times \mathbf{O}(k_n)$, it must have a finite upper bound. And that is also an upper bound for the function $f$. Finding a point with small derivatives {#sec:optimumpt} -------------------------------------- We first define some notations. Let $$X=\sum\limits_{i\in[n],j\in[k_i]} e^{t_{ij}}{\boldsymbol{x}}_{ij}{\boldsymbol{x}}_{ij}^T$$ be a matrix valued function of ${\boldsymbol{t}},R_1,R_2,\ldots,R_n$. Then $$f({\boldsymbol{t}},R_1,\ldots,R_n)=\langle{\boldsymbol{\gamma}},{\boldsymbol{t}}\rangle-\ln\det(X).$$ Note that $X$ is always a positive definite matrix, since for any ${\boldsymbol{w}}\neq{\boldsymbol{0}}$, $${\boldsymbol{w}}^TX{\boldsymbol{w}}=\sum_{i\in[n],j\in[k_i]}e^{t_{ij}}\langle{\boldsymbol{x}}_{ij},{\boldsymbol{w}}\rangle^2>0,$$ when ${\boldsymbol{x}}_{11},\ldots,\ldots,{\boldsymbol{x}}_{nk_n}$ span the entire space (implied by $V_1+V_2+\cdots+V_n={\mathbb{R}}^\ell$). Define $M$ to be the $\ell\times\ell$ full rank matrix satisfying $M^TM=X^{-1}$. We note that $M$ is also a function of ${\boldsymbol{t}},R_1,R_2,\ldots,R_n$. In a later part of the proof we will show that the linear map defined by $M$ satisfies the requirement in Theorem \[thm:barthe\] when ${\boldsymbol{t}}$, $R_1,R_2,\ldots,R_n$ take appropriate values. We first find an appropriate value of $(R_1,R_2,\ldots,R_n)=(R_1^*({\boldsymbol{t}}),R_2^*({\boldsymbol{t}}),\ldots,R_n^*({\boldsymbol{t}}))$ for every ${\boldsymbol{t}}\in{\mathbb{R}}^m$, and then find some ${\boldsymbol{t}}^*$ with specific properties. \[lem:optr\] For every ${\boldsymbol{t}}\in{\mathbb{R}}^m$, there exists $\big(R_1^*({\boldsymbol{t}}),R_2^*({\boldsymbol{t}}),\ldots,R_n^*({\boldsymbol{t}})\big)$ satisfying 1. $f\big({\boldsymbol{t}},R_1^*({\boldsymbol{t}}),R_2^*({\boldsymbol{t}}),\ldots,R_n^*({\boldsymbol{t}})\big)=\max_{R_1,R_2,\ldots,R_n}\big\{f({\boldsymbol{t}},R_1,R_2,\ldots,R_n)\big\}$. 2. For every $i\in[n]$, if $t_{ij}=t_{ij'}$ for some $j\neq j'\in[k_i]$, then $$\langle M{\boldsymbol{x}}_{ij},M{\boldsymbol{x}}_{ij'}\rangle=0,$$ where $[{\boldsymbol{x}}_{i1},\ldots,{\boldsymbol{x}}_{ik_i}]=[{\boldsymbol{v}}_{i1},\ldots,{\boldsymbol{v}}_{ik_i}]R_i^*({\boldsymbol{t}})$. The first condition can be satisfied by the compactness of $\mathbf{O}(k_1)\times \mathbf{O}(k_2)\times\cdots\times \mathbf{O}(k_n)$. We will show how to change $(R_1^*({\boldsymbol{t}}),R_2^*({\boldsymbol{t}}),\ldots,R_n^*({\boldsymbol{t}}))$, which already satisfies the first condition, so that it satisfies the second condition while preserving the first condition. Fix an $i\in[n]$ and partition the indices of $(t_{i1},t_{i2},\ldots,t_{ik_i})$ into equivalence classes $J_1,J_2,\ldots,J_b\subseteq[k_i]$ such that for $j,j'$ in the same class $t_{ij}=t_{ij'}$ and for $j,j'$ in different classes $t_{ij}\neq t_{ij'}$. We use $t_{J_r}$ to denote the value of $t_{ij}$ for $j\in J_r$, and $L_{J_r}$ to denote the matrix consisting of all columns ${\boldsymbol{x}}_{ij}$ with $j\in J_r$. The terms in $X$ that depend on $R_i$ are $$\sum_{r\in[b]}\left(e^{t_{J_r}}\sum_{j\in J_r}{\boldsymbol{x}}_{ij}{\boldsymbol{x}}_{ij}^T\right)=\sum_{r\in[b]}\left(e^{t_{J_r}}\cdot L_{J_r}L_{J_r}^T\right)=\sum_{r\in[b]}\left(e^{t_{J_r}}\cdot L_{J_r}Q_rQ_r^TL_{J_r}^T\right),$$ where $Q_r$ can be taken to be any $|J_r|\times|J_r|$ orthogonal matrix. This means that if we change $R_i^*({\boldsymbol{t}})$ to $R_i^*({\boldsymbol{t}})\operatorname{diag}(Q_1,\ldots,Q_b)$ (here $\operatorname{diag}(Q_1,\ldots,Q_b)$ denotes the matrix in which the submatrix with row and column indices $J_r$ is $Q_r$), or equivalently change $L_{J_r}$ to $L_{J_r}Q_r$ for every $r\in[b]$, the matrix $X$ does not change, hence $M$ and $f$ do not change, and the first condition is preserved as $f$ is still the maximum for the fixed ${\boldsymbol{t}}$. For every $r\in[b]$, we can find a $Q_r$ such that the columns of $ML_{J_r}Q_r$ are orthogonal (consider the singular value decomposition of $ML_{J_r}$). Change $R_i^*({\boldsymbol{t}})$ to $R_i^*({\boldsymbol{t}})\operatorname{diag}(Q_1,\ldots,Q_b)$ and the second condition is satisfied while preserving the first condition. Doing this for every $i$ we can obtain an $(R_1^*({\boldsymbol{t}}),R_2^*({\boldsymbol{t}}),\ldots,R_n^*({\boldsymbol{t}}))$ satisfying both conditions. From now on we use $R_1^*({\boldsymbol{t}}),R_2^*({\boldsymbol{t}}),\ldots,R_n^*({\boldsymbol{t}})$ to denote the matrices satisfying the conditions in Lemma \[lem:optr\]. \[lem:derivative\] For any $\varepsilon>0$, there exists ${\boldsymbol{t}}^*\in{\mathbb{R}}^m$ such that for every $i\in[n],j\in[k_i]$. $$\left|\frac{\partial f}{\partial t_{ij}}\Big({\boldsymbol{t}}^*,R_1^*({\boldsymbol{t}}^*),R_2^*({\boldsymbol{t}}^*),\ldots,R_n^*({\boldsymbol{t}}^*)\Big)\right|\leq\varepsilon.$$ This lemma follows immediately from the following more general lemma. \[lem:smallderiv\] Let $\mathcal{A}\subseteq{\mathbb{R}}^{h}$ ($h\in{\mathbb{Z}}^+$) be a compact set. Let $f:{\mathbb{R}}^m\times\mathcal{A}\mapsto{\mathbb{R}}$ and $y^*:{\mathbb{R}}^m\mapsto\mathcal{A}$ be functions satisfying the following properties: 1. $f({\boldsymbol{x}},y)$ is bounded and continuous on ${\mathbb{R}}^m\times\mathcal{A}$. 2. For every ${\boldsymbol{x}}\in{\mathbb{R}}^m$, $f({\boldsymbol{x}},y^*({\boldsymbol{x}}))=\max_{y\in\mathcal{A}}\{f({\boldsymbol{x}},y)\}$. 3. For every fixed $y\in\mathcal{A}$, $f({\boldsymbol{x}},y)$ as a function of ${\boldsymbol{x}}$ is differentiable on ${\mathbb{R}}^m$. Then, for every $\varepsilon>0$, there exists an ${\boldsymbol{x}}^*\in{\mathbb{R}}^m$ such that for every $i\in[m]$, $$\left|\frac{\partial f}{\partial x_i}\Big({\boldsymbol{x}}^*,y^*({\boldsymbol{x}}^*)\Big)\right|\leq\varepsilon.$$ We denote by $f^*({\boldsymbol{x}})=f\big({\boldsymbol{x}},y^*({\boldsymbol{x}})\big)$. For the sake of contradiction, assume that for any ${\boldsymbol{x}}\in{\mathbb{R}}^m$, there is an index $i\in[m]$ such that $$\label{eqn:bigderivative} \left|\frac{\partial f}{\partial x_i}\Big({\boldsymbol{x}},y^*({\boldsymbol{x}})\Big)\right|>\varepsilon.$$ In particular, there is a derivative greater than $\varepsilon$ at ${\boldsymbol{x}}={\boldsymbol{0}}$. Therefore there exists an ${\boldsymbol{x}}_0\neq{\boldsymbol{0}}$ such that $$f^*({\boldsymbol{x}}_0)-f^*({\boldsymbol{0}})\geq f\big({\boldsymbol{x}}_0,y^*({\boldsymbol{0}})\big)-f\big({\boldsymbol{0}},y^*({\boldsymbol{0}})\big)\geq0.9\varepsilon\cdot\|{\boldsymbol{x}}_0-{\boldsymbol{0}}\|=0.9\varepsilon\cdot\|{\boldsymbol{x}}_0\|.$$ Define $$g({\boldsymbol{x}},y)=f({\boldsymbol{x}},y)-f^*({\boldsymbol{0}})-0.9\varepsilon\cdot\|{\boldsymbol{x}}\|,$$ and $$\mathcal{G}=\big\{({\boldsymbol{x}},y)\in{\mathbb{R}}^m\times\mathcal{A}: g({\boldsymbol{x}},y)\geq0\big\}=g^{-1}\big([0,+\infty)\big).$$ We can see $\mathcal{G}\neq\emptyset$ by ${\boldsymbol{x}}_0\in\mathcal{G}$. By Property 1, $f({\boldsymbol{x}},y)$ is bounded and any $({\boldsymbol{x}},y)$ with sufficiently large $\|{\boldsymbol{x}}\|$ cannot be in $\mathcal{G}$. Hence $\mathcal{G}$ is bounded. Since $g({\boldsymbol{x}},y)$ is a continuous function by Property 2, the set $\mathcal{G}=g^{-1}\big([0,+\infty)\big)$ must be closed. Therefore $\mathcal{G}$ is compact. Thus we can find $$Z=\max_{({\boldsymbol{x}},y)\in\mathcal{G}}\big\{\|{\boldsymbol{x}}\|\big\}.$$ Pick $({\boldsymbol{x}}_1,y_1)\in\mathcal{G}$ with $\|{\boldsymbol{x}}_1\|=Z$. The point $({\boldsymbol{x}}_1,y_1)$ is in the compact set $$\mathcal{B}_Z=\big\{{\boldsymbol{x}}\in{\mathbb{R}}^m:\|{\boldsymbol{x}}\|=Z\big\}\times\mathcal{A}.$$ Let $\big({\boldsymbol{x}}_1^*,y^*({\boldsymbol{x}}_1^*)\big)\in\mathcal{B}_Z$ be any point where $f$ is maximized over $\mathcal{B}_Z$. By $({\boldsymbol{x}}_1,y_1)\in\mathcal{G}$, we have $$\label{eqn:xpping} f^*({\boldsymbol{x}}_1^*)-f^*({\boldsymbol{0}})\geq f({\boldsymbol{x}}_1,y_1)-f^*({\boldsymbol{0}})\geq0.9\varepsilon\cdot\|{\boldsymbol{x}}_1\|=0.9\varepsilon\cdot\|{\boldsymbol{x}}_1^*\|.$$ By (\[eqn:bigderivative\]), there must be an ${\boldsymbol{x}}_2\neq{\boldsymbol{x}}_1^*$ such that $$\label{eqn:xpppdef} f^*({\boldsymbol{x}}_2)-f^*({\boldsymbol{x}}_1^*)\geq f\big({\boldsymbol{x}}_2,y^*({\boldsymbol{x}}_1^*)\big)-f\big({\boldsymbol{x}}_1^*,y^*({\boldsymbol{x}}_1^*)\big)\geq0.9\varepsilon\cdot\|{\boldsymbol{x}}_2-{\boldsymbol{x}}_1^*\|.$$ Note that $f^*({\boldsymbol{x}}_2)$ is strictly greater than $f^*({\boldsymbol{x}}_1^*)$. By the maximality of $f\big({\boldsymbol{x}}_1^*,y^*({\boldsymbol{x}}_1^*)\big)$ on $\mathcal{B}_Z$, we can see $\|{\boldsymbol{x}}_2\|\neq Z$. There are two cases: 1. $\|{\boldsymbol{x}}_2\|<Z$. This implies that the maximum value of $f$ over $$\mathcal{B}_{\leq Z}=\big\{{\boldsymbol{x}}\in{\mathbb{R}}^m:\|{\boldsymbol{x}}\|\leq Z\big\}\times\mathcal{A}$$ is at least $f^*({\boldsymbol{x}}_2)>f^*({\boldsymbol{x}}_1^*)$. Say $f$ archives the maximum value over $\mathcal{B}_{\leq Z}$ at $({\boldsymbol{x}},y)=\big({\boldsymbol{x}}_3,y^*({\boldsymbol{x}}_3)\big)$. Then we have $\|{\boldsymbol{x}}_3\|<Z$ by the maximality of $f\big({\boldsymbol{x}}_1^*,y^*({\boldsymbol{x}}_1^*)\big)$ on $\mathcal{B}_Z$. And ${\boldsymbol{x}}={\boldsymbol{x}}_3$ must be a local maximum of $f\big({\boldsymbol{x}},y^*({\boldsymbol{x}}_3)\big)$ with $y=y^*({\boldsymbol{x}}_3)$ fixed. Therefore $$\frac{\partial f}{\partial x_i}\Big({\boldsymbol{x}}_3,y^*({\boldsymbol{x}}_3)\Big)=0\quad\forall i\in[m],$$ violating (\[eqn:bigderivative\]). 2. $\|{\boldsymbol{x}}_2\|>Z$. By (\[eqn:xpping\]) and (\[eqn:xpppdef\]), we have $$\begin{split} f^*({\boldsymbol{x}}_2)-f^*({\boldsymbol{0}}) & =f^*({\boldsymbol{x}}_2)-f^*({\boldsymbol{x}}_1^*)+f^*({\boldsymbol{x}}_1^*)-f^*({\boldsymbol{0}}) \\ &\geq0.9\varepsilon\cdot\|{\boldsymbol{x}}_2-{\boldsymbol{x}}_1^*\|+0.9\varepsilon\cdot\|{\boldsymbol{x}}_1^*\| \\ &\geq0.9\varepsilon\cdot\|{\boldsymbol{x}}_2\|. \end{split}$$ Therefore $\big({\boldsymbol{x}}_2,y^*({\boldsymbol{x}}_2)\big)\in\mathcal{G}$. By the definition of $Z$, there should be $\|{\boldsymbol{x}}_2\|\leq Z$, contradiction. Thus the lemma is proved. Proof of Theorem \[thm:barthe\] {#sec:barthepf} ------------------------------- We apply Lemma \[lem:derivative\] with $\varepsilon'=\varepsilon/m$ and obtain a ${\boldsymbol{t}}^*$. In the remaining proof we will use $X$, $M$ and ${\boldsymbol{x}}_{ij}$ ($i\in[n],j\in[k_i]$) to denote their values when ${\boldsymbol{t}}={\boldsymbol{t}}^*$ and $R_i=R_i^*({\boldsymbol{t}}^*)$ ($i\in[n]$). $\langle M{\boldsymbol{x}}_{ij},M{\boldsymbol{x}}_{ij'}\rangle=0$ for every $i\in[n]$ and $j\neq j'\in[k_i]$. We fix $i_0\in[n], j_0\neq j_0'\in[k_{i_0}]$ and prove $\langle M{\boldsymbol{x}}_{i_0j_0},M{\boldsymbol{x}}_{i_0j_0'}\rangle=0$. If $t_{i_0j_0}^*=t_{i_0j_0'}^*$, this is guaranteed by Lemma \[lem:optr\]. We only consider the case that $t_{i_0j_0}^*\neq t_{i_0j_0'}^*$. Let $\theta\in{\mathbb{R}}$ be a variable, and define ${\boldsymbol{x}}_{ij}'$ for $i\in[n]$, $j\in[k_i]$ as follows. $${\boldsymbol{x}}_{ij}'=\begin{cases} \cos\theta\cdot{\boldsymbol{x}}_{i_0j_0}-\sin\theta\cdot{\boldsymbol{x}}_{i_0j_0'} & \quad(i,j)=(i_0,j_0), \\ \sin\theta\cdot{\boldsymbol{x}}_{i_0j_0}+\cos\theta\cdot{\boldsymbol{x}}_{i_0j_0'} & \quad(i,j)=(i_0,j_0'), \\ {\boldsymbol{x}}_{ij} & \quad\text{otherwise}. \end{cases}$$ We consider the following function $h:{\mathbb{R}}\mapsto{\mathbb{R}}$, $$h(\theta)=\langle{\boldsymbol{\gamma}},{\boldsymbol{t}}^*\rangle-\ln\det\left(\sum_{i\in[n],j\in[k_i]}e^{t_{ij}^*}{\boldsymbol{x}}_{ij}'{{\boldsymbol{x}}_{ij}'}^T\right).$$ $h(\theta)$ has a maximum at $\theta=0$. Let $R(\theta)$ be the $k_{i_0}\times k_{i_0}$ orthogonal matrix obtained from the identity matrix by changing the $(j_0,j_0)$, $(j_0',j_0')$ entries to $\cos\theta$, the $(j_0,j_0')$ entry to $\sin\theta$, and the $(j_0',j_0)$ entry to $-\sin\theta$. We can see $R(0)$ is the identity matrix and $$[{\boldsymbol{x}}_{i_01}',\ldots,{\boldsymbol{x}}_{i_0k_{i_0}}']=[{\boldsymbol{x}}_{i_01},\ldots,{\boldsymbol{x}}_{i_0k_{i_0}}]R(\theta).$$ Therefore for all $\theta\in{\mathbb{R}}$. $$\begin{split} h(\theta) & = f\Big({\boldsymbol{t}}^*,R_1^*({\boldsymbol{t}}^*),\ldots,R_{i_0-1}^*({\boldsymbol{t}}^*),R_{i_0}^*({\boldsymbol{t}}^*)\cdot R(\theta),R_{i_0+1}^*({\boldsymbol{t}}^*),\ldots,R_n^*({\boldsymbol{t}}^*)\Big) \\ & \leq f\Big({\boldsymbol{t}}^*,R_1^*({\boldsymbol{t}}^*),\ldots,R_{i_0-1}^*({\boldsymbol{t}}^*),R_{i_0}^*({\boldsymbol{t}}^*),R_{i_0+1}^*({\boldsymbol{t}}^*),\ldots,R_n^*({\boldsymbol{t}}^*)\Big) \\ & = h(0). \end{split}$$ Thus the claim is proved. Using $\frac{d}{ds}\ln\det(A)=\operatorname{tr}(A^{-1}\frac{d}{ds}A)$ for invertible matrix $A$ (Theorem 4 in [@Lax07 Chapter 9]), we can calculate the derivative of $h$. $$\begin{split} \frac{d h}{d\theta}(0) = & -\operatorname{tr}\Big[X^{-1}\Big(e^{t_{i_0j_0}^*}\left.\frac{d}{d\theta}\right|_{\theta=0}{\boldsymbol{x}}_{i_0j_0}'{{\boldsymbol{x}}_{i_0j_0}'}^T+e^{t_{i_0j_0'}^*}\left.\frac{d}{d\theta}\right|_{\theta=0}{\boldsymbol{x}}_{i_0j_0'}'{{\boldsymbol{x}}_{i_0j_0'}'}^T\Big)\Big] \\ = & -\operatorname{tr}\Big[X^{-1}\Big(e^{t_{i_0j_0}^*}\left.\frac{d}{d\theta}\right|_{\theta=0}(\cos\theta\cdot{\boldsymbol{x}}_{i_0j_0}-\sin\theta\cdot{\boldsymbol{x}}_{i_0j_0'})(\cos\theta\cdot{\boldsymbol{x}}_{i_0j_0}-\sin\theta\cdot{\boldsymbol{x}}_{i_0j_0'})^T \\ & \hspace{1.4cm} +e^{t_{i_0j_0'}^*}\left.\frac{d}{d\theta}\right|_{\theta=0}(\sin\theta\cdot{\boldsymbol{x}}_{i_0j_0}+\cos\theta\cdot{\boldsymbol{x}}_{i_0j_0'})(\sin\theta\cdot{\boldsymbol{x}}_{i_0j_0}+\cos\theta\cdot{\boldsymbol{x}}_{i_0j_0'})^T\Big)\Big] \\ = & -e^{t_{i_0j_0}^*}\operatorname{tr}\Big[\left.\frac{d}{d\theta}\right|_{\theta=0}(\cos\theta\cdot M{\boldsymbol{x}}_{i_0j_0}-\sin\theta\cdot M{\boldsymbol{x}}_{i_0j_0'})(\cos\theta\cdot M{\boldsymbol{x}}_{i_0j_0}-\sin\theta\cdot M{\boldsymbol{x}}_{i_0j_0'})^T\Big] \\ & -e^{t_{i_0j_0'}^*}\operatorname{tr}\Big[\left.\frac{d}{d\theta}\right|_{\theta=0}(\sin\theta\cdot M{\boldsymbol{x}}_{i_0j_0}+\cos\theta\cdot M{\boldsymbol{x}}_{i_0j_0'})(\sin\theta\cdot M{\boldsymbol{x}}_{i_0j_0}+\cos\theta\cdot M{\boldsymbol{x}}_{i_0j_0'})^T\Big] \\ = & -e^{t_{i_0j_0}^*}\big[-2\cdot\langle M{\boldsymbol{x}}_{i_0j_0},M{\boldsymbol{x}}_{i_0j_0'}\rangle\big]-e^{t_{i_0j_0'}^*}\big[2\cdot\langle M{\boldsymbol{x}}_{i_0j_0},M{\boldsymbol{x}}_{i_0j_0'}\rangle\big] \\ = & 2(e^{t_{i_0j_0}^*}-e^{t_{i_0j_0'}^*})\cdot\langle M{\boldsymbol{x}}_{i_0j_0},M{\boldsymbol{x}}_{i_0j_0'}\rangle. \end{split}$$ Since $h(0)$ is the maximum, we have $\frac{d h}{d\theta}(0)=0$. By $t_{i_0j_0}^*\neq t_{i_0j_0'}^*$, the above equation implies $\langle M{\boldsymbol{x}}_{i_0j_0},M{\boldsymbol{x}}_{i_0j_0'}\rangle=0$. Finally we are able to prove Theorem \[thm:barthe\]. With a slight abuse of notation, we also use $M$ to denote the linear map defined by the matrix $M$. We show that $M$ satisfies the requirement in Theorem \[thm:barthe\]. Let ${\boldsymbol{u}}_{ij}=M{\boldsymbol{x}}_{ij}/\|M{\boldsymbol{x}}_{ij}\|$ ($i\in[n]$, $j\in[k_i]$). Then $\{{\boldsymbol{u}}_{i1},{\boldsymbol{u}}_{i2},\ldots,{\boldsymbol{u}}_{ik_i}\}$ is an orthonormal basis of $M(V_i)$, and $$\label{eqn:projmatrix} \operatorname{Proj}_{M(V_i)}=[{\boldsymbol{u}}_{i1},{\boldsymbol{u}}_{i2},\ldots,{\boldsymbol{u}}_{ik_i}]\left[\begin{array}{c}{\boldsymbol{u}}_{i1}^T \\ \vdots \\ {\boldsymbol{u}}_{ik_i}^T\end{array}\right]=\sum_{j=1}^{k_i}{\boldsymbol{u}}_{ij}{\boldsymbol{u}}_{ij}^T.$$ We define $$\varepsilon_{ij}=\frac{\partial f}{\partial t_{ij}}\Big({\boldsymbol{t}}^*,R_1^*({\boldsymbol{t}}^*),R_2^*({\boldsymbol{t}}^*),\ldots,R_n^*({\boldsymbol{t}}^*)\Big)\in[-\frac{\varepsilon}{m},\frac{\varepsilon}{m}].$$ Note that $\frac{d}{ds}\ln\det(A)=\operatorname{tr}(A^{-1}\frac{d}{ds}A)$ for invertible matrix $A$ (Theorem 4 in [@Lax07 Chapter 9]). We have $$\varepsilon_{ij}= p_i-\operatorname{tr}\left(X^{-1}e^{t_{ij}^*}{\boldsymbol{x}}_{ij}{\boldsymbol{x}}_{ij}^T\right)=p_i-e^{t_{ij}^*}\cdot\operatorname{tr}\left(M{\boldsymbol{x}}_{ij}{\boldsymbol{x}}_{ij}^TM^T\right)=p_i-e^{t_{ij}^*}\cdot\|M{\boldsymbol{x}}_{ij}\|^2.$$ By the definition of $X$ and $M$, $$M^{-1}(M^T)^{-1}=X=\sum_{i\in[n],j\in[k_i]}e^{t_{ij}^*}{\boldsymbol{x}}_{ij}{\boldsymbol{x}}_{ij}^T\quad\Longrightarrow\quad\sum_{i\in[n],j\in[k_i]}e^{t_{ij}^*}(M{\boldsymbol{x}}_{ij})(M{\boldsymbol{x}}_{ij})^T=I_{\ell\times\ell}.$$ Therefore $$\sum_{i\in[n],j\in[k_i]}(p_i-\varepsilon_{ij}){\boldsymbol{u}}_{ij}{\boldsymbol{u}}_{ij}^T=\sum_{i\in[n],j\in[k_i]}e^{t_{ij}^*}\|M{\boldsymbol{x}}_{ij}\|^2\left(\frac{M{\boldsymbol{x}}_{ij}}{\|M{\boldsymbol{x}}_{ij}\|}\right)\left(\frac{M{\boldsymbol{x}}_{ij}}{\|M{\boldsymbol{x}}_{ij}\|}\right)^T=I_{\ell\times\ell}.$$ By (\[eqn:projmatrix\]), $$\Big\|\sum_{i=1}^np_i\operatorname{Proj}_{M(V_i)}-I_{\ell\times\ell}\Big\|=\Big\|\sum_{i\in[n],j\in[k_i]}\varepsilon_{ij}{\boldsymbol{u}}_{ij}{\boldsymbol{u}}_{ij}^T\Big\|\leq\frac{\varepsilon}{m}\sum_{i\in[n],j\in[k_i]}\|{\boldsymbol{u}}_{ij}{\boldsymbol{u}}_{ij}^T\|\leq\varepsilon.$$ Thus Theorem \[thm:barthe\] is proved. A convenient form of Theorem \[thm:barthe\] ------------------------------------------- We give Theorem \[thm:barthec\] which is implied by Theorem \[thm:barthe\] and is the form that will be used in our proof. Before stating the theorem, we need to define [*admissible sets*]{} and [*admissible vectors*]{} as Definition \[def:adsv\], which have weaker requirements than admissible basis sets and admissible basis vectors (Definition \[def:adbasic\]) as they are not required to span the entire arrangement. \[def:adsv\] Given a list of vector spaces $\mathcal{V}=(V_1,V_2,\ldots,V_n)$ ($V_i\subseteq{\mathbb{R}}^\ell$), a set $H\subseteq [n]$ is called a [*$\mathcal{V}$-admissible set*]{} if $\dim(\sum_{i\in H}V_i)=\sum_{i\in H}\dim(V_i),$ i.e. if every space with index in $H$ has intersection $\{{\boldsymbol{0}}\}$ with the span of the other spaces with indices in $H$. A [*$\mathcal{V}$-admissible vector*]{} is any indicator vector ${\boldsymbol{1}}_H$ of some $\mathcal{V}$-admissible set $H$. \[thm:barthec\] Given a list of vector spaces $\mathcal{V}=(V_1,V_2,\ldots,V_n)$ ($V_i\subseteq{\mathbb{R}}^\ell$) and a vector ${\boldsymbol{p}}\in{\mathbb{R}}^n$ in the convex hull of all $\mathcal{V}$-admissible vectors. Then there exists an invertible linear map $M:{\mathbb{R}}^\ell\mapsto{\mathbb{R}}^\ell$ such that for any unit vector ${\boldsymbol{w}}\in{\mathbb{R}}^\ell$, $$\sum_{i=1}^np_i\|\operatorname{Proj}_{M(V_i)}({\boldsymbol{w}})\|^2\leq2,$$ where $\operatorname{Proj}_{M(V_i)}({\boldsymbol{w}})$ is the projection of ${\boldsymbol{w}}$ onto $M(V_i)$. Note that with a slight abuse of notation we use $\operatorname{Proj}_{M(V_i)}$ to denote both the projection matrix and the projection map. We use $V$ to denote $V_1+V_2+\cdots+V_n$. Let $d=\dim(V)$ and $\{{\boldsymbol{b}}_1,{\boldsymbol{b}}_2,\ldots,{\boldsymbol{b}}_d\}$ be some orthonormal basis of $V$. We construct $(\mathcal{V}',{\boldsymbol{p}}')$ satisfying the conditions in Theorem \[thm:barthe\] in the following 2 steps. 1. In this step, we construct $\widetilde{\mathcal{V}}$ and ${\boldsymbol{p}}'$ so that ${\boldsymbol{p}}'$ is in the convex hull of all $\widetilde{\mathcal{V}}$-admissible basis vectors. Define $V_{n+1}=\operatorname{span}\{{\boldsymbol{b}}_1\}$, $V_{n+2}=\operatorname{span}\{{\boldsymbol{b}}_2\}$, …, $V_{n+d}=\operatorname{span}\{{\boldsymbol{b}}_d\}$ and $$\widetilde{\mathcal{V}}=(V_1,V_2,\ldots,V_n,V_{n+1},V_{n+2},\ldots,V_{n+d}).$$ For every $\mathcal{V}$-admissible set $H\subseteq[n]$, we can see that $H$ is also $\widetilde{\mathcal{V}}$-admissible, and there is a subset $G\subseteq\{n+1,n+2,\ldots,n+d\}$ such that $H'=H\cup G$ is a $\widetilde{\mathcal{V}}$-admissible basis set. Assume $${\boldsymbol{p}}=\sum_{\mathcal{V}\text{-admissible }H}\mu_H{\boldsymbol{1}}_H,$$ where $\mu_H\in[0,1]$ and $\sum\mu_H=1$. We define $${\boldsymbol{p}}'=\sum_{\mathcal{V}\text{-admissible }H}\mu_H{\boldsymbol{1}}_{H'},$$ where $H'$ is the $\widetilde{\mathcal{V}}$-admissible basis set extended from $H$ as above. We can see that ${\boldsymbol{p}}$ is a prefix of ${\boldsymbol{p}}'$, and ${\boldsymbol{p}}'$ is in the convex hull of all $\widetilde{\mathcal{V}}$-admissible basis vectors. 2. In this step, we construct $\mathcal{V}'$ based on $\widetilde{\mathcal{V}}$ so that the vector spaces span the entire Euclidean space. We find an isomorphism linear map $P:V\mapsto{\mathbb{R}}^d$ such that $P({\boldsymbol{b}}_i)={\boldsymbol{e}}_i$ for $i\in[d]$, where $\{{\boldsymbol{e}}_1,{\boldsymbol{e}}_2,\ldots,{\boldsymbol{e}}_d\}$ is the standard basis of ${\mathbb{R}}^d$. Define $$\mathcal{V}'=\big(V_1',V_2',\ldots,V_{n+d}'\big)=\big(P(V_1),P(V_2),\ldots,P(V_{n+d})\big).$$ We can see that $V_1'+V_2'+\cdots+V_{n+d}'={\mathbb{R}}^d$ and ${\boldsymbol{p}}'$ is in the convex hull of all $\mathcal{V}'$-admissible basis vectors. Hence $(\mathcal{V}',{\boldsymbol{p}}')$ satisfy the conditions in Theorem \[thm:barthe\]. Apply Theorem \[thm:barthe\] on $(\mathcal{V}',{\boldsymbol{p}}')$ with $\varepsilon=1$. There exist an invertible linear map $M':{\mathbb{R}}^d\mapsto{\mathbb{R}}^d$ such that $$\Big\|\sum_{i=1}^{n+d}p_i'\operatorname{Proj}_{M'(V_i')}-I_{d\times d}\Big\|\leq1.$$ For every unit vector ${\boldsymbol{w}}'\in{\mathbb{R}}^d$, we have $$\begin{split} &1\geq{\boldsymbol{w}}^T\left(\sum_{i=1}^{n+d}p_i'\operatorname{Proj}_{M'(V_i')}-I_{d\times d}\right){\boldsymbol{w}}=\sum_{i=1}^{n+d}p_i'\|\operatorname{Proj}_{M'(V_i')}({\boldsymbol{w}})\|^2-1, \\ \Longrightarrow\quad&\sum_{i=1}^{n+d}p_i'\|\operatorname{Proj}_{M'(V_i')}({\boldsymbol{w}})\|^2\leq2. \end{split}$$ Note that the linear map $P$ defined in Step 2 only changes orthonormal basis. We find an invertible linear map $M:{\mathbb{R}}^\ell\mapsto{\mathbb{R}}^\ell$ such that $M({\boldsymbol{v}})=P^{-1}(M'(P({\boldsymbol{v}})))$ for every ${\boldsymbol{v}}\in V$. Then for every unit vector ${\boldsymbol{w}}\in V$, $$\sum_{i=1}^{n+d}p_i'\|\operatorname{Proj}_{M(V_i)}({\boldsymbol{w}})\|^2\leq2.$$ It is easy to see that the same inequality holds for every unit vector ${\boldsymbol{w}}\in{\mathbb{R}}^\ell$. Recall that in Step 1, ${\boldsymbol{p}}$ is a prefix of ${\boldsymbol{p}}'$. The theorem is proved because the above inequality is stronger than the required. Proof of the main Theorem {#sec:main} ========================= Theorem \[thm:main\] will follow from the following theorem using a simple recursive argument. \[thm:main\_\] Let $\mathcal{V}=(V_1,V_2,\ldots,V_n)$ ($V_i\in{\mathbb{R}}^\ell$) be a list of $k$-bounded vector spaces with an $(\alpha,\delta)$-system and $d=\dim(V_1+V_2+\cdots+V_n)$, then for any $\beta\in(0,1)$, at least one of these two cases holds: 1. $d\leq400\alpha k^3/(\beta\delta)$, 2. There is a sublist of $q\geq\delta n/(20\alpha)$ spaces $(V_{i_1},V_{i_2},\ldots,V_{i_q})$ such that there are nonzero vectors ${\boldsymbol{z}}_1\in V_{i_1},{\boldsymbol{z}}_2\in V_{i_2},\ldots,{\boldsymbol{z}}_q\in V_{i_q}$ with $$\dim({\boldsymbol{z}}_1,{\boldsymbol{z}}_2,\ldots,{\boldsymbol{z}}_q)\leq\beta d.$$ Initially let $\mathcal{V}^{(0)}=(V_1^{(0)},V_2^{(0)},\ldots,V_{n_0}^{(0)})=\mathcal{V}$, $\delta_0=\delta$ and $d_0=d$, where $n_0=n$ and $V_i^{(0)}=V_i$. Starting with $t=0$, $\mathcal{V}^{(t)}=(V_1^{(t)},V_2^{(t)},\ldots,V_{n_t}^{(t)})$ is a list of $k$-bounded vectors spaces with an $(\alpha,\delta_t)$-system and $d_t=\dim(V_1^{(t)}+V_2^{(t)}+\cdots+V_{n_t}^{(t)})$. We apply Theorem \[thm:main\_\] on $\mathcal{V}^{(t)}$. - If the first case of Theorem \[thm:main\_\] holds, i.e. $d_t\leq400\alpha k^3/(\beta\delta_t)$, terminate. - If the second case of Theorem \[thm:main\_\] holds, i.e. there exist ${\boldsymbol{z}}_1,{\boldsymbol{z}}_2,\ldots,{\boldsymbol{z}}_q$ from $q\geq\delta_tn_t/(20\alpha)$ spaces such that $\dim({\boldsymbol{z}}_1,{\boldsymbol{z}}_2,\ldots,{\boldsymbol{z}}_q)\leq\beta d$. We find a linear map $P:{\mathbb{R}}^\ell\mapsto{\mathbb{R}}^\ell$ whose kernel equals $\operatorname{span}\{{\boldsymbol{z}}_1,{\boldsymbol{z}}_2,\ldots,{\boldsymbol{z}}_q\}$. Define $$\mathcal{V}^{(t+1)}=(V_1^{(t+1)},V_2^{(t+1)},\ldots,V_{n_{t+1}}^{(t+1)})$$ as the list of nonzero spaces in $P(V_1^{(t)}),P(V_1^{(t)}),\ldots,P(V_{n_t}^{(t)})$. By Corollary \[cor:remove\], $\mathcal{V}^{(t+1)}$ has an $(\alpha,\delta_{t+1})$-system for $\delta_{t+1}=\delta_tn_t/n_{t+1}$. Let $t\leftarrow t+1$ and repeat the procedure. In the above procedure, we can see $\delta_tn_t=\delta_{t-1}n_{t-1}=\cdots=\delta_0n_0=\delta n$. Note that each step we map vectors from $q\geq\delta_tn_t/(20\alpha)=\delta n/(20\alpha)$ spaces to ${\boldsymbol{0}}$, hence $$\dim V_1^{(t+1)}+\dim V_2^{(t+1)}+\cdots+\dim V_{n_{t+1}}^{(t+1)}\leq\dim V_1^{(t)}+\dim V_2^{(t)}+\cdots+\dim V_{n_t}^{(t)}-\frac{\delta n}{20\alpha}.$$ Since initially $\dim V_1+\dim V_2+\cdots+\dim V_n\leq kn$, we must terminate after at most $$\frac{kn}{\delta n/(20\alpha)}=\frac{20\alpha k}{\delta}$$ steps. At the $t$-th step, $$d_t\geq(1-\beta)d_{t-1}\geq\cdots\geq(1-\beta)^td_0=(1-\beta)^td.$$ And if the $t$-th step is the last step we have $d_t\leq400\alpha k^3/(\beta\delta_t)\leq400\alpha k^3/(\beta\delta)$ by $\delta_t\geq\delta$ (implied by $\delta_tn_t=\delta n$ and $n_t\leq n$). Therefore $$d\leq\left(\frac{1}{1-\beta}\right)^{20\alpha k/\delta}\cdot\frac{400\alpha k^3}{\beta\delta}.$$ We assign $\beta=\min\{1/2,\delta/(\alpha k)\}$. It is easy to verify that $1/(1-\beta)^{\alpha k/\delta}\leq4$ in both cases $\delta/(\alpha k)<1/2$ and $\delta/(\alpha k)\geq1/2$. Therefore $$d\leq4^{20}\cdot\frac{400\alpha k^3}{\beta\delta}=O(\alpha^2k^4/\delta^2),$$ and Theorem \[thm:main\] is proved. Proof of Theorem \[thm:main\_\] – a special case ------------------------------------------------ In this subsection, we consider the case that all vector spaces are ‘well separated’. Two vector spaces $V,V'\subseteq{\mathbb{R}}^\ell$ are [*$\tau$-separated*]{} if $|\langle{\boldsymbol{u}},{\boldsymbol{u}}'\rangle|\leq1-\tau$ for any two unit vectors ${\boldsymbol{u}}\in V$ and ${\boldsymbol{u}}'\in V'$. We will use the following two simple lemmas about $\tau$-separated spaces. \[lem:smallcoef\] Given two vector spaces $V,V'\subseteq{\mathbb{R}}^\ell$ that are $\tau$-separated and let $B=\{{\boldsymbol{u}}_1,{\boldsymbol{u}}_2,\ldots,{\boldsymbol{u}}_{k_1}\}$ and $B'=\{{\boldsymbol{u}}_1',{\boldsymbol{u}}_2',\ldots,{\boldsymbol{u}}_{k_2}'\}$ be orthonormal bases for $V,V'$ respectively. For any unit vector ${\boldsymbol{u}}\in V+V'$, if we write ${\boldsymbol{u}}$ as $${\boldsymbol{u}}=\lambda_1{\boldsymbol{u}}_1+\lambda_2{\boldsymbol{u}}_2+\cdots+\lambda_{k_1}{\boldsymbol{u}}_{k_1}+\mu_1{\boldsymbol{u}}_1'+\mu_2{\boldsymbol{u}}_2'+\cdots+\mu_{k_2}{\boldsymbol{u}}_{k_2}',$$ then the coefficients satisfy $\lambda_1^2+\lambda_2^2+\cdots+\lambda_{k_1}^2+\mu_1^2+\mu_2^2+\cdots+\mu_{k_2}^2\leq\frac{1}{\tau}.$ Let ${\boldsymbol{v}}=\lambda_1{\boldsymbol{u}}_1+\lambda_2{\boldsymbol{u}}_2+\cdots+\lambda_{k_1}{\boldsymbol{u}}_{k_1}$ and ${\boldsymbol{w}}=\mu_1{\boldsymbol{u}}_1'+\mu_2{\boldsymbol{u}}_2'+\cdots+\mu_{k_2}{\boldsymbol{u}}_{k_2}'$. We have $$\begin{split} 1&=\|{\boldsymbol{u}}\|^2=\|{\boldsymbol{v}}+{\boldsymbol{w}}\|^2=\|{\boldsymbol{v}}\|^2+\|{\boldsymbol{w}}\|^2+2\langle{\boldsymbol{v}},{\boldsymbol{w}}\rangle\geq\|{\boldsymbol{v}}\|^2+\|{\boldsymbol{w}}\|^2-2(1-\tau)\|{\boldsymbol{v}}\|\|{\boldsymbol{w}}\| \\ &\geq\tau(\|{\boldsymbol{v}}\|^2+\|{\boldsymbol{w}}\|^2) \\ &=\tau(\lambda_1^2+\lambda_2^2+\cdots+\lambda_{k_1}^2+\mu_1^2+\mu_2^2+\cdots+\mu_{k_2}^2).\qedhere \end{split}$$ \[lem:spacetobasis\] Given two vector spaces $V,V'\subseteq{\mathbb{R}}^\ell$ and let $B=\{{\boldsymbol{u}}_1,{\boldsymbol{u}}_2,\ldots,{\boldsymbol{u}}_{k_1}\}$ be an orthonormal basis of $V$. If $V$ and $V'$ are not $\tau$-separated, there must exist $j\in[k_1]$ such that $\|\operatorname{Proj}_{V'}({\boldsymbol{u}}_j)\|^2\geq(1-\tau)^2/k_1$, where $\operatorname{Proj}_{V'}({\boldsymbol{u}}_j)$ is the projection of ${\boldsymbol{u}}_j$ onto $V'$. Let ${\boldsymbol{u}}\in V$, ${\boldsymbol{u}}'\in V'$ be unit vectors such that $|\langle{\boldsymbol{u}},{\boldsymbol{u}}'\rangle|>1-\tau$. Then $\|\operatorname{Proj}_{V'}({\boldsymbol{u}})\|\geq|\langle{\boldsymbol{u}},{\boldsymbol{u}}'\rangle|>1-\tau$. Suppose ${\boldsymbol{u}}=\lambda_1{\boldsymbol{u}}_1+\lambda_2{\boldsymbol{u}}_2+\cdots+\lambda_{k_1}{\boldsymbol{u}}_{k_1}$, where $\lambda_1^2+\lambda_2^2+\cdots+\lambda_{k_1}^2=1$. We have $$\begin{split} (1-\tau)^2 & <\|\operatorname{Proj}_{V'}({\boldsymbol{u}})\|^2\leq\Big(\sum_{j=1}^{k_1}|\lambda_j|\cdot\|\operatorname{Proj}_{V'}({\boldsymbol{u}}_j)\|\Big)^2\leq\Big(\sum_{j=1}^{k_1}\lambda_j^2\Big)\Big(\sum_{j=1}^{k_1}\|\operatorname{Proj}_{V'}({\boldsymbol{u}}_j)\|^2\Big) \\ & = \sum_{j=1}^{k_1}\|\operatorname{Proj}_{V'}({\boldsymbol{u}}_j)\|^2. \end{split}$$ Therefore there exists $j\in[k_1]$ such that $\|\operatorname{Proj}_{V'}({\boldsymbol{u}}_j)\|^2\geq(1-\tau)^2/k_1$. We will need the following lower bound for the rank of a diagonal dominating matrix. The same lemma for Hermitian matrices was proved in [@BDWY-pnas]. Here we change the proof slightly and show that the consequence also holds for an arbitrary matrix. \[lem:diagdom\] Let $D=(d_{ij})$ be a complex $m\times m$ matrix and $L,K$ be positive real numbers. If $d_{ii}=L$ for every $i\in[m]$ and $\sum_{i\neq j}|d_{ij}|^2\leq K$, then $\operatorname{rank}(D)\geq m-K/L^2$. Let $r$ be the rank of $D$. Consider the singular value decomposition of $D$, say $D=U\Sigma V$, where $U,V$ are unitary matrices and $\Sigma$ is a non-negative diagonal matrix. Let $\sigma_1,\sigma_2,\ldots,\sigma_r$ be the nonzero singular values on the diagonal of $\Sigma$. $$\begin{split} (mL)^2&=\operatorname{tr}(D)^2=\operatorname{tr}(U\Sigma V)^2=\operatorname{tr}(\Sigma(VU))^2\leq(\sigma_1+\cdots+\sigma_r)^2\leq r(\sigma_1^2+\cdots+\sigma_r^2) \\ &=r\|D\|_F^2\leq r(mL^2+K). \end{split}$$ Therefore $r\geq(mL)^2/(mL^2+K)=m^2/(m+K/L^2)\geq m-K/L^2$. The following theorem handles the ‘well separated case’ of Theorem \[thm:main\_\]. \[thm:sep\] Let $\mathcal{V}=(V_1,V_2,\ldots,V_n)$ ($V_i\in{\mathbb{R}}^\ell$) be a list of $k$-bounded vector spaces with an $(\alpha,\delta)$-system $\mathcal{S}=(S_1,S_2,\ldots,S_w)$ and $d=\dim(V_1+V_2+\cdots+V_n)$. If for every $j\in[w]$ and $\{i_1,i_2\}\subseteq S_j$, $V_{i_1}$ and $V_{i_2}$ are $\tau$-separated, then $d\leq\alpha k/(\tau\delta)$. Let $k_1,k_2,\ldots,k_n$ be the dimensions of $V_1,V_2,\ldots,V_n$, and $m=k_1+k_2+\cdots+k_n$. For every $i\in[n]$, fix $B_i=\{{\boldsymbol{u}}_{i1},{\boldsymbol{u}}_{i2},\ldots,{\boldsymbol{u}}_{i{k_i}}\}$ to be some orthonormal basis of $V_i$. We use $A$ to denote the $m\times\ell$ matrix whose rows are ${\boldsymbol{u}}_{11}^T,\ldots,\ldots,{\boldsymbol{u}}_{nk_n}^T$. We will bound $d=\operatorname{rank}(A)$ by constructing a high rank $m\times m$ matrix $D$ satisfying $DA=0$. For $s\in[m]$, we use $\psi(s)\in[n]$ to denote the number satisfying $$k_1+k_2+\cdots+k_{\psi(s)-1}+1\leq s\leq k_1+k_2+\cdots+k_{\psi(s)-1}+k_{\psi(s)}.$$ In other words, the $s$-th row of $A$ is a vector in $B_{\psi(s)}$. For every $s\in[m]$, there is a vector ${\boldsymbol{y}}_s\in{\mathbb{R}}^m$ satisfying ${\boldsymbol{y}}_s^TA={\boldsymbol{0}}^T$, $y_{ss}=\lceil\delta n\rceil$, and $\sum_{t\neq s}y_{st}^2\leq\alpha\lceil\delta n\rceil/\tau$. Say the $s$-th row of $A$ is ${\boldsymbol{u}}^T$, where ${\boldsymbol{u}}\in B_{\psi(s)}$. Let $J\subseteq[w]$ be a set of size $|J|=\lceil\delta n\rceil$ such that for every $j\in J$, $S_j$ contains $\psi(s)$. We construct a vector ${\boldsymbol{c}}_j$ for every $j\in J$ as following. - If $S_j$ contains 3 elements $\{\psi(s),i,i'\}$, we have $\lambda_1,\lambda_2,\ldots,\lambda_{k_i},\mu_1,\mu_2,\ldots,\mu_{k_{i'}}\in{\mathbb{R}}$ such that $${\boldsymbol{u}}-\lambda_1{\boldsymbol{u}}_{i1}-\lambda_2{\boldsymbol{u}}_{i2}-\cdots-\lambda_{k_i}{\boldsymbol{u}}_{ik_i}-\mu_1{\boldsymbol{u}}_{i'1}-\mu_2{\boldsymbol{u}}_{i'2}-\cdots-\mu_{k_{i'}}{\boldsymbol{u}}_{i'k_{i'}}={\boldsymbol{0}}.$$ We can obtain from this equation a vector ${\boldsymbol{c}}_j$ such that ${\boldsymbol{c}}_j^TA={\boldsymbol{0}}^T$, $c_{js}=1$, and by Lemma \[lem:smallcoef\] $$\sum_{t\neq s}c_{jt}^2=\lambda_1^2+\lambda_2^2+\cdots+\lambda_{k_i}^2+\mu_1^2+\mu_2^2+\cdots+\mu_{k_{i'}}^2\leq\frac{1}{\tau}.$$ - If $S_j$ contains 2 elements $\{\psi(s),i\}$, there exist $\lambda_1,\lambda_2,\ldots,\lambda_{k_i}$ with $\lambda_1^2+\lambda_2^2+\cdots+\lambda_{k_i}^2=1$ such that $${\boldsymbol{u}}-\lambda_1{\boldsymbol{u}}_{i1}-\lambda_2{\boldsymbol{u}}_{i2}-\cdots-\lambda_{k_i}{\boldsymbol{u}}_{ik_i}={\boldsymbol{0}}.$$ We can obtain from this equation a vector ${\boldsymbol{c}}_j$ such that ${\boldsymbol{c}}_j^TA={\boldsymbol{0}}^T$, $c_{js}=1$, and $$\sum_{t\neq s}c_{jt}^2=\lambda_1^2+\lambda_2^2+\cdots+\lambda_{k_i}^2=1\leq1/\tau.$$ In either case we obtain a ${\boldsymbol{c}}_j$ such that ${\boldsymbol{c}}_j^TA={\boldsymbol{0}}^T$, $c_{js}=1$ and $\sum_{t\neq s}c_{jt}^2\leq1/\tau$. We define $${\boldsymbol{y}}_s=\sum_{j\in J}{\boldsymbol{c}}_j.$$ We have ${\boldsymbol{y}}_s^TA={\boldsymbol{0}}^T$ and $y_{ss}=\lceil\delta n\rceil$. We consider $\sum_{t\neq s}y_{st}^2$. From the above construction of ${\boldsymbol{c}}_j$, we can see $c_{jt}\neq0$ ($t\neq s$) only when $\psi(t)\neq\psi(s)$ and $\{\psi(s),\psi(t)\}\subseteq S_j$. Hence for every $t\neq s$, there are at most $\alpha$ nonzero values in $\{c_{jt}\}_{j\in J}$. It follows that $$\sum_{t\neq s}y_{st}^2=\sum_{t\neq s}\left(\sum_{j\in J}c_{jt}\right)^2\leq\alpha\sum_{t\neq s}\left(\sum_{j\in J}c_{jt}^2\right)=\alpha\sum_{j\in J}\left(\sum_{t\neq s}c_{jt}^2\right)\leq\frac{\alpha\lceil\delta n\rceil}{\tau}.$$ Thus the claim is proved. Define $D$ to be the matrix consists of rows ${\boldsymbol{y}}_1^T,{\boldsymbol{y}}_2^T,\ldots,{\boldsymbol{y}}_m^T$. Then every entry on the diagonal of $D$ is $\lceil\delta n\rceil$, and the sum of squares of all entries off the diagonal is at most $\alpha\lceil\delta n\rceil m/\tau$. Apply Lemma \[lem:diagdom\] on $D$, and we have $$\operatorname{rank}(D)\geq m-\frac{\alpha\lceil\delta n\rceil m/\tau}{\lceil\delta n\rceil^2}=m-\frac{\alpha m}{\tau\lceil\delta n\rceil}\geq m-\frac{\alpha k}{\tau\delta}.$$ By $DA=0$, the rank of $A$ is $d\leq\alpha k/(\tau\delta)$. Proof of Theorem \[thm:main\_\] – general case ---------------------------------------------- Now we prove Theorem \[thm:main\_\]. We assume that the first case of Theorem \[thm:main\_\] does not hold, i.e. $d>400\alpha k^3/(\beta\delta)$. We will show the second case holds. \[lem:bigprob\] At least one of the following two cases holds: 1. The second case of Theorem \[thm:main\_\] holds, i.e. there exists a sublist of $q\geq\delta n/(20\alpha)$ spaces $(V_{i_1},V_{i_2},\ldots,V_{i_q})$ such that there are nonzero vectors ${\boldsymbol{z}}_1\in V_{i_1},{\boldsymbol{z}}_2\in V_{i_2},\ldots,{\boldsymbol{z}}_q\in V_{i_q}$ with $$\dim({\boldsymbol{z}}_1,{\boldsymbol{z}}_2,\ldots,{\boldsymbol{z}}_q)\leq\beta d.$$ 2. There exist a distribution $\mathcal{D}$ on $\mathcal{V}$-admissible sets and an $I\subseteq[n]$ with $|I|\geq(1-\delta/(10\alpha))n$ such that for every $i\in I$, $$\Pr_{H\sim\mathcal{D}}[i\in H]\geq\frac{\beta d}{4kn}.$$ We sample a $\mathcal{V}$-admissible set as follows: Initially let $F=\emptyset$. In each step we pick a space $V_{i_0}$ among $V_1,V_2,\ldots,V_n$ with $V_{i_0}\bigcap\sum_{i\in F}V_i=\{{\boldsymbol{0}}\}$, and add $i_0$ to $F$. If such a $V_{i_0}$ does not exist, the procedure terminates. Let $H$ be the final value of $F$. Clearly, $H$ is $\mathcal{V}$-admissible. Let $\mathcal{D}$ be the distribution of $H$. We will show that if there does not exist an $I\subseteq[n]$ such that $\mathcal{D}$ and $I$ satisfy the second case, the first case must hold. In the above random procedure, if it is possible that $\dim(\sum_{i\in H}V_i)\leq\beta d$, then there are nonzero vectors ${\boldsymbol{z}}_1\in V_1$, ${\boldsymbol{z}}_2\in V_2$, …, ${\boldsymbol{z}}_n\in V_n$ contained in $\sum_{i\in H}V_i$, which has dimension at most $\beta d$. Since $\delta/\alpha\leq3/2$ by Lemma \[lem:awd\], we have $n\geq\delta n/(20\alpha)$ and the lemma is proved. In the remaining proof we assume that $H$ always satisfies $\dim\left(\sum_{i\in H}V_i\right)>\beta d.$ This implies that there are always at least $\beta d/k$ elements in $H$. Fix $t=\lceil \beta d/(2k)\rceil<|H|$ (recall our assumption $d>400\alpha k^3/(\beta\delta)$), and let $V_{j_1},V_{j_2},\ldots,V_{j_t}$ be the first $t$ spaces. We assume that the second case of the lemma does not hold, i.e. there is a set $X$ of at least $\delta n/(10\alpha)$ $i$’s with $\Pr[i\in H]\leq \beta d/(4kn)\leq t/(2n)$. We will show the first case holds. For every $i\in X$, $\Pr\big[V_i\cap(V_{j_1}+V_{j_2}+\cdots+V_{j_t})\neq\{{\boldsymbol{0}}\}\big]\geq\frac{1}{2}$. The proof is similar to Claim 6.4 in [@DSW14]. There are 3 disjoint events $$\begin{aligned} E_1 & : & V_i\cap(V_{j_1}+V_{j_2}+\cdots+V_{j_t})=\{{\boldsymbol{0}}\}, \\ E_2 & : & i\in\{j_1,j_2,\ldots,j_t\},\text{ i.e. $i$ is picked in the first $t$ steps}, \\ E_3 & : & i\notin\{j_1,j_2,\ldots,j_t\}\text{ and }V_i\cap(V_{j_1}+V_{j_2}+\cdots+V_{j_t})\neq\{{\boldsymbol{0}}\}.\end{aligned}$$ As long as $i$ is not picked, the $s$-th ($s\in[t]$) element $j_s$, conditioned on $ E_1\cup E_2$, is sampled uniformly at random from $([n]\setminus J_s)\cup\{i\}$, where $J_s=\{j\in[n]:V_j\cap(V_i+V_{j_1}+V_{j_2}+\cdots+V_{j_{s-1}})\neq\{{\boldsymbol{0}}\}$. Therefore the probability that $V_i$ is not picked in the first $t$ steps conditioning on $E_1\cup E_2$ is $$\Pr[E_1\mid E_1\cup E_2]\leq(1-\frac{1}{n})(1-\frac{1}{n-1})\cdots(1-\frac{1}{n-t+1})=\frac{n-t}{n}.$$ Hence $t/(2n)\geq\Pr[i\in H]\geq\Pr[E_2]\geq(t/n)\Pr[E_1\cup E_2]$. It follows that $$\Pr\Big[V_i\cap(V_{j_1}+V_{j_2}+\cdots+V_{j_t})\neq\{{\boldsymbol{0}}\}\Big]\geq\Pr[E_3]=1-\Pr[E_1+E_2]\geq1-\frac{1}{2}=\frac{1}{2}.$$ Thus the claim is proved. Therefore the expected number of $i$’s in $X$ with $V_i\cap(V_{j_1}+V_{j_2}+\cdots+V_{j_t})\neq\{{\boldsymbol{0}}\}$ is at least $|X|/2\geq\delta n/(20\alpha)$. Each of these $V_i$’s has a nonzero vector contained in $V_{j_1}+V_{j_2}+\cdots+V_{j_t}$. The first case is proved by $\dim(V_{j_1}+V_{j_2}+\cdots+V_{j_t})\leq kt\leq\beta d$. To prove Theorem \[thm:main\_\], we only need to consider the second case in Lemma \[lem:bigprob\]. Let $p_i$ ($i\in[n]$) be the probability that $i$ is contained in $H\sim\mathcal{D}$, and $I\subseteq[n]$ be the set such that $|I|\geq(1-\delta/(10\alpha))n$ and $p_i\geq \beta d/(4kn)$ for every $i\in I$. We use $k_1,k_2,\ldots,k_n$ to denote the dimensions of $V_1,V_2,\ldots,V_n$. The vector ${\boldsymbol{p}}=(p_1,p_2,\ldots,p_n)$ is in the convex hull of $\mathcal{V}$-admissible vectors. For every $\mathcal{V}$-admissible set $H$, we use $q_H$ to denote the probability that $H$ is picked according to $\mathcal{D}$, and ${\boldsymbol{1}}_H$ to denote the $\mathcal{V}$-admissible vector corresponding to $H$. Then, $${\boldsymbol{p}}=(p_1,p_2,\ldots,p_n)=\sum_{\mathcal{V}\text{-admissible }H}q_H{\boldsymbol{1}}_H$$ and $p_i$ is exactly the probability that $i \in H$. We apply Theorem \[thm:barthec\] with the ${\boldsymbol{p}}=(p_1,p_2,\ldots,p_n)$, and obtain an invertible linear map $M:{\mathbb{R}}^\ell\mapsto{\mathbb{R}}^\ell$ such that for any unit vector ${\boldsymbol{w}}\in{\mathbb{R}}^\ell$, $$\sum_{i=1}^np_i\|\operatorname{Proj}_{V_i'}({\boldsymbol{w}})\|^2\leq2,$$ where $V_i'$ denotes $M(V_i)$. Since $p_i\geq \beta d/(4kn)$ for every $i\in I$, we have $$\label{eqn:bartheapp} \sum_{i\in I}\|\operatorname{Proj}_{V_i'}({\boldsymbol{w}})\|^2\leq\frac{8kn}{\beta d}.$$ We will reduce the problem to the special case discussed in the previous subsection. We say a pair $\{i_1,i_2\}\subseteq[n]$ is [*bad*]{} if $V_{i_1}',V_{i_2}'$ are not $0.5$-separated. Let $\mathcal{S}=(S_1,S_2,\ldots,S_w)$ be the $(\alpha,\delta)$-system of $\mathcal{V}$. By Lemma \[lem:linearmap\], $\mathcal{S}$ is also an $(\alpha,\delta)$-system of $\mathcal{V}'=(V_1',V_2',\ldots,V_n')$. We estimate the number of sets among $S_1,S_2,\ldots,S_w$ containing a bad pair. For every $i_0\in I$, there at most $\delta n/(10\alpha)$ values of $i\in I$ such that $V_{i_0}'$ and $V_i'$ are not $0.5$-separated. Let $\{{\boldsymbol{u}}_1,{\boldsymbol{u}}_2,\ldots,{\boldsymbol{u}}_{k_{i_0}}\}$ be an orthonormal basis of $V_{i_0}'$. For any $i$ that $V_{i_0}'$ and $V_i'$ are not $0.5$-separated, by Lemma \[lem:spacetobasis\], there must be $j\in[k_{i_0}]$ such that $$\|\operatorname{Proj}_{V_i'}({\boldsymbol{u}}_j)\|^2\geq\frac{1}{4k_{i_0}}\geq\frac{1}{4k}.$$ For every $j_0\in[k_{i_0}]$, we set ${\boldsymbol{w}}={\boldsymbol{u}}_{j_0}$ in inequality (\[eqn:bartheapp\]). The number of $i$’s that $\|\operatorname{Proj}_{V_i'}({\boldsymbol{u}}_{j_0})\|\geq1/(4k)$ is at most $$\frac{8kn}{\beta d}\left/\frac{1}{4k}\right.=\frac{32k^2n}{\beta d}.$$ Since there are $k_{i_0}\leq k$ values of $j_0\in[k_{i_0}]$, the number of $i$’s that $V_{i_0}'$ and $V_i'$ are not $0.5$-separated is at most $$k\cdot\frac{32k^2n}{\beta d}\leq\frac{32k^3n}{\beta d}\leq\frac{\delta n}{10\alpha}.$$ In the last inequality we used the assumption $d>400\alpha k^3/(\beta \delta)$. The number of bad pairs is at most $$|[n]\setminus I|\cdot n+|I|\cdot\frac{\delta n}{10\alpha}\leq\frac{\delta n^2}{10\alpha}+\frac{\delta n^2}{10\alpha}=\frac{\delta n^2}{5\alpha}.$$ We remove all $S_j$’s that contains a bad pair and use $\mathcal{S}'$ to denote the list of the remaining sets. Since each pair appears at most $\alpha$ times, we have removed at most $\delta n^2/5$ sets. Originally we have at least $\delta n^2/3$ sets by Lemma \[lem:awd\]. Now we have at least $\delta n^2/3-\delta n^2/5\geq\delta n^2/10$ sets. By Lemma \[lem:removebad\], there is a sublist $\mathcal{V}''=(V_{i_1}',V_{i_2}',\ldots,V_{i_q}')$ ($q\geq\delta n/(20\alpha)$) of $\mathcal{V}'$ and a sublist $\mathcal{S}''$ of $\mathcal{S}'$ such that $\mathcal{S}''$ is an $(\alpha,\delta/20)$-system of $\mathcal{V}''$. Since we have removed all bad pairs, $\mathcal{V}''$ and $\mathcal{S}''$ must satisfy the conditions of Theorem \[thm:sep\]. By Theorem \[thm:sep\], $$\dim(V_{i_1}'+V_{i_2}'+\cdots+V_{i_q}')\leq\frac{\alpha k}{0.5\cdot\delta/20}=\frac{40\alpha k}{\delta}\leq\beta d.$$ In the last inequality we used the assumption $d>400\alpha k^3/(\beta \delta)$. Recall that the linear map $M$ is invertible. So the space $V_{i_1}+V_{i_2}+\cdots+V_{i_q}$ has the same dimension as $V_{i_1}'+V_{i_2}'+\cdots+V_{i_q}'$. Therefore there are $q\geq\delta n/(20\alpha)$ spaces $V_{i_1},V_{i_2},\ldots,V_{i_q}$ within dimension $\beta d$. The second case of Theorem \[thm:main\_\] holds. In summary, under the assumption $d>400\alpha k^3/(\beta\delta)$ we have shown the second case of Theorem \[thm:main\_\] is always satisfied. Therefore Theorem \[thm:main\_\] is proved. $\Box$ [^1]: Department of Computer Science and Department of Mathematics, Princeton University, Princeton NJ. Email: `zeev.dvir@gmail.com`. [^2]: Department of Computer Science Princeton University, Princeton NJ. Email: `guangdah@cs.princeton.edu`. [^3]: One important difference is that LCC’s give rise to configurations where each point can repeat more than once.

--- abstract: 'We investigate the dynamics of two tunnel-coupled two-dimensional degenerate Bose gases. The reduced dimensionality of the clouds enables us to excite specific angular momentum modes by tuning the coupling strength, thereby creating striking patterns in the atom density profile. The extreme sensitivity of the system to the coupling and initial phase difference results in a rich variety of subsequent dynamics, including vortex production, complex oscillations in relative atom number and chiral symmetry breaking due to counter-rotation of the two clouds.' author: - 'T.W.A. Montgomery, R.G. Scott, I. Lesanovsky, and T.M. Fromhold' bibliography: - 'biblio.bib' date: '07/10/09' title: 'Spontaneous creation of non-zero angular momentum modes in tunnel-coupled two-dimensional degenerate Bose gases' --- Introduction ============ A major focus of cold-atom research is coupling multiple degenerate Bose gases (DBGs) using atom chips and optical lattices [@krugerRev; @reichelrev; @fortaghrev; @kasevich; @cornell; @kettnew; @kettnew2; @schmiedreview; @Hofferberth; @andreasdiff2; @mott; @JJExper; @kruger; @OberPRL]. The results provide stepping stones to future applications, such as interferometers or processors of quantum information. Since atom chips and optical lattices typically generate very high (kHz) trapping frequencies, there is growing interest in the dynamics of coupled one- and two-dimensional (1D and 2D) clouds [@savin; @annular; @bouchoule; @brandfluxons; @malomed]. This interest has also been fueled by the radically different physics that has been observed in lower-dimensional systems, such as the suppression of equilibration [@newtons], quasicondensation [@phasedefects], and the Kosterlitz-Thouless transition [@hello; @kruger07PRL; @kruger08NJP; @CornellPRL; @PhillipsPRL; @TapioPRL; @DalibardRev]. Since this body of work has uncovered such rich dynamics, it is natural to wonder how coupled lower-dimensional systems will behave, and whether they can reveal a crossover to 3D phenomena. Some recent work on 1D coupled rings has shown that the reduced dimensionality leads to unexpected effects, such as spontaneous population of rotating excitations and chiral symmetry breaking [@annular]. Further work is now needed to explore symmetry breaking in Josephson junctions, the boundaries of lower-dimensional physics, and to establish how double-well interferometers will perform in reduced dimensions. In this paper, we investigate the dynamics of coupled 2D disk-shaped DBGs. We find that the reduced dimensionality of the clouds has profound implications for the dynamics, because the instabilities of excited states observed in three dimensions are suppressed [@RScottInter; @RScottInter2]. Starting from an irrotational stationary state, and without introducing any stirring, we observe spontaneous occupation of low-lying excitations with non-zero angular momentum, mediated by the interatomic interactions [@annular; @spinchiral; @saito]. We use a linear stability analysis to predict which rotating Bogoliubov modes will be excited, and hence identify regimes where we can excite specific modes *alone* by tuning the coupling strength. This targeted selection of a single rotating mode creates striking oscillatory patterns in the atomic density profile. As we excite different rotating modes, we uncover a rich parameter space. Modes with a high angular momentum periodically grow and decay exponentially. The growth rates of these modes are extremely sensitive to changes in the coupling and interaction strengths, and in the relative phase of the two DBGs. The growth of modes with lower angular momentum becomes unstable due to collisions between rotating and non-rotating atoms, disrupting the internal structure of the clouds. This leads to a variety of subsequent dynamics such as vortex production, oscillations in relative atom number, and chiral symmetry breaking due to counter-rotation of the two clouds. System and Methodology ====================== ![(Color online) Schematic constant density surfaces of the upper (blue/light grey) and lower (red/dark grey) DBGs. Arrows show co-ordinate axes.[]{data-label="f0"}](fig1.eps){width="0.91\columnwidth"} The system consists of two 2D DBGs, referred to as the upper and lower DBGs, in the $z = + z_{0}$ and $- z_{0}$ planes respectively, as shown in Fig \[f0\]. The two DBGs are coupled by a tunnel junction created by the symmetric double well potential $V_{\text{dw}}(z)$. In the $x-y$ plane, the DBGs are contained by the harmonic trapping potential $V(r) = \frac{1}{2} m\omega^{2} r^{2}$, where $m$ is the mass of a single atom, $\omega$ is the trap frequency and $r=\sqrt{x^2+y^2}$. For $\left|z\right|\approx z_{0}$, $V_{\text{dw}}(z)$ can be approximated by the tight harmonic potential $V_{\text{sw}}^{\alpha/\beta}(z) = \frac{1}{2} m \lambda^2 \omega^2(z \mp z_{0})^2$, where $\lambda \gg 1$ to ensure that $\lambda\hbar\omega > \mu$. Consequently, atomic motion in the $z$ direction is frozen into the single-particle groundstate, and the DBG wavefunction becomes 2D [@2dness]. Hence, in the weak coupling limit [@JJTheo], we may represent the order parameter for the two 2D DBGs by the scalar complex field $$\label{eq_ansatz1} \psi = \zeta(z-z_{0})\chi^{\alpha}(\rho,\phi,\tau) + \zeta(z+z_{0})\chi^{\beta}(\rho,\phi,\tau)$$ where the superscripts $\alpha$ and $\beta$ refer to the upper and lower DBGs, $\zeta$ is the normalized single-particle harmonic groundstate of $V_{\text{sw}}^{\alpha/\beta}(z)$, $\phi$ is the azimuthal angle, and we have introduced the dimensionless time $\tau = \omega t$ and the dimensionless length $\rho = r/a_{\text{ho}}$, in which $a_{\text{ho}} = \sqrt{\hbar/m\omega}$. Substituting Eq. into the Gross-Pitaevskii equation results in two coupled equations for $\chi^{\alpha/\beta}(\rho,\phi,\tau)$ [@foot2] $$\label{eq_gp2d} \begin{array}{c c} i\partial_{\tau}\chi^{\alpha/\beta} & = -\left[\partial^2_{\rho} + \frac{1}{\rho}\partial_{\rho} + \frac{1}{\rho^2}\partial^2_{\phi} - \rho^2 +\mu\right]\chi^{\alpha/\beta}\\ & \ \ \ + \ \gamma|\chi^{\alpha/\beta}|^2\chi^{\alpha/\beta} - |\kappa|\chi^{\beta/\alpha} \end{array}$$ where the dimensionless quantities $\gamma = \left(8\pi\lambda\right)^{1/2}a_{0}/a_{\text{ho}}$ and $\kappa = \frac{1}{2} a^2_{\text{ho}}\int{\zeta(z+z_{0}) \left[\partial^2_{z}-2m V_{\text{dw}}(z)/\hbar\right] \zeta(z-z_{0})dz}$ represent the interaction and coupling energy respectively, and $a_{0}$ is the s-wave scattering length. At $\tau=0$, there are an equal number of atoms, $N_{0}$, in each well, and $\chi^{\alpha/\beta}$ is the non-rotating groundstate of $V(r)$, with chemical potential $\mu_{0}$. For this initial configuration, and finite $\kappa$, there are two possible stationary states of Eq. . These are the ground state, defined by $\chi^{\alpha}(r,\phi,0) = \chi^{\beta}(r,\phi,0)$, with chemical potential $\mu_{0} - |\kappa|$, and the excited asymmetric stationary state, henceforth referred to as the $\pi$-state, defined by $\chi^{\alpha}(r,\phi,0) = -\chi^{\beta}(r,\phi,0)$, with chemical potential $\mu_{0} + |\kappa|$ [@brandcomment]. Unsurprisingly, the ground state is stable for all coupling strengths. However, the $\pi$-state shows much richer behavior. In three dimensions, the phase discontinuity may bend, creating vortices via the well known snake instability [@RScottInter; @RScottInter2; @dutton; @anderson]. This process cannot occur in our system, because the reduced dimensionality of the disks precludes the movement of the phase discontinuity. In the following section, we perform a stability analysis to identify what excitations may occur. Stability analysis of the $\pi$-state ===================================== We perform a stability analysis of the stationary states by calculating the excitations of the system using the Bogoliubov ansatz [@stability] $$\label{eq_BogAns} \begin{array}{c c} \chi^{\alpha/\beta}(\rho,\phi,\tau) = \chi^{\alpha/\beta}(\rho,\phi,0) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \ \ \ \ \ \ \ \ \ \ \ \ + \left[ u^{\alpha/\beta}_{l}(\rho,\phi)e^{iE_{l} \tau} + v^{\alpha/\beta \; *}_{l}(\rho,\phi)e^{iE_{l}^{*} \tau} \right]. \end{array}$$ In the above equation, the $u$’s, $v$’s and mode energies $E_{l}$ have been explicitly labeled with their angular momentum quantum number, $l$, to derive the four coupled Bogoliuvbov-de Gennes equations $$\label{eq_bogeqns} \begin{array}{c c c } E_{\kappa,l} u^{\alpha/\beta}_{l} &=& -\left[\partial^2_{\rho} + \frac{1}{\rho}\partial_{\rho} - \frac{l^2}{\rho^2}\right]u_{l}^{\alpha/\beta} +2\epsilon|\chi^{\alpha/\beta}|^2 u_{l}^{\alpha/\beta}\\ & & -\epsilon|\chi^{\alpha/\beta}|^2 v_{-l}^{\alpha/\beta} -|\kappa|u_{l}^{\beta/\alpha}\\ -E_{\kappa,l} v^{\alpha/\beta}_{-l} &=& -\left[\partial^2_{\rho} + \frac{1}{\rho}\partial_{\rho} - \frac{l^2}{\rho^2}\right]v_{-l}^{\alpha/\beta} +2\epsilon|\chi^{\alpha/\beta}|^2 v_{-l}^{\alpha/\beta}\\ & & -\epsilon|\chi^{\alpha/\beta}|^2 u_{l}^{\alpha/\beta} -|\kappa|v_{-l}^{\beta/\alpha} \end{array}$$ where $\epsilon = \gamma N_{0}$ and $E_{\kappa,l}$ ($\equiv E_{l}$ in Eq. \[eq\_BogAns\]) is the energy of the Bogoliubov mode for a particular $\kappa$ and $l$. ![(Color) Upper plot: Imaginary component of ${E_{\kappa,l}}$ (scale shown at top) for angular momentum modes $|l|= 0$ to $4$ as a function of $|\kappa|$. Lower surface plots: three examples of excited modes with (a) $l = 4$ and no radial node ($|\kappa| = 1.7$), (b) $l = 1$ and no radial node ($|\kappa| = 0.56$), (c) $l = 1$ and a single radial node $|\kappa| = 1.35$. The vertical height shows the atomic density of the mode, and the colors denote its phase (blue = 0, green = $\pi$, red = $2\pi$).[]{data-label="f1"}](fig2.eps){width="1.0\columnwidth"} The upper panel in Fig. 2 is a gray-scale plot of the imaginary component of $E_{\kappa,l}$ for $|l| = 0$ to $4$ calculated as a function of $|\kappa|$, for a fixed interaction strength $\epsilon=7$. We are interested in complex eigenvalues because they indicate unstable modes which grow exponentially at a rate proportional to the imaginary component. The results are presented in terms of $|l|$ because each Bogoliubov mode has a degenerate pair with equal and opposite angular momentum. Each shaded region in the figure corresponds to a single mode with a particular angular momentum and radial excitation. To illustrate this, we pick three example couplings of $|\kappa|= \left\{1.7,0.56,1.35\right\}$, indicated in the upper panel of Fig. 1 by the vertical $\left\{\right.$dotted, solid, dashed$\left.\right\}$ lines, which pass through shaded regions labeled $\left\{\right.$(a), (b), (c)$\left.\right\}$. The modes corresponding to these shaded regions are shown, also labeled $\left\{\right.$(a), (b), (c)$\left.\right\}$, in the lower half of Fig. 2 as surface plots, whose vertical height denotes the atomic density of the mode, and the colors denote its phase. Mode (a) in the lower panel of Fig. 2 has four quanta of angular momentum ($l=4$), but no radial node. Mode (b) is similar to mode (a), but has only one quantum of angular momentum. As we move to larger values of $|\kappa|$ for a given $|l|$, we find regions of instability corresponding to modes with a greater radial excitation. For example, mode (c) has a higher radial excitation than mode (b) because it has an additional radial density node [@foot3]. The positions of the shaded regions in the upper panel of Fig. 2 can be understood by considering the limiting case as $\epsilon \to 0$ (no inter-atomic interactions). In this limit, the shaded regions shrink to points where the coupling strength, $\kappa$, matches the single-particle energies, $n+|l|/2$, where $n$ denotes the number of radial nodes in the corresponding wavefunction. Clearly, in this case, excitations can be degenerate. However, inter-atomic interactions destroy this degeneracy. In general, such interactions shift the shaded regions to higher energy. The size of this shift varies, so that there are certain values of $|\kappa|$ (for example, $|\kappa|= 1.7$, shown by the dotted line in the upper panel of Fig. 2) where only one mode is unstable, and hence we can tune the coupling to excite that mode *alone*. There are also values of $\kappa$ (for example, $|\kappa|= 2.15$) where no modes are unstable, meaning that the $\pi$-state is stable. In addition to shifting the positions of the shaded regions to higher $\kappa$, increasing the interactions from zero also broadens the unstable regions and increases the imaginary component of ${E_{\kappa,l}}$ within them. This illustrates that, although the tunnel coupling provides the energy for the growth of unstable modes, the scattering from the $\pi$-state into the excitation is mediated by the inter-atomic interactions. We find that $\epsilon=7$ is a suitable value to provide reasonable growth rates of excited modes, without causing the unstable regions to become so broad that they overlap, so that specific modes can still be accessed individually.\ Simulations of the dynamics =========================== To simulate the growth of the unstable modes, we must model inter-atomic scattering events not captured by the Gross-Pitaevskii equation. Hence we add non-zero angular momentum fluctuations of the form $\sum{C^{(\alpha/\beta)}_{l}r^{|l|}e^{-\rho^2/2}e^{i l\phi}}$ (the single-particle excitations) to $\chi^{\alpha/\beta}$ [@fluct], then evolve Eq. \[2\] in time. Here, $C^{(\alpha/\beta)}_{l}$ is a random complex number such that $\langle |C^{(\alpha/\beta)}_{l}| \rangle = 0$ and $\langle |C^{(\alpha/\beta)}| \rangle^{2} \approx 10^{-8}\mathcal{O}(N_{0})$. These fluctuations are small enough that they do not raise the temperature significantly from zero in our model. Periodic excitation of a rotating mode ($|\kappa| = 1.7$) --------------------------------------------------------- Figure 3 shows the results of a simulation for $|\kappa| = 1.7$. For this coupling, only a single mode with $|l| = 4$ \[shown by surface plot (a) in the lower panel of Fig. 2\] is unstable, as indicated by the vertical dotted line in the upper panel of Fig. 2. Initially, both clouds have a smooth density profile \[Fig. 3(a)\]. However, as we evolve in time, a macroscopic number of atoms are scattered from the $\pi$-state via the inter-atomic interactions into the $l=4$ mode. To preserve angular momentum, an equal number of atoms transfer into the degenerate $l=-4$ mode, producing the star-like interference pattern with eight density antinodes at $\tau \approx 95$ shown in Fig. 3(b). To quantify the growth of non-zero angular momentum modes, we introduce the quantity $P^{\alpha/\beta}_{l} = 1/2\pi|\int\chi^{\alpha/\beta}(e^{-il\phi} + e^{il\phi})\rho d\rho~d\phi|^2$, which is the projection of the upper or lower DBG onto modes with angular momentum $|l|$. We plot $\text{log}_{10}(P^{\alpha}_{4})$ in Fig. 3(d). The graph shows that the population of the $|l|=4$ mode increases linearly on the logarithmic plot, indicating exponential growth, reaching a maximum of approximately $N_{0}/3$ at $\tau\approx95$. At this point, the growth of the $l=4$ mode halts, and atoms begin transferring back into the $\pi$-state. The population of the $\pi$-state does not reach zero because the stability analysis presented in Fig. 2 assumes that all atoms are in the $\pi$-state, and is therefore no longer valid once a macroscopic number of atoms have entered an excited mode. Eventually, the DBG returns to its original configuration at $\tau \approx 160$ \[Fig. 3(c)\], at which point the process repeats. This periodic transfer can be maintained because, due to its large angular momentum, the excited mode’s peak density is far from the origin, meaning that there is little interaction with the remaining atoms in the $\pi$-state. Consequently, the system is able to supresses particle transfer between the two wells over a large time, as shown in Fig. 3(e), which plots the number imbalance $\eta = (N^{\alpha}-N^{\beta})/2N_{0}$ between the two wells during the simulation [@foot1]. The width of the unstable $l=4$ mode in Fig. 2 is very narrow, indicating that the formation of the star-like interference pattern is a sharply resonant process. Hence, the growth rate varies extremely sensitively with $\kappa$, reaching zero if $\kappa=1.7$ is changed by only $\sim 2\%$. Similarly, the system is also highly sensitive to the relative phases of the DBGs. For example, when the above simulation is run with an initial relative phase of $0.9 \pi$ between the two clouds, the star-like interference pattern is no longer observed. Moreover, an initial relative phase of $0.99 \pi$ causes a 10 $\%$ change in the growth rate of the star-like pattern. Since this predictable and macroscopic pattern is stable, it could be detected experimentally *in situ* using absorption imaging. Oscillations in relative atom number and vortex production ($|\kappa| = 0.56$) {#sec:JO} ------------------------------------------------------------------------------ ![(Color online) Simulation for $|\kappa| = 1.7$. (a)-(c) Atom density plot (black = high) of upper (first row) and lower (second row) DBG at (a) $\tau = 0$, (b) $\tau = 95$ and (c) $\tau = 105$. (d) Logarithmic plot of the projection $P^{\alpha}_{4}$ of the upper DBG onto angular momentum modes with $|l| = 4$. (e) Atom number imbalance, $\eta$, between upper and lower well.[]{data-label="f2"}](fig3.eps){width="1.0\columnwidth"} ![(Color online) Simulation for $|\kappa| = 0.56$. (a)-(c) Atom density plot (black = high) of upper (first row) and lower (second row) DBG at (a) $\tau = 0$, (b) $\tau = 25$ and (c) $\tau = 75$. The positions of vortices are indicated by the black (minus) plus, which denotes (anti) clockwise rotation. (d) Solid (dashed) line: logarithmic plot of the projection $P^{\alpha}_{1}$ ($P^{\alpha}_{2}$) of the upper DBG onto angular momentum modes with $|l| = 1$ ($|l| = 2$). (e) Atom number imbalance, $\eta$, between upper and lower well.[]{data-label="f3"}](fig4.eps){width="1.0\columnwidth"} We now tune $|\kappa|$ to $0.56$ to excite the lowest possible rotating mode, as shown by the vertical solid line in the upper panel of Fig. 2. Comparing the surface plots in the lower half of Fig. 2, we see that this mode \[Fig. 2(b)\] is similar to the one previously excited \[Fig. 2(a)\], except that it has only one quantum of angular momentum. Consequently, we now expect the formation of an interference pattern with two antinodes. Starting from the smooth atom density profile at $\tau = 0$ \[Fig. 4(a)\], such an interference pattern begins to form, but the two antinodes have very different peak densities \[Fig. 4(b)\]. The projection $P^{\alpha}_{1}$ shown by the solid curve in Fig. 4(d) reveals that, as expected from the stability analysis, the population of the $|l|=1$ modes initially grows exponentially. However, as the number of atoms in the $|l| = 1$ modes becomes macroscopic, there is also a significant rise in the projection $P^{\alpha}_{2}$ \[dashed curve in Fig. 4(d)\], which is not predicted by the stability analysis. We find these unexpected scattering processes when we excite *low* energy modes, which are concentrated close to the center of the trap, and hence collide more frequently with the remaining atoms in the $\pi$-state. Despite these unexpected scattering events, for $10 \lesssim \tau \lesssim 40$ we may still observe one oscillation in the population of the $|l| = 1$ mode in Fig. 4(d). However, at $\tau \approx 50$ the oscillation breaks down, and vortex pairs appear in the density profile, as shown in the lower panel of Fig. 4(c), in which the position and (anti)clockwise rotation of the vortex is indicated by the (minus) plus sign. At the same time, we observe the onset of regular, large amplitude ($\sim 0.8$) oscillations in the number imbalance $\eta$ \[plotted in Fig. 4(e)\] due to inter-well transfer. These dynamics reflect that, at this low $\kappa$, the system has insufficient energy to populate other rotating modes and, consequently, it redistributes its energy by performing oscillations in relative atom number. This behavior can alternatively be regarded as instability in the symmetric $l=0$ groundstate mode, which occurs within the shaded region labeled (d) in the upper panel of Fig. 2. Despite the apparent disruption and the macroscopic population of rotating modes in the simulations presented hitherto, the system has always conserved zero net angular momentum in *each* DBG, as illustrated, for example, by the presence of a vortex anti-vortex pair in Fig. 4(c). However the complete system, defined by Eq. , only requires that the *total* angular momentum about the $z$ axis be conserved, meaning that each DBG may break its initial chiral symmetry and obtain a net angular momentum, so that $\langle L^{\alpha}_{z} \rangle = -\langle L^{\beta}_{z} \rangle \neq 0$, where $\langle L^{\alpha/\beta}_{z} \rangle$ is the angular momentum of the upper$/$lower DBG. No such mode is predicted fom the stability analysis as the mechanism for spontanous decay of the $\pi$-state involves two atoms with zero initial angular momentum gaining equal and opposite angular momentum via an elastic collision. Since this process is mediated by the non-linear term in Eq. , it may only occur between atoms in the same well. Consequently, we conclude that the breaking of chiral symmetry is a secondary process, caused by tunneling of angular momentum between the wells, once rotating modes have become populated. We now present an example of this behavior. Counter-rotation of the two clouds ($|\kappa| = 1.35$) ------------------------------------------------------ {width="1.0\columnwidth"} We set $\kappa$ to $1.35$ (dashed line in Fig. 2) in order to excite $|l| = 1$ modes with additional radial energy \[shown in surface plot (c) in the lower panel of Fig. 2\]. Figure 2 reveals that an $|l| = 2$ mode is also unstable for this coupling, but its imaginary eigenvalue is much smaller (lighter gray in Fig. 2) than the corresponding value for the $|l| = 1$ modes. We would therefore expect the $|l| = 1$ modes to grow faster. Our simulation confirms this: the initially smooth density profile \[Fig. 5(a)\] transforms into an interference pattern \[Fig. 5(b)\] with two antinodes as $\phi$ is rotated through $2\pi$, and two antinodes in the $\rho$ direction. The population of this mode \[shown by the solid curve in Fig. 5(d)\] performs one oscillation for $15 \lesssim \tau \lesssim 45$, but by $\tau \approx 50$ the growing population of the $|l|=2$ modes \[shown by dashed line in Fig. 5(d)\] has become significant. The occupation of multiple rotating modes disrupts the DBGs’ internal structure, but eventually four rough density peaks, encircling the central peak, appear in the atomic density profile, revealing macroscopic population of the $|l|=2$ modes \[Fig. 5(c)\]. As this occurs, we observe the onset of oscillations in relative atom number, indicated by the periodic fluctuations in $\eta$ shown in Fig. 5(e). The frequency of the oscillations is higher than that shown in Fig. 4(e), because $\kappa$ has been increased [@JJTheo]. Unlike the case presented in section \[sec:JO\], this is accompanied by a transfer of angular momentum between the two DBGs, such that the two DBGs counter-rotate. We quantify this effect in the enlargement of the dashed area in Fig. 5(e), which shows not only $\eta$ (solid curve, left axis), but also $\langle L^{\alpha/\beta}_{z} \rangle$ (dashed$/$dotted curve, right axis). The curves illustrate that, although $\langle L^{\alpha}_{z} \rangle = -\langle L^{\beta}_{z} \rangle$, maintaining the net zero angular momentum of the *whole* system, $\langle L^{\alpha}_{z} \rangle$ and $\langle L^{\beta}_{z} \rangle$ perform periodic oscillations, with a maximum angular momentum difference of $1.6\hbar$. Numerically, we find that the lowest energy counter-rotating mode with $\langle L^{\beta}_{z} \rangle = -\langle L^{\alpha}_{z} \rangle = \hbar$ has an energy of $2N_0(\mu_{0} + 1.3)$. This explains why macroscopic occupation of a counter-rotating mode is not observed in the simulation presented in Fig 3, which has an energy of $2N_0(\mu_{0} + 0.56)$. Conclusion ========== In summary, we have shown that the well known decay mechanism of a three-dimensional $\pi$-state [@RScottInter; @RScottInter2; @dutton; @anderson] is suppressed in tunnel-coupled 2D DBGs. This results in a rich variety of physics, such as spontaneous rotation, vortex formation, and chiral symmetry breaking due to counter-rotation of the two clouds. Our paper builds on previous studies of coupled 1D rings [@annular], showing that the radial degree of freedom in the 2D disks plays an important role. Firstly, excitations in the radial ($r$) direction appear in the stability diagram, and may be selectively populated and experimentally observed. Secondly, the clouds may develop a complex internal structure, including vortex anti-vortex pairs. Thirdly, the radial position of the peak density, which is fixed in the case of the rings, depends on the angular momentum of the 2D cloud. As a consequence, disruption arises more readily when we populate modes with low angular momentum. Since this work is restricted to zero temperature, it remains an open question as to how coupling multiple 2D DBGs will change their dynamics at finite temperature. Finite temperature will introduce a variation of phase across the clouds, which might increase disruption, causing the counter-rotation to occur more readily. It is also possible that coupled systems may provide a link between 2D Kosterlitz-Thouless behavior and three-dimensional Bose-Einstein condensation. We note that these systems can be realized with current experimental techniques. Similar set ups have already been created by combining a harmonic trap with a one-dimensional optical lattice, to investigate quantum fluctuation-induced localization [@kasevich] and the Kosterlitz-Thouless transition [@hello]. Using this method, values of $\lambda$ up to $370$ have been achieved. A suitable choice of parameters would be, for example, $\omega = 2\pi\times30$Hz, $\lambda = 10$ and $\epsilon = 7$. In this case, $N_{0} \approx 200$ $^{87}$Rb atoms and the conditions for two-dimensionality are satisfied. These parameters are comparable to those in previous experiments [@kasevich]. Using optical lattices to create this system also offers the intriguing possibility of extending our work to spinor DBGs or arrays of coupled DBGs [@kasevich; @cornell; @thzrad], which might reveal a crossover from 2D behavior to three-dimensional band dynamics. Finally, we speculate that this system could potentially be developed as a sensor to detect tiny gravitational or electromagnetic forces, in a similar manner to the production of vortices between merged elongated DBGs [@kettnew; @RScottInter; @RScottInter2]. In principle, one of the DBGs could be exposed to a tiny force, causing a phase change of $0.01 \pi$, which could then be detected as a change in the growth rate of rotating modes. This work is funded by EPSRC-UK. We thank P. Kr[ü]{}ger for helpful discussions.

--- abstract: 'In a social network, agents are intelligent and have the capability to make decisions to maximize their utilities. They can either make wise decisions by taking advantages of other agents’ experiences through learning, or make decisions earlier to avoid competitions from huge crowds. Both these two effects, social learning and negative network externality, play important roles in the decision process of an agent. While there are existing works on either social learning or negative network externality, a general study on considering both these two contradictory effects is still limited. We find that the Chinese restaurant process, a popular random process, provides a well-defined structure to model the decision process of an agent under these two effects. By introducing the strategic behavior into the non-strategic Chinese restaurant process, in Part I of this two-part paper, we propose a new game, called Chinese Restaurant Game, to formulate the social learning problem with negative network externality. Through analyzing the proposed Chinese restaurant game, we derive the optimal strategy of each agent and provide a recursive method to achieve the optimal strategy. How social learning and negative network externality influence each other under various settings is also studied through simulations.' author: - '\' bibliography: - 'crg.bib' title: 'Chinese Restaurant Game - Part I: Theory of Learning with Negative Network Externality' --- Introduction ============ How agents in a network learn and make decisions is an important issue in numerous research fields, such as social learning in social networks, machine learning with communications among devices, and cognitive adaptation in cognitive radio networks. Agents make decisions in a network in order to achieve certain objectives. For example, one customer goes to the supermarket for an orange juice. He may need to choose one from dozens of brands. However, the agent’s knowledge on the market may be very limited due to the limited ability in observations or the external uncertainty in the market, which means that the customer may not know the quality of all orange juice in different brands. This limitation reduces the accuracy of the agent’s decision for his objective, e.g., to get the best orange juice of his taste. The limited knowledge of one agent can be expanded through learning. One agent may learn from some information sources, such as the decisions of other agents, the advertisements from some brands, or his experience in previous purchases. All the information can help the agent to construct a belief, which is mostly probabilistic, on the unknown state. In most cases, the accuracy of the agent’s decision can be greatly enhanced by taking into account the belief. A general learning and decision making process in a network can be described as follows. First, an agent collects information through available communication or observation methods and updates his belief on the uncertain states based on the collected information. Then, the agent estimates the expected rewards of certain actions according to the belief he constructed. Finally, the agent chooses the action that maximizes his reward. Let us consider a social network in an uncertain system state. The state has an impact on the agents’ rewards. When the impact is differential, i.e., one action results in a higher reward than other actions in one state but not in all states, the state information becomes critical for one agent to make the correct decision. In most social learning literatures, the state information is unknown to agents. Nevertheless, some signals related to the system state are revealed to the agents. These signals may be preserved in private or revealed to others. Then, the agents make their decisions sequentially, while their actions/signals may be fully or partially observed by other agents. Most of existing works [@bala1998learning; @golub2007naive; @acemoglu2011bayesian; @acemoglu2010opinion] study how the believes of agents are formed through learning in the sequential decision process, and how accurate the believes will be when more information is revealed. One popular assumption in traditional social learning literatures is that there is no network externality, i.e., the actions of subsequent agents do not influent the reward of the former agents. In such a case, agents will make their decisions purely based on their own believes without considering the actions of subsequent agents. This assumption greatly limits the potential applications of these existing works. The network externality, i.e., the influence of other agents’ behaviors on one agent’s reward, is a classic topic in economics. How the relations of agents influence an agent’s behavior is one of the major problems in coordinate game theory [@cooper1999coordination]. When the network externality is positive, the problem can be modeled as a coordination game, where agents seek the best common decisions to cooperate with others. When the externality is negative, it becomes an anti-coordination game, where agents try to avoid making the same decisions with others [@katz1986technology; @sandholm2005negative; @fagiolo2005endogenous]. In the literature, there are some works on combining the positive network externality with social learning, such as voting game [@wit1999social; @battaglini2005sequential; @ali2010observ] and investment game [@gale1995dynamic; @dasgupta2000social; @dasgupta2007coordination; @choi2011network]. In the voting game, an election with several candidates is hold, where voters have their own preferences on the candidates. The preference of a voter on the candidates is constructed by the voter’s belief on how the candidates can benefit him if winning the election. However, since the candidate can make efforts only when he wins the election, a voter’s vote depends not only on his own preference but also on the probability that the candidate wins the election. In such a case, the estimation and prediction on the decisions of other voters become critical in the voting game. A learning process is involved when the voting game is sequential, i.e., voters vote the candidates sequentially and the vote of each voter is known by others. In the sequential voting game, voters learn from the previous votes to update their believes on the candidates and the probability that the candidates win the election. In the investment game, there are multiple projects and investors, where each project has different probability of success and different payoff. One investor may invest one or several projects if his budget allows. If the project succeeds, he receives a payoff from the project. When more investors invest in the same project, the succeeding probability of the project increases, which benefits all investors investing this project. Note that in both voting and investment games, the agent’s decision has a positive effect on ones’ decisions. When one agent makes a decision, the subsequent agents are encouraged to make the same decision in two aspects: the probability that this action has the positive outcome increases due to this agent’s decision, and the potential reward of this action may be significantly large according to the belief of this agent. The combination of negative network externality with social learning, on the other hand, is difficult to analyze. When the network externality is negative, the game becomes an anti-coordination game, where one agent seeks the strategy that differs from others’ to maximize his own reward. Nevertheless, in such a scenario, the agent’s decision also contains some information about his belief on the uncertain system state, which can be learned by subsequent agents through social learning algorithms. Thus, subsequent agents may then realize that his choice is better than others, and make the same decision with the agent. Since the network externality is negative, the information leaked by the agent’s decision may impair the reward the agent can obtain in the game. Therefore, rational agents should take into account the possible reactions of subsequent players to maximize their own rewards. The negative network externality plays an important rule in many applications in different research fields, such as spectrum access in cognitive radio, storage service selection in cloud computing, and deal selection on Groupon in online social networking. In spectrum access problem, for instance, secondary users access the same spectrum need to share with each others. The more secondary users access the same channel, the less available access time for each of them. In storage service selection problem, the reliability and availability are affected by the number of subscribers. The more subscribers using the same service, the lower the service quality of the cloud storage platform. For the deal selection on Groupon website, some businesses may receive overwhelming number of customers under the discounted deal. The overwhelming number of customers has a negative network externality on the quality of the products. In these examples, the negative network externality degrades the utility of the agents making the same decision. Therefore, the agents should take into account the possibility of degraded utility, e.g., less access time, lower reliability, or lower service quality, when making the decisions. The aforementioned social learning approaches are mostly strategic, where agents are considered as players with bounded or unbounded rationality in maximizing their own rewards. Machine learning, which is another class of approaches for the learning problem, focuses on designing algorithms for making use of the past experience to improve the performance of similar tasks in the future [@mitchell1997machine]. Generally there exists some training data and the devices follow a learning method designed by the system designer to learn and improve the performance of some specific tasks. Most learning approaches studied in machine learning are non-strategic without the rationality on considering their own benefit. Such non-strategic learning approaches may not be applicable to the scenario where devices are rational and intelligent enough to choose actions to maximize their own benefits instead of following the rule designed by the system designer. Chinese restaurant process, which is introduced in non-parametric learning methods in machine learning [@aldous1985exchangeable], provides an interesting non-strategic learning method for unbounded number of objects. In Chinese restaurant process, there exists infinite number of tables, where each table has infinite number of seats. There are infinite number of customers entering the restaurant sequentially. When one customer enters the restaurant, he can choose either to share the table with other customers or to open a new table, with the probability being predefined by the process. Generally, if a table is occupied by more customers, then a new customer is more likely to join the table, and the probability that a customer opens a new table can be controlled by a parameter [@pitman1995exchangeable]. This process provides a systematic method to construct the parameters for modeling unknown distributions. By introducing the strategic behavior into the non-strategic Chinese restaurant process, we proposed a new game, called **Chinese Restaurant Game**, to formulate the social learning problem with negative network externality. Let us consider a Chinese restaurant with $K$ tables. There are $N$ customers sequentially requesting for seats from these $K$ tables for having their meals. One customer may request one of the tables in number. After requesting, he will be seating in the table he requested. We assume that all customers are rational, i.e., they prefer bigger space for a comfortable dining experience. Thus, one may be delighted if he has a bigger table. However, since all tables are available to all customers, he may need to share the table with others if multiple customers request for the same table. In such a case, the customer’s dining space reduces, due to which the dining experience is impaired. Therefore, the key issue in the proposed Chinese restaurant game is how the customers choose the tables to enhance their own dining experience. This model involves the negative network externality since the customer’s dining experience is impaired when others share the same table with him. Moreover, when the table size is unknown to the customers, but each of them receives some signals related to the table size, this game involves the learning process if customers can observe previous actions or signals. Such a theoretic Chinese restaurant game framework is very general and can be applied into many research areas, such as online social networks, wireless communication, and cloud computing, which will be discussed in Part II of this two-part paper [@wang2011crgpart2]. In the rest of the paper, we first provide detailed descriptions on the system model of Chinese restaurant game in Section \[sec\_sys\]. Then, we analyze the simultaneous game model to show how customers behave given the perfect knowledge on the table size in Section \[sec\_simul\]. Next, we study the sequential game model with perfect information to illustrate the advantage of playing first in Section \[sec\_seque\]. In Section \[sec\_gen\], we show the general Chinese restaurant game framework by analyzing the learning behaviors of customers under the negative network externality and uncertain system state. We provide a recursive method to construct the best response for customers, and discuss the simulation results in Section \[sec\_sim\]. Finally, we draw conclusions in Section \[sec\_con\]. Related Works ============= A closely-related strategic game model to our work is the global game [@carlsson1993global; @morris2001global]. In the global game, all agents, with limited knowledge on the system state and information hold by other agents, make decisions simultaneously. The agent’s reward in the game is determined by the system state and the number of agents making the same decision with him. The influence may be positive or negative depending on the type of network externality. An important characteristics of global game is that the equilibrium is unique, which simplifies the discussion on the possible outcome of the game. It draws great attentions in various research fields, such as financial crisis [@angeletos2006crises], sensor networks [@krishnamurthy2011decentra] and cognitive radio networks [@krishnamurthy2009decentra]. Since all players in the global game make decisions simultaneously, there is no learning involved in the global game. In recent years, several works [@dasgupta2000social; @angeletos2006signaling; @angeletos2007dynamic; @costain2007herding; @dasgupta2007coordination] make efforts to introduce the learning and signaling into the global game. Dasgupta’s first attempt was investigating a binary investment model, while one project will succeed only when enough number of agents invest in the project in [@dasgupta2000social]. Then, Dasgupta studied a two-period dynamic global game, where the agents have the options to delay their decisions in order to have better private information of the unknown state in [@dasgupta2007coordination]. Angeletos *et. al.* studied a specific dynamic global game called regime of changes [@angeletos2006signaling; @angeletos2007dynamic]. In the regime of changes game, each agent may propose an attack to the status quo, i.e., the current politic state of the society. When the collected attacks are large enough, the status quo is abandoned and all attackers receive positive payoffs. If the status quo does not change, the attackers receive negative payoffs. Angeletos *et. al.* first studied a signaling model with signals at the beginning of the game in [@angeletos2006signaling]. Then, they proposed a multiple stages dynamic game to study the learning behaviors of agents in the regime of change game in [@angeletos2007dynamic]. Costain provided a more general dynamic global game with an unknown binary state and a general utility function in [@costain2007herding]. The utility function includes information revelation, strategic complementarities, and payoff heterogeneity. To simplify the analysis, the positions of the agents in the game are assumed to be unknown. Nevertheless, most of these works study the multiplicity of equilibria in dynamic global game with simplified models, such as binary state or binary investment model. Moreover, the network externality they considered in their models are mostly positive. By proposing the Chinese restaurant game, we hereby provides a more general game-theoretic framework on studying the social learning in a network with negative network externality, which has many applications in various research fields. System Model {#sec_sys} ============ Let us consider a Chinese restaurant with $K$ tables numbered $1,2,...,K$ and $N$ customers labeled with $1,2,...,N$. Each customer requests for one table for having a meal. There may be multiple customers request for the same table. Each table has infinite seats, but may be in different size. We model the table sizes of a restaurant with two components: the restaurant state $\theta$ and the table size functions $\{R_1(\theta),R_2(\theta),...,R_K(\theta)\}$. The state $\theta$ represents an objective parameter, which may be changed when the restaurant is remodeled. The table size function $R_j(\theta)$ is fixed, i.e., the functions $\{R_1(\theta),R_2(\theta),...,R_K(\theta)\}$ will be the same every time the restaurant is remodeled. An example of $\theta$ is the order of existing tables. Suppose that the restaurant has two tables, one is of size $L$ and the other is of size $S$. Then, the owner may choose to number the large one as table $1$, and the small one as table $2$. The decision on the numbering can be modeled as $\theta \in \{1,2\}$, while the table size functions $R_1(\theta)$ and $R_2(\theta)$ are given as $R_1(1)=L$, $R_1(2)=S$, and $R_2(1)=S$, $R_2(2)=L$. Let $\Theta$ be the set of all possible state of the restaurant. In this example, $\Theta=\{1,2\}$. We formulate the table selection problem as a game, called **Chinese Restaurant Game**. We first denote $\mathbf{X} = \{1,...,K\}$ as the action set (tables) that a customer may choose, where $x_i \in A$ means that customer $i$ chooses the table $x_i$ for a seat. Then, the utility function of customer $i$ is given by $U(R_{x_i},n_{x_i})$, where $n_{x_i}$ is the number of customers choosing table $x_i$. According to our previous discussion, the utility function should be an increasing function of $R_{x_i}$, and a decreasing function of $n_{x_i}$. Note that the decreasing characteristic of $U(R_{x_i},n_{x_i})$ over $n_{x_i}$ can be regarded as the negative network externality effect since the degradation of the utility is due to the joining of other customers. Finally, let $\mathbf{n}=\{n_1,n_2,...,n_K\}$ be the numbers of customers on the $K$ tables, i.e., the grouping of customers in the restaurant. As mentioned above, the restaurant is in a state $\theta \in \Theta$. However, customers may not know the exact state $\theta$, i.e., they may not know the exact size of each table before requesting. Instead, they may have received some advertisements or gathered some reviews about the restaurant. The information can be treated as some kinds of signals related to the true state of the restaurant. In such a case, they can estimate $\theta$ through the available information, i.e., the information they know and/or gather in the game process. Therefore, we assume that all customers know the prior distribution of the state information $\theta$, which is denoted as $\mathbf{g_0} =\{g_{0,l}|g_{0,l}=Pr(\theta=l),~\forall l \in \Theta\}$. The signal each customer received $s_i \in S$ is generated from a predefined distribution $f(s|\theta)$. Belief on State --------------- In this subsection, we introduce the concept of belief to describe how the customers estimate the system state $\theta$. Since customers may make decisions sequentially, it is possible that the customers who make decisions later learn the signals from those customers who make decisions first. Let us denote the signals customer $i$ learned, excluding his own signal $s_i$, as $\mathbf{h_i}=\{s\}$. With the help of these signals $\mathbf{h_i}$, his own signal $s_i$, the prior distribution $\mathbf{g_0}$, and the conditional distribution $f(s|\theta)$, each customer $i$ can estimate the current system state in probability with the belief being defined as $$\mathbf{g_i} = \{g_{i,l}|g_{i,l}=Pr(\theta=l|\mathbf{h_i},s_i,\mathbf{g_0},f),~\forall l \in \Theta\} ~\forall i \in N .$$ According to the above definition, $g_{i,l}$ represents the probability that system state $\theta$ is equal to $l$ conditioning on the collected signals $\mathbf{h_i}$, received signal $s_i$, the prior probability $\mathbf{g_0}$, and the conditional distribution $f(s|\theta)$. Notice that in the social learning literature, the belief can be obtained through either non-Bayesian naive updating rule [@bala1998learning; @golub2007naive] or fully rational Bayesian rule [@acemoglu2011bayesian]. For the non-Bayesian naive updating rule, it is implicitly based on the assumption that customers are only limited rational and follows some predefined rules to compute their believes. Their capability to maximize their utilities is limited not only by the game structure and learned information, but also by the non-Bayesian naive updating rules. In the fully rational Bayesian rule, customers are fully rational and have the potential to optimize their actions without the restriction on the fixed belief updating rule. Since the customers we considered here are fully rational, they will follow the Bayesian rule to update their believes as follows: $$\label{eqn_g} g_{i,l}= \frac{g_{0,l}Pr(\mathbf{h_i},s_i|\theta=l)}{\sum_{l' \in \Theta}g_{0,l'}Pr(\mathbf{h_i},s_i|\theta=l')}.$$ Notice that the exact expression for belief updating depends on how the signals are generated and learned, which is generally affected by the conditional distribution $f(s|\theta)$ and the game structure. Simultaneous Game with Perfect Signal: How Negative Network Externality Affects {#sec_simul} =============================================================================== The first game structure we would like to discuss is the simultaneous game, in which all customers make decisions simultaneously, e.g., all agents arrive the restaurant at the same time. In such a scenario, there is no learning involved in the game since customers request the tables at the same time. By investigating this game model, we can have an initial understanding on how customers behave in the game. We start with a simple case where there are only two customers and two tables. In such a case, there are two possible system states, $\Theta=\{\theta_1, \theta_2\}$, indicating which table is larger. When the system state is $\theta_1$, $R_1(\theta_1)=L$ and $R_2(\theta_1)=S$ where $L \geq S$. On the other hand, if the system state is $\theta_2$, then $R_1(\theta_2)=L$ and $R_2(\theta_2)=S$. Moreover, in such a scenario, the signal is assumed to perfectly reveal the system state and indicate the exact amount of resource in each pool, e.g., $S=\{s_1,s_2\},~f(s_1|\theta=\theta_1)=1$, and $~f(s_2|\theta=\theta_2)=1$. Under such a signal structure, a customer can immediately know what the system state is when receiving the signal. The customer also knows that his opponent has the perfect signal of the true system state, i.e., the opponent also knows the exact size of each table. With such a simple setting, we temporally remove the uncertainty on the state to see how customers make decisions given the network externality. Given the opponent’s decision, which may be the larger table or smaller table, a rational customer should choose the table that can maximize his utility. If the decision made by the opponent is the smaller one, then the customer’s best choice is the larger one since $U(L,1) > U(S,1)$. However, if the opponent chooses the larger table, the customer’s choice depends on how severe the negative network externality is. If $U(L,2) > U(S,1)$, the customer will choose the same larger table. Otherwise, the smaller table will be chosen. The result shows that even both customers have already learned the true system state, i.e., which table is larger, they may not both choose the larger one in the equilibrium. Instead, the equilibrium depends on how severe the negative network externality is. If the externality results in an unacceptable penalty, then customers should choose different tables to avoid it. Best Response of Customers Under Perfect Signal ----------------------------------------------- Now let us consider the general scenario where there are $N$ customers, $K$ tables, and $L$ possible states $\Theta=\{\theta_1,...,\theta_L\}$. Here, we consider the perfect signals case, i.e., the system state $\theta$ and the sizes of tables $R_1(\theta),R_2(\theta),...,R_K(\theta)$ are known by all customers. The imperfect signal case will be discussed in Section \[sec\_gen\]. Since the customers are rational, their objectives in this game are to maximize their own utilities. However, since their utilities are determined by not only their own actions but also others’, the customers’ behaviors in the game are influenced by each other. A strategy describes how a player will play given any possible situations in the game. In the simultaneous Chinese restaurant game, the customer’s strategy should be a mapping from other customers’ table selections to his own table selection. Recalling that $n_j$ stands for the number of customers choosing table $j$. Let us denote $\mathbf{n_{-i}}=\{n_{-i,1},n_{-i,2},...,n_{-i,K}\}$ with $n_{-i,j}$ being the number of customers except customer $i$ choosing table $j$. Then, given $\mathbf{n_{-i}}$, a rational customer $i$ should choose the action as $$\label{eqn3} BE_i(\mathbf{n_{-i}},\theta) = \arg \max_{x \in A} U(R_x(\theta),n_{-i,x}+1).$$ The (\[eqn3\]) describes a special set of strategy called best response, which represents the optimal action of a customer that maximizes the utility given other customers’ actions. In the following, we give a formal definition of best response. Considering a game with $N$ players, each with an action space $A_i$ and a utility function $u_i(x_i,x_{-i})$, where $x_i$ is player $i$’s action and $x_{-i}$ is the actions of all players except player $i$. The best response of customer $i$ is $$BE_i(x_{-i}) = \arg \max_{x \in A_i} u_i(x,x_{-i}).$$ Nash Equilibrium Under Perfect Signal ------------------------------------- Nash equilibrium is a popular concept for predicting the outcome of a game with rational customers. Informally speaking, Nash equilibrium is an action profile, where each customer’s action is the best response to other customers’ actions in the profile. Since all customers use their best responses, none of them have the incentive to deviate from their actions. A formal definition of Nash equilibrium for the simultaneous game is given as follows. Nash equilibrium is the action profile $\mathbf{x^*}=\{x_1^*,x_2^*,...,x_N^*\}$ where $\forall i \in N$, $BE_i(x^*_{-i})=x_i^*$. According to the definition of Nash equilibrium, the sufficient and necessary condition of Nash equilibrium in the simultaneous Chinese restaurant game is stated in the following theorem. \[thm\_ne\_sim\_p\] Given the customer set $\{1,...,N\}$, the table set $\{1,...,K\}$, and the current system state $\theta$, for any Nash equilibrium of the simultaneous Chinese restaurant game with perfect signal, its equilibrium grouping $\mathbf{n^*}$ should satisfy the following conditions $$\label{eq_ne_sim_p} U(R_x(\theta),n^*_x) \geq U(R_y(\theta),n^*_y+1),\ \ if\ n^*_x > 0, \forall x,y \in \{1,...,K\}.$$ - Sufficient condition: suppose that the action profile of all players is $\mathbf{x}=\{x_1,...,x_N\}$ and such an action profile leads to the grouping $\mathbf{n^*}=\{n_1^*,...,n_K^*\}$ that satisfies (\[eq\_ne\_sim\_p\]). Without loss of generality, let us assume that customer $i$ chooses table $j$, i.e., $x_i=j$, then we have $$u_i(x_i,x_{-i}) = U(R_j(\theta),n^*_j).$$ If customer $i$ chooses any other table $k\neq j$, i.e., $x'_i=k\neq x_i=j$, then his utility becomes $$u'_i(x'_i,x_{-i}) = U(R_k(\theta),n^*_k+1).$$ Since $U(R_j(\theta),n^*_j)\geq U(R_k(\theta),n^*_k+1), \forall j, k$, we have $BE_i(x_{-i})=x_i, \forall i$. Therefore, $a=\{x_1,...,x_N\}$ is a Nash equilibrium. - Necessary condition: suppose that the Nash equilibrium $\mathbf{x^*}=\{x_1^*,x_2^*,...,x_N^*\}$ leads to the grouping $\mathbf{n^*}=\{n_1^*,...,n_K^*\}$. Without loss of generality, let us assume that customer $i$ chooses table $j$, i.e., $x_i^*=j$, then we have $$u_i(x_i^*,x^*_{-i}) = U(R_j(\theta),n^*_j) \text{ and } n^*_j > 0.$$ If customer $i$ chooses any other table $k\neq j$, i.e., $x'_i=k\neq x_i^*=j$, then his utility becomes $$u'_i(x'_i,x^*_{-i}) = U(R_k(\theta),n^*_k+1).$$ Since $\mathbf{x^*}=\{x_1^*,x_2^*,...,x_N^*\}$ is a Nash equilibrium, we have $BE_i(x^*_{-i})=x_i^*, \forall i$, i.e., $$U(R_j(\theta),n^*_j) \geq U(R_k(\theta),n^*_k+1),\ \ if\ n^*_j > 0, \forall j,k \in \{1,...,K\}.$$ From Theorem \[thm\_ne\_sim\_p\], we can see that, at Nash equilibrium, one customer’s utility would never become higher by deviating to another table. Moreover, any deviation to another table will degrade the utility of all customers in that table due to the negative network externality. The (\[eq\_ne\_sim\_p\]) also implies that customers may eventually have different utilities even the tables they choose have the same size. A simple example would be a three-customer restaurant with two tables in exact same size. Since there are three customers, at the Nash equilibrium, one of the table must be chosen by two customers while the other table is occupied only by one customer. Uniqueness of Equilibrium Grouping ---------------------------------- Obviously, there will be more than one Nash equilibrium since we can always exchange the actions of any two customers in one Nash equilibrium to build a new Nash equilibrium without violating the sufficient and necessary condition shown in (\[eq\_ne\_sim\_p\]). Nevertheless, the equilibrium grouping $\mathbf{n^*}$ may be unique as stated in the following Theorem. \[thm\_exist\_unique\_ne\_sim\_p\] There exists a Nash equilibrium in the simultaneous game with perfect signal. If the inequality in (\[eq\_ne\_sim\_p\]) strictly holds for all $x,y \in \{1,...,K\}$, then the equilibrium grouping $\mathbf{n}^*=\{n_1^*,...,n_K^*\}$ is unique. Since the signals are prefect, the current system state $\theta$ is known by all customers. In the following, we first propose a greedy algorithm to construct a Nash equilibrium and then show the uniqueness of the equilibrium grouping when the inequality strictly holds. Since exchanging the actions of any two customers at the Nash equilibrium will lead to another Nash equilibrium, the Nash equilibrium is generally not unique. Without loss of generality, in the proposed greedy algorithm, the customers choose their actions sequentially with customer $i$ being the $i-th$ customer choosing the action. We let customers choose their actions in the myopic way, i.e., they choose the tables that can maximize their current utilities purely based on what they have observed. Let $\mathbf{n_i}=\{n_{i,1},n_{i,2},...,n_{i,K}\}$ with $\sum_{j=1}^{K} n_{i,j}=i-1$ be the grouping observed by customer $i$. Then, customer $i$ will choose the myopic action given by $$BE^{myopic}_i(\mathbf{n_i},\theta)=\arg \max_{x \in A} U(R_x(\theta),n_{i,x}+1).$$ Let $\mathbf{x^*}=\{x_1^*,x_2^*,...,x_N^*\}$ be the output action set of the proposed greedy algorithm and $\mathbf{n^*}=\{n_1^*,n_2^*,...,n_K^*\}$ be the corresponding grouping. For any table $j$ with $n^*_j > 0$, suppose customer $k$ is the last customer choosing table $j$. Then, we have $$\label{eqn12} U(R_j(\theta),n_{k,j}+1) \geq U(R_{j'}(\theta),n_{k,j'}+1), \forall j' \in \{1,...,K\}.$$ Since customer $k$ is the last customer choosing table $j$, we have $n^*_j=n_{k,j}+1$ and $n^*_{j'} \geq n_{k,j'}$. Then, according to (\[eqn12\]), $\forall j' \in \{1,...,K\}$, we have $$\label{eqn13} U(R_j(\theta),n^*_{j})= U(R_j(\theta),n_{k,j}+1) \geq U(R_{j'}(\theta),n_{k,j'}+1) \geq U(R_{j'}(\theta),n^*_{j'}+1),$$ where the last inequality comes from the fact that $U(\cdot)$ is a decreasing function in terms of $n$. Note that (\[eqn13\]) holds for all $j,j' \in \{1,...,K\}$ with $n^*_j > 0$, i.e., $U(R_j(\theta),n^*_{j}) \geq U(R_{j'}(\theta),n^*_{j'}+1),\ \forall j,j' \in \{1,...,K\}\ with\ n^*_j > 0$. According to Theorem \[thm\_ne\_sim\_p\] and (\[eqn13\]), the output action set $\mathbf{x^*}=\{x_1^*,x_2^*,...,x_N^*\}$ from the proposed greedy algorithm is a Nash equilibrium. Next, we would like to prove by contradiction that if the inequality in (\[eq\_ne\_sim\_p\]) strictly holds, the equilibrium grouping $\mathbf{n}^*=\{n_1^*,...,n_K^*\}$ is unique. Suppose that there exists another Nash equilibrium with equilibrium goruping $\mathbf{n'}=\{n'_1,...,n'_K\}$, where $n'_j \neq n^*_j$ for some $j \in \{1,...,K\}$. Since both $\mathbf{n}^*$ and $\mathbf{n'}$ are equilibrium groupings, we have $\sum_{j=1}^K n'_j = \sum_{j=1}^K n^*_j = N$. In such a case, there exists two table $x$ and $y$ with $n'_x > n^*_x$ and $n'_y < n^*_y$. Then, since $\mathbf{n^*}$ is an equilibrium grouping, we have $$\label{eqn15} U(R_y(\theta),n^*_y) > U(R_x(\theta),n^*_x+1).$$ Since $n'_x > n^*_x$, $n'_y < n^*_y$, and $U(\cdot)$ is a deceasing function of $n$, we have $$\label{eqn17} U(R_x(\theta),n^*_x) > U(R_x(\theta),n^*_x+1) \geq U(R_x(\theta),n'_x),$$ and $$\label{eqn18} U(R_y(\theta),n'_y) > U(R_y(\theta),n'_y+1) \geq U(R_y(\theta),n^*_y).$$ Since $\mathbf{n'}$ is also an equilibrium grouping, we have $$\label{eqn19} U(R_x(\theta),n'_x) \geq U(R_y(\theta),n'_y+1).$$ According to (\[eqn17\]), (\[eqn18\]), and (\[eqn19\]) we have $$U(R_x(\theta),n^*_x+1) \geq U(R_x(\theta),n'_x) \geq U(R_y(\theta),n'_y+1) \geq U(R_y(\theta),n^*_y),$$ which contradicts with (\[eqn15\]). Therefore, the equilibrium grouping $\mathbf{n}^*$ is unique when the inequality in (\[eq\_ne\_sim\_p\]) strictly holds. Sequential Game with Perfect Signal: The Advantage of Playing First {#sec_seque} =================================================================== In the previous section, we have studied how negative network externality affects the action of each customer and found that a balance will finally be achieved among tables such that there will be no overwhelming requests for one table. However, we also find that some customers may have higher utilities at the Nash equilibrium. In this section, we extend the Chinese restaurant game into a sequential game model, where customers choose their actions in a pre-determined order. We assume that every customer can observe all the actions chosen before him, but cannot change the action once chosen. In this sequential Chinese restaurant game, customers make decisions sequentially with a predetermined order known by all customers, e.g., waiting in a line of the queue outside of the restaurant. Without loss of generality, in the rest of this paper, we assume the order is the same as the customer’s number, i.e., the order of customer $i$ is $i$. We assume every customer knows the decisions of the customers who make decisions before him, i.e., customer $i$ knows the decisions of customers $\{1,...,i-1\}$. Let $\mathbf{n_i}=\{n_{i,1},n_{i,2},...,n_{i,K}\}$ be the current grouping, i.e., the number of customers choosing table $\{1,2,...,K\}$ before customer $i$. The $\mathbf{n_i}$ roughly represents how crowded each tables is when customer $i$ enters the restaurant. Notice that $\mathbf{n_i}$ may not be equal to $\mathbf{n}$, which is the final grouping that determines customers’ utilities. A table with only few customers may eventually be chosen by many customers at the end of the game. The best response function of customer $i$ is $$BE^{se}_i(\mathbf{n_i},\theta)=\arg \max_x U(R_x(\theta),n_x(\mathbf{n_i})),$$ where $n_x(\mathbf{n_i})$ denotes the expected number of customers choosing table $x$ given $\mathbf{n_i}$. The problem here is how to predict the decisions of the remaining customers given the current observation $\mathbf{n_i}$ and state $\theta$. Subgame Perfect Nash Equilibrium and Advantage of Playing First --------------------------------------------------------------- In this subsection, we will study the possible equilibria of the sequential Chinese restaurant game. In particular, we will study the subgame perfect Nash equilibrium. A subgame is a part of the original game. In our sequential Chinese restaurant game, any game process begins from player $i$, given all possible actions before player $i$, could be a subgame. A formal definition of subgame is given as follows. A subgame in the sequential Chinese restaurant game is consisted of two elements: 1) It begins from customer $i$; 2) The current grouping before customer $i$ is $\mathbf{n_i}=\{n_{i,1},...,n_{i,K}\}$ with $\sum_{j=1}^{K} n_{i,j} = i-1$. With the definition of subgame, a subgame perfect Nash equilibrium is defined as follows. A Nash equilibrium is a subgame perfect Nash equilibrium if and only if it is a Nash equilibrium for any subgame. With the concept of subgame perfect Nash equilibrium, we can refine the number of Nash equilibria in the original game. We would like to show the subgame perfect Nash equilibrium in the sequential Chinese restaurant game by constructing one. Given a subgame, the corresponding equilibrium grouping and the best responses of agents in the subgame can be derived through two functions as follows. Let $EG(\mathbf{X_s},N_s)$ be the function that generates the equilibrium grouping for a table set $\mathbf{X_s}$ and number of customers $N_s$. The equilibrium grouping is given by (\[eq\_ne\_sim\_p\]) with $\mathbf{X}$ being replaced by $\mathbf{X_s}$ and $N$ being replaced by $N_s$. Notice that $\mathbf{X_s}$ could be any subset of the total table set $\{1,...,K\}$, and $N_s$ is less or equal to $N$. We will prove that the output of $EG(\cdot)$ is the corresponding equilibrium grouping in the subgame in Lemma \[lem\_eg\_subgame\]. Then, let $PC(\mathbf{X_s},\mathbf{n_s},N_s)$, where $\mathbf{n_s}$ denotes the current grouping observed by the customers, be the algorithm that generates the set of available tables given the current grouping $\mathbf{n_s}$ in the subgame. The algorithm removes the tables which have been already over-requested, i.e., the tables that already occupied by more than the expected number of customers in the equilibrium grouping. The procedures of implementing $PC(\mathbf{X_s},\mathbf{n_s},N_s)$ are described as follows: 1. Initialize: $\mathbf{X_o} = \mathbf{X_s}$, $N_t = N_s$ 2. $\mathbf{X_t} = \mathbf{X_o}$, $\mathbf{n^e}=EG(\mathbf{X_t},N_t)$, $\mathbf{X_o} = \{x|x \in \mathbf{X_t}, n^e_j \geq n_{s,j}\}$, $N_t = N_s - \sum_{x \in \mathbf{X_s} \setminus \mathbf{X_o}} n_{s,x}$. 3. If $\mathbf{X_o} \neq \mathbf{X_t}$, go back to step 2. 4. Output $\mathbf{X_o}$. As shown in the following Lemma, the $PC(\mathbf{X_s},\mathbf{n_s},N_s)$ will never remove the tables that are best choices of the customers. \[lem\_pg\_remove\] Given a subgame with current grouping $\mathbf{n_s}$, current available table set $\mathbf{X_s}$, and the number of players $N_s$, if table $j \not\in \mathbf{X'_s}=PC(\mathbf{X_s},\mathbf{n_s},N_s)$, then there exists at least one table $j' \in \mathbf{X'_s}$ such that $U(R_{j'}(\theta),n'_{j'}) \geq U(R_{j}(\theta),n_{s,j})$. Let $\mathbf{n^*} = EG(N_s,X_s)$. Since table $j$ is removed by $PC(\mathbf{X_s},\mathbf{n_s},N_s)$, the inequality $n_{s,j} > n^*_j$ should be hold. However, since $n_{s,j} > n^*_j$, the equilibrium grouping $\mathbf{n^*}$ is impossible to be reached in the Nash equilibrium of this subgame. Assuming the Nash equilibrium of this subgame is $\mathbf{n'}$, we have $n'_j \geq n_{s,j} > n^*_j$, which means $\sum_{k \in X_s, k \neq j} n'_k < \sum_{k \in X_s, k \neq j} n^*_k$. Therefore, there exists a $j' \in \mathbf{X_s} \setminus \{j\}$ such that $n'_{j'} < n^*_{j'}$. Since $\mathbf{n^*}$ is an equilibrium grouping, we have $$\label{eqn23} U(R_{j'}(\theta),n^*_{j'}) \geq U(R_{j}(\theta),n^*_{j}+1).$$ According to the above discussions, we have $$\label{eqn24} U(R_{j'}(\theta),n'_{j'}) > U(R_{j'}(\theta),n^*_{j'}) \geq U(R_{j}(\theta),n^*_{j}+1) \geq U(R_j(\theta),n_{s,j}) \geq U(R_j(\theta),n'_j) $$ where the the first inequality is due to $n'_{j'} < n^*_{j'}$, the second inequality is due to (\[eqn23\]), and the last two inequalities are due to $n'_j \geq n_{s,j} > n^*_j$. According to (\[eqn24\]), there exists at least one table $j'$ that can give the customer a higher utility than table $j$ in this subgame. Therefore, table $j$ is never the best response of the customer. Now, we propose a method to construct a subgame perfect Nash equilibrium. This equilibrium will satisfy the equilibrium grouping in (\[eq\_ne\_sim\_p\]). For each customer $i$, his strategy is described as follows: $$\label{eq_be_seq} BE^{se*}_i(\mathbf{n_i},\theta)=\arg \max_{x \in \mathbf{X^{i,cand}},n_{i,x} < n^{i,cand}_x} U(R_x(\theta),n^{i,cand}_x),$$ where $$\begin{aligned} \mathbf{X^{i,cand}} = PC(\mathbf{X},\mathbf{n_i},N), \\ N^{i,cand} = N - \sum_{x \in X \setminus X^{i,cand}} n_{i,x}, \\ \mathbf{n^{i,cand}} = EG(\mathbf{X^{i,cand}},N^{i,cand}).\end{aligned}$$ In Lemma \[lem\_eg\_subgame\], we show that the above strategy results in the equilibrium grouping in any subgame. \[lem\_eg\_subgame\] Given the available table set $X_s=PC(\mathbf{X},\mathbf{n_s},N)$, $N_s = N - \sum_{x \in X \setminus X_s} n_{s,x}$, the proposed strategy shown in (\[eq\_be\_seq\]) leads to an equilibrium grouping $\mathbf{n^*_s} = EG(\mathbf{X_s},N_s)$. We prove this by contradiction. Let $\mathbf{n}=\{n_j| j \in X_s\}$ be the final grouping after all customers choose their tables according to (\[eq\_be\_seq\]). Suppose that $\mathbf{n} \neq \mathbf{n^*_s}=EG(X_s,N_s)$, then there exists some tables $j$ that $n_j > n^*_{s,j}$. Let table $j$ be the first table that exceeds $n_{s,j}$ in this sequential subgame. Since $n_j > n^*_{s,j}$, there are at least $n^*_{s,j}+1$ customers choosing table $j$. Suppose the $n^*_{s,j}+1$-th customer choosing table $j$ is customer $i$. Let $\mathbf{n_i}=\{n_{i,1},n_{i,2},...,n_{i,K}\}$ be the current grouping observed by customer $i$ before he chooses the table. Since customer $i$ is the $n^*_{s,j}+1$-th customer choosing table $j$, we have $n_{i,j}=n^*_{s,j}.$ Since table $j$ is the first table exceeding $\mathbf{n^*_s}$ after customer $i$’s choice, we have $$n_{i,x} \leq n^*_{s,x}~\forall x \in X_s.$$ According to the definition of $PC(\cdot)$, none of the tables will be removed from candidates. Thus, $X^{i,cand}=X_s$ and $N^{i,cand}=N_s$. We have $$\mathbf{n^{i,cand}} = EG(\mathbf{X^{i,cand}},N^{i,cand}) = EG(X_s,N_s) = \mathbf{n^*_s}.$$ However, according to (\[eq\_be\_seq\]), the customer $i$ should not choose table $j$ since $n_{i,j}=n^*_{s,j}=n^{i,cand}_j$. This contradicts with our assumption that customer $i$ is the $n^*_{s,j}+1$-th customer choosing table $j$. Thus, the strategy (\[eq\_be\_seq\]) should lead to the equilibrium grouping $\mathbf{n^*_s}=EG(X_s,N_s)$. Note that Lemma \[lem\_eg\_subgame\] also shows that the final grouping of the sequential game should be $\mathbf{n^*}=EG(X,N)$ if all customers follow the proposed strategy in (\[eq\_be\_seq\]). With Lemma \[lem\_eg\_subgame\], we show the existence of subgame perfect Nash equilibrium with the following Theorem. \[thm\_exist\_ne\_seq\_p\] Given customer set $\{1,...,N\}$, table set $\mathbf{X}=\{1,...,K\}$, and the current state $\theta$, there always exists a subgame perfect Nash equilibrium with the corresponding equilibrium grouping $\mathbf{n^*}=\{n_1^*,...,n_K^*\}$ satisfying the conditions in (\[eq\_ne\_sim\_p\]). We would like to show that the proposed strategy in (\[eq\_be\_seq\]) forms a Nash equilibrium. Suppose customer $i$ chooses table $j$ in his round according to (\[eq\_be\_seq\]). Then, customer $i$’s utility is $u_i=U(R_j(\theta),n^*_j)$ since based on Lemma \[lem\_eg\_subgame\], the equilibrium grouping $\mathbf{n^*}$ will be reached at the end of the game. If customer $i$ is the last customer, i.e, $i=N$, and chooses another table $j' \neq j$ in his round, then his utility becomes $U(R_{j'}(\theta),n^*_{j'}+1)$. However, according to (\[eq\_ne\_sim\_p\]), we have $$u^*_j = U(R_{j}(\theta),n^*_j) \geq U(R_{j'}(\theta),n^*_{j'}+1) .$$ Thus, choosing table $j$ is never worse than choosing table $j'$ for customer $N$. If that customer $i$ is not the last customer, and he chooses table $j'$ instead of table $j$ in his round. Since all customers before customer $i$ follows (\[eq\_be\_seq\]), we have $n_{i,j} \leq n^*_{j}~\forall j \in \mathbf{X}$. Otherwise, $\mathbf{n^*}$ cannot be reached, which contradicts with Lemma \[lem\_eg\_subgame\]. If $n_{i,j'} < n^*_{j'}$, we have $n_{i+1,j'} \leq n^*_{j'}$. In addition, we have $n_{i+1,j} = n_{i,j} \leq n^*_{j}~\forall j \in \mathbf{X} \setminus \{j'\}$, since other tables are not chosen by customer $i$. Thus, $\mathbf{X^{i+1,cand}}=PC(X,\mathbf{n_{i+1}},N=X)$ and $N^{i,cand}=N$. According to Lemma \[lem\_eg\_subgame\], the final grouping should be $\mathbf{n^*}=EG(X,N)$. Thus, the new utility of customer $i$ becomes $u'_i = U(R_{j'}(\theta),n^*_{j'})$. However, according to (\[eq\_be\_seq\]), we have $$u_i=U(R_j(\theta),n^*_j) =\arg \max_{x \in \mathbf{X},n_{i,x} < n^*_x} U(R_x(\theta),n^*_x) \geq U(R_{j'}(\theta),n^*_{j'}) = u'_i.$$ Thus, choosing table $j'$ never gives customer $i$ a higher utility. If $n_{i,j'} = n^*_{j'}$, and the final grouping is $\mathbf{n'}=\{n'_1,n'_2,...,n'_K\}$. Since customer $i$ chooses table $j'$ when $n_{i,j'} = n^*_{j'}$, we have $n'_{j'} \geq n_{i+1,j'} = n_{i,j'}+1 = n^*_{j'}+1$. Thus, we have $$u_i = U(R_{j}(\theta),n^*_j) \geq U(R_{j'}(\theta),n^*_{j'}+1) \geq U(R_{j'}(\theta),n'_{j'}) = u'_i,~\forall j' \in X,$$ where the first inequality comes from the equilibrium grouping condition in (\[eq\_ne\_sim\_p\]), and the second inequality comes from the fact that $U(R,n)$ is decreasing over $n$ and $n'_{j'} \geq n^*_{j'}+1$. Thus, under both cases, choosing table $j'$ is never better than choosing table $j$. We conclude that $\{BE^{se*}_i(\cdot)\}$ in (\[eq\_be\_seq\]) forms a Nash equilibrium, where the grouping being the equilibrium grouping $\mathbf{n^*}$. Finally, we show that the proposed strategy forms a Nash equilibrium in every subgame. In Lemma \[lem\_pg\_remove\], we show that if the table $j$ is removed by $PC(X,\mathbf{n_s},N)$, it is never the best response of all remaining customers. Thus, we only need to consider the remaining table candidates $X_s = PC(X,\mathbf{n_s},N)$ in the subgame. Then, with Lemma \[lem\_eg\_subgame\], we show that for every possible subgame with corresponding $\mathbf{X_s}$, the equilibrium grouping $\mathbf{n^*_s}=EG(X_s,N_s)$ will be achieved at the end of the subgame. Moreover, the above proof shows that if the equilibrium grouping $\mathbf{n_s}$ will be achieved at the end of the subgame, $BE^{se*}_i(\cdot)$ is the best response function. Therefore, the proposed strategies indeed form a Nash equilibrium in every subgame, i.e., we have a subgame perfect Nash equilibrium. In the proof of the subgame perfect Nash equilibrium, we observe that sequential game structure brings advantages for those customers making decisions at the beginning of the game. According to (\[eq\_be\_seq\]), customers who make decisions early can choose the table that provides the largest utility in the equilibrium. When the number of customers choosing that table reaches equilibrium number, the second best table will be chosen by subsequent customers until it is full again. For the last customer, he has no choice but to choose the worst one. Imperfect Signal Model: How Learning Evolves {#sec_gen} ============================================ In Section \[sec\_seque\], we have showed that in the sequential game with perfect signal, customers choosing first have the advantages for getting better tables and thus higher utilities. However, such a conclusion may not be true when the signals are not perfect. When there are uncertainties on the table sizes, customers who arrive first may not choose the right tables, due to which their utilities may be lower. Instead, customers who arrive later may eventually have better chances to get the better tables since they can collect more information to make the right decisions. In other words, when signals are not perfect, learning will occur and may result in higher utilities for customers choosing later. Therefore, there is a trade-off between more choices when playing first and more accurate signals when playing later. In this section, we would like to study this trade-off by discussing the imperfect signal model. In the imperfect signal model, we assume that the system state $\theta \in \Theta=\{1,2,...,L\}$ is unknown to all $N$ customers. The sizes of $K$ tables can be expressed as functions of $\theta$, which are denoted as $R_1(\theta),R_2(\theta),...,R_K(\theta)$. The prior probability of $\theta$, $\mathbf{g_0}=\{g_{0,1},g_{0,2},...,g_{0,K}\}$ with $g_{0,l}=Pr(\theta=l)$, is assumed to be known by all customers. Moreover, each customer receives a private signal $s_i \in S$, which follows a p.d.f $f(s|\theta)$. Here, we assume $f(s|\theta)$ is public information to all customers. When conditioning on the system state $\theta$, the signals received by the customers are uncorrelated. In this sequential Chinese restaurant game with imperfect signal model, the customers make decisions sequentially with the decision orders being their numbers. After a customer $i$ made his decision, he cannot change his mind in any subsequent time and his decision and signal are revealed to all other customers. We assume customers are fully rational, which means that they follow the Bayesian learning method to learn the true state and choose their strategies to maximize their own utilities. Since signals are revealed sequentially, the customers who make decisions later can collect more information for better estimations of the system state. When a new signal is revealed, all customers follow the Bayesian rule to update their believes based on their current believes. Derived from (\[eqn\_g\]), we have the following Bayesian belief updating function $$\label{eqn33} g_{i,l} = \frac{g_{i-1,l}f(s_i|\theta=l)}{\sum_{w \in \Theta} {g_{i-1,w}f(s_i|\theta=w)}}.$$ Based on the updating rule in (\[eqn33\]), customer $i$ can update his belief when a new signal is revealed. Best Response of Customers -------------------------- Since the customers are rational, they will choose the action to maximize their own expected utility conditioning on the information they collect. Let $\mathbf{n_i} = \{n_{i,1},n_{i,2},...,n_{i,K}\}$ be the current grouping observed by customer $i$ before he chooses the table, where $n_{i,j}$ is the number of customers choosing table $j$ before customer $i$. Then, let $\mathbf{h_i}=\{s_1,s_2,...,s_{i-1}\}$ be the history of revealed signals before customer $i$. In such a case, the best response of customer $i$ can be written as $$\label{eqn34} x_i = BE_i(\mathbf{n_i},\mathbf{h_i},s_i) = \arg\max_j E[u_i(R_j(\theta),n_j)|\mathbf{n_i}, \mathbf{h_i}, s_i].$$ From (\[eqn34\]), we can see that when estimating the expected utility in the best response function, there are two key terms needed to be estimated by the customer: the system state $\theta$ and the final grouping $\mathbf{n}=\{n_1,n_2,...,n_K\}$. The system state $\theta$ is estimated using the concept of belief denoted as $\mathbf{g_i}=\{g_{i,1},g_{i,2},...,g_{i,L}\}$ with $g_{i,l}=Pr(\theta=l|\mathbf{h_i},s_i)$. Since the information on the system state $\theta$ in $\mathbf{n_i}$ is fully revealed by $\mathbf{h_i}$, given $\mathbf{h_i}$, $\mathbf{g_i}$ is independent with $\mathbf{n_i}$. Therefore, given the customer’s belief $\mathbf{g_i}$, the expected utility of customer $i$ choosing table $j$ becomes $$\label{eqn35} E[u_i(R_j(\theta),n_j)|\mathbf{n_i},\mathbf{h_i},s_i,x_i=j]=\sum_{w \in \Theta} {g_{i,w} E[u_i(R_j(w),n_j)|\mathbf{n_i},\mathbf{h_i},s_i,x_i=j,\theta=w]}.$$ Note that the decisions of customers $i+1,...,N$ are unknown to customer $i$ when customer $i$ makes the decision. Therefore, a close-form solution to (\[eqn35\]) is generally impossible and impractical. In this paper, we purpose a recursive approach to compute the expected utility. Recursive Form of Best Response {#sec_gen_recur} ------------------------------- Let $BE_{i+1}(\mathbf{n_{i+1}},h_{i+1},s_{i+1})$ be the best response function of customer $i+1$. Then, according to $BE_{i+1}(\mathbf{n_{i+1}},\mathbf{\mathbf{h_{i+1}}},s_{i+1})$, the signal space $S$ can be partitioned into $S_{i+1,1},...,S_{i+1,K}$ subspaces with $$\label{eqn36} S_{i+1,j}(\mathbf{n_{i+1}},\mathbf{h_{i+1}}) = \{s | s \in S, BE_{i+1}(\mathbf{n_{i+1}},\mathbf{h_{i+1}},s) = j\},~~ \forall j\in\{1,...,K\}.$$ Based on (\[eqn36\]), we can see that, given $\mathbf{n_{i+1}}$ and $\mathbf{h_{i+1}}$, $BE_{i+1}(\mathbf{n_{i+1}},\mathbf{h_{i+1}},s_{i+1})=j$ if and only if $s_{i+1} \in S_{i+1,j}$. Therefore, the decision of customer $i+1$ can be predicted according to the signal distribution $f(s|\theta)$ given by $$\label{eqn37} Pr(x_{i+1}=j|\mathbf{n_{i+1}},\mathbf{h_{i+1}})= \int_{s \in S_{i+1,j}(\mathbf{n_{i+1}},\mathbf{h_{i+1}})} {f(s)ds}.$$ Let us define $m_{i,j}$ as the number of customers choosing table $j$ after customer $i$ (including customer $i$ himself). Then, we have $n_j=n_{i,j}+m_{i,j}$, where $n_j$ denotes the final number of customers choosing table $j$ at the end of the game. Moreover, according to the definition of $m_{i,j}$, we have $$\label{eqn38} m_{i,j}=\left\{ \begin{array}{ll} 1+m_{i+1,j}, & \hbox{$x_i=j$;} \\ m_{i+1,j}, & \hbox{else.} \end{array} \right.$$ The recursive relation of $m_{i,j}$ in (\[eqn38\]) will be used in the following to get the recursive form of the best response function. We first derive the recursive form of the distribution of $m_{i,j}$, i.e., $Pr(m_{i,j}=X|\mathbf{n_{i}},\mathbf{h_{i}},s_{i},x_{i})$ can be expressed as a function of $Pr(m_{i+1,j}=X|\mathbf{n_{i+1}},\mathbf{h_{i+1}},s_{i+1},x_{i+1}=j,\theta=l),~\forall~l \in \Theta,~0 \leq j \leq K$, as follows: $$\begin{aligned} \label{eqn39} \!\!\!\!\!\!\!\!\!\!\!\!&&\!\!\!\!\!\!\!\!\!\!\!\!Pr(m_{i,j}=X|\mathbf{n_i},\mathbf{h_i},s_i,x_i,\theta=l) =\left\{ \begin{array}{ll} Pr(m_{i+1,j}=X-1|\mathbf{n_i},\mathbf{h_i},s_i,x_i,\theta=l), & \hbox{$x_i=j$,} \\ Pr(m_{i+1,j}=X|\mathbf{n_i},\mathbf{h_i},s_i,x_i,\theta=l), & \hbox{$x_i \neq j$,} \end{array} \right. \\ \!\!\!\!\!\!\!\!\!\!\!\!&&\!\!\!\!\!\!\!\!\!\!\!\!=\!\!\!\left\{\!\!\!\! \begin{array}{ll} \sum_{u \in \{1,...,K\}} \int_{s \in S_{i+1,u}(\mathbf{n_{i+1}},\mathbf{h_{i+1}})} \nonumber {Pr(m_{i+1,j}=X-1|\mathbf{n_{i+1}},\mathbf{h_{i+1}},s_{i+1}=s,x_{i+1}=u, \theta=l)} f(s|\theta=l) ds, &\!\!\! \hbox{$x_i=j$,} \\ \sum_{u \in \{1,...,K\}} \int_{s \in S_{i+1,u}(\mathbf{n_{i+1}},\mathbf{h_{i+1}})} {Pr(m_{i+1,j}=X|\mathbf{n_{i+1}},\mathbf{h_{i+1}},s_{i+1}=s,x_{i+1}=u, \theta=l)} f(s|\theta=l) ds, &\!\!\! \hbox{$x_i \neq j$,} \end{array} \right.\end{aligned}$$ where $\mathbf{h_{i+1}}$ and $\mathbf{n_{i+1}}$ can be obtained using $$\label{eq_h_plus} \mathbf{h_{i+1}}=\{h_{i},s_i\} \text{ and } \mathbf{n_{i+1}}=\{n_{i+1,1},...,n_{i+1,K}\},$$ with $$\label{eq_n_plus} n_{i+1,k}=\left\{ \begin{array}{ll} n_{i,k}+1, & \mbox{if $x_i=k$}, \\ n_{i,k}, & \mbox{otherwise}. \\ \end{array} \right.$$ Based on (\[eqn39\]), $Pr(m_{i,j}=X|\mathbf{n_{i}},\mathbf{h_{i}},s_{i},x_{i},\theta=l)$ can be recursively calculated. Therefore, we can calculate the expected utility $E[u_i(R_j(\theta),n_j)|\mathbf{n_i}, \mathbf{h_i}, s_i]$ as $$\label{eq_exp_util_recur} E[u_i(R_j(\theta),n_j)|\mathbf{n_i}, \mathbf{h_i}, s_i] = \sum_{l \in \Theta} \sum_{x=0}^{N-i+1} g_{i,l} Pr(m_{i,j}=x|\mathbf{n_i},\mathbf{h_i},s_i,x_i=j,\theta=l)u_i(R_j(l),n_{i,j}+x).$$ Finally, the best response function of customer $i$ can be derived by $$\label{eqn44} BE_i(\mathbf{n_i},\mathbf{h_i},s_i) = \arg\max_j \sum_{l \in \Theta} \sum_{x=0}^{N-i+1} g_{i,l}Pr(m_{i,j}=x|\mathbf{n_i},\mathbf{h_i},s_i,x_i=j,\theta=l)u_i(R_j(l),n_{i,j}+x).$$ With the recursive form, the best response function of all customers can be obtained using backward induction. The best response function of the last customer $N$ can be found as $$\label{eqn45} BE_N(\mathbf{n_N},h_N,s_N) = \arg\max_j \sum_{l \in \Theta} g_{N,l} u_N(R_j(l),n_{N,j}+1).$$ Note that $Pr(m_{N,j}=X|\mathbf{n_{N}},\mathbf{h_{N}},s_{N},x_{N},\theta)$ can be easily derived as follows: $$\label{eqn46} Pr(m_{N,j}=1|\mathbf{n_{N}},\mathbf{h_{N}},s_{N},x_{N},\theta)= \left\{ \begin{array}{ll} 1, & \mbox{if $x_{N} = j$}, \\ 0, & \mbox{otherwise}. \\ \end{array} \right.$$ Simulation {#sec_sim} ========== In this section, we verify the proposed recursive best response and corresponding equilibrium. We simulate a Chinese restaurant with two tables $\{1,2\}$ and two possible states $\theta \in \{1,2\}$. When $\theta=1$, the size of table $1$ is $R_1(1)=100$ and the size of table $2$ is $R_2(1)=40$. When $\theta=2$, $R_1(2)=40$ and $R_2(2)=100$. The state is uniformly randomly chosen at the beginning of each simulation with probability $0.5$. The number of customers is fixed. Each customer receives a randomly generated signal $s_i$ at the beginning of the simulation. While conditioning on the system state, the signals customers received are independent. The signal distribution $f(s|\theta)$ is given by $$Pr(s=1|\theta=1) = Pr(s=2|\theta=2) = p~,~Pr(s=2|\theta=1) = Pr(s=1|\theta=2) = 1-p,$$ where $p \geq 0.5$ can be regarded as the quality of signals. When the signal quality $p$ is closer to $1$, the signal is more likely to reflect the true state $\theta$. With the signals, customers make their decisions sequentially. After the $i$-th customer makes his choice, he reveals his decision and signal to other customers. The game ends after the last customer made his decision. Then, the utility of the customer $i$ choosing table $j$ is given by $U_i(R_j(\theta),n_j)=\frac{R_j}{n_j}$, where $n_j$ is the number of customers choosing table $j$ at the end of the game. Optimality of Proposed Best Response ------------------------------------ We first verify the optimality of the best response. We study the $5$-customer scheme with the same settings in the previous simulation. We assume that all customers except customer $2$ apply their best response strategies while customer $2$ chooses the table opposite to his best response strategy with a miss probability $p^{mis}$. We assume the signal quality $p \in [0.5,1]$ and $p^{mis} \in [0,1]$ in the simulations. From Fig. \[fig\_sim\_check\_2\], we can see that when $p^{mis}$ increases, the expected utility of customer $2$ always decreases for any signal quality $p$. This confirms the optimality of the proposed best response function. We also observe that when customer $2$ has a positive $p^{mis}$, at least one of the other customers will have a better average utility. Using the expected utility increase of customer $3$ shown in Fig. \[fig\_sim\_check\_3\] as an example, when customer $2$ makes a mistake in choosing the table, customer $3$ benefits from the mistake of customer $2$ under most signal qualities. Advantage of Playing Positions vs. Signal Quality ------------------------------------------------- Next, we investigate how the decision order and quality of signals affect the utility of customers. We follow the same settings in previous simulations except the table sizes. We fix the size of one table as $100$. The size of the other table is $r\times100$, where $r$ is the ratio of the table sizes. In the simulations, we assume the ratio $r \in [0,1]$. When the ratio $r=1$, two tables are identical, but the utility of choosing each table may have different utility since we may have odd customers. When $r=0$, one table has a size of $0$, which means a customer has a positive utility only when he chooses the correct table. We first simulate a 3-customer scheme. From Fig. \[fig\_sim\_adv\_3\], we can see that the advantage of customers making decisions at different order is significantly affected by both the signal quality and the table size ratio. As shown in Fig. \[fig\_sim\_adv\_3\_max\], when the signal quality is high and the table size ratio is low, customer $3$ has the largest expected utility. For other regions, customer $1$ has the largest expected utility. This phenomenon can be explained as follows. When the ratio is lower than $\frac{1}{3}$, all customers desire the larger table since even all of them select the larger one, each of them still have a utility larger than choosing the smaller one. In such a case, customers who choose late would have advantages since they have collected more signals and have a higher probability to identify the large table. On one hand, when the signal quality is low, even the third customer cannot form a strong belief on the true state. In such a case, the expected size of each table becomes less significantly, and customers’ decisions rely more on the negative network externality effect, i.e., how crowded of each table. When the first two customers choose the same table, customer $3$ is more likely to choose the other table to avoid the negative network externality. On the other hand, when the signal quality is high, customer $3$ is likely to form a strong belief on the true state and will choose the table according to the signals he collected. Therefore, when signal quality is high and the table size ratio is low, customer $3$ has the advantage of getting a higher utility in the Chinese restaurant game. Nevertheless, due to the complicated game structure in Chinese restaurant game, the effect of signal quality and table size ratio is generally non-linear. As shown in Fig. \[fig\_sim\_adv\_5\], when the number of customers increases to $5$, similar to the 3-customer scheme, customer $5$ has the largest utility when the signal quality is high and the table size ratio is low, while customer $1$ has the largest utility when the signal quality is low and the table size ratio is high. However, we observe that in some cases, customer $3$ becomes the one with largest utility. The reasons behind this phenomenon is as follows. In these cases, we observe that the expected number of customers in the larger table is $3$, and this table provides the customers a larger utility then the other one at the equilibrium. Therefore, customers would try to identify this table and choose it according to their own believes. Since customer $3$ collects more signals than customer $1$ and $2$, he is more likely to choose the correct table. Moreover, since he is the third customer to choose a table, this table is always available to him. Therefore, customer $3$ has the largest expected utility in these cases. Note that the expected table size is determined by both the signal quality and the table size ratio. Generally, when the signal quality is low, a customer is less likely to construct a strong belief on the true state, i.e., the expected table sizes of both tables are similar. This suggests that a lower signal quality has a similar effect on the expected table size as a higher table size ratio. Our arguments are supported by the concentric-like structure shown in Fig. \[fig\_sim\_adv\_5\]. The same arguments can be applied to the 10-customer scheme, which is shown in Fig. \[fig\_sim\_adv\_10\]. We can observe the similar concentric-like structure. Additionally, we observe that when the table size ratio increases, the order of customer who has the largest utility in the peaks decreases from $10$ to $5$. This is consistent with our arguments since when the table size ratio increases, the equilibrium number of customers in the large table decreases from $10$ to $5$. This also explains why customer $1$ does not have the largest utility when the table size ratio is high. In this case, the equilibrium number of customers in the large table is $5$, and the large table provides higher utilities to customers in the equilibrium. Since customer $5$ can collect more signals than previous customers, he has better knowledge on the table size than customer 1 to 4. Moreover, since customer $5$ is the fifth one to choose the table, he always has the opportunity to choose the large table. In such a case, customer 5 is the one with the largest expected utility when the table size ratio is high. Next, we discuss two specific scenarios: the resource pool scenario with $r=0.4$ and available/unavailable scenarios with $r=0$. In resource pool scenario, the table size of the second table is $40$. Users act sequentially and rationally to choose these two tables to maximize their utilities. In available/unavailable scenario, the second table size is $0$, which means that a customer has positive utility only when he chooses the right table. For both scenarios, we examine the schemes with the number of customers $N=3$ and $N=5$. From Fig. \[fig\_sim\_pool\], we can see that in the resource pool scenario with $r=0.4$, customer $1$ on average has significant higher utility, which is consistent with the result in Fig. \[fig\_sim\_adv\_5\]. Using 5-customer scheme shown in Fig. \[fig\_sim\_pool\_5\] as an example, the advantage of playing first becomes significant when signal quality is very low ($p < 0.6$), or the signal quality is high ($p > 0.7$). We also find that customer $5$ has the lowest average utility for most signal quality $p$. We may have a clearer view on this in the 3-customer scheme. We list the best response of customers given the received signals in Fig. \[tab\_be\_pool\_3\]. We observe that when signal quality $p$ is large, both customer $1$ and $2$ follow the signals they received to choose the tables. However, customer $3$ does not follows his signal if the first two customers choose the same table. Instead, customer $3$ will choose the table that is still empty. In this case, although customer $3$ know which table is larger, he does not choose that table since it has been occupied by the first two customers. The network externality effect dominates the learning advantage in this case. However, when $p$ is low, the best response of customer $1$ is opposite, i.e., he will choose the table that is indicated as the smaller one by the signal he received. At the first glance, the best response of customer $1$ seems to be unreasonable. However, such a strategy is indeed customer $1$’s best response considering the expected equilibrium in this case. According to Theorem \[thm\_exist\_ne\_seq\_p\], if perfect signals ($p=1$) are given, the large table should be chosen by customer $1$ and $2$ since the utility of large table is $100/2=50$ is larger than the that of the small table, which is $40/1=40$, in the equilibrium. However, when the imperfect signals are given, customers choose the tables based on the expected table sizes. When signal quality is low, the uncertainty on the table size is large, which leads to similar expected table sizes for both tables. In such a case, customer $1$ favors the smaller table because it can provide a higher expected utility, compared with sharing with another customer in the larger table. In the available/unavailable scenario, as shown in Fig. \[fig\_sim\_avail\], the advantage of customer $1$ in playing first becomes less significant. Using 5-customer scheme shown in Fig. \[fig\_sim\_avail\_5\] as an example, when signal quality $p$ is larger than $0.6$, customer $5$ has the largest average utility and customer $1$ has smallest average utility. Such a phenomenon is because customers should try their best on identifying the available table when $r=0$. Learning from previous signals gives the later customers a significant advantage in this case. Nevertheless, we observe that the best responses of later customers are not necessary always choosing the table that is more likely to be available. We use the 3-customer as an illustrative example. We list the best response of all customers given the received signals in Fig. \[tab\_be\_avail\_3\]. When the signal quality is pretty low ($p = 0.55$), we have the same best response as the one in resource pool scenario, where the network externality effect still plays a significant role. Using $(s_1,s_2,s_3)=(2,2,1)$ as an example, even customer $3$ finds that table $2$ is more likely to be available, his best response is still choosing table $1$ since table $2$ is already chosen by both customer $1$ and $2$, and the expected utility of choosing table $1$ with only himself is higher than that of choosing table $2$ with other two customers. As the signal quality $p$ becomes high, e.g., $p = 0.9$, customer $3$ will choose the table according to all signals $s_1,s_2,s_3$ he collected. The belief constructed by the signals are now strong enough to overcome the loss in the network externality effect, which makes him choose the table that is more likely to be available. Conclusion {#sec_con} ========== In this paper, we proposed a new game, called Chinese Restaurant Game, by combining the strategic game-theoretic analysis and non-strategic machine learning technique. The proposed Chinese restaurant game can provide a new general framework for analyzing the strategic learning and predicting behaviors of rational agents in a social network with negative network externality. By conducting the analysis on the proposed game, we derived the optimal strategy for each agent and provided a recursive method to achieve the equilibrium. The tradeoff between two contradictory advantages, which are making decisions earlier for choosing better tables and making decisions later for learning more accurate believes, is discussed through simulations. We found that both the signal quality of the unknown system state and the table size ratio affect the expected utilities of customers with different decision orders. Generally, when the signal quality is low and the table size ratio is high, the advantage of playing first dominates the benefit from learning. On the contrary, when the signal quality is high and the table size ratio is low, the advantage of playing later for better knowledge on the true state increases the expected utility of later agents.

--- author: - 'M. H. D. van der Wiel' - 'D. A. Naylor' - 'G. Makiwa' - 'M. Satta' - 'A. Abergel' bibliography: - '../../literature/allreferences.bib' title: 'Three-dimensional distribution of hydrogen fluoride gas toward NGC6334I and I(N)[^1]' --- [ The HF molecule has been proposed as a sensitive tracer of diffuse interstellar gas, while at higher densities its abundance could be influenced heavily by freeze-out onto dust grains. ]{} [ We investigate the spatial distribution of a collection of absorbing gas clouds, some associated with the dense, massive star-forming core NGC6334I, and others with diffuse foreground clouds elsewhere along the line of sight. For the former category, we aim to study the dynamical properties of the clouds in order to assess their potential to feed the accreting protostellar cores. ]{} [ We use far-infrared spectral imaging from the SPIRE iFTS to construct a map of HF absorption at 243  in a 6$\times$35 region surrounding NGC6334 I and I(N). ]{} [ The combination of new, spatially fully sampled, but spectrally unresolved mapping with a previous, single-pointing, spectrally resolved HF signature yields a three-dimensional picture of absorbing gas clouds in the direction of NGC6334. Toward core I, the HF equivalent width matches that of the spectrally resolved observation. At angular separations $\gtrsim$20 from core I, the HF absorption becomes weaker, consistent with three of the seven components being associated with this dense star-forming envelope. Of the remaining four components, two disappear beyond $\sim$1 distance from the NGC6334 filament, suggesting that these clouds are spatially associated with the star-forming complex. Our data also implies a lack of gas phase HF in the envelope of core I(N). Using a simple description of adsorption onto and desorption from dust grain surfaces, we show that the overall lower temperature of the envelope of source I(N) is consistent with freeze-out of HF, while it remains in the gas phase in source I. ]{} [ We use the HF molecule as a tracer of column density in diffuse gas ($n_\mathrm{H}$$\approx$$10^2$–$10^3$ ), and find that it may uniquely trace a relatively low density portion of the gas reservoir available for star formation that otherwise escapes detection. At higher densities prevailing in protostellar envelopes ($\gtrsim$$10^4$ ), we find evidence of HF depletion from the gas phase under sufficiently cold conditions. ]{} Introduction {#sec:intro} ============ The hydrogen fluoride molecule, HF, was first observed in the interstellar medium by @neufeld1997b with the [*Infrared Space Observatory*]{} [[*ISO*]{}, @kessler1996]. While [*ISO*]{} had a wavelength range that encompassed only the $J$=2–1 rotational transition of HF, the next observatory able to observe HF – the [*Herschel*]{} Space Observatory [@pilbratt2010] – covered longer far-infrared wavelengths, and it thus opened up access to the ground-state rotational transition, $J$=1–0, at 1232.48 GHz (243.24 ). [*Herschel*]{} has observed HF in absorption along many lines of sight, both inside the Galaxy [@neufeld2010b; @sonnentrucker2010; @sonnentrucker2015; @philips2010; @kirk2010; @monje2011a; @emprechtinger2012; @lopez-sepulcre2013a; @goicoechea2013] and in nearby extragalactic objects [@rangwala2011; @kamenetzky2012; @rosenberg2014a; @monje2014]. HF absorption has even been detected with ground-based observatories: @monje2011c have made use of the substantial redshift of the Cloverleaf quasar at $z$=2.56 shifting the HF 1–0 line into the submillimeter window attainable with the CSO on Mauna Kea, and @kawaguchi2016 detect it in the $z$=0.89 absorber toward PKS1830$-$211, using ALMA in the Chilean Atacama desert. Because of its large dipole moment and high Einstein $A$ coefficient for radiative decay, rotational states $J$$\neq$0 of HF only become significantly populated in highly energetic conditions. It is for this reason that HF has been clearly detected in emission in a mere handful of cases: in the inner region of an AGB star’s envelope [IRC+10216, @agundez2011], in the Orion Bar photodissociation region [@vandertak2012a], and in an external galaxy harboring an actively accreting black hole [Mrk231, @vanderwerf2010]. Atomic fluoride, F, has a unique place in the interstellar chemistry of simple molecules. It is the only element which, simultaneously, (1) is mainly neutral because of its ionization potential $>$13.6 eV, (2) reacts exothermically with  – unlike *any* other neutral atom – to form its neutral diatomic hydride HF, and (3) lacks an efficient chemical pathway to produce its hydride cation HF$^+$ due to the strongly endothermic nature of the reaction with H${_3}^+$. We refer to @neufeld2009b, references therein, and the comprehensive review by @gerin2016 for more details on the chemistry of HF and a comparison with other hydride molecules. For the reasons listed above, chemical models predict that essentially all interstellar F is locked in HF molecules [@zhu2002; @neufeld2005], which has been confirmed by observations across a wide range of atomic and molecular ISM conditions [e.g., @sonnentrucker2010; @sonnentrucker2015]. With recent experimental results by @tizniti2014 showing that, especially at low temperatures approaching 10 K, the reaction F +  $\rightarrow$ HF + H proceeds somewhat slower than earlier assumptions, chemical models are now able to reproduce HF/ ratios of $\sim$, measured most directly by @indriolo2013a, and observed to be rather stable across different sightlines. Interferometric observations show that CF$^+$, the next most abundant F-bearing species after HF, has an abundance roughly two orders of magnitude lower than HF, both inside our Galaxy [@liszt2015b] and in an extragalactic absorber [@muller2016]. As for destruction of HF, the most efficient processes are UV photodissociation and reactions with C$^+$, but both of these are unable to drive the majority of fluoride out of HF, due to shielding, already at modest depths of $A_V>0.1$ [@neufeld2005]. Because of the constant HF/ abundance ratio and the high probability that HF molecules are in the rotational ground state, measurements of HF $J$=0$\rightarrow$1 absorption provide a straightforward proxy of  column density. This has led to the suggestion that, at least in diffuse gas, HF absorption is a more reliable tracer of total gas column density than the widely used carbon monoxide (CO) rotational *emission* lines, and is more sensitive than CH or  absorption [e.g., @gerin2016]. Apart from the uncertain and variable CO abundance, local excitation conditions have a profound effect on the level populations of CO, complicating the conversion from observed line strength of a particular CO transition to  column density [@bolatto2013]. The greatest gas-phase CO abundance variations occur in dense, cold regions where CO freezes out onto surfaces of dust grains, proven by observed CO abundances decreasing in the gas phase and increasing in the ice phase as conditions get colder [e.g., @jorgensen2005a; @pontoppidan2005a]. In addition, the particular fraction of the neutral ISM that is in the diffuse/translucent phase is inconspicuous in CO [@bolatto2013], but is detectable using hydride absorption lines. Of course, for absorption line studies, one relies on lines of sight with sufficiently strong continuum background, for example those toward dense star-forming clouds. Such restrictions do not apply for emission line tracers. Besides CO rotational lines, fine structure line emission due to atomic C and the C$^+$ and N$^+$ ions has been used as a tracer of (diffuse) gas throughout the Galaxy [e.g., @langer2014a; @velusamy2014b; @gerin2015a; @goicoechea2015b; @goldsmith2015]. For all these tracers, however, the conversion to  column density depends strongly on physical properties such as ionization fraction and excitation conditions. Based on the above arguments, HF absorption measurements are a good tracer of overall gas column density. However, as addressed for example by @philips2010 and @emprechtinger2012, HF itself may suffer from freeze-out effects as occurs with other interstellar molecules. While studies have been done on the interaction of  with HF as a polluting agent in the Earth’s atmosphere [@girardet2001], the density and temperature conditions needed for HF adsorption onto dust grains have not been studied in astrophysical contexts so far. Any freeze-out of interstellar HF will obfuscate the direct connection between HF absorption depth and  column density described above. The well-known progression of pre- and protostellar stages for stars with masses similar to the Sun [@shu1977] is not applicable for high-mass stars ($\gtrsim8$ ). In the latter category, protostellar hydrogen fusion starts while accretion from the surrounding gas envelope is still ongoing [@palla1993]. In the ‘competitive accretion’ scenario, multiple massive protostars in a clustered environment are fed from the same gas reservoir [@bonnell2001b]. For high-mass protostars, material can continuously be added to the gravitationally bound circumstellar envelope which provides the reservoir for further accretion onto the protostar. It is therefore important, particularly for regions of *high-mass* star formation, to study not just the gravitationally bound circumstellar envelopes, but also the dynamical properties of surrounding gas clouds. Especially for the latter component, simple hydride molecules have the potential to reveal gas reservoirs to which emission lines of ‘traditional’ tracer species, such as CO, , and CS, are insensitive due to their relatively high critical densities. In this paper, we investigate two envelopes of (clusters of) protostars embedded in the molecular cloud as well as lower density clouds surrounding the dense complex. The filamentary, star-forming cloud NGC 6334, at a distance of 1.35 kpc [@wu2014], harbors a string of dense cores, identified in the far-infrared by roman numerals I through VI [@mcbreen1979], with an additional source identified $\sim$2 north of source I, later named ‘I(N)’ [@gezari1982]. The larger scale NGC6334 filament has an  column density of $>$  even at positions away from the embedded cores [@russeil2013]. @zernickel2013 observed the velocity structure of NGC6334 at 0.15 pc resolution. These authors explain the velocity profile along the filament with a cylindrical model collapsing along its longest axis under the influence of gravity. In this paper we study specifically the region of $\sim$2.4$\times$1.6 pc surrounding the embedded cores and . Source I is host to an ultra-compact  region, designated source ‘F’ in a 6 cm radio image of the cloud [@rodriguez1982]. Based on multi-wavelength dust continuum measurements, studies by @sandell2000 and @vandertak2013a have independently determined that the mass of source I(N) exceeds that of sister source I by a factor of $\sim$2–5, but the ratio of their bolometric luminosities is 30–140 in favor of source I, due to the markedly lower temperature for source I(N). As expected for a warm (up to $\sim$100 K), dense, massive star-forming core, NGC6334 I is extremely rich in molecular lines, spectacularly demonstrated by the 4300 lines detected in the 480–1907 GHz spectral survey by @zernickel2012[^2]. The differences between the two neighboring cores all suggest that core I is in a more evolved stage of star formation than core I(N). Both cores have been studied with radio and (sub)millimeter interferometer observatories, showing that each separates into several subcores at arcsecond resolution [i.e., at scales $\lesssim$0.01 pc, @hunter2006; @hunter2014; @brogan2009]. To probe gas clouds in front of the NGC6334 complex, absorption measurements have been obtained in lines of several hydrides. Spatially extended OH hyperfine line absorption at was observed toward the NGC6334 filament by @brooks2001. The spectrally resolved mapping observations from the Australia Telescope Compact Array allowed these authors to ascribe particular velocity components of the absorption to a foreground cloud close to NGC6334 and other components to clouds with even larger angular extent. @vanderwiel2010 used the Heterodyne Instrument for the Far-Infrared [HIFI, @degraauw2010] onboard [*Herschel*]{} to study the spectral profile of the rotational ground state lines of CH at 532 and 537 GHz, and found four distinct absorption components overlapping with the velocity range of OH absorption, and one single emission component emanating in core NGC6334 I itself. At 1232.5 GHz in the same spectral survey, @emprechtinger2012 find the exact same four absorbing clouds in the HF rotational ground state, and invoke three velocity components to explain the hot core component. While the hot core component(s) appear in emission in CH, they are in absorption in HF, because CH 1$\rightarrow$0 has a lower Einstein $A$ coefficient than HF 1$\rightarrow$0 (see above). ---------------- ------------- --------------- ----------------- ------------- ------------ --------- ------- Observation Target name Right Ascension Declination Jiggle Gain observation ID date (J2000) (J2000) pattern mode 1342214827 2011-02-26 NGC6334 I 17205415 $-$3547074 4$\times$4 Nominal 11491 1342214841 2011-02-27 NGC6334 I(N) 17205609 $-$3545073 4$\times$4 Nominal 11491 1342251326 2012-09-24 NGC6334 I 17205415 $-$3547074 4$\times$4 Bright 9605 1342214828 2011-02-26 NGC6334 ‘INT’ 17204810 $-$3545429 Stare Nominal 915 1342214829 2011-02-26 NGC6334 ‘OFF’ 17203904 $-$3543435 Stare Nominal 915 ---------------- ------------- --------------- ----------------- ------------- ------------ --------- ------- The CH and HF signatures were observed toward NGC6334 I with the high spectral resolution spectrometer HIFI in single point mode [@degraauw2010]; its single pixel receiver did not provide any spatial information. In a *map* of CH or HF absorption covering the region surrounding source I, one would expect to see a disentanglement of the different spatial extent of each velocity component as illustrated in Fig. \[fig:HFabscomp\]. Toward core I, the velocity resolved HF absorption signature, with a total equivalent width, $\int (1-I_\mathrm{norm}) \mathrm{d}V$, of 16 , was modeled with seven components. At positions away from , but still on the NGC6334 filament, the equivalent width should diminish to 11  representing the four foreground clouds, while at positions off of the cores and the filament, only the two foreground components that are more extended than the dense molecular cloud should be visible, and the equivalent width should drop to 3 . This paper presents results from [*Herschel*]{} SPIRE iFTS spectral mapping observations toward a 6$\times$35 region surrounding cores I and I(N) in the NGC6334 star-forming complex. The observations are described in Sect. \[sec:obs\] and the resulting map of HF absorption depth is discussed in Sect. \[sec:obsresults\]. The signal is interpreted in Sect. \[sec:analysis\], both in the context of foreground clouds and in that of freeze-out conditions in the dense cores. Conclusions are summarized in Sect. \[sec:conclusions\]. Observations and data reduction {#sec:obs} =============================== The spectral mapping observations used in this work were obtained as part of the ‘evolution of interstellar dust’ guaranteed time program [@abergel2010] with the Spectral and Photometric Imaging Receiver [SPIRE, @griffin2010] on board the [*Herschel*]{} space observatory [@pilbratt2010]. SPIRE’s imaging Fourier Transform Spectrometer (iFTS) provides a jiggling observing mode that uses its 54 detectors to obtain Nyquist sampled spatial maps, covering the entire frequency range of the Spectrometer Long Wavelength (SLW, 447–1018 GHz) and the Spectrometer Short Wavelength (SSW, 944–1568 GHz) bands. The spectral resolution of 1.2 GHz corresponds to a resolving power $\nu/\Delta\nu\approx10^3$, roughly 300  at the frequency of the HF 1–0 transition, 1232.5 GHz (243 ). Three partly overlapping, fully sampled SPIRE iFTS 4$\times$4 jiggle observations were performed in a total of nine hours of observing time, two centered on NGC6334 I and the third on NGC6334 I(N). A fourth, sparsely sampled observation, centered $\sim$2 northwest of core I, has considerable overlap with the combined area covered by the three other observations. This fourth observation is treated as an extra jiggle position in the gridding process described below. Finally, a fifth observation, also sparsely sampled, is centered 45 northwest of core I and its footprint therefore has no overlap with our mapped area. At each jiggle position, four repeated scans of the FTS mechanism were executed in high spectral resolution mode. Details of the observations are summarized in Table \[t:obs\]. The placement of the different pointings described here is shown in Fig. \[fig:FTSfootprints\] in the Appendix. After inspection of the initial observations from February 2011, some detectors were found to suffer from saturation due to the bright emission toward source I. The observation toward source I was therefore repeated in ‘bright source mode’ [@lu2014b] in September 2012 to obtain well calibrated spectra toward the brightest position. The majority of detector/jiggle combinations in the original observation point toward less bright regions and are therefore still useful in constructing the final map. The above SPIRE iFTS observations are processed with the ‘extended source’ pipeline in HIPE 12.1.0 and the `spire_cal_12_3` calibration tree, which includes the outer ring of partly vignetted detectors [@fulton2016]. The pipeline is interrupted at the pre-cube stage, before spectra from individual jiggle positions are gridded onto a rectangular spatial pattern. The spectrum for each jiggle position and each detector is visually inspected. Discarding all spectra that show excessive noise and/or irregular continuum shape (resulting from partial saturation) results in filtering 28 of the total of 919 SLW spectra (3%) and 135 of the 1696 SSW spectra (8%). The final processing step is the combination of the individual positions from each of the 19 (SLW) or 37 (SSW) detectors of each of the 55 jiggle positions into spectral cubes with square spatial pixels in Right Ascension and Declination coordinates. Due to the complex frequency dependence of the beam size and shape [@makiwa2013], the native angular resolution of SPIRE iFTS observations varies non-monotonously across its frequency range. We use the convolution gridding scheme, which weighs each input value according to the distance from the target pixel center by means of a differential Gaussian kernel, with the aim of obtaining a cube with a constant effective reference beam of 43 FWHM. The pixel grid is identical for SLW and SSW, with square pixels measuring 175$\times$175. Gaussian convolution only results in a completely Gaussian target beam if the original beam is also well represented by a Gaussian shape. Such is the case for the entire SSW band and for low frequencies in SLW, but not for SLW frequencies between $\sim$700 and 1018 GHz [@makiwa2013]. Results {#sec:obsresults} ======= The spectral cubes, as constructed in Sect. \[sec:obs\], show a smooth dust continuum superposed with spectrally unresolved lines. The dominant line signal in the cube arises from the ladder of rotational transitions of CO ($J$=4–3 to 13–12). Early versions of the CO intensity maps of NGC 6334, based on subsets of the SPIRE iFTS observations used here, were presented in @naylor2013 and @makiwa2013osafts. To retain the highest possible spectral resolution, we use unapodized FTS data, in which any unresolved spectral line has a Sinc-shaped profile. It is therefore important that one carefully fits and subtracts the Sinc-shaped profiles of nearby bright lines, to avoid any remaining sidelobes of strong lines affecting the apparent profile of the weak absorption line under study. This work focuses on absorption signatures of two hydride molecules, for which we use two spectral sections of the data cubes: 1132–1332 GHz from SSW and 760–935 GHz from SLW, chosen specifically to include the two CO emission lines closest to HF 1–0 at 1232.48 GHz [rotational spectroscopy by @nolt1987] and  1–0 at 835.08 GHz [rotational transition frequency measured by @pearson2006]. We construct a script in HIPE [@ott2010], derived from one of the post-pipeline analysis scripts provided by the SPIRE iFTS working group [@polehampton2015], to fit a third order polynomial for the continuum simultaneously with the following lines: CO at 1152.0 and 1267.0 GHz (SSW), and CO at 806.7 and 921.8 and \[\] at 809.3 GHz (SLW). Knowing that the intrinsic width of CO lines in this region is only a few  [@zernickel2012], lines are spectrally unresolved by SPIRE iFTS, and we adopt for each spectral line a Sinc profile with a fixed peak-to-first-zero-crossing width of 1.18 GHz. We then divide the observed spectrum by the fit of continuum and two/three Sinc lines to obtain a continuum-normalized spectrum, $I_\mathrm{norm}$. This process, illustrated in the top panel of Fig. \[fig:fittingprocess\], is repeated for each spatial pixel in the cube. In the resulting continuum-normalized cube around 1232 GHz, we fit two Sinc functions to the absorption profile of HF 0$\leftarrow$1 at 1232.48 GHz and the nearby emission line of  2$_{2,0}$$\rightarrow$2$_{1,1}$ at 1228.79 GHz (Fig. \[fig:fittingprocess\], middle panel). In the continuum-normalized cube around 835 GHz, we fit three Sinc functions to the absorption profile of  0$\leftarrow$1 at 835.08 GHz and  emission lines at 771.18 and 881.27 GHz. The maps of equivalent width of the HF and  absorption depth, in units of Hz, are multiplied by the ratio of the speed of light, c, and the observed frequency, $\nu_\mathrm{obs}$, to convert to  units: c/$\nu_\mathrm{obs}$= Hz$^{-1}$ for HF, and  Hz$^{-1}$ for . Line fits are rejected if the signal-to-noise ratio is lower than 2 and/or the fitted line center is more than half a resolution element from the line’s expected frequency at $-8.3$   [@vanderwiel2010]. The resulting map of HF equivalent width in Fig. \[fig:HFabsmap\] reveals the spatial distribution of the HF absorption feature, detected in 81% (signal-to-noise$>$2) or 72% (signal-to-noise$>$3) of all the pixels in the 6$\times$35 map coverage. See also the signal-to-noise map in Fig. \[fig:SNmapHF\]. Uncertainties are calculated from the spectral rms noise in the continuum-normalized cubes in two 20 GHz ranges surrounding the HF absorption line. For , the frequency ranges for calculating noise are composed of a 10 GHz section below and a 30 GHz section above the  frequency, to avoid incorporating residual from the fit to the blended CO 7–6 and \[\] $^3$P$_2$–$^3$P$_1$ lines at 806 and 809 GHz. The noise on the equivalent width is obtained by multiplying the unitless spectral rms with the instrumental line width of 1.18 GHz $\times$ c/$\nu_\mathrm{obs}$. Noise values are variable across the maps, in the range 1.2–3.4  for HF and 1.4–7  for . We do not include the following contributions to the uncertainty. Firstly, any multiplicative effects such as those of the absolute intensity calibration [@benielli2014; @swinyard2014] are divided out by normalizing the spectra to the local continuum. Secondly, additive uncertainties in the continuum level offsets are $\sim$  for SLW and $\sim$  for SSW [@swinyard2014; @hopwood2015]. These values are negligible compared to the brightness of the continuum in our cube, which exceeds the offset uncertainty by a factor of a few hundred even in the faintest outer regions. Spectra from the ‘OFF’ observation, pointed just northwest of the mapped area shown in Fig. \[fig:HFabsmap\], are also inspected at the HF frequency, but no convincing detections are found. The spectra from individual detectors in the OFF observation exhibit rms noise values between 4 and 10 , with a median of 6 . This noise is considerably higher than that in the cube pixels based on the other four observations combined, in which each pixel encompasses at least four, but typically more than eight individual detector pointings. The lack of HF absorption detections in the OFF position is thus consistent at the 2-$\sigma$ level with an HF absorption depth $\lesssim$12  in the area 3–6 northwest of source I, i.e., absorption depths could be anywhere in the range shown in our mapped area, except the central 40 around source I itself where the strongest absorption is seen. To rule out contamination of the HF signature by other spectral lines within SPIRE’s spectral resolution element, we inspect the high spectral resolution spectrum toward the position of the chemically rich core NGC6334 I [@zernickel2012], observed as part of the spectral survey key program CHESS [@ceccarelli2010] using the HIFI spectrometer [@degraauw2010; @roelfsema2012]. The only spectral lines detected by HIFI in a frequency span of $\pm$2 GHz around the HF frequency are four marginally detected methanol emission lines (A. Zernickel, private communication, Jun. 2014) together amounting to $<$0.4  in equivalent width. The possible methanol contaminations for the measured HF absorption depth are therefore contained within the uncertainty for our HF equivalent widths quoted above. The effect of the  emission line at 1229 GHz (see also above) could be more significant: at the brightest position, toward core I, the water line is as bright as one third of the deepest HF absorption. As described above and shown in Fig. \[fig:fittingprocess\], the effects of the water line on the HF absorption profile are taken into account by applying a simultaneous fit of these two lines, separated in frequency by three times the SPIRE instrumental line width. We also detect the signature of  1$\leftarrow$0 absorption at 835.08 GHz in the spectral map from the SLW array. We refrain from interpreting its signal here for the following reasons. Firstly, the signal-to-noise ratio in the  absorption map is much lower than that in the HF map (see Fig. \[fig:SNmapHF\] and \[fig:SNmapCHplus\]), resulting in signal-to-noise$>$2 detections in only 46% of the mapped pixels. For completeness, the distribution of detected  absorption is shown in the Appendix in Fig. \[fig:SNmapCHplus\]. Importantly, around the position of source I, there is no confident detection to be compared with heterodyne observations from @zernickel2012. Only a few isolated pixels near that position have detections of , but at a signal-to-noise of $<$3. Secondly, the spectrally resolved HIFI spectrum toward source I (A. Zernickel, private communication, Jun. 2014) show seven distinct emission line features due to methanol at frequencies within 0.6 GHz of the  line, i.e., half of the SPIRE spectral resolution. The combined intensity of these lines is sufficient to compensate more than half of the  absorption observed by HIFI, and they therefore severely contaminate the spectrally unresolved profile of  absorption in the SPIRE spectrum. In fact, these methanol emission lines could be the cause of the weak detection at the position of core I with SPIRE’s modest spectral resolution. Thirdly, the  transition falls in a frequency range in which the beam profile of the SPIRE iFTS is non-Gaussian in shape [@makiwa2013], complicating the map convolution and the interpretation of any spatial structure. Our SPIRE spectroscopic data also show evidence for detections of NH$_2$ at 952.6 and 959.5 GHz, and NH at 974.5 GHz and at 1000 GHz. However, compared to HF, it is more challenging for astrochemical models to explain the observed abundances of N-bearing hydrides [e.g., @persson_cm2012], and the analysis of line features is complicated by hyperfine structure within rotational transitions [cf. spectrally resolved detections with HIFI by @zernickel2012]. For these two reasons, we refrain from interpreting the NH and NH$_2$ absorption depth maps in this paper, but for completeness they are shown in Figs. \[fig:SNmapNH\] and \[fig:SNmapNH2\]. The OH$^+$ doublet at 909.05 and 909.16 GHz is also within the frequency range covered by the SPIRE iFTS, but the bright methanol emission line at 909.07 GHz apparent in the HIFI spectrum published by @zernickel2012 would make analysis of the blended OH$^+$ signature in the SPIRE spectrum impossible. Analysis and discussion {#sec:analysis} ======================= HF optical depth, equivalent width, and column density {#sec:eqwidth} ------------------------------------------------------ An absorption line is saturated when its depth reaches zero, i.e., absorbing all continuum photons in any specific spectral channel. HIFI observations seem to show that the combined absorption feature of HF due to the NGC6334 I envelope, hot cores, and the foreground cloud at $-3.0$  is saturated between  of $-7$ and $-4$  (Fig. \[fig:HFabscomp\]a). The individual components, however, do not reach 100% absorption. Of all seven components, the cloud at $+6.5$  comes closest to having saturated absorption in HF. As already noted by @emprechtinger2012, even this component is only marginally optically thick, evidence for which is provided by the line width of the same component in the optically thin tracer CH [@vanderwiel2010] which is the same (1.5 ) as for HF. All other components are believed to be optically thin [@emprechtinger2012]. Absorption line depth is converted into optical depth using $I_\mathrm{norm}=\mathrm{e}^{-\tau_\nu}$. With the caution of one of the seven components being marginally saturated, we take the optically thin limit to relate optical depth integrated over the line profile, , to column density, $N_\mathrm{HF}$, following @neufeld2010b: $$\label{eq:tautocolumn} \int \tau_{\nu,\mathrm{HF}} \mathrm{d}V = \frac{A_\mathrm{ul} g_\mathrm{u} \lambda^3}{8\pi g_\mathrm{l}} N_\mathrm{HF} = \powm{4.16}{-13} \, [\mathrm{cm}^2 \, \mathrm{km}\, \mathrm{s}^{-1}]\ N_\mathrm{HF} ,$$ where $A_\mathrm{ul}$ is the Einstein $A$ coefficient for spontaneous emission, s$^{-1}$ for HF 1–0 [@pickett1998], $g_\mathrm{u}$=3 and $g_\mathrm{l}$=1 are the statistical weights of rotational levels $J$=1 and $J$=0, respectively, and $\lambda$ is the wavelength of the transition, 243.24 . In addition to the optically thin limit, Eq. (\[eq:tautocolumn\]) assumes that all HF molecules are in the rotational ground state, a fair assumption given its high $A_\mathrm{ul}$. The conversion from $I_\mathrm{norm}$ to $\tau_\nu$ follows a linear relation for low values of $\tau_\nu$: $$\begin{aligned} \nonumber \tau_\nu & = & -\ln(I_\mathrm{norm}) \\ & \approx & 1-I_\mathrm{norm} \qquad [\mathrm{for}\ I_\mathrm{norm} \approx 1]. \end{aligned}$$ This relation holds to within $\sim$10% for $\tau_\nu<0.2$ (line absorbs up to 20% of the continuum), but $\tau_\nu/(1-I_\mathrm{norm})$ is already 1.2 at 40% absorption, rising to 2 at 80% absorption. The integrated optical depth is therefore systematically underestimated for a spectrally unresolved line that is smeared out over a velocity range wider than its intrinsic profile.[^3] Since the HF line is spectrally unresolved in our SPIRE spectra, a column density derived from these observations would merely constitute a lower limit to the true column density. Contrary to the optical depth, the absorption depth integrated over the line profile, i.e., the equivalent width of the absorbed ‘area’ below $I_\mathrm{norm}$=1, is conserved regardless of the spectral resolution. This is confirmed by the matching equivalent width values measured by HIFI and SPIRE toward core I: is 15.7 and 16.4 , respectively, with uncertainty margins of $\sim$1  in both cases. In the remainder of this paper, we analyze HF absorption depth measured with SPIRE based exclusively on the conserved quantity, equivalent width, . When deriving optical depths and column densities, we rely exclusively on spectrally resolved profiles such as that in @emprechtinger2012. Distribution of HF absorbing clouds toward NGC6334 {#sec:absclouds} -------------------------------------------------- The range of HF equivalent width values in the map shown in Fig. \[fig:HFabsmap\] can be divided into three regimes: (a) $>$12 , only occurring toward the position of core I; (b) 8–12 , spatially consistent with the larger scale filament in which cores I and I(N) are embedded; and (c) $\lesssim$5 , exclusively localized at projected distances $>$0.6 pc from the cores and the connecting filament. The non-detection of HF in the ‘OFF’ observation (see Sect. \[sec:obsresults\]), sparsely sampling the area just northwest of our map, is consistent with regimes (c) or (b). ### Distinguishing foreground from dense star-forming gas {#sec:distinguishforeground} We interpret the three regimes in the context of the velocity resolved HF spectrum published by @emprechtinger2012, who identified seven individual physical components responsible for the HF absorption toward NGC6334 I: the dense envelope at =$-6.5$ , two compact subcores at $-6.0$ and $-8.0$ , and four foreground layers at $-3.0$, $0.0$, $+6.5$, and $+8.0$ . The spectral signature of each of these components is reproduced in our Fig. \[fig:HFabscomp\]a. Regime (a) requires all seven components to explain the total equivalent width of HF. The two other panels in Fig. \[fig:HFabscomp\] represent adaptations of the model from @emprechtinger2012 with progressively fewer absorption components taken into account. In Fig. \[fig:HFabscomp\]b, regime (b) is explained by the superposition of four absorbing foreground clouds, discarding the components associated with the envelope and subcores of NGC6334 I. We highlight that the HF absorption depth observed toward the dense star-forming envelope I(N) is consistent with regime (b). Differences in HF content between the dense envelopes I and I(N) are discussed in more detail in Sect. \[sec:freezeout\]. Finally, in Fig. \[fig:HFabscomp\]c, we show that regime (c) is consistent with a model composed of just two specific foreground clouds, namely those at $+6.5$ and $+8.0$  [@vanderwiel2010; @emprechtinger2012]. In contrast with the detailed study of the HF profile in @emprechtinger2012 and in this work, the spectral survey paper by @zernickel2012, analyzing $\sim$4300 individual spectral line features, uses a simplified model which explains the HF absorption with only three components. The two approaches are not inconsistent, but the latter paper groups components together as: (1) the NGC6334 I envelope and two subcores, (2) two foreground clouds that are kinematically close to the NGC6334 complex, and (3) two other foreground clouds with larger  offsets [cf. Table 1 in @emprechtinger2012]. In this three-component model, regime (a) would be explained by groups (1)+(2)+(3), regime (b) by groups (2)+(3), and regime (c) by only group (3). The combination of our HF absorption depth map with the previous, single-pointing, velocity resolved HF spectrum now reveals a three-dimensional picture of the layers of absorbing gas toward the NGC6334 complex. Our interpretation of the relative positions of the foreground layers is sketched in Fig. \[fig:geometrysketch\]. With the exception of the direction toward core I, the HF absorption depth at all other positions in the map can be explained by (a subset of) the four extended foreground clouds. ### Relation of foreground clouds to NGC6334 {#sec:foregroundrelation} @vanderwiel2010 used a single-pointing [*Herschel*]{} HIFI observation of CH rotational line absorption, coupled with results from OH hyperfine line absorption measurements at radio wavelengths [@brooks2001], to suggest that the clouds at =$-3.0$ and $+0.0$  are associated with the NGC6334 complex, while the remaining two velocity components, at $+6.5$ and $+8.0$ , originate in foreground clouds farther away from the NGC6334 complex. This interpretation is consistent with the first two components being seen exclusively in regime (b) of the HF equivalent width map and the latter two components being spread over a more extended area of regimes (b) and (c) combined. Therefore, the spatial distribution of HF absorption measured with SPIRE iFTS supports the previous hypothesis that the $-3.0$ and $+0.0$  clouds are associated with NGC6334, since they have a spatial morphology that closely follows the dense molecular cloud traced by the 250  dust emission (Fig. \[fig:HFabsmap\]), i.e., a region roughly in east-west extent and stretching along the north-south direction. The vertical extent of the ‘related’ foreground clouds depicted in Fig. \[fig:geometrysketch\] is drawn directly from the observed east-west extent of these component in the plane of the sky. Besides the diatomic hydride species HF, CH, and OH, a subset of our absorbing clouds also exhibit absorption lines due to  [@emprechtinger2010; @vandertak2013a],  [@ossenkopf2010b], and H$_2$Cl$^+$ [@lis2010a]. Absorption components in OH$^+$ and  detected toward both cores by @indriolo2015a peak at =$-2$ and $+3$ . The former could be a blend of the $-3.0$ and $+0.0$  clouds seen in CH and HF, but the latter is inconsistent with any of our components. These lines were all detected in observations with the single-pixel, high spectral resolution [*Herschel*]{} HIFI spectrometer, in some cases toward both individual protostellar cores. It is interesting that ionized species,  and H$_2$Cl$^+$, are only detected at  values that match the $-3$ and $+0$  clouds. Chemical models tailored to halogen hydrides [@neufeld2009b] indicate that cation species become abundant under the influence of strong UV radiation. We hypothesize that the presence of  and H$_2$Cl$^+$ is further evidence for the physical proximity of these two clouds to the massive protostars and the  region embedded in one of the dense cores. Another ionized species that has been detected in spectrally resolved observations toward NGC6334 is . While a velocity resolved observation of HF exists only toward the position of core I,  1–0 has been observed with HIFI toward both cores I and I(N), the latter as part of the WISH program [@vandishoeck2011; @benz2016]. Toward source I, the  profile looks very similar to the HF profile, amounting to a total equivalent width of $\sim$20 . In comparison, the  observation toward core I(N)[^4] reveals an equivalent width of only $\sim$13 . The majority (85%) of the reduced absorption toward core I(N) relative to core I falls in the $[-15,5]$ range, as expected if the missing components are the envelope and subcore components at =$-6.5$, $-6.0$, and $-8.0$  (cf. Fig. \[fig:HFabscomp\]), i.e., those components that occur only in regime (a) in our HF absorption map. As mentioned above in Sect. \[sec:obsresults\], the SPIRE iFTS spectral cube has too low signal-to-noise near 835 GHz to make a meaningful comparison with the  signature detected by this instrument. The foreground clouds at =$+6.5$ and $+8.0$ , supposedly unrelated to the NGC6334 dense filament [@brooks2001; @vanderwiel2010], have a combined HF equivalent width that is entirely consistent with the observation in this work that regime (c) is spatially extended beyond the dense filament. An unrelated set of foreground cloud(s) (the two leftmost clouds in Fig. \[fig:geometrysketch\]) are likely to have a larger angular extent than the background continuum source (dashed lines in Fig. \[fig:geometrysketch\]), despite a possibly modest linear size. This explains why a minimum level of HF equivalent width of $\sim$3  is detected not just toward the NGC6334 filament, but throughout the extent of our 6$\times$35 map (Figs. \[fig:HFabsmap\] and \[fig:SNmapHF\]). All four foreground clouds detected in HF and CH are redshifted ($\geq$$-3$ ) with respect to the  of the part of NGC6334 cloud near cores I and I(N) (around $-5$ , based on  observations by @zernickel2013). Thus, the absorbing gas clouds are moving *toward* NGC6334, instead of following the Galactic rotation, which at $\ell=351\degr$ yields exclusively negative line-of-sight velocities for sources between Sun and NGC6334, i.e., approaching the standard of rest of stars in the Solar neighborhood. We therefore conclude that, in addition to the gas flows within the dense gas *along* the filament’s long axis [@zernickel2013], gas may also be accreting onto the filament in the perpendicular direction. We also note that, whereas @indriolo2015a [their Sect. 3.7 and Table 5] put the  and  absorption clouds toward both cores I and I(N) at the distance of 1.35 kpc of the NGC6334 cloud, their location along the sight line toward NGC6334 is in fact poorly constrained. Recognizing that there must be peculiar motions at play, deviating from the ‘rigid’ Galactic rotation curve, these clouds could in fact be anywhere between the local arm of the Milky Way and the Sagittarius arm that harbors the NGC6334 complex. For the two foreground clouds related to NGC6334, at $-3$ and $+0$ , we calculate their total mass by multiplying the sum of their column densities with a rough estimate of the area covered by this component of 160$\times$270 (regime (b) in Fig. \[fig:HFabsmap\]), corresponding to 1.5 pc$^2$ at the distance of NGC6334. Depending on the choice of fiducial  tracer, either taking $N_\mathrm{CH}$ from @vanderwiel2010 and $N_\mathrm{CH}$/$N_\mathrm{H_2}$= from @sheffer2008, or $N$(HF) from @emprechtinger2012 and $N_\mathrm{HF}$/$N_\mathrm{H_2}$= from @indriolo2013a, this calculation yields a mass in the range 37–98  or 69–191 , respectively. From symmetry arguments, a similar reservoir of additional gas is expected to lie behind the NGC6334 dense cloud. This means that a significant total mass of several hundred  could be on its way to accreting onto the dense cloud NGC6334 near ($<$0.3 pc) the embedded cores I and I(N). This gas reservoir has escaped detection so far, because it does not appear in traditional gas tracers such as CO, , and CS. While there is evidence that the $-3.0$ and $+0.0$  foreground clouds are closer to NGC6334 than the $+6.5$ and $+8.0$  clouds, there is no direct metric of the geometrical distance along the line of sight from each cloud to the dense filament and cores. Therefore, we refrain from speculating about accretion time scales of even the ‘related’ clouds, since this would rely on unsupported assumptions on relative distances. ### Spatial distribution of HF toward other Galactic sight lines The only other published work of a spatial map of HF absorption so far is that toward by @etxaluze2013, using data also obtained with [*Herschel*]{} SPIRE iFTS. A direct comparison of measured absorption line depths is complicated by the choice of @etxaluze2013 to present signal in terms of integrated optical depth, apparently without taking into account the systematic underestimation of optical depths derived from spectrally unresolved measurements, as discussed in Sect. \[sec:eqwidth\]. Nonetheless, it appears that the total HF absorption toward Sgr B2(M) is about an order of magnitude stronger than toward NGC6334 I. @etxaluze2013 find a variation of HF absorption depth of only a factor $\sim$2 across the $\sim$25 mapped area, significantly less variation than in our fully sampled map toward NGC6334 I. Any intrinsic variation may have been partially smoothed by the interpolation process that was applied in @etxaluze2013 to construct a map from the spatially undersampled SPIRE iFTS observation. More importantly, the line of sight toward Sgr B2, close to the Galactic Center, crosses many more spiral arms than that toward NGC6334. Evidence of this is found for example by @qin2010, who detect a total of 31 individual velocity components in CH rotational ground state absorption toward Sgr B2(M). The 30 foreground clouds, associated with the various intervening Galactic arms, amount to a total CH column density of  , with Sgr B2(M) itself adding a component of only   [@qin2010]. This is consistent with many sheets of foreground gas together creating a roughly uniform cover of absorbing gas spanning at least a few arcminutes on the sky. Comparatively little additional absorption is contributed by the massive molecular cloud Sgr B2 and the embedded cores in the background, which explains the lack of variation in HF absorption depth seen toward Sgr B2 by @etxaluze2013. In contrast, our map of HF absorption toward NGC6334 reveals a mix of components due to foreground clouds and the star-forming envelope and cores within NGC6334, which we are able to disentangle owing to the complementary, velocity resolved spectra of CH and HF obtained with [*Herschel*]{} HIFI [@vanderwiel2010; @emprechtinger2012]. In addition, we highlight the discovery by @lopez-sepulcre2013a of a foreground cloud toward the intermediate-mass star-forming core , based on single-pointing HIFI spectra of HF and other hydride molecules. Several oxygen-bearing hydrides show absorption exclusively at a blueshifted velocity relative to the background protostellar core. The same foreground cloud was later also identified in H$_2$Cl$^+$ by @kama2015a. The HF profile shows absorption at the same blueshifted velocity, but shows additional evidence for a second absorption component that matches the  of the protostellar core [@lopez-sepulcre2013a]. Instead of kinematical and morphological arguments such as those used in this work to determine the physical location of foreground clouds toward NGC6334, @lopez-sepulcre2013a use detailed photochemical modeling to infer proximity of their OMC-2 foreground gas to a source of copious far-UV radiation. With that radiation source assumed to be the trapezium cluster of OB stars, it is concluded that the absorbing slab is physically connected to OMC-1. In an attempt to study the spatial distribution of this absorbing OMC-1 ‘fossil’ slab, we have searched archival SPIRE iFTS data toward ([*Herschel*]{} observation ID 1342214847) for signatures of HF at 1232.5 GHz (and  at 835.1 GHz), but find no detections in any of the 37 (and 19) SSW (and SLW) detectors in the $\sim$3 footprint. Freeze-out of HF in envelopes of dense cores {#sec:freezeout} -------------------------------------------- The HF equivalent width of 15.9$\pm$1.4  measured at the position of NGC6334 I in our SPIRE map is explained in Sect. \[sec:absclouds\] by invoking the superposition of four absorption components in the foreground and three associated to the dense core I itself [see Fig. \[fig:HFabscomp\], and @emprechtinger2012]. A striking feature of our HF absorption map is the lack of additional absorption toward the position of core I(N), where the HF equivalent width of only 10.9$\pm$1.1  can be explained by the four foreground clouds alone, adding up to 10.5  (Fig. \[fig:HFabscomp\]b). In contrast, adding even just an envelope component similar to that of core I sums up to 13.7  (not counting the two subcore components), which is inconsistent with the observation toward core I(N). Since the envelope of core I(N) is more massive than that of core I, but has a similar size [see model fits in @vandertak2013a], the total gas column density toward core I(N) should be higher. Therefore, the lack of HF absorption associated to the I(N) core is not due to the difference in total () column. Instead, we hypothesize that HF is primarily frozen out onto dust grains in core I(N), while HF is in the gas phase in core I. To support the hypothesis of HF being depleted from the gas phase in core I(N), we set up a rudimentary model based on the following ingredients. We take the spherically symmetric physical structure, i.e., radial profiles of density and temperature, of the envelopes of NGC6334 I and I(N) as fitted to submillimeter dust continuum maps and the far-infrared / submillimeter spectral energy distribution [@vandertak2013a]. We then calculate, at every radial point, the timescales for adsorption (freeze-out) and desorption (evaporation) of HF molecules onto dust grains. Following, e.g., @rodgers2003 and @jorgensen2005a, we assume that thermal desorption is the dominant mechanism that drives molecules from the grain surface back into the gas phase, and are left with the balance between adsorption rate: $$\lambda(n_\mathrm{H}, T_\mathrm{gas}) = \powm{4.55}{-18} \left( \frac{T_\mathrm{gas}}{m_\mathrm{HF}} \right)^{0.5} n_\mathrm{H} \qquad [\mathrm{s}^{-1}] , \label{eq:freezerate}$$ and desorption rate: $$\xi(T_\mathrm{dust}) = \nu_\mathrm{vib} \exp\left( -\frac{E_\mathrm{b,HF}}{k\,T_\mathrm{dust}} \right) \qquad [\mathrm{s}^{-1}] . \label{eq:desorbrate}$$ Here,  and  are the temperatures of gas and dust, assumed to be equal as in the modeling of @vandertak2013a, $m_\mathrm{HF}$ is the molecular weight of HF (20), $n_\mathrm{H}$ is the density of hydrogen nuclei, $\nu_\mathrm{vib}$ is the vibrational frequency of the HF molecule in its binding site, for which we adopt $10^{13}$ s$^{-1}$, $k$ is the Boltzmann constant, and $E_\mathrm{b,HF}$ is the binding energy of HF to the dust grain surface. It has previously been inferred by @philips2010 that a density of $\sim$$10^5$  allows HF to condense onto dust grains, whereas densities of $\sim$$10^3$  more typical for diffuse gas are too low for HF freeze-out to occur. In our case, the density $n_\mathrm{H}$ – which incidentally exceeds $10^5$  at almost all radii in the envelopes of I and I(N) – enters directly into Equation \[eq:freezerate\] to govern the adsorption rate. In this work we consider multiple versions of the desorption timescale, because the binding energy in the exponent of Equation \[eq:desorbrate\] is heavily dependent on the type of grain surface. Unlike for more common molecular species such as CO [e.g., @bisschop2006; @noble2012], the desorption behavior of HF from astrophysically relevant grain surfaces has not been studied experimentally, so we rely on theoretical calculations. Typical interstellar dust grains, especially those embedded in cold, star-forming regions, are covered in one or multiple layers of ice consisting of various molecules, mainly , CO, and  [for a recent review, see @boogert2015]. We collect binding energy values for several types of grain surfaces in Table \[t:Ebinding\]. For CO and  ice covered grains we adopt calculated binding energies from the literature [@chen2006; @rivera-rivera2012], while for hydrogenated bare silicate grains and  ice covered grains, these values result from original ab initio chemical calculations performed for this work. -------------------------------- ------------------- --------------------- ------- Type of grain surface $E_\mathrm{b,HF}$ $E_\mathrm{b,HF}/k$ Ref. (kJ/mol) ($10^3$ K) Hydrogenated crystaline silica 9.2 1.1 \[1\]  ice on amorphous silica 53 6.3 \[1\] CO ice on amorphous silica 8.9 1.07 \[2\]  ice on amorphous silica 9.3 1.12 \[3\] -------------------------------- ------------------- --------------------- ------- : Binding energies for HF onto various surfaces.[]{data-label="t:Ebinding"} To calculate the HF binding energies in the first two rows of Table \[t:Ebinding\] (with a bare grain and with  ice), we carry out quantum calculations within the Kohn-Sham implementation of Density Functional Theory using the Quantum Espresso Simulation Package [@giannozzi2009]. Perdew-Burke-Ernzerhof exchange-correlation functional ultrasoft pseudopotentials are used. KS valence states are expanded in a plane-wave basis set with a cutoff at 340 eV for the kinetic energy. The self-consistency of the electron density is obtained with the energy threshold set to $10^{-5}$ eV. Calculations are performed using the primitive unit cell containing a total number of 46 atoms for bare hydrogenated silica, and 54 atoms for hydrogenated silica covered with one layer of  ice. The geometry optimization is used within the conjugate gradients scheme, with a threshold of 0.01 eV$\AA^{-1}$ on the Hellmann-Feynman forces on all atoms; the Si atoms of the bottom layers are fixed at their bulk values. The binding energy of HF with the SiH terminus of hydrogenated crystalline silica is based on calculations for the hydroxylated alpha-quartz (001) surface. The binding energy of HF with one layer of  ice on amorphous hydrogenated silicate is estimated by assuming that the the most common structure in this case is the HF molecule interacting with a  molecule bonded to silanol (SiOH), which is the most abundant surface group in amorphous silica [@ewing2014]. The binding energies of HF with CO and  ice (last two rows of Table \[t:Ebinding\]) are taken from calculations by @rivera-rivera2012 and @chen2006, respectively. These authors performed calculations for molecules in the gas phase. We consider the gas phase binding energies of HF with CO and  to be similar to those of HF with CO and  ices adsorbed on an inert surface such as that of hydroxylated amorphous silica. This approximation is based on the weak interactions of these ices with hydroxylated silica and within the CO and  molecular solids, so that the electronic density of CO and  in solid form is not significantly altered with respect to their state in the gas phase. Hence, for the aim of the present work, the binding energy of the HF molecule with CO or  as calculated in the gas phase is applicable for the condensed phase. The situation is notably different for interactions with  in the gas or adsorbed form, because of its stronger interaction with silica and HF. For HF interacting with  ice, we use binding energies from our own calculations described above. With the physical structure of both envelopes, the equations for the adsorption/desorption balance, and the binding energy values, the ‘freeze-out’ region within each envelope is calculated in Fig. \[fig:HFphasediagram\]. Defining $t_\mathrm{freeze} = 1/\lambda$ and $t_\mathrm{desorb}=1/\xi$, HF molecules will deplete from the gas phase in the region of the envelope wherever $t_\mathrm{desorb}$ &gt; $t_\mathrm{freeze}$. Numerical simulations by @das2016 suggest that, within a mixed-composition ice layer, the abundances of CO and  are enhanced compared to  in high-density ($\gtrsim10^5$ ) environments. At such densities applicable for the protostellar envelopes studied here, we thus expect the HF binding energy to lie close to, but slightly above that of pure CO or  ice, and the true desorption time scale line in Fig. \[fig:HFphasediagram\] therefore somewhat to the left of the dash-dotted line for CO ice. In this case, HF is expected to stay frozen onto grain surfaces at a broad range of radii in core I(N): cumulative mass $\sim$0.1 to 1, i.e., 90% of the mass, where the temperature is $\lesssim$20 K. Core I(N) is overall colder than core I both in the envelope (blue solid lines in Fig. \[fig:HFphasediagram\]) and in the embedded subcores [@hunter2006]. For the comparatively warmer envelope of core I, the lines for the desorption timescale of HF from CO/ ice fall off the scale on the right hand side of the axes, leaving no freeze-out zone in this envelope. This could explain why HF is seen in the gas phase in core I, but not in core I(N). In an alternative scenario in which the dust grains are covered in pure  ice – or the desorption characteristics of a mixed mantle are dominated by that of  ice [@collings2004] – the binding energy of HF with the ice would be greatly increased (see Table \[t:Ebinding\]). In this case, the HF freeze-out zone would expand to cover $>$98% of the mass for *both* envelopes, and our observations should have revealed no gas phase HF in either of the cores. Our observations are therefore inconsistent with pure  ice coating on the grains. Instead, our interpretation relies on a significant part of the ice coating to consist of CO and/or  molecules. In principle, the composition of ice coatings do not need to be the same in the two neighboring envelopes. Particularly, given the lower temperatures of core I(N), there is higher probability that a significant amount of  is frozen out in that envelope. This, again, would enhance the binding energy of HF onto the ice-covered dust grains in the envelope of core I(N), which may help to explain the lack of gas-phase HF observed toward source I(N). Conversely, if additional mechanisms for desorption, e.g., induced by (UV) photons or cosmic rays, would be taken into account, $\xi$ would increase, and the freeze-out region would be pushed to larger radii in the envelopes. Particularly UV photodesorption could have a different effect in one envelope compared to the other, because core I is more evolved and contains an  region. An increased total desorption rate would have no effect on HF freeze-out in the envelope of core I, in which HF is already completely in the gas phase, but would reduce the size of the freeze-out zone for envelope I(N). If, however, thermal desorption as expressed in Equation \[eq:desorbrate\] is the dominant desorption mechanism and the binding energy of HF onto the ice surface is close to that of CO or  ice (Table \[t:Ebinding\]), our model predicts significant freeze-out of HF in core I(N), and none in core I. It is important to note that, indeed, the temperature and density conditions under which HF remains frozen onto dust grains depend greatly on the exact composition and mixing of the ice mantle and therefore on the chemical history. It has already been recognized for example for the CO molecule that the freeze-out temperature ‘threshold’ can vary considerably from one object to another [@qi2015b]. Conclusions {#sec:conclusions} =========== In this work we present a map of HF absorption toward the northern end of the molecular cloud NGC6334, containing two well studied massive star-forming cores I and I(N). Although in the original definition of the observing program it was not anticipated that hydride absorption lines would be found within these data, the discovery space provided by the enormous frequency coverage of the [*Herschel*]{} SPIRE iFTS instrument has made this study possible. Such wide coverage in the far-infrared/submillimeter is only attainable with broadband FTS spectrometers [see @naylor2013 for a review]. The absorption line of HF is detected in 80% of our mapped area, although it is spectrally unresolved by SPIRE. By complementing the new SPIRE iFTS data with existing, single pointing, high spectral resolution spectra from the [*Herschel*]{} HIFI instrument [@vanderwiel2010; @emprechtinger2012], we construct a three-dimensional picture of gas clouds in front of and inside the massive star-forming filament NGC6334. We find that our observations are consistent with a scenario of four individual foreground clouds on the line of sight toward NGC6334 I and I(N), two of which are unrelated to the star-forming complex (Sect. \[sec:distinguishforeground\]). The other two clouds are posited to be close to the dense molecular filament based on their spatial morphology. Their velocities are such that they are moving toward the star-forming cloud and could be adding several hundreds of solar masses of gas to the dense filament and the embedded cores in which massive star formation is already ongoing (Sect. \[sec:foregroundrelation\]). This component of gas is detected in rotational lines of diatomic hydride molecules, but had been unseen in studies of traditional dense gas tracers. In fact, using such a tracer, , @zernickel2013 have inferred that the roughly cylindrically shaped NGC6334 filament is collapsing along its longest axis. Our work now indicates that accretion may also be ongoing in the perpendicular (radial) direction. Future studies of (competitively) accreting high-mass star-forming cores may need to take into account this additional low-density phase of the gas reservoir. Finally, in Sect. \[sec:freezeout\] we explain why HF is observed in the gas phase toward core I, but appears completely absent in core I(N). For this purpose, we use a simple description of adsorption and desorption time scales for HF interacting with dust grain surfaces, depending on the (radially variable) density and temperature. Since interactions of HF with interstellar-like dust grains have not been studied in the laboratory, we adopt binding energy values for different types of grain surfaces from theoretical calculations from the literature as well as from original work first presented in this paper. The conclusion is that the lower temperature of core I(N) compared to core I could lead to freeze-out of HF exclusively in the former, but only if the binding energy of HF onto the grain surface is governed by that of CO or  ice on a silicate surface. In this case, at the densities relevant in the envelope of source I(N) ($>$ ), we find that HF freezes out in the region of the envelope where the temperature is below $\sim$20 K, rather similar to the freeze-out temperature often adopted for CO. In contrast, if  is the dominant constituent in the ice mantles, our model predicts that HF should have been frozen out at all radii in the envelopes of both sources I(N) and I. Since we observe a significant amount of HF in the gas phase in source I, this scenario is inconsistent with our data. Summarizing, this work uses HF as a sensitive tracer for (molecular) gas at relatively low densities that may be contributing mass to star forming cores. The HF signature reveals a gas reservoir that is inconspicuous in traditional dense gas tracers such as CO. In addition, we show that gas phase HF in higher density environments ($>$$10^5$ ) is extremely sensitive to interactions with dust grains and will be depleted significantly at low dust temperatures. The research of MHDvdW at the University of Lethbridge was supported by the Canadian Space Agency (CSA) and the Natural Sciences and Engineering Research Council of Canada (NSERC), and at the University of Copenhagen by the Lundbeck Foundation. Research at the Centre for Star and Planet Formation is funded by the Danish National Research Foundation and the University of Copenhagen’s programme of excellence.\ SPIRE has been developed by a consortium of institutes led by Cardiff University (UK) and including Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC, UKSA (UK); and NASA (USA). HIPE is a joint development by the Herschel Science Ground Segment Consortium, consisting of ESA, the NASA Herschel Science Center, and the HIFI, PACS and SPIRE consortia.\ This research has made use of NASA’s Astrophysics Data System Bibliographic Services. The graphical representations of the results in this paper were created using APLpy, an open-source plotting package for Python hosted at , Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration [-@astropy2013]), and the matplotlib plotting library [@matplotlib2007].\ The authors are grateful to Alexander Zernickel for providing and discussing excerpts of the CHESS spectral survey data, to Raquel Monje for providing the HF spectral model component profiles, and to Floris van der Tak for providing the physical structure models of the two envelopes in electronic table format. We thank Tommaso Grassi, Wing-Fai Thi, Jes Jørgensen, Søren Frimann, and Mihkel Kama for discussions. Complementary figures ===================== Maps displaying the signal-to-noise ratio of the detections of the HF and  lines in each spatial pixel of our spectral cube (Sect. \[sec:obsresults\]) are shown in Fig. \[fig:SNmapHF\] and \[fig:SNmapCHplus\]. The colored contours in Fig. \[fig:SNmapCHplus\] show that  absorption is only confidently detected (signal-to-noise $>$ 5) in the northeastern section of the map. In addition, although the signal from nitrogen species is not interpreted in this paper, maps of line absorption depth due to NH and NH$_2$ are shown in Figs. \[fig:SNmapNH\] and \[fig:SNmapNH2\]. The continuum-normalized cube for these lines is created – analogous to those for HF and  in Sect. \[sec:obsresults\] – by fitting the continuum and the CO 9–8 line in the 945–1055 GHz section of the SSW cube. Absorption lines of NH (at 974.47 and 999.98 GHz) and two of NH$_2$ (at 952.57 and 959.50 GHz) are then fitted simultaneously with emission lines due to  9–8 at 991.3 GHz and at 987.9 GHz. [^1]: [*Herschel*]{} is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA. [^2]: See the introduction section of this reference for a list of earlier spectral survey work on NGC6334 I. [^3]: The logarithm operator gives a disproportionally higher weight to more strongly absorbed channels. For example, a 20  wide feature with constant 80% absorption ($I_\mathrm{norm}$=0.20) and a smeared-out version of 320  wide at 5% absorption ($I_\mathrm{norm}$=0.95) both have the same equivalent width, = 16 . The latter, however, yields a $\int \tau_\nu \mathrm{d}V$ that is smaller by a factor 2. The effect is larger yet for line profiles that come even closer to full absorption. [^4]: Estimate based on the HIFI spectrum downloaded from the Science Archive, observation ID 1342214306, processed with HIPE pipeline version 13.

--- abstract: 'A versatile and efficient variational approach is developed to solve in- and out-of-equilibrium problems of generic quantum spin-impurity systems. Employing the discrete symmetry hidden in spin-impurity models, we present a new canonical transformation that completely decouples the impurity and bath degrees of freedom. Combining it with Gaussian states, we present a family of many-body states to efficiently encode nontrivial impurity-bath correlations. We demonstrate its successful application to the anisotropic and two-lead Kondo models by studying their spatiotemporal dynamics and universal behavior in the correlations, relaxation times and the differential conductance. We compare them to previous analytical and numerical results. In particular, we apply our method to study new types of nonequilibrium phenomena that have not been studied by other methods, such as long-time crossover in the ferromagnetic easy-plane Kondo model. The present approach will be applicable to a variety of unsolved problems in solid-state and ultracold-atomic systems.' author: - Yuto Ashida - Tao Shi - Mari Carmen Bañuls - 'J. Ignacio Cirac' - Eugene Demler bibliography: - 'reference.bib' title: Solving quantum impurity problems in and out of equilibrium with variational approach --- Understanding out-of-equilibrium dynamics of quantum many-body systems has become one of the central problems in physics. Recent experimental developments in diverse fields such as ultracold atoms [@EM11; @CM12; @FuT15; @KAM16; @RL17], mesoscopic physics [@DFS02; @THE11; @LC11; @IZ15; @MMD17], molecular electronics [@BDN11], and carbon nanotubes [@MA07; @CSJ12] have posed new theoretical questions for studying many-body dynamics driven by external fields or fast changes in the Hamiltonian. Quantum spin-impurity models (SIM), such as the famous Kondo model [@KJ64], constitute a paradigmatic class of many-body systems which lie at the heart of many strongly correlated systems. Their nonequilibrium dynamics underly transport phenomena in mesoscopic systems [@GL88; @NTK88; @LW02; @SF99; @WWG00; @RMP07; @KAV11] and non-Fermi liquid behavior in heavy fermion materials [@HAC97; @LH07; @SQ10], and give theoretical foundation for the real-time formulation of dynamical mean-field theory (DMFT) [@GA96]. The ground-state properties of SIM are now well established by perturbative renormalization group (RG) [@APW70], numerical renormalization group (NRG) [@WKG75] and the Bethe ansatz [@PBW80; @AND80; @ANL81; @NK81; @AN83; @SP89]. The challenging and fascinating question of out-of-equilibrium dynamics has recently come under active investigations in theory [@AFB05; @AFB06; @AFB07; @AFB08; @RD08; @JE10; @LB14; @WSR04; @SP04; @AHKA06; @DSLG08; @SH08; @WA09; @HMF09; @HMF10; @NHTM17; @NM15; @DB17; @STL08; @WP09; @SM09; @WP10; @CG13; @NP99; @KA00; @AR03; @HA08; @KM01; @PM10; @HA09L; @HA09B; @CT11; @BS14; @FS15; @BCZ17; @LF96; @LF98; @SA98; @LD05; @VR13; @SG14; @MM13; @BCJ16] and experiments [@RL17; @DFS02; @THE11; @LC11; @IZ15; @MMD17]. Examples include time-dependent NRG [@AFB05; @AFB06; @AFB07; @AFB08; @RD08; @JE10; @LB14], density-matrix renormalization group (DMRG) [@WSR04; @SP04; @AHKA06; @DSLG08; @SH08; @WA09; @HMF09; @HMF10; @NHTM17], time evolving block decimation (TEBD) [@NM15; @DB17], real-time Monte Carlo [@STL08; @WP09; @SM09; @WP10; @CG13], perturbative RG [@NP99; @KA00; @AR03; @HA08; @KM01; @PM10], flow equation method [@HA09L; @HA09B; @CT11], coherent-state expansion [@BS14; @FS15; @BCZ17], and exact analyses [@LF96; @LF98; @SA98; @LD05; @VR13; @SG14; @MM13; @BCJ16]. Despite the rich variety of methods, they often become increasingly costly at long times due to, e.g., artifacts of the logarithmic discretization [@RA12] or large entanglement in the time-evolved state [@SU11]. Some of them can only determine the dynamics of the impurity but not that of the bath, or are restricted to particular parameter regimes. Moreover, it remains a major challenge to apply them to generic SIM beyond the simplest Kondo models. These challenges motivate the search for new approaches to quantum impurity systems. ![\[fig\_aniso\] (a) Ground-state impurity-bath spin correlation $\chi^{z}_{l}$ of the anisotropic Kondo model. (a,inset) The RG phase diagram and the parameters $(j_\parallel,j_\perp)$ corresponding to I $(-0.5,0.2)$ (blue square) in the ferromagnetic phase (FM), II $(0.5,0.2)$ (red triangle) and III $(-1.85,2)$ (brown circle) in the antiferromagnetic phase (AFM). (b) Quench dynamics of the impurity magnetization $\langle\hat{\sigma}_{\rm imp}^{z}(t)\rangle$. (c) The corresponding spatiotemporal dynamics of correlations $\chi^{z}_{l}(t)$ in I FM phase, II AFM phase, III easy-plane FM regime and IV the same as in III but on a different scale. System size is $L=400$. ](fig_aniso_arxiv.pdf){width="86mm"} In this Letter, introducing a new canonical transformation, we present a widely applicable variational approach to study in- and out-of-equilibrium properties of generic SIM. Besides the ability to efficiently capture the correct impurity-bath correlations and the conductance behavior, it reveals previously unexplored nonequilibrium dynamics such as ferromagnetic (FM) to antiferromagnetic (AFM) crossover (see the panels III and IV in Fig. \[fig\_aniso\]c) in the FM easy-plane Kondo model. Such long-time spatiotemporal dynamics is difficult (if not impossible) to obtain in other approaches. Our versatile variational approach will pave the way towards solving interesting novel problems in both solid-state and ultracold-atomic systems. [*Canonical transformation.—*]{} We first formulate our approach in the most general way as it is applicable to a wide class of SIM. The difficulty in SIM stems from the need to treat the strong entanglement between the impurity and bath. Here we introduce a new canonical transformation that completely decouples the impurity spin and bath degrees of freedom. We consider the Hamiltonian [ $$\begin{aligned} \label{totalH} \hat{H}=\hat{H}_{\rm bath}+\hat{H}_{\rm int}+\hat{H}_{\rm imp},\end{aligned}$$ ]{} where $\hat{H}_{\rm bath}=\sum_{lm\alpha}\hat{\Psi}^{\dagger}_{l\alpha}h_{lm}\hat{\Psi}_{m\alpha}$ describes an arbitrary single-particle Hamiltonian, with fermionic or bosonic creation (annihilation) operator $\hat{\Psi}^{\dagger}_{l\alpha}$ ($\hat{\Psi}_{l\alpha}$) for the $l$-th bath mode with spin $\alpha$. For simplicity, we consider a noninteracting spin-1/2 bath with $\alpha=\uparrow,\downarrow$ [^1]. The Hamiltonian $\hat{H}_{\rm int}=\hat{\bf s}_{\rm imp}\cdot\hat{\bf \Sigma}$ represents a generic interaction between the impurity and the bath with $\hat{\bf s}_{\rm imp}=\hat{\boldsymbol \sigma}_{\rm imp}/2$ being the impurity spin-1/2 operator. We define the bath-spin operator including couplings as $\hat{\Sigma}^{\gamma}=\sum_{l}g_{l}^{\gamma}\hat{\sigma}_{l}^{\gamma}/2$ with $\hat{\sigma}_{l}^{\gamma}=\sum_{\alpha\beta}\hat{\Psi}_{l\alpha}^{\dagger}\sigma^\gamma_{\alpha\beta}\hat{\Psi}_{l\beta}$. The interaction strengths $g_{l}^{\gamma}$ are arbitrary and can be anisotropic and long-range. We also include the impurity Hamiltonian as $\hat{H}_{\rm imp}=-h_z\hat{s}_{\rm imp}^{z}$. Paradigmatic examples having the interaction form $\hat{H}_{\rm int}$ include the Kondo-type Hamiltonians [@KJ64] where the coupling $g_{l}^\gamma$ is local, and the central spin model [@JS03] where an interaction is long-range while $\hat{H}_{\rm bath}$ is frozen. To construct the canonical transformation, we observe that the Hamiltonian has a parity symmetry, $[\hat{H},\hat{\mathbb P}]=0$, with $\hat{\mathbb{P}}=\hat{\sigma}_{\rm imp}^{z}\hat{\mathbb P}_{\rm bath}$. Here, $\hat{\mathbb P}_{\rm bath}=e^{(i\pi/2)(\sum_{l}\hat{\sigma}^{z}_{l}+\hat{N})}$ is the parity operator acting on the bath, where $\hat{N}$ is the total particle number. The symmetry follows from the fact that $\hat{H}$ is invariant under the transformation $\hat{\mathbb P}^{-1}\,\hat{O}\,\hat{\mathbb P}$, which rotates the entire system around $z$ axis by $\pi$, i.e., transforms both impurity and bath spins as $\hat{\sigma}^{x,y}\to -\hat{\sigma}^{x,y}$. Our aim is to employ a parity conservation to find the disentangling transformation $\hat{U}$ satisfying $\hat{U}^{\dagger}\hat{\mathbb P}\hat{U}=\hat{\sigma}_{\rm imp}^{x}$ such that the impurity spin turns out to be a conserved quantity in the transformed frame. We can construct such a unitary transformation as [ $$\begin{aligned} \label{canonical} \hat{U}=\exp\left[\frac{i\pi}{4}\hat{\sigma}_{\rm imp}^{y}\hat{\mathbb P}_{\rm bath}\right]=\frac{1}{\sqrt{2}}\left(1+i\hat{\sigma}_{\rm imp}^{y}\hat{\mathbb P}_{\rm bath}\right),\end{aligned}$$ ]{} where we use $\hat{\mathbb P}_{\rm bath}^2=1$. This leaves $\hat{H}_{\rm bath}$ invariant, while it maps the interaction onto $\hat{\tilde{H}}_{\rm int}=\hat{U}^{\dagger}\hat{H}_{\rm int}\hat{U}$: [ $$\begin{aligned} \label{transformed} \hat{\tilde{H}}_{\rm int}\!=\!\hat{s}_{\rm imp}^x \hat{\Sigma}^{x}+\hat{\mathbb P}_{\rm bath}\left(-i\hat{\Sigma}^y/2+\hat{s}^x_{\rm imp}\hat{\Sigma}^z\right),\end{aligned}$$ ]{} and $\hat{H}_{\rm imp}$ onto $\hat{\tilde{H}}_{\rm imp}=-h_z\hat{s}_{\rm imp}^{x}\hat{\mathbb P}_{\rm bath}$. Remarkably, the impurity spin now commutes with the transformed Hamiltonian $[\hat{\tilde{H}},\hat{s}_{\rm imp}^x]=0$ and is thus completely decoupled from the bath degrees of freedom. The construction of $\hat{U}$ holds true for arbitrary conserved parity operator and can be readily applied to a variety of SIM, including two-impurity systems [@SP18]. [*Variational approach.—*]{} We combine the transformation (\[canonical\]) with fermionic Gaussian states [@WC12; @CVK10] and introduce variational states to efficiently encode nonfactorizable impurity-bath correlations. A Gaussian state for the bath, $|\Psi_{\rm b}\rangle$, is completely determined by its covariance matrix $\Gamma$ [@WC12]: [ $$\begin{aligned} (\Gamma)_{\xi l\alpha,\eta m\beta}=\frac{i}{2}\langle\Psi_{\rm b}|[\hat{\psi}_{\xi,l\alpha},\hat{\psi}_{\eta,m\beta}]|\Psi_{\rm b}\rangle,\end{aligned}$$ ]{} where we introduce the Majorana operators $\hat{\psi}_{1,l\alpha}=\hat{\Psi}^{\dagger}_{l\alpha}+\hat{\Psi}_{l\alpha}$ and $\hat{\psi}_{2,l\alpha}=i(\hat{\Psi}^{\dagger}_{l\alpha}-\hat{\Psi}_{l\alpha})$. For the total system, we construct states of the form $|\Psi_{\rm tot}\rangle=\hat{U}|+_{x}\rangle_{\rm imp}|\Psi_{\rm b}\rangle$ with $\Gamma$ as variational parameters. Employing the time-dependent variational principle [@JR79; @ST17], we obtain the imaginary- and real-time evolution equations for $\Gamma$: [ $$\begin{aligned} \label{imag} \frac{d\Gamma}{d\tau}&=&-{\cal H}-\Gamma {\cal H}\Gamma,\\\label{real} \frac{d\Gamma}{dt}&=&{\cal H}\Gamma-\Gamma {\cal H},\end{aligned}$$ ]{} where ${\cal H}=4\delta E/\delta\Gamma$ is the functional derivative of the mean energy $E=\langle\Psi_{\rm tot}|\hat{H}|\Psi_{\rm tot}\rangle$ [@YA18F]. The variational ground state can be obtained in the limit $\tau\to\infty$ in the imaginary-time evolution (\[imag\]). In contrast, Eq. (\[real\]) allows us to calculate the real-time dynamics of SIM. [*Equilibrium properties.—*]{} As a paradigmatic example, we first apply our approach to the anisotropic Kondo model: [ $$\begin{aligned} \hat{H}&=&-t_{\rm h}\sum_{l=-L}^{L}\left(\hat{c}^{\dagger}_{l,\alpha}\hat{c}_{l+1,\alpha}+{\rm h.c.}\right)\nonumber\\ &&\!+\!\frac{J_{\perp}}{4}\!\!\sum_{\gamma=x,y}\!\!\hat{\sigma}^{\gamma}_{\rm imp}\hat{c}_{0,\alpha}^{\dagger}\sigma^{\gamma}_{\alpha\beta}\hat{c}_{0,\beta}\!+\!\frac{J_\parallel}{4}\hat{\sigma}^{z}_{\rm imp}\hat{c}_{0,\alpha}^{\dagger}\sigma^{z}_{\alpha\beta}\hat{c}_{0,\beta},\end{aligned}$$ ]{} where $\hat{c}_{l,\alpha}^{\dagger}$ ($\hat{c}_{l,\alpha}$) creates (annihilates) a fermion with position $l$ and spin $\alpha$, the summations over $\alpha,\beta$ are contracted. We denote the dimensionless Kondo couplings as $j_{\parallel,\perp}=\rho_{\rm F}J_{\parallel,\perp}$ with $\rho_{\rm F}=1/(2\pi t_{\rm h})$ being the density of states at the Fermi energy. We choose the unit $t_{\rm h}=1$ hereafter. The anisotropic Kondo model exhibits a quantum phase transition [@LAJ87] between FM and AFM phases as shown in the RG phase diagram [@APW70] in the inset of Fig. \[fig\_aniso\]a. In the main panel, we show the ground-state impurity-bath spin correlations $\chi^{z}_{l}=\langle\hat{\sigma}^{z}_{\rm imp}\hat{\sigma}_{l}^{z}\rangle/4$ in three different regimes. The FM results at I (blue square) and AFM results at II (red triangle) indicate the formation of the triplet and singlet pairs of the impurity and bath spins, respectively. Importantly, our method also correctly reveals the AFM nature at III (brown circle) that is close to the phase boundary. ![\[fig\_gs\] Ground-state properties of the Kondo model. (a) Screening length $\xi_{\rm K}$ plotted for different Kondo coupling $j=\rho_{\rm F}J$ and thresholds $f$. The dashed lines indicate the scaling $\xi_{\rm K}\propto e^{1/j}$. (inset) The Kondo length $\xi_{\rm K}$ is extracted as a length scale in which a fraction $1-f$ of total antiferromagnetic correlations is contained. (b) Spin correlations plotted in the dimensionless unit of $\xi_{\rm K}$ for $f=0.05$, collapsing onto the universal curve. The dashed lines indicate the scaling $l^{-1}$ ($l^{-2}$) in short (long) distance. System size is $L=400$.](fig_gs.pdf){width="86mm"} As a critical test of our approach, we extract the Kondo screening length $\xi_{\rm K}$ in the variational ground state and test the universal behavior in the SU(2)-symmetric case $j=j_\parallel=j_\perp>0$. We determine $\xi_{\rm K}$ as the length scale below which most of the Kondo screening cloud is developed [@AH09; @NM15]. Specifically, we introduce a threshold $f$ for the integrated antiferromagnetic correlations $\Sigma_{\rm AF}(l)=\sum_{|m|=0,2,4\ldots}^{l}\chi_{m}$ (Fig. \[fig\_gs\]a, inset) with $\chi_{m}=\langle\hat{\boldsymbol \sigma}_{\rm imp}\cdot\hat{\boldsymbol \sigma}_{m}\rangle/4$, and extract $\xi_{\rm K}$ from the implicit relation: $f=1-\Sigma_{\rm AF}(\xi_{\rm K}(f))/\Sigma_{\rm AF}(L)$ [^2]. Figure \[fig\_gs\]a plots the extracted $\xi_{\rm K}(f)$ against the inverse Kondo coupling $1/j$ for different $f$. The results agree with the nonperturbative scaling $\xi_{\rm K}\propto T_{\rm K}^{-1}\propto e^{1/j}$ [@HAC97] independent of the choice of $f$. As a further test, we plot $\chi_l$ in units of the extracted $\xi_{\rm K}$ (Fig. \[fig\_gs\]b). Remarkably, all the results for different Kondo couplings $j$ collapse onto the same universal curve and show the crossover from $l^{-1}$ to $l^{-2}$ decay at $l/\xi_{\rm K}\sim 1$ [@IH78; @BV98; @HT06]. To avoid finite-size and lattice effects, here we set $j$ large enough such that $\xi_{\rm K}\ll L$ while it is kept small enough so that $\xi_{\rm K}$ is still larger than the lattice constant. To meet the former condition, we ensure that the sum rule $\sum_l\chi_{l}=-3/4$ [@BL07] is satisfied with an error below $0.5$%. [*Out-of-equilibrium dynamics.—*]{} We now apply our approach to study out-of-equilibrium dynamics. To be concrete, we analyze the quench dynamics starting from the initial state $|\!\!\uparrow\rangle_{\rm imp}|{\rm FS}\rangle$, where $|{\rm FS}\rangle$ represents the half-filled Fermi sea of the bath. Previously, using the bosonization mapping between the Kondo model and the spin-boson model [@LAJ87], the relaxation dynamics have been studied by NRG [@AFB06] and the bosonic Gaussian states combined with a unitary transformation [@ST17]. While the latter has been specifically designed to the spin-boson model, our approach is applicable to generic SIM. Moreover, in the previous methods one had to use strictly linear dispersion and to introduce an artificial cut-off energy. Hence, one of distinctive features in our approach is that it can be applied without relying on the bosonization and thus allows for a quantitative comparison with an experimental system. This is particularly important in light of recent experimental developments in simulating dynamics of SIM [@RL17; @DFS02; @THE11; @LC11; @IZ15; @MMD17; @BDN11; @KM17]. Figures \[fig\_aniso\]b,c show the magnetization dynamics $\langle\hat{\sigma}_{\rm imp}^{z}(t)\rangle$ and spatiotemporal spreading of spin correlations $\chi^{z}_{l}(t)$ after the quench. As shown in the panels I and II in Fig. \[fig\_aniso\]c, spin correlations develop FM and AFM correlations after passing through the “light cone" created by AFM and FM ballistic spin waves, respectively. These AFM (FM) spin waves result from the excess spin in the generation of the triplet (singlet) pair around the impurity. As shown in Fig. \[fig\_aniso\]b, the magnetization eventually relaxes to a value close to zero in the AFM phase, indicating the formation of the Kondo singlet, while the value remains finite in the FM phase. The dynamics associate with the fast oscillations having period characterized by the bandwidth $2\pi/{\cal D}$ with ${\cal D}=4t_{\rm h}$ and $\hbar=1$ (see e.g., Fig. \[fig\_aniso\]b inset). These fast oscillations originate from high-energy excitations of a particle from the bottom of the band [@KM12] and were absent in the bosonized treatments. Most interestingly, at the point III in easy-plane FM regime ($|J_\parallel|<|J_\perp|$), spin correlations exhibit the distinct crossover dynamics from FM to AFM (panel III in Fig. \[fig\_aniso\]c). As shown in the closeup panel IV, the initial development of FM correlations leads to the emission of ballistic AFM spin waves while the subsequent crossover to AFM associates with the repeated emissions of FM spin waves. The origin of such crossover can be understood from the nonmonotonic RG flows in this regime (Fig. \[fig\_aniso\]a, inset), where short (long) time dynamics is governed by the high (low) energy physics characterized by FM (AFM) coupling $J_\parallel$ ($J_\perp$). Here, the real time effectively plays the role of the inverse RG scale [@NP99]. The predicted spatiotemporal dynamics can be readily tested with site-resolved measurements as allowed by quantum gas microscopy [@EM11; @CM12; @FuT15; @KAM16]. ![\[fig\_dy\] Relaxation time scales $\tau$ in (a) correlation and (b) magnetization plotted for different Kondo coupling $j=\rho_{\rm F}J$. (Insets) The relaxation times are extracted by fitting $\Sigma_{\rm AF}(L)(t)$ and $\langle\hat{\sigma}_{\rm imp}^{z}(t)\rangle$ in long-time regime with the function $a+b e^{-t/c}$. The dashed lines in the main panels indicate the fitted lines, showing nonperturbative scaling $\ln\tau\propto 1/j$. System size is $L=400$.](fig_dy.pdf){width="86mm"} As a critical test, we study the nonperturbative scalings of the relaxation time scales $\tau_{\rm corr}$ for the integrated correlations $\Sigma_{\rm AF}(L,t)$ and $\tau_{\rm mag}$ for the impurity magnetization $\langle\hat{\sigma}^{z}_{\rm imp}(t)\rangle$ in the SU(2)-symmetric case. After the quench, each observable eventually relaxes to its steady-state value and we extract the relaxation times by fitting the tale dynamics with an exponential function (Fig. \[fig\_dy\]a,b inset). The main panels show that within numerical errors the relaxation times for both observables show the nonperturbative dependence $\tau_{\rm corr}\propto e^{1/j}$ and $\tau_{\rm mag}\propto e^{2/j}$. The observed different scalings agree with the TEBD results [@NM15]: $ \tau_{\rm corr}\propto e^{(1.5\pm 0.2)/j}$ and $\tau_{\rm mag}\propto e^{(1.9\pm 0.2)/j}$ (a rather large deviation in $\tau_{\rm corr}$ has been attributed to the difficulty of taking the adiabatic limit in the Anderson model). [*Transport dynamics.—*]{} ![\[fig\_trans\] Differential conductance $G$ with varying (a) magnetic field $h_z/T_{\rm K}$, (b) Kondo coupling $j$, and (c) bias potential $V/T_{\rm K}$. In (a), we show the obtained results (black open circles) and the asymptotic scalings with the infinite-bandwidth approximation (blue dashed lines). The inset magnifies the low-field behavior. Kondo temperature $T_{\rm K}$ is extracted from the magnetic susceptibility $\chi=1/(4T_{\rm K})$. System size is $L=200$ for each lead and we use (a) $j=0.35$ and $V=0$, (b) $h_z=0$ and $V=0.8t_{\rm h}$, and (c) $h_z=0$ and $j=0.4$.](fig_trans.pdf){width="86mm"} We finally apply our approach to the two-lead Kondo model [@KA00] that is relevant to experiments in mesoscopic systems. We consider the Hamiltonian [ $$\begin{aligned} \label{two-leads} \hat{H}_{\rm two}&=&\sum_{l\eta}\biggl[-t_{\rm h}\bigl(\hat{c}^{\dagger}_{l,\alpha\eta}\hat{c}_{l+1,\alpha\eta}\!+\!{\rm h.c.}\bigr)+eV_{\eta}\,\hat{c}^{\dagger}_{l,\alpha\eta}\hat{c}_{l,\alpha\eta}\biggr]\nonumber\\ &&+\frac{J}{4}\sum_{\eta\eta'}\hat{\boldsymbol{\sigma}}_{\rm imp}\cdot\hat{c}_{0,\alpha\eta}^{\dagger}\boldsymbol{\sigma}_{\alpha\beta}\hat{c}_{0,\beta\eta'},\end{aligned}$$ ]{} where $\eta={\rm L,R}$ denotes the left (L) or right (R) lead. We set the bias potential $V_{\rm L,R}$ of each lead to be $V_{\rm L,R}=\pm V/2$. The initial condition is $|\!\!\uparrow\rangle_{\rm imp}|{\rm FS}\rangle_{\rm L}|{\rm FS}\rangle_{\rm R}$ with $|{\rm FS}\rangle_{\rm L,R}$ being the half-filled Fermi sea of each lead. We then quench the Hamiltonian (\[two-leads\]) and study the dynamics of the current $I(t)$ between the two leads: [ $$\begin{aligned} I(t)=\frac{ie}{4\hbar}J\sum_{\alpha\beta}\left[\langle\hat{\boldsymbol \sigma}_{\rm imp}\cdot\hat{c}_{0\alpha {\rm L}}^{\dagger}\boldsymbol{\sigma}_{\alpha\beta}\,\hat{c}_{0\beta {\rm R}}\rangle-{\rm h.c.}\right].\end{aligned}$$ ]{} After the quench, the current eventually reaches its quasi-steady value. We determine the differential conductance $G=d\overline{I}/dV$ from the steady current $\overline{I}(V)$obtained by taking time average [@AHKA06; @DSLG08; @HMF09; @JE10]. Applying a magnetic field $h_z$, we confirm the quadratic behavior $G_{0}(1-c_{B}(h_z/T_{\rm K})^2)$ with the correct coefficient $c_{B}=\pi^2/16$ at low field and the logarithmic behavior $\pi^2G_{0}/(16\ln^2(h_z/T_{\rm K}))$ at high field, where $G_0$ is the conductance at the zero field [@AR01; @ACH05; @GL05; @SE09; @MC09; @KAV11; @MC15; @FM17; @OA18; @OA182] (Fig. \[fig\_trans\]a). In contrast, if we change the Kondo coupling $j$, we expect the nonmonotonic behavior because $G$ is trivially zero at $j=0$, while it should degrade in $j\to\infty$ due to the formation of the bound state tightly localized at the impurity site, which prevents other electrons from approaching the junction. Figure \[fig\_trans\]b confirms this nonmonotonic dependence of $G$ against the Kondo coupling $j$. Different from two-channel systems [@MAK12; @MAK16; @MAK17], the nonmonotonicity originates from intrinsically finite bandwidth in the lattice model and is absent in the conventional infinite-bandwidth treatment [@AR01; @SN162]. Figure \[fig\_trans\]c shows the nonlinear conductance behavior at finite bias $V$. Two remarks are in order. Firstly, the numerical error due to current fluctuation in time obscures minuscule changes of $G$ in $V\ll T_{\rm K}$, making it difficult to precisely test the quadratic behavior [@AHKA06; @DSLG08; @HMF09; @JE10] in the perturbative regime. This can be worked out if we implement our approach in a different way based on the linear response theory. Secondly, in $V\gg T_{\rm K}$ the bias eventually becomes comparable to the finite bandwidth (and to the Fermi energy) and calculations of current and conductance are no longer faithful. This is a common limitation in real-space calculations [@HMF09; @JE10] and can be avoided if one uses the momentum basis of bath modes and specify the linear dispersion with a large bandwidth. Yet, we emphasize that the present implementation is already reliable (at least) in the intermediate regime $V\sim T_{\rm K}$. [*Discussions.—*]{} A simple entanglement-based argument can give insights into the success of our approach. On the one hand, our variational approach considers the following family of states: [ $$\begin{aligned} |\Psi_{\rm tot}\rangle&=&\hat{U}|+_{x}\rangle_{\rm imp}|\Psi_{\rm b}\rangle\nonumber\\ &=&|\!\uparrow~\!\!\rangle_{\rm imp}\hat{\mathbb P}_{+}|\Psi_{\rm b}\rangle+|\!\downarrow~\!\!\rangle_{\rm imp}\hat{\mathbb P}_{-}|\Psi_{\rm b}\rangle, \label{tot_entanglement}\end{aligned}$$ ]{} where $|\Psi_{\rm b}\rangle$ is a Gaussian state and $\hat{{\mathbb P}}_{\pm}=(1\pm\hat{{\mathbb P}}_{\rm bath})/2$. On the other hand, a recent study [@YC17] has shown that most of the entanglement in the Kondo singlet takes place with just one specific single-particle state, leading to the approximative expression originally suggested by Yosida [@KY66]: [ $$\begin{aligned} \label{kondo_entanglement} |\Psi_{\rm Kondo}\rangle\!=\!\frac{1}{\sqrt 2}\!\left(|\!\uparrow~\!\!\rangle_{\rm imp}\,\hat{d}^{\dagger}_{\downarrow}|{\rm FS}\rangle\!-\!|\!\downarrow~\!\!\rangle_{\rm imp}\,\hat{d}^{\dagger}_{\uparrow}|{\rm FS}\rangle\right),\end{aligned}$$ ]{} where $\hat{d}^{\dagger}_{\sigma}=\sum_{l}d_{l}\hat{c}_{l\sigma}^{\dagger}$ is the dominant single-particle state. In fact, Eq. (\[kondo\_entanglement\]) belongs to our family of variational states (\[tot\_entanglement\]) as shown by the choice $|\Psi_{\rm b}\rangle=(\hat{d}^{\dagger}_{\downarrow}-\hat{d}^{\dagger}_{\uparrow})|{\rm FS}\rangle/\sqrt{2}$. This observation indicates the ability of our variational state to efficiently encode the most significant part of the impurity-bath entanglement. Yet, we stress that our variational states go beyond the simple ansatz (\[kondo\_entanglement\]) since they take into account general Gaussian states instead of the trivial Fermi sea. Such a flexibility is crucial to obtain quantitatively accurate results [@YA18F]. In summary, we presented a versatile and efficient variational approach to study in- and out-of-equilibrium physics of SIM. Despite its simplicity, we demonstrated in the anisotropic and two-lead Kondo models that the variational states successfully capture the correct correlations and conductance behavior. In particular, it has already found applications to revealing previously unexplored physics such as the long-time crossover dynamics. Further details can be found in the accompanying paper [@YA18F], where the full expression of the functional derivative $\cal H$ and the benchmark results with the matrix-product-state calculations are presented. The present approach should be applicable to a variety of interesting unsolved problems in both solid-state and ultracold-atomic systems. For instance, our approach can be readily generalized to bosonic systems [@FGM04; @FS06; @FFM11; @FT15], the Anderson model and multiple impurities [@SP18], which will be published elsewhere. Another promising direction is an extension of our approach to multi-channel systems [@RMP07; @IZ15; @MAK12; @MAK16; @MAK17] and the central spin model [@JS03]. A generalization to finite temperatures is possible by using Gaussian density matrices. Including the phase factor, it is also possible to calculate the spectral function [@AR03; @WA09]. It is particularly interesting to test the maximally fast information scrambling [@MJ16] in the non-Fermi liquid phase of the multi-channel Kondo models [@DB17]. On another front, the proposed variational approach could be applied as a basis for a new type of impurity solver for DMFT [@GA96]. #### Acknowledgements.— {#acknowledgements. .unnumbered} We acknowledge Carlos Bolech Gret, Adrian E. Feiguin, Shunsuke Furukawa, Leonid Glazman, Vladimir Gritsev, Masaya Nakagawa and Achim Rosch for fruitful discussions. Y.A. acknowledges support from the Japan Society for the Promotion of Science through Program for Leading Graduate Schools (ALPS) and Grant No. JP16J03613, and Harvard University for hospitality. T.S. acknowledges the Thousand-Youth-Talent Program of China. J.I.C. is supported by the ERC QENOCOBA under the EU Horizon 2020 program (grant agreement 742102). E.D. acknowledges support from Harvard-MIT CUA, NSF Grant No. DMR-1308435, AFOSR Quantum Simulation MURI, AFOSR grant number FA9550-16-1-0323, the Humboldt Foundation, and the Max Planck Institute for Quantum Optics. [^1]: A generalization of our variational approach to interacting bath having arbitrary bath spin is straightforward. [^2]: Here we sum the correlations over even sites only to avoid cancellations from ferromagnetic contributions on odd sites and obtain a better accuracy of fitting procedure to extract Kondo length $\xi_{\rm K}$

--- abstract: 'An initial-boundary value problem for the time-fractional diffusion equation is discretized in space using continuous piecewise-linear finite elements on a polygonal domain with a re-entrant corner. Known error bounds for the case of a convex polygon break down because the associated Poisson equation is no longer $H^2$-regular. In particular, the method is no longer second-order accurate if quasi-uniform triangulations are used. We prove that a suitable local mesh refinement about the re-entrant corner restores second-order convergence. In this way, we generalize known results for the classical heat equation due to Chatzipantelidis, Lazarov, Thomée and Wahlbin.' author: - Kim Ngan Le - William McLean - Bishnu Lamichhane bibliography: - 'nonconvexrefs.bib' title: 'Finite element approximation of a time-fractional diffusion problem in a non-convex polygonal domain[^1]' --- Introduction ============ In a standard model of subdiffusion [@KlafterSokolov2011], each particle undergoes a continuous-time random walk with a common waiting-time distribution that obeys a power law. Consequently, the mean-square displacement of a particle is proportional to $t^\alpha$ with $0<\alpha<1$, and the macroscopic concentration $u(x,t)$ of the particles satisfies the time-fractional diffusion equation $$\label{eq: fpde} \partial_t u-\partial_t^{1-\alpha}K\nabla^2u=f(x,t).$$ Here, $\partial_t=\partial/\partial t$ and $\nabla^2$ denotes the spatial Laplacian. The fractional time derivative is of Riemann–Liouville type: $$\partial_t^{1-\alpha}v(x,t)=\frac{\partial}{\partial t}\int_0^t \omega_\alpha(t-s)v(x,s)\,ds,\qquad \omega_\alpha(t)=\frac{t^{\alpha-1}}{\Gamma(\alpha)} \quad\text{for~$t>0$.}$$ If no sources or sinks are present, then the inhomogeneous term $f$ is identically zero. We assume for simplicity that the generalized diffusivity $K$ is a positive constant, and that the fractional PDE  holds for $x$ in a polygonal domain $\Omega\subseteq{\mathbb{R}}^2$ subject to homogeneous Dirichlet boundary conditions, with the initial condition $$\label{eq: ic} u(x,0)=u_0(x)\quad\text{for $x\in\Omega$.}$$ In the limiting case when $\alpha\to1$, the fractional PDE  reduces to the classical heat equation that arises when the diffusing particles instead undergo Brownian motion. Consider a spatial discretization of the preceding initial-boundary value problem using continuous piecewise-linear finite elements to obtain a semidiscrete solution $u_h$. The behaviour of $u_h$ is well understood if $\Omega$ is convex [@JinLazarovZhou2013; @McLeanThomee2010]: in this case, for general initial data $u_0\in L_2(\Omega)$ and an appropriate choice of $u_h(0)$, $$\|u_h(t)-u(t)\|\le Ct^{-\alpha}h^2\|u_0\|,\qquad0<t\le T,$$ whereas for smoother initial data $u_0\in H^2(\Omega)$, $$\|u_h(t)-u(t)\|\le Ch^2\|u_0\|_{H^2(\Omega)},\qquad0\le t\le T,$$ where  $\|\cdot\|=\|\cdot\|_{L_2(\Omega)}$. The error analysis establishing these bounds relies on the $H^2$-regularity property of the associated elliptic equation in $\Omega$, namely, that if $$\label{eq: elliptic bvp} -K\nabla^2u=f\quad\text{in~$\Omega$,}\quad \text{with $u=0$ on~$\partial\Omega$,}$$ then $u\in H^2(\Omega)$ with $\|u\|_{H^2(\Omega)}\le C\|f\|$. In the present work, our aim is to study $u_h$ in the case when $\Omega$ is not convex. Since the above $H^2$-regularity breaks down, we can no longer expect $O(h^2)$ convergence if the finite element mesh is quasi-uniform. Our results generalize those of Chatzipantelidis, Lazarov, Thomée and Wahlbin [@ChatzipantelidisEtAl2006] for the heat equation (the limiting case $\alpha=1$) to the fractional-order case ($0<\alpha<1$). Our method of analysis relies on Laplace transformation, extending the approach of McLean and Thomee [@McLeanThomee2010] for the fractional order problem on a convex domain. (0,0) – (0.4,0) arc \[radius=0.4, start angle=0, end angle=296.565\] – (0,0); (0,0) – (1,0) – (1,1) – (-1,1) – (-1,-1) – (0.5,-1) – (0,0); (-1.25,0) – (1.25,0); (0,-1.25) – (0,1.25); (0.4,-0.025) – (0.4,0.025); at (0.4,0) [$r_0$]{}; (0,0) circle \[radius=[0.015]{}\]; at (0,0) [$p_0=p_6$]{}; (1,0) circle \[radius=[0.015]{}\]; at (1,0) [$p_1$]{}; (1,1) circle \[radius=[0.015]{}\]; at (1,1) [$p_2$]{}; (-1,1) circle \[radius=[0.015]{}\]; at (-1,1) [$p_3$]{}; (-1,-1) circle \[radius=[0.015]{}\]; at (-1,-1) [$p_4$]{}; (0.5,-1) circle \[radius=[0.015]{}\]; at (0.5,-1) [$p_5$]{}; at (0.5,0) [$\Gamma_0$]{}; at (1,0.5) [$\Gamma_1$]{}; at (0,1) [$\Gamma_2$]{}; at (-1,0) [$\Gamma_3$]{}; at (-0.25,-1) [$\Gamma_4$]{}; at (0.25,-0.5) [$\Gamma_5$]{}; To focus on the essential difficulty, we assume that $\Omega$ has only a single re-entrant corner with angle $\pi/\beta$ for $1/2<\beta<1$. Without loss of generality, we assume that this corner is located at the origin and that, for some $r_0>0$, the intersection of $\Omega$ with the open disk $|x|<r_0$ is described in polar coordinates by $$\label{eq: 0 nbhd} 0<r<r_0\quad\text{and}\quad 0<\theta<\pi/\beta,$$ as illustrated in Figure \[fig: Omega\]. We denote the vertices of $\Omega$ by $p_0=(0,0)$, $p_1$, $p_2$, …, $p_J=p_0$, and the $j$th side by $$\Gamma_j=(p_j,p_{j+1}) =\{\,(1-\sigma)p_j+\sigma p_{j+1}: 0<\sigma<1\,\} \quad\text{for $0\le j\le J-1$.}$$ Section \[sec: elliptic\] summarizes some key facts about the singular behaviour of the solution to the elliptic problem . In Section \[sec: FEM\], we describe a family of shape-regular triangulations ${\mathcal{T}}_h$ (indexed by the mesh parameter $h$) that depend on a local refinement parameter $\gamma\ge1$. The elements near the origin have sizes of order $h^\gamma$, so the ${\mathcal{T}}_h$ are quasi-uniform if $\gamma=1$ but become more highly refined with increasing $\gamma$. Our error bounds will be stated in terms of the quantity $$\label{eq: epsilon} \epsilon(h,\gamma)=\begin{cases} h^{\gamma\beta}/\sqrt{\gamma^{-1}-\beta},&1\le\gamma<1/\beta,\\ h\sqrt{\log(1+h^{-1})},&\gamma=1/\beta,\\ h/\sqrt{\beta-\gamma^{-1}},&\gamma>1/\beta, \end{cases}$$ which ranges in size from $O(h^\beta)$ when $\gamma=1$ (the quasiuniform case) down to $O(h)$ when $\gamma>1/\beta$. We briefly review results for the finite element approximation of the elliptic problem, needed for our subsequent analysis: the error in $H^1(\Omega)$ is of order $\epsilon(h,\gamma)$, and the error in $L_2(\Omega)$ is of order $\epsilon(h,\gamma)^2$. Section \[sec: time-dep\] gathers together some pertinant facts about the solution of the time-dependent problem  and its Laplace transform. Next, in Section \[sec: semidiscrete\], we introduce the semidiscrete finite element solution $u_h(t)$ of the time-dependent problem, and see that its stability properties mimic those of $u(t)$. In Section \[sec: error\] we study first the homogeneous equation (i.e., the case $f=0$), showing that the error in $L_2(\Omega)$ is of order $t^{-\alpha}\epsilon(h,\gamma)^2$ when $u_0\in L_2(\Omega)$. For smoother initial data, the $L_2$-error is of order $\epsilon(h,\gamma)^2$ uniformly for $0\le t\le T$. We also prove that for the inhomogeneous equation ($f\ne0$) with vanishing initial data ($u_0=0$), the error in $L_2(\Omega)$ is of order $t^{1-\alpha}\epsilon(h,\gamma)^2$. Thus, by choosing the mesh refinement parameter $\gamma>1/\beta$ we can restore second-order convergence in $L_2(\Omega)$. Section \[sec: alt bc\] outlines briefly how these results are affected by different choices of the boundary conditions. We conclude in Section \[sec: numerical\] with some numerical examples that illustrate our theoretical error bounds. Singular behaviour in the elliptic problem {#sec: elliptic} ========================================== In the weak formulation of the elliptic boundary-value problem  we introduce the Sobolev space $$V={\widetilde H}^1(\Omega)=H^1_0(\Omega)$$ and seek $u\in V$ satisfying $$a(u,v)={\langlef,v\rangle}\quad\text{for all $v\in V$,}$$ where $$\label{eq: a(u,v)} a(u,v)=K\int_\Omega\nabla u\cdot\nabla v\,dx \quad\text{and}\quad {\langlef,v\rangle}=\int_\Omega fv\,dx.$$ Here, $f$ may belong to the dual space $V^*=H^{-1}(\Omega)$ if ${\langlef,v\rangle}$ is interpreted as the duality pairing on $V^*\times V$. Since $a(u,v)$ is bounded and coercive on ${\widetilde H}^1(\Omega)$, there exists a unique weak solution $u$, and $$\label{eq: Htilde u} \|u\|_{{\widetilde H}^1(\Omega)}\le C\|f\|_{H^{-1}(\Omega)}.$$ To understand the difficulty created by the re-entrant corner, we separate variables in polar coordinates to construct the functions $$u_n^\pm(x)=r^{\pm n\beta}\sin(n\beta\theta)\quad \text{for $x=(r\cos\theta,r\sin\theta)$ and $n=1$, $2$, $3$, \dots,}$$ satisfying $$\label{eq: Laplacian in sector} \nabla^2 u_n^\pm=0\quad \text{for $0<r<\infty$ and $0<\theta<\pi/\beta$,}$$ with $u_n^\pm=0$ if $\theta=0$ or $\theta=\pi/\beta$. Introducing a $C^\infty$ cutoff function $\eta$ such that $$\eta(x)=1\quad\text{for $|x|\le r_0/2$} \quad\text{and}\quad \eta(x)=0\quad\text{for $|x|\ge r_0$,}$$ we find that $$\eta u_n^+\in{\widetilde H}^1(\Omega) \quad\text{but}\quad \eta u_n^-\notin{\widetilde H}^1(\Omega) \quad\text{for all $n\ge1$,}$$ and that $$\eta u_n^+\in H^2(\Omega)\quad\text{for all $n\ge2$,} \quad\text{but}\quad \eta u_1^+\notin H^2(\Omega).$$ Now consider the function $f=-K\nabla^2(\eta u_1^+)$. The choice of $\eta$ means that $f(x)=0$ for $|x|\le r_0/2$, and consequently $f$ is $C^\infty$ on $\overline{\Omega}$. Nevertheless, the (unique weak) solution of , namely $u=\eta u_1^+$, fails to belong to $H^2(\Omega)$. Put $A=-K\nabla^2$ and $$\label{eq: V2 def} V^2=H^2(\Omega)\cap{\widetilde H}^1(\Omega) =\{\,v\in H^2(\Omega):\text{$v=0$ on $\partial\Omega$}\,\}.$$ The bounded linear operator defined by the restriction $$A|_{V^2}:V^2\to L_2(\Omega)$$ is one-one and has closed range. See Grisard [@Grisvard1992 Section 2.3]. Our task now is to identify the orthogonal complement in $L_2(\Omega)$ of the range $${\mathcal{R}}=\{\,f\in L_2(\Omega): \text{$f=Au$ for some $u\in V^2$}\,\}.$$ To this end, we define in the usual way the Hilbert space $$L_2(\Omega,A)=\{\,\phi\in L_2(\Omega): A\phi\in L_2(\Omega)\,\}$$ with the graph norm $\|\phi\|_{L_2(\Omega,A)}^2 =\|\phi\|^2+\|A\phi\|^2$. Let $\partial_n$ denote the outward normal derivative operator. It can be shown that the trace map $\phi\mapsto\bigl(\phi|_{\Gamma_j},\partial_n\phi|_{\Gamma_j}\bigr)$, has unique extensions from $C^1(\overline{\Omega})$ to bounded linear operators [@Grisvard1992 Theorems 1.4.2 and 1.5.2] $$H^2(\Omega)\to H^{3/2}(\Gamma_j)\times H^{1/2}(\Gamma_j) \quad\text{and}\quad L_2(\Omega,A)\to {\widetilde H}^{-1/2}(\Gamma_j)\times{\widetilde H}^{-3/2}(\Gamma_j),$$ and that the second Green identity holds in the form [@Grisvard1992 Theorem 1.5.3] $$\int_\Omega\bigl[(Au)v-u(Av)\bigr]\,dx=\sum_{j=0}^{J-1} K\bigl[{\langleu,\partial_nv\rangle}_{\Gamma_j} -{\langle\partial_nu,v\rangle}_{\Gamma_j}\bigr]$$ for $u\in H^2(\Omega)$ and $v\in L_2(\Omega,A)$. Hence, $${\langleAu,\phi\rangle}={\langleu,A\phi\rangle}\quad \text{if $u\in V^2$, $\phi\in L_2(\Omega,A)$ and $\phi|_{\Gamma_j}=0$ for all $j$,}$$ implying that ${\mathcal{R}}$ is orthogonal in $L_2(\Omega)$ to the closed subspace $${\mathcal{N}}=\{\,\phi\in L_2(\Omega,A): \text{$A\phi=0$ in $\Omega$, and $\phi|_{\Gamma_j}=0$ for every~$j$}\,\}.$$ Notice that ${\mathcal{N}}\cap{\widetilde H}^1(\Omega)=\{0\}$ because if $f=0$ then the unique weak solution of  in ${\widetilde H}^1(\Omega)$ is $u=0$. The Hilbert space $L_2(\Omega)$ is the orthogonal direct sum of ${\mathcal{R}}$ and ${\mathcal{N}}$, and $\dim{\mathcal{N}}=1$ (assuming $\Omega$ has only a single re-entrant corner). See [@Grisvard1992 Theorem 2.3.7]. Thus, given any $f\in L_2(\Omega)$, the unique weak solution $u\in{\widetilde H}^1(\Omega)$ of  belongs to $H^2(\Omega)$ if and only if $f\perp{\mathcal{N}}$. For general $f$, the following holds. \[thm: singular term\] There exists $q\in{\mathcal{N}}$ (depending only on $\Omega$ and $\eta$) such that if $f\in L_2(\Omega)$ then the weak solution $u$ of  satisfies $u-{\langlef,q\rangle}\eta u_1^+\in V^2$ with $$\bigl\|u-{\langlef,q\rangle}\eta u_1^+\bigr\|_{H^2(\Omega)}\le C\|f\|.$$ Choose any nonzero $\phi\in{\mathcal{N}}$. Since $\eta u_1^+\in L_2(\Omega, A)$ but $\eta u_1^+\notin V^2$, we have ${\langleA(\eta u_1^+),\phi\rangle}\ne0$ and may therefore define $q=c\phi\in{\mathcal{N}}$ by letting $c=1/{\langleA(\eta u_1^+),\phi\rangle}$, so that ${\langleA(\eta u_1^+),q\rangle}=1$. Define $$u_1=u-{\langlef,q\rangle}\eta u_1^+\in{\widetilde H}^1(\Omega),$$ and observe that $u_1$ satisfies $Au_1=f_1$ where $f_1=f-{\langlef,q\rangle}A(\eta u_1^+)$. Since ${\langlef_1,q\rangle}=0$, the we deduce that $u_1\in H^2(\Omega)$ and $$\|u_1\|_{H^2(\Omega)}\le C\|f_1\|\le C\|f\|+C|{\langlef,q\rangle}|\le C\|f\|,$$ because $A(\eta u_1^+)\in C^\infty(\overline{\Omega})$ and $q\in L_2(\Omega)$. Finite element approximation {#sec: FEM} ============================ Consider a family ${\mathcal{T}}_h$ of shape-regular triangulations of $\Omega$, indexed by the maximum element diameter $h$. For each element $\triangle\in{\mathcal{T}}_h$, let $$h_\triangle={\operatorname{diam}}(\triangle)\quad\text{and}\quad r_\triangle={\operatorname{dist}}(0,\triangle),$$ and suppose that for some $\gamma\ge1$, $$\label{eq: h gamma} chr_\triangle^{1-1/\gamma}\le h_\triangle \le Chr_\triangle^{1-1/\gamma} \quad\text{whenever $h^\gamma\le r_\triangle\le1$,}$$ with $$\label{eq: h gamma 0} ch^\gamma\le h_\triangle\le Ch^\gamma \quad\text{whenever $r_\triangle\le h^\gamma$.}$$ Thus, if $\gamma=1$ then the mesh is globally quasiuniform, but for $\gamma>1$ the element diameter decreases from order $h$, when $r_\triangle\ge1$, to order $h^\gamma$, when $r_\triangle\le h^\gamma$. Such triangulations are widely used for elliptic problems on domains with re-entrant corners; see for instance Apel et al. [@ApelSaendigWhiteman1996 Section 3]. For each triangulation ${\mathcal{T}}_h$, we let $V_h$ denote the corresponding space of continuous piecewise-linear functions that vanish on $\partial\Omega$, so that $V_h\subseteq V={\widetilde H}^1(\Omega)$. Since the bilinear form  is bounded and coercive on $V$, there exists a unique finite element solution $u_h\in V_h$ defined by $$\label{eq: uh weak} a(u_h,v)={\langlef,v\rangle}\quad\text{for all $v\in V_h$.}$$ This solution is stable in ${\widetilde H}^1(\Omega)$, $$\label{eq: Htilde uh} \|u_h\|_{{\widetilde H}^1(\Omega)}\le C\|f\|_{H^{-1}(\Omega)},$$ and satisfies the quasi-optimal error bound $$\label{eq: quasioptimal} \|u_h-u\|_{{\widetilde H}^1(\Omega)} \le C\min_{v\in V_h}\|v-u\|_{{\widetilde H}^1(\Omega)}.$$ Let $\Pi_h:C(\overline{\Omega})\to V_h$ denote the nodal interpolation operator, define the seminorm $$|v|_{m,\Omega}=\biggl(\sum_{j_1+j_2=m}\int_\Omega |\partial^jv(x)|^2\,dx\biggr)^{1/2},$$ where $\partial^j=\partial_{x_1}^{j_1}\partial_{x_2}^{j_2}$, and recall the standard interpolation error bounds [@Ciarlet2002] $$\label{eq: Pi_h} |v-\Pi_hv|_{m,\triangle}\le Ch_\triangle^{2-m}|v|_{2,\triangle}, \qquad m\in\{0,1\}.$$ The next theorem reflects the influence of the singular behaviour of $u$ and the local mesh refinement parameter $\gamma$ on the accuracy of the approximation $u\approx\Pi_h u$. \[thm: Pi\_h\] If $f\in L_2(\Omega)$ then the solution $u\in V$ of the elliptic problem  satisfies $$\|u-\Pi_h u\|\le Ch\epsilon(\gamma,h)\|f\| \quad\text{and}\quad \|u-\Pi_h u\|_{{\widetilde H}^1(\Omega)}\le C\epsilon(\gamma,h)\|f\|,$$ where $\epsilon(h,\gamma)$ is given by . We use Theorem \[thm: singular term\] to split $u$ into singular and regular parts, $$u={u_{\operatorname{s}}}+{u_{\operatorname{r}}},\qquad {u_{\operatorname{s}}}={\langlef,q\rangle}\eta u_1^+,\qquad {u_{\operatorname{r}}}\in H^2(\Omega),$$ with $\|{u_{\operatorname{r}}}\|_{H^2(\Omega)}\le C\|f\|$, leading to a corresponding decomposition of the interpolation error, $$u-\Pi_h u=({u_{\operatorname{s}}}-\Pi_h{u_{\operatorname{s}}})+({u_{\operatorname{r}}}-\Pi_h{u_{\operatorname{r}}}).$$ We see from  that $$\|{u_{\operatorname{r}}}-\Pi_h{u_{\operatorname{r}}}\|\le Ch^2|{u_{\operatorname{r}}}|_{2,\Omega} \le Ch^2\|f\|\le Ch\epsilon(h,\gamma)\|f\|$$ and $$|{u_{\operatorname{r}}}-\Pi_h{u_{\operatorname{r}}}|_{1,\Omega}\le Ch|{u_{\operatorname{r}}}|_{2,\Omega} \le Ch\|f\|\le C\epsilon(h,\gamma)\|f\|,$$ so it suffices to consider ${u_{\operatorname{s}}}-\Pi_h{u_{\operatorname{s}}}$. Note that $|\partial^j{u_{\operatorname{s}}}(x)|\le C\|f\||x|^{\beta-|j|}$ for any multi-index $j$, because $u_1^+$ is homogeneous of degree $\beta$. We partition the triangulation into three subsets, $$\begin{gathered} {\mathcal{T}}_h^1=\{\,\triangle\in{\mathcal{T}}_h:r_\triangle<h^\gamma\,\},\qquad {\mathcal{T}}_h^2=\{\,\triangle\in{\mathcal{T}}_h:h^\gamma\le r_\triangle<1\,\},\\ {\mathcal{T}}_h^3=\{\,\triangle\in{\mathcal{T}}_h:r_\triangle\ge1\,\},\end{gathered}$$ and write $$|{u_{\operatorname{s}}}-\Pi_h{u_{\operatorname{s}}}|_{1,\Omega}^2=S_1+S_2+S_3 \quad\text{where}\quad S_p=\sum_{\triangle\in{\mathcal{T}}_h^p}|{u_{\operatorname{s}}}-\Pi_h{u_{\operatorname{s}}}|_{1,\triangle}^2.$$ If $r_\triangle<h^\gamma$, then $|{u_{\operatorname{s}}}-\Pi_h{u_{\operatorname{s}}}|_{1,\triangle}\le|{u_{\operatorname{s}}}|_{1,\triangle}+ |\Pi_h{u_{\operatorname{s}}}|_{1,\triangle}$, and we estimate separately $$|{u_{\operatorname{s}}}|_{1,\triangle}^2\le C\|f\|^2\int_\triangle|x|^{2(\beta-1)}\,dx$$ and, using , $$|\Pi_h{u_{\operatorname{s}}}|_{1,\triangle}^2 \le Ch_\triangle^{-2}|\Pi_hu_s|_{0,\triangle}^2 \le Ch^{-2\gamma}\|f\|^2\int_\triangle|x|^{2\beta}\,dx.$$ Since $|x|\le r_\triangle+h_\triangle\le Ch^\gamma$ for $x\in\triangle$, we see that $$\label{eq: S1} S_1\le C\|f\|^2\int_{|x|\le Ch^\gamma}|x|^{2(\beta-1)}\,dx +Ch^{-2\gamma}\|f\|^2\int_{|x|\le Ch^\gamma}|x|^{2\beta}\,dx \le Ch^{2\gamma\beta}\|f\|^2.$$ If $h^\gamma\le r_\triangle<1$, then gives $$|{u_{\operatorname{s}}}-\Pi_h{u_{\operatorname{s}}}|_{1,\triangle}^2\le Ch_\triangle^2\|f\|^2 \int_\triangle|x|^{2(\beta-2)}\,dx,$$ and our assumption  on the mesh implies that for $x\in\triangle$, $$h_\triangle|x|^{\beta-2} \le Chr_\triangle^{1-1/\gamma}|x|^{\beta-2} =Ch\biggl(\frac{r_\triangle}{|x|}\biggr)^{1-1/\gamma} |x|^{\beta-1-1/\gamma} \le Ch|x|^{\beta-1-1/\gamma}$$ so $$\label{eq: S2} \begin{aligned} S_2&\le Ch^2\|f\|^2\int_{h^\gamma\le|x|\le 1+h} |x|^{2(\beta-1-1/\gamma)}\,dx\\ &\le Ch^2\|f\|^2\int_{h^\gamma}^{1+h}r^{2(\beta-1/\gamma)-1}\,dr \le C\epsilon(h,\gamma)^2\|f\|^2. \end{aligned}$$ In the remaining case $r_\triangle\ge1$, putting $R=\sup\{\,|x|:x\in\Omega\,\}$ we have $1\le|x|\le R$ for $x\in\triangle$, and thus $$\label{eq: S3} S_3\le\sum_{\triangle\in{\mathcal{T}}_h^3} Ch_\triangle^2\|f\|^2\int_\triangle\,dx \le Ch^2\|f\|^2\int_{1\le|x|\le R}\,dx\le Ch^2\|f\|^2.$$ Together, – show that $|{u_{\operatorname{s}}}-\Pi_h{u_{\operatorname{s}}}|_{1,\Omega}\le C\epsilon(h,\gamma)\|f\|$. A similar argument shows $$\sum_{\triangle\in{\mathcal{T}}_h^1}|{u_{\operatorname{s}}}-I_h{u_{\operatorname{s}}}|_{0,\triangle}^2 \le C\|f\|^2\int_0^{Ch^\gamma}h^{2\gamma\beta}r\,dr \le Ch^{2\gamma(\beta+1)}\|f\|^2$$ and $$\begin{aligned} \sum_{\triangle\in{\mathcal{T}}_h^2\cup{\mathcal{T}}_h^3} |{u_{\operatorname{s}}}-I_h{u_{\operatorname{s}}}|_{0,\triangle}^2 &\le Ch^4\|f\|^2\int_{h^\gamma}^{R} r^{2(\beta-1/\gamma)-1}r^{2(1-1/\gamma)}\,dr \le Ch^2\epsilon(h,\gamma)^2\|f\|^2.\end{aligned}$$ Hence, $\|{u_{\operatorname{s}}}-\Pi_h{u_{\operatorname{s}}}\|\le Ch\epsilon(h,\gamma)\|f\|$ and the desired bounds follow. \[thm: elliptic error\] If $f\in L_2(\Omega)$ then the finite element solution $u_h\in V_h$ of the elliptic problem  satisfies $$\|u_h-u\|\le C\epsilon(\gamma,h)^2\|f\| \quad\text{and}\quad \|u_h-u\|_{{\widetilde H}^1(\Omega)}\le C\epsilon(\gamma,h)\|f\|,$$ where $\epsilon(h,\gamma)$ is given by . The bound in ${\widetilde H}^1(\Omega)$ follows at once from  and Theorem \[thm: Pi\_h\]. The error bound in $L_2(\Omega)$ is proved via the usual duality argument. In fact, given any $\phi\in L_2(\Omega)$, the dual variational problem $$a(w,\psi)={\langlew,\phi\rangle}\quad\text{for all $w\in{\widetilde H}^1(\Omega)$,}$$ has a unique solution $\psi\in{\widetilde H}^1(\Omega)$. Since the bilinear form $a$ is symmetric, the preceding estimate for $u-\Pi_hu$ carries over, with $\phi$ playing the role of $f$, to yield $\|\psi-\Pi_h\psi\|_{{\widetilde H}^1(\Omega)} \le C\epsilon(h,\gamma)\|\phi\|$. Thus, $$\begin{aligned} |{\langleu_h-u,\phi\rangle}|&=|a(u_h-u,\psi)|=|a(u_h-u,\psi-\Pi_h\psi)|\\ &\le C\|u_h-u\|_{{\widetilde H}^1(\Omega)} \|\psi-\Pi_h\psi\|_{{\widetilde H}^1(\Omega)} \le C\epsilon(h,\gamma)^2\|f\|\|\phi\|,\end{aligned}$$ implying that $\|u_h-u\|\le C\epsilon(h,\gamma)^2\|f\|$. The time-dependent problem {#sec: time-dep} ========================== We may view $A=-K\nabla^2$ as an unbounded operator on $L_2(\Omega)$ with domain $V^2$ given by . Since the associated bilinear form  is symmetric and coercive, and since the inclusion ${\widetilde H}^1(\Omega)\subseteq L_2(\Omega)$ is compact, there exists a complete orthonormal sequence of eigenfunctions $\phi_1$, $\phi_2$, $\phi_3$, …and corresponding real eigenvalues $\lambda_1$, $\lambda_2$, $\lambda_3$, …with $\lambda_j\to\infty$ as $j\to\infty$. Thus, $$A\phi_n=\lambda_n\phi_n\quad\text{and}\quad {\langle\phi_m,\phi_n\rangle}=\delta_{mn}\quad \text{for all $m$, $n\in\{1,2,3,\ldots\}$,}$$ and we may assume that $0<\lambda_1\le\lambda_2\le\lambda_3\le\cdots$. Moreover, $$(zI-A)^{-1}:L_2(\Omega)\to L_2(\Omega)$$ is a bounded linear operator for each complex number $z$ not in the spectrum ${\operatorname{spec}}(A)=\{\,\lambda_1,\lambda_2,\lambda_3,\ldots\}$, and given any $\theta_0\in(0,\pi)$ we have a resolvent estimate in the induced operator norm [@LeGiaMcLean2011 Lemma 1], $$\label{eq: resolvent} \|(zI-A)^{-1}\|\le\frac{1+2/\lambda_1}{\sin\theta_0}\,\frac{1}{1+|z|} \quad\text{for $|\arg z|>\theta_0$.}$$ Define the Laplace transform $\hat f={\mathcal{L}}f$ of a suitable $f:[0,\infty)\to L_2(\Omega)$ by $$\hat f(z)=({\mathcal{L}}f)(z)={\mathcal{L}}\{f(t)\}_{t\to z} =\int_0^\infty e^{-zt}f(t)\,dt,$$ for $\Re z$ sufficiently large. Since ${\mathcal{L}}\{\partial_t^{1-\alpha}f\}_{t\to z}=z^{1-\alpha}\hat f(z)$, a formal calculation implies that the fractional diffusion equation  transforms to an elliptic problem (with complex coefficients) for $\hat u(z)$, $$z\hat u(z)+z^{1-\alpha}A\hat u(z)=u_0+\hat f(z),$$ and so $$\label{eq: u hat} \hat u(z)=z^{\alpha-1}(z^\alpha I+A)^{-1}\bigl(u_0+\hat f(z)\bigr).$$ The boundary condition $u(t)=0$ on $\partial\Omega$ tranforms to give $\hat u(z)=0$ on $\partial\Omega$. Using ${\mathcal{L}}\{t^{p\alpha}/\Gamma(1+p\alpha)\}_{t\to z}=z^{-1-p\alpha}$ we find that for $\lambda>0$ and $|z|>\lambda^{-1/\alpha}$, $$z^{\alpha-1}(z^\alpha+\lambda)^{-1} =z^{-1}\sum_{p=0}^\infty\bigl(-\lambda z^{-\alpha}\bigr)^p ={\mathcal{L}}\biggl\{\sum_{p=0}^\infty \frac{(-\lambda t^\alpha)^p}{\Gamma(1+p\alpha)} \biggr\}_{t\to z} ={\mathcal{L}}\{E_\alpha(-\lambda t^\alpha)\},$$ where $E_\alpha$ is the Mittag–Leffler function. The inequalities $0\le E_\alpha(-t)\le1$ for $0\le t<\infty$ imply that the sum $$\label{eq: E(t)} {\mathcal{E}}(t)v=\sum_{n=1}^\infty E_\alpha(-\lambda_nt^\alpha) {\langlev,\phi_n\rangle}\phi_n$$ defines a bounded linear operator ${\mathcal{E}}(t):L_2(\Omega)\to L_2(\Omega)$ satisfying $$\label{eq: ||E(t)||} \|{\mathcal{E}}(t)v\|\le\|v\|\quad\text{for $0\le t<\infty$.}$$ Thus, for each eigenfunction $\phi_n$, $$z^{\alpha-1}(z^\alpha I+A)^{-1}\phi_n =z^{\alpha-1}(z^\alpha+\lambda_n)^{-1}\phi_n ={\mathcal{L}}\{E_\alpha(-\lambda_nt^\alpha)\phi_n\}_{t\to z},$$ and we conclude that $$\label{eq: E hat} {\widehat{\mathcal{E}}}(z)=z^{\alpha-1}(z^\alpha I+A)^{-1}.$$ Writing  as $\hat u(z)={\widehat{\mathcal{E}}}(z)u_0+{\widehat{\mathcal{E}}}(z)\hat f(z)$ then yields a Duhamel formula, $$\label{eq: u(t) mild} u(t)={\mathcal{E}}(t)u_0+\int_0^t{\mathcal{E}}(t-s)f(s)\,ds\quad\text{for $t>0$,}$$ that serves to define the *mild solution* of our initial-boundary value problem for . In particular, ${\mathcal{E}}(t)$ is the solution operator for the homogeneous problem ($f\equiv0$) with initial data $u_0\in L_2(\Omega)$. Also, the bound  immediately implies a stability estimate in $L_2(\Omega)$ for a general, locally integrable $f:[0,\infty)\to L_2(\Omega)$, namely $$\|u(t)\|\le\|u_0\|+\int_0^t\|f(s)\|\,ds\quad\text{for $t>0$.}$$ The semidiscrete finite element solution {#sec: semidiscrete} ======================================== Let $P_h$ denote the orthoprojector $L_2(\Omega)\to V_h$, that is, $P_hv\in V_h$ satisfies ${\langleP_hv,w\rangle}={\langlev,w\rangle}$ for all $v\in L_2(\Omega)$ and $w\in V_h$. There exists a unique linear operator $A_h:V_h\to V_h$ such that $${\langleA_hv,w\rangle}=a(v,w)\quad\text{for all $v$, $w\in V_h$,}$$ and the operator equation $A_hu_h=P_hf$ is equivalent to the variational equation  used to define the finite element solution $u_h\in V_h$ of the elliptic problem . Denote the number of degrees of freedom by $N=\dim V_h$ and equip $V_h$ with the norm induced from $L_2(\Omega)$. The finite element space $V_h$ has an orthonormal basis of eigenfunctions $\Phi_1$, $\Phi_2$, …, $\Phi_N$ with corresponding real eigenvalues $\Lambda_1$, $\Lambda_2$, …, $\Lambda_N$. Thus, $$A_h\Phi_n=\Lambda_n\Phi_n\quad\text{and}\quad {\langle\Phi_m,\Phi_n\rangle}=\delta_{mn}\quad \text{for $m$, $n\in\{1,2,\ldots,N\}$,}$$ and we may assume that $0<\Lambda_1\le\Lambda_2\le\cdots\le\Lambda_N$. Moreover, the resolvent $$(zI-A_h)^{-1}:V_h\to V_h$$ exists for every $z\notin{\operatorname{spec}}(A_h)=\{\Lambda_1,\Lambda_2,\ldots,\Lambda_N\}$, and we have the an estimate corresponding to : $$\label{eq: resolvent h} \|(zI-A_h)^{-1}\|\le\frac{1+2/\Lambda_1}{\sin\theta_0} \,\frac{1}{1+|z|}\quad\text{for $|\arg z|>\theta_0$.}$$ Note that $\lambda_1\le\Lambda_1$ so this bound is uniform in $h$. The first Green identity yields the variational formulation for , $$\label{eq: u(t) variational} {\langle\partial_tu,v\rangle}+a(\partial_t^{1-\alpha}u,v)={\langlef(t),v\rangle} \quad\text{for all $v\in{\widetilde H}^1(\Omega)$ and $t>0$,}$$ so we define the finite element solution $u_h:[0,\infty)\to V_h$ by $$\label{eq: uh(t) variational} {\langle\partial_tu_h,v\rangle}+a(\partial_t^{1-\alpha}u_h,v)={\langlef(t),v\rangle} \quad\text{for all $v\in V_h$ and $t>0$,}$$ with $u_h(0)=u_{0h}$, where $u_{0h}\in V_h$ is a suitable approximation to the initial data $u_0$. Thus, the vector of nodal values ${\mathbf{U}}(t)$ satisfies the integro-differential equation in ${\mathbb{R}}^N$, $$\label{eq: MOL} {\mathbf{M}}\partial_t{\mathbf{U}}+{\mathbf{S}}\partial_t^{1-\alpha} {\mathbf{U}}={\mathbf{F}}(t) ,$$ where ${\mathbf{M}}$ and ${\mathbf{S}}$ denote the $N\times N$ mass and stiffness matrices, respectively, and ${\mathbf{F}}(t)$ denotes the load vector. In the limiting case as $\alpha\to1$, when  becomes the heat equation, we see that reduces to the usual system of (stiff) ODEs arising in the method of lines. The variational equation  is equivalent to $$\partial_tu_h+\partial_t^{1-\alpha}A_hu_h=P_hf(t) \quad\text{for $t>0$.}$$ Taking Laplace transforms as in Section \[sec: time-dep\], we find that $$\hat u_h(z)=z^{\alpha-1}(z^\alpha I+A_h)^{-1} \bigl(u_{0h}+P_h\hat f(z)\bigr)$$ and thus $$u_h(t)={\mathcal{E}}_h(t)u_{0h}+\int_0^t{\mathcal{E}}_h(t-s)P_hf(s)\,ds \quad\text{for $t>0$,}$$ where $${\mathcal{E}}_h(t)v=\sum_{n=1}^N E_\alpha(-\Lambda_n t^\alpha) {\langlev,\Phi_n\rangle} \Phi_n.$$ In the same way as  we have $$\label{eq: ||Eh(t)||} \|{\mathcal{E}}_h(t)v\|\le\|v\|\quad\text{for $t>0$~and $v\in v_h$,}$$ implying that the finite element solution is stable in $L_2(\Omega)$, $$\label{eq: uh stable} \|u_h(t)\|\le\|u_{0h}\|+\int_0^t\|f(s)\|\,ds.$$ For convenience, we put $$B(z)=(z^\alpha I+A)^{-1} \quad\text{and}\quad B_h(z)=(z^\alpha I+A_h)^{-1},$$ which satisfy the following bounds. \[lem: B(z)\] If $|\arg z^\alpha|<\pi-\theta_0$, then 1. $\|B(z)v\|\le C\|v\|/(1+|z|^\alpha)$ and $\|B(z)v\|_{{\widetilde H}^1(\Omega)}\le C\|v\|$ for $v\in L_2(\Omega)$; 2. $\|B_h(z)v\|\le C\|v\|/(1+|z|^\alpha)$ and $\|B_h(z)v\|_{{\widetilde H}^1(\Omega)}\le C\|v\|$ for $v\in V_h$. First let $v\in L_2(\Omega)$. The resolvent estimate  immediately implies the desired bounds for $w(z)=B(z)v$ in $L_2(\Omega)$. To estimate the norm of $w(z)$ in ${\widetilde H}^1(\Omega)$, observe that $Aw(z)=v-z^\alpha w(z)$ so by , $$\|w(z)\|_{{\widetilde H}^1(\Omega)}\le C\|v-z^\alpha w(z)\|_{H^{-1}(\Omega)} \le C\|v-z^\alpha B(z)v\|\le C\|v\|.$$ When $v\in V_h$, the estimates for $B_h(z)v$ follow in the same way from  and . Since ${\mathcal{L}}\{{\mathcal{E}}(t)\phi_n\}_{t\to z} =z^{\alpha-1}(z^\alpha+\lambda_n)^{-1}\phi_n=z^{\alpha-1}B(z)\phi_n$, the Laplace inversion formula implies that $$\begin{aligned} {\mathcal{E}}(t)\phi_n&=\lim_{M\to\infty}\biggl(\frac{1}{2\pi i} \int_{1-iM}^{1+iM} e^{zt}z^{\alpha-1}(z^\alpha+\lambda_n)^{-1}\,dz\biggr)\phi_n =\frac{1}{2\pi i}\int_\Gamma e^{zt}z^{\alpha-1}B(z)\phi_n\,dz,\end{aligned}$$ for $t>0$ and for any contour $\Gamma$ that begins at $\infty$ in the third quadrant, ends at $\infty$ in the second quadrant and avoids the negative real axis. The factor $e^{zt}$ is exponentially small as $\Re z\to-\infty$, so and Lemma \[lem: B(z)\] ensure that $$\label{eq: E integral} {\mathcal{E}}(t)v=\frac{1}{2\pi i}\int_\Gamma e^{zt}z^{\alpha-1}B(z)v\,dz \quad\text{for $t>0$ and $v\in L_2(\Omega)$,}$$ where the integral over $\Gamma$ is absolutely convergent in $L_2(\Omega)$. Likewise, ${\mathcal{L}}\{{\mathcal{E}}_h(t)\Phi_n\}_{t\to z} =z^{\alpha-1}(z^\alpha+\Lambda_n)^{-1}\Phi_n=z^{\alpha-1}B_h(z)\Phi_n$ and we have a corresponding integral representation $$\label{eq: Eh integral} {\mathcal{E}}_h(t)v=\frac{1}{2\pi i}\int_\Gamma e^{zt}z^{\alpha-1} B_h(z)v\,dz\quad\text{for $t>0$ and $v\in V_h$.}$$ Error bounds {#sec: error} ============ The homogeneous equation ------------------------ We now consider the error $u_h(t)-u(t)$ in the case $f\equiv0$. The main difficulty will be to estimate the difference $$\label{eq: E(t) error} {\mathcal{E}}_h(t)P_hu_0-{\mathcal{E}}(t)u_0=\frac{1}{2\pi i}\int_\Gamma e^{zt}z^{\alpha-1}G_h(z)u_0\,dz,$$ where, by  and , $G_h(z)=B_h(z)P_h-B(z)$. We begin by estimating $G_h(z)v$. \[lem: Gh(z)\] If $v\in L_2(\Omega)$ and $|\arg z^\alpha|<\pi-\theta_0$, then $$\|G_h(z)v\|\le C\epsilon(h,\gamma)^2\|v\| \quad\text{and}\quad \|G_h(z)v\|_{{\widetilde H}^1(\Omega)}\le C(1+|z|^\alpha)\epsilon(h,\gamma)\|v\|.$$ Given $v\in L_2(\Omega)$, let $w(z)=B(z)v\in V$ so that $Aw(z)=v-z^\alpha w(z)$, and let $w_h(z)\in V_h$ be the solution of $$A_hw_h(z)=P_h[v-z^\alpha w(z)].$$ In this way, $P_hv=z^\alpha P_h w(z)+A_hw_h(z)=(z^\alpha I+A_h)w_h(z) -z^\alpha[w_h(z)-P_hw(z)]$, and thus $B_h(z)P_hv=w_h(z)-z^\alpha B_h(z)[w_h(z)-P_hw(z)]$, implying that $$\label{eq: Gh(z)} G_h(z)v=w_h(z)-w(z)-z^\alpha B_h(z)P_h[w_h(z)-w(z)].$$ Lemma \[lem: B(z)\] shows that $\|w(z)\|\le C\|v\|/(1+|z|^\alpha)$ so $\|v-z^\alpha w(z)\|\le C\|v\|$. By applying Theorem \[thm: elliptic error\], with $w(z)$ and $v-z^\alpha w(z)$ playing the roles of $u$ and $f$, respectively, we deduce that $$\|w_h(z)-w(z)\|\le C\epsilon(h,\gamma)^2\|v\| \quad\text{and}\quad \|w_h(z)-w(z)\|_{{\widetilde H}^1(\Omega)}\le C\epsilon(h,\gamma)\|v\|.$$ The result now follows from  after another application of Lemma \[lem: B(z)\]. \[thm: non-smooth u0\] Assume that $f\equiv0$ and $u_0\in L_2(\Omega)$. Then the mild solution $u(t)={\mathcal{E}}(t)u_0$ and its finite element approximation $u_h(t)={\mathcal{E}}_h(t)u_{0h}$ satisfy the error bounds $$\|u_h(t)-u(t)\|\le\|u_{0h}-P_hu_0\| +Ct^{-\alpha}\epsilon(h,\gamma)^2\|u_0\|$$ and $$\|u_h(t)-u(t)\|_{{\widetilde H}^1(\Omega)} \le Ct^{-\alpha}\|u_{0h}-P_hu_0\| +C(t^{-2\alpha}+t^{-\alpha})\epsilon(h,\gamma)\|u_0\|$$ for $t>0$, where $\epsilon(h,\gamma)$ is given by . We split the error into two terms, $$u_h(t)-u(t)={\mathcal{E}}_h(t)\bigl(u_{0h}-P_hu_0\bigr) +\bigl[{\mathcal{E}}_h(t)P_hu_0-{\mathcal{E}}(t)u_0\bigr].$$ It follows from  that $\bigl\|{\mathcal{E}}_h(t)\bigl(u_{0h}-P_hu_0\bigr)\bigr\| \le\|u_{0h}-P_hu_0\|$, and to estimate the second term we use the integral representation  with $\Gamma=\Gamma_+-\Gamma_-$, where $\Gamma_\pm$ is the contour $z=se^{\pm i3\pi/4}$ for $0<s<\infty$. Applying Lemma \[lem: Gh(z)\] and making an obvious substitution, we find that $$\begin{aligned} \bigl\|{\mathcal{E}}_h(t)P_hu_0-{\mathcal{E}}(t)u_0\bigr\|&\le C\epsilon(h,\gamma)^2\|u_0\| \int_0^\infty e^{-st/\sqrt{2}}s^\alpha\,\frac{ds}{s}\\ &=C\epsilon(h,\gamma)^2\|u_0\|t^{-\alpha} \int_0^\infty e^{-s/\sqrt{2}}s^\alpha\,\frac{ds}{s},\end{aligned}$$ which proves the first error bound of the theorem. Choosing $\Gamma=\Gamma_+-\Gamma_-$ in the integral representation  of ${\mathcal{E}}_h(t)v$, and using Lemma \[lem: B(z)\], we have for $v\in V_h$, $$\|{\mathcal{E}}_h(t)v\|_{{\widetilde H}^1(\Omega)}\le C\int_0^\infty e^{-st/\sqrt{2}} s^\alpha\|v\|\,\frac{ds}{s} =Ct^{-\alpha}\|v\|\int_0^\infty e^{-s/\sqrt{2}}s^\alpha\, \frac{ds}{s}\le Ct^{-\alpha}\|v\|,$$ so in particular, when $v=u_{0h}-P_hu_0$, $$\bigl\|{\mathcal{E}}_h(t)\bigl(u_{0h}-P_hu_0\bigr)\bigr\|_{{\widetilde H}^1(\Omega)} \le Ct^{-\alpha}\|u_{0h}-P_hu_0\|.$$ Finally, using  and Lemma \[lem: Gh(z)\] again, $$\begin{gathered} \bigl\|{\mathcal{E}}_h(t)P_hu_0-{\mathcal{E}}(t)u_0\bigr\|_{{\widetilde H}^1(\Omega)} \le C\epsilon(h,\gamma)\|u_0\| \int_0^\infty e^{-st/\sqrt{2}}s^\alpha(1+s^\alpha)\,\frac{ds}{s}\\ =C\epsilon(h,\gamma)\|u_0\|t^{-2\alpha}\int_0^\infty e^{-s/\sqrt{2}}s^\alpha(t^\alpha+s^\alpha)\,\frac{ds}{s} \le C\epsilon(h,\gamma)(t^{-\alpha}+t^{-2\alpha})\|u_0\|,\end{gathered}$$ proving the second error estimate of the theorem. When $u_0$ is sufficiently regular we obtain an error bound that is uniform in $t$. The proof uses the Ritz projector $R_h:{\widetilde H}^1(\Omega)\to V_h$, defined by $$\label{eq: Ritz R_h} a(R_hu,v)=a(u,v)\quad\text{for all $v\in V_h$,}$$ and relies on the regularity estimate [@McLean2010 Theorem 4.4] $$\label{eq: Au_t reg} t^\alpha\|A\partial_tu\|\le Ct^{\sigma\alpha-1}\|A^\sigma u_0\| \quad\text{for $0<t\le T$ and $0\le\sigma\le2$.}$$ \[thm: u0 smooth\] Assume that $f\equiv0$ and $0<\delta\le1$. If $A^{1+\delta}u_0\in L_2(\Omega)$, then $$\|u_h(t)-u(t)\|\le\|u_{0h}-R_hu_0\| +C\delta^{-1}\epsilon(h,\gamma)^2\|A^{1+\delta}u_0\| \quad\text{for $0\le t\le T$.}$$ To begin with, we permit $f\ne0$, and in the usual way, decompose the error as $$u_h(t)-u(t)=\vartheta(t)+\varrho(t),$$ where $\vartheta(t)=u_h(t)-R_hu(t)\in V_h$ and $\varrho(t)=R_hu(t)-u(t)$. Furthermore, since $\varrho(t)=\varrho(0)+\int_0^t\partial_t\varrho(s)\,ds$ it follows that $$\|\varrho(t)\|\le\|\varrho(0)\|+\int_0^t\|\partial_t\varrho(s)\|\,ds.$$ By , if $v\in V_h$ then $${\langle\partial_t\vartheta,v\rangle}+a(\partial_t^{1-\alpha}\vartheta,v) ={\langlef,v\rangle}-{\langle\partial_tR_hu,v\rangle}-a(\partial_t^{1-\alpha}R_hu,v),$$ and using the definition of the Ritz projector  followed by , we have $$a(\partial_t^{1-\alpha}R_hu,v) =a(\partial_t^{1-\alpha}u,v)={\langlef,v\rangle}-{\langle\partial_tu,v\rangle},$$ so $$\label{eq: theta} {\langle\partial_t\vartheta,v\rangle}+a(\partial_t^{1-\alpha}\vartheta,v) =-{\langle\partial_t\varrho,v\rangle}.$$ In other words, $\vartheta$ is the finite element solution of the fractional diffusion problem with source term $-\partial_t\varrho(t)$. Thus, the stability estimate  gives $$\|\vartheta(t)\|\le\|\vartheta(0)\| +\int_0^t\|\partial_t\varrho(s)\|\,ds.$$ Theorem \[thm: elliptic error\] implies that $\|v-R_hv\|\le C\epsilon(h,\gamma)^2\|Av\|$, so $$\begin{aligned} \|u_h(t)-u(t)\|&\le\|\vartheta(0)\|+\|\varrho(0)\|+2\int_0^t \|\partial_t\varrho(s)\|\,ds\\ &\le\|u_{0h}-R_hu_0\|+C\epsilon(h,\gamma)^2\biggl(\|Au_0\| +\int_0^t\|A\partial_tu(s)\|\,ds\biggr),\end{aligned}$$ and, assuming now that $f\equiv0$, we use  to bound the integral by $C\delta^{-1}\|A^{1+\delta}u_0\|$. Intermediate regularity of $u_0$ ensures a milder growth of the error as $t\to0$. Assume that $f\equiv0$ and $0<\delta\le1$. If $0<\theta<1$ and $u_{0h}=P_hu_0$, then $$\|u_h(t)-u(t)\|\le C\delta^{-\theta}t^{-\alpha(1-\theta)} \epsilon(h,\gamma)^2\|A^{(1+\delta)\theta}u_0\| \quad\text{for $0<t\le T$.}$$ Since $\|u_{0h}-R_hu_0\|=\|P_h(u_0-R_hu_0)\|\le\|u_0-R_hu_0\| \le C\epsilon(h,\gamma)^2\|Au_0\|$, we see that $$\|u_h(t)-u(t)\|\le Ct^{-\alpha}\epsilon(h,\gamma)^2\|u_0\|$$ and $$\|u_h(t)-u(t)\| \le C\delta^{-1}\epsilon(h,\gamma)^2\|A^{1+\delta}u_0\|.$$ By interpolation, $\|u_h(t)-u(t)\|\le C\bigl( t^{-\alpha}\epsilon(h,\gamma)^2\bigr)^{1-\theta} \bigl(\delta^{-1}\epsilon(h,\gamma)^2\bigr)^\theta \|A^{(1+\delta)\theta}u_0\|$. The inhomogeneous equation -------------------------- When $u_0=0$ and $f\ne0$, we readily adapt the proof of Theorem \[thm: u0 smooth\] to show the following error bound; see also McLean and Thomée [@McLeanThomee2010 Lemma 4.1]. Instead of , we now rely on the regularity result [@McLean2010 Theorem 4.1] $$\label{eq: Au reg} \|A{\mathcal{E}}(t)v\|\le Ct^{-\alpha}\|v\|\quad\text{for $0<t\le T$.}$$ \[thm: u0=0\] If $u_0=u_{0h}=0$ then $$\|u_h(t)-u(t)\|\le Ct^{1-\alpha}\epsilon(h,\gamma)^2\biggl( \|f(0)\|+\int_0^t\|\partial_tf(s)\|\,ds\biggr) \quad\text{for $t>0$.}$$ The equation  for $\vartheta$ holds for a general $f$, and now $\vartheta(0)=\varrho(0)=0$, so stability of the finite element solution implies that $$\|u_h(t)-u(t)\|\le2\int_0^t\|\partial_t\varrho(s)\|\,ds \le C\epsilon(h,\gamma)^2\int_0^t\|A\partial_tu(s)\|\,ds.$$ Since $$Au(s)=A\int_0^s{\mathcal{E}}(s-\tau)f(\tau)\,d\tau =\int_0^sA{\mathcal{E}}(\tau)f(s-\tau)\,d\tau,$$ we have $$A\partial_tu(s)=A{\mathcal{E}}(s)f(0) +\int_0^sA{\mathcal{E}}(\tau)\partial_tf(s-\tau)\,d\tau,$$ so using , $$\|A\partial_tu(s)\|\le Cs^{-\alpha}\|f(0)\| +\int_0^s \tau^{-\alpha}\|\partial_tf(s-\tau)\|\,d\tau.$$ Thus, $$\int_0^t\|A\partial_tu(s)\|\,ds\le Ct^{1-\alpha}\|f(0)\| +\int_0^t\int_0^s\tau^{-\alpha}\|\partial_tf(s-\tau)\|\,d\tau\,ds$$ and the double integral equals $$\int_0^t\int_0^s(s-\tau)^{-\alpha}\|\partial_tf(\tau)\|\,d\tau\,ds =\int_0^t\|\partial_tf(\tau)\|(t-\tau)^{1-\alpha}\,d\tau,$$ implying the desired estimate. Alternative boundary conditions {#sec: alt bc} =============================== Neumann boundary conditions --------------------------- Separation of variables in polar coordinates yields the functions $$u_n^\pm=r^{\pm n\beta}\cos(n\beta\theta) \quad\text{for $n=1$, $2$, $3$, \dots}$$ satisfying with $\partial_\theta u_n^\pm=0$ if $\theta=0$ or $\theta=\pi/\beta$. In addition, for $n=0$ we find $u_0^+=1$ and $u_0^-=\log r$, and can readily check that $$\label{eq: un H1} \eta u_n^+\in H^1(\Omega) \quad\text{but}\quad \eta u_n^-\notin H^1(\Omega) \quad\text{for all $n\ge0$,}$$ and that $\eta u^+_n\in H^2(\Omega)$ iff $n\ne1$. If we impose a homogeneous Neumann boundary condition $\partial_n u=0$ on $\partial\Omega$, then our results are essentially unchanged, but the fact that $A=-K\nabla^2$ now possesses a zero eigenvalue complicates the analysis [@MustaphaMcLean2011 Section 4]. Mixed boundary conditions ------------------------- The functions $$u_n^\pm=r^{(n-\tfrac12)\beta}\sin(n-\tfrac12)\beta\theta \quad\text{for $n=1$, $2$, $3$, \dots}$$ satisfy with $u_n^\pm=0$ if $\theta=0$ and $\partial_\theta u_n^\pm=0$ if $\theta=\pi/\beta$. Once again, holds, for all $n\ge1$, however now we have $$\eta u_n^+\in H^2(\Omega)\quad\text{for all $n\ge3$,} \quad\text{but}\quad \eta u_1^+, \eta u_2^+\notin H^2(\Omega),$$ assuming $1/2<\beta<1$. A new feature is that $\eta u_1^+\notin H^2(\Omega)$ also when $1\le\beta<2$, that is, for an interior angle between $\pi/2$ and $\pi$, in which case $\Omega$ is in fact convex. The proof of Theorem 3.1 must be modified by replacing $\beta$ with $\beta/2$, and replacing $\epsilon(h,\gamma)$ with $$\label{eq: epsmix} {\epsilon_{\text{mix}}}(h,\gamma)=\begin{cases} h^{\gamma\beta/2}/\sqrt{\gamma^{-1}-\beta/2},& 1\le\gamma<2/\beta,\\ h\sqrt{\log(1+h^{-1})},&\gamma=2/\beta,\\ h/\sqrt{\beta/2-\gamma^{-1}},&\gamma>2/\beta, \end{cases}$$ provided the interior angles at $p_1$, $p_2$, …, $p_{J-1}$ are all less than or equal to $\pi/2$. We may then proceed as for Dirichlet boundary conditions (since all the eigenvalues of $A$ are strictly positive), with $\epsilon(h,\gamma)$ replaced by ${\epsilon_{\text{mix}}}(h,\gamma)$ in our error estimates. Numerical experiments {#sec: numerical} ===================== We consider two problems posed on a domain of the form $$\Omega=\{\,(r\cos\theta,r\sin\theta): \text{$0<r<1$ and $0<\theta<\pi/\beta$}\,\},$$ with $\beta=2/3$. Although $\Omega$ is not a polygon, the additional error in $u_h$ due to approximation of the curved part of $\partial\Omega$ is of order $h^2$ in $L_2(\Omega)$, and hence our error bounds should apply unchanged. To fix the time scale for the solutions of the fractional diffusion equation , we choose the generalized diffusivity $K$ so that the smallest eigenvalue of $A=-K\nabla^2$ equals $1$. Figure \[fig: meshes\] shows two successive meshes out of a sequence satisfying our assumptions  and for $\gamma=1/\beta=3/2$; notice that these meshes are not nested. The mesh generation code takes a specified $h_*$ and $\gamma$ and produces a triangulation with maximum element diameter $h$ equivalent to $h_*$. All source files were written in Julia 0.4 [@Julia] with some calls to Gmsh 2.10.1 [@Gmsh], and all computations performed on a desktop PC with 16GB of RAM and an Intel Core i7-4770 CPU. ![Meshes with $h_*=2^{-3}$ (left) and $h_*=2^{-4}$ (right) from a sequence satisfying  and for $\gamma=3/2$.[]{data-label="fig: meshes"}](mesh3-crop.eps "fig:") ![Meshes with $h_*=2^{-3}$ (left) and $h_*=2^{-4}$ (right) from a sequence satisfying  and for $\gamma=3/2$.[]{data-label="fig: meshes"}](mesh4-crop.eps "fig:") For the time integration, we use a technique [@McLeanThomee2010; @WeidemanTrefethen2007] based on a quadrature approximation to the Laplace inversion formula, $$u_h(t)=\frac{1}{2\pi i}\int_\Gamma e^{zt}\hat u_h(z)\,dz =\frac{1}{2\pi i}\int_{-\infty}^\infty e^{z(\xi)t}\hat u\bigl(z(\xi)\bigr)z'(\xi) \,d\xi,$$ where the contour $\Gamma$ has the parametric representation $$z(\xi)=\mu\bigl(1-\sin(\delta-i\xi)\bigr) \quad\text{for $-\infty<\xi<\infty$,}$$ with $\delta=1.1721\,0423$ and $\mu=4.4920\,7528\,M/t$ for given $t>0$ and a chosen positive integer $M$. Therefore, the contour $\Gamma$ is the left branch of an hyperbola with asymptotes $y=\pm(x-\mu)\cot\delta$ for $z=x+iy$. Putting $$z_j=z(\xi_j),\quad z'_j=z'(\xi_j),\quad \xi_j=j\,{\Delta\xi},\quad{\Delta\xi}=\frac{1.0817\,9214}{M},$$ we define $$U_{M,h}(t)=\frac{{\Delta\xi}}{2\pi i}\sum_{j=-M}^Me^{z_jt} \hat u_h(z_j)z'_j\approx u_h(t).$$ To compute $\hat u_h(z_j)$ we solve the (complex) finite element equations $$z_j^\alpha{\langle\hat u_h(z_j),\chi\rangle}+a\bigl(\hat u_h(z_j),\chi\bigr) =z_j^{\alpha-1}{\langleu_{0h}+\hat f(z_j),\chi\rangle},\quad\chi\in V_h,$$ and since we choose real $u_{0h}$ and $f$, it follows that $\hat u_h(z_{-j})=\hat u_h(\bar z_j)=\overline{\hat u_h(z_j)}$ so the number of elliptic solves needed to evaluate $U_{M,h}(t)$ is $M+1$, not $2M+1$. An error bound for the quadrature error $\|U_{M,h}(t)-u_h(t)\|$ includes a decay factor $10.1315^{-M}$, and we observe in practice that the overall error $\|U_{M,h}(t)-u(t)\|$ is dominated by the finite element error $\|u_h(t)-u(t)\|$ for modest values of $M$. In the computations reported below we use $M=8$ to compute $U_{M,h}(t)\approx u_h(t)$, and choose $u_{0h}=P_hu_0$ for the discrete initial data. ![Behaviour of the $L_2$-error $\|u_h(t)-u(t)\|$ for Example 1 when $t=1$ — quasiuniform versus locally refined triangulations.[]{data-label="fig: locally-refined"}](Figure8_2.eps) Example 1 {#example-1 .unnumbered} --------- In our first example, we use $\alpha=1/2$ and choose $u_0$ and $f$ so that the solution of the initial-boundary value problem for  is $$u(x,y,t)=\bigl(1+\omega_{\alpha+1}(t)\bigr) r^\beta(1-r)\sin(\beta\theta).$$ In view of  and , the singular behaviour of $u$ as $r\to0$ or $t\to0$ is typical for such problems. Figure \[fig: locally-refined\] compares the behaviour of the $L_2$-error at $t=1$ for quasi-uniform ($\gamma=1$) and locally-refined ($\gamma=1/\beta=3/2$) triangulations. From Theorems \[thm: non-smooth u0\] and \[thm: u0=0\], we expect errors of order $\epsilon(h,1)^2=h^{2\gamma\beta}=h^{4/3}$ and $\epsilon(h,3/2)^2=h^2\log^2(1+h^{-1})$, respectively. The number of degrees of freedom is of order $h^{-2}$ in both cases, so in Figure \[fig: locally-refined\] we expect the corresponding error curves to be straight lines with gradients $-2/3$ and $-1$, which are in fact close to the observed values $-0.7249$ and $-0.9707$, respectively, as determined by simple linear least squares fits. ![The $L_2$-error as a function of $t$ for Example 2 with $\alpha=1/2$ and $\gamma=2/\beta$.[]{data-label="fig: t dependence"}](Figure8_3.eps) ---------- -------- ----------- ------- ----------- ------- ----------- ------- $h_*$ error rate error rate error rate $2^{-4}$ 1957 1.465e-03 1.485e-03 1.452e-03 $2^{-5}$ 7593 3.673e-04 1.996 3.723e-04 1.996 3.640e-04 1.997 $2^{-6}$ 29771 9.471e-05 1.955 9.597e-05 1.956 9.380e-05 1.956 $2^{-7}$ 117039 2.420e-05 1.969 2.451e-05 1.970 2.391e-05 1.972 $2^{-8}$ 466089 6.059e-06 1.998 6.119e-06 2.002 5.931e-06 2.011 ---------- -------- ----------- ------- ----------- ------- ----------- ------- : $L_2$-Errors and empirical convergence rates (powers of $h$) for Example 2 when $t=1$, with $\gamma=2/\beta$ and different choices of $\alpha$. (Recall that $N$ denotes the number of degrees of freedom in the finite element triangulation.)[]{data-label="tab: mixed"} Example 2 {#example-2 .unnumbered} --------- In our second example, we impose mixed boundary conditions: homogeneous Dirichlet for $\theta=0$ or $r=1$, and homogeneous Neumann for $\theta=\pi/\beta$. As the initial data we choose the first eigenfunction of $A=-K\nabla^2$, $$u_0(x,y)=J_{\beta/2}(\omega r)\sin(\tfrac12\beta\theta),$$ where $\omega$ is the first positive zero of the Bessel function $J_{\beta/2}$. We put $f=0$ so (recalling that our choice of $K$ means that the corresponding eigenvalue equals 1) the solution is $u(x,y,t)={\mathcal{E}}(t)u_0=E_\alpha(-t^\alpha)u_0(x,y)$, and choose $\gamma=2/\beta=3$ so that ${\epsilon_{\text{mix}}}(h,\gamma)$ is of order $h\log(1+h^{-1})$; see . Since $A^ru_0\propto u_0\in L_2(\Omega)$ for all $r>0$, we conclude from Theorem \[thm: u0 smooth\] that the $L_2$-error $\|u_h(t)-u(t)\|$ is of order $h^2\log^2(1+h^{-1})$ uniformly for $0\le t\le T$. Figure \[fig: t dependence\] confirms this behaviour in the case $\alpha=1/2$. Finally, Table \[tab: mixed\] shows that at a fixed positive time $t=1$ the $L_2$-error does not vary much with $\alpha$. [^1]: This work was supported by the Australian Research Council grant DP140101193.

--- abstract: 'Recent empirical studies have confirmed the key roles of complex contagion mechanisms such as memory, social reinforcement, and decay effects in information diffusion and behavior spreading. Inspired by this fact, we here propose a new agent–based model to capture the whole picture of the joint action of the three mechanisms in information spreading, by quantifying the complex contagion mechanisms as stickiness and persistence, and carry out extensive simulations of the model on various networks. By numerical simulations as well as theoretical analysis, we find that the stickiness of the message determines the critical dynamics of message diffusion on tree-like networks, whereas the persistence plays a decisive role on dense regular lattices. In either network, the greater persistence can effectively make the message more invasive. Of particular interest is that our research results renew our previous knowledge that messages can spread broader in networks with large clustering, which turns out to be only true when they can inform a non-zero fraction of the population in the limit of large system size.' author: - Pengbi Cui - Ming Tang - Zhixi Wu title: 'Message spreading in networks with stickiness and persistence: Large clustering does not always facilitate large-scale diffusion' --- Over the last few years, many empirical works [@contagion; @decay3; @science; @attention; @contagion2; @origin1; @origin2] or practical model [@zhou; @model2] have identified the strong relevance of complex contagion mechanisms such as memory effect, social reinforcement and decay effects to information diffusion or behavior spreading. On account of memory effect, the previous contact activities can affect the current spreading process [@memory; @attention]. Specifically, individual’s selection of message items can be naturally expedited by the increasing frequencies of the same choices of other people if they find the items interesting or crucial enough [@science; @zhou; @decay]. This is usually interpreted as the results of social reinforcement [@rein1; @model2; @theory2]. On the other hand, there are an increasing amount of new messages an individual is facing every day in modern real life, whereas the attention and processing abilities of people are finite and saturated [@attention; @decay; @attention1]. The novelty of a message usually trend to fade with time and hence the attention people pay to it, which is normally described as decay effects [@burst2; @attention; @contagion; @decay]. It is shown that the social reinforcement effect could be weakened or even counterbalanced by decay effects [@attention; @decay; @contagion2]. Although the competition between social reinforcement and decay effects has been emphasized and used as a guideline to measure the natural time scale that attention fades away [@decay], to our best knowledge few works have been attempted to model explicitly the competition and memory effect, and study deeply how it shapes the spreading of information on complex networks. Here we want to point out that the three mentioned effects in information spreading are quite different from those have been considered in the studies on Naming Game (NG) and Category Game (CG), since either NG or CG is a two-step multi-state negotiation process [@ng1; @ng2; @cg1; @cg2], whereas information spreading is not. First, herein memory effect performs as the storing of the times of contact of people with recipients of information [@science; @zhou], rather than the possible words (or names) for the object (or a category) in NG (or CG) [@ng1; @ng2; @cg1; @cg2]. Second, decay effect in information spreading reflects the decay of people’s interest or attention in a message owing to the competition with other news or stories [@attention; @decay], contrary to the NG (CG) in which it means the decrease of the number of different words used in the system (or average number of words per category) [@ng1; @ng2; @cg1; @cg2]. Third, unlike the phenomenon that an hearing would have more opportunities to add (or remain) one word only if more selected speakers try to transmit the same one to it [@ng1; @ng2; @cg1; @cg2], the reinforcement effect in information diffusion indicates the more simple situation that the more neighbors adopting the message, the higher likelihood an individual following them [@science; @contagion2]. Next, the big challenge we are confronted with is the possibility of modeling and studying the message spreading along with both social reinforcement and decay effects based on the memory effect. Recent researches [@origin1] have shown that the variation in the ways that different information spread is attributed to not only the stickiness – the probability of information adoption is mainly dominated by the first few exposures [@origin2; @origin1], but also the persistence – the relative extent to which more repeated exposures to the message continue to have durative effect. Similar results especially the exposure response behaviors were also confirmed by a lot of empirical studies [@contagion2; @origin2; @attention]. The two mechanisms, stickiness and persistence, thus enable us to quantitatively study the joint action of the three effects together. At the same time, the structures of complex social systems can be characterized by complex networks, on which many spreading activities may take place, ranging from the spreading of epidemics [@tang1; @tang2; @tang3; @tang4], the diffusion of behaviors and news [@science; @zhou], to the promotion of technique innovations [@innovation], etc. Consequently, motivated by the empirical studies [@science; @zhou; @attention; @contagion2; @origin1; @origin2; @decay] mentioned above, we propose a new agent-based model offering an opportunity to explore the impact of social reinforcement and decaying effects quantified by stickiness and persistence on the message (information) diffusion on various networks. In the presence of strong decay effects, we find that a message is more likely to outbreak (i.e., it can reach a non-zero fraction of the population in the thermodynamic limit) on the tree-like networks such as scale-free (SF) networks and Erd[ő]{}s-R[é]{}nyi (ER) random networks rather than on the regular lattices (RLs). Specifically, a message can spread broader in the RLs than that in the tree-like networks only if it can outbreak. The critical behaviors of the diffusion process can be reasonably estimated by the bond-percolation theory considering spatial correlations of the underlying networks through which message diffuses. In addition, we develop a verification approximation, whose solutions confirm well the non-negligible role of the dynamical correlations between transmission events in the RLs. Results {#results .unnumbered} ======= Here, we first carry out extensive simulations for the agent-based model of message diffusion on square lattice. We then compare the simulation results with the predictions from the analytical bond percolation theory and verification method involving time correlations of the spreading events. Finally we extend our model and analytical methods to other networks such as RLs, SF networks, and ER networks to validate the robustness of our findings. **Message diffusion on square lattice.** \[square\] We first consider the message diffusion on a square lattice of size $N=L\times L$ with periodic boundary conditions. The message starts spreading from the center node (selected as the seed), while all the others are in the susceptible state (i.e., they hear nothing about the message). ![**The time evolution of spatial patterns,** for two different values of stickiness (a1) $a=0.40$, (a2) $a=0.45$ (bottom panel) where $b=0$; and for two values of persistence (b1) $b=-1.00$, (b2) $b=1.00$ (bottom panel) where $a=0.45$. Red sites represent recovered or alerted nodes, bright green sites represent infected ones, and blue sites denote susceptible nodes. Other parameters are chosen as $n_{s}=2$ and $N=101\times 101$.[]{data-label="spatial"}](spatial.jpeg){width="\textwidth"} To intuitively grasp the roles of stickiness and persistence, we begin by presenting the time evolution of spatial patterns of message spreading in Fig. \[spatial\]. The message with stickiness $a=0.40$ (see the Methods for the precise definitions of $a$, $b$ and other parameters) spreads in an irregular manner (see Fig. \[spatial\](a1)). By comparison, the message with a slightly stronger stickiness $a=0.45$ diffuses outward to susceptible areas in a quasi-circular manner with a broader rim of informed (infected) individuals (see Fig. \[spatial\](a2)). This indicates that messages with different strength of stickiness could give rise quite different spreading patterns and behaviors. Figs. \[spatial\](b1) (b2) and Supplementary Fig. S1 show that the persistence $b$ also affects considerably the whole spreading size, by governing the number of the isolated susceptible islands (blue domains surrounded by red areas, the emergence of these islands arises from the fact that continually increasing number of infected neighbors fail to infect those individuals owing to small persistence). The above arguments suggest that the stickiness and persistence have great but different influences on the spreading of message. ![**The evolution of proportions of the transmission events.** The parameters are chosen at (a) subcritical point $a=0.20$, $b=0.20$; (b) critical point $a=0.35$, $b=0.20$; and (c) supercritical point $a=0.43$, $b=0.20$.[]{data-label="alround"}](alpharound.jpeg){width="\textwidth"} We next explore the critical behaviors of message diffusion for various stickiness and persistence, by providing a quantitative assessment of the message burstiness. By means of the theory of non-equilibrium phase transition in statistical physics which has been successfully generalized to study the epidemic dynamics, previous studies [@variance1; @variance2] on spreading dynamics have shown that the fluctuation of the order parameter is divergent at the critical point. We thus use the procedure proposed in [@variance2] to numerically determine the critical areas. To be more specific, a series of variabilities $v(a,b)$ are firstly obtained as $$\label{eq:variance} v(a,b)=\frac{\sqrt{<\rho^{2}_{R}(a,b)>-<\rho_{R}(a,b)>^{2}}}{<\rho_{R}(a,b)>},$$ where $\rho_{R}$ and $v(a,b)$ denote, respectively, the density of recovered individuals in the population and the relative standard deviation (RSD) at parameter point $(a,b)$. There exists a maximum variability $v_{max}(b)$ for each value of $b$ when varying $a$ from $0$ to $0.5$, and the values of $v_{max}(b)$ with $b\in[-1~1]$ can be used as the numerical estimation of the threshold position. Although a bond-percolation process can be mapped to the SIR model [@prl; @bond], its extension to the current model is not straightforward. First, in our model, the transmission probability that a susceptible individual approves the message varies with the times he has received it from his infected neighbors (i.e., the number of informed neighbor he has had). Second, the time correlations between different transmission events $E_{n}$ [@prl] (the transmission event $E_{n}(t)$ represents that an individual, who has received the message at least once before, successfully approves the message when he has received it another $n$ times until time $t$). To confirm the existence of the time correlation, in Fig. \[alround\] we compare the dynamic proportions of the four transmission events ($\alpha_{n}(a,b,t)=\frac{\omega_{n}(a,b,t)}{\sum^{\langle k\rangle -1}_{m=0}\omega_{m}(a,b,t)}$) obtained from numerical simulations with those predicted by the bond percolation considering spatial correlations ($\beta_{n}(a,b,t)=\frac{q_{n}T_{n}(a,b)}{\sum^{\langle k\rangle -1}_{m=0}q_{m}T_{m}(a,b)}$ and see the Methods section for definitions of $q_{n}$ and $T_{n}(a,b)$) for three parameter points $(a,b)$. Here, $\omega_{n}(a,b,t)$ represents the occurrence frequency of $E_{n}(t)$ obtained from numerical simulations, and $\langle k\rangle$ denotes the average degree of the networks. In Figs. \[alround\] (b) and (c), we see that $\alpha_{0}(t)$ ($\alpha_{2}(t)$) is greater (less) than $\beta_{0}(t)$ ($\beta_{2}(t)$) during the spreading process, whereas $\alpha_{1}(t)$ ($\alpha_{3}(t)$) is equal or close to $\beta_{1}(t)$ ($\beta_{3}(t)$). The reason is that the existence of time correlations of the transmission events $E_{n}(t)$ can determine whether the subsequent events $E_{m}(t)$ ($m>n$) happen or not in the spreading process. If $E_{n}(t)$ does not happen, the events $E_{m}(t)$ ($m>n$) will probably happen; otherwise $E_{m}(t)$ ($m>n$) will never happen since an informed individual transmits the message only one time and then becomes recovered (i.e., completely ignores the message) forever. Consequently, $E_{0}$ ($E_{2}$) contributes more (less) than predicted by percolation theory to the spreading course (also see Supplementary Fig. S2). To overcome the challenge, we develop a verification approximation involving the time correlations of the transmission events, besides the spatial correlations originating from the spatial structure of the lattice [@prl]. It is necessary to mention that the discrete-time synchronous transmission of the message enables us to avoid concerning about the additional synergistic effect [@synergy]. Moreover, Figs. \[alround\](b) and (c) show that $\alpha_{n}(a,b,t)$ is dynamically stable, which shows that this correlation always exists. It allows us to adopt average values of the four indices $\alpha_{n}(a,b,t)$ at the critical regions for the verification approximation. ![**The phase diagram of the message spreading.** The real numerical critical boundaries (crosses) are obtained by Eq.  for various $n_{s}$ on square lattice of size $N = 101\times 101$. For comparison, analytical boundary (black line) and verification boundary (dash black line) for $n_{s}=2$ are also shown (the cases for $n_{s}=3,~4,~5$ do not allow us to analytically identify the critical lines). Herein, we select a narrow parameter ranges $b\in[-1,~1]$ and $a\in[0.32,~0.36]$ containing the numerical critical boundary for the calculation of verification threshold (see more detailed method in the Methods section).[]{data-label="latthre"}](trelattice.jpg){width="\textwidth"} Based on the proposed methods in the Methods section and Eq. , we yield both the analytical prediction and the verification threshold for $n_{s}=2$, plus the numerical results for various $n_{s}$, as depicted in Fig. \[latthre\]. We note that the numerical thresholds stay at $a\approx 0.32$, which is mainly determined by the stickiness of message (i.e., the parameter $a$), regardless of the values of $b$ and $n_{s}$. This means that most informed individuals are actually infected by their first one or two infected neighbors (also see Supplementary Figs. S2–S6). Furthermore, the numerical estimations are fairly reproduced by the bond-percolation theory. Comparing the analytical boundary, the verification approximation involving both spatial and time correlations gives a higher accurate estimation than the bond-percolation method considering only spatial correlations (the dashed black line is clearly closer to the numeric markers than the black solid line is). **Message diffusion on regular lattice networks and regular random networks.** \[hm\] Centola’s work [@science] concludes that social behaviors can spread farther and faster across clustered-lattice networks than across corresponding regular random networks (RRNs), owing to the strong social reinforcement induced by clustered ties. RRNs are networks that all nodes have exactly the same degree while links are randomly distributed among nodes, avoiding self-connections and multiple connections. To check whether the findings by Centola are still fulfilled for information diffusion, we further investigate our model defined on the RLs (Hexagonal network and Moore network) and the RRNs. ![[]{data-label="lhgap"}](lhgap.jpg){width="\textwidth"} To get a comparison, we present in Fig. \[lhgap\] the differences of the final size of recovered population on the two networks with the same average degree. The blue areas characterize the parameter regions where the conclusion of Centola’s experiment (that the information spreads farther across the RLs than across the corresponding RRNs) does not hold. The violation is attributed to the presence of strong decay effects ($b<0$) in the vicinity of the critical regions. Specifically, the outbreaks of message can happen more easily in the RRNs than that in the RLs for negative persistence owing to that strong decay effects outcompetes the weak reinforcement effect (also see Supplementary Fig. S7, strong reinforcement effect (decay effects) is reflected by large stickiness and/or positive persistence (negative persistence) in our model). As $a$ and $b$ get larger, things turn out differently, the message is able to seize a larger population in the RLs, which is accordant with the anticipation of Centola’s experiment. This means that high level of clustering created by redundant ties that linked each node’s neighbors to one another in the RLs strengthens the reinforcement effect, and hence facilitates the diffusion of the message [@science]. Moreover, larger $n_{s}$ improves the performance of stickiness in facilitating the message diffusion, making social reinforcement effects be the most prominent for the RLs. Consequently, the blue areas shrink with increasing $n_{s}$. Thus, the above differences investigated indicate that network topology and $n_{s}$ (which can be regarded as one of the intrinsic characteristics of the message) simultaneously determine the effects of stickiness and persistence on the spreading dynamics. ![**The phase diagrams of the message spreading on RLs and RRNs.** The underlying networks are (a) RRN with $\left\langle k\right\rangle=6$, (b) RRN with $\left\langle k\right\rangle=8$, (c) Hexagonal lattice, and (d) Moore lattice. The predictions from the bond-percolation theory (solid lines) and verification approximations (dashed lines) for $n_{s}=2$ are illustrated to compare with the simulation data (markers) for various $n_{s}$. To get the verification trajectory on Hexagonal lattice (and Moore lattice), we select two parameter regions containing the numerical thresholds (see more detailed method in the Methods section). One region is $b\in[-0.3~1.0]$ and $a\in[0.06~0.24]$; the other one is $b\in[-0.3~1.0]$ and $a\in[0.01~0.19]$ in Hexagonal lattice. One region is $b\in[-1.0~-0.3]$ and $a\in[0.23~0.24]$; the other one is $b\in[-1.0~0.3]$ and $a\in[0.15~0.18]$ in Moore lattice.[]{data-label="rhthre"}](rhthre.jpg){width="\textwidth"} Using Eq. , we obtain the numerical thresholds for various values of $n_{s}$. According to the methods described in the Methods section, we can yield the analytical (theoretical) thresholds of message diffusion on both the RRNs and the RLs, in addition to the verification thresholds on the RLs for $n_{s}=2$ (Supplementary Fig. S10 and Fig. 11 show that the time correlations between the transmission events in Hexagonal lattice and Moore lattice are noticeable). In the case of RRNs (Figs. \[rhthre\](a) and (b)), we observe that the positions of the critical boundaries are mainly determined by both the stickiness and $\left\langle k\right\rangle$ instead of the persistence and $n_{s}$, on account of weaker social reinforcement [@science; @prl] resulting from the low clustering coefficient. Unlike the case of RRNs, the theoretical analysis by means of bond-percolation theory gives rather accurate predications of the position of the threshold with strong persistence, but not for negative persistence (i.e, with the presence of strong decay effects) on the RLs. When the underlying networks for message diffusion are Hexagonal lattice and Moore lattice, more available edges for message spreading will further strengthen the role of persistence in message diffusion, especially with stronger social reinforcement (positive $b$). That enables all transmission events involve in the diffusion (see Supplementary Figs. S10–S13), so that $\beta_{n}(a,b,t)$ gets close to $\alpha_{n}(a,b,t)$ (see Supplementary Figs. S10 and S11). On the other hand, the trajectories obtained by verification approximation are in good agreement with the simulations, from which one can conclude that the effect of time correlations of transmission events is indeed general on the RLs. Additionally, the message steps forward to arrive in half of the neighbors of the same host on the RLs by flowing through almost $\frac{\left\langle k\right\rangle}{2}$ edges connecting it. This makes the message with positive persistence ($b>0$) outbreak more easily in the presence of denser local connections, and the persistence thus imposes a greater influence on outbreaks of message on RLs with larger average degree for small $n_{s}$ ( $\frac{n_{s}}{\left\langle k\right\rangle}<\frac{1}{2}$). The message also has reached a saturation state when the subsequent events $E_{i}$ ($i>\frac{\left\langle k\right\rangle}{2}$) happen (see Supplementary Figs. S14–S21). Also in RLs, higher $n_{s}$ limits the effect of persistence, and the phase transitions are determined by not only the topologies of the networks but also the stickiness and persistence of the message. In addition, the reinforcement effect begins to work as the message is bursting and prevailing on both the RLs and RRNs. Therefore, the results for positive persistence near the thresholds (see Supplementary Figs. S12–S24) elucidate the actual phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"), where the majority do not believe the message as a truth until at least three neighbors have tried to transmit this message to them. **Message diffusion on SF networks and ER networks.** \[eb\] To further check the robustness of our above findings, we finally investigate our message diffusion model on SF and ER networks with $\langle k\rangle=6, 8, 10, 12$. Since two or more transmission events fail to last over the long time at the critical points (see Supplementary Fig. S24), we do not take the time correlations into consideration in theoretical analysis. Compared with the simulation data, the analytical results for SF networks and ER networks with different average degrees show that the theoretical approaches are already sufficient to give fairly precise expressions of the outbreaks of message completely determined by the stickiness $a$ (see Supplementary Fig. S25). Nevertheless, the persistence also partly boosts its impact on the size of message diffusion as $\left\langle k\right\rangle$ gets larger (see Supplementary Fig. S26). ![**The differences of spreading sizes between the RLs and SF (ER) networks ($\rho_{SF/ER}-\rho_{RL}$).** (a,b) the difference $\rho_{SF}-\rho_{RL}$, (c,d) the difference $\rho_{ER}-\rho_{RL}$. (a,c) The average degrees of the networks are $\left\langle k\right\rangle=6$ (a,c), and $\left\langle k\right\rangle=8$ (b,d), respectively. Here, $\rho_{SF}$ ($\rho_{ER}$, $\rho_{RL}$) represents the recovered density in SF networks (ER networks, RLs). Other parameter: $n_{s}=2$.[]{data-label="brhmgap"}](brhmgap.jpg){width="\textwidth"} Fig. \[brhmgap\] displays evident differences of the final spreading sizes on the RLs and SF (ER) networks with the same degree. The spreading sizes are larger in SF (ER) than those in the RLs at the parameter regions where the message has already outbroken on SF (ER) but not yet on the RLs, owing to the hubs and shorter shortest paths in SF (ER) [@shortest] (see Supplementary Fig. S25 for the critical boundaries of message on SF and ER). Just for strong persistence, denser local connectivity of the RLs can make an invasion take place more easily. As $a$ and $b$ increase beyond the critical boundaries displayed in Fig. \[rhthre\], the message can capture a larger population again on the RLs, despite of the existence of hubs the short characteristic path length in the SF (ER) networks. The reason is that very smaller clustering coefficient gives rise to weak social reinforcement effect [@science; @report], which again leads to weak performance of persistence in promoting the spread of message on the SF (ER) networks. Our results indicate that the role of reinforcement effect is more important than that of hubs or shortest paths in facilitating message spreading only when the message outbreaks on the RLs. Otherwise, especially for the presence of decay effects (negative persistence) the facilitation of hubs and shortest paths to the diffusion on SF (ER) cannot be neglected. In addition, the results for positive persistence on SF and ER networks can also be treated as evidence of the mechanism “Three men make a tiger” (see Supplementary Fig. S27 and Fig. S28). Discussion {#con .unnumbered} ========== In conclusion, taking into account social reinforcement and decay effects based on memory effect in reality, we have proposed a new agent-based message spreading model with stickiness and persistence, and carried out extensive computer simulations of our model on various types of networks. By means of the relative standard deviation method and the bond percolation theory involving the spatial correlations, we are able to determine numerically and analytically the positions of critical boundaries. Moreover, the remarkable accuracy of verification approximation involving the time correlations between different transmission events validates the wide existence of such correlations for the message diffusion on regular lattices. Our preliminary results show that in RLs, the persistence depends greatly on the position of inflection point $n_{s}$ and average degree $\left\langle k\right\rangle$ of the underlying networks, and begins to play a pivotal role in the spreading process with increasing $\left\langle k\right\rangle$ owning to the emergent large clustering coefficient [@science]. Stronger social reinforcement arising from larger clustering coefficient leads readily to stronger infectivity of the message, which can invade a great number of susceptible individuals in the RLs, confirming the conclusion of Centola’s work. By comparison, in tree-like networks such as RRNs, SF and ER, the critical thresholds of message diffusion are only dominated by the stickiness, and both the hubs and the short characteristic path length facilitate the outbreaks of message in the presence of decay effects. It worth emphasizing that the results presented in this paper has successfully substantiated the phenomenon “Three men make a tiger“ (or ”A lie, if repeated often enough, will be accepted as truth"). Placing our study in the context of social media, the hub nodes actually play a role of broadcasters, advertisements and so on, which are very important for the large scale spreading of information. However, the intrinsic contents of messages and their adaptation to hosts are extremely relevant in determining the message diffusion. In particular, the results for the message diffusion on the SF and ER networks constitute a proof that exposure to mass media can favour the outbreaks of behaviors, news or messages, despite the decay effects, only if the stickiness of the message is large enough. On the other hand, the RLs are efficient in taking advantage of social reinforcement effects to promote the global spreading of the message, owning to more cluster ties (or local pressures) that function as form of ’initial groups’ or ’small groups’ through interpersonal communication or ’machine-interpersonal communication’ [@fed]. It is also verified by our research that the local, personal communication is irreplaceable to lead to propagation of message despite of the developed media industry today, from the perspective of communication. Recent studies have started considering the memory effect [@science; @zhou], social reinforcement and decay effects [@attention] in information spreading. The mechanisms were yet investigated in isolation. Our work is the first attempt to account explicitly for the three key mechanisms together, and to evaluate the joint action of them by quantifying their effects as stickiness and persistence. It provides a quantitative guideline for future social experiments for message spreading. In reality, the ways in which information spread may be very complicated. In the present study, we do not capture the difference of individual-level preference [@leader] that might have also influenced their decisions to adopt one message. For example, in an online social network such as Twitter, individuals may prefer different hashtags, and significant variations in the ways that the hashtags on different topics spread were observed [@origin2]. In addition, one need to concern about the diverse cultural and societal backgrounds [@culture] which would lead to the different styles by which individuals contact with the medium, or even hamper communications among different groups of members [@fed]. Moreover, the volatilities of complex contagion of controversial topics, psychological status of individuals [@media] reveal that the status of the individuals in the communication systems are time dependent, which should be addressed in future research. Methods {#methods .unnumbered} ======= **Message spreading model with stickiness and persistence.** The message spreading model is implemented on a network consisting of $N$ nodes and $E$ edges, where the nodes represent the individuals in a population and the edges the social interactions among them, through which information propagates. Each individual is allowed to be in one of three states at each time step: (i) Susceptible (or uninformed) state—the individual has not yet heard the message or is aware of the news but not willing to transmit it. (ii) Infected state—the individual catches the message and forwards it to all his nearest neighbors. (iii) Recovered state—the individual will never transmit the message any more after having transmitted it once before. Specifically, the information propagation is modeled in a probabilistic framework at individual level [@zhou; @model2]. A susceptible individual $i$ will adopt the message with a probability $\lambda_{n^{o}}(a,b)$, given that he has heard it from his informed neighbors $n^{o}$ times (i.e., $n^{o}$ informed neighbors he has had), plus the first time. In detail, $n^{o}=n+1$ when individual $i$ has owned at least one informed neighbors, otherwise $n^{o}=0$. Here $\lambda_{n^{o}}(a,b)$ is a linear piecewise function of the times he has received message from his informed neighbors: $$\begin{aligned} \label{eq:lambda} &\lambda_{n^{o}}(a,b) = \begin{cases} \min\{an^{o}, 1\}, \quad 0\leq n\leq n_{s};\\ \min\{bn^{o}+n_{s}(a-b),1\}, \quad n_{s}< n\leq k_{i}; \end{cases}\\ \notag &\text{and} \quad \lambda_{n^{o}}(a,b) = 0, \qquad \text{if} \quad \lambda_{n^{o}}(a,b)<0; \end{aligned}$$ where $k_{i}$ is the degree of node $i$. $n_s$ is the inflection point beyond which persistence is the dominant factor for the infection probability of message, and the parameters $a$ (stickiness) and $b$ (persistence) characterize how $\lambda_{n^{o}}(a,b)$ change with $n^{o}$, as illustrated in Fig. \[model\]. Since empirical data [@science; @origin2; @origin1] have shown that social reinforcement sets in such that initial exposures generally increase infection probability, the parameter $a$ should be non-negative when $n^{o}\leq n_{s}$. For $n^{o}> n_{s}$, the competition between social reinforcement and decay effects, characterized by the parameter $b$, will be taken into account for the message adoption. If the reinforcement is strong enough the individual will be more likely to adopt the message with increasing $n^{o}$ ($b>0$). Otherwise, even if many infected neighbors try to transmit the information to the focal individual $i$, the multiple exposures will lead to a decreased probability for the information adoption ($b<0$). In the present study, we set $b\in[-1,1]$ and $a\in(0, 0.5]$ so that the spreading dynamics of the message can be comprehensively investigated. For simplicity, we do not consider the diversity of individuals’ response to the message, and all individuals behave identically with the same values of parameters $a$ and $b$. ![**The adopting probability of message as a function of $n^{o}$.** The degree of stickiness and persistence are quantified as $an_{s}$ and $b$, respectively. $n_{s}$ denotes the position of inflection point.[]{data-label="model"}](model.jpeg){width="\textwidth"} We perform Monte Carlo (MC) simulations with synchronous updating of the states of all the individuals. Each MC step consists of the following three procedures: (i) All susceptible individuals decide whether or not to adopt the message with probability $\lambda_{n^{o}}(a,b)$; (ii) If an individual adopts the message, he will try to transmit what he has approved to all his nearest susceptible neighbors in the next step, and then becomes recovered immediately; (iii) Otherwise, the susceptible individuals will wait to repeat the procedure (i) in the following MC steps. The above elementary spreading processes are repeated $T^{'}_{S}=500$ steps until there are no infected individuals anymore in the population. **Theoretical analysis of the model.** For the occurrence probability of transmission event $E_{n}$, $T_{n}(a,b)= 1-e^{-\lambda_{n+1}(a,b)\tau}\notag = 1-e^{-\lambda_{n+1}(a,b)}$ with $\tau =1$ [@newman]. We consider the spatial correlations that affect the process of diffusion, but do not yet influence the critical behaviour of the message spreading [@prl]. The transition point from susceptible to infected phase is determined by [@prl; @newman] $$\begin{aligned} T_{C}=\left\langle T\right\rangle, \label{eq:threshold}\end{aligned}$$ where $\left\langle T\right\rangle$ and $T_{C}$ are, respectively, the mean transmissibility and the critical topology-dependent bond-percolation threshold. On the other hand, the mean transmissibility can be gotten as $$\begin{aligned} \left\langle T\right\rangle & = & \sum^{\left\langle k\right\rangle -1}_{n=0}q_{n}T_{n}(a,b), \label{eq:meanthreshold}\end{aligned}$$ where $q_{n}=\binom{\left\langle k\right\rangle-1}{n}p^{n}(1-p)^{\left\langle k\right\rangle-n-1}$ is the probability that the recipient has other $n$ ($n=0, 1, 2, 3$) infected neighbors except for the one chosen beforehand when considering the spatial correlations ($p$ is the probability that one nearest neighbor of the focal individual is in infected state). Combine Eqs. , , , and $T_{n}(a,b) = 1-e^{-\lambda_{n+1}(a,b)}$, the mean transmissibility for the discrete case reads as $$\begin{aligned} \label{eq:percolation2} \left\langle T\right\rangle & = \sum^{\left\langle k\right\rangle-1}_{n=0}q_{n}(1-e^{-\lambda_{n+1}(a,b)}) \notag \\ & = \sum^{\left\langle k\right\rangle -1}_{n=0}q_{n}-\sum^{n_{s}-2}_{n=0}q_{n}e^{-\lambda_{n+1}(a,b)}-\sum^{\left\langle k\right\rangle -1}_{n=n_{s}-1}q_{n}e^{-\lambda_{n+1}(a,b)} \\ & = 1-\sum^{n_{s}-2}_{n=0}q_{n}e^{-(n+1)a}-e^{-n_{s}a}\sum^{\left\langle k\right\rangle -1}_{n=n_{s}-1}\binom{\left\langle k\right\rangle -1}{n}e^{-(n-1)b}.\notag\end{aligned}$$ Then we have $$\begin{aligned} \theta_{2}e^{-2a}+\theta_{1}e^{-a}+\theta_{0}=0, \label{eq:lattper}\end{aligned}$$ where $\theta_{2}=\sum^{\left\langle k\right\rangle-1}_{n=1} \frac{(\left\langle k\right\rangle-1)!}{n!(\left\langle k\right\rangle-n-1)!} p^{n}(1-p)^{\left\langle k\right\rangle-n-1} e^{(1-n)b}$, $\theta_{1}=(1-p)^{\left\langle k\right\rangle-1}$ and $\theta_{0}=T_{C}-1=-\frac{1}{2}$. The positive root is selected as the theoretical prediction $$a(b)=\ln(\frac{2\theta_{2}}{\sqrt{\theta^{2}_{1}-4\theta_{2}\theta_{0}}-\theta_{1}}). \label{eq:lattre}$$ In SF networks, the critical point $T_{C}$ beyond which the message can reach a finite faction of the population can be obtained [@newman] as $T_{c}=\frac{\left\langle k\right\rangle}{\left\langle k^{2}\right\rangle -\left\langle k\right\rangle}$. Since RRNs and ER networks are generated by connecting randomly selected pair of nodes, $T_{c}=\frac{1}{\left\langle k\right\rangle-1}$. Due to the randomness of connections in these networks, each edge of a neighbor of one host can probably connected to any other $N-2$ individuals. Therefore, the transmission of message from an informed neighbor to the susceptible host (i.e., message flows through the edge in the system) will happen only if there is at least one infected individual in the left $N-2$ ones to transmit the message to the recipient. In other words, the probability that the message flow can reach the susceptible host through the edges connecting to the recipient is $p=\frac{1}{N-2}$ for the three networks (i. e., the RRNs, SF networks, and ER networks). Also, two or more transmission events fail to last over the long time at the critical points (see Supplementary Fig. S8, Fig. S9, and Figs. S25), so the time correlations are ruled out in the theoretical analysis for the cases of RRNs, SF networks, and ER networks. In the regular lattices, $p=\frac{1}{\left\langle k\right\rangle-1}$ since each one has exact $\langle k\rangle$ specific neighbors. Specifically, $p=\frac{1}{3}\approx0.333$ for square lattice with von Neumann neighborhood, $p=\frac{1}{5}$ for Hexagonal lattice, and $p=\frac{1}{7}\approx0.143$ for the lattice with Moore neighborhood. The bond percolation threshold $T_{C}=\frac{1}{2}$ for a square lattice, $T_{C}=0.347$ for Hexagonal lattice, and $T_{C}=0.232$ for square lattice with Moore neighborhood [@prl; @tc]. **Verification approximation of information threshold.** In analogous to $q_{n}(a,b)$, we first set $Q^{'}_{n}(a,b) (n=0,~1,~2,~3)$. According to Eq. , $Q^{'}_{n}(a,b) (n=0,~1,~2,~3)$ can be calculated numerically by counting the relative number of successful attacks (i.e, transmission events) $\alpha_{n}(a,b)$ from infected neighbors to hosts for given values of $a$ and $b$ and equating this to $\alpha_{n}(a,b)=\frac{\sum_{t=0}^{T^{'}_{S}}\omega_{n}(a,b,t)}{\sum^{\langle k\rangle -1}_{m=0}\sum^{T^{'}_{S}}_{t=0}\omega_{m}(a,b,t)}=\frac{Q^{'}_{n}(a,b)T_{i}(a,b)}{<T(a,b)>}$. In other words, $\alpha_{n}(a,b)$ represents numerically the proportion of $E_{n}$ occurring at parameter point $(a,b)$. Furthermore, $Q^{'}_{n}(a,b)$ can be normalized as $$Q_{n}(a,b)= \frac{Q^{'}_{n}(a,b)}{\sum^{\langle k\rangle -1}_{m=0}Q^{'}_{m}(a,b)}= \frac{\frac{\alpha_{n}(a,b)<T(a,b)>}{T_{n}(a,b)}} {\sum^{\langle k\rangle -1}_{m=0}\frac{\alpha_{m}(a,b)<T(a,b)>}{T_{m}(a,b)}}= \frac{\frac{\alpha_{n}(a,b)}{T_{n}(a,b)}} {\sum^{\langle k\rangle -1}_{m=0}\frac{\alpha_{m}(a,b)}{T_{m}(a,b)}}.$$ $Q_{n}(a,b)$ actually represents the probability that the recipient owns $n$ ($n=0, 1, 2, 3$) infected neighbors expect for the preselected informed one, involving both the spatial and time correlations of the message diffusion. Herein, we select different parameter regions containing the numerical critical boundaries for corresponding lattices with $n_{s}=2$ for the calculation of $Q_{n}(a,b)$. We average all non-zero $Q_{n}(a,b)$ in the selected parameter ranges for the expected indices $Q_{n}$, and substitute them into the equation $$\begin{aligned} T_{C} = \left\langle T\right\rangle=\sum^{\langle k\rangle-1}_{n=0}Q_{n}T_{n}(a,b)\end{aligned}$$ to get the verification thresholds. Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported by the National Natural Science Foundation of China under Grants No. 11135001 and No. 11105025, and by the Fundamental Research Funds for the Central Universities under Grant No. lzujbky-2014-28. [0]{} Centola, D. $\&$ Macy, M. Complex contagions and the weakness of long ties. [*Am. J. Sociol.*]{} **113**, 702–734 (2007). Portrat, S., Barrouillet, P. $\&$ Camos, V. Time-related decay or interference-based forgetting in working memory. [*J. Exp. Psychol. Learn Mem. Cogn.*]{} **34**, 1561 (2008). Centola, D. The spread of behavior in an online social network experiment. [*Science*]{} **329**, 1194 (2010). Weng, L., Flammini, A., Vespignani, A. $\&$ Menczer, F. Competition among memes in a world with limited attention. [*Sci. Rep.*]{} **2**, 335 (2012). Hodas, N. O. $\&$ Lerman, K. The Simple Rules of Social Contagion. [*Sci. Rep.*]{} **4**, 4343 (2014). Cosley, D., Huttenlocher, D. P., Kleinberg, J. M., Lan, X.-Y. $\&$ Suri, S. Sequential influence models in social networks. In [*Proc. 4th Int. AAAI Conf. on Weblogs and Social Media*]{}, ICWSM$^{^{,}}$10, (ACM, New York, NY, USA, 2010). Romero, D. M., Meeder, B. $\&$ Kleinberg, J. Differences in the mechanics of information diffusion across topics: Idioms, political hashtags, and complex contagion on twitter. In [*Proc. 20th Int. Conf. on World Wide Web*]{}, WWW$^{^{,}}$11, 695–704 (ACM, New York, NY, USA, 2011). Lu, L., Chen, D. B. $\&$ Zhou, T. Small world yields the most effective information spreading. [*New J. Phys.*]{} **13**, 123005 (2011). Krapivsky, P. L., Redner, S. $\&$ Volovik, D. Reinforcement-driven spread of innovations and fads. [*J. Stat. Mech-Theory E.*]{} **12**, 1104 (2011). McQuail, D. $\&$ Windahl, S. [*Communication Models for the Study of Mass Communications*]{} (Longman, London, 1993). Wu, F. $\&$ Huberman, B. A. Novelty and collective attention. [*Proc. Nat. Acad. Sci.*]{} **104**, 17599–17601 (2007). Centola, D., Egu[í]{}luz, V. M. $\&$ Macy, M. W. Cascade dynamics of complex propagation. Physica A **374**, 449–456 (2007). Watts, D. J. $\&$ Dodds, P. S. [*Influentials, networks, and public opinion formation.*]{} [*J. Consum. Res.*]{} **34**, 441-458 (2007). Romero, D. M., Galuba, W., Asur, S. $\&$ Huberman, B. A. Influence and passivity in social media. In [*Proc. 20th Int. Conf. on World Wide Web*]{}, WWW$^{^{,}}$11, 113-114 (ACM, New York, NY, USA, 2011). Crane, R. $\&$ Sornette, D. Robust dynamic classes revealed by measuring the response function of a social system. [*Proc. Nat. Acad. Sci.*]{} **105**, 15649–15653 (2008). Dall’Asta, L., Baronchelli, A., Barrat, A. $\&$ Loreto, V. Nonequilibrium dynamics of language games on complex networks. [*Phys. Rev. E*]{} **74**, 036105 (2006). Barrat, A., Baronchelli, A., Dall’Asta, L. $\&$ Loreto, V. Agreement dynamics on interaction networks with diverse topologies. [*Chaos*]{} **17**, 026111 (2007). Puglisi, A., Baronchelli, A. $\&$ Loreto, V. Cultural route to the emergence of linguistic categories. [*Proc. Nat. Acad. Sci.*]{} **105**, 7936–7940 (2008). Baronchelli, A., Gong, T., Puglisi, A. $\&$ Loreto, V. Modeling the emergence of universality in color naming patterns. [*Proc. Nat. Acad. Sci.*]{} **107**, 2403-2407 (2010). Tang, M., Liu, L. $\&$ Liu, Z.-H. Influence of dynamical condensation on epidemic spreading in scale-free networks. [*Phys. Rev. E*]{} **79**, 016108 (2009). Tang, M., Liu, Z.-H. $\&$ Li, B.-W. Epidemic spreading by objective traveling. [*Europhys. Lett.*]{} **87**, 18005 (2009). Zhao, Z.-D., Liu, Y. $\&$ Tang, M. Epidemic Variability in Hierarchical Geographical Networks with Human Activity Patterns. [*Chaos*]{} **22**, 023150 (2012). Gong, K., Tang, M., Yang, H. $\&$ Shang, M.-S. Variability of contact process in complex networks. [*Chaos*]{} **21**, 043130 (2011). Montanari, A. $\&$ Saberi, A. The spread of innovations in social networks. [*Proc. Nat. Acad. Sci.*]{} **107**, 20196-20201 (2010). Ferreira, S. C., Castellano, C. $\&$ Pastor-Satorras, R. Epidemic thresholds of the susceptible-infected-susceptible model on networks: A comparison of numerical and theoretical results. [*Phys. Rev. E*]{} **86**, 041125 (2012). Marro, J. $\&$ Dickman, R. [*Nonequilibrium Phase Transitions Lattice Models*]{} (Cambridge University Press, New York, 2005). Pe$^{'}$rez-Reche, F. J., Ludlam, J. J., Taraskin, S. N. $\&$ Gilligan, C. A. Synergy in spreading processes: from exploitative to explorative foraging strategies. [*Phys. Rev. Lett.*]{} **106**, 218701 (2011). Newman, M. [*Networks: An introduction.*]{} (Oxford University Press, Oxford, 2010). Taraskin, S. N. $\&$ P[é]{}rez-Reche, F. J., Effects of variable-state neighborhoods for spreading synergystic processes on lattices. [*Phys. Rev. E*]{} **88**, 062815 (2013). Noah, E. Theoretical foundations for centrality measures. [*Am. J. Sociol.*]{} **96**, 1478–1504 (1991). Gonzalez-Bailon, S., Borge-Holthoefer, J., Rivero, A. $\&$ Moreno, Y. The Dynamics of Protest Recruitment through an Online Network. [*Sci. Rep.*]{} **1**, 197 (2011). Turow, J. [*Media Today: An Introduction to Mass Communication*]{} (Taylor $\&$ Francis, 2011). Katz, E. $\&$ Lazarsfeld, P. F. [*Personal influence: The part played by people in the flow of mass communications*]{} (Transaction Publishers, 2006). Samovar, L. A. Porter, R. E. $\&$ McDaniel, E. R. [*Communication between cultures*]{} (Cengage Learning, 2009). Biagi, S. [*Media impact: An introduction to mass media*]{} (Wadsworth Publishing Company, 2011). Newman, M. Spread of epidemic disease on networks. [*Phys. Rev. E*]{} **66**, 016128 (2002). Sykes, M. F. $\&$ Essam, J. W. Exact critical percolation probabilities for site and bond problems in two dimensions. [*J. Math. Phys.*]{} **5**, 1117–1127 (2004). Message diffusion on square lattice {#square} =================================== We first present additional results to support the arguments for message diffusion on square lattice. In Fig. S\[lattice1470\], we can observe that the size of message spreading depends more on the stickiness of the message (values of $a$ with fixed $n_{s}$) than on the persistence $b$. We observe that the information can reach the vast majority of population (more than $80\%$) for $a\gtrsim 0.45$. Fig. S\[laalpha\] shows the dependence of the size of recovered population on the parameters $a$ and $b$. For $a\lesssim0.3$, $\alpha_{0}\approx 0.7$, whose value is much larger than the other three indices $\alpha_{n}$ $(n=1,2,3)$, which indicates that the information has not yet outbreak. As $a$ increases, transmission events $E_{m}$ ($m>0$) contribute a lot to the message spreading, hinting the large scale outbreak of the message. Accordingly, $\alpha_{0}$ ($\alpha_{n}$) decreases (increases) sharply, as shown in Figs. S\[laalpha\](b)–(d). In addition, it is found that $E_{0}$ and $E_{1}$ (depending more closely on $a$) constitute most of the transmission events, whereas $E_{2}$ and $E_{3}$ rarely occurs during the spreading process. To provide support for the real phenomena “three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"), in what follows we investigate the changes of the accumulative indices $\eta_{i}(a,b)$ by evaluating verification indices $\alpha_{i}$ as $\eta_{i}=\sum_{j<i}\alpha_{j}$ ($i=1,2,3,4$). Here the accumulative indices $\eta_{i}$ represents the proportions of individuals that have adopted the message when they heard it from at most $i$ informed neighbors. Like the results in Fig. S\[laalpha\], high values of $\eta_{1}$ reflect that the occurrences of $E_{0}$ and $E_{1}$ account for most of the transmission events. In addition, the results in Figs. S\[lbar2\](b)(c) show that the vast majority will accept a message as truth if it is mentioned or reported by at least two or three neighbors, which supports the mechanisms “three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). Moreover, it also indicates that the reinforcement begins to work as the information is bursting and prevailing on the square lattice, where the growth of $\eta_{i}$ keep increasing with $n_{i}$ (Fig. S\[lbar2\](b)). To make the point clear, we plot the global graphs of the four accumulative indices $\eta_{i}(a,b)$ for different values of inflection point ($n_{s}$) in Fig. S\[lasatu2\], Fig. S\[lasatu3\], and Fig. S\[lasatu4\], respectively. Similar behaviors can be detected in the three figures. The results, especially for positive persistence, also reveal the more important roles of $E_{0}$ and $E_{1}$, and simultaneously provide the evidence for the real phenomenon “three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). Message diffusion on random regular networks (RRNs) and regular lattice networks (RLs) {#regular} ====================================================================================== In this section, we present additional results to support the arguments for message diffusion on RRNs and RLs. The sizes of message diffusion are presented in Fig. S\[rhinfor\] as a function of $a$ and $b$. In the presence of social reinforcement ($b>0$), we can observe that the message can more easily invade and reach the majority of the population in larger parameter regions beyond the thresholds on RLs, by comparing with that on RRNs with the same average degree. The denser regular lattices (Moore lattice in Fig. S\[rhinfor\](b) (d)) are much more efficient in promoting information spreading [@science] when the message outbreaks. The reason is that there are more local clustering links can be used for transmission with smaller $n_{s}$ [@science; @zhou; @report] and positive persistence. Instead, the message can more easily diffuse in the RRNs in the presence of strong decay effects ($b<0$). The results summarized in Fig. S\[r6alpharound\] and Fig. S\[r8alpharound\] show that the peaks of $E_{i}(t)$ ($i=0, 1, 2, 3$) brings about the peaks of subsequent transmission event $E_{i+1}(t)$, which indicates that transmission events $E_{i}(t)$ with $i>2$ fail to last stably simultaneously, even at the critical points throughout the spreading processes. There are thus no time correlations among different events. The time correlations of different transmission events can be completely ruled out in estimating the critical behaviors of the message spreading. By comparison, the occurrences of transmission events in Hexagonal lattices and Moore lattices can last stably for a long time at critical points (see Fig. S\[halpharound2\](b) and Fig. S\[malpharound2\](b)). This suggests the existence of the time correlations among different transmission events $E_{i}(t)$. Besides, the observed huge disparities between $\alpha_{i}(a,b,t)$ and the corresponding $\beta_{i}(a,b,t)$ illustrate that the time correlations among transmission events in the both lattices are considerable. As observed in Fig. S\[hbar2\](b)(c), Fig. S\[mbar2\](b)(c), and the figures ranging from Fig. S\[hsatu2\] to Fig. S\[msatu5\], the message captures the vast majority of the population until $E_{i}(t)$ ($i>2$) happens when message outbreaks and prevails. And the spreading reaches a saturation state for the case where $\frac{n_{s}}{\left\langle k\right\rangle}>\frac{1}{2}$, which means that the transmission events $E_{i}$ ($i>\frac{1}{2}\left\langle k\right\rangle$) rarely happen in the spreading process. Therefore, the results in Fig. S\[hbar2\]–Fig. S\[msatu5\] can be regarded as the evidence of the mechanism “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"), in addition to the results for positive persistence illustrated in Fig. S\[r6satu2\], and Fig. S\[r8satu2\]. Message diffusion on ER and SF networks ======================================= In this section, we provide additional results to support the arguments for message diffusion on ER networks and SF networks. We observe in Fig. S\[baalpharound\] that transmission events $E_{i}(t)$ ($i>2$) fail to last simultaneously and stably over the long time at the critical points throughout the spreading process. Therefore, the time correlations need not to be taken into consideration in theoretical analysis. We have found the same phenomena in the ER networks and the SF with other average degrees. As illustrated in Fig. S\[ae8thre\], the theoretical estimations are sufficient to give fairly precise value of the thresholds. It is apparent that the critical behaviors of the message spreading are completely determined by stickiness ($a$) of message. Moreover, in comparison with the case of ER networks, the analytical solutions are in better agreement with the simulation in SF networks, attributing to shorter shortest paths and hubs [@path1; @bapath; @shortest]. More interestingly, both the analytical boundaries and the numerical thresholds are shifting to left with average degree $\left\langle k\right\rangle$, instead of size of the populations as stressed in [@hub1]. This demonstrates that message can easily reach and infect a larger amount of susceptible individuals through paths from those high-degree vertices whose links increase rapidly with $\left\langle k\right\rangle$, although they can be infected only once. In Fig. S\[beinfor\], we find that the persistence also boosts its reasonable impact on the spreading as $\left\langle k\right\rangle$ gets larger. It implies that hub nodes of larger size and shorter shortest paths in the networks can transmit the information more efficiently [@path1; @hub1; @shortest]. It is in accordance with the conclusion in [@degree] that higher degrees and densities are relevant factors in improving the global spread of information. It should be noted that the both ER and SF networks with small average degrees (such as $\langle k \rangle =6, 8$) are tree-like, with few short loops, indicating that the critical transmissibility $T_{C}$ can be derived from equation $T_{c} = \frac{\left\langle k\right\rangle}{\left\langle k^{2}\right\rangle -\left\langle k\right\rangle}$ [@degree; @ther1]. More specifically, $T_{c} = \frac{1}{\left\langle k\right\rangle -1}$ for ER networks. Qualitatively, the contributions of the infection events $E_{n}~(n>\left\langle k\right\rangle-2)$ to the spread are insignificant (see Fig. S\[er6satu\] and Fig. S\[ba6satu\]). Therefore we neglect the contributions of the transmission events $E_{l}(t)$ ($l>\langle k\rangle$) to message spreading, and further assume that the following relationship $T_{C} = \left\langle T\right\rangle = \sum^{\left\langle k\right\rangle-1}_{n=0}q_{n}(1-e^{-\lambda_{n+1}(a,b)})$ is always satisfied in estimating the analytical thresholds. Similar to what has been demonstrated in Sec. \[regular\], both Fig. S\[er6satu\] (ER networks) and Fig. S\[ba6satu\] (SF networks) show that the results nearby the critical points for $b>0$ can also be considered as the evidence of the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). [0]{} Centola, D. The spread of behavior in an online social network experiment. [*Science*]{} **329**, 1194–1197 (2010). Lu, L., Chen, D. B. $\&$ Zhou, T. Small world yields the most effective information spreading. [*New Journal of Physics*]{} **13**, 123005 (2011). Gonzalez-Bailon, S., Borge-Holthoefer, J., Rivero, A. $\&$ Moreno, Y. The Dynamics of Protest Recruitment through an Online Network. [*Sci. Rep.*]{} **1**, 197 (2011). Valdez, L., Macri, P. $\&$ Braunstein, L. Intermittent social distancing strategy for epidemic control. [*Phys. Rev. E*]{} **85**, 036108 (2012). Callaway, D., Newman, M., Strogatz, S. $\&$ Watts, D. Network robustness and fragility: Percolation on random graphs. [*Phys. Rev. Lett.*]{} **85**, 5468–5471 (2000). Barrat, A. Barthlemy, M. $\&$ Vespignani, A. [*Dynamical processes on complex networks*]{} (Cambridge University Press, New York, 2008). Kalisky, T., Cohen, R., Mokryn, O., Dolev, D., Shavitt, Y. $\&$ Havlin, S. Tomography of scale-free networks and shortest path trees. [*Phys. Rev. E*]{} **74**, 066108 (2006). Friedkin, N. Theoretical foundations for centrality measures. [*American journal of Sociology*]{} 1478–1504 (1991). Castellano, C. $\&$ Pastor-Satorras, R. Thresholds for epidemic spreading in networks, [*Phys. Rev. Lett.*]{} **105**, 218701 (2010). {#figures .unnumbered} ![**The densities of recovered individuals as a function of $a$ and $b$.** Each data point is obtained by averaging $100$ independent realizations. The other parameters are $n_{s}=2$ and $L=101$.[]{data-label="lattice1470"}](latticeinfor.jpeg){width="\textwidth"} ![**The four indices (a) $\alpha_{0}(a,b)$, (b) $\alpha_{1}(a,b)$, (c) $\alpha_{2}(a,b)$, and (d) $\alpha_{3}(a,b)$ as a function of $a$ and $b$.** The other parameters are token as $n_{s}=2$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. As message outbreaks ($a\gtrsim 0.32$), it is clear in (a) and (b) that the transmission events $E_{0}$ and $E_{1}$ contribute the most to the whole spreading process. []{data-label="laalpha"}](laalpha2.jpeg){width="\textwidth"} ![**The four accumulative indices $\eta_{i}(a,b)$ in square latttice.** $n_{i}$ denotes the number of informed neighbors an individual has had at most when it approves the message. Analysis is performed at (a) subcritical point $a=0.20$, $b=0.20$; (b) critical point $a=0.33$, $b=0.20$; and (c) supercritical point $a=0.45$, $b=0.20$. The other parameters are token as $n_{s}=2$ and $L=101$. Results are obtained by averaging $100$ independent realizations. No matter which case, the diffusion of the message owes much to $E_{0}(t)$ and $E_{1}(t)$. In (b) and (c), the value of $\eta_{3}$ approaches to $1$ rather than $\eta_{i}$ ($i<4$) and $\eta_{i}$ ($i<3$), and the gaps between $\eta_{1}$ and $\eta_{2}$ are obvious. The results in (b) and (c) can be considered as an evidence of the emergence of “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth") – the vast majority of the population accept the message as truth only when it is repeated more than two times in their ears.[]{data-label="lbar2"}](labar2.jpeg){width="\textwidth"} ![**The four accumulative indices $\eta_{0}$ (a), $\eta_{1}$ (b), $\eta_{2}$ (c), and $\eta_{3}$ (d) as a function of $a$ and $b$.** The other parameters are token as $n_{s}=2$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>90\%$ ($i\geq 2$) rather than $\eta_{1}$, where transmission events $E_{i}(t)$ ($i\geq 2$) contribute more than $60\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results with positive persistence nearby the threshold also provide the evidence for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth").[]{data-label="lasatu2"}](lasatu2.jpeg){width="\textwidth"} ![**The four accumulative indices $\eta_{0}$ (a), $\eta_{1}$ (b), $\eta_{2}$ (c), and $\eta_{3}$ (d) as a function of $a$ and $b$.** The other parameters are token as $n_{s}=4$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>90\%$ ($i\geq 2$) rather than $\eta_{1}$, where transmission events $E_{i}(t)$ ($i\geq 2$) contribute more than $60\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results with positive persistence nearby the threshold also provide the evidence for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth").[]{data-label="lasatu4"}](lasatu3.jpeg){width="\textwidth"} ![**The four accumulative indices $\eta_{0}$ (a), $\eta_{1}$ (b), $\eta_{2}$ (c), and $\eta_{3}$ (d) as a function of $a$ and $b$.** The other parameters are token as $n_{s}=4$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>90\%$ ($i\geq 2$) rather than $\eta_{1}$, where transmission events $E_{i}(t)$ ($i\geq 2$) contribute more than $60\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results with positive persistence nearby the threshold also provide the evidence for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth").[]{data-label="lasatu4"}](lasatu4.jpeg){width="\textwidth"} ![**The densities of recovered individuals as a function of $a$ and $b$.** The other parameters are token as $n_{s}=2$, $N=1000$ for the RRNs, and $L=101$ for the RLs. The scales of spreading on RLs ((a) (b)) are compared with that in RRNs ((a) (b)). The degrees of the networks are $<k>=6$ (a, c) and $<k>=8$ (b, d), respectively. Each data is obtained by averaging $100$ independent realizations. By comparing (a) with (c), for $b>0$, it is clear in (c) that the message can more easily outbreak and capture a larger population, in contrast to what is observed in (a), owing to the function of social reinforcement effects. However, more individuals accept the message in RRNs for $b<0$, indicating the advantage of the RRNs in facilitating the diffusion of message in the presence of strong decay effects. As expected, the same conclusion can also be reached by comparing the plots in (b) with that in (d). []{data-label="rhinfor"}](rhinfor.jpeg){width="\textwidth"} ![**The evolution of proportions of the transmission events.** The evolution of indices $\alpha_{i}(a,b,t)$ from simulation (solid lines) and $\beta_{i}(a,b,t)$ from prediction of percolation theory (dashed lines) on the RRN with $\langle k\rangle=8$ are presented. Three different cases are considered here: (a) the information vanishes for $a=0.10$, $b=0.20$, (b) it outbreaks for $a=0.13$, $b=0.20$; and prevails for $a=0.30$, $b=0.20$ (c).The other parameters are token as $n_{s}=2$ and $N=1000$. It can be observed that the occurrences of all the transmission events fail to last stably throughout the whole spreading process. The time correlations among the transmission events can thus be neglected in estimating the critical behavior of the message spreading.[]{data-label="r8alpharound"}](r6alpharound.jpeg){width="\textwidth"} ![**The evolution of proportions of the transmission events.** The evolution of indices $\alpha_{i}(a,b,t)$ from simulation (solid lines) and $\beta_{i}(a,b,t)$ from prediction of percolation theory (dashed lines) on the RRN with $\langle k\rangle=8$ are presented. Three different cases are considered here: (a) the information vanishes for $a=0.10$, $b=0.20$, (b) it outbreaks for $a=0.13$, $b=0.20$; and prevails for $a=0.30$, $b=0.20$ (c).The other parameters are token as $n_{s}=2$ and $N=1000$. It can be observed that the occurrences of all the transmission events fail to last stably throughout the whole spreading process. The time correlations among the transmission events can thus be neglected in estimating the critical behavior of the message spreading.[]{data-label="r8alpharound"}](r8alpharound.jpeg){width="\textwidth"} ![**The evolution of proportions of the transmission events.** The evolution of indices $\alpha_{i}(a,b,t)$ from simulation (solid lines) and $\beta_{i}(a,b,t)$ from prediction of percolation theory (dashed lines) on the Moore lattice are presented. Three different cases are considered here: (a) the information vanishes for $a=0.05$, $b=0.20$; (b) it outbreaks for $a=0.10$, $b=0.20$ and prevails for $a=0.30$, $b=0.20$ (c). The other parameters are token as $n_{s}=2$ and $L=101$. In comparison with the dynamic behaviors of corresponding RRNs shown in Fig. S\[r8alpharound\], the occurrences of transmission events in Moore lattice can last stably for over $450$ MCs at the critical point, hence the time correlation among the events cannot be neglected in estimating the critical behavior of the message spreading. The inconsistencies between $\alpha_{i}(a,b,t)$ and the corresponding $\beta_{i}(a,b,t)$ have been chosen to demonstrate the existence of the time correlations between different transmisssion events $E_{i}(t)$. The phenomenon $\alpha_{i}(t)>0$ in (b) indicates that almost all transmission events involve in the spreading process at the critical point. In addition, $\beta_{i}(t)$ gets close to corresponding $\alpha_{i}(t)$ in (b) and (c).[]{data-label="malpharound2"}](halpharound2.jpeg){width="\textwidth"} ![**The evolution of proportions of the transmission events.** The evolution of indices $\alpha_{i}(a,b,t)$ from simulation (solid lines) and $\beta_{i}(a,b,t)$ from prediction of percolation theory (dashed lines) on the Moore lattice are presented. Three different cases are considered here: (a) the information vanishes for $a=0.05$, $b=0.20$; (b) it outbreaks for $a=0.10$, $b=0.20$ and prevails for $a=0.30$, $b=0.20$ (c). The other parameters are token as $n_{s}=2$ and $L=101$. In comparison with the dynamic behaviors of corresponding RRNs shown in Fig. S\[r8alpharound\], the occurrences of transmission events in Moore lattice can last stably for over $450$ MCs at the critical point, hence the time correlation among the events cannot be neglected in estimating the critical behavior of the message spreading. The inconsistencies between $\alpha_{i}(a,b,t)$ and the corresponding $\beta_{i}(a,b,t)$ have been chosen to demonstrate the existence of the time correlations between different transmisssion events $E_{i}(t)$. The phenomenon $\alpha_{i}(t)>0$ in (b) indicates that almost all transmission events involve in the spreading process at the critical point. In addition, $\beta_{i}(t)$ gets close to corresponding $\alpha_{i}(t)$ in (b) and (c).[]{data-label="malpharound2"}](malpharound2.jpeg){width="\textwidth"} ![**The eight accumulative indices $\eta_{i}(a,b)$ in Moore lattice.** $n_{i}$ denotes the number of informed neighbors an individual has had at most when it approves the message. The parameters are chosen at three selected parameter points: (a) subcritical point $a=0.10$, $b=0.20$; (b) critical point $a=0.18$, $b=0.20$ and (c) supercritical point $a=0.40$, $b=0.20$. The other parameters are token as $n_{s}=2$ and $L=101$. Results are obtained by averaging $100$ independent realizations. In (b), the gaps between $\eta_{i}$ and $\eta_{i+1}$ ($i<4$) are apparent, which indicates that almost all transmission events involve in the spreading process. In (b) and (c), the value of $\eta_{4}$ or $\eta_{3}$ approach to $1$, rather than $\eta_{i}$ ($i<6$) and $\eta_{i}$ ($i<4$). That can be considered as an evidence of the emergence of “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth") – the vast majority of the population accept the message as truth only when it is repeated more than two times in their ears.[]{data-label="mbar2"}](hbar2.jpeg){width="\textwidth"} ![**The eight accumulative indices $\eta_{i}(a,b)$ in Moore lattice.** $n_{i}$ denotes the number of informed neighbors an individual has had at most when it approves the message. The parameters are chosen at three selected parameter points: (a) subcritical point $a=0.10$, $b=0.20$; (b) critical point $a=0.18$, $b=0.20$ and (c) supercritical point $a=0.40$, $b=0.20$. The other parameters are token as $n_{s}=2$ and $L=101$. Results are obtained by averaging $100$ independent realizations. In (b), the gaps between $\eta_{i}$ and $\eta_{i+1}$ ($i<4$) are apparent, which indicates that almost all transmission events involve in the spreading process. In (b) and (c), the value of $\eta_{4}$ or $\eta_{3}$ approach to $1$, rather than $\eta_{i}$ ($i<6$) and $\eta_{i}$ ($i<4$). That can be considered as an evidence of the emergence of “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth") – the vast majority of the population accept the message as truth only when it is repeated more than two times in their ears.[]{data-label="mbar2"}](mbar2.jpeg){width="\textwidth"} ![ **The six accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), and $\eta_{6}$ (f) as a function of $a$ and $b$ for Hexagonal lattice.** The other parameters are token as $n_{s}=3$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions with positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i\geq 2$) contribute more than $80\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results with positive persistence nearby the threshold also provide the evidence for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). []{data-label="hsatu3"}](hsatu2.jpeg){width="\textwidth"} ![ **The six accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), and $\eta_{6}$ (f) as a function of $a$ and $b$ for Hexagonal lattice.** The other parameters are token as $n_{s}=3$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions with positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i\geq 2$) contribute more than $80\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results with positive persistence nearby the threshold also provide the evidence for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). []{data-label="hsatu3"}](hsatu3.jpeg){width="\textwidth"} ![**The six accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), and $\eta_{6}$ (f) as a function of $a$ and $b$ for Hexagonal lattice.** The other parameters are token as $n_{s}=5$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i\geq 2$) contribute more than $80\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results nearby the threshold also provide the evidence for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). []{data-label="hsatu5"}](hsatu4.jpeg){width="\textwidth"} ![**The six accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), and $\eta_{6}$ (f) as a function of $a$ and $b$ for Hexagonal lattice.** The other parameters are token as $n_{s}=5$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i\geq 2$) contribute more than $80\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results nearby the threshold also provide the evidence for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). []{data-label="hsatu5"}](hsatu5.jpeg){width="\textwidth"} ![**The eight accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), $\eta_{6}$ (f), $\eta_{7}$ (g), and $\eta_{8}$ (h) as a function of $a$ and $b$ for Moore lattice.** The other parameters are token as $n_{s}=3$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i> 2$) contribute more than $70\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results with positive nearby the threshold also provide the evidences for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). []{data-label="msatu3"}](msatu2.jpeg){width="\textwidth"} ![**The eight accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), $\eta_{6}$ (f), $\eta_{7}$ (g), and $\eta_{8}$ (h) as a function of $a$ and $b$ for Moore lattice.** The other parameters are token as $n_{s}=3$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i> 2$) contribute more than $70\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results with positive nearby the threshold also provide the evidences for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). []{data-label="msatu3"}](msatu3.jpeg){width="\textwidth"} ![**The eight accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), $\eta_{6}$ (f), $\eta_{7}$ (g), and $\eta_{8}$ (h) as a function of $a$ and $b$ for Moore lattice.** The other parameters are token as $n_{s}=5$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i> 2$) contribute more than $70\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results nearby the threshold also provide the evidences for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). []{data-label="msatu5"}](msatu4.jpeg){width="\textwidth"} ![**The eight accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), $\eta_{6}$ (f), $\eta_{7}$ (g), and $\eta_{8}$ (h) as a function of $a$ and $b$ for Moore lattice.** The other parameters are token as $n_{s}=5$ and $L=101$. Each data point is obtained by averaging $100$ independent realizations. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i> 2$) contribute more than $70\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results nearby the threshold also provide the evidences for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). []{data-label="msatu5"}](msatu5.jpeg){width="\textwidth"} ![**The eight accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), $\eta_{6}$ (f), $\eta_{7}$ (g), and $\eta_{8}$ (h) as function of $a$ and $b$ for RRs with $\langle k \rangle=8$.** The other parameters are token as $n_{s}=5$ and $N=10000$. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i> 2$) contribute more than $60\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results with positive persistence nearby the threshold also provide the evidences for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). []{data-label="r8satu2"}](r6satu2.jpeg){width="\textwidth"} ![**The eight accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), $\eta_{6}$ (f), $\eta_{7}$ (g), and $\eta_{8}$ (h) as function of $a$ and $b$ for RRs with $\langle k \rangle=8$.** The other parameters are token as $n_{s}=5$ and $N=10000$. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i> 2$) contribute more than $60\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results with positive persistence nearby the threshold also provide the evidences for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). []{data-label="r8satu2"}](r8satu2.jpeg){width="\textwidth"} \[be\] ![**The evolution of proportions of the transmission events.** The evolution of indices $\alpha_{i}(a,b,t)$ from simulation (solid lines) and $\beta_{i}(a,b,t)$ from predications of percolation theory (dashed lines) are presented for SF with $\left\langle k\right\rangle=8$. Three difference cases are considered here: (a) the information vanishes for $a=0.10$, $b=0.20$, (b) it outbreaks for $a=0.19$, $b=0.20$ and prevails for $a=0.30$, $b=0.20$ (c). The other parameters are token as $n_{s}=5$ and $N=10000$. It can be observed that the occurrences of all the transmission events fail to last simultaneously and stably in the whole spreading process, departing from what exhibited in Fig. S\[malpharound2\] for moore lattice. The time correlations among the transmission events can thus be neglected in estimating the critical behaviors of the message spreading. []{data-label="baalpharound"}](ba6alpharound2.jpeg){width="\textwidth"} ![**Locus of thresholds of message diffusion on SF networks and ER networks**. Respectively, analytical solutions (solid lines) for four SF networks (top panel) and four ER networks (bottom panel) with different average degrees are plotted to compare with the corresponding exact numerical data (markers). The other parameters are token as $n_{s}=2$ and $N=10000$. Each numerical data point is obtained by averaging $1000$ independent realizations. And (a)(e) $\left\langle k\right\rangle=6$, (b)(f) $\left\langle k\right\rangle=8$, (c)(h) $\left\langle k\right\rangle=10$ and (d)(i) $\left\langle k\right\rangle=12$. Both the simulations and the analytical predications show that the critical behaviors of the spreading are dominated by the stickiness of the message ($b$).[]{data-label="ae8thre"}](ae8thre.jpeg){width="\textwidth"} ![**The densities of recovered individuals as function of $a$ and $b$,** on SF networks (top panel) and Erdos-Renyi networks (bottom panel) with different average degrees. The other parameters are token as $n_{s}=2$ and $N=10000$. Each numerical data point is obtained by averaging $100$ independent realizations. Precisely, (a)(e) $\left\langle k\right\rangle =6$, (b)(f) $\left\langle k\right\rangle =8$, (c)(h) $\left\langle k\right\rangle=10$ and (d)(i) $\left\langle k\right\rangle =12$. Although the critical behaviors of the message spreading on SF and ER are dominated by the stickiness, the persistence of the message (i.e., $b$) has a reasonable impact on the sizes of message spreading, especially at the parameter regions nearby $b=0$. []{data-label="beinfor"}](beinfor.jpeg){width="\textwidth"} ![**The eight accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), $\eta_{6}$ (f), $\eta_{7}$ (g) $\eta_{8}$ (h) as function of $a$ and $b$ for ER networks with $\langle k \rangle=6$.** The other parameters are token as $n_{s}=2$ and $N=10000$. Each numerical data point is obtained by averaging $100$ independent realizations. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i> 2$) contribute more than $70\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results for $b>0$ nearby the threshold also provide the evidences for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). Similar results are obtained for SF networks with the same average degree. []{data-label="ba6satu"}](ba8satu.jpeg){width="\textwidth"} ![**The eight accumulative indices $\eta_{1}$ (a), $\eta_{2}$ (b), $\eta_{3}$ (c), $\eta_{4}$ (d), $\eta_{5}$ (e), $\eta_{6}$ (f), $\eta_{7}$ (g) $\eta_{8}$ (h) as function of $a$ and $b$ for ER networks with $\langle k \rangle=6$.** The other parameters are token as $n_{s}=2$ and $N=10000$. Each numerical data point is obtained by averaging $100$ independent realizations. In the wide parameter regions for positive persistence, especially nearby the critical boundary (threshold), only $\eta_{i}>80\%$ ($i\geq \frac{\langle k \rangle}{2}$) rather than $\eta_{j}$ ($j<3$), where transmission events $E_{i}(t)$ ($i> 2$) contribute more than $70\%$ of the scale of message spreading (i.e., the spreading thus reaches a saturation level). It also implies that the majority of the population accept the message as truth only if the message is repeated more than three times in their ears. Therefore, the results for $b>0$ nearby the threshold also provide the evidences for the real phenomenon “Three men make a tiger" (or “A lie, if repeated often enough, will be accepted as truth"). Similar results are obtained for SF networks with the same average degree. []{data-label="ba6satu"}](er8satu.jpeg){width="\textwidth"}

--- abstract: 'We consider a two dimensional colloidal dispersion of soft-core particles driven by a one dimensional stochastic flashing ratchet that induces a time averaged directed particle current through the system. It undergoes a non-equilibrium melting transition as the directed current approaches a maximum associated with a resonance of the ratcheting frequency with the relaxation frequency of the system. We use extensive molecular dynamics simulations to present a detailed phase diagram in the ratcheting rate- mean density plane. With the help of numerically calculated structure factor, solid and hexatic order parameters, and pair correlation functions, we show that the non-equilibrium melting is a continuous transition from a quasi-long ranged ordered solid to a hexatic phase. The transition is mediated by the unbinding of dislocations, and formation of compact and string-like defect clusters.' author: - Shubhendu Shekhar Khali - Dipanjan Chakraborty - Debasish Chaudhuri title: 'Structure-dynamics relationship in ratcheted colloids: Resonance melting, dislocations, and defect clusters' --- § ł Ł ø \#1 Introduction {#sec:introduction} ============ A class of non-equilibrium driven systems called pump models are particularly intriguing due to their following property. They involve periodic forces, in time and space, that vanish under spatio-temporal averaging but still drives an overall directed current [@Julicher1997a; @Astumian2002; @Hanggi2009; @Reimann2002; @Brouwer1998; @Citro2003; @Jain2007; @Chaudhuri2011; @Chaudhuri2015; @Chaudhuri2015f]. This is achieved via the breaking of time-reversal symmetry through, e.g., a phase lag between spatially non-local drives [@Brouwer1998; @Jain2007; @Chaudhuri2011], or breaking of space inversion symmetry of the external potential profile [@Julicher1997a; @Reimann2002; @Astumian2002; @Hanggi2009]. Most of the biological processes generating directed motion involve reaction cycles and utilize some variant of this principle. Natural examples involve ion-pumps, e.g., the Na$^+$, K$^+$-ATPase pumps, and molecular motors [@Gadsby2009], e.g., Kinesin or myosin moving on polymeric tracks of microtubules or F-actins, respectively [@Reimann2002]. The flashing ratchet model has been used to describe molecular motor locomotion [@Julicher1997a]. In experiments on colloids, ratcheting could be generated using optical [@Faucheux1995; @Lopez2008], magnetic [@Tierno2010; @Tierno2012] or electrical fields [@Rousselet1994; @Leibler1994; @Marquet2002]. Most of the studies on pump models focused on systems of non-interacting particles, restricted to one dimension, with a few exceptions that analyzed the impact of interaction on molecular motors [@Derenyi1995; @Derenyi1996], collective properties of particle pumps [@Jain2007; @Marathe2008; @Chaudhuri2011; @Chaudhuri2015; @Chaudhuri2015f], and in ratchet models [@Savelev2004; @Pototsky2010; @Savelev2003; @Hanggi2009]. In a recent study, we used an asymmetric periodic potential that switches between an [*on*]{} and [*off*]{} state in a stochastic manner to drive a directed current of particles in a two dimensional (2d) dispersion of sterically stabilized colloids [@Chakraborty2014], focusing on the frequency and density dependence of the ratcheted current. With the change in the rate of ratcheting, the time- averaged directed current carried by the colloids show a resonance with the system’s relaxation frequency [@Chakraborty2014]. The current shows a non-monotonic dependence on density as well. This change in the dynamical properties, as we show in this paper, is closely related to the associated structural changes, e.g., the solid melts near the resonance frequency. In the limit of extremely high switching frequency, higher than the inherent relaxation time of the colloids, the system can only respond to essentially a time- averaged potential profile. In addition, if one considers the limit of vanishing asymmetry in the potential profile, the scenario becomes equivalent to that of the re-entrant laser induced melting transition (RLIM) [@Chowdhury1985; @Wei1998; @Frey1999; @Chaudhuri2006], in which a high- density colloidal liquid undergoes solidification followed by melting, as the strength of a commensurate external periodic potential is increased. This is an equilibrium phase transition of the Kosterlitz-Thouless type [@Frey1999; @Chaudhuri2006], and is described in terms of unbinding of a specific type of dislocations, allowed by the potential profile. In this paper we consider an asymmetric ratcheting of soft-core particles, and investigate structural transitions associated with the change in dynamical behavior of the system, observed in terms of its current carrying capacity. Using a large scale molecular dynamics simulation, we obtain the phase diagram in the density- ratcheting rate plane, showing melting from a solid to hexatic phase. We find a re-entrant solid- hexatic- solid transition with changing ratcheting frequency. The transitions are associated with a non-monotonic variation of the mean directed current. As we demonstrate in detail, the non-equilibrium melting is a continuous transition from a quasi- long ranged ordered (QLRO) solid to a hexatic phase, and is mediated by the formation of topological defects. The dominant defect types generated at the solid melting are dislocations, and compact or string-like defect clusters. {width="\linewidth"} In \[sec:model\_simulation\] we present the model and details of numerical simulations. In \[sec:results\_discussion\] we discuss the detailed phase diagram, explaining the properties of the different non-equilibrium phases. The associated variation of driven directed current with driving frequency and density is shown in \[ssec:current\]. In this section we establish the relation of changing particle current to the non-equilibrium phase transitions. This is followed by a detailed analysis of the melting transitions in terms of the order parameters presented in \[ssec:reentrant\_transition\]. In the following three subsections, the phase- transitions are further characterized in terms of the distribution functions of order parameters, correlation functions, and formation of topological defects. We finally conclude presenting a discussion and outlook in \[sec:outlook\]. Model and Simulation Details {#sec:model_simulation} ============================ We consider a two dimensional system of a repulsively interacting colloidal suspension of $N$ particles in a volume $A=L_x L_y$. The mean inter-particle separation in this system $a^2=\sqrt{3} \rho/2$ is set by the particle density $\rho = N/A$. We assume that the colloids repel each other via a shifted soft-core potential $U(r)=\e \,[(\sigma/r)^{12}-2^{-12}]$ when the inter-particle separation $r< r_c r_c=2 \sigma$, and $U(r)=0$ otherwise. The units of energy and length scales are set by $\e$, $\s$ respectively. The system evolves under an asymmetric ratchet potential $U_{\rm ext}(x,y,t)=V(t) \left[\sin \left(2 \pi y/\lambda \right)+\alpha \sin \left(4 \pi y/\lambda \right) \right]$, where the time-dependent strength $V(t)$ switches between $\e$ and $0$ stochastically with a rate $f$. The two sinusoidal terms in the above expression of $U_{\rm ext}$ with $\a=0.2$ maintains the asymmetric shape of the potential profile. When it assumes a triangular lattice structure, the separation between consecutive lattice planes in the system is $a_y=\sqrt{3} a/2$. We have chosen the periodicity of the external potential $\lambda = a_y$, commensurate with the mean lattice spacing. In the absence of the external potential, the soft core solid is expected to undergo a two stage solid- hexatic- liquid transition [@Kosterlitz1973; @Halperin1978; @Young1979; @Kapfer:2015ca], with the solid melting point at $\r \s^2 \approx 1.01$. In the presence of a time- independent potential profile with $V(t)=U_0$ and $\a=0$, the system undergoes RLIM with increase in $U_0$ [@Wei1998; @Frey1999; @Chaudhuri2006]. At $U_0=\e$, the laser induced melting point of the soft-core solid is $\r \s^2 = 0.95$ [@Chaudhuri2006]. We perform molecular dynamics simulations of the system in the presence of an external ratcheting potential using the standard leap-frog algorithm [@Frenkel2002] with a time-step $\d t = 0.001\,\t$ where $\t= \s \sqrt{m/\e}$ is the characteristic time scale. We use $m=1$. The temperature of the system is kept constant at $T =1.0 \e/\kb$ using a Langevin thermostat characterized by an isotropic friction $\g = 1/\t$. At each step a trial move is performed to switch $V(t)$ between $0$ and $\e$, and accepted with a probability $f \, \d t$. In this paper, we present the results for a large system of $N=262144$ particles. We discard simulations over initial $10^7$ steps to ensure achievement of steady state, and the analyses are performed collecting data over further $10^7$ steps. Results and Discussion {#sec:results_discussion} ====================== Phase diagram {#ssec:phase_diagram} ------------- In \[fig:phase\_diagram\] we present the detailed phase diagram along with local density profiles of the 2D system of mono-dispersed ratcheted colloids. The system displays a solid and a density modulated hexatic phase, controlled by the dimensionless density $\r \s^2$ and ratcheting rate $f \t$. The color codes in \[fig:phase\_diagram\]($a$) denote the values of mean solid order parameter. The two dashed lines with open symbols signify the two solid melting boundaries at small and high frequencies. The dash-dotted line with filled squares denotes the inverse of relaxation time-scales $f_s$ of the system at a given density. For ratcheting rates slower than this time-scale, the solid and hexatic order can [*equilibrate*]{} to the instantaneous external potential and follow its change. The details of the calculation of such relaxation times are discussed in \[appendix:relaxtion\]. We characterize the various phases using the structure factor, the pair correlation function, the solid and hexatic order parameters, and their distribution functions. Further details of such characterization are presented in later sections. As is shown in \[fig:phase\_diagram\]($a$), the system remains in a QLRO triangular lattice solid phase at the highest frequencies, if the ambient density permits. As the frequency decreases, the solid melts into a hexatic phase, below the dashed line through open circles (\[fig:phase\_diagram\]). As is shown later, the melting is a continuous transition and happens via a proliferation of topological defects, including the unbinding of dislocation pairs. The molten phase is a hexatic displaying a unimodal distribution of local- hexatic order parameter. This excludes any possibility of phase coexistence, indicating a continuous transition. As the frequency of the drive is decreased further, below the equilibrium relaxation time, the system starts to follow the time variation of the external potential. As a result, the time-averaged properties turn out to be approximately a superposition of the properties of the equilibrium states in the presence ($V(t)=\e$) and absence ($V(t)=0$) of external potential profile. At high densities ($\r \s^2 \gtrsim 1.026$) the solid phase is stabilized even at low ratcheting frequencies like $f\t=0.01$. The melting boundary of this solid is shown by the open $\triangledown$ and dashed line. In \[fig:phase\_diagram\]($b$)-($h$) we present the time-averaged local density profiles $\r(x,y)$ at two mean densities $\r \s^2=1.04,\, 0.98$. We show the results over local cross- sections of area $10\s \times 10\s$, for better visibility. At $\r \s^2=1.04$, the density profile shows triangular lattice structure at $f \t=0.01$ (\[fig:phase\_diagram\]($b$)). At $f \t=1$ the solid melts into a phase with density modulation along $y$-axis, the direction of ratcheting drive (\[fig:phase\_diagram\]($c$)). The melting is quantified later in terms of vanishingly small solid order. At $f \t=10$, a local triangular lattice- like pattern reappears, albeit the solid order remains small (\[fig:phase\_diagram\]($d$)). The system shows a more compact triangular lattice solid at $f\t=30$ associated with appearance of significant solid order (\[fig:phase\_diagram\]($e$)). As is turns out, this order decreases with the system size in a power law manner, identifying a QLRO solid (see \[fig:scaling\_solid\_order\_parameter\]). As we show later, the density modulated phases of the system at $f \t=1$ and $10$ share similar amount of hexatic order (see \[fig:solid\_op\_freq\_density\]). In the top and left axes around the $\r(x,y)$ plots, we show the linear density profiles $\r_x$ (ordinate) and $\r_y$ (abscissa) measured along the horizontal and vertical dashed white lines indicated in the $\r(x,y)$ plots. The clean density modulations in $\r_x$ captures the spontaneous emergence of the solid- like order in that direction. Due to the shape of the triangular lattice solid, $\r_y$ shows large followed by tiny peaks along the white line perpendicular to the lattice planes, e.g., in \[fig:phase\_diagram\]($b$) and ($e$). Note that this feature is not shared by the other local density plots after the solid melts, shown in \[fig:phase\_diagram\]. The strong density modulation $\r_y$ in the hexatic phase is induced directly by the external potential minima. This phase lacks density modulation in $\r_x$, as is shown in \[fig:phase\_diagram\]($c$). However, as we shall quantify later, it shows significant hexatic orientation order. Similar characterization at the lower density $\r \s^2 = 0.98$ are presented in \[fig:phase\_diagram\]($f$)-($h$). A clear solid- like triangular lattice order is observed at $f\t=30$ (\[fig:phase\_diagram\]($h$)). \[fig:phase\_diagram\]($g$) shows local weak triangular lattice like pattern that gets smeared within the hexatic phase. As we show later in Sec.-\[ssec:reentrant\_transition\], this phase in \[fig:phase\_diagram\]($f$) and ($g$) indeed display a vanishingly small solid order, in the presence of a reasonably large hexatic order. The system remains in the molten phase at the intermediate frequency range, between $0.1 \leq f \tau \leq 10$. Note that the densities considered here are relatively large with respect to the equilibrium melting point in the absence of external potential. As we demonstrate in the following, the non-equilibrium melting discussed in this section is associated with an increase in directed particle current carried by the system under the flashing ratchet drive. Direct particle current out of alternating drive {#ssec:current} ------------------------------------------------ The inherent asymmetry of the ratchet potential, added with the stochastic switching drives a time- and space- averaged directed particle current in the system, $$\label{eq:current_def} \left\langle j_y \right\rangle=\frac{1}{\tau_m}\frac{1}{A}\int^{\tau_m} \upd_t\int^{L_x}\upd_x\int^{L_y}\upd_y ~j_y(x,y,t),$$ where, $\tau_m$ is an integral multiple of $1/f$, the mean switching time of the flashing ratchet potential. ![(color online ) [(a)]{} Variation of the scaled particle current along the direction of the drive as a function $f/\nu$ for different densities as indicated in the legend. The solid line is plot of the function $(f/\nu)/\left(1+(f/\nu)^2\right)$ as given by \[eq:current\]. [(b)]{} Plot of the current amplitude $\rho \kappa v_0$ as a function of density. The solid line is a fit to the data using the functional form $\kappa D_0 \rho^{3/2}(1-\rho/\r_c)$ with $\kappa D_0$ and $\rho_c$ as fitting parameters with values $\kappa D_0 \approx 0.22$ and $\rho_c \s^2 \approx 1.004$. []{data-label="fig:fluxplot"}](flux_amp_combi.pdf){width="\linewidth"} The competition between the intrinsic relaxation of the system and the external drive leads to a resonance in the mean particle current. At low frequencies the particle current increases linearly with the driving frequency $f$, achieves a maximum around $f=\nu$, and beyond this decays as $f^{-1}$. The behaviour of the current in the whole frequency range can be captured by the simple ansatz [@Chakraborty2014] $$\label{eq:current} \left\langle j_y \right\rangle = \kappa \frac{\nu f}{\nu^2+f^2} \rho v_0,$$ where, $\nu$ is the intrinsic relaxation frequency, $v_0 = \nu \l$ intrinsic velocity, and $\kappa$ is a proportionality constant. In the high density regime, where, the mean-free path of the particles is small, the diffusive time scale $\tau_D$ to travel the typical distance $\lambda$ is set by $\tau_D=\lambda^2/D(\rho)$. Here $D(\rho)$ denotes the density- dependent tagged particle diffusivity, which we assume to decrease linearly with density, $D(\rho)=D_0(1-\rho/\rho_c)$ [@Lahtinen2001; @Falck2004]. Here $D_0=\kb T/\gamma$ is the bare diffusivity. The commensurate external potential ensures that $\lambda^2 \sim 1/\rho$, and consequently the intrinsic relaxation frequency takes the form $\nu=\rho D(\rho)$. The intrinsic velocity scale is set by $v_0=\lambda/\tau_D= \rho^{1/2}D(\rho)$. Substituting for $v_0$ and $\nu$ in \[eq:current\], the density and frequency dependent current takes the form $$\langle j_y \rangle =\frac{f D_0^2}{D_0^2 \rho^2 (1-\rho/\rho_c)^2+f^2} \rho^{5/2} (1-\rho/\rho_c)^2. \label{eq:current_expression}$$ The resonance in the particle current appears at the ratcheting rate $f=\nu=D_0 \rho (1-\rho/\rho_c)$ and the density dependent amplitude take the form $\kappa \rho v_0 = \kappa D_0 \rho^{3/2}(1-\rho/\rho_c)$. ![(color online ) Plot of the mean-square displacement at density $ \rho \s^2 = 1.04$ for frequencies $f \tau=0.01$ (green), $1.0$ (red) and $30.0$ (blue). The displacements in $\la \D x^2 \ra$ are shown with open symbols and that in $\la \D y^2 \ra$ are represented with filled symbols. Typical trajectories of particles at densities $\rho\sigma^2=1.04$ ($b$) and $0.98$ ($c$) are shown for three different frequencies $f\tau=0.01$, $1$ and $30$. At high frequencies the particles get localized, whereas at very low frequencies localization happens only at densities above $\rho_c$. In the intermediate frequencies, at both densities the trajectories span a length scale of $\approx 20\,\s - 30\,\sigma$. []{data-label="fig:msdplot"}](msd_rho104 "fig:"){width="0.7\linewidth"} ![(color online ) Plot of the mean-square displacement at density $ \rho \s^2 = 1.04$ for frequencies $f \tau=0.01$ (green), $1.0$ (red) and $30.0$ (blue). The displacements in $\la \D x^2 \ra$ are shown with open symbols and that in $\la \D y^2 \ra$ are represented with filled symbols. Typical trajectories of particles at densities $\rho\sigma^2=1.04$ ($b$) and $0.98$ ($c$) are shown for three different frequencies $f\tau=0.01$, $1$ and $30$. At high frequencies the particles get localized, whereas at very low frequencies localization happens only at densities above $\rho_c$. In the intermediate frequencies, at both densities the trajectories span a length scale of $\approx 20\,\s - 30\,\sigma$. []{data-label="fig:msdplot"}](traj_new "fig:"){width="\linewidth"} \[fig:fluxplot\]($a$) shows data collapse of particle currents when plotted as $(\kappa \rho v_0)^{-1} \langle j_y \rangle $ against the dimensionless variable $f/\nu$. The current maximizes at the resonance frequency of $f=\nu$. \[fig:fluxplot\]($b$) shows the limit of validity of the approximate form $v_0 = D_0 \rho^{1/2}(1-\rho/\rho_c)$. A comparison of \[fig:phase\_diagram\] and \[fig:fluxplot\]$a$ shows the relationship between the structure and dynamics. For example, at the resonance frequency, the system melts in order to carry the largest directed current. The dynamics at the local scale can be better appreciated by examining trajectories of individual particles. First we consider the system at high density $\rho \sigma^2=1.04$. Both the components of the displacement fluctuations in \[fig:msdplot\]($a$) show an initial ballistic part $\sim t^2$ due to the inertial nature of the dynamics. Both in the low and high frequency limits, this crosses over to a sub-diffusive regime ($\sim t^\nu$ with $\nu<1$). The test particles show localized motion within cages formed by neighbours (see \[fig:msdplot\]($b$)). The intermediate frequency driving at $f\t=1$ shows a long time diffusive behavior in $x$, $\la \D x^2 \ra \sim t$, and almost ballistic motion in $y$, $\la \D y^2 \ra \sim t^\nu$ with $1 < \nu \lesssim 2$ (\[fig:msdplot\]($a$)). The corresponding particle trajectories display long excursions in the $y$-direction, as is displayed in \[fig:msdplot\]($b$). \[fig:msdplot\]($c$) shows typical particle trajectories at low densities, $\r \s^2 = 0.98$. They get localized only at the highest driving frequencies $f \t=30$. At small and intermediate frequencies they show long excursions which are extended mainly in the $y$-direction, the direction of ratchet drive. The localization of trajectories is related to the low directed current carried by the system, and its ordering into a solid phase. Similarly, the extended particle trajectories are related to the melting of the solid and the presence of relatively large directed current. ![Plots of the static structure factor $\la \psi_{\bf q} \ra$ for the densities $\rho \sigma^2=1.04$ ($a$-$c$) and $0.98$ (figures $d$-$f$). The three columns correspond to three different frequencies $f \tau=0.01$ ($a$ and $d$), $1.00$ ($b$ and $e$) and $30$ ($c$ and $f$).[]{data-label="fig:struc"}](skplane_driven_sys_plot.png){width="8.6cm"} Non-equilibrium melting {#ssec:reentrant_transition} ----------------------- For a more quantitative analysis, we turn our attention to the phase diagram \[fig:phase\_diagram\]($a$) and consider the phase behavior along constant frequency, and constant density lines. The structure factor, $\la \psi_{\bf q} \ra = (1/N)\,\la \rho_{\bf q}\, \rho_{-\bf q} \ra$ (see \[fig:struc\]) with $\rho_{\bf q}=\sum_{j=1}^{N} e^{-{\mathbbm{i}}{\bf q} \cdot \bf{r}_j}$ and $\rho^*_{\bf q}=\rho_{-\bf q}$, can clearly distinguish between a solid, hexatic, liquid and a modulated liquid phase [@Chaikin2012]. In the solid phase $\la \psi_{\bf q} \ra$ shows a characteristic six fold symmetry with peaks at $\G_1=(0,\pm 2 \pi /a_y)$ and $\G_2=(\pm 2\pi/a,\pm \pi/a_y)$ reflecting the underlying triangular lattice structure (see \[fig:struc\]($a$),($c$),($f$)). The six intensity maxima broaden along the constant radius $q=2\pi/a$ circle in a hexatic (\[fig:struc\]($b$),($e$)). In a simple liquid with spherical symmetry, the broadening extends to overlap forming a characteristic ring structure. On the other hand, in a modulated liquid phase, $\la \psi_{\bf q} \ra$ is expected to show two bright spots at $\G_1$, in addition to the ring structure characterizing a simple liquid (see \[fig:struc\]($b$),($e$)). Note that the presence of the external periodic potential in the present context induces an explicit symmetry breaking by imposing density modulations in $y$-direction, ensuring $\la \psi_{\G_1} \ra> \la \psi_{\G_2} \ra$. The other four quasi Bragg peaks at $\G_2$, e.g., at the highest ratcheting frequencies, identify the appearance of the quasi long ranged positional order (QLRO). We use their arithmatic mean as the measure of solid order parameter $\la \psi_{\G_2} \ra$. The QLRO in the solid phase is explicitly demonstrated using the system size dependence of $\la \psi_{\G_2} \ra$ shown in \[fig:scaling\_solid\_order\_parameter\]. The calculations are performed over sub-blocks of sizes $\ell_x \times \ell_y = \ell^2$. Within both the high and low frequency solid regions, $\la \psi_{\G_2}(\ell) \ra \sim \ell^{-\nu}$ where $\nu < 1/3$, the value of the exponent expected at the equilibrium KTHNY (Kosterlitz- Thouless- Halperin- Nelson- Young) melting [@Kosterlitz1973; @Halperin1978; @Young1979]. In this case, the value of the exponent $\nu$ depends on the mean density and ratcheting frequency. ![System size dependence of the solid order parameter. The ratio of mean order parameter measured over blocks of size $\ell= (\ell_x \times \ell_y)^{1/2}$ with respect to that measured over the whole system, $\la \psi_{\mathbf{G}_2}(\ell)\ra/\la \psi_{\mathbf{G}_2}(L) \ra$ decays with $\ell/L$ with power law $(\ell/L)^{-\nu}$. The data is shown for different densities as indicated in the legend. The solid and open symbols denote results at frequency $f\t=30$ and $f\t=0.01$, respectively. The solid black line is a plot of the power-law $(\ell/L)^{-1/3}$ expected at the equilibrium KTHNY melting point.[]{data-label="fig:scaling_solid_order_parameter"}](psiG2_vs_boxL_combi_3.pdf){width="6cm"} ![ Dependence of the ($a$)solid order parameter $\langle \psi_{\bf{G}_2} \rangle $ and ($b$)hexatic order parameter $\langle \psi_6 \rangle$ as a function of frequency is plotted for different densities as indicated in the legend on the top. The equilibrium melting point $\la \psi^{m}_{\bf{G}_2} \ra = 0.376$ is denoted by the dashed lines in ($a$). The variations of ($c$)$\langle \psi_{\bf{G}_2} \rangle$ and ($d$)$\langle \psi_6 \rangle$ are shown as a function of the mean density for three driving frequencies $f\tau = 0.01$, $1$ and $30$ as indicated in the legends. []{data-label="fig:solid_op_freq_density"}](op_combi_2.pdf){width="\linewidth"} The phase behaviors are further characterized by following the change in the hexatic bond orientational order $\la \psi_6 \ra = (1/N) \la | \sum_{i=1}^N \ \vec \psi^i_6|^2 \ra$, where we define the local hexatic order $\vec \psi^i_6 = \sum_{k=1}^{n_v} (\ell_k/\ell) \exp({\mathbbm{i}}6 \phi_{ik} )$ utilizing the $n_v$ Voronoi neighbors of the $i$-th test particle. Here $\phi_{ik}$ is the angle subtended by the bond between the $i$-th particle and its $k$-th Voronoi neighbor. In this definition we used the weighted average over the weight factor $\ell_k/\ell$ such that $\ell = \sum_{k=1}^{n_v} \ell_k$ and $\ell_k$ denotes the length of the Voronoi edge corresponding to the $k$-th topological neighbor [@Mickel2013]. In \[fig:solid\_op\_freq\_density\]($a$) and ($b$) we show the variations of the solid and hexatic orders, $\langle \psi_{\G_2} \rangle$ and $\langle \psi_6 \rangle$, as a function of the ratcheting frequency keeping the density of the system fixed. \[fig:solid\_op\_freq\_density\]($c$) and ($d$) show similar plots, but as a function of the density, keeping the driving frequency fixed. Both the order parameters show non-monotonic variation with frequency. The large variation of the solid order parameter $\la \psi_{\G_2} \ra$ with $f\t$ signifies melting, followed by a [*re-entrant solidification*]{}. Here we use the value of the solid order parameter at the equilibrium melting point, $\la \psi^m_{\G_2} \ra \approx 0.376$ (\[appendix:PT\]), to identify the boundary of solid phase, $\la \psi_{\G_2}\ra \ge \la \psi^m_{\G_2} \ra$. As we show later, the density-density correlation changes from a power law to an exponential decay across this melting. As \[fig:solid\_op\_freq\_density\]($a$) shows, at all the densities considered the system remains in a solid phase at the highest driving frequencies. With reduction of frequency below $f\t \approx 10$, the system melts. At densities $\r \s^2 \gtrsim 1.01$ the system re-solidifies as the frequency is lowered further below $f\t \approx 0.1$. At intermediate to high frequencies, whether the system remains in a solid or fluid phase is essentially determined by the driving frequency, and not the mean density of the system (see \[fig:solid\_op\_freq\_density\]($c$)). Only at the lowest driving frequencies, one finds density dependent solidification. The hexatic order parameter $\la \psi_6\ra$ shows similar variations, albeit with lesser magnitude (see \[fig:solid\_op\_freq\_density\]($b$), ($d$)). At driving frequencies much larger than the inverse relaxation times, the system responds to only a time-integrated potential profile. The corresponding behavior is similar to that in the presence of a time-independent commensurate potential. For equilibrium melting of the free system, and melting in the presence of a time-independent periodic potential commensurate with the density, see \[fig:op\_free\_lif\] presented in \[appendix:PT\]. ![Probability distributions of the local solid and hexatic order parameters $\psi^l_{\bf{G}_2}$ and $|\vec \psi_6^i|^2$. The distribution functions ${\cal P}(\psi^l_{\bf{G}_2})$ at densities $\rho \sigma^2=0.98$($a$) and $\rho \sigma^2=1.04$($b$). The local solid order parameter is determined over subsystems of length $\ell = (\ell_x \times \ell_y)^{1/2}$, where $\ell_x/L_x=\ell_y/L_y=1/14$. The distribution functions of local hexatic order ${\cal P}(|\vec \psi_6^i|^2)$ at $\rho \sigma^2=0.98$($c$) and $1.04$($d$), corresponding to four representative ratcheting frequencies as indicated in the legends.[]{data-label="fig:solid_op_hist_sq"}](lpsiG2_hist_driven_sys_combi.pdf "fig:"){width="\linewidth"} ![Probability distributions of the local solid and hexatic order parameters $\psi^l_{\bf{G}_2}$ and $|\vec \psi_6^i|^2$. The distribution functions ${\cal P}(\psi^l_{\bf{G}_2})$ at densities $\rho \sigma^2=0.98$($a$) and $\rho \sigma^2=1.04$($b$). The local solid order parameter is determined over subsystems of length $\ell = (\ell_x \times \ell_y)^{1/2}$, where $\ell_x/L_x=\ell_y/L_y=1/14$. The distribution functions of local hexatic order ${\cal P}(|\vec \psi_6^i|^2)$ at $\rho \sigma^2=0.98$($c$) and $1.04$($d$), corresponding to four representative ratcheting frequencies as indicated in the legends.[]{data-label="fig:solid_op_hist_sq"}](lpsi6_hist_all_snaps_combi.pdf "fig:"){width="\linewidth"} ![ (Color Online) The pair correlation functions $g(x,0)-g(\infty,0)$ calculated at the mean densities $\rho \sigma^2=0.98$ ($a$) and $1.04$ ($b$) for the frequencies indicated in the legends. []{data-label="fig:gx_plot"}]({gx_combi_3}.pdf){width="0.65\linewidth"} Continuous transition: Distribution of order parameters ------------------------------------------------------- We probe the order of the phase transitions using the distribution of the local solid and hexatic order parameters. In determining the local solid order, we divided the simulation box into sub-boxes of size $\ell = (\ell_x \times \ell_y)^{1/2}$ with $\ell_x/L_x=\ell_y/L_y=1/14$. $\psi^l_{\G_2}$ is then calculated using the definition of $\psi_{\bf q}$ restricted within these sub-boxes. For the local hexatic order parameter, we calculate $|\rho^i_6|^2$ for all the particles. The distribution functions of these quantities, ${\cal P}(\psi^l_{\G_2})$ and ${\cal P}(|\rho_{6}|^2)$ are plotted in \[fig:solid\_op\_hist\_sq\]. They remain unimodal at all points of the phase diagram. At low densities, $\r \s^2 = 0.98$ in \[fig:solid\_op\_hist\_sq\]($a$), the maximum of ${\cal P}(\psi^l_{\G_2})$ appears at an order parameter corresponding to the solid phase only at the highest frequencies. With decreasing frequency the peak shifts towards lower values, signifying melting below $f\t = 10$ and remain low at the lowest frequencies. The unimodal nature of the distribution function, as the peak shifts to lower values, signifies the absence of any metastable state across the transition, a characteristic of continuous transitions. At high densities, e.g., $\r \s^2 = 1.04$ in \[fig:solid\_op\_hist\_sq\]($b$), corresponding to the re-entrant transition, the peak of the distribution ${\cal P}(\psi^l_{\G_2})$ shifts from a high to low and back to high values as the ratcheting rate decreases from the highest frequencies. As before, the unimodal nature of the distribution corresponds to a continuous melting transition. The solid melts to a hexatic phase, characterized by the finite hexatic order. However, a further melting of the hexatic is not observed as the frequency is varied. In the density-frequency range bounded between the two dashed lines with open inverted triangles and open circles denoted in \[fig:phase\_diagram\] ($a$), the system remains in a hexatic phase. This is corroborated by the distribution of the local hexatic order ${\cal P}( |\rho_6|^2)$ shown in \[fig:solid\_op\_hist\_sq\]($c$) and ($d$) corresponding to densities $\rho \sigma^{2}=0.98$ and $1.04$, respectively. The uni-modal nature of the distribution with a roughly unchanged peak position and a fat tail persists throughout the frequency range. The peak of the distribution does not shift. However, it is important to note that deep inside the hexatic phase, near $f \t = 1$, a significant fraction of the system displays vanishing hexatic order. This is more prominent at lower densities (see \[fig:solid\_op\_hist\_sq\]($c$)). As we show in a later section, local dip in the hexatic order is associated with the formation of grain boundaries. Melting of solid: Correlation functions {#ssec:pair_correlation} --------------------------------------- The pair correlation functions $g(x,y)$ capture the solid melting (see \[appendix:pair\_correlation\_function\] for further details). A density modulation is externally induced in the system along the $y$-direction by the ratcheting drive, breaking the translational symmetry in that direction, explicitly. To study the spontaneous symmetry breaking, here we focus on the $x$-component of the two point correlation functions $g(x,y)$. The component of the correlation $g(x,0)-g(\infty,0)$ along the axis perpendicular to the direction of the ratcheting drive are shown in \[fig:gx\_plot\]. This provides a more conclusive evidence to the nature of the phases in different density and frequency regimes. At low densities, as we have shown before, the solid order exists only at the highest frequencies. At such frequencies we find an algebraic decay of the correlation along the $x$ direction, signifying the QLRO (see \[fig:gx\_plot\]($a$)). At intermediate and low frequencies the system melts. This is captured by the exponential decay of the correlation with correlation length $\sim 10\sigma$ (see \[fig:gx\_plot\]($a$)). The scenario changes at higher densities. At high frequency, as before, we again find a solid phase, with the correlation exhibiting algebraic decay corresponding to the QLRO (\[fig:gx\_plot\]($b$)). At high densities, one obtains another solid phase at the low ratcheting frequencies. This also shows algebraic decay of correlations signifying QLRO (see $f\t =0.01$ graph in \[fig:gx\_plot\]($b$)). The power law $x^{-1/3}$ shown in the figures denote the expected correlation at the onset of the equilibrium KTHNY melting. At the intermediate driving frequencies, $f\t=1$ in $\r\s^2=1.04$ system, the correlation shows exponential decay with a correlation length $\sim 10\,\s$, similar to the behavior observed in the low density regime (\[fig:gx\_plot\]($b$)). The change from algebraic to exponential decay is utilized to identify the solid melting points in the phase diagram, as is detailed further in \[appendix:pair\_correlation\_function\]. Defect formation ---------------- The equilibrium melting of the two- dimensional solid within the continuous KTHNY melting scenario is known to proceed by the unbinding of dislocation pairs into free dislocations. To identify such topological defects, we first obtain the coordination number $n_v$ of each particle in the system counting the number of its Voronoi neighbours. In a perfect triangular lattice $n_v=6$ for all particles. We follow $n_v \neq 6$ particles to identify the $n_v$-fold defects. Even within the solid phase, fluctuations of bound quartets of $5-7-5-7$ defects (bound dislocation pairs) keep appearing. They form dislocations by dissociating into separate $5-7$ and $7-5$ non-neighboring pairs. Presence of a finite fraction of particles associated with dislocations characterize the hexatic phase. The system shows dislocation formation as the solid melts. Moreover, we find defect clusters larger than quartets that are either compact or string-like (grain boundary) [@Qi:2014]. All the dominant defect types observed in our simulations are indicated in \[fig:defect\_types\]($f$). Their typical configurations in a sub-volume of size $100 \s \times 100 \s$ at $\r \s^2=1.04$ and different ratcheting frequencies are shown in \[fig:defect\_types\]($a$)-($d$). In these figures, the colors associated to particles indicate the number of topological neighbors they have, $n_v=4\,$(purple), $5\,$(red), $6\,$(green), $7\,$(blue), $8\,$(yellow). Clearly, defect formation is suppressed at both the extremities of the ratcheting frequency. It increases significantly in the intermediate frequency regime associated with solid melting (\[fig:defect\_types\] ($b$)). The relative fraction of different defect types also vary with the driving frequency. In the highest frequency solids, only bound quartets (bound dislocation pairs) are observed in \[fig:defect\_types\] $(d)$. As the solid melts with decreasing frequency, dislocations and defect clusters start to appear and eventually dominate over the quartets in the system (\[fig:defect\_types\] $(b)$ and ($c$)). The string-like defects remain extended along the $y$-direction, the direction of particle current under ratcheting. Such a connected string of defects is shown in \[fig:defect\_types\] $(b)$ and has been highlighted in \[fig:defect\_types\] $(e)$, which shows Voronoi diagram of a region containing the connected string of defects. The color code in each Voronoi cell denotes the amount of hexatic order, and the arrows denote the corresponding hexatic orientations. At the location of the connected clusters of defects, the local hexatic order is low, and shows a hexatic orientation approximately orthogonal to the neighboring defect-free regions. With further lowering of the ratcheting frequency below $f_s$, the defect fraction decreases strongly. This description is quantified by focusing on the time evolution of defect fractions. We first consider the evolution of the total fraction of all the topological defects, the percentage of particles having non-six Voronoi neighbors $N_d = (1 - n_6/N)\times 100$, where $n_6$ denotes the total number of particles with $n_v=6$ (\[fig:defect\_dynamics\] $(a)$). Clearly, the largest value of $N_d$ with the strongest fluctuations appear at the intermediate frequencies. The defect formation gets dramatically suppressed in the solid phase corresponding to the high ratcheting frequencies. At the lowest frequencies ($f\t=0.01$ in \[fig:defect\_dynamics\] $(a)$), $N_d$ remains relatively low and follows the switching of external potential. The mean value $\la N_d \ra$ remains less than $4\%$ and varies non-monotonically with $f\t$ (\[fig:defect\_dynamics\] $(b)$). It shows a maximum at the resonance frequency corresponding to the largest directed current, relating formation of topological defects with carrying capacity of particle current in the system. Further insight into the structure- dynamics relations can be obtained by following the behavior of different defect fractions separately. For this purpose, the percentage fraction $d_f$ of a defect type is defined as $d_f=(n_d/N) \times 100$, where $n_d$ is the total number of $n_v\neq 6$ particles that may contribute to either a quartet, a dislocation, a cluster, or a disclination as described above. The time averaged percentage fractions of these topological defects as a function of the driving frequency is shown in \[fig:defect\_dynamics\] $(c)$. They exhibit a similar non-monotonic behavior as $N_d$ and the mean particle current. In the high frequency solid, the dominant defects are the quartets. As the frequency is decreased, the melting of the solid is mediated by the unbinding of these quartets into dislocations. The dislocation fraction becomes larger than that of quartets. More importantly, at the resonance melting, the formation of defect clusters dominate (\[fig:defect\_dynamics\]$(c)$). The fraction of disclinations remain relatively insignificant (less than $0.03\%$), about two orders of magnitude smaller than that of the defect clusters. This is consistent with the fact that the hexatic does not melt within these parameter regimes. Discussion {#sec:outlook} ========== In conclusion, using a large scale simulation involving $262144$ particles, we have presented a detailed study of a ratcheted two-dimensional colloidal suspension, focusing on the structure- dynamics relationship. The mean directed particle current driven by the ratchet exhibits a resonance behavior. Associated with this, the solid melts to hexatic providing a mechanism allowing directed transport. The system exhibits a rich non-equilibrium phase diagram as a function of the driving frequency and mean density. At high densities, we found a re-entrant melting transition as a function of ratcheting frequency. The different phases are characterized by the spatially resolved density profile, the density-density correlation function, the structure factor, the solid and hexatic order parameters, and their distribution functions. The role of the defects in the phase transition has been investigated in detail. The solid- melting is associated with formation of dislocations, but unlike the equilibrium two- dimensional melting, this non-equilibrium melting is dominated by the formation of defect clusters, connected strings of defects that remain oriented largely along the direction of the ratcheting drive. Remarkably, the driven hexatic does not melt to a fluid within the studied range of density and ratcheting drive. Our detailed predictions regarding the variation of particle current and associated phase transitions can be verified using colloidal particles and optical [@Faucheux1995] or magnetic ratcheting [@Tierno2012] in a suitable laser trapping setup [@Wei1998]. The impact of changing degree of potential asymmetry on the dynamics and phase behavior remains an interesting future direction of study. Acknowledgements {#acknowledgements .unnumbered} ================ DC thanks ICTS-TIFR, Bangalore for an associateship, and SERB, India, for financial support through grant number EMR/2016/001454. Relaxation under external potential {#appendix:relaxtion} =================================== ![Relaxation dynamics of the solid and hexatic order parameters at $\r \s^2 =1.04$($a$) and $\r \s^2 =0.98$($b$), respectively. The time scales for relaxation are $f_s^{-1} \approx 6\,\t$($a$) and $f_s^{-1}\approx 70\,\t$($b$).[]{data-label="fig:relaxation_frequency"}](relaxation_combi_new2.pdf){width="50.00000%"} ![Plot of the solid order parameter $\la \psi_{\bf{G}_2}\ra$ and the hexatic order parameter $\la \psi_6 \ra$ for the equilibrium phase transition and the laser induced freezing transition. The green dashed line denotes solid melting point $\la \psi_{{\bf G}_2}^m \ra=0.376$, and the black dash-dotted line denotes the hexatic melting point $\la \psi_6 \ra=0.06$.[]{data-label="fig:op_free_lif"}](ops_appen2_plot.pdf){width="50.00000%"} In this section, we show the results for the relaxation time-scales of the solid and the hexatic order parameters after withdrawing an external potential commensurate with the system density under which the system is initially equilibrated. This time-scale at different densities are determined from separate simulations. The initial equilibration is performed under a time-independent external potential of the form given in \[sec:model\_simulation\] with $V(t)=\epsilon$ over $10^7$ simulation steps. Thereafter, the external potential is removed and the evolution of the solid and the hexatic order parameters are measured over time. In \[fig:relaxation\_frequency\] we show the time evolution of these quantities at mean densities $\rho \sigma^2=1.04$ (figure $a$) and $0.98$ (figure $b$). At densities higher than the equilibrium melting point $\r_m \s^2 \approx 1.01$, the solid and the hexatic order parameters, $ \psi_{\G_2}$ and $\psi_6$, decay to finite values, indicating order even in the absence of external potential (\[fig:relaxation\_frequency\]($a$)). In contrast, at densities $\r < \r_m$, the solid and the hexatic order parameters vanish with time (\[fig:relaxation\_frequency\]($b$)). The time-scale of such decay, indicated in \[fig:relaxation\_frequency\], gives the estimate of the relevant relaxation time. If a time-independent external potential switches with a rate slower than this time-scale, the system will have enough time to [*equilibrate*]{} to instantaneous potential profiles. The relaxation frequency $f_s$ is inverse of this time-scale, and has been indicated by the dash-dotted line through $\blacksquare$ symbols in the phase diagram \[fig:phase\_diagram\]. Equilibrium Phase Transition {#appendix:PT} ============================ One way we identified the solid melting in the main text is by choosing an appropriate cutoff value of the solid order parameter $\la \psi_{\G_2} \ra$. For this, we used the equilibrium melting point in the absence of external potential. To demonstrate this, we perform separate molecular dynamics simulations of particles interacting via the soft-core potential. The temperature of the system is kept fixed at $T=1.0 \e/\kb$ using a Langevin heat bath. The corresponding variation of $\la \psi_{\G_2}\ra$ and $\la \psi_6 \ra$ with the mean density is shown in \[fig:op\_free\_lif\]($a$). The order parameters change continuously from low to high values. The solid melting point is separately determined by following the pressure-density curve and the density correlation function as in Ref. (data not shown). This melting point is found at density $\rho_m \sigma^2 \approx 1.014$, where the solid order parameter $\la \psi_{\G_2}^m\ra \approx 0.376$. The hexatic order remains significantly large even at lower densities. The melting of hexatic to liquid is identified from the change in correlation of the hexatic order parameter $\la g_6(r) \ra$, which transforms from an algebraic decay in the hexatic phase to an exponential decay in the liquid. The hexatic- melting is obtained at $\rho \sigma^2 \approx 0.998$, where $\la \psi_6 \ra \approx 0.06$. In \[fig:op\_free\_lif\]($b$) we show a similar plot for the two order parameters but in the presence of a time- independent potential of the form $U_{ext}(x,y)$ given in \[sec:model\_simulation\] with $V(t)=\epsilon$. In the presence of this potential, the hexatic does not melt in the regime of $\r \s^2 \geq 0.94$. The external potential maintains a significant hexatic order, with a value greater than $\la \psi_6 \ra =0.51$, although the solid order does drop below $0.376$ at $\r \s^2 \approx 0.96$. ![ Pair correlation functions in the driven 2D colloidal suspension at densities $\rho\sigma^2=1.04$ in the top panels ($a$-$c$) and $\rho\sigma^2$=0.98 in the bottom panels ($d$-$f$) corresponding to three different frequencies $f\tau=0.01$ (left column), $f\tau=1$ (middle column) and $f\tau=30$ (right column). []{data-label="fig:gxy_plot"}](gxy_combi_1.png){width="0.95\linewidth"} Non-equilibrium melting and pair correlation {#appendix:pair_correlation_function} ============================================ The two dimensional pair correlation functions $g(x,y) = \la \r(x,y) \r(0,0)\ra/ \la \r \ra^2$ at $\rho \sigma^2=0.98$ and $\rho \sigma^2=1.04$ are shown in \[fig:gxy\_plot\] at the three representative frequencies: low ($f\tau=0.01$), intermediate ($f\tau=1$) and high frequency ($f\tau=30$). The figures show the correlations over a length scale of $\pm 9\s$. While the clear contrast in \[fig:gxy\_plot\]($a$), ($c$) and ($f$) demonstrate the triangular lattice symmetry, the local diffused approximately triangular structures in the other $g(x,y)$ figures are characteristic of the hexatic phase. {width="\textwidth"} The component of the pair correlation function along the minima of the potential $g(x,0)-g(\infty,0)$ changes from a power-law to exponential decay with changing ratcheting frequency, identifying the melting point of a quasi- long ranged ordered solid. In \[fig:gx\_trans\], $(a)$–$(h)$ show the high-frequency melting, while $(i)$–$(p)$ show possible melting at low frequencies. At $\r \s^2 \lesssim 1.03$ and low frequencies, the decay of the pair correlation function remains always exponential, identifying an absence of transition. A crossover to an algebraic decay in $g(x,0)-g(x,\infty)$ appears at $\r \s^2 \gtrsim 1.03$, resulting in solid- hexatic transition points. The phase boundaries displayed in \[fig:phase\_diagram\] ($a$) are consistent with the transition points obtained from this analysis. @ifundefined [39]{} \[1\][\#1]{} \[1\] \[2\] \[3\] \[3\] F. Julicher, A. Ajdari and J. Prost, *Reviews of Modern Physics*, 1997, **69**, 1269–1282 R. D. Astumian and P. Hänggi, *Physics Today*, 2002, **55**, 33 P. Hänggi, *Reviews of Modern Physics*, 2009, **81**, 387–442 P. Reimann, *Physics Reports*, 2002, **361**, 57–265 P. Brouwer, *Phys. Rev. B*, 1998, **58**, R10135–R10138 R. Citro, N. Andrei and Q. Niu, *Phys. Rev. B*, 2003, **68**, 165312 K. Jain, R. Marathe, A. Chaudhuri and A. Dhar, *Phys. Rev. Lett.*, 2007, **99**, 190601 D. Chaudhuri and A. Dhar, *EPL (Europhysics Letters)*, 2011, **94**, 30006 D. Chaudhuri, A. Raju and A. Dhar, *Phys. Rev. E*, 2015, **91**, 050103 D. Chaudhuri, *J. Phys. Conf. Ser.*, 2015, **638**, 012011 D. C. Gadsby, A. Takeuchi, P. Artigas and N. Reyes, *Philos. Trans. R. Soc. B Biol. Sci.*, 2009, **364**, 229–238 L. Faucheux, L. Bourdieu, P. Kaplan and A. Libchaber, *Physical Review Letters*, 1995, **74**, 1504–1507 B. Lopez, N. Kuwada, E. Craig, B. Long and H. Linke, *Physical Review Letters*, 2008, **101**, 220601 P. Tierno, P. Reimann, T. H. Johansen and F. Sagués, *Physical Review Letters*, 2010, **105**, 230602 P. Tierno, *Physical Review Letters*, 2012, **109**, 198304 J. Rousselet, L. Salome, A. Ajdari and J. Prost, *Nature*, 1994, **370**, 446 S. Leibler, *Nature*, 1994, **370**, 412 C. Marquet, A. Buguin, L. Talini and P. Silberzan, *Physical Review Letters*, 2002, **88**, 168301 I. Derényi and T. Vicsek, *Physical review letters*, 1995, **75**, 374 I. Derényi and A. Ajdari, *Physical Review E*, 1996, **54**, R5–R8 R. Marathe, K. Jain and A. Dhar, *J. Stat. Mech. Theory Exp.*, 2008, **2008**, P11014 S. Savel’ev, F. Marchesoni and F. Nori, *Phys. Rev. E*, 2004, **70**, 061107 A. Pototsky, A. J. Archer, M. Bestehorn, D. Merkt, S. Savel’ev and F. Marchesoni, *Phys. Rev. E*, 2010, **82**, 030401 S. Savel’ev, F. Marchesoni and F. Nori, *Phys. Rev. Lett.*, 2003, **91**, 010601 D. Chakraborty and D. Chaudhuri, *Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, 2015, **91**, 050301(R) A. Chowdhury, B. J. Ackerson and N. A. Clark, *Phys. Rev. Lett.*, 1985, **55**, 833 Q.-H. Wei, C. Bechinger, D. Rudhardt and P. Leiderer, *Physical Review Letters*, 1998, **81**, 2606–2609 E. Frey, D. R. Nelson and L. Radzihovsky, *Phys. Rev. Lett.*, 1999, **83**, 2977 D. Chaudhuri and S. Sengupta, *Physical Review E*, 2006, **73**, 11507 J. M. Kosterlitz and D. J. Thouless, *J. Phys. C*, 1973, **6**, 1181 B. I. Halperin and D. R. Nelson, *Phys. Rev. Lett.*, 1978, **41**, 121–124 A. P. Young, *Phys. Rev. B*, 1979, **19**, 1855 S. C. Kapfer and W. Krauth, *Physical Review Letters*, 2015, **114**, 035702–5 D. Frenkel and B. Smit, *[Understanding molecular simulation: from algorithms to applications]{}*, Academic press, NY, 2002 J. Lahtinen, T. Hjelt, T. Ala-Nissila and Z. Chvoj, *Physical Review E*, 2001, **64**, 021204 E. Falck and J. Lahtinen, *The European Physical Journal E*, 2004, **13**, 267 P. M. Chaikin and T. C. Lubensky, *Principles of Condensed Matter Physics*, Cambridge University Press, Cambridge, 2012 W. Mickel, S. C. Kapfer, G. E. Schr[ö]{}der-Turk and K. Mecke, *J. Chem. Phys.*, 2013, **138**, 044501 W. Qi, A. P. Gantapara and M. Dijkstra, *Soft Matter*, 2014, **10**, 5449–5457

--- abstract: 'The two-dimensional $\mathcal{N}=(2,2)$ Wess–Zumino (WZ) model with a cubic superpotential is numerically studied with a momentum-cutoff regularization that preserves supersymmetry. A numerical algorithm based on the Nicolai map is employed and the resulting configurations have no autocorrelation. This system is believed to flow to an $\mathcal{N}=(2,2)$ superconformal field theory (SCFT) in the infrared (IR), the $A_2$ model. From a finite-size scaling analysis of the susceptibility of the scalar field in the WZ model, we determine $1-h-\Bar{h}=0.616(25)(13)$ for the conformal dimensions $h$ and $\Bar{h}$, while $1-h-\Bar{h}=0.666\dots$ for the $A_2$ model. We also measure the central charge in the IR region from a correlation function between conserved supercurrents and obtain $c=1.09(14)(31)$ ($c=1$ for the $A_2$ model). These results are consistent with the conjectured emergence of the $A_2$ model, and at the same time demonstrate that numerical studies can be complementary to analytical investigations for this two-dimensional supersymmetric field theory.' address: - 'Graduate School of Science, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima-ku, Tokyo 171-8501, Japan' - 'Theoretical Research Division, RIKEN Nishina Center, Wako 2-1, Saitama 351-0198, Japan' author: - Syo Kamata - Hiroshi Suzuki bibliography: - '&lt;your-bib-database&gt;.bib' title: 'Numerical simulation of the $\mathcal{N}=(2,2)$ Landau–Ginzburg model' --- Supersymmetry,Non-perturbative study,Landau–Ginzburg model,Nicolai map Introduction ============ It is believed that the infrared (IR) limit of the two-dimensional $\mathcal{N}=(2,2)$ Wess–Zumino model[^1] (2D $\mathcal{N}=(2,2)$ WZ model) with a quasi-homogeneous superpotential[^2] is a non-trivial $\mathcal{N}=(2,2)$ superconformal field theory (SCFT) [@Kastor:1988ef; @Vafa:1988uu; @Lerche:1989uy; @Howe:1989qr; @Cecotti:1989jc; @Howe:1989az; @Cecotti:1989gv; @Cecotti:1990kz; @Witten:1993jg]. See Section 19.4 of Ref. [@Polchinski:1998rr] and Section 14.4 of Ref. [@Hori:2003ic] for reviews. This Landau–Ginzburg (LG) description [@Zamolodchikov:1986db] of $\mathcal{N}=(2,2)$ SCFT is a remarkable non-perturbative phenomenon in field theory and physically, for example, provides a basis for application of the gauged linear sigma model [@Witten:1993yc] to the Calabi-Yau compactification. Although the emergence of SCFT has been tested in various ways, it is very difficult to confirm this phenomenon directly in correlation functions, because the 2D WZ model is strongly coupled in low energies. Application of conventional numerical techniques (such as the lattice) would not be straightforward either, because supersymmetry (SUSY) must be essential in the above non-perturbative dynamics. In a recent interesting paper [@Kawai:2010yj], Kawai and Kikukawa revisited this problem and they computed non-perturbatively some correlation functions in the 2D WZ model by employing a lattice formulation of Ref. [@Kikukawa:2002as]. They considered the 2D $\mathcal{N}=(2,2)$ WZ model with a massless cubic superpotential $$W(\Phi)=\frac{\lambda}{3}\Phi^3, \label{eq:(1.1)}$$ which, according to the conjectured correspondence, should provide a LG description of a pair of the $\mathcal{N}=2$ $c=1$ minimal models, where one is left-moving and the other is right-moving (the so-called $A_2$ model). In the IR limit, the scalar field in the WZ model $A$ is identified with a chiral primary field in the $A_2$ model with the conformal dimensions $(h,\Bar{h})=(1/6,1/6)$ and $U(1)$ charges $(q,\Bar{q})=(1/3,1/3)$. (The complex conjugate $A^*$ is identified with an anti-chiral primary field with $(h,\Bar{h})=(1/6,1/6)$ and $(q,\Bar{q})=(-1/3,-1/3)$.) The authors of Ref. [@Kawai:2010yj] obtained finite-size scalings of scalar two-point functions which are remarkably consistent with the above SCFT correspondence, thus demonstrated the power of a lattice formulation of this supersymmetric field theory.[^3] In this paper, motivated by the success of Ref. [@Kawai:2010yj], we study the 2D $\mathcal{N}=(2,2)$ WZ model with massless cubic superpotential  numerically. We employ a non-perturbative formulation advocated in Ref. [@Kadoh:2009sp] that uses a simple momentum cutoff regularization. Although there is an issue concerning the locality in this formulation, the restoration of an expected locality property can be shown at least within perturbation theory [@Kadoh:2009sp]. This formulation possesses very nice symmetry properties: it exactly preserves full SUSY, translational invariance, and linear internal symmetries such as the $R$-symmetry. We believe that these nice symmetry properties are especially useful in defining Noether currents in the regularized framework. In fact, by defining conserved supercurrents and identifying the component of the superconformal currents in the IR limit, we numerically measure the central charge of the system in the IR region: Together with a measurement of the conformal dimension, this forms a main result of the present paper. Throughout this paper, Greek indices from the middle of the alphabet, $\mu$, $\nu$, … run over $0$ and $1$. Greek indices from the beginning $\alpha$, $\beta$, … are for spinor indices and run over $1$ and $2$. Repeated indices are *not* summed over unless explicit summation symbol is indicated. We extensively use the complex coordinates defined by $$z\equiv x_0+ix_1,\qquad\Bar{z}\equiv x_0-ix_1,$$ and $$\partial_z\equiv\frac{1}{2}(\partial_0-i\partial_1),\qquad \partial_{\Bar{z}}\equiv\frac{1}{2}(\partial_0+i\partial_1).$$ Conjugate momenta are defined by $$p_z\equiv\frac{1}{2}(p_0-ip_1),\qquad p_{\Bar{z}}\equiv\frac{1}{2}(p_0+ip_1).$$ Two-dimensional gamma matrices are defined by $$\gamma_0\equiv\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad \gamma_1\equiv\begin{pmatrix}0&i\\-i&0\end{pmatrix},$$ and $$\gamma_z\equiv\begin{pmatrix}0&1\\0&0\end{pmatrix},\qquad \gamma_{\Bar{z}}\equiv\begin{pmatrix}0&0\\1&0\end{pmatrix}.$$ Supersymmetric formulation of the 2D $\mathcal{N}=(2,2)$ WZ model ================================================================= We start by recapitulating the formulation of Ref. [@Kadoh:2009sp]. We suppose that the system is defined in a finite box with a physical size $L_0\times L_1$. The Fourier modes $\Tilde{f}(p)$ of a periodic function in the box are defined by $$f(x)=\frac{1}{L_0L_1}\sum_pe^{ipx}\Tilde{f}(p),\qquad \Tilde{f}(p)=\int d^2x\,e^{-ipx}f(x),$$ where the momentum $p$ takes discrete values $$p_\mu=\frac{2\pi}{L_\mu}\,n_\mu,\qquad n_\mu=0,\pm1,\pm2,\dots,$$ and by the definition, $$\Tilde{f^*}(p)=\Tilde{f}(-p)^*,$$ where the left-hand side denotes the Fourier transformation of the complex conjugate of $f(x)$, $f(x)^*$. In the present formulation [@Kadoh:2009sp], we restrict the momentum $p$ by an ultraviolet (UV) cutoff $\Lambda$, $$-\Lambda\leq p_\mu\leq\Lambda,\qquad\text{for $\mu=0$, $1$}. \label{eq:(2.4)}$$ We parametrize this UV cutoff by a “lattice spacing” $a$, $$\Lambda\equiv\frac{\pi}{a}, \label{eq:(2.5)}$$ although we do not assume an underlying spacetime lattice structure in this paper (see below). Throughout this paper, all dimensionful quantities are measured in units of the lattice spacing. In particular, if we define the number of lattice points $N_\mu$ by $$L_\mu=N_\mu a,$$ then imposing Eq.  implies $$-\frac{N_\mu}{2}\leq n_\mu\leq\frac{N_\mu}{2},\qquad\text{for $\mu=0$, $1$},$$ and thus the number of points in the momentum grid is given by $\sum_p1=(N_0+1)(N_1+1)$. In the present formulation of the 2D $\mathcal{N}=(2,2)$ WZ model, the partition function is defined by $$\mathcal{Z}\equiv \int\prod_{-\Lambda\leq p_\mu\leq\Lambda} \left[d\Tilde{A}(p)\,d\Tilde{A^*}(p) \prod_{\alpha=1}^2d\Tilde{\psi}_\alpha(p) \prod_{\Dot{\alpha}=\Dot{1}}^{\Dot{2}}d\Tilde{\Bar{\psi}}_{\Dot{\alpha}}(p) \,d\Tilde{F}(p)\,d\Tilde{F^*}(p)\right]e^{-S}, \label{eq:(2.8)}$$ where $A$, $(\psi_\alpha,\Bar{\psi}_{\Dot{\alpha}})$, and $F$ are the scalar, fermion and auxiliary fields, respectively. In this expression, the action $S$ is simply the action of the continuum WZ model in terms of Fourier modes: $$\begin{aligned} S&=\frac{1}{L_0L_1}\sum_p \Biggl[4p_z\Tilde{A^*}(-p)p_{\Bar{z}}\Tilde{A}(p) -\Tilde{F^*}(-p)\Tilde{F}(p) \notag\\ &\qquad\qquad\qquad{} -\Tilde{F^*}(-p)*W'(\Tilde{A})^*(p) -\Tilde{F}(-p)*W'(\Tilde{A})(p) \notag\\ &\qquad\qquad\qquad{} +(\Tilde{\Bar{\psi}}_{\Dot{1}},\Tilde{\psi}_2)(-p) \begin{pmatrix} 2ip_z&W''(\Tilde{A})^**\\ W''(\Tilde{A})*&2ip_{\Bar{z}}\\ \end{pmatrix} \begin{pmatrix} \Tilde{\psi}_1\\ \Tilde{\Bar{\psi}}_{\Dot{2}}\\ \end{pmatrix}(p) \Biggr], \label{eq:(2.9)}\end{aligned}$$ where the holomorphic function $W(A)$ is the superpotential and $*$ denotes the convolution $$\left(\Tilde{\varphi}_1*\Tilde{\varphi}_2\right)(p)\equiv \frac{1}{L_0L_1}\sum_q \Tilde{\varphi}_1(q)\Tilde{\varphi}_2(p-q).$$ Products contained in $W'(\Tilde{A})$ and $W''(\Tilde{A})$ are also understood as convolutions. Since action  is identical to that in the continuum theory, the regularized theory  is manifestly invariant under all symmetries that are consistent with the momentum restriction . This is the case for symmetry transformations that act linearly on field variables. Thus, in the present formulation, SUSY, translational invariance, and the $R$-symmetry (if it exists) are exactly preserved. One can derive Ward–Takahashi (WT) identities associated with these symmetries in a regularized framework. What is sacrificed in the present formulation, on the other hand, is locality. One sees that the kinetic terms and the interaction terms are quite non-local in the configuration space when the UV cutoff $\Lambda$ is finite. In fact, when both integers $N_0$ and $N_1$ are odd, the present formulation is nothing but a two-dimensional version of a lattice formulation of the four-dimensional WZ model studied in Ref. [@Bartels:1983wm] that is based on the SLAC derivative [@Drell:1976bq; @Drell:1976mj]. A detailed analysis of a one-dimensional version (i.e., quantum mechanics) can be found in Ref. [@Bergner:2009vg]. Thus there is an issue of whether the present formulation reproduces an IR physics expected in the original target theory (i.e., whether it belongs to the same universality class or not). Although one can show [@Kadoh:2009sp] that within perturbation theory the expected locality is restored in the limit $\Lambda\to\infty$ for two- and three-dimensional WZ models, the validity of the present formulation at the non-perturbative level is not obvious *a priori*. We believe that our results in this paper will provide an affirmative indication regarding this question of locality. As a side remark, we note that the present formulation cannot be regarded as a spacetime lattice formulation when either $N_0$ or $N_1$ is even. Fourier modes $\Tilde{f}(p)$ of a function $f(x)$ defined on a spacetime lattice is periodic in the Brillouin zone $\Tilde{f}(p+(\pi/a)\Hat{\mu})=\Tilde{f}(p)$, where $\Hat{\mu}$ denotes a unit vector in the $\mu$ direction. However, the combination $p_\mu\Tilde{f}(p)$ appearing in the action breaks this periodicity at the boundary of the Brillouin zone and this cannot be regarded as a Fourier transformation of a function defined on a lattice. (When $N_\mu$ are odd, there is no Fourier mode on the boundary of the Brillouin zone.) Throughout this paper we set $N_\mu$ even and this means that we lose a connection with a spacetime lattice formulation; we have to regard our configuration space as continuous (with a limited resolution). Still, Eq.  provides a regularized partition function and can serve as a starting point for non-perturbative study. Simulation algorithm based on the Nicolai map ============================================= After integrating over the auxiliary fields $\Tilde{F}$ and $\Tilde{F^*}$ in Eq. , the partition function becomes $$\mathcal{Z}\equiv \int\prod_{-\Lambda\leq p_\mu\leq\Lambda} \left[d\Tilde{A}(p)\,d\Tilde{A^*}(p) \prod_{\alpha=1}^2d\Tilde{\psi}_\alpha(p) \prod_{\Dot{\alpha}=\Dot{1}}^{\Dot{2}} d\Tilde{\Bar{\psi}}_{\dot\alpha}(p) \right]e^{-S},$$ where $$\begin{aligned} S&=\frac{1}{L_0L_1}\sum_p \biggl[\Tilde{N^*}(-p)\Tilde{N}(p) \notag\\ &\qquad\qquad\qquad{}+ (\Tilde{\Bar{\psi}}_{\dot1},\Tilde{\psi}_2)(-p) \begin{pmatrix} 2ip_z&W''(\Tilde{A})^**\\ W''(\Tilde{A})*&2ip_{\Bar{z}}\\ \end{pmatrix} \begin{pmatrix} \Tilde{\psi}_1\\ \Tilde{\Bar{\psi}}_{\dot2}\\ \end{pmatrix}(p) \Biggr].\end{aligned}$$ In this expression, $\Tilde{N}(p)$ is a function that specifies the Nicolai map [@Nicolai:1979nr; @Nicolai:1980jc; @Cecotti:1981fu; @Parisi:1982ud; @Cecotti:1982ad], $$\begin{aligned} &\Tilde{N}(p)\equiv2ip_z\Tilde{A}(p)+W'(\Tilde{A})^*(p), \\ &\Tilde{N^*}(-p)=\Tilde{N}(p)^* =-2ip_{\Bar{z}}\Tilde{A^*}(-p)+W'(\Tilde{A})(-p).\end{aligned}$$ For example, for cubic superpotential , the explicit form of the function $\Tilde{N}(p)$ is given by $$\begin{aligned} \Tilde{N}(p)&= i(p_0-ip_1)\Tilde{A}(p) +\lambda\frac{1}{L_0L_1}\sum_q\Tilde{A^*}(q)\Tilde{A^*}(p-q) \notag\\ &=i(p_0-ip_1)\Tilde{A}(p) +\lambda\frac{1}{L_0L_1}\sum_q\Tilde{A}(q)^*\Tilde{A}(-p-q)^*.\end{aligned}$$ We then note $$S=\frac{1}{L_0L_1}\sum_p \left[\Tilde{N^*}(-p)\Tilde{N}(p) +(\Tilde{\Bar{\psi}}_{\dot1},\Tilde{\psi}_2)(-p) \begin{pmatrix} \frac{\partial\Tilde{N}(p)}{\partial\Tilde{A}(p)} &\frac{\partial\Tilde{N}(p)}{\partial\Tilde{A^*}(p)}*\\ \frac{\partial\Tilde{N^*}(p)}{\partial\Tilde{A}(p)}* &\frac{\partial\Tilde{N^*}(p)}{\partial\Tilde{A^*}(p)}\\ \end{pmatrix} \begin{pmatrix} \Tilde{\psi}_1\\ \Tilde{\Bar{\psi}}_{\dot2}\\ \end{pmatrix}(p) \right],$$ and therefore, after integrating over fermion fields, $$\mathcal{Z}\equiv \int\prod_{-\Lambda\leq p_\mu\leq\Lambda} \left[d\Tilde{A}(p)\,d\Tilde{A^*}(p)\right] \exp\left[\frac{1}{L_0L_1}\sum_p\Tilde{N^*}(-p)\Tilde{N}(p)\right] \det\frac{\partial(\Tilde{N},\Tilde{N^*})}{\partial(\Tilde{A},\Tilde{A^*})}.$$ In this partition function, we may change integration variables from $(\Tilde{A}(p),\Tilde{A^*}(p))$ to $(\Tilde{N}(p),\Tilde{N^*}(p))$. Then the Jacobian associated with this change of variables precisely cancels the absolute value of the fermion determinant. In this way, we arrive at the expression $$\begin{aligned} \mathcal{Z}&= \int\prod_{-\Lambda\leq p_\mu\leq\Lambda} \left[d\Tilde{N}(p)\,d\Tilde{N^*}(p)\right] \exp\left[-\frac{1}{L_0L_1}\sum_p\Tilde{N}(p)^*\Tilde{N}(p)\right] \notag\\ &\qquad\qquad\qquad\qquad\qquad\qquad{} \times \sum_i\left.\operatorname{sign}\det \frac{\partial(\Tilde{N},\Tilde{N^*})} {\partial(\Tilde{A},\Tilde{A^*})}\right| _{\Tilde{A}=\Tilde{A}_i,\Tilde{A^*}=\Tilde{A^*}_i}, \label{eq:(3.8)}\end{aligned}$$ where $\Tilde{A}(p)_i$ and $\Tilde{A^*}(p)_i$ ($i=1$, $2$, …) are solutions of $$\begin{aligned} 2ip_z\Tilde{A}(p)+W'(\Tilde{A})(-p)^*-\Tilde{N}(p)&=0, \label{eq:(3.9)} \\ -2ip_{\Bar{z}}\Tilde{A}(p)^*+W'(\Tilde{A})(-p)-\Tilde{N}(p)^*&=0. \label{eq:(3.10)}\end{aligned}$$ Representation  for the partition function (and a similar representation for expectation values of observables) gives rise to the following simulation algorithm [@Beccaria:1998vi; @Kawai:2010yj]: - Generate Gaussian random numbers $(\Tilde{N}(p),\Tilde{N}(p)^*)$. - Find (numerically) all the solutions $(\Tilde{A}(p)_i,\Tilde{A}(p)^*_i)$ ($i=1$, $2$, …) of Eqs.  and . - Then compute the sums $$\sum_i\left.\operatorname{sign}\det \frac{\partial(\Tilde{N},\Tilde{N^*})} {\partial(\Tilde{A},\Tilde{A^*})}\right| _{\Tilde{A}=\Tilde{A}_i,\Tilde{A^*}=\Tilde{A^*}_i},$$ and $$\sum_i\left. \operatorname{sign}\det \frac{\partial(\Tilde{N},\Tilde{N^*})} {\partial(\Tilde{A},\Tilde{A^*})}\, \mathcal{O}(\Tilde{A},\Tilde{A^*}) \right| _{\Tilde{A}=\Tilde{A}_i,\Tilde{A^*}=\Tilde{A^*}_i},$$ where $\mathcal{O}(\Tilde{A},\Tilde{A^*})$ is an observable. - Repeat the steps from (i) and take an average over the Gaussian random numbers. The expectation value of an observable $\mathcal{O}$ is given by $$\left\langle\mathcal{O}\right\rangle =\frac{1}{\Delta} \frac{\left\langle \sum_i\left. \operatorname{sign}\det \frac{\partial(\Tilde{N},\Tilde{N^*})} {\partial(\Tilde{A},\Tilde{A^*})}\, \mathcal{O}(\Tilde{A},\Tilde{A^*}) \right| _{\Tilde{A}=\Tilde{A}_i,\Tilde{A^*}=\Tilde{A^*}_i}\right\rangle_N} {\left\langle1\right\rangle_N}, \label{eq:(3.13)}$$ where $\langle\cdot\rangle_N$ denotes the average over the Gaussian random numbers and $$\Delta\equiv \frac{\left\langle \sum_i\left. \operatorname{sign}\det \frac{\partial(\Tilde{N},\Tilde{N^*})} {\partial(\Tilde{A},\Tilde{A^*})}\right| _{\Tilde{A}=\Tilde{A}_i,\Tilde{A^*}=\Tilde{A^*}_i}\right\rangle_N} {\left\langle1\right\rangle_N}. \label{eq:(3.14)}$$ As Eq.  shows, in the present algorithm the expectation value of an observable $\langle\mathcal{O}\rangle$ is given by the ratio of two expectation values with respect to the Gaussian random numbers. In this paper, we apply a simple propagation of error rule to the ratio to estimate the statistical error in $\langle\mathcal{O}\rangle$. This procedure must be good enough because the statistical error in the denominator of the ratio, $\Delta$, is generally quite small (see below). Since the above simulation algorithm is based on the generation of the Gaussian random numbers, there is no autocorrelation among generated configurations and there is no critical slowing down; this is an overwhelming advantage of the present simulation algorithm.[^4] Another interesting feature of this algorithm is that the “normalized partition function” $\Delta$ in Eq.  can be argued to give the Witten index $\operatorname{tr}(-1)^F$ [@Witten:1982df].[^5] For the massive free theory $W(\Phi)=m\Phi^2/2$, one easily sees that $\Delta=1$ and reproduces the correct Witten index $\operatorname{tr}(-1)^F=1$. Thus assuming that the proportionality constant between the partition function defined by the functional integral and $\operatorname{tr}(-1)^F$ is independent of the interaction, we have $\Delta=\operatorname{tr}(-1)^F$. It is known that for a homogeneous superpotential $W(\Phi)=\lambda\Phi^n/n$, $\operatorname{tr}(-1)^F=n-1$. Therefore, the relation $\Delta=n-1$ provides a quantitative test of the simulation [@Kawai:2010yj]. A disadvantage of the present simulation algorithm is that it is not *a priori* clear how many solutions $(\Tilde{A}(p)_i,\Tilde{A}(p)^*_i)$ that Eqs.  and  have. Thus we cannot be completely sure whether we have found all the solutions or not. Another limitation of the algorithm is that it is applicable only to supersymmetric boundary conditions; so it cannot explore physics with the finite temperature, for example. Simulation parameters ===================== In this paper, we fix the coupling constant in Eq.  to be $$a\lambda=0.3.$$ This is the same choice of the coupling constant as Ref. [@Kawai:2010yj], provided that our parameter $a$ in Eq.  is identified with the lattice spacing in Ref. [@Kawai:2010yj]. The size of the momentum grid $N_0\times N_1$ is varied from $16\times16$ to $36\times36$. For each value of $N_0\times N_1$, we generated $1280$ configurations of the Gaussian random numbers $(\Tilde{N}(p),\Tilde{N}(p)^*)$. We then solved Eqs.  and  by using the Newton–Raphson (NR) method [@NR3]. The solution $(\Tilde{A}(p),\Tilde{A}(p)^*)$ depends on the initial guess in the NR method and, as the initial guess, we used $100$ random configurations $(\Tilde{A}(p),\Tilde{A}(p)^*)$ generated with Gaussian random numbers with a unit variant. We judged that the convergence of the NR method is achieved when the maximum norm of the residue of Eqs.  and  becomes smaller than $10^{-13}$. Two obtained solutions were regarded identical when the the maximum norm of the difference of the solutions is smaller than $10^{-11}$. In this way, we obtained configurations tabulated in Tables \[table:1\] and \[table:2\]. The total amount of computational time was $34\,307.7\,\text{core}\cdot\text{hour}$ on the Intel Xeon $2.93\text{GHz}$. $N_0=N_1$ $16$ $18$ $20$ $22$ $24$ $26$ ----------------- ---------- ---------- ------------- ---------- ------------- ------------- $(+,+)$ $1276$ $1273$ $1275$ $1271$ $1271$ $1273$ $(-,+,+,+)$ $4$ $7$ $5$ $9$ $9$ $7$ $(+)$ $0$ $0$ $0$ $0$ $0$ $0$ $(+,+,+)$ $0$ $0$ $0$ $0$ $0$ $0$ $(-,+,+,+,+)$ $0$ $0$ $0$ $0$ $0$ $0$ $(-,-,+,+,+,+)$ $0$ $0$ $0$ $0$ $0$ $0$ $\Delta$ $2$ $2$ $2$ $2$ $2$ $2$ $\delta$ \[%\] $0.2(2)$ $0.1(1)$ ${-}0.2(1)$ $0.1(1)$ ${-}0.1(1)$ ${-}0.1(1)$ : Classification of obtained configurations. $\Delta$ is given by Eq.  and $\delta$ is defined by Eq. .[]{data-label="table:1"} $N_0=N_1$ $28$ $30$ $32$ $34$ $36$ ----------------- ------------- ------------ ------------ ------------- ------------ $(+,+)$ $1264$ $1261$ $1250$ $1254$ $1221$ $(-,+,+,+)$ $16$ $17$ $24$ $17$ $26$ $(+)$ $0$ $1$ $2$ $6$ $31$ $(+,+,+)$ $0$ $0$ $4$ $2$ $2$ $(-,+,+,+,+)$ $0$ $1$ $0$ $0$ $0$ $(-,-,+,+,+,+)$ $0$ $0$ $0$ $1$ $0$ $\Delta$ $2$ $2.000(1)$ $2.002(2)$ $1.997(2)$ $1.977(4)$ $\delta$ \[%\] ${-}0.0(1)$ $0.1(2)$ $0.0(2)$ ${-}0.1(3)$ $0.0(5)$ : Classification of obtained configurations (continued). $\Delta$ is given by Eq.  and $\delta$ is defined by Eq. .[]{data-label="table:2"} In Tables \[table:1\] and \[table:2\], configurations are classified according to the number of associated solutions $(\Tilde{A}(p)_i,\Tilde{A}(p)^*_i)$ and the associated sign of the Jacobian [@Kawai:2010yj]. For example, an entry in a row with the symbol $(-,+,+,+)$ implies that there was a configuration of $(\Tilde{N}(p),\Tilde{N}(p)^*)$ for which we have four different solutions $(\Tilde{A}(p)_i,\Tilde{A}(p)^*_i)$ ($i=1$, $2$, $3$, $4$) and the sign of the Jacobian is negative for one of them and it is positive for other three. Thus, we can read off $\Delta$ in Eq.  from this table. We see that the expected relation $\Delta=2$ for cubic superpotential  actually holds almost within one percent even for the worst case, for $N_0\times N_1=36\times36$. This fact suggests that even if our root-finding code missed some of solutions of Eqs.  and , they are precious few. SUSY WT identities ================== WT identities associated with exact symmetries in the formulation provide a consistency check for the numerical simulation. In particular, since SUSY is an exact symmetry in the present formulation, SUSY WT identities must hold even with a *finite* UV cutoff. This aspect is quite different from conventional lattice formulations in which one hopes that SUSY is restored only in the continuum limit. Thus an explicit confirmation of SUSY WT identities in the present formulation is interesting in its own right. SUSY transformations in the 2D $\mathcal{N}=(2,2)$ WZ model are given by $Q_1$, $Q_2$, $Q_{\Dot{1}}$, and $Q_{\Dot{2}}$. In terms of Fourier modes, they are given by $$\begin{aligned} &Q_1\Tilde{\Bar{\psi}}_{\Dot{1}}(p) =-2\sqrt{2}ip_{\Bar{z}}\Tilde{A^*}(p), & &Q_1\Tilde{A^*}(p)=0, \notag\\ &Q_1\Tilde{F^*}(p) =2\sqrt{2}ip_{\Bar{z}}\Tilde{\Bar{\psi}}_{\Dot{2}}(p), & &Q_1\Tilde{\Bar{\psi}}_{\Dot{2}}(p)=0, \notag\\ &Q_1\Tilde{A}(p) =\sqrt{2}\Tilde{\psi}_1(p), & &Q_1\Tilde{\psi}_1(p)=0, \notag\\ &Q_1\Tilde{\psi}_2(p) =\sqrt{2}\Tilde{F}(p), & &Q_1\Tilde{F}(p)=0, \label{eq:(5.1)}\end{aligned}$$ for $Q_1$, $$\begin{aligned} &Q_2\Tilde{\Bar{\psi}}_{\Dot{2}}(p) =-2\sqrt{2}ip_z\Tilde{A^*}(p), & &Q_2\Tilde{A^*}(p)=0, \notag\\ &Q_2\Tilde{F^*}(p) =-2\sqrt{2}ip_z\Tilde{\Bar{\psi}}_{\Dot{1}}(p), & &Q_2\Tilde{\Bar{\psi}}_{\Dot{1}}(p)=0, \notag\\ &Q_2\Tilde{A}(p) =\sqrt{2}\Tilde{\psi}_2(p), & &Q_2\Tilde{\psi}_2(p)=0, \notag\\ &Q_2\Tilde{\psi}_1(p) =-\sqrt{2}\Tilde{F}(p), & &Q_2\Tilde{F}(p)=0, \label{eq:(5.2)}\end{aligned}$$ for $Q_2$ $$\begin{aligned} &\Bar{Q}_{\Dot{1}}\Tilde{\psi}_1(p) =-2\sqrt{2}ip_{\Bar{z}}\Tilde{A}(p), & &\Bar{Q}_{\Dot{1}}\Tilde{A}(p)=0, \notag\\ &\Bar{Q}_{\Dot{1}}\Tilde{F}(p) =-2\sqrt{2}ip_{\Bar{z}}\Tilde{\psi}_2(p), & &\Bar{Q}_{\Dot{1}}\Tilde{\psi}_2(p)=0, \notag\\ &\Bar{Q}_{\Dot{1}}\Tilde{A^*}(p) =\sqrt{2}\Tilde{\Bar{\psi}}_{\Dot{1}}(p), & &\Bar{Q}_{\Dot{1}}\Tilde{\Bar{\psi}}_{\Dot{1}}(p)=0, \notag\\ &\Bar{Q}_{\Dot{1}}\Tilde{\Bar{\psi}}_{\Dot{2}}(p) =-\sqrt{2}\Tilde{F^*}(p), & &\Bar{Q}_{\Dot{1}}\Tilde{F^*}(p)=0, \label{eq:(5.3)}\end{aligned}$$ for $\Bar{Q}_{\Dot{1}}$, $$\begin{aligned} &\Bar{Q}_{\Dot{2}}\Tilde{\psi}_2(p) =-2\sqrt{2}ip_z\Tilde{A}(p), & &\Bar{Q}_{\Dot{2}}\Tilde{A}(p)=0, \notag\\ &\Bar{Q}_{\Dot{2}}\Tilde{F}(p) =2\sqrt{2}ip_z\Tilde{\psi}_1(p), & &\Bar{Q}_{\Dot{2}}\Tilde{\psi}_1(p)=0, \notag\\ &\Bar{Q}_{\Dot{2}}\Tilde{A^*}(p) =\sqrt{2}\Tilde{\Bar{\psi}}_{\Dot{2}}(p), & &\Bar{Q}_{\Dot{2}}\Tilde{\Bar{\psi}}_{\Dot{2}}(p)=0, \notag\\ &\Bar{Q}_{\Dot{2}}\Tilde{\Bar{\psi}}_{\Dot{1}}(p) =\sqrt{2}\Tilde{F^*}(p), & &\Bar{Q}_{\Dot{2}}\Tilde{F^*}(p)=0, \label{eq:(5.4)}\end{aligned}$$ for $\Bar{Q}_{\Dot{2}}$. A simple one-point SUSY WT identity is given by [@Catterall:2001fr] $$0=\left\langle S\right\rangle =\left\langle S_B\right\rangle +\left(\sum_p1-2\sum_p1\right) =\left\langle S_B\right\rangle-(N_0+1)(N_1+1), \label{eq:(5.5)}$$ where $S_B$ is the bosonic part of the action after integrating over the auxiliary field, giving by $$S_B\equiv\frac{1}{L_0L_1}\sum_p\Tilde{N^*}(-p)\Tilde{N}(p).$$ In Eq. , the first equality holds because the action $S$ in Eq.  can be written as $$\begin{aligned} S&=-\frac{1}{2}Q\frac{1}{L_0L_1}\sum_p \Bigl\{ \Bigl[2ip_z\Tilde{A^*}(-p)+\Tilde{F^*}(-p)+2W'(\Tilde{A})* \Bigr]\Tilde{\psi}_1(p) \notag\\ &\qquad\qquad\qquad\qquad{} +\Bigl[-2ip_{\Bar{z}}\Tilde{A}(p)+\Tilde{F}(p)+2W'(\Tilde{A})^** \Bigr]\Tilde{\Bar{\psi}}_{\Dot{2}}(-p) \Bigr\},\end{aligned}$$ where $Q\equiv-(\Bar{Q}_{\Dot{1}}+Q_2)/\sqrt{2}$ is a nilpotent SUSY transformation. Therefore $\langle S\rangle=0$ provided SUSY is not spontaneously broken, as the non-zero Witten index in the present system shows. One can then exactly evaluate the expectation value of the parts of the action which are quadratic in the auxiliary field and the fermion field; this gives the second equality in Eq. . Since $\sum_p1=(N_0+1)(N_1+1)$, we have the last equality in Eq. . That is, we have $$\delta\equiv\frac{\left\langle S_B\right\rangle}{(N_0+1)(N_1+1)}-1=0. \label{eq:(5.8)}$$ This relation provides a quantitative test of the simulation [@Catterall:2001fr; @Kawai:2010yj].[^6] We show $\delta$ in Tables \[table:1\] and \[table:2\] for each value of $N_0\times N_1$ and see that relation  holds within 0.5%. As for a two-point SUSY WT identity, from the relation $\langle Q_1(\Tilde{A}(p)\Tilde{\Bar{\psi}}_{\Dot{1}}(-p))\rangle=0$, we consider $$p_0\left\langle\Tilde{A}(p)\Tilde{A^*}(-p)\right\rangle =-\operatorname{Im}\left\langle\Tilde{\psi}_1(p)\Tilde{\Bar{\psi}}_{\Dot{1}}(-p)\right\rangle. \label{eq:(5.9)}$$ In Fig. \[fig:1\], we plotted both sides of this relation for our coarsest grid ($N_0\times N_1=16\times16$) as a function of $ap_0$; both sides coincide within the statistical error. ![The left-hand side (“bosonic”) and the right-hand side (“fermionic”) of Eq.  as a function of $ap_0$ along the line $ap_1=0$; $N_0\times N_1=16\times16$.[]{data-label="fig:1"}](Figure_1.eps){width="120mm"} ![The ratio of the left-hand side and the right-hand side of Eq.  as a function of $ap_0$ along the line $ap_1=0$ (the origin $p=0$ is excluded from the plot). The cases of $N_0\times N_1=16\times16$ and $N_0\times N_1=36\times36$ are plotted.[]{data-label="fig:2"}](Figure_2.eps){width="120mm"} For more quantitative comparison, we plotted in Fig. \[fig:2\] the ratio of the left-hand side and the right-hand side again as a function of $ap_0$. (The error in the ratio was estimated by a simple propagation of error rule.) We plotted the cases of $N_0\times N_1=16\times16$ and $N_0\times N_1=36\times36$. For both cases of the momentum grid, the identity holds within $\sim5\%$, a good indication of SUSY. Similarly, from $\langle Q_2(\Tilde{F^*}(p)\Tilde{\psi}_1(-p))\rangle=0$, we have[^7] $$\left\langle\Tilde{F}(p)\Tilde{F^*}(-p)\right\rangle =-p_1 \operatorname{Re}\left\langle\Tilde{\psi}_1(p)\Tilde{\Bar{\psi}}_{\Dot{1}}(-p)\right\rangle +p_0 \operatorname{Im}\left\langle\Tilde{\psi}_1(p)\Tilde{\Bar{\psi}}_{\Dot{1}}(-p) \right\rangle. \label{eq:(5.10)}$$ The results for this identity are depicted in Figs. \[fig:3\] and \[fig:4\].[^8] ![The left-hand side (“bosonic”) and the right-hand side (“fermionic”) of Eq.  along the line $ap_1=0$; $N_0\times N_1=16\times16$.[]{data-label="fig:3"}](Figure_3.eps){width="120mm"} ![The ratio of the left-hand side and the right-hand side of Eq.  as a function of $ap_0$ along the line $ap_1=0$ (the origin $p=0$ is excluded from the plot). The cases of $N_0\times N_1=16\times16$ and $N_0\times N_1=36\times36$ are plotted.[]{data-label="fig:4"}](Figure_4.eps){width="120mm"} Having obtained these encouraging results concerning SUSY in our numerical simulation, let us proceed to study physical questions. Conformal dimension of a chiral primary field ============================================= As noted in the Introduction, in the IR limit, the scalar field $A$ in the WZ model with cubic potential  is expected to behave as a chiral primary field with the conformal dimensions $(h,\Bar{h})=(1/6,1/6)$. For such a field, the two-point function will behave as $$\left\langle A(x)A^*(0)\right\rangle\propto \frac{1}{z^{2h}\Bar{z}^{2\Bar{h}}}, \qquad\text{for $|x|$ large}. \label{eq:(6.1)}$$ Then assuming that $A$ is spinless, $h=\Bar{h}$, the anomalous dimension $h+\Bar{h}=2h$ could be extracted from the susceptibility of the scalar field $\chi_\phi$, defined by [@Kawai:2010yj] $$\chi_\phi\equiv \frac{1}{a^2}\int_{L_0L_1}d^2x\left\langle A(x)A^*(0)\right\rangle =\frac{1}{a^2L_0L_1}\left\langle\left|\Tilde{A}(0)\right|^2\right\rangle, \label{eq:(6.2)}$$ as $$\chi_\phi\propto \int_{L_0L_1}d^2x\,\frac{1}{(x^2)^{2h}} \propto(L_0L_1)^{1-h-\Bar{h}},\qquad\text{for $L_\mu$ large}. \label{eq:(6.3)}$$ Thus $\ln(\chi_\phi)$ would be a linear function of $\ln(L_0L_1)$ for large $L_\mu$ and $1-h-\Bar{h}$ is obtained by the slope of the linear function [@Kawai:2010yj]. In Fig. \[fig:5\], we plot $\ln(\chi_\phi)$ as a function of $\ln(a^{-2}L_0L_1)$ (recall that in our present simulation the lattice spacing is fixed to $a\lambda=0.3$). From the plot, we see that a fit by a linear function would be good for large sizes as $a^{-2}L_0L_1\gtrsim24\times24$. ![$\ln(\chi_\phi)$ as a function of $\ln(a^{-2}L_0L_1)$. No UV subtraction is made. The broken line is a linear fit with $1-h-\Bar{h}=0.603$.[]{data-label="fig:5"}](Figure_5.eps){width="120mm"} Therefore, we applied a linear $\chi^2$ fit by using data from $N_0\times N_1=24\times24$ to $N_0\times N_1=36\times36$. To estimate a systematic error associated with this choice of fitting range, we also carried out a linear fit using data from $N_0\times N_1=26\times26$ to $N_0\times N_1=36\times36$. See Table \[table:3\] $n_0\times n_1$ Fitting range of $N_0\times N_1$ $\chi^2/\text{d.o.f.}$ $1-h-\Bar{h}$ ----------------- ----------------------------------- ------------------------ --------------- $0\times0$ from $24\times24$ to $36\times36$ $0.904$ $0.603(19)$ $0\times0$ from $26\times26$ to $36\times36$ $1.088$ $0.609(25)$ $3\times3$ from $24\times24$ to $36\times36$ $0.910$ $0.624(20)$ $3\times3$ from $26\times26$ to $36\times36$ $1.108$ $0.629(26)$ : Linear $\chi^2$ fit used for a determination of the anomalous dimension in Eq. . $n_0\times n_1$ denotes the UV subtraction region defined in Eq. .[]{data-label="table:3"} In Eq. , the integral over $x$ is performed for all $x$ including the coincidence point $x=0$. This might be physically unnatural because the coincidence point $x=0$ would suffer from ambiguity associated with the UV regularization. On the other hand, true long-distance physics should be independent of such a UV ambiguity. To avoid such UV ambiguity and to estimate how much our determination depends on the UV prescription, we could define the susceptibility with the UV part subtracted [@Kawai:2010yj]: $$\begin{aligned} \chi_\phi&\equiv \frac{1}{a^2}\int_{L_0L_1-n_0n_1a^2}d^2x\left\langle A(x)A^*(0)\right\rangle \notag\\ &=\frac{1}{a^2L_0L_1}\left\langle\left|\Tilde{A}(0)\right|^2\right\rangle \notag\\ &\qquad{} -\frac{1}{a^2(L_0L_1)^2}\sum_p \frac{2}{p_0}\sin\left(\frac{p_0an_0}{2}\right) \frac{2}{p_1}\sin\left(\frac{p_1an_1}{2}\right) \left\langle\left|\Tilde{A}(p)\right|^2\right\rangle, \label{eq:(6.4)}\end{aligned}$$ where the integral has been defined by extracting a region $a^2n_0\times n_1$ containing the coincidence point $x=0$; in this expression it is understood that $(2/p_\mu)\sin(p_\mu an_\mu/2)=an_\mu$ for $p_\mu=0$. Figure \[fig:6\] is the result with the UV subtraction with $n_0\times n_1=3\times3$ (this is a choice identical to Ref. [@Kawai:2010yj]). ![$\ln(\chi_\phi)$ as a function of $\ln(a^{-2}L_0L_1)$. UV subtraction  with $n_0\times n_1=3\times3$ is made. The broken line is a linear fit with $1-h-\Bar{h}=0.624$.[]{data-label="fig:6"}](Figure_6.eps){width="120mm"} Our linear fits are summarized in Table \[table:3\]. By adopting the average of the second and the third rows in Table \[table:3\] as the central value and estimating a systematic error associated with the fitting range and the UV ambiguity by variations in Table \[table:3\], we quote $$1-h-\Bar{h}=0.616(25)(13). \label{eq:(6.5)}$$ This value is somewhat smaller than the value obtained in Ref. [@Kawai:2010yj], but still consistent with the expected exact value $1-h-\Bar{h}=2/3=0.666\dots$ within $1.3\sigma$. Central charge from a supercurrent correlator ============================================= The central charge $c$, defined by the Virasoro anomaly $$\left\langle T_{zz}(x)T_{zz}(0)\right\rangle =\frac{1}{8\pi^2}\frac{c/2}{z^4}, \label{eq:(7.1)}$$ where $T_{zz}$ is the holomorphic part of the energy-momentum tensor, is a fundamental quantity that characterizes a conformal field theory (CFT). It is thus of great interest whether we can measure this quantity $c$ in the IR region from the present numerical simulation and obtain further support for the conjectured correspondence between the WZ model and SCFT. A salient feature of the present formulation is its nice symmetry properties, including translational invariance. Thus, we may define the energy-momentum tensor as a Noether current associated with the translational invariance in a regularized framework. In an $\mathcal{N}=2$ SCFT, the central charge $c$ appears also in the correlation function between the holomorphic $U(1)$ currents: $$\left\langle J_z(x)J_z(0)\right\rangle =\frac{1}{8\pi^2}\frac{c/3}{z^2}. \label{eq:(7.2)}$$ In the LG description, the $U(1)$ symmetry in the $\mathcal{N}=2$ SCFT is identified with the $U(1)_R$ symmetry of the WZ model with a (quasi-)homogeneous superpotential (see below) and this $U(1)_R$ is also exactly preserved in the present formulation. We can thus define a conserved $U(1)_R$ current whose appropriate component is identified with $J_z$. In preliminary numerical study of the above two-point functions of *bosonic* operators ($T_{zz}$ for Eq.  and $J_z$ for Eq. ), however, we found that signals are generally very noisy and reliable fits for the central charge would be rather difficult.[^9] So, in what follows, we adopt a different approach that employs a two-point function of the $\mathcal{N}=2$ superconformal currents: $$\left\langle G_z^+(x)G_z^-(0)\right\rangle =\frac{1}{8\pi^2}\frac{2c/3}{z^3}. \label{eq:(7.3)}$$ Since the operators $G_z^\pm$ are *fermionic*, there is no “disconnected diagram” which could contribute to Eq.  and we expect a clear signal from numerical simulation. In fact the following results seem to be in accord with this naive expectation. Now, since our present formulation possesses exact SUSY, we can define a conserved supercurrent. To obtain this, we consider the localized SUSY transformation (for a generic field $\varphi$) $$\delta\Tilde{\varphi}(p) =\frac{1}{L_0L_1}\sum_q\left[ \sum_{\alpha=1}^2 \Tilde{\xi}^\alpha(q)Q_\alpha\Tilde{\varphi}(p-q) -\sum_{\Dot{\alpha}=\Dot{1}}^{\Dot{2}} \Tilde{\Bar{\xi}}^{\Dot{\alpha}}(q)\Bar{Q}_{\Dot{\alpha}}\Tilde{\varphi}(p-q) \right],$$ where $\Tilde{\xi}^\alpha(q)$ and $\Tilde{\Bar{\xi}}^{\Dot{\alpha}}(q)$ are non-constant Grassmann parameters and $Q_\alpha$ and $\Bar{Q}_{\Dot{\alpha}}$ are defined by Eqs. –. Then from the variation of the action, we define supercurrents $\Tilde{S}_\mu^\pm(p)$ and $\Tilde{\Bar{S}}_\mu^\pm(p)$ as $$\begin{aligned} \delta S&\equiv\frac{1}{L_0L_1} \sum_p(-2)\sum_\mu\Bigl[ \Tilde{\Bar{\xi}}^{\Dot{2}}(-p)(-ip_\mu)\Tilde{S}_\mu^+(p) +\Tilde{\xi}^2(-p)(-ip_\mu)\Tilde{S}_\mu^-(p) \notag\\ &\qquad\qquad\qquad\qquad\qquad{} +\Tilde{\Bar{\xi}}^{\Dot{1}}(-p)(-ip_\mu)\Tilde{\Bar{S}}_\mu^+(p) +\Tilde{\xi}^1(-p)(-ip_\mu)\Tilde{\Bar{S}}_\mu^-(p) \Bigr]. \label{eq:(7.5)}\end{aligned}$$ By construction, above supercurrents satisfy SUSY WT identities, such as $$p_\mu\left\langle\Tilde{S}_\mu^+(p) \Tilde{\varphi}_1(q_1)\dots\Tilde{\varphi}_n(q_n) \right\rangle =\frac{i}{2}\sum_{i=1}^n\left\langle \Tilde{\varphi}_1(q_1)\dots\Bar{Q}_{\Dot{2}}\Tilde{\varphi}_i(q_i+p) \dots\Tilde{\varphi}_n(q_n) \right\rangle.$$ This identity would imply that (according to the standard argument) the current $\Tilde{S}_\mu^+$ is a correctly normalized operator. In Eq. , the explicit form of the supercurrents are given by $$\begin{aligned} &\Tilde{S}_z^+(p) =\frac{1}{L_0L_1}\sum_q\sqrt{2}i(p-q)_z \Tilde{A}(p-q)\Tilde{\Bar{\psi}}_{\Dot{2}}(q), \label{eq:(7.7)}\\ &\Tilde{S}_{\Bar{z}}^+(p) =\frac{1}{L_0L_1}\sum_q \frac{1}{\sqrt{2}}W'(\Tilde{A})(p-q)\Tilde{\psi}_1(q), \label{eq:(7.8)}\\ &\Tilde{S}_z^-(p) =-\frac{1}{L_0L_1}\sum_q\sqrt{2}i(p-q)_z \Tilde{A^*}(p-q)\Tilde{\psi}_2(q), \label{eq:(7.9)}\\ &\Tilde{S}_{\Bar{z}}^-(p) =\frac{1}{L_0L_1}\sum_q \frac{1}{\sqrt{2}}W'(\Tilde{A})^*(p-q)\Tilde{\Bar{\psi}}_{\Dot{1}}(q), \label{eq:(7.10)}\end{aligned}$$ and $$\begin{aligned} &\Tilde{\Bar{S}}_{\Bar{z}}^+(p) =\frac{1}{L_0L_1}\sum_q\sqrt{2}i(p-q)_{\Bar{z}} \Tilde{A}(p-q)\Tilde{\Bar{\psi}}_{\Dot{1}}(q), \label{eq:(7.11)}\\ &\Tilde{\Bar{S}}_z^+(p) =-\frac{1}{L_0L_1}\sum_q \frac{1}{\sqrt{2}}W'(\Tilde{A})(p-q)\Tilde{\psi}_2(q), \label{eq:(7.12)}\\ &\Tilde{\Bar{S}}_{\Bar{z}}^-(p) =-\frac{1}{L_0L_1}\sum_q\sqrt{2}i(p-q)_{\Bar{z}} \Tilde{A^*}(p-q)\Tilde{\psi}_1(q), \label{eq:(7.13)}\\ &\Tilde{\Bar{S}}_z^-(p) =-\frac{1}{L_0L_1}\sum_q \frac{1}{\sqrt{2}}W'(\Tilde{A})^*(p-q)\Tilde{\Bar{\psi}}_{\Dot{2}}(q). \label{eq:(7.14)}\end{aligned}$$ A Noether current is, however, always ambiguous as one can change the definition as $\Tilde{S}_z^\pm(p)\to\Tilde{S}_z^\pm(p)+p_z\Tilde{X}^\pm(p)$ and $\Tilde{S}_{\Bar{z}}^\pm(p)\to\Tilde{S}_{\Bar{z}}^\pm(p)-p_{\Bar{z}}\Tilde{X}^\pm(p)$ by using certain combinations $\Tilde{X}^\pm(p)$ without affecting the conservation law, $p_{\Bar{z}}\Tilde{S}_z^\pm(p)+p_z\Tilde{S}_{\Bar{z}}^\pm(p)=0$. To fix this ambiguity, we required in Eqs.  to  that $$\Tilde{S}_{\Bar{z}}^+=\Tilde{S}_{\Bar{z}}^- =\Tilde{\Bar{S}}_z^+=\Tilde{\Bar{S}}_z^-=0, \label{eq:(7.15)}$$ when $W'=0$. The WZ model with $W'=0$ is a massless free theory that itself is an $\mathcal{N}=(2,2)$ SCFT. For this system it is natural to adopt a supercurrent that obeys the gamma-traceless condition, $$\sum_\mu\gamma_\mu \begin{pmatrix} \Tilde{\Bar{S}}_\mu^-\\\Tilde{S}_\mu^+ \end{pmatrix}= \sum_\mu\gamma_\mu \begin{pmatrix} \Tilde{\Bar{S}}_\mu^+\\\Tilde{S}_\mu^- \end{pmatrix}=0, \label{eq:(7.16)}$$ because this condition is a super-partner of the traceless condition $T_{z\Bar{z}}=0$ [@Ferrara:1974pz] that is usually assumed in CFT. Eq.  is nothing but Eq.  in components. Thus we had a natural definition of the supercurrents. We then postulate a correspondence between the components of above supercurrents and the holomorphic part of the superconformal currents in the IR limit. That is, $$\Tilde{S}_z^+(p)\to\Tilde{G}_z^+(p),\qquad \Tilde{S}_z^-(p)\to\Tilde{G}_z^-(p). \label{eq:(7.17)}$$ Our reasoning for this correspondence is as follows: - The conservation law of the supercurrents in the coordinate space yields $\partial_{\Bar{z}}S_z^\pm(x)+\partial_zS_{\Bar{z}}^\pm(x)=0$. In the IR limit, the derivative of the superpotential $W'(A)$ is expected to become irrelevant (see, for example, Section 14.4 of Ref. [@Hori:2003ic]) so $S_{\Bar{z}}^\pm(x)$, from explicit forms  and , could be neglected in the IR limit. Then the conservation law implies that $S_z^\pm(x)$ are holomorphic functions, a correct property of $G_z^\pm(x)$. - The WZ model with a homogeneous superpotential $W=\lambda\Phi^n/n$ possesses the $U(1)_R$ symmetry $$\begin{aligned} &A\to e^{(2/n)i\theta}A,\qquad \psi_\alpha\to e^{-(1-2/n)i\theta}\psi_\alpha, \notag\\ &\Bar{\psi}_{\Dot{\alpha}}\to e^{(1-2/n)i\theta}\Bar\psi_{\Dot{\alpha}},\qquad F\to e^{-(2-2/n)i\theta}F, \label{eq:(7.18)}\end{aligned}$$ which is, in the IR, identified with a $U(1)$ symmetry generated by a *sum* of two $U(1)$ charges $(q,\Bar{q})$ in the left- and right-moving $\mathcal{N}=2$ superconformal algebras (see, for example, Section 19.4 of Ref. [@Polchinski:1998rr]). Thus, in Eq. , the $U(1)_R$ charge of the scalar field $A$ is assigned $2/n$, because $A$ is identified with a chiral primary field in the $A_{n-1}$ model with the $U(1)$ charges $(q,\Bar{q})=(1/n,1/n)$. Then, assuming that $\Tilde{S}_z^+$ in Eq.  possesses $U(1)$ charges $(q,0)$ in the IR, we see from the $U(1)$ charge assignment in Eq.  that $q=+1$, a correct $U(1)$ charge of $G_z^+$. Similarly, for $\Tilde{S}_z^-$ in Eq. , we have $q=-1$, a correct $U(1)$ charge of $G_z^-$. - The overall normalization of the supercurrents has been fixed such that Eq.  is reproduced under correspondence  for the massless free theory ($W'=0$). This theory itself is an $\mathcal{N}=(2,2)$ SCFT whose left-moving sector possesses the central charge $c=3$ (one complex scalar and two Majorana-Weyl fermions). Thus, under identification , our procedure is as follows: we numerically compute the two-point function of the supercurrents in the momentum space $$\left\langle\Tilde{S}_z^+(p)\Tilde{S}_z^-(-p)\right\rangle. \label{eq:(7.19)}$$ Then, in view of correspondence , we compare this function in the IR region[^10] $|p|\lesssim\lambda$ with the two-point function of the superconformal currents in the momentum space $$\begin{aligned} &L_0L_1\int d^2x\,e^{-ipx} \left\langle G_z^+(x)G_z^-(0)\right\rangle \notag\\ &=L_0L_1\frac{-ic}{48\pi}\frac{\partial^3}{\partial p_{\Bar{z}}^3} \frac{p^2}{\delta^2}K_2(|p|\delta) \xrightarrow{|p|\delta\to0} L_0L_1\frac{ic}{24\pi}\frac{p_z^2}{p_{\Bar{z}}}, \label{eq:(7.20)}\end{aligned}$$ where we have used Eq.  that is, strictly speaking, an expression valid only on $\mathbb{R}^2$. In deriving the last expression, we regularized the singularity in the integrand at $x=0$ by setting $1/z^3=(\Bar{z})^3/(x^2)^3\to(\Bar{z})^3/(x^2+\delta^2)^3$. Note that the last low-momentum behavior (i.e., the IR physics) in Eq.  is independent of the regularization parameter $\delta$. For the above computation, we used only data with the finest grid $N_0\times N_1=36\times36$. In Fig. \[fig:7\], we depicted the real part of correlation function  as a function of $ap_0$ along the line $ap_1=\pi/18\sim0.1745$. ![The real part of the correlation function  as a function of $ap_0$ along the line $ap_1=\pi/18\sim0.1745$; $N_0\times N_1=36\times36$. The broken line is the real part of function  with $c=1$.[]{data-label="fig:7"}](Figure_7.eps){width="120mm"} The broken line is the real part of function  with $c=1$; it agrees well with the data for $|ap_0|\leq\pi/18\sim0.1745<0.3$, a good indication for $c\sim1$ in the IR region. Fig. \[fig:8\] is the same as Fig. \[fig:7\], but for the imaginary part. ![The imaginary part of the correlation function  as a function of $ap_0$ along the line $ap_1=\pi/18\sim0.1745$; $N_0\times N_1=36\times36$. The broken line is the imaginary part of function  with $c=1$.[]{data-label="fig:8"}](Figure_8.eps){width="120mm"} The broken line is the imaginary part of function  with $c=1$; this time it agrees well with the data for $|ap_0|\leq2\pi/18\sim0.349 \simeq0.3$. Our fit for $c$ proceeds as follows: We first consider the real part of the ratio of correlation function  to the function $L_0L_1(i/24\pi)p_z^2/p_{\Bar{z}}$ appearing in Eq. . For illustration, the real part of this ratio along several constant $ap_1$ lines are depicted in Figs. \[fig:9\], \[fig:10\] and \[fig:11\]. ![The real part of the ratio of the correlation function  and the function $L_0L_1(i/24\pi)p_z^2/p_{\Bar{z}}$ as a function of $ap_0$ along the line $ap_1=0$; $N_0\times N_1=36\times36$. The origin $p=0$ is excluded from the plot because $p_z^2/p_{\Bar{z}}$ is singular at the origin.[]{data-label="fig:9"}](Figure_9.eps){width="120mm"} ![The real part of the ratio of the correlation function  and the function $L_0L_1(i/24\pi)p_z^2/p_{\Bar{z}}$ as a function of $ap_0$ along the line $ap_1=\pi/18\sim0.1745$; $N_0\times N_1=36\times36$.[]{data-label="fig:10"}](Figure_10.eps){width="120mm"} ![The real part of the ratio of the correlation function  and the function $L_0L_1(i/24\pi)p_z^2/p_{\Bar{z}}$ as a function of $ap_0$ along the line $ap_1=2\pi/18\sim0.349$; $N_0\times N_1=36\times36$.[]{data-label="fig:11"}](Figure_11.eps){width="120mm"} We see that actually the ratio is close to $1$ around the origin in the momentum space $|p|\sim0$. Then, to use data points with a fixed energy scale, we define a fitting region $R(b_1,b_2)$ on the momentum grid as illustrated in Fig. \[fig:12\]. (180,180)(0,-10) (90,90)[(1,0)[90]{}]{} (90,90)[(-1,0)[90]{}]{} (90,90)[(0,1)[90]{}]{} (90,90)[(0,-1)[90]{}]{} (65,65)(5,0)[10]{}[(1,1)[5]{}]{} (65,70)(0,5)[9]{}[(1,1)[5]{}]{} (70,110)(5,0)[9]{}[(1,1)[5]{}]{} (110,70)(0,5)[8]{}[(1,1)[5]{}]{} (110,70)[(0,-1)[30]{}]{} (115,75)[(0,-1)[60]{}]{} (97,36)[$b_1$]{} (99,10)[$b_2$]{} (90,90)[(180,180)]{} (0,0)(0,5)[37]{}[(0,0)(5,0)[37]{}]{} (90,90)[(40,40)]{} (90,90)[(50,50)]{} (90,50)[(1,0)[20]{}]{} (90,50)[(-1,0)[0]{}]{} (90,24)[(1,0)[25]{}]{} (90,24)[(-1,0)[0]{}]{} Then we carry out a $\chi^2$ fit by a constant for data points in the fitting region $R(b_1,b_2)$. Since the IR region is characterized by the condition $|ap|\lesssim a\lambda=0.3$ and the lowest non-zero momentum in the present $N_0\times N_1=36\times36$ momentum grid is $ap_\mu=\pi/18\sim0.1745$, we regard the fitting region $R(1,1)$, or at most $R(1,2)$, as the IR region; see Table \[table:4\]. Fitting region Number of data points $\chi^2/\text{d.o.f.}$ $c$ ---------------- ----------------------- ------------------------ ------------ $R(1,1)$ $8$ $1.93$ $1.09(14)$ $R(1,2)$ $24$ $3.50$ $1.40(8)$ : Results of a $\chi^2$ fit used for a determination of the central charge $c$ in .[]{data-label="table:4"} From the numbers in the table, we estimate $$c=1.09(14)(31), \label{eq:(7.21)}$$ as the central charge in the IR region. Our estimate reproduces the conjectured value $c=1$ very well. Although strictly speaking expression  is meaningful only for very small momenta and the LG description is expected to be valid only for the low-energy region $|ap|\lesssim a\lambda=0.3$, we found that it is nevertheless interesting to repeat the above procedure for “intermediate” energy $|ap|\gtrsim a\lambda=0.3$ or even “high” energy $|ap|\sim\pi$ regions. Figures \[fig:13\] and \[fig:14\] is a result of the fitting in an “intermediate” energy region $|ap|\sim0.7$. ![The real part of the correlation function  as a function of $ap_0$ along the line $ap_1=4\pi/18\sim0.698$; $N_0\times N_1=36\times36$. The broken line is the real part of function  with $c=1.95$, that was obtained by a constant fit in the fitting region $R(4,4)$ ($\chi^2/\text{d.o.f.}=1.58$).[]{data-label="fig:13"}](Figure_13.eps){width="120mm"} ![The imaginary part of the correlation function  as a function of $ap_0$ along the line $ap_1=4\pi/18\sim0.698$; $N_0\times N_1=36\times36$. The broken line is the imaginary part of function  with $c=1.95$.[]{data-label="fig:14"}](Figure_14.eps){width="120mm"} We observe a rather good fit with an “effective $c$”, $c\sim2$. We may repeat such a fit by changing the parameter $b$ in $R(b,b)$ which roughly corresponds to the energy scale. The result of such fits as a function of $b$ are shown is Fig. \[fig:15\]. ![“Effective $c$” obtained from a constant fit in the fitting region $R(b,b)$.[]{data-label="fig:15"}](Figure_15.eps){width="120mm"} It is interesting that, going from the UV side to the IR side, the plot goes from $c\sim3$, the central charge corresponding to the massless free theory, and decreases to $c\sim1$, the central charge of the $A_2$ model. Although one should not take this plot so seriously because $\chi^2$ becomes quite large for large $b$ (for example, $\chi^2/\text{d.o.f.}\sim19$ for $b=8$), it is still interesting to note an analogue of the Zamolodchikov $C$ function [@Zamolodchikov:1986gt] in this plot; the $C$ function monotonically decreases along the renormalization group flow from UV to IR and coincides with the central charge at a fixed point. It is an interesting problem to clarify the underlying reason for this similarity of Fig. \[fig:15\] with the $C$ function. Conclusion ========== In this paper, we carried out a non-perturbative numerical study of low-energy physics of the 2D WZ model with the massless cubic superpotential. We obtained the critical exponent $1-h-\Bar{h}$ and the central charge $c$ which are consistent with the conjectured emergence of a non-trivial $\mathcal{N}=(2,2)$ SCFT, thus provided further support for the LG description. Our results indicate that our supersymmetric non-perturbative formulation of the WZ model is working, although there has been an issue concerning the locality in this formulation. For further improvement of these results, better observables which yield less systematic errors should be investigated. Also, we want to generalize the present analysis to higher critical LG models. We would like to thank Michael G. Endres, Kazuo Fujikawa, Masafumi Fukuma, Daisuke Kadoh, Hikaru Kawai, Akitsugu Miwa, and Tsuneo Uematsu for discussions and comments. We are quite indebted to Hiroki Kawai and Yoshio Kikukawa for discussions and detailed explanation on the simulation in Ref. [@Kawai:2010yj] and to Michael G. Endres for a careful reading of the manuscript. Our numerical calculations were carried out by using the RIKEN Integrated Cluster of Clusters (RICC) facility. The work of H.S. is supported in part by a Grant-in-Aid for Scientific Research, 22340069 and 23540330. [00]{} J. Wess and B. Zumino, Nucl. Phys.  B [**70**]{} (1974) 39. D. A. Kastor, E. J. Martinec and S. H. Shenker, Nucl. Phys.  B [**316**]{} (1989) 590. C. Vafa and N. P. Warner, Phys. Lett.  B [**218**]{} (1989) 51. W. Lerche, C. Vafa and N. P. Warner, Nucl. Phys.  B [**324**]{} (1989) 427. P. S. Howe and P. C. West, Phys. Lett.  B [**223**]{} (1989) 377. S. Cecotti, L. Girardello and A. Pasquinucci, Nucl. Phys.  B [**328**]{} (1989) 701. P. S. Howe and P. C. West, Phys. Lett.  B [**227**]{} (1989) 397. S. Cecotti, L. Girardello and A. Pasquinucci, Int. J. Mod. Phys.  A [**6**]{} (1991) 2427. S. Cecotti, Int. J. Mod. Phys.  A [**6**]{} (1991) 1749. E. Witten, Int. J. Mod. Phys.  A [**9**]{} (1994) 4783 \[arXiv:hep-th/9304026\]. J. Polchinski, “String theory. Vol. 2: Superstring theory and beyond,” [*Cambridge, UK: Univ. Pr. (1998) 531 p*]{} K. Hori [*et al.*]{}, “Mirror symmetry,” [*Providence, USA: AMS (2003) 929 p*]{} A. B. Zamolodchikov, Sov. J. Nucl. Phys.  [**44**]{} (1986) 529 \[Yad. Fiz.  [**44**]{} (1986) 821\]. E. Witten, Nucl. Phys.  B [**403**]{} (1993) 159 \[arXiv:hep-th/9301042\]. H. Kawai and Y. Kikukawa, Phys. Rev.  D [**83**]{} (2011) 074502 \[arXiv:1005.4671 \[hep-lat\]\]. Y. Kikukawa and Y. Nakayama, Phys. Rev.  D [**66**]{} (2002) 094508 \[arXiv:hep-lat/0207013\]. M. Beccaria, G. Curci and E. D’Ambrosio, Phys. Rev.  D [**58**]{} (1998) 065009 \[arXiv:hep-lat/9804010\]. S. Catterall and S. Karamov, Phys. Rev.  D [**65**]{} (2002) 094501 \[arXiv:hep-lat/0108024\]. J. Giedt, Nucl. Phys.  B [**726**]{} (2005) 210 \[arXiv:hep-lat/0507016\]. G. Bergner, T. Kaestner, S. Uhlmann and A. Wipf, Annals Phys.  [**323**]{} (2008) 946 \[arXiv:0705.2212 \[hep-lat\]\]. T. Kästner, G. Bergner, S. Uhlmann, A. Wipf and C. Wozar, Phys. Rev.  D [**78**]{} (2008) 095001 \[arXiv:0807.1905 \[hep-lat\]\]. F. Synatschke-Czerwonka, T. Fischbacher and G. Bergner, Phys. Rev.  D [**82**]{} (2010) 085003 \[arXiv:1006.1823 \[hep-th\]\]. N. Sakai and M. Sakamoto, Nucl. Phys.  B [**229**]{} (1983) 173. K. Fujikawa, Phys. Rev.  D [**66**]{} (2002) 074510 \[arXiv:hep-lat/0208015\]. D. Kadoh and H. Suzuki, Phys. Lett.  B [**684**]{} (2010) 167 \[arXiv:0909.3686 \[hep-th\]\]. J. Bartels and J. B. Bronzan, Phys. Rev.  D [**28**]{} (1983) 818. S. D. Drell, M. Weinstein and S. Yankielowicz, Phys. Rev.  D [**14**]{} (1976) 487. S. D. Drell, M. Weinstein and S. Yankielowicz, Phys. Rev.  D [**14**]{} (1976) 1627. G. Bergner, JHEP [**1001**]{} (2010) 024 \[arXiv:0909.4791 \[hep-lat\]\]. H. Nicolai, Phys. Lett.  B [**89**]{} (1980) 341. H. Nicolai, Nucl. Phys.  B [**176**]{} (1980) 419. S. Cecotti and L. Girardello, Phys. Lett.  B [**110**]{} (1982) 39. G. Parisi and N. Sourlas, Nucl. Phys.  B [**206**]{} (1982) 321. S. Cecotti and L. Girardello, Nucl. Phys.  B [**226**]{} (1983) 417. E. Witten, Nucl. Phys.  B [**202**]{} (1982) 253. I. Kanamori, Nucl. Phys.  B [**841**]{} (2010) 426 \[arXiv:1006.2468 \[hep-lat\]\]. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, “Numerical Recipes. The Art of Scientific Computing, 3rd Edition,” [*New York, USA: Cambridge University Press (2007) 1235 p*]{} D. Kadoh and H. Suzuki, Phys. Lett.  B [**696**]{} (2011) 163 \[arXiv:1011.0788 \[hep-lat\]\]. S. Ferrara and B. Zumino, Nucl. Phys.  B [**87**]{} (1975) 207. A. B. Zamolodchikov, JETP Lett.  [**43**]{} (1986) 730 \[Pisma Zh. Eksp. Teor. Fiz.  [**43**]{} (1986) 565\]. [^1]: This system is obtained by a dimensional reduction of the four-dimensional Wess–Zumino model [@Wess:1974tw] from four dimensions to two dimensions. [^2]: A polynomial $W(\Phi)$ of variables $\Phi_I$ ($I=1$, $2$, …, $N$) is called quasi-homogeneous when there exist some weights $\omega_I$ such that $W(\Phi_I\to\Lambda^{\omega_I}\Phi_I) =\Lambda W(\Phi)$. [^3]: For preceding numerical simulations of the 2D $\mathcal{N}=(2,2)$ WZ model with a *massive* cubic superpotential $W(\Phi)=m\Phi^2/2+\lambda\Phi^3/3$, see Refs. [@Beccaria:1998vi; @Catterall:2001fr; @Giedt:2005ae; @Bergner:2007pu; @Kastner:2008zc; @Synatschke:2010jn]. See also Refs. [@Sakai:1983dg; @Fujikawa:2002pa] for theoretical background. [^4]: We would like thank Martin Lüscher for bringing our attention to this point. [^5]: With conventional simulation algorithms, one needs more elaborate method to compute the Witten index; see Ref. [@Kanamori:2010gw]. [^6]: Lattice formulations adopted in Refs. [@Catterall:2001fr] and [@Kawai:2010yj] and more generally lattice formulations based on the Nicolai map [@Sakai:1983dg; @Beccaria:1998vi; @Catterall:2001fr; @Kikukawa:2002as; @Fujikawa:2002pa; @Giedt:2005ae; @Bergner:2007pu; @Kastner:2008zc] possess exact invariance under the above nilpotent $Q$ transformation. See also Ref. [@Kadoh:2010ca] and references therein. [^7]: In the Wess–Zumino model, the integration over the auxiliary field is defined by an analytic continuation, and according to this prescription, the correlation function is expressed as $$\left\langle\Tilde{F}(p)\Tilde{F^*}(-p))\right\rangle =\left\langle \left[-\Tilde{N}(p)+2ip_z\Tilde{A}(p)\right] \left[-\Tilde{N^*}(-p)-2ip_{\Bar{z}}\Tilde{A^*}(-p)\right] \right\rangle-L_0L_1,$$ in terms of a correlation function of the Gaussian random fields and the scalar fields. The last term gives a negative contribution, and despite its appearance, the left-hand side can be negative as our plot shows. [^8]: Since $\langle\Tilde{F}(0)\Tilde{F^*}(0)\rangle= -\langle Q_2(\Tilde{\psi}_1(0)\Tilde{F^*}(0))\rangle/\sqrt{2}$, $\langle\Tilde{F}(p)\Tilde{F^*}(-p)\rangle$ evaluated at $p=0$ must vanish if SUSY is not spontaneously broken. Our plot in Fig. \[fig:3\] is consistent with this expectation. [^9]: We stress that this preliminary study was quite incomplete. In particular, we have not seriously considered the necessity of subtracting some disconnected parts from correlation functions. Clearly additional study is needed to conclude something about the use of those correlation functions in computation of the central charge. [^10]: Note that the coupling constant $\lambda$ in Eq.  is a unique physical dimensionful parameter in the present system on $\mathbb{R}^2$.